# Tipler - Physics, 5 Ed. -- Complete Solutions

# Tipler - Physics, 5 Ed. -- Complete Solutions - Chapter 1...

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Unformatted text preview: Chapter 1 Systems of Measurement Conceptual Problems *1 • Determine the Concept The fundamental physical quantities in the SI system include mass, length, and time. Force, being the product of mass and acceleration, is not a fundamental quantity. (c) is correct. 2 • Picture the Problem We can express and simplify the ratio of m/s to m/s2 to determine the final units. m 2 s = m ⋅ s = s and (d ) is correct. m m ⋅s 2 s Express and simplify the ratio of m/s to m/s2: 3 • Determine the Concept Consulting Table 1-1 we note that the prefix giga means 109. (c ) is correct. 4 • Determine the Concept Consulting Table 1-1 we note that the prefix mega means 106. (d ) is correct. *5 • Determine the Concept Consulting Table 1-1 we note that the prefix pico means 10−12. (a ) is correct. 6 • Determine the Concept Counting from left to right and ignoring zeros to the left of the first nonzero digit, the last significant figure is the first digit that is in doubt. Applying this criterion, the three zeros after the decimal point are not significant figures, but the last zero is significant. Hence, there are four significant figures in this number. (c) is correct. 1 2 Chapter 1 7 • Determine the Concept Counting from left to right, the last significant figure is the first digit that is in doubt. Applying this criterion, there are six significant figures in this number. (e) is correct. 8 • Determine the Concept The advantage is that the length measure is always with you. The disadvantage is that arm lengths are not uniform; if you wish to purchase a board of ″two arm lengths″ it may be longer or shorter than you wish, or else you may have to physically go to the lumberyard to use your own arm as a measure of length. 9 • (a) True. You cannot add ″apples to oranges″ or a length (distance traveled) to a volume (liters of milk). (b) False. The distance traveled is the product of speed (length/time) multiplied by the time of travel (time). (c) True. Multiplying by any conversion factor is equivalent to multiplying by 1. Doing so does not change the value of a quantity; it changes its units. Estimation and Approximation *10 •• Picture the Problem Because θ is small, we can approximate it by θ ≈ D/rm provided that it is in radian measure. We can solve this relationship for the diameter of the moon. Express the moon’s diameter D in terms of the angle it subtends at the earth θ and the earth-moon distance rm: D = θ rm Find θ in radians: θ = 0.524° × Substitute and evaluate D: D = (0.00915 rad )(384 Mm ) 2π rad = 0.00915 rad 360° = 3.51 × 106 m Systems of Measurement 3 *11 •• Picture the Problem We’ll assume that the sun is made up entirely of hydrogen. Then we can relate the mass of the sun to the number of hydrogen atoms and the mass of each. Express the mass of the sun MS as the product of the number of hydrogen atoms NH and the mass of each atom MH: M S = NHM H Solve for NH: NH = MS MH Substitute numerical values and evaluate NH: NH = 1.99 × 1030 kg = 1.19 × 1057 1.67 × 10 −27 kg 12 •• Picture the Problem Let P represent the population of the United States, r the rate of consumption and N the number of aluminum cans used annually. The population of the United States is roughly 3×108 people. Let’s assume that, on average, each person drinks one can of soft drink every day. The mass of a soft-drink can is approximately 1.8 ×10−2 kg. (a) Express the number of cans N used annually in terms of the daily rate of consumption of soft drinks r and the population P: N = rP∆t Substitute numerical values and approximate N: ⎛ 1can ⎞ 8 N =⎜ ⎜ person ⋅ d ⎟ 3 × 10 people ⎟ ⎝ ⎠ ⎛ d⎞ × (1 y )⎜ 365.24 ⎟ ⎜ y⎟ ⎝ ⎠ ( ≈ 1011 cans (b) Express the total mass of aluminum used per year for soft drink cans M as a function of the number of cans consumed and the mass m per can: M = Nm ) 4 Chapter 1 Substitute numerical values and evaluate M: (c) Express the value of the aluminum as the product of M and the value at recycling centers: ( )( M = 1011 cans/y 1.8 × 10−2 kg/can ) ≈ 2 × 109 kg/y Value = ($1 / kg )M ( = ($1 / kg ) 2 × 109 kg/y ) = $2 × 10 / y 9 = 2 billion dollars/y 13 •• Picture the Problem We can estimate the number of words in Encyclopedia Britannica by counting the number of volumes, estimating the average number of pages per volume, estimating the number of words per page, and finding the product of these measurements and estimates. Doing so in Encyclopedia Britannica leads to an estimate of approximately 200 million for the number of words. If we assume an average word length of five letters, then our estimate of the number of letters in Encyclopedia Britannica becomes 109. (a) Relate the area available for one letter s2 and the number of letters N to be written on the pinhead to the area of the pinhead: Solve for s to obtain: Substitute numerical values and evaluate s: Ns 2 = π 4 d 2 where d is the diameter of the pinhead. s= πd 2 4N ⎡ s= (b) Express the number of atoms per letter n in terms of s and the atomic spacing in a metal datomic: n= Substitute numerical values and evaluate n: n= ⎣ 2 cm ⎞⎤ ⎛ ⎟ in ⎠⎥ ⎝ ⎦ ≈ 10−8 m 9 4 10 1 π ⎢(16 in )⎜ 2.54 ( ) s d atomic 10 −8 m ≈ 20 atoms 5 × 10 −10 atoms/m *14 •• Picture the Problem The population of the United States is roughly 3 × 108 people. Assuming that the average family has four people, with an average of two cars per Systems of Measurement 5 family, there are about 1.5 × 108 cars in the United States. If we double that number to include trucks, cabs, etc., we have 3 × 108 vehicles. Let’s assume that each vehicle uses, on average, about 12 gallons of gasoline per week. (a) Find the daily consumption of gasoline G: Assuming a price per gallon P = $1.50, find the daily cost C of gasoline: (b) Relate the number of barrels N of crude oil required annually to the yearly consumption of gasoline Y and the number of gallons of gasoline n that can be made from one barrel of crude oil: Substitute numerical values and estimate N: ( ) G = 3×108 vehicles (2 gal/d ) = 6 ×108 gal/d ( ) C = GP = 6 × 108 gal/d ($1.50 / gal) = $9 × 108 / d ≈ $1 billion dollars/d N= Y G∆t = n n (6 ×10 N= ) gal/d (365.24 d/y ) 19.4 gal/barrel 8 ≈ 1010 barrels/y 15 •• Picture the Problem We’ll assume a population of 300 million (fairly accurate as of September, 2002) and a life expectancy of 76 y. We’ll also assume that a diaper has a volume of about half a liter. In (c) we’ll assume the disposal site is a rectangular hole in the ground and use the formula for the volume of such an opening to estimate the surface area required. (a) Express the total number N of disposable diapers used in the United States per year in terms of the number of children n in diapers and the number of diapers D used by each child in 2.5 y: N = nD Use the daily consumption, the number of days in a year, and the estimated length of time a child is in diapers to estimate the number of diapers D required per child: D= 3 diapers 365.24 d × × 2.5 y d y ≈ 3 × 103 diapers/child 6 Chapter 1 Use the assumed life expectancy to estimate the number of children n in diapers: ⎛ 2 .5 y ⎞ 6 n=⎜ ⎜ 76 y ⎟ 300 × 10 children ⎟ ⎝ ⎠ 7 ≈ 10 children Substitute to obtain: N = 107 children ( ( ( ) × 3 × 10 diapers/child 3 ) ) ≈ 3 × 1010 diapers (b) Express the required landfill volume V in terms of the volume of diapers to be buried: Substitute numerical values and evaluate V: V = NVone diaper ( ) V = 3 × 1010 diapers (0.5 L/diaper ) ≈ 1.5 × 107 m 3 (c) Express the required volume in terms of the volume of a rectangular parallelepiped: V = Ah Solve and evaluate h: V 1.5 × 107 m 3 A= = = 1.5 × 106 m 2 10 m h Use a conversion factor to express this area in square miles: A = 1.5 × 106 m 2 × 1 mi 2 2.590 km 2 ≈ 0.6 mi 2 16 ••• Picture the Problem The number of bits that can be stored on the disk can be found from the product of the capacity of the disk and the number of bits per byte. In part (b) we’ll need to estimate (i) the number of bits required for the alphabet, (ii) the average number of letters per word, (iii) an average number of words per line, (iv) an average number of lines per page, and (v) a book length in pages. (a) Express the number of bits Nbits as a function of the number of bits per byte and the capacity of the hard disk Nbytes: N bits = N bytes (8 bits/byte) = (2 × 109 bytes)(8 bits/byte) = 1.60 × 1010 bits Systems of Measurement (b) Assume an average of 8 letters/word and 8 bits/character to estimate the number of bytes required per word: 8 7 bits characters bits ×8 = 64 character word word bytes =8 word words bytes bytes ×8 = 4800 page word page Assume 10 words/line and 60 lines/page: 600 Assume a book length of 300 pages and approximate the number bytes required: 300pages × 4800 Divide the number of bytes per disk by our estimated number of bytes required per book to obtain an estimate of the number of books the 2-gigabyte hard disk can hold: N books = bytes = 1.44 × 106 bytes page 2 × 109 bytes 1.44 × 106 bytes/book ≈ 1400 books *17 •• Picture the Problem Assume that, on average, four cars go through each toll station per minute. Let R represent the yearly revenue from the tolls. We can estimate the yearly revenue from the number of lanes N, the number of cars per minute n, and the $6 toll per car C. R = NnC = 14 lanes × 4 min h d $6 cars × 60 × 24 × 365.24 × = $177M min h d y car Units 18 • Picture the Problem We can use the metric prefixes listed in Table 1-1 and the abbreviations on page EP-1 to express each of these quantities. (a) (c) 1,000,000 watts = 10 watts 6 3 × 10 −6 meter = 3 µm = 1 MW (d) (b) −3 0.002gram = 2 × 10 g = 2 mg 30,000 seconds = 30 × 103 s = 30 ks 8 Chapter 1 19 • Picture the Problem We can use the definitions of the metric prefixes listed in Table 1-1 to express each of these quantities without prefixes. (c) (a) 40 µW = 40 × 10 W = 0.000040 W 3 MW = 3 × 106 W = 3,000,000 W (b) (d) −6 −9 4 ns = 4 × 10 s = 0.000000004 s 25 km = 25 × 103 m = 25,000 m *20 • Picture the Problem We can use the definitions of the metric prefixes listed in Table 1-1 to express each of these quantities without abbreviations. (a) 10 −12 boo = 1 picoboo (e) 106 phone = 1megaphone (b) 10 9 low = 1 gigalow (f) 10 −9 goat = 1 nanogoat (c) 10 −6 phone = 1 microphone (g) 1012 bull = 1 terabull (d) 10 −18 boy = 1 attoboy 21 •• Picture the Problem We can determine the SI units of each term on the right-hand side of the equations from the units of the physical quantity on the left-hand side. (a) Because x is in meters, C1 and C2t must be in meters: C1 is in m; C2 is in m/s (b) Because x is in meters, ½C1t2 must be in meters: C1 is in m/s 2 (c) Because v2 is in m2/s2, 2C1x must be in m2/s2: C1 is in m/s 2 (d) The argument of trigonometric function must be dimensionless; i.e. without units. Therefore, because x C1 is in m; C2 is in s −1 Systems of Measurement 9 is in meters: (e) The argument of an exponential function must be dimensionless; i.e. without units. Therefore, because v is in m/s: C1 is in m/s; C2 is in s −1 22 •• Picture the Problem We can determine the US customary units of each term on the right-hand side of the equations from the units of the physical quantity on the left-hand side. (a) Because x is in feet, C1 and C2t must be in feet: C1 is in ft; C2 is in ft/s (b) Because x is in feet, ½C1t2 must be in feet: C1 is in ft/s 2 (c) Because v2 is in ft2/s2, 2C1x must be in ft2/s2: C1 is in ft/s 2 (d) The argument of trigonometric function must be dimensionless; i.e. without units. Therefore, because x is in feet: C1 is in ft; C2 is in s −1 (e) The argument of an exponential function must be dimensionless; i.e. without units. Therefore, because v is in ft/s: C1 is in ft/s; C2 is in s −1 Conversion of Units 23 • Picture the Problem We can use the formula for the circumference of a circle to find the radius of the earth and the conversion factor 1 mi = 1.61 km to convert distances in meters into distances in miles. (a) The Pole-Equator distance is one-fourth of the circumference: c = 4 × 107 m 10 Chapter 1 (b) Use the formula for the circumference of a circle to obtain: c 4 × 10−7 m R= = = 6.37 × 106 m 2π 2π (c) Use the conversion factors 1 km = 1000 m and 1 mi = 1.61 km: c = 4 × 107 m × 1 km 1 mi × 3 10 m 1.61km = 2.48 × 104 mi and R = 6.37 × 106 m × 1 km 1 mi × 3 10 m 1.61 km = 3.96 × 103 mi 24 • Picture the Problem We can use the conversion factor 1 mi = 1.61 km to convert speeds in km/h into mi/h. Find the speed of the plane in km/s: v = 2(340 m/s ) = 680 m/s m ⎞ ⎛ 1 km ⎞ ⎛ s⎞ ⎛ = ⎜ 680 ⎟ ⎜ 3 ⎟ ⎜ 3600 ⎟ ⎜ 10 m ⎟ s ⎠⎝ h⎠ ⎝ ⎠⎝ = 2450 km/h Convert v into mi/h: km ⎞ ⎛ 1 mi ⎞ ⎛ ⎟ v = ⎜ 2450 ⎟⎜ h ⎠ ⎜ 1.61 km ⎟ ⎝ ⎠ ⎝ = 1520 mi/h *25 • Picture the Problem We’ll first express his height in inches and then use the conversion factor 1 in = 2.54 cm. 12 in + 10.5 in = 82.5 in ft Express the player’s height into inches: h = 6 ft × Convert h into cm: h = 82.5 in × 2.54 cm = 210 cm in 26 • Picture the Problem We can use the conversion factors 1 mi = 1.61 km, 1 in = 2.54 cm, and 1 m = 1.094 yd to complete these conversions. Systems of Measurement (a) 100 11 km km 1 mi mi = 100 × = 62.1 h h 1.61km h 1in = 23.6 in 2.54 cm (b) 60 cm = 60 cm × (c) 100 yd = 100 yd × 1m = 91.4 m 1.094 yd 27 • Picture the Problem We can use the conversion factor 1.609 km = 5280 ft to convert the length of the main span of the Golden Gate Bridge into kilometers. Convert 4200 ft into km: 4200 ft = 4200 ft × 1.609 km = 1.28 km 5280 ft *28 • Picture the Problem Let v be the speed of an object in mi/h. We can use the conversion factor 1 mi = 1.61 km to convert this speed to km/h. Multiply v mi/h by 1.61 km/mi to convert v to km/h: v mi mi 1.61 km =v × = 1.61v km/h h h mi 29 • Picture the Problem Use the conversion factors 1 h = 3600 s, 1.609 km = 1 mi, and 1 mi = 5280 ft to make these conversions. (a) 1.296 × 105 km ⎛ km ⎞ ⎛ 1 h ⎞ km = ⎜1.296 × 105 2 ⎟ ⎜ 2 ⎟ ⎜ 3600 s ⎟ = 36.0 h ⋅ s h h ⎠⎝ ⎝ ⎠ km ⎞ ⎛ 1 h ⎞ km ⎛ ⎟ (b) 1.296 × 10 2 = ⎜1.296 × 105 2 ⎟ ⎜ h ⎠ ⎜ 3600 s ⎟ h ⎝ ⎝ ⎠ 5 2 ⎛ 103 m ⎞ m ⎜ ⎜ km ⎟ = 10.0 s 2 ⎟ ⎝ ⎠ (c) 60 mi ⎛ mi ⎞ ⎛ 5280 ft ⎞ ⎛ 1 h ⎞ ft = ⎜ 60 ⎟ ⎜ ⎜ 1 mi ⎟ ⎜ 3600 s ⎟ = 88.0 s ⎟⎜ ⎟ h ⎝ h ⎠⎝ ⎠⎝ ⎠ (d) 60 mi ⎛ mi ⎞ ⎛ 1.609 km ⎞ ⎛ 103 m ⎞ ⎛ 1 h ⎞ m = ⎜ 60 ⎟ ⎜ ⎜ 1 mi ⎟ ⎜ km ⎟ ⎜ 3600 s ⎟ = 26.8 s ⎟⎜ ⎟ ⎟⎜ h ⎝ h ⎠⎝ ⎠⎝ ⎠ ⎠⎝ 12 Chapter 1 30 • Picture the Problem We can use the conversion factor 1 L = 1.057 qt to convert gallons into liters and then use this gallons-to-liters conversion factor to convert barrels into cubic meters. ⎛ 4 qt ⎞ ⎛ 1 L ⎞ ⎟ = 3.784 L ⎟⎜ ⎟⎜ ⎟ ⎝ gal ⎠ ⎝ 1.057 qt ⎠ (a) 1gal = (1gal)⎜ ⎜ 3 −3 ⎛ 42 gal ⎞ ⎛ 3.784 L ⎞ ⎛ 10 m ⎞ 3 ⎟⎜ ⎟⎜ ⎜ gal ⎟ ⎜ L ⎟ = 0.1589 m ⎟ ⎝ barrel ⎠ ⎝ ⎠⎝ ⎠ (b) 1 barrel = (1 barrel)⎜ 31 • Picture the Problem We can use the conversion factor given in the problem statement and the fact that 1 mi = 1.609 km to express the number of square meters in one acre. Multiply by 1 twice, properly chosen, to convert one acre into square miles, and then into square meters: ⎛ 1mi 2 ⎞ ⎛ 1609 m ⎞ 1acre = (1acre)⎜ ⎜ 640 acres ⎟ ⎜ mi ⎟ ⎟ ⎠ ⎝ ⎠⎝ 2 = 4050 m 2 32 •• Picture the Problem The volume of a right circular cylinder is the area of its base multiplied by its height. Let d represent the diameter and h the height of the right circular cylinder; use conversion factors to express the volume V in the given units. (a) Express the volume of the cylinder: Substitute numerical values and evaluate V: V = 1 πd 2 h 4 V = 1 π (6.8 in ) (2 ft ) 4 2 ⎛ 1ft ⎞ = π (6.8 in ) (2 ft )⎜ ⎟ ⎜ 12 in ⎟ ⎝ 2 2 1 4 = 0.504 ft 3 3 (b) Use the fact that 1 m = 3.281 ft to convert the volume in cubic feet into cubic meters: ⎛ 1m ⎞ V = 0.504 ft ⎜ ⎜ 3.281 ft ⎟ ⎟ ⎝ ⎠ (c) Because 1 L = 10−3 m3: ⎛ 1L V = 0.0143m 3 ⎜ − 3 3 ⎜ 10 m ⎝ ( 3 ) = 0.0143 m 3 ( ) ⎞ ⎟ = 14.3 L ⎟ ⎠ Systems of Measurement 13 *33 •• Picture the Problem We can treat the SI units as though they are algebraic quantities to simplify each of these combinations of physical quantities and constants. (a) Express and simplify the units of v2/x: (m s )2 m (b) Express and simplify the units of x a: (c) Noting that the constant factor 1 2 has no units, express and simplify the units of 1 2 = m2 m = 2 2 m⋅s s m = s2 = s m/s 2 ( ) ⎛m⎞ 2 ⎛m⎞ 2 ⎜ 2 ⎟(s ) = ⎜ 2 ⎟ s = m ⎝s ⎠ ⎝s ⎠ at 2 : Dimensions of Physical Quantities 34 • Picture the Problem We can use the facts that each term in an equation must have the same dimensions and that the arguments of a trigonometric or exponential function must be dimensionless to determine the dimensions of the constants. (a) x = C1 + C2 t L L T T L (b) x = 1 2 C1 t 2 L T2 T2 L (d) x = C1 cos C2 L (e) v = L T L C1 L T t 1 T T exp( −C2 t) 1 T T (c) v 2 = 2 C1 x L T2 L 2 L T2 35 •• Picture the Problem Because the exponent of the exponential function must be dimensionl the dimension of λ must be T −1. 14 Chapter 1 *36 •• Picture the Problem We can solve Newton’s law of gravitation for G and substitute the dimensions of the variables. Treating them as algebraic quantities will allow us to express the dimensions in their simplest form. Finally, we can substitute the SI units for the dimensions to find the units of G. Fr 2 m1m2 Solve Newton’s law of gravitation for G to obtain: G= Substitute the dimensions of the variables: ML 2 ×L L3 T2 G= = M2 MT 2 Use the SI units for L, M, and T: Units of G are m3 kg ⋅ s 2 37 •• Picture the Problem Let m represent the mass of the object, v its speed, and r the radius of the circle in which it moves. We can express the force as the product of m, v, and r (each raised to a power) and then use the dimensions of force F, mass m, speed v, and radius r to obtain three equations in the assumed powers. Solving these equations simultaneously will give us the dependence of F on m, v, and r. Express the force in terms of powers of the variables: F = mavb r c Substitute the dimensions of the physical quantities: ⎛L⎞ MLT −2 = M a ⎜ ⎟ Lc ⎝T ⎠ Simplify to obtain: MLT −2 = M a Lb+cT − b Equate the exponents to obtain: a = 1, b + c = 1, and −b = −2 Solve this system of equations to obtain: a = 1, b = 2, and c = −1 Substitute in equation (1): F = mv 2 r −1 = m b v2 r Systems of Measurement 15 38 •• Picture the Problem We note from Table 1-2 that the dimensions of power are ML2/T3. The dimensions of mass, acceleration, and speed are M, L/T2, and L/T respectively. Express the dimensions of mav: [mav] = M × From Table 1-2: L L ML2 × = 3 T2 T T [P ] = ML 3 2 T Comparing these results, we see that the product of mass, acceleration, and speed has the dimensions of power. 39 •• Picture the Problem The dimensions of mass and velocity are M and L/T, respectively. We note from Table 1-2 that the dimensions of force are ML/T2. Express the dimensions of momentum: [mv] = M × L = ML T From Table 1-2: T [F ] = ML 2 T Express the dimensions of force multiplied by time: [Ft ] = ML × T = ML 2 T T Comparing these results, we see that momentum has the dimensions of force multiplied by time. 40 •• Picture the Problem Let X represent the physical quantity of interest. Then we can express the dimensional relationship between F, X, and P and solve this relationship for the dimensions of X. Express the relationship of X to force and power dimensionally: Solve for [ X ] : [F ][X ] = [P] [X ] = [P] [F ] 16 Chapter 1 Substitute the dimensions of force and power and simplify to obtain: Because the dimensions of velocity are L/T, we can conclude that: ML2 T3 [X ] = ML = L T 2 T [P ] = [F ][v] Remarks: While it is true that P = Fv, dimensional analysis does not reveal the presence of dimensionless constants. For example, if P = πFv , the analysis shown above would fail to establish the factor of π. *41 •• Picture the Problem We can find the dimensions of C by solving the drag force equation for C and substituting the dimensions of force, area, and velocity. Fair Av 2 Solve the drag force equation for the constant C: C= Express this equation dimensionally: [C ] = [Fair ]2 [A][v] Substitute the dimensions of force, area, and velocity and simplify to obtain: ML 2 [C ] = T 2 = M L3 2⎛L⎞ L ⎜ ⎟ ⎝T ⎠ 42 •• Picture the Problem We can express the period of a planet as the product of these factors (each raised to a power) and then perform dimensional analysis to determine the values of the exponents. c T = Cr a G b M S Express the period T of a planet as c the product of r a , G b , and M S : where C is a dimensionless constant. Solve the law of gravitation for the constant G: Fr 2 G= m1m2 Express this equation dimensionally: [F ][r ]2 [G ] = [m1 ][m2 ] (1) Systems of Measurement Substitute the dimensions of F, r, and m: ML 2 × (L ) 2 L3 T = [G ] = M ×M MT 2 Noting that the dimension of time is represented by the same letter as is the period of a planet, substitute the dimensions in equation (1) to obtain: ⎛ L3 ⎞ c T = (L ) ⎜ ⎜ MT 2 ⎟ (M ) ⎟ ⎝ ⎠ Introduce the product of M 0 and L0 in the left hand side of the equation and simplify to obtain: M 0 L0T 1 = M c −b La +3bT −2b Equate the exponents on the two sides of the equation to obtain: 17 0 = c – b, 0 = a + 3b, and 1 = –2b Solve these equations simultaneously to obtain: Substitute in equation (1): b a a = 3 , b = − 1 , and c = − 1 2 2 2 − T = Cr 3 2G −1 2 M S 1 2 = C r3 2 GM S Scientific Notation and Significant Figures *43 • Picture the Problem We can use the rules governing scientific notation to express each of these numbers as a decimal number. (a) 3 × 10 4 = 30,000 (c) 4 × 10 −6 = 0.000004 (b) 6.2 × 10 −3 = 0.0062 (d) 2.17 × 105 = 217,000 44 • Picture the Problem We can use the rules governing scientific notation to express each of these measurements in scientific notation. (a) 3.1GW = 3.1 × 109 W (c) 2.3 fs = 2.3 × 10 −15 s 18 Chapter 1 (b) 10 pm = 10 × 10 −12 m = 10 −11 m (d) 4 µs = 4 × 10 −6 s 45 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) The number of significant figures in each factor is three; therefore the result has three significant figures: (b) Express both terms with the same power of 10. Because the first measurement has only two digits after the decimal point, the result can have only two digits after the decimal point: (c) We’ll assume that 12 is exact. Hence, the answer will have three significant figures: (d) Proceed as in (b): (1.14)(9.99 × 104 ) = 1.14 × 105 (2.78 × 10 ) − (5.31 × 10 ) −8 −9 = (2.78 − 0.531) × 10−8 = 2.25 × 10−8 12π = 8.27 × 103 4.56 × 10 −3 ( ) 27.6 + 5.99 × 10 2 = 27.6 + 599 = 627 = 6.27 × 10 2 46 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) Note that both factors have four significant figures. (b) Express the first factor in scientific notation and note that both factors have three significant figures. (200.9)(569.3) = 1.144 × 105 (0.000000513)(62.3 × 107 ) ( )( = 5.13 × 10 −7 62.3 × 107 = 3.20 × 10 2 ) Systems of Measurement 19 ( ) = (2.841 × 10 ) + (5.78 × 10 ) (c) Express both terms in scientific notation and note that the second has only three significant figures. Hence the result will have only three significant figures. 28401 + 5.78 × 104 (d) Because the divisor has three significant figures, the result will have three significant figures. 63.25 = 1.52 × 104 −3 4.17 × 10 4 4 = (2.841 + 5.78) × 104 = 8.62 × 104 *47 • Picture the Problem Let N represent the required number of membranes and express N in terms of the thickness of each cell membrane. Express N in terms of the thickness of a single membrane: N= 1in 7 nm Convert the units into SI units and simplify to obtain: N= 1in 2.54 cm 1m 1 nm × × × −9 7 nm in 100 cm 10 m = 4 × 106 48 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) Both factors and the result have three significant figures: (b) Because the second factor has three significant figures, the result will have three significant figures: (c) Both factors and the result have three significant figures: (d) Write both terms using the same power of 10. Note that the result will have only three significant figures: (2.00 × 10 )(6.10 × 10 ) = −2 4 (3.141592)(4.00 × 105 ) = 1.22 × 103 1.26 × 106 2.32 × 103 = 2.00 × 10−5 8 1.16 × 10 (5.14 × 10 ) + (2.78 × 10 ) = (5.14 × 10 ) + (0.278 × 10 ) 3 2 3 = (5.14 + 0.278) × 103 = 5.42 × 103 3 20 Chapter 1 (e) Follow the same procedure used in (d): (1.99 × 10 ) + (9.99 × 10 ) = (1.99 × 10 ) + (0.000000999 × 10 ) −5 2 2 2 = 1.99 × 102 *49 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) The second factor and the result have three significant figures: 3.141592654 × (23.2 ) = 1.69 × 103 2 (b) We’ll assume that 2 is exact. Therefore, the result will have two significant figures: 2 × 3.141592654 × 0.76 = 4.8 (c) We’ll assume that 4/3 is exact. Therefore the result will have two significant figures: 4 π × (1.1)3 = 5.6 3 (d) Because 2.0 has two significant figures, the result has two significant figures: (2.0)5 3.141592654 = 10 General Problems 50 • Picture the Problem We can use the conversion factor 1 mi = 1.61 km to convert 100 km/h into mi/h. Multiply 100 km/h by 1 mi/1.61 km to obtain: 100 km km 1 mi = 100 × h h 1.61km = 62.1 mi/h *51 • Picture the Problem We can use a series of conversion factors to convert 1 billion seconds into years. Multiply 1 billion seconds by the appropriate conversion factors to convert into years: Systems of Measurement 109 s = 109 s × 21 1h 1day 1y × × = 31.7 y 3600 s 24 h 365.24 days 52 • Picture the Problem In both the examples cited we can equate expressions for the physical quantities, expressed in different units, and then divide both sides of the equation by one of the expressions to obtain the desired conversion factor. 3 × 108 m/s = 1.61 × 103 m/mi 5 1.86 × 10 mi/h (a) Divide both sides of the equation expressing the speed of light in the two systems of measurement by 186,000 mi/s to obtain: 1= (b) Find the volume of 1.00 kg of water: Volume of 1.00 kg = 103 g is 103 cm3 Express 103 cm3 in ft3: ⎛ ⎞ ⎛ ⎞ (10 cm ) ⎜ 1in ⎟ ⎜ 1ft ⎟ ⎜ 2.54 cm ⎟ ⎜ 12 in ⎟ ⎝ ⎠ ⎝ ⎠ = 0.0353 ft 3 Relate the weight of 1 ft3 of water to the volume occupied by 1 kg of water: 1.00 kg lb = 62.4 3 3 0.0353 ft ft Divide both sides of the equation by the left-hand side to obtain: lb ft 3 = 2.20 lb/kg 1= 1.00 kg 0.0353 ft 3 m ⎞ ⎛ 1 km ⎞ ⎛ ⎟ = ⎜1.61 × 103 ⎟⎜ mi ⎠ ⎜ 103 m ⎟ ⎝ ⎝ ⎠ = 1.61 km/mi 3 3 3 62.4 53 •• Picture the Problem We can use the given information to equate the ratios of the number of uranium atoms in 8 g of pure uranium and of 1 atom to its mass. Express the proportion relating the number of uranium atoms NU in 8 g of pure uranium to the mass of 1 atom: 1atom NU = 8 g 4.0 × 10−26 kg 22 Chapter 1 ⎛ ⎞ 1atom N U = (8 g )⎜ ⎜ 4.0 × 10 −26 kg ⎟ ⎟ ⎝ ⎠ Solve for and evaluate NU: = 2.0 × 10 23 54 •• Picture the Problem We can relate the weight of the water to its weight per unit volume and the volume it occupies. Express the weight w of water falling on the acre in terms of the weight of one cubic foot of water, the depth d of the water, and the area A over which the rain falls: lb ⎞ ⎛ w = ⎜ 62.4 3 ⎟ Ad ft ⎠ ⎝ Find the area A in ft2: ⎛ 1 mi 2 ⎞ ⎛ 5280 ft ⎞ A = (1acre)⎜ ⎜ 640 acre ⎟ ⎜ mi ⎟ ⎟ ⎠ ⎝ ⎠⎝ 4 2 = 4.356 × 10 ft 2 Substitute numerical values and evaluate w: ⎛ 1ft ⎞ lb ⎞ ⎛ 5 w = ⎜ 62.4 3 ⎟ 4.356 × 10 4 ft 2 (1.4 in ) ⎜ ⎜ 12 in ⎟ = 3.17 × 10 lb ⎟ ft ⎠ ⎝ ⎝ ⎠ ( ) 55 •• Picture the Problem We can use the definition of density and the formula for the volume of a sphere to find the density of iron. Once we know the density of iron, we can use these same relationships to find what the radius of the earth would be if it had the same mass per unit volume as iron. m V (a) Using its definition, express the density of iron: ρ= Assuming it to be spherical, express the volume of an iron nucleus as a function of its radius: V = 4 π r3 3 Substitute to obtain: ρ= 3m 4π r 3 (1) Systems of Measurement Substitute numerical values and evaluate ρ: ρ= ( 4π (5.4 × 10 ) m) 3 9.3 × 10 −26 kg −15 3 = 1.41 × 1017 kg/m 3 (b) Because equation (1) relates the density of any spherical object to its mass and radius, we can solve for r to obtain: r=3 3m 4πρ Substitute numerical values and evaluate r: r=3 3 5.98 × 10 24 kg = 216 m 4π 1.41 × 1017 kg/m 3 ( ( ) ) 56 •• Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) Because all of the factors have two significant figures, the result will have two significant figures: (5.6 × 10 ) (0.0000075) −5 2.4 × 10 −12 (5.6 × 10 ) (7.5 × 10 ) = −5 −6 2.4 × 10 −12 = 1.8 × 10 2 (b) Because the factor with the fewest significant figures in the first term has two significant figures, the result will have two significant figures. Because its last significant figure is in the tenth’s position, the difference between the first and second term will have its last significant figure in the tenth’s position: (c) Because all of the factors have two significant figures, the result will have two significant figures: (14.2) (6.4 × 107 )(8.2 × 10−9 ) − 4.06 = 7.8 − 4.06 = 3.4 (6.1 × 10 ) (3.6 × 10 ) (3.6 × 10 ) −6 2 4 3 −11 1 2 = 2.9 × 108 23 24 Chapter 1 (d) Because the factor with the fewest significant figures has two significant figures, the result will have two significant figures. (0.000064)1 3 (12.8 × 10 )(490 × 10 ) (6.4 × 10 ) = (12.8 × 10 ) (490 × 10 ) −1 1 2 −3 −5 1 3 −1 1 2 −3 = 0.45 *57 •• Picture the Problem We can use the relationship between an angle θ, measured in radians, subtended at the center of a circle, the radius R of the circle, and the length L of the arc to answer these questions concerning the astronomical units of measure. S R (a) Relate the angle θ subtended by an arc of length S to the distance R: θ= Solve for and evaluate S: S = Rθ (1) ⎛ 1 min ⎞ = (1 parsec)(1s )⎜ ⎜ 60 s ⎟ ⎟ ⎝ ⎠ ⎛ 1° ⎞ ⎛ 2π rad ⎞ ×⎜ ⎜ 60 min ⎟ ⎜ 360° ⎟ ⎟ ⎠ ⎝ ⎠⎝ = 4.85 × 10 −6 parsec (b) Solve equation (1) for and evaluate R: R= = S θ 1.496 × 1011 m ⎜ (1s ) ⎛ 1min ⎞ ⎛ 1° ⎞ ⎛ 2π rad ⎞ ⎜ 60 s ⎟ ⎜ 60 min ⎟ ⎜ 360° ⎟ ⎟⎜ ⎟ ⎠ ⎝ ⎠⎝ ⎠⎝ = 3.09 × 1016 m (c) Relate the distance D light travels in a given interval of time ∆t to its speed c and evaluate D for ∆t = 1 y: D = c∆t ⎛ m⎞ s⎞ ⎛ = ⎜ 3 × 108 ⎟ (1 y )⎜ 3.156 × 107 ⎟ ⎜ s⎠ y⎟ ⎝ ⎝ ⎠ = 9.47 × 1015 m Systems of Measurement (d) Use the definition of 1 AU and the result from part (c) to obtain: 25 ⎛ ⎞ 1 AU 1c ⋅ y = 9.47 × 1015 m ⎜ ⎜ 1.496 × 1011 m ⎟ ⎟ ⎝ ⎠ ( ) = 6.33 × 10 4 AU (e) Combine the results of parts (b) and (c) to obtain: ( 1 parsec = 3.08 × 1016 m ) ⎛ ⎞ 1c ⋅ y ×⎜ ⎜ 9.47 × 1015 m ⎟ ⎟ ⎝ ⎠ = 3.25 c ⋅ y 58 •• Picture the Problem Let Ne and Np represent the number of electrons and the number of protons, respectively and ρ the critical average density of the universe. We can relate these quantities to the masses of the electron and proton using the definition of density. m N e me = V V (a) Using its definition, relate the required density ρ to the electron density Ne/V: ρ= Solve for Ne/V: Ne ρ = V me Substitute numerical values and evaluate Ne/V: 6 × 10−27 kg/m 3 Ne = 9.11 × 10−31 kg/electron V (1) = 6.59 × 103 electrons/m 3 (b) Express and evaluate the ratio of the masses of an electron and a proton: me 9.11 × 10−31 kg = = 5.46 × 10 −4 −27 mp 1.67 × 10 kg Rewrite equation (1) in terms of protons: Np Divide equation (2) by equation (1) to obtain: Np V = ρ mp (2) V = me or N p = me ⎛ N e ⎞ ⎜ ⎟ Ne V mp ⎝ V ⎠ mp V 26 Chapter 1 Substitute numerical values and use the result from part (a) to evaluate Np/V: Np V ( = 5.46 × 10 −4 ) ( × 6.59 × 103 protons/m 3 ) = 3.59 protons/m 3 *59 •• Picture the Problem We can use the definition of density to relate the mass of the water in the cylinder to its volume and the formula for the volume of a cylinder to express the volume of water used in the detector’s cylinder. To convert our answer in kg to lb, we can use the fact that 1 kg weighs about 2.205 lb. Relate the mass of water contained in the cylinder to its density and volume: m = ρV Express the volume of a cylinder in terms of its diameter d and height h: V = Abase h = Substitute to obtain: m=ρ Substitute numerical values and evaluate m: Convert 5.02 × 107 kg to tons: π 4 π 4 d 2h d 2h ⎛π ⎞ 2 m = 103 kg/m 3 ⎜ ⎟ (39.3 m ) (41.4 m ) ⎝4⎠ 7 = 5.02 × 10 kg ( ) m = 5.02 × 107 kg × 2.205 lb 1 ton × 2000 lb kg = 55.4 × 103 ton The 50,000 − ton claim is conservative. The actual weight is closer to 55,000 tons. 60 ••• Picture the Problem We’ll solve this problem two ways. First, we’ll substitute two of the ordered pairs in the given equation to obtain two equations in C and n that we can solve simultaneously. Then we’ll use a spreadsheet program to create a graph of log T as a function of log m and use its curve-fitting capability to find n and C. Finally, we can identify the data points that deviate the most from a straight-line plot by examination of the graph. Systems of Measurement 27 1st Solution for (a) (a) To estimate C and n, we can n apply the relation T = Cm to two arbitrarily selected data points. We’ll use the 1st and 6th ordered pairs. This will produce simultaneous equations that can be solved for C and n. T1 = Cm1n and n T6 = Cm6 Divide the second equation by the first to obtain: n T6 Cm6 ⎛ m6 ⎞ = =⎜ ⎟ T1 Cm12 ⎜ m1 ⎟ ⎝ ⎠ Substitute numerical values and solve for n to obtain: 1.75 s ⎛ 1 kg ⎞ ⎟ =⎜ 0.56 s ⎜ 0.1kg ⎟ ⎝ ⎠ n n or 3.125 = 10n ⇒ n = 0.4948 and so a ″judicial″ guess is that n = 0.5. Substituting this value into the second equation gives: 0 T5 = Cm5 .5 so 1.75 s = C(1 kg ) 0.5 Solving for C gives: C = 1.75 s/kg 0.5 2nd Solution for (a) Take the logarithm (we’ll arbitrarily use base 10) of both sides of T = Cmn and simplify to obtain: ( ) log(T ) = log Cm n = log C + log m n = n log m + log C which, we note, is of the form y = mx + b . Hence a graph of log T vs. log m should be linear with a slope of n and a log Tintercept log C. The graph of log T vs. log m shown below was created using a spreadsheet program. The equation shown on the graph was obtained using Excel’s ″Add Trendline″ function. (Excel’s ″Add Trendline″ function uses regression analysis to generate the trendline.) 28 Chapter 1 0.4 0.3 log T = 0.4987log m + 0.2479 log T 0.2 0.1 0.0 -0.1 -0.2 -0.3 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 log m n = 0.499 Comparing the equation on the graph generated by the Add Trendline function to log (T ) = n log m + log C , we and observe: or C = 10 0.2479 = 1.77 s/kg1 2 ( ) T = 1.77 s/kg1 2 m 0.499 (b) From the graph we see that the data points that deviate the most from a straight-line plot are: m = 0.02 kg, T = 0.471 s, and m = 1.50 kg, T = 2.22 s From the graph we see that the points generated using the data pairs (b) (0.02 kg, 0.471 s) and (0.4 kg, 1.05 s) deviate the most from the line representing the best fit to the points plotted on the graph. Remarks: Still another way to find n and C is to use your graphing calculator to perform regression analysis on the given set of data for log T versus log m. The slope yields n and the y-intercept yields log C. 61 ••• Picture the Problem We can plot log T versus log r and find the slope of the best-fit line to determine the exponent n. We can then use any of the ordered pairs to evaluate C. Once we know n and C, we can solve T = Crn for r as a function of T. Systems of Measurement (a) Take the logarithm (we’ll arbitrarily use base 10) of both sides of T = Crn and simplify to obtain: 29 ( ) log(T ) = log Cr n = log C + log r n = n log r + log C Note that this equation is of the form y = mx + b . Hence a graph of log T vs. log r should be linear with a slope of n and a log T -intercept log C. The graph of log T versus log r shown below was created using a spreadsheet program. The equation shown on the graph was obtained using Excel’s ″Add Trendline″ function. (Excel’s ″Add Trendline″ function uses regression analysis to generate the trendline.) 1.0 0.8 y = 1.5036x + 1.2311 log T 0.6 0.4 0.2 0.0 -0.2 -0.4 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 log r From the regression analysis we observe that: n = 1.50 and C = 101.2311 = 17.0 y/(Gm ) 32 ( or T = 17.0 y/(Gm ) 32 (b) Solve equation (1) for the radius of the planet’s orbit: ⎛ ⎞ T ⎟ r =⎜ 32 ⎟ ⎜ 17.0 y / (Gm ) ⎝ ⎠ Substitute numerical values and evaluate r: ⎞ ⎛ 6.20 y ⎟ r =⎜ ⎜ 17.0 y/(Gm )3 2 ⎟ ⎠ ⎝ )r 1.50 (1) 23 23 = 0.510 Gm 30 Chapter 1 *62 ••• Picture the Problem We can express the relationship between the period T of the pendulum, its length L, and the acceleration of gravity g as T = CLa g b and perform dimensional analysis to find the values of a and b and, hence, the function relating these variables. Once we’ve performed the experiment called for in part (b), we can determine an experimental value for C. (a) Express T as the product of L and g raised to powers a and b: Write this equation in dimensional form: T = CLa g b (1) where C is a dimensionless constant. [T ] = [L] a [g ] b b Noting that the symbols for the dimension of the period and length of the pendulum are the same as those representing the physical quantities, substitute the dimensions to obtain: ⎛ L ⎞ T =L⎜ 2⎟ ⎝T ⎠ Because L does not appear on the left-hand side of the equation, we can write this equation as: L0T 1 = La +bT −2b Equate the exponents to obtain: Solve these equations simultaneously to find a and b: Substitute in equation (1) to obtain: (b) If you use pendulums of lengths 1 m and 0.5 m; the periods should be about: (c) Solve equation (2) for C: a a + b = 0 and − 2b = 1 a = 1 and b = − 1 2 2 T = CL1 2 g −1 2 = C T (1 m ) = 2 s and T (0.5 m ) = 1.4 s C =T g L L g (2) Systems of Measurement Evaluate C with L = 1 m and T = 2 s: 9.81 m/s 2 = 6.26 ≈ 2π 1m T = 2π Substitute in equation (2) to obtain: C = (2 s ) L g 63 ••• Picture the Problem The weight of the earth’s atmosphere per unit area is known as the atmospheric pressure. We can use this definition to express the weight w of the earth’s atmosphere as the product of the atmospheric pressure and the surface area of the earth. w A Using its definition, relate atmospheric pressure to the weight of the earth’s atmosphere: P= Solve for w: w = PA Relate the surface area of the earth to its radius R: A = 4π R 2 Substitute to obtain: w = 4π R 2 P Substitute numerical values and evaluate w: 2 ⎛ 103 m ⎞ ⎛ 39.37 in ⎞ ⎛ lb ⎞ 19 w = 4π (6370 km ) ⎜ km ⎟ ⎜ m ⎟ ⎜14.7 in 2 ⎟ = 1.16 × 10 lb ⎟ ⎠ ⎠ ⎝ ⎝ ⎠ ⎝ 2 2 31 32 Chapter 1 Chapter 2 Motion in One Dimension Conceptual Problems 1 • Determine the Concept The "average velocity" is being requested as opposed to "average speed". The average velocity is defined as the change in position or displacement divided by the change in time. The change in position for any "round trip" is zero by definition. So the average velocity for any round trip must also be zero. vav = ∆y ∆t vav = ∆y 0 = = 0 ∆t ∆t *2 • Determine the Concept The important concept here is that "average speed" is being requested as opposed to "average velocity". Under all circumstances, including constant acceleration, the definition of the average speed is the ratio of the total distance traveled (H + H) to the total time elapsed, in this case 2H/T. (d ) is correct. Remarks: Because this motion involves a round trip, if the question asked for "average velocity," the answer would be zero. 3 • Determine the Concept Flying with the wind, the speed of the plane relative to the ground (vPG) is the sum of the speed of the wind relative to the ground (vWG) and the speed of the plane relative to the air (vPG = vWG + vPA). Flying into or against the wind the speed relative to the ground is the difference between the wind speed and the true air speed of the plane (vg = vw – vt). Because the ground speed landing against the wind is smaller than the ground speed landing with the wind, it is safer to land against the wind. 4 • Determine the Concept The important concept here is that a = dv/dt, where a is the acceleration and v is the velocity. Thus, the acceleration is positive if dv is positive; the acceleration is negative if dv is negative. (a) Let’s take the direction a car is moving to be the positive direction: Because the car is moving in the direction we’ve chosen to be positive, its velocity is positive (dx > 0). If the car is braking, then its velocity is decreasing (dv < 0) and its acceleration (dv/dt) is negative. (b) Consider a car that is moving to Because the car is moving in the direction 33 34 Chapter 2 the right but choose the positive direction to be to the left: opposite to that we’ve chosen to be positive, its velocity is negative (dx < 0). If the car is braking, then its velocity is increasing (dv > 0) and its acceleration (dv/dt) is positive. *5 • Determine the Concept The important concept is that when both the acceleration and the velocity are in the same direction, the speed increases. On the other hand, when the acceleration and the velocity are in opposite directions, the speed decreases. (a) (b) Because your velocity remains negative, your displacement must be negative. Define the direction of your trip as the negative direction. During the last five steps gradually slow the speed of walking, until the wall is reached. (c) A graph of v as a function of t that is consistent with the conditions stated in the problem is shown below: 0 v (m/s) -1 -2 -3 -4 -5 0 0.5 1 1.5 2 2.5 t (s) 6 • Determine the Concept True. We can use the definition of average velocity to express the displacement ∆x as ∆x = vav∆t. Note that, if the acceleration is constant, the average velocity is also given by vav = (vi + vf)/2. 7 • Determine the Concept Acceleration is the slope of the velocity versus time curve, a = dv/dt; while velocity is the slope of the position versus time curve, v = dx/dt. The speed of an object is the magnitude of its velocity. Motion in One Dimension 35 (a) True. Zero acceleration implies that the velocity is constant. If the velocity is constant (including zero), the speed must also be constant. (b) True in one dimension. Remarks: The answer to (b) would be False in more than one dimension. In one dimension, if the speed remains constant, then the object cannot speed up, slow down, or reverse direction. Thus, if the speed remains constant, the velocity remains constant, which implies that the acceleration remains zero. (In more than one-dimensional motion, an object can change direction while maintaining constant speed. This constitutes a change in the direction of the velocity.) Consider a ball moving in a circle at a constant rotation rate. The speed (magnitude of the velocity) is constant while the velocity is tangent to the circle and always changing. The acceleration is always pointing inward and is certainly NOT zero. *8 •• Determine the Concept Velocity is the slope of the position versus time curve and acceleration is the slope of the velocity versus time curve. See the graphs below. 7 6 position (m) 5 4 3 2 1 0 0 5 10 15 20 25 15 20 25 time (s) 3 2 acceleration (m/s ) 2 1 0 -1 -2 -3 -4 0 5 10 time (s) 36 Chapter 2 1.0 0.8 velocity (m/s) 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0 5 10 15 20 25 time (s) 9 • Determine the Concept False. The average velocity is defined (for any acceleration) as the change in position (the displacement) divided by the change in time vav = ∆x ∆t . It is always valid. If the acceleration remains constant the average velocity is also given by vav = vi + vf 2 Consider an engine piston moving up and down as an example of non-constant velocity. For one complete cycle, vf = vi and xi = xf so vav = ∆x/∆t is zero. The formula involving the mean of vf and vi cannot be applied because the acceleration is not constant, and yields an incorrect nonzero value of vi. 10 • Determine the Concept This can occur if the rocks have different initial speeds. Ignoring air resistance, the acceleration is constant. Choose a coordinate system in which the origin is at the point of release and upward is the positive direction. From the constant-acceleration equation y = y0 + v0t + 1 at 2 2 we see that the only way two objects can have the same acceleration (–g in this case) and cover the same distance, ∆y = y – y0, in different times would be if the initial velocities of the two rocks were different. Actually, the answer would be the same whether or not the acceleration is constant. It is just easier to see for the special case of constant acceleration. *11 •• Determine the Concept Neglecting air resistance, the balls are in free fall, each with the same free-fall acceleration, which is a constant. At the time the second ball is released, the first ball is already moving. Thus, during any time interval their velocities will increase by exactly the same amount. What can be said about the speeds of the two balls? The first ball will always be moving faster than the second ball. This being the case, what happens to the separation of the two balls while they are both Motion in One Dimension 37 falling? Their separation increases. (a ) is correct. 12 •• Determine the Concept The slope of an x(t) curve at any point in time represents the speed at that instant. The way the slope changes as time increases gives the sign of the acceleration. If the slope becomes less negative or more positive as time increases (as you move to the right on the time axis), then the acceleration is positive. If the slope becomes less positive or more negative, then the acceleration is negative. The slope of the slope of an x(t) curve at any point in time represents the acceleration at that instant. The slope of curve (a) is negative and becomes more negative as time increases. Therefore, the velocity is negative and the acceleration is negative. The slope of curve (b) is positive and constant and so the velocity is positive and constant. Therefore, the acceleration is zero. The slope of curve (c) is positive and decreasing. Therefore, the velocity is positive and the acceleration is negative. The slope of curve (d) is positive and increasing. Therefore, the velocity and acceleration are positive. We need more information to conclude that a is constant. The slope of curve (e) is zero. Therefore, the velocity and acceleration are zero. (d ) best shows motion with constant positive acceleration. *13 • Determine the Concept The slope of a v(t) curve at any point in time represents the acceleration at that instant. Only one curve has a constant and positive slope. (b ) is correct. 14 • Determine the Concept No. The word average implies an interval of time rather than an instant in time; therefore, the statement makes no sense. *15 • Determine the Concept Note that the ″average velocity″ is being requested as opposed to the ″average speed.″ 38 Chapter 2 Yes. In any roundtrip, A to B, and back to A, the average velocity is zero. ∆x ∆xAB + ∆xBA = ∆t ∆t ∆x + (− ∆xBA ) 0 = AB = ∆t ∆t vav (A→B→A ) = = 0 On the other hand, the average velocity between A and B is not generally zero. vav (A→B ) = ∆xAB ≠ 0 ∆t Remarks: Consider an object launched up in the air. Its average velocity on the way up is NOT zero. Neither is it zero on the way down. However, over the round trip, it is zero. 16 • Determine the Concept An object is farthest from the origin when it is farthest from the time axis. In one-dimensional motion starting from the origin, the point located farthest from the time axis in a distance-versus-time plot is the farthest from its starting point. Because the object’s initial position is at x = 0, point B represents the instant that the object is farthest from x = 0. (b) is correct. 17 • Determine the Concept No. If the velocity is constant, a graph of position as a function of time is linear with a constant slope equal to the velocity. 18 • Determine the Concept Yes. The average velocity in a time interval is defined as the displacement divided by the elapsed time vav = ∆x ∆t . The fact that vav = 0 for some time interval, ∆t, implies that the displacement ∆x over this interval is also zero. Because the instantaneous velocity is defined as v = lim ∆t →0 (∆x / ∆t ) , it follows that v must also be zero. As an example, in the following graph of x versus t, over the interval between t = 0 and t ≈ 21 s, ∆x = 0. Consequently, vav = 0 for this interval. Note that the instantaneous velocity is zero only at t ≈ 10 s. Motion in One Dimension 39 600 500 x (m) 400 300 200 100 0 0 5 10 15 20 t (s) 19 •• Determine the Concept In the one-dimensional motion shown in the figure, the velocity is a minimum when the slope of a position-versus-time plot goes to zero (i.e., the curve becomes horizontal). At these points, the slope of the position-versus-time curve is zero; therefore, the speed is zero. (b) is correct. *20 •• Determine the Concept In one-dimensional motion, the velocity is the slope of a position-versus-time plot and can be either positive or negative. On the other hand, the speed is the magnitude of the velocity and can only be positive. We’ll use v to denote velocity and the word “speed” for how fast the object is moving. (a) curve a: v(t 2 ) < v(t1 ) (b) curve a: speed (t 2 ) < speed (t1 ) curve c: v(t 2 ) > v(t1 ) curve c: speed(t 2 ) < speed(t1 ) curve b: v(t 2 ) = v(t1 ) curve d: v(t 2 ) < v(t1 ) curve b: speed(t 2 ) = speed(t1 ) curve d: speed(t 2 ) > speed(t1 ) 21 • Determine the Concept Acceleration is the slope of the velocity-versus-time curve, a = dv/dt, while velocity is the slope of the position-versus-time curve, v = dx/dt. (a) False. Zero acceleration implies that the velocity is not changing. The velocity could be any constant (including zero). But, if the velocity is constant and nonzero, the particle must be moving. (b) True. Again, zero acceleration implies that the velocity remains constant. This means that the x-versus-t curve has a constant slope (i.e., a straight line). Note: This does not necessarily mean a zero-slope line. 40 Chapter 2 22 • Determine the Concept Yes. If the velocity is changing the acceleration is not zero. The velocity is zero and the acceleration is nonzero any time an object is momentarily at rest. If the acceleration were also zero, the velocity would never change; therefore, the object would have to remain at rest. Remarks: It is important conceptually to note that when both the acceleration and the velocity have the same sign, the speed increases. On the other hand, when the acceleration and the velocity have opposite signs, the speed decreases. 23 • Determine the Concept In the absence of air resistance, the ball will experience a constant acceleration. Choose a coordinate system in which the origin is at the point of release and the upward direction is positive. The graph shows the velocity of a ball that has been thrown straight upward with an initial speed of 30 m/s as a function of time. Note that the slope of this graph, the acceleration, is the same at every point, including the point at which v = 0 (at the top of its flight). Thus, vtop of flight = 0 and atop of flight = − g . 30 20 v (m/s) 10 0 -10 -20 -30 0 1 2 3 4 5 6 t (s) The acceleration is the slope (–g). 24 • Determine the Concept The "average speed" is being requested as opposed to "average velocity." We can use the definition of average speed as distance traveled divided by the elapsed time and expression for the average speed of an object when it is experiencing constant acceleration to express vav in terms of v0. The average speed is defined as the total distance traveled divided by the change in time: total distance traveled total time H + H 2H = = T T vav = Motion in One Dimension 41 v0 + 0 H = 1 2 2T Find the average speed for the upward flight of the object: vav,up = Solve for H to obtain: H = 1 v0T 4 Find the average speed for the downward flight of the object: vav,down = Solve for H to obtain: H = 1 v0T 4 Substitute in our expression for vav to obtain: 2( 1 v0T ) v 4 = 0 T 2 Because v0 ≠ 0 , the average speed is not 0 + v0 H = 1 2 2T vav = zero. Remarks: 1) Because this motion involves a roundtrip, if the question asked for ″average velocity″, the answer would be zero. 2) Another easy way to obtain this result is take the absolute value of the velocity of the object to obtain a graph of its speed as a function of time. A simple geometric argument leads to the result we obtained above. 25 • Determine the Concept In the absence of air resistance, the bowling ball will experience constant acceleration. Choose a coordinate system with the origin at the point of release and upward as the positive direction. Whether the ball is moving upward and slowing down, is momentarily at the top of its trajectory, or is moving downward with ever increasing velocity, its acceleration is constant and equal to the acceleration due to gravity. (b) is correct. 26 • Determine the Concept Both objects experience the same constant acceleration. Choose a coordinate system in which downward is the positive direction and use a constantacceleration equation to express the position of each object as a function of time. Using constant-acceleration equations, express the positions of both objects as functions of time: xA = x0, A + v0t + 1 gt 2 2 and xB = x0, B + v0t + 1 gt 2 2 where v0 = 0. Express the separation of the two objects by evaluating xB − xA: xB − xA = x0,B − x0.A = 10 m and (d ) is correct. *27 •• Determine the Concept Because the Porsche accelerates uniformly, we need to look for a graph that represents constant acceleration. We are told that the Porsche has a constant acceleration that is positive (the velocity is increasing); therefore we must look for a velocity-versus-time curve with a positive constant slope and a nonzero intercept. 42 Chapter 2 (c ) is correct. *28 •• Determine the Concept In the absence of air resistance, the object experiences constant acceleration. Choose a coordinate system in which the downward direction is positive. Express the distance D that an object, released from rest, falls in time t: Because the distance fallen varies with the square of the time, during the first two seconds it falls four times the distance it falls during the first second. D = 1 gt 2 2 (a ) is correct. 29 •• Determine the Concept In the absence of air resistance, the acceleration of the ball is constant. Choose a coordinate system in which the point of release is the origin and upward is the positive y direction. The displacement of the ball halfway to its highest point is: Using a constant-acceleration equation, relate the ball’s initial and final velocities to its displacement and solve for the displacement: ∆y = ∆ymax 2 2 2 v 2 = v0 + 2a∆y = v0 − 2 g∆y Substitute v0 = 0 to determine the maximum displacement of the ball: ∆ymax = − Express the velocity of the ball at half its maximum height: 2 2 v 2 = v0 − 2 g∆y = v0 − 2 g 2 v0 v2 = 0 2(− g ) 2 g 2 = v0 − g∆ymax Solve for v: ∆ymax 2 2 v v2 2 = v0 − g 0 = 0 2g 2 2 v0 ≈ 0.707v0 2 and (c ) is correct. v= 30 • Determine the Concept As long as the acceleration remains constant the following constant-acceleration equations hold. If the acceleration is not constant, they do not, in general, give correct results except by coincidence. x = x0 + v0t + 1 at 2 2 v = v0 + at 2 v 2 = v0 + 2a∆x vav = vi + vf 2 Motion in One Dimension 43 (a) False. From the first equation, we see that (a) is true if and only if the acceleration is constant. (b) False. Consider a rock thrown straight up into the air. At the "top" of its flight, the velocity is zero but it is changing (otherwise the velocity would remain zero and the rock would hover); therefore the acceleration is not zero. (c) True. The definition of average velocity, vav = ∆x ∆t , requires that this always be true. *31 • Determine the Concept Because the acceleration of the object is constant, the constantacceleration equations can be used to describe its motion. The special expression for average velocity for constant acceleration is vav = vi + vf . (c ) is correct. 2 32 • Determine the Concept The constant slope of the x-versus-t graph tells us that the velocity is constant and the acceleration is zero. A linear position versus time curve implies a constant velocity. The negative slope indicates a constant negative velocity. The fact that the velocity is constant implies that the acceleration is also constant and zero. (e ) is correct. 33 •• Determine the Concept The velocity is the slope of the tangent to the curve, and the acceleration is the rate of change of this slope. Velocity is the slope of the positionversus-time curve. A parabolic x(t) curve opening upward implies an increasing velocity. The acceleration is positive. (a ) is correct. 34 •• Determine the Concept The acceleration is the slope of the tangent to the velocity as a function of time curve. For constant acceleration, a velocity-versus- time curve must be a straight line whose slope is the acceleration. Zero acceleration means that slope of v(t) must also be zero. (c ) is correct. 35 •• Determine the Concept The acceleration is the slope of the tangent to the velocity as a function of time curve. For constant acceleration, a velocity-versus- time curve must be a straight line whose slope is the acceleration. The acceleration and therefore the slope can be positive, negative, or zero. (d ) is correct. 36 •• Determine the Concept The velocity is positive if the curve is above the v = 0 line (the t axis), and the acceleration is negative if the tangent to the curve has a negative slope. Only graphs (a), (c), and (e) have positive v. Of these, only graph (e) has a negative slope. (e ) is correct. 44 Chapter 2 37 •• Determine the Concept The velocity is positive if the curve is above the v = 0 line (the t axis), and the acceleration is negative if the tangent to the curve has a negative slope. Only graphs (b) and (d) have negative v. Of these, only graph (d) has a negative slope. (d ) is correct. 38 •• Determine the Concept A linear velocity-versus-time curve implies constant acceleration. The displacement from time t = 0 can be determined by integrating vversus-t — that is, by finding the area under the curve. The initial velocity at t = 0 can be read directly from the graph of v-versus-t as the v-intercept; i.e., v(0). The acceleration of the object is the slope of v(t) . The average velocity of the object is given by drawing a horizontal line that has the same area under it as the area under the curve. Because all of these quantities can be determined (e ) is correct. *39 •• Determine the Concept The velocity is the slope of a position versus time curve and the acceleration is the rate at which the velocity, and thus the slope, changes. Velocity (a) Negative at t0 and t1. (b) Positive at t3, t4, t6, and t7. (c) Zero at t2 and t5. Acceleration (a) Negative at t4. (b) Positive at t2 and t6. (c) Zero at t0, t1, t3, t5, and t7. The acceleration is positive at points where the slope increases as you move toward the right. 40 •• Determine the Concept Acceleration is the slope of a velocity-versus-time curve. (a) Acceleration is zero and constant while velocity is not zero. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 t 2 2.5 3 Motion in One Dimension (b) Acceleration is constant but not zero. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 2 2.5 3 2 2.5 3 2 2.5 3 t (c) Velocity and acceleration are both positive. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 t (d) Velocity and acceleration are both negative. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 t 45 46 Chapter 2 (e) Velocity is positive and acceleration is negative. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 2 2.5 3 2 2.5 3 t (f) Velocity is negative and acceleration is positive. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 t Motion in One Dimension 47 (g) Velocity is momentarily zero at the intercept with the t axis but the acceleration is not zero. 3 2 v 1 0 -1 -2 -3 0 0.5 1 1.5 2 2.5 3 t 41 •• Determine the Concept Velocity is the slope and acceleration is the slope of the slope of a position-versus-time curve. Acceleration is the slope of a velocity- versus-time curve. (a) For constant velocity, x-versus-t must be a straight line; v-versus-t must be a horizontal straight line; and a-versus-t must be a straight horizontal line at a = 0. (b) For velocity to reverse its direction x-versus-t must have a slope that changes sign and vversus-t must cross the time axis. The acceleration cannot remain zero at all times. (a), (f), and (i) are the correct answers. (c) and (d) are the correct answers. (c) For constant acceleration, x-versus-t must be a straight line or a parabola, v-versus-t must be a straight line, and a-versus-t must be a horizontal straight line. (a), (d), (e), (f), (h), and (i) are the correct answers. (d) For non-constant acceleration, x-versus-t must not be a straight line or a parabola; v-versus-t must not be a straight line, or a-versus-t must not be a horizontal straight line. (b), (c), and (g) are the correct answers. 48 Chapter 2 For two graphs to be mutually consistent, the curves must be consistent with the definitions of velocity and acceleration. Graphs (a) and (i) are mutually consistent. Graphs (d) and (h) are mutually consistent. Graphs (f) and (i) are also mutually consistent. Estimation and Approximation 42 • Picture the Problem Assume that your heart beats at a constant rate. It does not, but the average is pretty stable. (a) We will use an average pulse rate of 70 bpm for a seated (resting) adult. One’s pulse rate is defined as the number of heartbeats per unit time: Pulse rate = and # of heartbeats Time # of heartbeats = Pulse rate × Time The time required to drive 1 mi at 60 mph is (1/60) h or 1 min: # of heartbeats = (70 beats/min )(1 min ) (b) Express the number of heartbeats during a lifetime in terms of the pulse rate and the life span of an individual: # of heartbeats = Pulse rate × Time = 70 beats Assuming a 95-y life span, calculate the time in minutes: Time = (95 y )(365.25 d/y )(24 h/d )(60 min/ h ) = 5.00 × 107 min Substitute numerical values and evaluate the number of heartbeats: ( ) # of heartbeats = (70 beats / min ) 5.00 × 107 min = 3.50 ×109 beats *43 •• Picture the Problem In the absence of air resistance, Carlos’ acceleration is constant. Because all the motion is downward, let’s use a coordinate system in which downward is positive and the origin is at the point at which the fall began. (a) Using a constant-acceleration equation, relate Carlos’ final velocity to his initial velocity, acceleration, and distance fallen and solve for his final velocity: Substitute numerical values and evaluate v: 2 v 2 = v0 + 2a∆y and, because v0 = 0 and a = g, v = 2 g∆y v = 2(9.81 m/s 2 )(150 m ) = 54.2 m/s Motion in One Dimension (b) While his acceleration by the snow is not constant, solve the same constant- acceleration equation to get an estimate of his average acceleration: 2 v 2 − v0 a= 2∆y Substitute numerical values and evaluate a: 49 − 54 m/s 2 a= = −1.20 × 103 m/s 2 (1.22m ) 2 ( ) 2 = − 123 g Remarks: The final velocity we obtained in part (a), approximately 121 mph, is about the same as the terminal velocity for an "average" man. This solution is probably only good to about 20% accuracy. 44 •• Picture the Problem Because we’re assuming that the accelerations of the skydiver and the mouse are constant to one-half their terminal velocities, we can use constantacceleration equations to find the times required for them to reach their ″upper-bound″ velocities and their distances of fall. Let’s use a coordinate system in which downward is the positive y direction. (a) Using a constant-acceleration equation, relate the upper-bound velocity to the free-fall acceleration and the time required to reach this velocity: Solve for ∆t: Substitute numerical values and evaluate ∆t: Using a constant-acceleration equation, relate the skydiver’s distance of fall to the elapsed time ∆t: Substitute numerical values and evaluate ∆y: (b) Proceed as in (a) with vupper bound = 0.5 m/s to obtain: vupper bound = v0 + g∆t or, because v0 = 0, vupper bound = g∆t ∆t = ∆t = vupper bound g 25 m/s = 2.55 s 9.81m/s 2 ∆y = v0 ∆t + 1 a(∆t ) 2 2 or, because v0 = 0 and a = g, ∆y = 1 g (∆t ) 2 2 ∆y = ∆t = 1 2 2 2 = 31.9 m 0.5 m/s = 0.0510 s 9.81m/s 2 and ∆y = (9.81m/s ) (2.55 s) 1 2 (9.81m/s )(0.0510 s) 2 2 = 1.27 cm 50 Chapter 2 45 •• Picture the Problem This is a constant-acceleration problem. Choose a coordinate system in which the direction Greene is running is the positive x direction. During the first 3 s of the race his acceleration is positive and during the rest of the race it is zero. The pictorial representation summarizes what we know about Greene’s race. Express the total distance covered by Greene in terms of the distances covered in the two phases of his race: 100 m = ∆x01 + ∆x12 Express the distance he runs getting to his maximum velocity: ∆x01 = v0 ∆t01 + 1 a01 (∆t01 ) = 1 a(3 s ) 2 2 Express the distance covered during the rest of the race at the constant maximum velocity: ∆x12 = vmax ∆t12 + 1 a12 (∆t12 ) 2 Substitute for these displacements and solve for a: 100 m = 1 a(3 s ) + a(3 s )(6.79 s ) 2 2 2 2 = (a∆t01 )∆t12 = a(3 s )(6.79 s ) 2 and a = 4.02 m/s 2 *46 •• Determine the Concept This is a constant-acceleration problem with a = −g if we take upward to be the positive direction. At the maximum height the ball will reach, its speed will be near zero and when the ball has just been tossed in the air its speed is near its maximum value. What conclusion can you draw from the image of the ball near its maximum height? To estimate the initial speed of the ball: Because the ball is moving slowly its blur is relatively short (i.e., there is less blurring). Motion in One Dimension 51 a) Estimate how far the ball being tossed moves in 1/30 s: The ball moves about 3 ball diameters in 1/30 s. b) Estimate the diameter of a tennis ball: The diameter of a tennis ball is approximately 5 cm. c) Now one can calculate the approximate distance the ball moved in 1/30 s: Distance traveled = (3 diameters) × (5 cm/diameter ) = 15 cm d) Calculate the average speed of the tennis ball over this distance: Average speed = e) Because the time interval is very short, the average speed of the ball is a good approximation to its initial speed: f) Finally, use the constantacceleration equation 2 v 2 = v0 + 2a∆y to solve for and 15 cm = 450 cm/s 1 s 30 = 4.50 m/s v0 = 4.5 m/s 2 − v0 − (4.5 m/s ) = = 1.03 m 2a 2 − 9.81 m/s 2 2 ∆y = ( ) evaluate ∆y: Remarks: This maximum height is in good agreement with the height of the higher ball in the photograph. *47 •• Picture the Problem The average speed of a nerve impulse is approximately 120 m/s. Assume an average height of 1.7 m and use the definition of average speed to estimate the travel time for the nerve impulse. Using the definition of average speed, express the travel time for the nerve impulse: ∆t = ∆x vav Substitute numerical values and evaluate ∆t: ∆t = 1.7 m = 14.2 ms 120 m/s Speed, Displacement, and Velocity 48 • Picture the Problem Think of the electron as traveling in a straight line at constant speed and use the definition of average speed. 52 Chapter 2 (a) Using its definition, express the average speed of the electron: Average speed = Solve for and evaluate the time of flight: ∆t = distance traveled time of flight ∆s = ∆t ∆s 0.16 m = Average speed 4 × 107 m s = 4 × 10−9 s = 4.00 ns (b) Calculate the time of flight for an electron in a 16-cm long current carrying wire similarly. ∆t = ∆s 0.16 m = Average speed 4 × 10 −5 m s = 4 × 103 s = 66.7 min *49 • Picture the Problem In this problem the runner is traveling in a straight line but not at constant speed - first she runs, then she walks. Let’s choose a coordinate system in which her initial direction of motion is taken as the positive x direction. (a) Using the definition of average velocity, calculate the average velocity for the first 9 min: vav = ∆x 2.5 km = = 0.278 km / min ∆t 9 min (b) Using the definition of average velocity, calculate her average speed for the 30 min spent walking: vav = ∆x − 2.5 km = ∆t 30 min = − 0.0833 km / min (c) Express her average velocity for the whole trip: vav = (d) Finally, express her average speed for the whole trip: Average speed = ∆xround trip ∆t = 0 = 0 ∆t distance traveled elapsed time 2(2.5 km) = 30 min + 9 min = 0.128 km / min 50 • Picture the Problem The car is traveling in a straight line but not at constant speed. Let the direction of motion be the positive x direction. (a) Express the total displacement of the car for the entire trip: ∆x total = ∆x1 + ∆x2 Motion in One Dimension Find the displacement for each leg of the trip: 53 ∆x1 = vav ,1∆t1 = (80 km/h )(2.5 h ) = 200 km and ∆x2 = vav , 2 ∆t2 = (40 km/h )(1.5 h ) = 60.0 km Add the individual displacements to get the total displacement: (b) As long as the car continues to move in the same direction, the average velocity for the total trip is given by: ∆xtotal = ∆x1 + ∆x2 = 200 km + 60.0 km = 260 km vav ≡ 260 km ∆xtotal = ∆ttotal 2.5 h + 1.5 h = 65.0 km h 51 • Picture the Problem However unlikely it may seem, imagine that both jets are flying in a straight line at constant speed. (a) The time of flight is the ratio of the distance traveled to the speed of the supersonic jet. tsupersonic = = sAtlantic speedsupersonic 5500 km 2(0.340 km/s )(3600 s/h ) = 2.25 h (b) The time of flight is the ratio of the distance traveled to the speed of the subsonic jet. tsubsonic = = sAtlantic speed subsonic 5500 km 0.9(0.340 km/s )(3600 s/h ) = 4.99 h (c) Adding 2 h on both the front and the back of the supersonic trip, we obtain the average speed of the supersonic flight. (d) Adding 2 h on both the front and the back of the subsonic trip, we obtain the average speed of the subsonic flight. speed av, supersonic = 5500 km 2.25 h + 4.00 h = 880 km h speed av, subsonic = 5500 km 5.00 h + 4.00 h = 611 km h 54 Chapter 2 *52 • Picture the Problem In free space, light travels in a straight line at constant speed, c. (a) Using the definition of average speed, solve for and evaluate the time required for light to travel from the sun to the earth: average speed = s t and t= s 1.5 × 1011 m = average speed 3 × 108 m/s = 500 s = 8.33 min (b) Proceed as in (a) this time using the moon-earth distance: (c) One light-year is the distance light travels in a vacuum in one year: t= 3.84 ×108 m = 1.28 s 3×108 m/s 1 light - year = 9.48 ×1015 m = 9.48 ×1012 km ( ) = 9.48 ×1012 km (1 mi/1.61 km ) = 5.89 ×10 mi 12 53 • Picture the Problem In free space, light travels in a straight line at constant speed, c. (a) Using the definition of average speed (equal here to the assumed constant speed of light), solve for the time required to travel the distance to Proxima Centauri: (b) Traveling at 10-4c, the delivery time (ttotal) will be the sum of the time for the order to reach Hoboken and the time for the pizza to be delivered to Proxima Centauri: t= distance traveled 4.1×1016 m = speed of light 3×108 m s = 1.37 ×108 s = 4.33 y t total = torder to be sent to Hoboken + torder to be delivered 4.1×1013 km = 4.33 y + − 4 10 3 × 108 m s ( )( ) = 4.33 y + 4.33×106 y ≈ 4.33×106 y Since 4.33 × 10 6 y >> 1000 y, Gregor does not have to pay. 54 • Picture the Problem The time for the second 50 km is equal to the time for the entire journey less the time for the first 50 km. We can use this time to determine the average speed for the second 50 km interval from the definition of average speed. Using the definition of average speed, find the time required for the total journey: t total = total distance 100 km = = 2h average speed 50 km h Motion in One Dimension Find the time required for the first 50 km: t1st 50 km = Find the time remaining to travel the last 50 km: t2nd 50 km = t total − t1st 50 km = 2 h − 1.25 h Finally, use the time remaining to travel the last 50 km to determine the average speed over this distance: 55 Average speed 2nd 50 km 50 km = 1.25 h 40 km h = 0.75 h = distance traveled2nd 50 km time2nd 50 km = 50 km = 66.7 km h 0.75 h *55 •• Picture the Problem Note that both the arrow and the sound travel a distance d. We can use the relationship between distance traveled, the speed of sound, the speed of the arrow, and the elapsed time to find the distance separating the archer and the target. Express the elapsed time between the archer firing the arrow and hearing it strike the target: Express the transit times for the arrow and the sound in terms of the distance, d, and their speeds: ∆t = 1s = ∆tarrow + ∆tsound ∆tarrow = varrow = d 40 m/s = d 340 m/s and ∆tsound = Substitute these two relationships in the expression obtained in step 1 and solve for d: d d vsound d d + = 1s 40 m/s 340 m/s and d = 35.8 m 56 •• Picture the Problem Assume both runners travel parallel paths in a straight line along the track. (a) Using the definition of average speed, find the time for Marcia: distance run Marcia' s speed distance run = 1.15 (John' s speed ) 100 m = = 14.5 s 1.15 (6 m s ) tMarcia = 56 Chapter 2 xJohn = (6 m s )(14.5 s ) = 87.0 m Find the distance covered by John in 14.5 s and the difference between that distance and 100 m: and Marcia wins by (b) Using the definition of average speed, find the time required by John to complete the 100-m run: tJohn = 100 m − 87 m = 13.0 m distance run 100 m = = 16.7 s John' s speed 6 m s Marsha wins by 16.7 s – 14.5 s = 2.2 s Alternatively, the time required by John to travel the last 13.0 m is (13 m)/(6 m/s) = 2.17 s 57 • Picture the Problem The average velocity in a time interval is defined as the displacement divided by the time elapsed; that is vav = ∆x / ∆t . (a) ∆xa = 0 vav = 0 (b) ∆xb = 1 m and ∆tb = 3 s vav = 0.333 m/s (c) ∆xc = –6 m and ∆tc = 3 s vav = − 2.00 m/s (d) ∆xd = 3 m and ∆td = 3 s vav = 1.00 m/s 58 •• Picture the Problem In free space, light travels in a straight line at constant speed c. We can use Hubble’s law to find the speed of the two planets. ( )( (a) Using Hubble’s law, calculate the speed of the first galaxy: va = 5 × 10 22 m 1.58 × 10 −18 s −1 (b) Using Hubble’s law, calculate the speed of the second galaxy: vb = 2 × 10 25 m 1.58 × 10 −18 s −1 (c) Using the relationship between distance, speed, and time for both galaxies, determine how long ago they were both located at the same place as the earth: ) = 7.90 × 10 4 m/s ( )( = 3.16 × 107 m/s r r 1 = = v rH H = 6.33 × 1017 s = 20.1× 109 y t= = 20.1 billion years ) Motion in One Dimension 57 *59 •• Picture the Problem Ignoring the time intervals during which members of this relay time get up to their running speeds, their accelerations are zero and their average speed can be found from its definition. Using its definition, relate the average speed to the total distance traveled and the elapsed time: vav = Express the time required for each animal to travel a distance L: tcheetah = distance traveled elapsed time tfalcon = L vcheetah L vfalcon , , and tsailfish = Express the total time, ∆t: L vsailfish ⎛ 1 1 1 ⎞ ⎟ ∆t = L⎜ + + ⎟ ⎜v ⎝ cheetah vfalcon vsailfish ⎠ Use the total distance traveled by the relay team and the elapsed time to calculate the average speed: vav = 3L = 122 km/h ⎛ ⎞ 1 1 1 L⎜ ⎜ 113 km/h + 161 km/h + 105 km/h ⎟ ⎟ ⎝ ⎠ Calculate the average of the three speeds: Averagethree speeds = 113 km/h + 161 km/h + 105 km/h 3 = 126 km/h = 1.03vav 60 •• Picture the Problem Perhaps the easiest way to solve this problem is to think in terms of the relative velocity of one car relative to the other. Solve this problem from the reference frame of car A. In this frame, car A remains at rest. Find the velocity of car B relative to car A: vrel = vB – vA = (110 – 80) km/h = 30 km/h Find the time before car B reaches car A: ∆t = Find the distance traveled, relative to the road, by car A in 1.5 h: d = (1.5 h )(80 km/h ) = 120 km 45 km ∆x = = 1.5 h vrel 30 km/h 58 Chapter 2 *61 •• Picture the Problem One way to solve this problem is by using a graphing calculator to plot the positions of each car as a function of time. Plotting these positions as functions of time allows us to visualize the motion of the two cars relative to the (fixed) ground. More importantly, it allows us to see the motion of the two cars relative to each other. We can, for example, tell how far apart the cars are at any given time by determining the length of a vertical line segment from one curve to the other. (a) Letting the origin of our coordinate system be at the intersection, the position of the slower car, x1(t), is given by: x1(t) = 20t where x1 is in meters if t is in seconds. Because the faster car is also moving at a constant speed, we know that the position of this car is given by a function of the form: x2(t) = 30t + b We know that when t = 5 s, this second car is at the intersection (i.e., x2(5 s) = 0). Using this information, you can convince yourself that: b = −150 m Thus, the position of the faster car is given by: x2 (t ) = 30t − 150 One can use a graphing calculator, graphing paper, or a spreadsheet to obtain the graphs of x1(t) (the solid line) and x2(t) (the dashed line) shown below: 350 300 x (m) 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 t (s) (b) Use the time coordinate of the intersection of the two lines to determine the time at which the second car overtakes the first: From the intersection of the two lines, one can see that the second car will "overtake" (catch up to) the first car at t = 15 s . Motion in One Dimension 59 (c) Use the position coordinate of the intersection of the two lines to determine the distance from the intersection at which the second car catches up to the first car: From the intersection of the two lines, one can see that the distance from the (d) Draw a vertical line from t = 5 s to the red line and then read the position coordinate of the intersection of this line and the red line to determine the position of the first car when the second car went through the intersection: From the graph, when the second car passes the intersection, the first car was intersection is 300 m . 100 m ahead . 62 • Picture the Problem Sally’s velocity relative to the ground (vSG) is the sum of her velocity relative to the moving belt (vSB) and the velocity of the belt relative to the ground (vBG). Joe’s velocity relative to the ground is the same as the velocity of the belt relative to the ground. Let D be the length of the moving sidewalk. Express D in terms of vBG (Joe’s speed relative to the ground): Solve for vBG: Express D in terms of vBG + vSG (Sally’s speed relative to the ground): Solve for vSG: Express D in terms of vBG + 2vSB (Sally’s speed for a fast walk relative to the ground): D = (2 min ) vBG vBG = D 2 min D = (1 min )(vBG + vSG ) ⎛ D ⎞ = (1 min )⎜ ⎜ 2 min + vSG ⎟ ⎟ ⎝ ⎠ vSG = D D D − = 1min 2 min 2 min ⎛ D 2D ⎞ + D = tf (vBG + 2vSB ) = tf ⎜ ⎜ 2 min 2 min ⎟ ⎟ ⎝ ⎠ ⎛ 3D ⎞ = tf ⎜ ⎜ 2 min ⎟ ⎟ ⎝ ⎠ Solve for tf as time for Sally's fast walk: tf = 2 min = 40.0 s 3 60 Chapter 2 63 •• Picture the Problem The speed of Margaret’s boat relative to the riverbank ( vBR ) is the sum or difference of the speed of her boat relative to the water ( vBW ) and the speed of the water relative to the riverbank ( vWR ), depending on whether she is heading with or against the current. Let D be the distance to the marina. Express the total time for the trip: t tot = t1 + t 2 Express the times of travel with the motor running in terms of D, vWR t1 = and vBW : and vBW t2 = Express the time required to drift distance D and solve for vWR : t3 = D = 4h − vWR vBW D + vWR D = 8h vWR and vWR = From t1 = 4 h, find vBW : Solve for t2: Add t1 and t2 to find the total time: D 8h vBW = D D D 3D + vWR = + = 4h 4h 8h 8h t2 = vBW D D = = 2h 3D D + vWR + 8h 8h t tot = t1 + t 2 = 6 h Acceleration 64 • Picture the Problem In part (a), we can apply the definition of average acceleration to find aav. In part (b), we can find the change in the car’s velocity in one second and add this change to its velocity at the beginning of the interval to find its speed one second later. (a) Apply the definition of average acceleration: ∆v 80.5 km/h − 48.3 km/h = 3.7 s ∆t km = 8.70 h ⋅s aav = Motion in One Dimension Convert to m/s2: m ⎞⎛ 1h ⎞ ⎛ ⎟ aav = ⎜ 8.70 × 103 ⎟⎜ h ⋅ s ⎠⎜ 3600 s ⎟ ⎝ ⎝ ⎠ = 2.42 m/s 2 (b) Express the speed of the car at the end of 4.7 s: v(4.7 s ) = v(3.7 s ) + ∆v1s Find the change in the speed of the car in 1 s: km ⎞ ⎛ ∆v = aav ∆t = ⎜ 8.70 ⎟(1s ) h ⋅s ⎠ ⎝ = 8.70 km/h Substitute and evaluate v(4.7 s): v(4.7 s ) = 80.5 km/h + 8.7 km/h = 80.5 km/h + ∆v1s = 89.2 km/h 65 • Picture the Problem Average acceleration is defined as aav = ∆v/∆t. The average acceleration is defined as the change in velocity divided by the change in time: aav = ∆v (− 1m/s ) − (5m/s ) = (8s ) − (5s ) ∆t = − 2.00 m/s 2 66 •• Picture the Problem The important concept here is the difference between average acceleration and instantaneous acceleration. (a) The average acceleration is defined as the change in velocity divided by the change in time: aav = ∆v/∆t Determine v at t = 3 s, t = 4 s, and t = 5 s: v(3 s) = 17 m/s v(4 s) = 25 m/s v(5 s) = 33 m/s Find aav for the two 1-s intervals: aav(3 s to 4 s) = (25 m/s – 17 m/s)/(1 s) = 8 m/s2 and aav(4 s to 5 s) = (33 m/s – 25 m/s)/(1 s) = 8 m/s2 The instantaneous acceleration is defined as the time derivative of the velocity or the slope of the velocityversus-time curve: a= dv = 8.00 m/s 2 dt 61 62 Chapter 2 (b) The given function was used to plot the following spreadsheet-graph of v-versus-t: 35 30 25 v (m/s) 20 15 10 5 0 -5 -10 0 1 2 3 4 5 t (s) 67 •• Picture the Problem We can closely approximate the instantaneous velocity by the average velocity in the limit as the time interval of the average becomes small. This is important because all we can ever obtain from any measurement is the average velocity, vav, which we use to approximate the instantaneous velocity v. (a) Find x(4 s) and x(3 s): x(4 s) = (4)2 – 5(4) + 1 = –3 m and x(3 s) = (3)2 – 5(3) + 1 = −5 m Find ∆x: ∆x = x(4 s) – x(3 s) = (–3 m) – (–5 m) = 2m Use the definition of average velocity: vav = ∆x/∆t = (2 m)/(1 s) = 2 m/s (b) Find x(t + ∆t): x(t + ∆t) = (t + ∆t)2 − 5(t + ∆t) + 1 = (t2 + 2t∆t + (∆t)2) – 5(t + ∆t) + 1 Express x(t + ∆t) – x(t) = ∆x: ∆x = (2t − 5)∆t + (∆t )2 where ∆x is in meters if t is in seconds. (c) From (b) find ∆x/∆t as ∆t → 0: ∆x (2t − 5)∆t + (∆t ) = ∆t ∆t = 2t − 5 + ∆t 2 and Motion in One Dimension 63 v = lim ∆t →0 (∆x / ∆t ) = 2t − 5 where v is in m/s if t is in seconds. Alternatively, we can take the derivative of x(t) with respect to time to obtain the instantaneous velocity. v(t ) = dx(t ) dt = ( ) d at 2 + bt + 1 dt = 2at + b = 2t − 5 *68 •• Picture the Problem The instantaneous velocity is dx dt and the acceleration is dv dt . Using the definitions of instantaneous velocity and acceleration, determine v and a: v= and a= Substitute numerical values for A and B and evaluate v and a: [ dx d = At 2 − Bt + C = 2 At − B dt dt dv d = [2 At − B ] = 2 A dr dt ( ) (16 m/s ) t − 6m/s v = 2 8m/s 2 t − 6 m/s = and 2 ( ) a = 2 8 m/s 2 = 16.0 m/s 2 69 •• Picture the Problem We can use the definition of average acceleration (aav = ∆v/∆t) to find aav for the three intervals of constant acceleration shown on the graph. (a) Using the definition of average acceleration, find aav for the interval AB: Find aav for the interval BC: Find aav for the interval CE: aav, AB = 15 m/s − 5 m/s = 3.33 m/s 2 3s aav, BC = 15 m/s − 15 m/s = 0 3s aav, CE = − 15 m/s − 15m/s = − 7.50m/s 2 4s (b) Use the formulas for the areas of trapezoids and triangles to find the area under the graph of v as a function of t. 64 Chapter 2 ∆x = (∆x )A→B + (∆x )B→C + (∆x )C→D + (∆x )D→E = 1 2 (5 m/s + 15 m/s)(3 s ) + (15 m/s)(3 s) + 1 (15 m/s)(2 s) + 1 (−15 m/s)(2 s) 2 2 = 75.0 m (c) The graph of displacement, x, as a function of time, t, is shown in the following figure. In the region from B to C the velocity is constant so the x- versus-t curve is a straight line. 100 x (m) 80 60 40 20 0 0 2 4 6 8 10 t (s) (d) Reading directly from the figure, we can find the time when the particle is moving the slowest. At point D, t = 8 s, the graph crosses the time axis; therefore, v = 0. Constant Acceleration and Free-Fall *70 • Picture the Problem Because the acceleration is constant (–g) we can use a constantacceleration equation to find the height of the projectile. Using a constant-acceleration equation, express the height of the object as a function of its initial velocity, the acceleration due to gravity, and its displacement: 2 v 2 = v0 + 2a∆y Solve for ∆ymax = h: Because v(h) = 0, h= From this expression for h we see that the maximum height attained is proportional to the square of the launch speed: 2 − v0 v2 = 0 2(− g ) 2 g 2 h ∝ v0 Motion in One Dimension h2v 0 (2v0 )2 = and Therefore, doubling the initial speed gives four times the height: 65 (a ) is correct. ⎛ v2 ⎞ = 4⎜ 0 ⎟ = 4hv0 ⎜ 2g ⎟ 2g ⎝ ⎠ 71 • Picture the Problem Because the acceleration of the car is constant we can use constantacceleration equations to describe its motion. ( ) (a) Uing a constant-acceleration equation, relate the velocity to the acceleration and the time: v = v0 + at = 0 + 8 m s 2 (10 s ) (b) sing a constant-acceleration equation, relate the displacement to the acceleration and the time: a ∆x = x − x0 = v0t + t 2 2 Substitute numerical values and evaluate ∆x: ∆x = 1 2 8 m s 2 (10 s ) = 400 m 2 vav = ∆x 400 m = = 40.0 m/s ∆t 10 s (c) Use the definition of vav: = 80.0 m s ( ) Remarks: Because the area under a velocity-versus-time graph is the displacement of the object, we could solve this problem graphically. 72 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement: Solve for and evaluate the displacement: 2 v 2 = v0 + 2 a ∆x ∆x = 2 v 2 − v0 (152 − 52 )m 2 s 2 = 2a 2 (2 m s 2 ) = 50.0 m *73 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement: 2 v 2 = v0 + 2 a ∆x 66 Chapter 2 2 v 2 − v0 2 ∆x Solve for the acceleration: a= Substitute numerical values and evaluate a: (15 a= 2 ) − 102 m 2 s 2 = 15.6 m s 2 2(4 m ) 74 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement: Solve for and evaluate v: 2 v 2 = v0 + 2 a ∆x (1 m s )2 + 2 (4 m v= ) s 2 (1 m ) = 3.00 m/s Using the definition of average acceleration, solve for the time: t= ∆v 3 m s − 1 m s = = 0.500 s aav 4 m s2 75 •• Picture the Problem In the absence of air resistance, the ball experiences constant acceleration. Choose a coordinate system with the origin at the point of release and the positive direction upward. (a) Using a constant-acceleration equation, relate the displacement of the ball to the acceleration and the time: Setting ∆y = 0 (the displacement for a round trip), solve for and evaluate the time required for the ball to return to its starting position: (b) Using a constant-acceleration equation, relate the final speed of the ball to its initial speed, the acceleration, and its displacement: Solve for and evaluate H: (c) Using the same constantacceleration equation with which we began part (a), express the displacement as a function of time: ∆y = v0t + 1 at 2 2 tround trip = 2v0 2(20 m/s ) = = 4.08 s g 9.81m/s 2 2 2 vtop = v0 + 2a∆y or, because vtop = 0 and a = −g, 2 0 = v 0 + 2(− g )H 2 (20 m s ) = 20.4 m v0 = 2 g 2 9.81m s 2 2 H= ( ∆y = v0t + 1 at 2 2 ) Motion in One Dimension Substitute numerical values to obtain: ⎛ 9.81 m/s 2 ⎞ 2 ⎟t 15 m = (20 m/s )t − ⎜ ⎜ ⎟ 2 ⎝ ⎠ Solve the quadratic equation for the times at which the displacement of the ball is 15 m: The solutions are t = 0.991s 67 (this corresponds to passing 15 m on the way up) and t = 3.09 s (this corresponds to passing 15 m on the way down). 76 •• Picture the Problem This is a multipart constant-acceleration problem using two different constant accelerations. We’ll choose a coordinate system in which downward is the positive direction and apply constant-acceleration equations to find the required times. (a) Using a constant-acceleration equation, relate the time for the slide to the distance of fall and the acceleration: Solve for t1: Substitute numerical values and evaluate t1: ∆y = y − y0 = h − 0 = v0t1 + 1 at12 2 or, because v0 = 0, h = 1 at12 2 t1 = 2h g t1 = 2(460 m ) = 9.68 s 9.81 m s 2 v1 = v0 + a1t1 (b) Using a constant-acceleration equation, relate the velocity at the bottom of the mountain to the acceleration and time: or, because v0 = 0 and a1 = g, Substitute numerical values and evaluate v1: v1 = 9.81 m s 2 (9.68 s ) = 95.0 m s (c) Using a constant-acceleration equation, relate the time required to stop the mass of rock and mud to its average speed and the distance it slides: Because the acceleration is constant: Substitute to obtain: v1 = gt1 ( ) ∆t = ∆x vav vav = v1 + vf v1 + 0 v1 = = 2 2 2 ∆t = 2∆x v1 68 Chapter 2 Substitute numerical values and evaluate ∆t: ∆t = 2(8000 m ) = 168 s 95.0 m s *77 •• Picture the Problem In the absence of air resistance, the brick experiences constant acceleration and we can use constant-acceleration equations to describe its motion. Constant acceleration implies a parabolic position-versus-time curve. y = y 0 + v 0 t + 1 (− g )t 2 2 (a) Using a constant-acceleration equation, relate the position of the brick to its initial position, initial velocity, acceleration, and time into its fall: ( ) = 6 m + (5 m s ) t − 4.91 m s 2 t 2 ( ) The following graph of y = 6 m + (5 m s )t − 4.91 m s 2 t 2 was plotted using a spreadsheet program: 8 7 6 y (m) 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 t (s) (b) Relate the greatest height reached by the brick to its height when it falls off the load and the additional height it rises ∆ymax: h = y0 + ∆ymax Using a constant-acceleration equation, relate the height reached by the brick to its acceleration and initial velocity: 2 2 vtop = v0 + 2(− g )∆ymax Solve for ∆ymax: Substitute numerical values and evaluate ∆ymax: or, because vtop = 0, 2 0 = v0 + 2(− g )∆ymax ∆ymax = ∆ymax = 2 v0 2g (5 m s ) 2 ( 2 9.81 m s 2 ) = 1.27 m Motion in One Dimension Substitute to obtain: 69 h = y0 + ∆ymax = 6 m + 1.27 m = 7.27 m Note: The graph shown above confirms this result. (c) Using the quadratic formula, solve for t in the equation obtained in part (a): ⎛−g⎞ 2 − v0 ± v0 − 4⎜ ⎟(− ∆y ) ⎝ 2 ⎠ t= ⎛−g⎞ 2⎜ ⎟ ⎝ 2 ⎠ ⎛ v ⎞⎛ 2 g (∆y ) ⎞ ⎟ = ⎜ 0 ⎟⎜1 ± 1 − 2 ⎜ g ⎟⎜ v0 ⎟ ⎝ ⎠⎝ ⎠ With ybottom = 0 and yo = 6 m or ∆y = –6 m, we obtain: t = 1.73 s and t = –0.708 s. Note: The second solution is nonphysical. (d) Using a constant-acceleration equation, relate the speed of the brick on impact to its acceleration and displacement, and solve for its speed: v = 2 gh Substitute numerical values and evaluate v: v = 2 9.81 m s 2 (7.27 m ) = 11.9 m s ( ) 78 •• Picture the Problem In the absence of air resistance, the acceleration of the bolt is constant. Choose a coordinate system in which upward is positive and the origin is at the bottom of the shaft (y = 0). (a) Using a constant-acceleration equation, relate the position of the bolt to its initial position, initial velocity, and fall time: ybottom = 0 Solve for the position of the bolt when it came loose: y0 = −v0t + 1 gt 2 2 Substitute numerical values and evaluate y0: y0 = −(6 m s )(3 s ) + 1 (9.81 m s 2 )(3 s ) 2 (b) Using a constant-acceleration equation, relate the speed of the bolt to its initial speed, acceleration, and fall time: v = v0 + at = y0 + v0t + 1 (− g )t 2 2 2 = 26.1 m 70 Chapter 2 Substitute numerical values and evaluate v : ( ) v = 6 m s − 9.81 m s 2 (3s ) = −23.4 m s and v = 23.4 m s *79 •• Picture the Problem In the absence of air resistance, the object’s acceleration is constant. Choose a coordinate system in which downward is positive and the origin is at the point of release. In this coordinate system, a = g and y = 120 m at the bottom of the fall. Express the distance fallen in the last second in terms of the object’s position at impact and its position 1 s before impact: ∆ylast second = 120 m − y1s before impact (1) Using a constant-acceleration equation, relate the object’s position upon impact to its initial position, initial velocity, and fall time: y = y0 + v0t + 1 gt 2 2 Solve for the fall time: Substitute numerical values and evaluate tfall: or, because y0 = 0 and v0 = 0, 2 y = 1 gtfall 2 tfall = 2y g tfall = 2(120 m ) = 4.95 s 9.81 m/s 2 (9.81 m/s )(3.95 s ) We know that, one second before impact, the object has fallen for 3.95 s. Using the same constantacceleration equation, calculate the object’s position 3.95 s into its fall: y (3.95 s) = Substitute in equation (1) to obtain: ∆ylast second = 120 m − 76.4 m = 43.6 m 1 2 2 2 = 76.4 m 80 •• Picture the Problem In the absence of air resistance, the acceleration of the object is constant. Choose a coordinate system with the origin at the point of release and downward as the positive direction. Using a constant-acceleration equation, relate the height to the initial and final velocities and the acceleration; solve for the height: 2 vf2 = v0 + 2a∆y or, because v0 = 0, h= vf2 2g (1) Motion in One Dimension Using the definition of average velocity, find the average velocity of the object during its final second of fall: vav = vf -1s + vf 2 = 71 ∆y 38 m = = 38 m s ∆t 1s Express the sum of the final velocity and the velocity 1 s before impact: vf -1s + vf = 2(38 m s ) = 76 m s From the definition of acceleration, we know that the change in velocity of the object, during 1 s of fall, is 9.81 m/s: ∆v = vf − vf -1s = 9.81 m s Add the equations that express the sum and difference of vf – 1 s and vf and solve for vf: Substitute in equation (1) and evaluate h: 76 m s + 9.81m s = 42.9 m s 2 vf = h= (42.9 m s )2 ( 2 9.81 m s 2 )= 93.8 m *81 • Picture the Problem In the absence of air resistance, the acceleration of the stone is constant. Choose a coordinate system with the origin at the bottom of the trajectory and the upward direction positive. Let vf -1 2 be the speed one-half second before impact and vf the speed at impact. Using a constant-acceleration equation, express the final speed of the stone in terms of its initial speed, acceleration, and displacement: 2 vf2 = v0 + 2a∆y Solve for the initial speed of the stone: v0 = vf2 + 2 g∆y Find the average speed in the last half second: vav = vf -1 2 + vf 2 = 90 m s = (1) ∆xlast half second 45 m = 0.5 s ∆t and vf -1 2 + vf = 2(90 m s ) = 180 m s Using a constant-acceleration equation, express the change in speed of the stone in the last half second in terms of the acceleration and the elapsed time; solve for the change in its speed: ∆v = vf − vf -1 2 = g∆t ( ) = 9.81 m s 2 (0.5 s ) = 4.91 m s 72 Chapter 2 Add the equations that express the sum and difference of vf – ½ and vf and solve for vf: vf = Substitute in equation (1) and evaluate v0: 180 m s + 4.91m s = 92.5 m s 2 v0 = (92.5m s )2 + 2(9.81m ) s 2 (− 200m ) = 68.1 m s Remarks: The stone may be thrown either up or down from the cliff and the results after it passes the cliff on the way down are the same. 82 •• Picture the Problem In the absence of air resistance, the acceleration of the object is constant. Choose a coordinate system in which downward is the positive direction and the object starts from rest. Apply constant-acceleration equations to find the average velocity of the object during its descent. Express the average velocity of the falling object in terms of its initial and final velocities: Using a constant-acceleration equation, express the displacement of the object during the 1st second in terms of its acceleration and the elapsed time: vav = v0 + vf 2 ∆y1st second gt 2 = = 4.91 m = 0.4 h 2 Solve for the displacement to obtain: Using a constant-acceleration equation, express the final velocity of the object in terms of its initial velocity, acceleration, and displacement: h = 12.3 m Substitute numerical values and evaluate the final velocity of the object: vf = 2 9.81m s 2 (12.3 m ) = 15.5 m s Substitute in the equation for the average velocity to obtain: 2 vf2 = v0 + 2 g∆y or, because v0 = 0, vf = 2 g∆y ( vav = ) 0 + 15.5 m s = 7.77 m s 2 83 •• Picture the Problem This is a three-part constant-acceleration problem. The bus starts from rest and accelerates for a given period of time, and then it travels at a constant velocity for another period of time, and, finally, decelerates uniformly to a stop. The pictorial representation will help us organize the information in the problem and develop our solution strategy. Motion in One Dimension 73 (a) Express the total displacement of the bus during the three intervals of time. ∆xtotal = ∆x(0 → 12 s ) + ∆x(12 s → 37 s ) Using a constant-acceleration equation, express the displacement of the bus during its first 12 s of motion in terms of its initial velocity, acceleration, and the elapsed time; solve for its displacement: ∆x(0 → 12 s ) = v0t + 1 at 2 2 + ∆x(37 s → end ) or, because v0 = 0, ∆x(0 → 12 s ) = 1 at 2 = 108 m 2 ( ) Using a constant-acceleration equation, express the velocity of the bus after 12 seconds in terms of its initial velocity, acceleration, and the elapsed time; solve for its velocity at the end of 12 s: v12 s = v0 + a0→12 s ∆t = 1.5 m/s 2 (12 s ) During the next 25 s, the bus moves with a constant velocity. Using the definition of average velocity, express the displacement of the bus during this interval in terms of its average (constant) velocity and the elapsed time: ∆x(12 s → 37 s ) = v12 s ∆t = (18 m/s )(25 s ) Because the bus slows down at the same rate that its velocity increased during the first 12 s of motion, we can conclude that its displacement during this braking period is the same as during its acceleration period and the time to brake to a stop is equal to the time that was required for the bus to accelerate to its cruising speed of 18 m/s. Hence: ∆x(37 s → 49s ) = 108 m Add the displacements to find the distance the bus traveled: ∆x total = 108 m + 450 m + 108 m = 18 m/s = 450 m = 666 m 74 Chapter 2 (b) Use the definition of average velocity to calculate the average velocity of the bus during this trip: vav = ∆xtotal 666 m = = 13.6 m s ∆t 49 s Remarks: One can also solve this problem graphically. Recall that the area under a velocity as a function-of-time graph equals the displacement of the moving object. *84 •• Picture the Problem While we can solve this problem analytically, there are many physical situations in which it is not easy to do so and one has to rely on numerical methods; for example, see the spreadsheet solution shown below. Because we’re neglecting the height of the release point, the position of the ball as a function of time is given by y = v0t − 1 gt 2 . The formulas used to calculate the quantities in the columns are 2 as follows: Cell Content/Formula Algebraic Form B1 20 v0 B2 9.81 g B5 0 t B6 B5 + 0.1 t + ∆t C6 $B$1*B6 − 0.5*$B$2*B6^2 v0t − 1 gt 2 2 (a) A 1 2 3 4 5 6 7 44 45 46 B t (s) 0.0 0.1 0.2 C m/s m/s^2 height (m) 0.00 1.95 3.80 3.9 4.0 4.1 3.39 1.52 −0.45 v0 = 20 g = 9.81 The graph shown below was generated from the data in the previous table. Note that the maximum height reached is a little more than 20 m and the time of flight is about 4 s. Motion in One Dimension 75 25 height (m) 20 15 10 5 0 0 1 2 3 4 t (s ) (b) In the spreadsheet, change the value in cell B1 from 20 to 10. The graph should automatically update. With an initial velocity of 10 m/s, the maximum height achieved is approximately 5 m and the time-of-flight is approximately 2 s. 6 5 height (m) 4 3 2 1 0 0.0 0.5 1.0 1.5 2.0 t (s) *85 •• Picture the Problem Because the accelerations of both Al and Bert are constant, constant-acceleration equations can be used to describe their motions. Choose the origin of the coordinate system to be where Al decides to begin his sprint. (a) Using a constant-acceleration equation, relate Al's initial velocity, his acceleration, and the time to reach the end of the trail to his ∆x = v0t + 1 at 2 2 76 Chapter 2 displacement in reaching the end of the trail: Substitute numerical values to obtain: 35 m = (0.75 m/s)t + 1 (0.5 m/s 2 )t 2 2 Solve for the time required for Al to reach the end of the trail: t = 10.4 s (b) Using constant-acceleration equations, express the positions of Bert and Al as functions of time. At the instant Al turns around at the end of the trail, t = 0. Also, x = 0 at a point 35 m from the end of the trail: x Bert = x Bert,0 + (0 .75 m/s ) t and xAl = xAl,0 − (0.85 m/s ) t = 35 m − (0.85 m/s ) t Calculate Bert’s position at t = 0. At that time he has been running for 10.4 s: xBert,0 = (0.75 m/s )(10.4 s ) = 7.80 m Because Bert and Al will be at the same location when they meet, equate their position functions and solve for t: 7.80 m + (0.75 m/s )t = 35 m − (0.85 m/s )t and t = 17.0 s To determine the elapsed time from when Al began his accelerated run, we need to add 10.4 s to this time: tstart = 17.0 s + 10.4 s = 27.4 s (c) Express Bert’s distance from the end of the trail when he and Al meet: d end of trail = 35 m − xBert,0 Substitute numerical values and evaluate dend of trail: d end of trail = 35 m − 7.80 m − d Bert runs until he meets Al − (17 s) (0.75 m/s ) = 14.5 m 86 •• Picture the Problem Generate two curves on one graph with the first curve representing Al's position as a function of time and the second curve representing Bert’s position as a function of time. Al’s position, as he runs toward the end of the trail, is given by xAl = v0t + 1 aAlt 2 and Bert’s position by xBert = x0, Bert + vBertt . Al’s position, once he’s 2 reached the end of the trail and is running back toward Bert, is given by xAl = xAl,0 + vAl (t − 10.5 s ) . The coordinates of the intersection of the two curves give the time and place where they meet. A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell Content/Formula Algebraic Form Motion in One Dimension B1 B2 B3 B10 C10 0.75 0.50 −0.85 B9 + 0.25 $B$1*B10 + 0.5*$B$2*B10^2 C52 $C$51 + $B$3*(B52 − $B$51) xAl,0 + vAl (t − 10.5 s ) F10 $F$9 + $B$1*B10 77 x0, Bert + vBertt v0 aAl t t + ∆t v0t + 1 aAlt 2 2 (b) and (c) 1 2 3 4 5 6 7 8 9 10 11 A B v0 = 0.75 a(Al) = 0.5 v(Al) = −0.85 C m/s m/s^2 m/s D E F t (s) x (m) x (m) 0.00 0.25 0.50 Al 0.00 0.20 0.44 Bert 0.00 0.19 0.38 49 50 51 52 53 54 55 56 10.00 10.25 10.50 10.75 11.00 11.25 11.50 11.75 32.50 33.95 35.44 35.23 35.01 34.80 34.59 34.38 7.50 7.69 7.88 8.06 8.25 8.44 8.63 8.81 119 120 121 122 27.50 27.75 28.00 28.25 20.99 20.78 20.56 20.35 20.63 20.81 21.00 21.19 127 128 129 29.50 29.75 30.00 19.29 19.08 18.86 22.13 22.31 22.50 *Al reaches end of trail and starts back toward Bert The graph shown below was generated from the spreadsheet; the positions of both Al and Bert were calculated as functions of time. The dashed curve shows Al’s position as a function of time for the two parts of his motion. The solid line that is linear from the origin shows Bert’s position as a function of time. 78 Chapter 2 40 Position on trail (m) 35 30 25 Al 20 Bert 15 10 5 0 0 5 10 15 20 25 30 t (s) Note that the spreadsheet and the graph (constructed from the spreadsheet data) confirm the results in Problem 85 by showing Al and Bert meeting at about 14.5 m from the end of the trail after an elapsed time of approximately 28 s. 87 •• Picture the Problem This is a two-part constant-acceleration problem. Choose a coordinate system in which the upward direction is positive. The pictorial representation will help us organize the information in the problem and develop our solution strategy. (a) Express the highest point the rocket reaches, h, as the sum of its displacements during the first two stages of its flight: h = ∆x1st stage + ∆x 2nd stage Using a constant-acceleration equation, express the altitude reached in the first stage in terms of the rocket’s initial velocity, acceleration, and burn time; solve for the first stage altitude: x1st stage = x0 + v0t + 1 a1st staget 2 2 Using a constant-acceleration equation, express the velocity of the rocket at the end of its first stage in terms of its initial velocity, acceleration, and displacement; calculate its end-of-first-stage velocity: v1st stage = v0 + a1st staget = 1 (20 m/s 2 )(25 s) 2 2 = 6250 m = (20 m/s 2 )(25 s) = 500 m/s Motion in One Dimension Using a constant-acceleration equation, express the final velocity of the rocket during the remainder of its climb in terms of its shut-off velocity, free-fall acceleration, and displacement; solve for its displacement: 79 2 2 vhighest point = vshutoff + 2a2 nd stage ∆y2nd stage and, because vhighest point = 0, 2 − vshutoff (500 m s )2 = − 2g 2(9.81 m s 2 ) ∆y 2 nd stage = = 1.2742 ×10 4 m Substitute in the expression for the total height to obtain: h = 6250 m + 1.27 × 10 4 m = 19.0 km (b) Express the total time the rocket is in the air in terms of the three segments of its flight: ∆ttotal = ∆tpowered climb + ∆t2nd segment + ∆tdescent Express ∆t2nd segment in terms of the rocket’s displacement and average velocity: Substitute numerical values and evaluate ∆t2nd segment: Using a constant-acceleration equation, relate the fall distance to the descent time: Solve for ∆tdescent: Substitute numerical values and evaluate ∆tdescent: = 25 s + ∆t2nd segment + ∆tdescent ∆t2nd segment = Displacement Average velocity 1.2742 × 10 4 m ∆t2nd segment = = 50.97 s ⎛ 0 + 500 m/s ⎞ ⎟ ⎜ 2 ⎝ ⎠ 2 1 ∆y = v0t + 2 g (∆tdescent ) or, because v0 = 0, ∆y = 1 g (∆tdescent ) 2 2 ∆tdescent = ∆t descent 2∆y g ( ) 2 1.90 × 10 4 m = = 62.2 s 9.81 m/s 2 Substitute and calculate the total time the rocket is in the air: ∆t = 25 s + 50.97 s + 62.2 s = 138 s (c) Using a constant-acceleration equation, express the impact velocity of the rocket in terms of its initial downward velocity, acceleration under free-fall, and time of descent; solve for its impact velocity: vimpact = v0 + g∆tdescent = 2 min 18 s and, because v0 = 0, ( ) vimpact = g∆t = 9.81 m/s 2 (62.2 s ) = 610 m/s 80 Chapter 2 88 •• Picture the Problem In the absence of air resistance, the acceleration of the flowerpot is constant. Choose a coordinate system in which downward is positive and the origin is at the point from which the flowerpot fell. Let t = time when the pot is at the top of the window, and t + ∆t the time when the pot is at the bottom of the window. To find the distance from the ledge to the top of the window, first find the time ttop that it takes the pot to fall to the top of the window. Using a constant-acceleration equation, express the distance y below the ledge from which the pot fell as a function of time: Express the position of the pot as it reaches the top of the window: Express the position of the pot as it reaches the bottom of the window: Subtract ybottom from ytop to obtain an expression for the displacement ∆ywindow of the pot as it passes the window: Solve for ttop: y = y0 + v0t + 1 at 2 2 Since a = g and v0 = y0 = 0, y = 1 gt 2 2 2 ytop = 1 gt top 2 ybottom = 1 g (t top + ∆t window ) 2 where ∆twindow = t top − tbottom 2 = t top Substitute numerical values and evaluate ttop: t top Substitute this value for ttop to obtain the distance from the ledge to the top of the window: [ g [2t 2 ∆ywindow = 1 g (t top + ∆t window ) − t top 2 1 2 2 top ∆t window + (∆t window ) 2∆ywindow 2 − (∆t window ) g = 2∆t window 2(4 m ) 2 − (0.2 s ) 2 9.81 m/s = = 1.839 s 2(0.2 s ) ytop = 1 (9.81 m/s 2 )(1.939 s) 2 = 18.4 m 2 *89 •• Picture the Problem The acceleration of the glider on the air track is constant. Its average acceleration is equal to the instantaneous (constant) acceleration. Choose a coordinate system in which the initial direction of the glider’s motion is the positive direction. Using the definition of acceleration, express the average acceleration of the glider in terms of the glider’s velocity change and the elapsed time: 2 a = aav = ∆v ∆t Motion in One Dimension Using a constant-acceleration equation, express the average velocity of the glider in terms of the displacement of the glider and the elapsed time: Solve for and evaluate the initial velocity: vav = ∆x v0 + v = 2 ∆t v0 = 2∆x 2(100 cm ) −v = − (−15 cm/s) ∆t 8s = 40.0 cm/s Substitute this value of v0 and evaluate the average acceleration of the glider: a= − 15 cm/s − (40 cm/s) 8s = − 6.88 cm/s 2 90 •• Picture the Problem In the absence of air resistance, the acceleration of the rock is constant and its motion can be described using the constant-acceleration equations. Choose a coordinate system in which the downward direction is positive and let the height of the cliff, which equals the displacement of the rock, be represented by h. Using a constant-acceleration equation, express the height h of the cliff in terms of the initial velocity of the rock, acceleration, and time of fall: 81 ∆y = v0t + 1 at 2 2 or, because v0 = 0, a = g, and ∆y = h, h = 1 gt 2 2 Using this equation, express the displacement of the rock during the a) first two-thirds of its fall, and 2 3 b) its complete fall in terms of the time required for it to fall this distance. h = 1 g (t + 1s ) 2 Substitute equation (2) in equation (1) to obtain a quadratic equation in t: t2 – (4 s)t – 2 s2 = 0 Solve for the positive root: t = 4.45 s Evaluate ∆t = t + 1 s: ∆t = 4.45 s + 1 s = 5.45 s Substitute numerical values in equation (2) and evaluate h: h= h = 1 gt 2 2 (1) 2 1 2 (2) (9.81m/s )(5.45 s ) 2 2 = 146 m 82 Chapter 2 91 ••• Picture the Problem Assume that the acceleration of the car is constant. The total distance the car travels while stopping is the sum of the distances it travels during the driver’s reaction time and the time it travels while braking. Choose a coordinate system in which the positive direction is the direction of motion of the automobile and apply a constant-acceleration equation to obtain a quadratic equation in the car’s initial speed v0. (a) Using a constant-acceleration equation, relate the velocity of the car to its initial velocity, acceleration, and displacement during braking: Solve for the distance traveled during braking: 2 v 2 = v0 + 2a∆xbrk or, because the final velocity is zero, 2 0 = v0 + 2a∆x brk ∆xbrk = − 2 v0 2a Express the total distance traveled by the car as the sum of the distance traveled during the reaction time and the distance traveled while slowing down: ∆xtot = ∆xreact + ∆xbrk Rearrange this quadratic equation to obtain: 2 v0 − 2a∆treact v0 + 2a∆xtot = 0 Substitute numerical values and simplify to obtain: 2 v0 − 2 − 7 m/s 2 (0.5 s )v0 2 v0 = v0 ∆treact − 2a ( ) + 2(− 7 m/s )(4 m) = 0 2 or 2 v0 + (7 m/s )v0 − 56 m 2 / s 2 = 0 Solve the quadratic equation for the positive root to obtain: v0 = 4.7613558 m/s Convert this speed to mi/h:: ⎛ 1 mi/h ⎞ v 0 = (4.7613558 m/s ) ⎜ ⎜ 0.477 m/s ⎟ ⎟ ⎝ ⎠ = 10.7 mi/h (b) Find the reaction-time distance: ∆xreact = v0 ∆treact = (4.76 m/s)(0.5 s) = 2.38 m Express and evaluate the ratio of the reaction distance to the total distance: ∆xreact 2.38 m = = 0.595 ∆xtot 4m Motion in One Dimension 83 92 •• Picture the Problem Assume that the accelerations of the trains are constant. Choose a coordinate system in which the direction of the motion of the train on the left is the positive direction. Take xo = 0 as the position of the train on the left at t = 0. ( = (0.7 m s )t ) Using a constant-acceleration equation, relate the distance the train on the left will travel before the trains pass to its acceleration and the time-to-passing: Using a constant-acceleration equation, relate the position of the train on the right to its initial velocity, position, and acceleration: xL = 1 aLt 2 = 1 1.4m s 2 t 2 2 2 Equate xL and xR and solve for t: 0.7t 2 = 40 − 1.1t 2 and t = 4.71s Find the position of the train initially on the left, xL, as they pass: xL = 1 (1.4 m/s 2 )(4.71 s) 2 = 15.6 m 2 2 2 xR = 40 m − 1 aR t 2 2 ( ) = 40 m − 1 2.2m s 2 t 2 2 Remarks: One can also solve this problem by graphing the functions for xL and xR. The coordinates of the intersection of the two curves give one the time-to-passing and the distance traveled by the train on the left. 93 •• Picture the Problem In the absence of air resistance, the acceleration of the stones is constant. Choose a coordinate system in which the downward direction is positive and the origin is at the point of release of the stones. Using constant-acceleration equations, relate the positions of the two stones to their initial positions, accelerations, and time-of-fall: x1 = 1 gt 2 2 and x2 = 1 g (t − 1.6 s) 2 2 Express the difference between x1 and x2: x1 − x2 = 36 m Substitute for x1 and x2 to obtain: 36 m = 1 gt 2 − 1 g (t − 1.6 s ) 2 2 2 84 Chapter 2 Solve this equation for the time t at which the stones will be separated by 36 m: t = 3.09 s Substitute this result in the expression for x2 and solve for x2: x2 = 1 2 (9.81 m s )(3.09 s − 1.6 s) 2 2 = 10.9 m *94 •• Picture the Problem The acceleration of the police officer’s car is positive and constant and the acceleration of the speeder’s car is zero. Choose a coordinate system such that the direction of motion of the two vehicles is the positive direction and the origin is at the stop sign. Express the velocity of the car in terms of the distance it will travel until the police officer catches up to it and the time that will elapse during this chase: vcar = d caught t car Letting t1 be the time during which she accelerates and t2 the time of travel at v1 = 110 km/h, express the time of travel of the police officer: tofficer = t1 + t2 Convert 110 km/h into m/s: v1 = (110 km/h )(103 m/km) (1 h/3600 s) = 30.6 m/s Express and evaluate t1: t1 = v1 amotorcycle = 30.6 m/s = 4.94 s 6.2 m/s 2 Express and evaluate d1: d1 = 1 v1t1 = 1 (30.6 m/s)(4.94 s) = 75.6 m 2 2 Determine d2: d 2 = d caught − d1 = 1400 m − 75.6 m = 1324.4 m Express and evaluate t2: Express the time of travel of the car: t2 = d 2 1324.4 m = = 43.3 s v1 30.6 m/s tcar = 2.0 s + 4.93 s + 43.3 s = 50.2 s Motion in One Dimension Finally, find the speed of the car: vcar = d caught tcar = 85 1400 m = 27.9 m/s 50.2 s ⎛ 1 mi/h ⎞ = (27.9 m/s )⎜ ⎟ ⎜ 0.447 m/s ⎟ ⎠ ⎝ = 62.4 mi/h 95 •• Picture the Problem In the absence of air resistance, the acceleration of the stone is constant. Choose a coordinate system in which downward is positive and the origin is at the point of release of the stone and apply constant-acceleration equations. Using a constant-acceleration equation, express the height of the cliff in terms of the initial position of the stones, acceleration due to gravity, and time for the first stone to hit the water: h = 1 gt12 2 Express the displacement of the second stone when it hits the water in terms of its initial velocity, acceleration, and time required for it to hit the water. 2 d 2 = v02t2 + 1 gt2 2 Because the stones will travel the same distances before hitting the water, equate h and d2 and solve for t. where t2 = t1 – 1.6 s. 2 gt12 = v02t2 + 1 gt2 2 1 2 or 1 2 (9.81m/s )t = (32m/s)(t − 1.6s ) + (9.81m/s ) (t − 1.6s ) 2 2 1 1 1 2 2 2 1 Solve for t1 to obtain: t1 = 2.37 s Substitute for t1 and evaluate h: h = 1 (9.81 m/s 2 )(2.37 s) 2 = 27.6 m 2 96 ••• Picture the Problem Assume that the acceleration of the passenger train is constant. Let xp = 0 be the location of the passenger train engine at the moment of sighting the freight train’s end; let t = 0 be the instant the passenger train begins to slow (0.4 s after the passenger train engineer sees the freight train ahead). Choose a coordinate system in which the direction of motion of the trains is the positive direction and use constantacceleration equations to express the positions of the trains in terms of their initial positions, speeds, accelerations, and elapsed time. (a) Using constant-acceleration equations, write expressions for the positions of the front of the passenger train and the rear of the xp = (29 m/s )(t + 0.4 s ) − 1 at 2 2 xf = (360 m ) + (6 m/s )(t + 0.4 s) where xp and xf are in meters if t is in 86 Chapter 2 freight train, xp and xf, respectively: seconds. at 2 − (23 m/s )t + 350.8 m = 0 Equate xf = xp to obtain an equation for t: 1 2 Find the discriminant D = B2 − 4AC of this equation: ⎛a⎞ 2 D = (23 m/s ) − 4⎜ ⎟(350.8 m ) ⎝2⎠ The equation must have real roots if it is to describe a collision. The necessary condition for real roots is that the discriminant be greater than or equal to zero: If (23 m/s)2 – a (701.6 m) ≥ 0, then (b) Express the relative speed of the trains: vrel = vpf = vp − vf Repeat the previous steps with a = 0.754 m/s2 and a 0.8 s reaction time. The quadratic equation that guarantees real roots with the longer reaction time is: 1 2 ( ) a ≤ 0.754 m/s 2 (1) (0.754 m/s )t − (23 m/s)t 2 2 + 341.6 m = 0 Solve for t to obtain the collision times: t = 25.6 s and t = 35.4 s Note that at t = 35.4 s, the trains have already collided; therefore this root is not a meaningful solution to our problem. Note: In the graph shown below, you will see why we keep only the smaller of the two solutions. Now we can substitute our value for t in the constant-acceleration equation for the passenger train and solve for the distance the train has moved prior to the collision: xp = (29 m/s)(25.6 s + 0.8 s) – ( 0.377 m/s2)(25.6 s)2 = 518 m Find the speeds of the two trains: vp = vop + at = (29 m/s) + (–0.754 m/s2)(25.5 s) = 9.77 m/s and vf = vof = 6 m/s Substitute in equation (1) and evaluate the relative speed of the trains: vrel = 9.77 m/s - 6.00 m/s = 3.77 m/s The graph shows the location of both trains as functions of time. The solid straight line is for the constant velocity freight train; the dashed curves are for the passenger train, with reaction times of 0.4 s for the lower curve and 0.8 s for the upper curve. Motion in One Dimension 87 700 600 x (m) 500 400 0.4 s reaction time 300 Freight train 200 0.8 s reaction time 100 0 0 10 20 30 40 t (s) Remarks: A collision occurs the first time the curve for the passenger train crosses the curve for the freight train. The smaller of two solutions will always give the time of the collision. 97 • Picture the Problem In the absence of air resistance, the acceleration of an object near the surface of the earth is constant. Choose a coordinate system in which the upward direction is positive and the origin is at the surface of the earth and apply constantacceleration equations. Using a constant-acceleration equation, relate the velocity to the acceleration and displacement: or, because v = 0 and a = −g, Solve for the height to which the projectile will rise: h = ∆y = Substitute numerical values and evaluate h: 2 v 2 = v0 + 2a∆y 2 0 = v0 − 2 g∆y h= 2 v0 2g (300 m/s)2 ( 2 9.81 m/s 2 )= 4.59 km *98 • Picture the Problem This is a composite of two constant accelerations with the acceleration equal to one constant prior to the elevator hitting the roof, and equal to a different constant after crashing through it. Choose a coordinate system in which the upward direction is positive and apply constant-acceleration equations. (a) Using a constant-acceleration equation, relate the velocity to the acceleration and displacement: 2 v 2 = v0 + 2 a ∆y or, because v = 0 and a = −g, 2 0 = v0 − 2 g ∆y 88 Chapter 2 Solve for v0: v0 = 2 g∆y Substitute numerical values and evaluate v0: v0 = 2 9.81 m/s 2 10 4 m = 443 m/s (b) Find the velocity of the elevator just before it crashed through the roof: vf = 2 × 443 m/s = 886 m/s Using the same constantacceleration equation, this time with v0 = 0, solve for the acceleration: v 2 = 2a∆y Substitute numerical values and evaluate a: (886 m/s)2 a= 2(150 m ) ( )( ) = 2.62 × 103 m/s 2 = 267 g 99 •• Picture the Problem Choose a coordinate system in which the upward direction is positive. We can use a constant-acceleration equation to find the beetle’s velocity as its feet lose contact with the ground and then use this velocity to calculate the height of its jump. Using a constant-acceleration equation, relate the beetle’s maximum height to its launch velocity, velocity at the top of its trajectory, and acceleration once it is airborne; solve for its maximum height: Because vhighest point = 0: Now, in order to determine the beetle’s launch velocity, relate its time of contact with the ground to its acceleration and push-off distance: Substitute numerical values and 2 evaluate vlaunch : Substitute to find the height to which the beetle can jump: 2 2 vhighest point = vlaunch + 2a∆yfree fall 2 = vlaunch + 2(− g )h h= 2 vlaunch 2g 2 2 vlaunch = v0 + 2a∆ylaunch or, because v0 = 0, 2 vlaunch = 2a∆ylaunch ( )( 2 vlaunch = 2(400 ) 9.81 m/s 2 0.6 × 10 −2 m = 47.1 m 2 /s 2 h= 2 vlaunch 47.1m 2 /s 2 = = 2.40 m 2g 2 9.81m/s2 ( ) ) Motion in One Dimension Using a constant-acceleration equation, relate the velocity of the beetle at its maximum height to its launch velocity, free-fall acceleration while in the air, and time-to-maximum height: Solve for tmax height: For zero displacement and constant acceleration, the time-of-flight is twice the time-to-maximum height: 89 v = v0 + at or vmax. height = vlaunch − gtmax. height and, because vmax height = 0, 0 = vlaunch − gtmax. height tmax height = vlaunch g tflight = 2tmax. height = = 2vlaunch g 2(6.86 m/s ) = 1.40 s 9.81 m/s 2 100 • Picture the Problem Because its acceleration is constant we can use the constantacceleration equations to describe the motion of the automobile. Using a constant-acceleration equation, relate the velocity to the acceleration and displacement: Solve for the acceleration a: Substitute numerical values and evaluate a: 2 v 2 = v0 + 2 a ∆x or, because v = 0, 2 0 = v0 + 2 a ∆x 2 − v0 a= 2∆x [ − (98 km h ) (103 m km )(1 h 3600 s ) a= 2(50m ) = − 7.41 m s 2 Express the ratio of a to g and then solve for a: a − 7.41m s 2 = = −0.755 g 9.81m s 2 and a = − 0.755 g Using the definition of average acceleration, solve for the stopping time: aav = Substitute numerical values and evaluate ∆t: ∆t = ∆v ∆v ⇒ ∆t = aav ∆t (− 98 km h )(103 m km )(1h = 3.67 s − 7.41m s 2 3600 s ) 2 90 Chapter 2 *101 •• Picture the Problem In the absence of air resistance, the puck experiences constant acceleration and we can use constant-acceleration equations to describe its position as a function of time. Choose a coordinate system in which downward is positive, the particle starts from rest (vo = 0), and the starting height is zero (y0 = 0). Using a constant-acceleration equation, relate the position of the falling puck to the acceleration and the time. Evaluate the y-position at successive equal time intervals ∆t, 2∆t, 3∆t, etc: −g (∆t )2 = − g (∆t )2 2 2 −g (2∆t )2 = − g (4)(∆t )2 y2 = 2 2 −g (3∆t )2 = − g (9)(∆t )2 y3 = 2 2 −g (4∆t )2 = − g (16)(∆t )2 y4 = 2 2 etc. Evaluate the changes in those positions in each time interval: ⎛−g⎞ 2 ∆y10 = y1 − 0 = ⎜ ⎟ (∆t ) ⎝ 2 ⎠ ⎛−g⎞ 2 ∆y21 = y2 − y1 = 3⎜ ⎟(∆t ) = 3∆y10 ⎝ 2 ⎠ ⎛−g⎞ 2 ∆y32 = y3 − y2 = 5⎜ ⎟(∆t ) = 5∆y10 ⎝ 2 ⎠ ⎛−g⎞ 2 ∆y43 = y4 − y3 = 7⎜ ⎟(∆t ) = 7 ∆y10 ⎝ 2 ⎠ etc. y1 = 102 •• Picture the Problem Because the particle moves with a constant acceleration we can use the constant-acceleration equations to describe its motion. A pictorial representation will help us organize the information in the problem and develop our solution strategy. Using a constant-acceleration equation, find the position x at t = 6 s. To find x at t = 6 s, we first need to find v0 and x0: x = x0 + v0t + 1 at 2 2 Motion in One Dimension Using the information that when t = 4 s, x = 100 m, obtain an equation in x0 and v0: x(4 s ) = 100 m ( ) 91 = x0 + v0 (4 s ) + 1 3 m/s 2 (4 s ) 2 2 or x0 + (4 s )v0 = 76 m ( ) Using the information that when t = 6 s, v = 15 m/s, obtain a second equation in x0 and v0: v(6 s ) = v0 + 3 m/s 2 (6 s ) Solve for v0 to obtain: v0 = −3 m/s Substitute this value for v0 in the previous equation and solve for x0: x0 = 88 m Substitute for x0 and v0 and evaluate x at t = 6 s: x(6 s ) = 88 m + (− 3 m/s ) (6 s ) + 1 (3 m/s 2 ) (6 s ) = 124 m 2 2 *103 •• Picture the Problem We can use constant-acceleration equations with the final velocity v = 0 to find the acceleration and stopping time of the plane. (a) Using a constant-acceleration equation, relate the known velocities to the acceleration and displacement: Solve for a: Substitute numerical values and evaluate a: (b) Using a constant-acceleration equation, relate the final and initial speeds of the plane to its acceleration and stopping time: Solve for and evaluate the stopping time: 2 v 2 = v0 + 2 a ∆x a= 2 2 v 2 − v0 − v0 = 2 ∆x 2 ∆x a= − (60 m s ) = − 25.7 m s 2 2(70 m ) 2 v = v0 + a∆t ∆t = v − v0 0 − 60 m s = = 2.33 s a − 25.7 m/s 2 104 •• Picture the Problem This is a multipart constant-acceleration problem using three different constant accelerations (+2 m/s2 for 20 s, then zero for 20 s, and then –3 m/s2 until the automobile stops). The final velocity is zero. The pictorial representation will help us organize the information in the problem and develop our solution strategy. 92 Chapter 2 Add up all the displacements to get the total: ∆x 03 = ∆x01 + ∆x12 + ∆x 23 Using constant-acceleration formulas, find the first displacement: ∆x01 = v0t1 + 1 a01t12 2 The speed is constant for the second displacement. Find the displacement: ∆x12 = v1 (t2 − t1 ) = 0 + 1 (2 m/s 2 )(20 s) 2 = 400 m 2 where v1 = v0 + a01t1 = 0 + a01t1 and ∆x12 = a01t1 (t2 − t1 ) = (2 m/s 2 )(20 s)(20 s) = 800 m Find the displacement during the braking interval: 2 2 v3 = v2 + 2a23∆x23 where v2 = v1 = a01t1 and v3 = 0 and 02 − (a01t1 ) − [(2 m/s) (20 s)] ∆x23 = = 2a23 2 − 3m s 2 2 ( 2 ) = 267m ∆x03 = ∆x01 + ∆x12 + ∆x23 = 1467 m Add the displacements to get the total: = 1.47 km Remarks: Because the area under the curve of a velocity-versus-time graph equals the displacement of the object experiencing the acceleration, we could solve this problem by plotting the velocity as a function of time and finding the area bounded by it and the time axis. *105 •• Picture the Problem Note: No material body can travel at speeds faster than light. When one is dealing with problems of this sort, the kinematic formulae for displacement, velocity and acceleration are no longer valid, and one must invoke the special theory of relativity to answer questions such as these. For now, ignore such subtleties. Although the formulas you are using (i.e., the constant- acceleration equations) are not quite correct, your answer to part (b) will be wrong by about 1%. (a) This part of the problem is an exercise in the conversion of units. Make use of the fact that 1 c⋅y = 9.47×1015 m and 1 y = 3.16×107 s: ( ⎞ ⎛ 3.16 × 10 7 s ⎛ 1c ⋅ y ⎜ g = 9.81m/s 2 ⎜ ⎟ ⎜ 9.47 × 1015 m ⎟ ⎜ (1 y )2 ⎠⎝ ⎝ ( ) ) 2 ⎞ ⎟ = 1.03c ⋅ y / y 2 ⎟ ⎠ Motion in One Dimension (b) Let t1/2 represent the time it takes to reach the halfway point. Then the total trip time is: t = 2 t1/2 Use a constant- acceleration equation to relate the half-distance to Mars ∆x to the initial speed, acceleration, and half-trip time t1/2 : 93 ∆x = v0t + 1 at12 2 2 Because v0 = 0 and a = g: The distance from Earth to Mars at closest approach is 7.8 × 1010 m. Substitute numerical values and evaluate t1/2 : Substitute for t1/2 in equation (1) to obtain: (1) t1 / 2 = 2∆x a t1 / 2 = 2 3.9 × 1010 m = 8.92 × 104 s 2 9.81 m/s ( ) t = 2(8.92 × 10 4 s ) = 1.78 × 105 s ≈ 2 d Remarks: Our result in part (b) seems remarkably short, considering how far Mars is and how low the acceleration is. 106 • Picture the Problem Because the elevator accelerates uniformly for half the distance and uniformly decelerates for the second half, we can use constant-acceleration equations to describe its motion Let t1/2 = 40 s be the time it takes to reach the halfway mark. Use the constant-acceleration equation that relates the acceleration to the known variables to obtain: Solve for a: Substitute numerical values and evaluate a: ∆y = v0t + 1 at 2 2 or, because v0 = 0, ∆y = 1 at 2 2 a= 2 ∆y t12/ 2 a= 2( 1 )(1173 ft )(1 m/3.281ft ) 2 = 0.223 m/s 2 2 (40 s ) = 0.0228 g 107 •• Picture the Problem Because the acceleration is constant, we can describe the motions of the train using constant-acceleration equations. Find expressions for the distances traveled, separately, by the train and the passenger. When are they equal? Note that the train is accelerating and the passenger runs at a constant minimum velocity (zero acceleration) such that she can just catch the train. 94 Chapter 2 1. Using the subscripts ″train″ and ″p″ to refer to the train and the passenger and the subscript ″c″ to identify ″critical″ conditions, express the position of the train and the passenger: xtrain,c (tc ) = Express the critical conditions that must be satisfied if the passenger is to catch the train: vtrain,c = vp, c 2. Express the train’s average velocity. 3. Using the definition of average velocity, express vav in terms of xp,c and tc. 4. Combine steps 2 and 3 and solve for xp,c. 5. Combine steps 1 and 4 and solve for tc. atrain 2 tc 2 and xp,c (tc ) = vp,c (tc − ∆t ) and xtrain, c = xp, c vav (0 to tc ) = vav ≡ xp,c = 0 + vtrain, c vtrain, c = 2 2 ∆x 0 + xp, c xp, c = = ∆t 0 + tc tc vtrain,ctc 2 vp,c (tc − ∆t ) = vtrain,ctc 2 or tc − ∆t = tc 2 and tc = 2 ∆t = 2 (6 s) = 12 s 6. Finally, combine steps 1 and 5 and solve for vtrain, c. vp, c = vtrain,c = atrain tc = (0.4 m/s 2 )(12 s ) = 4.80 m/s The graph shows the location of both the passenger and the train as a function of time. The parabolic solid curve is the graph of xtrain(t) for the accelerating train. The straight dashed line is passenger's position xp(t) if she arrives at ∆t = 6.0 s after the train departs. When the passenger catches the train, our graph shows that her speed and that of the train must be equal ( vtrain, c = vp, c ). Do you see why? Motion in One Dimension 95 50 45 40 Train 35 Passenger x (m) 30 25 20 15 10 5 0 0 4 8 12 16 t (s) 108 ••• Picture the Problem Both balls experience constant acceleration once they are in flight. Choose a coordinate system with the origin at the ground and the upward direction positive. When the balls collide they are at the same height above the ground. Using constant-acceleration equations, express the positions of both balls as functions of time. At the ground y = 0. The conditions at collision are that the heights are equal and the velocities are related: Express the velocities of both balls as functions of time: Substituting the position and velocity functions into the conditions at collision gives: yA = h − 1 gt 2 2 and yB = v0t − 1 gt 2 2 yA = yB and v A = − 2v B vA = − gt and vB = v0 − gt h − 1 gtc2 = v0tc − 1 gtc2 2 2 and − gtc = −2(v0 − gtc ) where tc is the time of collision. We now have two equations and two unknowns, tc and v0. Solving the equations for the unknowns gives: tc = Substitute the expression for tc into the equation for yA to obtain the height at collision: ⎛ 2h ⎞ 2h yA = h − 1 g ⎜ ⎟ = 2 ⎜ ⎟ 3 ⎝ 3g ⎠ 2h 3 gh and v0 = 3g 2 96 Chapter 2 Remarks: We can also solve this problem graphically by plotting velocity- versustime for both balls. Because ball A starts from rest its velocity is given by v A = − gt . Ball B initially moves with an unknown velocity vB0 and its velocity is given by v B = v B0 − gt . The graphs of these equations are shown below with T representing the time at which they collide. The height of the building is the sum of the sum of the distances traveled by the balls. Each of these distances is equal to the magnitude of the area ″under″ the corresponding v-versus-t curve. Thus, the height of the building equals the area of the parallelogram, which is vB0T. The distance that A falls is the area of the lower triangle, which is (1/3) vB0T. Therefore, the ratio of the distance fallen by A to the height of the building is 1/3, so the collision takes place at 2/3 the height of the building. 109 ••• Picture the Problem Both balls are moving with constant acceleration. Take the origin of the coordinate system to be at the ground and the upward direction to be positive. When the balls collide they are at the same height above the ground. The velocities at collision are related by vA = 4vB. Using constant-acceleration equations, express the positions of both balls as functions of time: yA = h − 1 gt 2 2 and y B = v0t − 1 gt 2 2 The conditions at collision are that the heights are equal and the velocities are related: Express the velocities of both balls as functions of time: yA = yB and vA = 4vB vA = − gt and v B = v0 − gt Motion in One Dimension Substitute the position and velocity functions into the conditions at collision to obtain: h − 1 gtc2 = v0t c − 1 gt c2 2 2 and − gtc = 4(v0 − gtc ) where tc is the time of collision. We now have two equations and two unknowns, tc and v0. Solving the equations for the unknowns gives: tc = Substitute the expression for tc into the equation for yA to obtain the height at collision: ⎛ 4h ⎞ h yA = h − 1 g ⎜ ⎟ = 2 ⎜ ⎟ 3 ⎝ 3g ⎠ 4h 3gh and v 0 = 3g 4 *110 •• Determine the Concept The problem describes two intervals of constant acceleration; one when the train’s velocity is increasing, and a second when it is decreasing. (a) Using a constant-acceleration equation, relate the half-distance ∆x between stations to the initial speed v0, the acceleration a of the train, and the time-to-midpoint ∆t: Solve for ∆t: Substitute numerical values and evaluate the time-to-midpoint ∆t: ∆x = v0 ∆t + 1 a (∆t ) 2 2 or, because v0 = 0, ∆x = 1 a (∆t ) 2 2 ∆t = 2∆x a ∆t = 2(450 m ) = 30.0 s 1 m/s 2 Because the train accelerates uniformly and from rest, the first part of its velocity graph will be linear, pass through the origin, and last for 30 s. Because it slows down uniformly and at the same rate for the second half of its journey, this part of its graph will also be linear but with a negative slope. The graph of v as a function of t is shown below. 97 98 Chapter 2 30 25 v (m/s) 20 15 10 5 0 0 10 20 30 40 50 60 t (s) (b) The graph of x as a function of t is obtained from the graph of v as a function of t by finding the area under the velocity curve. Looking at the velocity graph, note that when the train has been in motion for 10 s, it will have traveled a distance of 1 2 (10 s )(10 m/s) = 50 m and that this distance is plotted above 10 s on the following graph. 900 800 700 x (m) 600 500 400 300 200 100 0 0 10 20 30 40 50 60 t (s) Selecting additional points from the velocity graph and calculating the areas under the curve will confirm the graph of x as a function of t that is shown. 111 •• Picture the Problem This is a two-stage constant-acceleration problem. Choose a coordinate system in which the direction of the motion of the cars is the positive direction. The pictorial representation summarizes what we know about the motion of the speeder’s car and the patrol car. Motion in One Dimension Convert the speeds of the vehicles and the acceleration of the police car into SI units: km km 1h =8 × = 2.22 m/s 2 , h ⋅s h ⋅ s 3600 s km km 1h 125 = 125 × = 34.7 m/s , h h 3600 s 8 and 190 km km 1h = 190 × = 52.8 m/s h h 3600 s (a) Express the condition that determines when the police car catches the speeder; i.e., that their displacements will be the same: ∆xP,02 = ∆xS,02 Using a constant-acceleration equation, relate the displacement of the patrol car to its displacement while accelerating and its displacement once it reaches its maximum velocity: ∆xP,02 = ∆xP,01 + ∆xP,12 Using a constant-acceleration equation, relate the displacement of the speeder to its constant velocity and the time it takes the patrol car to catch it: ∆xS,02 = vS,02 ∆t02 Calculate the time during which the police car is speeding up: 99 = ∆xP,01 + vP ,1 (t 2 − t1 ) = (34.7 m/s ) t 2 ∆t P,01 = = ∆vP,01 vP ,1 − vP, 0 = aP,01 aP,01 52.8 m/s − 0 = 23.8 s 2.22 m/s 2 100 Chapter 2 Express the displacement of the patrol car: 2 ∆xP,01 = vP,0 ∆tP,01 + 1 aP ,01∆tP , 01 2 ( ) = 0 + 1 2.22 m/s 2 (23.8 s ) 2 2 = 629 m ∆xP,02 = ∆xP,01 + ∆xP,12 Equate the displacements of the two vehicles: = ∆xP,01 + vP ,1 (t 2 − t1 ) = 629 m + (52.8 m/s ) (t2 − 23.8 s) (34.7 m/s) t2 = 629 m + (52.8 m/s)(t2 – 23.8 s) Solve for the time to catch up to obtain: ∴ t 2 = 34.7 s ∆xS,02 = vS,02 ∆t02 = (34.7 m/s )(34.7 s ) (b) The distance traveled is the displacement, ∆x02,S, of the speeder during the catch: = 1.20 km (c) The graphs of xS and xP are shown below. The straight line (solid) represents xS(t) and the parabola (dashed) represents xP(t). 1400 1200 Speeder x (m) 1000 Officer 800 600 400 200 0 0 10 20 30 40 t (s) 112 •• Picture the Problem The accelerations of both cars are constant and we can use constant-acceleration equations to describe their motions. Choose a coordinate system in which the direction of motion of the cars is the positive direction, and the origin is at the initial position of the police car. (a) The collision will not occur if, during braking, the displacements of the two cars differ by less than 100 m. ∆xP − ∆xS < 100 m. Motion in One Dimension 101 Using a constant-acceleration equation, relate the speeder’s initial and final speeds to its displacement and acceleration and solve for the displacement: or, because vs = 0, ∆xs = 2 − v0,s 2as Substitute numerical values and evaluate ∆xs: ∆xS = − (34.7 m/s ) = 100 m 2 − 6 m/s 2 2 vs2 = v0,s + 2as ∆xs 2 ( ) Using a constant-acceleration equation, relate the patrol car’s initial and final speeds to its displacement and acceleration and solve for the displacement: or, assuming vp = 0, Substitute numerical values and evaluate ∆xp: − (52.8 m/s ) ∆xP = = 232 m 2 − 6 m/s 2 Finally, substitute these displacements into the inequality that determines whether a collision occurs: 232 m − 100 m = 132 m Because this difference is greater than (b) Using constant-acceleration equations, relate the positions of both vehicles to their initial positions, initial velocities, accelerations, and time in motion: xS = 100 m + (34.7 m/s )t − 3 m/s 2 t 2 Equate these expressions and solve for t: 2 2 vp = v0, p + 2ap ∆xp ∆xp = 2 − v0, p 2ap 2 ( ) 100 m, the cars collide . ( and ( ) ) xP = (52.8 m/s )t − 3 m/s 2 t 2 100 m + (34.7 m/s) t – (3 m/s2) t2 = (52.8 m/s) t – (3m/s2) t2 and t = 5.52 s (c) If you take the reaction time into account, the collision will occur sooner and be more severe. 113 •• Picture the Problem Lou’s acceleration is constant during both parts of his trip. Let t1 be the time when the brake is applied; L1 the distance traveled from t = 0 to t = t1. Let tfin be the time when Lou's car comes to rest at a distance L from the starting line. A pictorial representation will help organize the given information and plan the solution. 102 Chapter 2 (a) Express the total length, L, of the course in terms of the distance over which Lou will be accelerating, ∆x01, and the distance over which he will be braking, ∆x12: L = ∆x01 + ∆x12 Express the final velocity over the first portion of the course in terms of the initial velocity, acceleration, and displacement; solve for the displacement: 2 v12 = v0 + 2a01∆x01 Express the final velocity over the second portion of the course in terms of the initial velocity, acceleration, and displacement; solve for the displacement: Substitute for ∆x01 and ∆x12 to obtain: or, because v0 = 0, ∆x01 = L1, and a01 = a, ∆x01 = L1 = 2 v2 = v12 + 2a12 ∆x12 or, because v2 = 0 and a12 = −2a, v12 L1 ∆x12 = = 4a 2 L = ∆x01 + ∆x12 = L1 + 1 L1 = 3 L1 2 2 and L1 = (b) Using the fact that the acceleration was constant during both legs of the trip, express Lou’s average velocity over each leg: Express the time for Lou to reach his maximum velocity as a function of L1 and his maximum velocity: 2 3 L vav, 01 = vav,12 = ∆t01 = and t= vmax 2 ∆x01 2 L1 = vav,01 vmax ∆t01 ∝ L1 = Having just shown that the time required for the first segment of the trip is proportional to the length of the segment, use this result to express ∆t01 (= t1) in terms tfin: 2 v12 vmax = 2a 2a 2 3 fin t 2 L 3 Motion in One Dimension 103 114 •• Picture the Problem There are three intervals of constant acceleration described in this problem. Choose a coordinate system in which the upward direction (shown to the left below) is positive. A pictorial representation will help organize the details of the problem and plan the solution. (a) The graphs of a(t) (dashed lines) and v(t) (solid lines) are shown below. v (m/s) and a (m/s^2) 20 0 -20 -40 Velocity Acceleration -60 -80 0 2 4 6 8 10 12 14 16 t (s) (b) Using a constant-acceleration equation, express her speed in terms of her acceleration and the elapsed time; solve for her speed after 8 s of fall: v1 = v0 + a01t1 (c) Using the same constantacceleration equation that you used in part (b), determine the duration of her constant upward acceleration: v2 = v1 + a12 ∆t12 v −v − 5 m/s − (− 78.5 m/s ) ∆ t12 = 2 1 = a12 15 m/s 2 ( ) = 0 + 9.81 m/s 2 (8 s ) = 78.5 m/s = 4.90 s (d) Find her average speed as she slows from 78.5 m/s to 5 m/s: v1 + v2 78.5 m/s + 5 m/s = 2 2 = 41.8 m/s vav = 104 Chapter 2 ∆y12 = vav ∆t12 = (41.8 m/s )(4.90 s) Use this value to calculate how far she travels in 4.90 s: = 204 m She travels 204 m while slowing down. (e) Express the total time in terms of the times for each segment of her descent: t total = ∆t 01 + ∆t12 + ∆t 23 We know the times for the intervals from 0 to 1 and 1 to 2 so we only need to determine the time for the interval from 2 to 3. We can calculate ∆t23 from her displacement and constant velocity during that segment of her descent. ∆y23 = ∆ytotal − ∆y01 − ∆y12 Add the times to get the total time: t total = t01 + t12 + t23 ⎛ 78.5 m/s ⎞ = 575 m − ⎜ ⎟(8 s ) − 204 m 2 ⎝ ⎠ = 57.0 m = 8 s + 4.9 s + 57.0 m 5 m/s = 24.3 s (f) Using its definition, calculate her average velocity: vav = ∆x − 1500 m = = − 7.18 m/s ∆t 209 s Integration of the Equations of Motion *115 • Picture the Problem The integral of a function is equal to the "area" between the curve for that function and the independent-variable axis. (a) The graph is shown below: 35 30 v (m/s) 25 20 15 10 5 0 0 1 2 3 t (s) 4 5 Motion in One Dimension 105 The distance is found by determining the area under the curve. You can accomplish this easily because the shape of the area under the curve is a trapezoid. Alternatively, we could just count the blocks and fractions thereof. (b) To find the position function x(t), we integrate the velocity function v(t) over the time interval in question: A = (36 blocks)(2.5 m/block) = 90 m or ⎛ 33 m/s + 3 m/s ⎞ A=⎜ ⎟(5 s − 0 s ) = 90 m 2 ⎝ ⎠ There are approximately 36 blocks each having an area of (5 m/s)(0.5 s) = 2.5 m. t x(t ) = ∫ v(t ') dt ' 0 t [( ) = ∫ 6 m/s 2 t '+(3 m/s ) dt ' 0 and ( ) x(t ) = 3 m/s 2 t 2 + (3 m/s )t Now evaluate x(t) at 0 s and 5 s respectively and subtract to obtain ∆x: ∆x = x(5 s ) − x(0 s ) = 90 m − 0 m = 90.0 m 116 • Picture the Problem The integral of v(t) over a time interval is the displacement (change in position) during that time interval. The integral of a function is equivalent to the "area" between the curve for that function and the independent-variable axis. Count the grid boxes. (a) Find the area of the shaded gridbox: Area = (1 m/s )(1 s ) = 1 m per box (b) Find the approximate area under curve for 1 s ≤ t ≤ 2 s: ∆x1 s to 2 s = 1.2 m Find the approximate area under curve for 2 s ≤ t ≤ 3 s: ∆x2 s to 3 s = 3.2 m (c) Sum the displacements to obtain the total in the interval 1 s ≤ t ≤ 3 s: ∆x1 s to 3 s = 1.2 m + 3.2 m = 4.4 m ∆x1 s to 3 s 4.4 m = = 2.20 m/s ∆t1 s to 3 s 2s Using its definition, express and evaluate vav: vav = (d) Because the velocity of the particle is dx/dt, separate the dx = 0.5m/s3 dt ( so ) 106 Chapter 2 variables and integrate over the interval 1 s ≤ t ≤ 3 s to determine the displacement in this time interval: ( x ∆x1s→3s = ∫ dx' = 0.5m/s 3s 3 x0 )∫ t ' 2 dt ' 1s 3s ⎡ t ′3 ⎤ = 0.5m/s ⎢ ⎥ = 4.33m ⎣ 3 ⎦1s ( 3 ) This result is a little smaller than the sum of the displacements found in part (b). Calculate the average velocity over the 2-s interval from 1 s to 3 s: Calculate the initial and final velocities of the particle over the same interval: Finally, calculate the average value of the velocities at t = 1 s and t = 3 s: vav (1s− 3s) = ∆x1s− 3s ∆t1s− 3s = 4.33m = 2.17m/s 2s ( ) v(3 s ) = (0.5 m/s )(3 s ) v(1s ) = 0.5 m/s 3 (1s ) = 0.5 m/s 2 2 3 = 4.5 m/s v(1 s) + v(3 s) 0.5 m/s + 4.5 m/s = 2 2 = 2.50 m/s This average is not equal to the average velocity calculated above. Remarks: The fact that the average velocity was not equal to the average of the velocities at the beginning and the end of the time interval in part (d) is a consequence of the acceleration not being constant. *117 •• Picture the Problem Because the velocity of the particle varies with the square of the time, the acceleration is not constant. The displacement of the particle is found by integration. dx(t ) dt Express the velocity of a particle as the derivative of its position function: v(t ) = Separate the variables to obtain: dx(t ) = v(t )dt Express the integral of x from xo = 0 to x and t from t0 = 0 to t: Substitute for v(t′) to obtain: x(t ) = t t0 =0 x(t ) = x (t ) t0 = 0 ∫ dx' = ∫ v(t ') dt ' ∫ [(7 m/s )t ' −(5 m/s)]dt ' t 3 t0 =0 = ( 7 3 ) 2 m/s 3 t 3 − (5 m/s ) t Motion in One Dimension 107 118 •• Picture the Problem The graph is one of constant negative acceleration. Because vx = v(t) is a linear function of t, we can make use of the slope-intercept form of the equation of a straight line to find the relationship between these variables. We can then differentiate v(t) to obtain a(t) and integrate v(t) to obtain x(t). Find the acceleration (the slope of the graph) and the velocity at time 0 (the v-intercept) and use the slopeintercept form of the equation of a straight line to express vx(t): Find x(t) by integrating v(t): a = −10 m/s 2 v x (t ) = 50 m/s + (−10 m/s 2 )t [( ) x(t ) = ∫ − 10 m/s 2 t + 50 m/s dt ( ) = (50 m/s )t − 5 m/s 2 t 2 + C Using the fact that x = 0 when t = 0, evaluate C: Substitute to obtain: ( ) 0 = (50 m/s )(0) − 5 m/s 2 (0) + C and C=0 ( 2 ) x(t ) = (50 m/s ) t − 5 m/s 2 t 2 Note that this expression is quadratic in t and that the coefficient of t2 is negative and equal in magnitude to half the constant acceleration. Remarks: We can check our result for x(t) by evaluating it over the 10-s interval shown and comparing this result with the area bounded by this curve and the time axis. 119 ••• Picture the Problem During any time interval, the integral of a(t) is the change in velocity and the integral of v(t) is the displacement. The integral of a function equals the "area" between the curve for that function and the independent-variable axis. (a) Find the area of the shaded grid box in Figure 2-37: Area = (0.5 m/s2)(0.5 s) (b) We start from rest (vo = 0) at t = 0. For the velocities at the other times, count boxes and multiply by the 0.25 m/s per box that we found in part (a): Examples: v(1 s) = (3.7 boxes)[(0.25 m/s)/box] = 0.250 m/s per box = 0.925 m/s v(2 s) = (12.9 boxes)[(0.25 m/s)/box] = 3.22 m/s and 108 Chapter 2 v(3 s) = (24.6 boxes)[(0.25 m/s)/box] = 6.15 m/s (c) The graph of v as a function of t is shown below: 7 6 v (m/s) 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 t (s) Area = (1.0 m/s)(1.0 s) = 1.0 m per box Count the boxes under the v(t) curve to find the distance traveled: x(3 s ) = ∆x(0 → 3 s ) = (7 boxes )[(1.0 m ) / box ] = 7.00 m 120 •• Picture the Problem The integral of v(t) over a time interval is the displacement (change in position) during that time interval. The integral of a function equals the "area" between the curve for that function and the independent-variable axis. Because acceleration is the slope of a velocity versus time curve, this is a non-constant-acceleration problem. The derivative of a function is equal to the "slope" of the function at that value of the independent variable. (a) To obtain the data for x(t), we must estimate the accumulated area under the v(t) curve at each time interval: Find the area of a shaded grid box in Figure 2-38: A = (1 m/s)(0.5 s) = 0.5 m per box. We start from rest (vo = 0) at to= 0. For the position at the other times, count boxes and multiply by the 0.5 m per box that we found above. Remember to add the offset from the origin, xo = 5 m, and that boxes below the v = 0 line are counted as negative: Examples: ⎛ 0.5 m ⎞ x(3 s ) = (25.8 boxes )⎜ ⎟ + 5m ⎝ box ⎠ = 17.9 m ⎛ 0.5 m ⎞ x(5 s ) = (48.0 boxes )⎜ ⎟ + 5m ⎝ box ⎠ = 29.0 m Motion in One Dimension 109 ⎛ 0.5 m ⎞ x(10 s ) = (51.0 boxes )⎜ ⎟ ⎝ box ⎠ ⎛ 0.5 m ⎞ − (36.0 boxes )⎜ ⎟ + 5m ⎝ box ⎠ = 12.5 m A graph of x as a function of t follows: 35 30 x (m) 25 20 15 10 5 0 0 2 4 6 8 10 t (s) (b) To obtain the data for a(t), we must estimate the slope (∆v/∆t) of the v(t) curve at each time. A good way to get reasonably reliable readings from the graph is to enlarge several fold: Examples: a (1s ) = v(1.25 s ) − v(0.75 s ) 0 .5 s 4.9 m/s − 3.0 m/s = 3.8 m/s 2 0.5 s v(6.25 s ) − v(5.75 s ) a (6 s ) = 0 .5 s = = − 1.7 m/s − 0.4 m/s = −4.2 m/s 2 0.5 s 110 Chapter 2 A graph of a as a function of t follows: 6 4 a (m/s^2) 2 0 -2 -4 -6 0 2 4 6 8 10 t (s) *121 •• Picture the Problem Because the position of the body is not described by a parabolic function, the acceleration is not constant. Select a series of points on the graph of x(t) (e.g., at the extreme values and where the graph crosses the t axis), draw tangent lines at those points, and measure their slopes. In doing this, you are evaluating v = dx/dt at these points. Plot these slopes above the times at which you measured the slopes. Your graph should closely resemble the following graph. 8 6 4 2 v 0 -2 -4 -6 -8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t Select a series of points on the graph of v(t) (e.g., at the extreme values and where the graph crosses the t axis), draw tangent lines at those points, and measure their slopes. In doing this, you are evaluating a = dv/dt at these points. Plot these slopes above the times at which you measured the slopes. Your graph should closely resemble the graph shown below. Motion in One Dimension 111 15 10 5 a 0 -5 -10 -15 0 0.5 1 1.5 t 122 •• Picture the Problem Because the acceleration of the rocket varies with time, it is not constant and integration of this function is required to determine the rocket’s velocity and position as functions of time. The conditions on x and v at t = 0 are known as initial conditions. (a) Integrate a(t) to find v(t): v(t ) = ∫ a(t ) dt = b ∫ t dt = 1 bt 2 + C 2 where C, the constant of integration, can be determined from the initial conditions. Integrate v(t) to find x(t): [ x(t ) = ∫ v(t ) dt = ∫ 1 bt 2 + C dt 2 = 1 bt 3 + Ct + D 6 where D is a second constant of integration. Using the initial conditions, find the constants C and D: v(0 ) = 0 ⇒ C = 0 and x(0 ) = 0 ⇒ D = 0 ∴ x(t ) = 1 bt 3 6 (b) Evaluate v(5 s) and x(5 s) with C = D = 0 and b = 3 m/s2: v(5 s ) = and x(5 s ) = ( ) ( ) 1 2 3m/s 2 (5s ) = 37.5 m/s 2 1 3 3m/s 2 (5s ) = 62.5 m 6 123 •• Picture the Problem The acceleration is a function of time; therefore it is not constant. The instantaneous velocity can be determined by integration of the acceleration and the average velocity from the displacement of the particle during the given time interval. 112 Chapter 2 v (t ) (a) Because the acceleration is the derivative of the velocity, integrate the acceleration to find the instantaneous velocity v(t). dv a(t ) = ⇒ v(t ) = ∫ dv' = ∫ a(t ')dt ' dt v0 =0 t0 = 0 Calculate the instantaneous velocity using the acceleration given. v(t ) = 0.2 m/s ( t ) ∫ t ' dt ' t 3 t0 =0 and v(t ) = (0.1 m/s3 )t 2 (b) To calculate the average velocity, we need the displacement: Because the velocity is the derivative of the displacement, integrate the velocity to find ∆x. x (t ) t dx v(t ) ≡ ⇒ x(t ) = ∫ dx' = ∫ v(t ')dt ' dt x0 = 0 t0 =0 ( x(t ) = 0.1m/s3 ) ∫ t' t 2 ( dt ' = 0.1m/s3 t0 =0 )t3 3 and ∆x = x(7 s) − x(2 s) ⎡ (7 s )3 − (2 s)3 ⎤ = 0.1m/s3 ⎢ ⎥ 3 ⎣ ⎦ = 11.2 m ( Using the definition of the average velocity, calculate vav. ) vav = ∆x 11.2 m = = 2.23 m/s ∆t 5s 124 • Determine the Concept Because the acceleration is a function of time, it is not constant. Hence we’ll need to integrate the acceleration function to find the velocity as a function of time and integrate the velocity function to find the position as a function of time. The important concepts here are the definitions of velocity, acceleration, and average velocity. (a) Starting from to = 0, integrate the instantaneous acceleration to obtain the instantaneous velocity as a function of time: dv dt it follows that From a = v t v0 0 ∫ dv' = ∫ (a0 + bt ')dt ' and v = v0 + a0t + 1 bt 2 2 (b) Now integrate the instantaneous velocity to obtain the position as a function of time: From v = dx dt it follows that Motion in One Dimension 113 x t x0 t0 = 0 ∫ dx' = ∫ v(t ') dt ' b ⎞ ⎛ = ∫ ⎜ v9 + a0t '+ t '2 ⎟dt ' 2 ⎠ t0 ⎝ t and x = x0 + v0t + 1 a0t 2 + 1 bt 3 6 2 (c) The definition of the average velocity is the ratio of the displacement to the total time elapsed: vav ≡ ∆x x − x0 v0t + 1 a0t 2 + 1 bt 3 2 6 = = t − t0 t ∆t and vav = v0 + 1 a0t + 1 bt 2 6 2 Note that vav is not the same as that due to constant acceleration: (v ) constant acceleration av v0 + v 2 v + v0 + a0t + 1 bt 2 2 = 0 2 1 = v0 + 2 a0t + 1 bt 2 4 = ( ) ≠ vav General Problems 125 ••• Picture the Problem The acceleration of the marble is constant. Because the motion is downward, choose a coordinate system with downward as the positive direction. The equation gexp = (1 m)/(∆t)2 originates in the constant-acceleration equation ∆x = v0 ∆t + 1 a (∆t ) . Because the motion starts from rest, the displacement of the 2 2 marble is 1 m, the acceleration is the experimental value gexp, and the equation simplifies to gexp = (1 m)/(∆t)2. Express the percent difference between the accepted and experimental values for the acceleration due to gravity: Using a constant-acceleration equation, express the velocity of the marble in terms of its initial velocity, acceleration, and displacement: % difference = g accepted − g exp g accepted 2 vf2 = v0 + 2a∆y or, because v0 = 0 and a = g, vf2 = 2 g∆y Solve for vf: vf = 2 g∆y Let v1 be the velocity the ball has reached when it has fallen 0.5 cm, v1 = 2 9.81 m/s 2 (0.005 m ) = 0.313 m/s ( ) 114 Chapter 2 and v2 be the velocity the ball has reached when it has fallen 0.5 m to obtain. and Using a constant-acceleration equation, express v2 in terms of v1, g and ∆t: v 2 = v1 + g∆t ( ) v2 = 2 9.81 m/s 2 (0.5 m ) = 3.13 m/s ∆t = v2 − v1 g Substitute numerical values and evaluate ∆t: ∆t = 3.13 m/s − 0.313 m/s = 0.2872 s 9.81m/s 2 Calculate the experimental value of the acceleration due to gravity from gexp = (1 m)/(∆t)2: g exp = Solve for ∆t: Finally, calculate the percent difference between this experimental result and the value accepted for g at sea level. 1m = 12.13 m/s 2 2 (0.2872 s ) % difference = 9.81 m/s 2 − 12.13 m/s 2 9.81 m/s 2 = 23.6 % *126 ••• Picture the Problem We can obtain an average velocity, vav = ∆x/∆t, over fixed time intervals. The instantaneous velocity, v = dx/dt can only be obtained by differentiation. (a) The graph of x versus t is shown below: 8 6 v (m/s) 4 2 0 -2 -4 -6 0 5 10 15 20 25 30 35 t (s) (b) Draw a tangent line at the origin and measure its rise and run. Use this ratio to obtain an approximate value for the slope at the origin: The tangent line appears to, at least approximately, pass through the point (5, 4). Using the origin as the second point, Motion in One Dimension 115 ∆x = 4 cm – 0 = 4 cm and Therefore, the slope of the tangent line and the velocity of the body as it passes through the origin is approximately: ∆t = 5 s – 0 = 5 s v(0) = rise ∆x 4 cm = = = 0.800 cm/s run ∆t 5s (c) Calculate the average velocity for the series of time intervals given by completing the table shown below: t0 (s) 0 0 0 0 0 0 t (s) 6 3 2 1 0.5 0.25 ∆t (s) 6 3 2 1 0.5 0.25 x0 (cm) 0 0 0 0 0 0 x (cm) 4.34 2.51 1.71 0.871 0.437 0.219 ∆x (cm) 4.34 2.51 1.71 0.871 0.437 0.219 vav=∆x/∆t (m/s) 0.723 0.835 0.857 0.871 0.874 0.875 (d) Express the time derivative of the position: dx = Aω cos ωt dt Substitute numerical values and dx = Aω cos 0 = Aω dt = (0.05 m ) 0.175 s −1 evaluate dx at t = 0: dt ( ) = 0.875 cm/s (e) Compare the average velocities from part (c) with the instantaneous velocity from part (d): As ∆t, and thus ∆x, becomes small, the value for the average velocity approaches that for the instantaneous velocity obtained in part (d). For ∆t = 0.25 s, they agree to three significant figures. 127 ••• Determine the Concept Because the velocity varies nonlinearly with time, the acceleration of the object is not constant. We can find the acceleration of the object by differentiating its velocity with respect to time and its position function by integrating the velocity function. The important concepts here are the definitions of acceleration and velocity. (a) The acceleration of the object is the derivative of its velocity with respect to time: a= dv d = [v max sin (ωt )] dt dt = ω v max cos (ωt ) 116 Chapter 2 Because a varies sinusoidally with time it is not constant. (b) Integrate the velocity with respect to time from 0 to t to obtain the change in position of the body: x t x0 t0 ∫ dx' = ∫ [v max sin (ωt ')]dt ' and ⎡− v ⎤ x − x0 = ⎢ max cos (ωt ')⎥ ⎦0 ⎣ ω t = − vmax cos (ωt ) + vmax [1 − cos(ωt )] ω or x = x0 + ω vmax ω Note that, as given in the problem statement, x(0 s) = x0. 128 ••• Picture the Problem Because the acceleration of the particle is a function of its position, it is not constant. Changing the variable of integration in the definition of acceleration will allow us to determine its velocity and position as functions of position. (a) Because a = dv/dt, we must integrate to find v(t). Because a is given as a function of x, we’ll need to change variables in order to carry out the integration. Once we’ve changed variables, we’ll separate them with v on the left side of the equation and x on the right: Integrate from xo and vo to x and v: a= ( ) dv dv dx dv = = v = 2 s−2 x dt dx dt dx or, upon separating variables, ( ) vdv = 2 s −2 xdx ∫ v'dv' = ∫ (2 s )x'dx' v x v0 = 0 x0 and −2 ( )( 2 2 v 2 − v0 = 2 s −2 x 2 − x0 ) Solve for v to obtain: 2 2 v = v0 + (2 s − 2 )(x 2 − x0 ) Now set vo = 0, xo = 1 m, x = 3 m, b =2 s–2 and evaluate the speed: v = ± (2 s − 2 ) (3 m ) − (1m ) [ and v = 4.00m/s 2 2 Motion in One Dimension 117 (b) Using the definition of v, separate the variables, and integrate to get an expression for t: v( x ) = dx dt and t 0 dx′ x x0 ∫ dt ' = ∫ v(x′) To evaluate this integral we first must find v(x). Show that the acceleration is always positive and use this to find the sign of v(x). a = (2 s–2 )x and x0 = 1 m. x0 is positive, so a0 is also positive. v0 is zero and a0 is positive, so the object moves in the direction of increasing x. As x increases the acceleration remains positive, so the velocity also remains positive. Thus, (2 s )(x −2 v= Substitute (2 s )(x −2 2 ) 2 − x0 for v and evaluate the integral. (It can be found in standard integral tables.) t ∫ v(x′) 0 x0 dx′ x = ∫ (2 s )(x′ −2 x0 = = Evaluate this expression with xo = 1 m and x = 3 m to obtain: 2 − x0 ) . dx′ x t = ∫ dt' = 2 x 1 (2 s −2 )∫ x0 1 (2 s ) −2 2 2 − x0 ) dx′ 2 x′2 − x0 ⎛ x + x2 − x2 0 ln⎜ ⎜ x0 ⎝ ⎞ ⎟ ⎟ ⎠ t = 1.25 s 129 ••• Picture the Problem The acceleration of this particle is not constant. Separating variables and integrating will allow us to express the particle’s position as a function of time and the differentiation of this expression will give us the acceleration of the particle as a function of time. (a) Write the definition of velocity: v= dx dt We are given that x = bv, where b = 1 s. Substitute for v and separate variables to obtain: dx x dx = ⇒ dt = b dt b x Integrate and solve for x(t): t ⎛ x⎞ dx′ ⇒ (t − t0 ) = b ln⎜ ⎟ ⎜x ⎟ x′ ⎝ 0⎠ x0 x ∫ dt ' = b ∫ t0 and 118 Chapter 2 x(t ) = x0e (b) Differentiate twice to obtain v(t) and a(t): v= (t − t 0 ) / b (t −t0 ) / b dx 1 = x0 e dt b and a= Substitute the result in part (a) to obtain the desired results: (t −t0 ) / b dv 1 = 2 x0 e dt b v(t ) = 1 x(t ) b and 1 x(t ) b2 a(t ) = so 1 1 v(t ) = 2 x(t ) b b a(t ) = Because the numerical value of b, expressed in SI units, is one, the numerical values of a, v, and x are the same at each instant in time. 130 ••• Picture the Problem Because the acceleration of the rock is a function of time, it is not constant. Choose a coordinate system in which downward is positive and the origin at the point of release of the rock. Separate variables in a(t) = dv/dt = ge−bt to obtain: Integrate from to = 0, vo = 0 to some later time t and velocity v: dv = ge − bt dt v t 0 0 v = ∫ dv' = ∫ ge −bt ' dt ' = = ( ) ( g 1 − e −bt = vterm 1 − e −bt b where g b vterm = Separate variables in ( v = dy dt = vterm 1 − e − bt ) to ( ) dy = vterm 1 − e − bt dt obtain: Integrate from to = 0, yo = 0 to some later time t and position y: [ ] g −bt ' e −b −bt ' ∫ dy' = ∫ vterm (1 − e )dt ' y t 0 0 ) t 0 Motion in One Dimension 119 t ⎡ 1 ⎤ y = vterm ⎢t '+ e −bt ' ⎥ ⎣ b ⎦0 = vtermt − ( vterm 1 − e −bt b ) This last result is very interesting. It says that throughout its free-fall, the object experiences drag; therefore it has not fallen as far at any given time as it would have if it were falling at the constant velocity, vterm. On the other hand, just as the velocity of the object asymptotically approaches vterm, the distance it has covered during its free-fall as a function of time asymptotically approaches the distance it would have fallen if it had fallen with vterm throughout its motion. y (tlarge ) → vtermt − v → vtermt b This should not be surprising because in the expression above, the first term grows linearly with time while the second term approaches a constant and therefore becomes less important with time. *131 ••• Picture the Problem Because the acceleration of the rock is a function of its velocity, it is not constant. Choose a coordinate system in which downward is positive and the origin is at the point of release of the rock. Rewrite a = g – bv explicitly as a differential equation: dv = g − bv dt Separate the variables, v on the left, t on the right: dv = dt g − bv Integrate the left-hand side of this equation from 0 to v and the righthand side from 0 to t: v t dv' ∫ g − bv' = ∫ dt ' 0 0 and 1 ⎛ g − bv ⎞ ⎟=t − ln⎜ b ⎜ g ⎟ ⎝ ⎠ Solve this expression for v. Finally, differentiate this expression with respect to time to obtain an expression for the acceleration and ( ) v= g 1 − e − bt b a= dv = ge −bt dt 120 Chapter 2 complete the proof. 132 ••• Picture the Problem The skydiver’s acceleration is a function of her velocity; therefore it is not constant. Expressing her acceleration as the derivative of her velocity, separating the variables, and then integrating will give her velocity as a function of time. (a) Rewrite a = g – cv2 explicitly as a differential equation: dv = g − cv 2 dt Separate the variables, with v on the left, and t on the right: dv = dt g − cv 2 Eliminate c by using c = g : 2 vT dv dv = g ⎡ ⎛ v g − 2 v2 g ⎢1 − ⎜ vT ⎜ ⎢ ⎝ vT ⎣ or dv ⎛ v 1− ⎜ ⎜v ⎝ T ⎞ ⎟ ⎟ ⎠ 2 ⎞ ⎟ ⎟ ⎠ ∫ The integral can be found in integral tables: vT tanh −1 (v / vT ) = gt 0 t dv' ⎛ v' 1− ⎜ ⎜v ⎝ T ⎤ ⎥ ⎥ ⎦ = gdt Integrate the left-hand side of this equation from 0 to v and the righthand side from 0 to t: v 2 ⎞ ⎟ ⎟ ⎠ 2 = g ∫ dt ' = gt 0 or tanh -1 (v / vT ) = ( g / vT )t Solve this equation for v to obtain: ⎛g v = vT tanh⎜ ⎜v ⎝ T Because c has units of m−1, and g has units of m/s2, (cg)−1/2 will have units of time. Let’s represent this expression with the time-scale factor T: i.e., T = (cg)−1/2 ⎞ ⎟t ⎟ ⎠ = dt Motion in One Dimension 121 The skydiver falls with her terminal velocity when a = 0. Using this definition, relate her terminal velocity to the acceleration due to gravity and the constant c in the acceleration equation: 2 0 = g − cvT and vT = Convince yourself that T is also equal to vT/g and use this relationship to eliminate g and vT in the solution to the differential equation: g c ⎛t⎞ v(t ) = vT tanh⎜ ⎟ ⎝T ⎠ (b) The following table was generated using a spreadsheet and the equation we derived in part (a) for v(t). The cell formulas and their algebraic forms are: Cell Content/Formula Algebraic Form D1 56 vT D2 5.71 T B7 B6 + 0.25 t + 0.25 C7 $B$1*TANH(B7/$B$2) ⎛t⎞ vT tanh⎜ ⎟ ⎝T ⎠ A 1 2 3 4 5 6 7 8 9 10 54 55 56 57 58 59 B vT = 56 T= 5.71 C m/s s time (s) 0.00 0.25 0.50 0.75 1.00 v (m/s) 0.00 2.45 4.89 7.32 9.71 12.00 12.25 12.50 12.75 13.00 13.25 54.35 54.49 54.61 54.73 54.83 54.93 D E 122 Chapter 2 60 50 v (m/s) 40 30 20 10 0 0 2 4 6 8 10 12 14 t (s) Note that the velocity increases linearly over time (i.e., with constant acceleration) for about time T, but then it approaches the terminal velocity as the acceleration decreases. Chapter 3 Motion in Two and Three Dimensions Conceptual Problems *1 • Determine the Concept The distance traveled along a path can be represented as a sequence of displacements. Suppose we take a trip along some path and consider the trip as a sequence of many very small displacements. The net displacement is the vector sum of the very small displacements, and the total distance traveled is the sum of the magnitudes of the very small displacements. That is, r r r r total distance = ∆r0,1 + ∆r1, 2 + ∆r2,3 + ... + ∆rN −1, N where N is the number of very small displacements. (For this to be exactly true we have to take the limit as N goes to infinity and each displacement magnitude goes to zero.) Now, using ″the shortest distance between two points is a straight line,″ we have r r r r r ∆r0, N ≤ ∆r0,1 + ∆r1, 2 + ∆r2,3 + ... + ∆rN −1, N , r where ∆r0, N is the magnitude of the net displacement. Hence, we have shown that the magnitude of the displacement of a particle is less than or equal to the distance it travels along its path. 2 • Determine the Concept The displacement of an object is its final position vector minus r r r its initial position vector ( ∆r = rf − ri ). The displacement can be less but never more than the distance traveled. Suppose the path is one complete trip around the earth at the equator. Then, the displacement is 0 but the distance traveled is 2πRe. 123 124 Chapter 3 3 • Determine the Concept The important distinction here is that average velocity is being requested, as opposed to average speed. The average velocity is defined as the displacement divided by the elapsed time. r r ∆r 0 vav = = =0 ∆t ∆t The displacement for any trip around the track is zero. Thus we see that no matter how fast the race car travels, the average velocity is always zero at the end of each complete circuit. What is the correct answer if we were asked for average speed? The average speed is defined as the distance traveled divided by the elapsed time. vav ≡ total distance ∆t For one complete circuit of any track, the total distance traveled will be greater than zero and the average is not zero. 4 • False. Vectors are quantities with magnitude and direction that can be added and subtracted like displacements. Consider two vectors that are equal in magnitude and oppositely directed. Their sum is zero, showing by counterexample that the statement is false. 5 • Determine the Concept We can answer this question by expressing the relationship r between the magnitude of vector A and its component AS and then using properties of the cosine function. Express AS in terms of A and θ : AS = A cosθ Take the absolute value of both sides of this expression: ⎜AS⎜ = ⎜A cosθ ⎜ = A⎜cosθ ⎜ and ⎜cosθ ⎜= AS A Motion in One and Two Dimensions 125 Using the fact that 0 < ⎟cosθ ⎜≤ 1, substitute for⎟cosθ ⎜to obtain: 0< AS ≤ 1 or 0 < AS ≤ A A No. The magnitude of a component of a vector must be less than or equal to the magnitude of the vector. If the angle θ shown in the figure is equal to 0° or multiples of 180°, then the magnitude of the vector and its component are equal. *6 • Determine the Concept The diagram r shows a vector A and its components Ax and Ay. We can relate the magnitude of r A is related to the lengths of its components through the Pythagorean theorem. r 2 2 Suppose that A is equal to zero. Then A2 = Ax + Ay = 0. 2 2 But Ax + Ay = 0 ⇒ Ax = Ay = 0. No. If a vector is equal to zero, each of its components must be zero too. 7 • r r Determine the Concept No. Consider the special case in which B = − A . r r r r r If B = − A ≠ 0, then C = 0 and the magnitudes of the components of A and B are r larger than the components of C . *8 • Determine the Concept The instantaneous acceleration is the limiting value, as ∆t r r approaches zero, of ∆v ∆t . Thus, the acceleration vector is in the same direction as ∆v . False. Consider a ball that has been thrown upward near the surface of the earth and is slowing down. The direction of its motion is upward. The diagram shows the ball’s velocity vectors at two instants of r time and the determination of ∆v . r Note that because ∆v is downward so is the acceleration of the ball. 126 Chapter 3 9 • Determine the Concept The instantaneous acceleration is the limiting value, as ∆t r r approaches zero, of ∆v ∆t and is in the same direction as ∆v . r Other than through the definition of a, the instantaneous velocity and acceleration vectors are unrelated. Knowing the direction of the velocity at one instant tells one nothing about how the velocity is changing at that instant. (e) is correct. 10 • Determine the Concept The changing velocity of the golf ball during its flight can be understood by recognizing that it has both horizontal and vertical components. The nature of its acceleration near the highest point of its flight can be understood by analyzing the vertical components of its velocity on either side of this point. At the highest point of its flight, the ball is still traveling horizontally even though its vertical velocity is momentarily zero. The figure to the right shows the vertical components of the ball’s velocity just before and just after it has reached its highest point. The change in velocity during this short interval is a non-zero, downward-pointing vector. Because the acceleration is proportional to the change in velocity, it must also be nonzero. (d ) is correct. Remarks: Note that vx is nonzero and vy is zero, while ax is zero and ay is nonzero. 11 • Determine the Concept The change in the velocity is in the same direction as the acceleration. Choose an x-y coordinate system with east being the positive x direction and north the positive y direction. r r Given our choice of coordinate system, the x component of a is negative and so v will r r decrease. The y component of a is positive and so v will increase toward the north. (c) is correct. *12 • r Determine the Concept The average velocity of a particle, vav , is the ratio of the particle’s displacement to the time required for the displacement. r (a) We can calculate ∆r from the given information and ∆t is known. (a ) is correct. r (b) We do not have enough information to calculate ∆v and cannot compute the Motion in One and Two Dimensions 127 particle’s average acceleration. (c) We would need to know how the particle’s velocity varies with time in order to compute its instantaneous velocity. (d) We would need to know how the particle’s velocity varies with time in order to compute its instantaneous acceleration. 13 •• Determine the Concept The velocity vector is always in the direction of motion and, thus, tangent to the path. (a) The velocity vector, as a consequence of always being in the direction of motion, is tangent to the path. (b) A sketch showing two velocity vectors for a particle moving along a path is shown to the right. 14 • Determine the Concept An object experiences acceleration whenever either its speed changes or it changes direction. The acceleration of a car moving in a straight path at constant speed is zero. In the other examples, either the magnitude or the direction of the velocity vector is changing and, hence, the car is accelerated. (b) is correct. *15 • r r Determine the Concept The velocity vector is defined by v = dr / dt , while the r r acceleration vector is defined by a = dv / dt. (a) A car moving along a straight road while braking. (b) A car moving along a straight road while speeding up. (c) A particle moving around a circular track at constant speed. 16 • Determine the Concept A particle experiences accelerated motion when either its speed or direction of motion changes. A particle moving at constant speed in a circular path is accelerating because the 128 Chapter 3 direction of its velocity vector is changing. If a particle is moving at constant velocity, it is not accelerating. 17 •• Determine the Concept The acceleration vector is in the same direction as the change in r velocity vector, ∆v . (a) The sketch for the dart thrown upward is shown to the right. The acceleration vector is in the direction of the change in the r velocity vector ∆v . (b) The sketch for the falling dart is shown to the right. Again, the acceleration vector is in the direction of the change in the r velocity vector ∆v . (c) The acceleration vector is in the direction of the change in the velocity vector … and hence is downward as shown the right: *18 •• Determine the Concept The acceleration vector is in the same direction as the change in r velocity vector, ∆v . The drawing is shown to the right. 19 •• Determine the Concept The acceleration vector is in the same direction as the change in r velocity vector, ∆v . The sketch is shown to the right. Motion in One and Two Dimensions 129 20 • Determine the Concept We can decide what the pilot should do by considering the speeds of the boat and of the current. Give up. The speed of the stream is equal to the maximum speed of the boat in still water. The best the boat can do is, while facing directly upstream, maintain its position relative to the bank. (d ) is correct. *21 • Determine the Concept True. In the absence of air resistance, both projectiles experience the same downward acceleration. Because both projectiles have initial vertical velocities of zero, their vertical motions must be identical. 22 • Determine the Concept In the absence of air resistance, the horizontal component of the projectile’s velocity is constant for the duration of its flight. At the highest point, the speed is the horizontal component of the initial velocity. The vertical component is zero at the highest point. (e) is correct. 23 • Determine the Concept In the absence of air resistance, the acceleration of the ball depends only on the change in its velocity and is independent of its velocity. As the ball moves along its trajectory between points A and C, the vertical component of its velocity decreases and the change in its velocity is a downward pointing vector. Between points C and E, the vertical component of its velocity increases and the change in its velocity is also a downward pointing vector. There is no change in the horizontal component of the velocity. (d ) is correct. 24 • Determine the Concept In the absence of air resistance, the horizontal component of the velocity remains constant throughout the flight. The vertical component has its maximum values at launch and impact. (a) The speed is greatest at A and E. (b) The speed is least at point C. (c) The speed is the same at A and E. The horizontal components are equal at these points but the vertical components are oppositely directed. 25 • Determine the Concept Speed is a scalar quantity, whereas acceleration, equal to the rate of change of velocity, is a vector quantity. (a) False. Consider a ball on the end of a string. The ball can move with constant speed 130 Chapter 3 (a scalar) even though its acceleration (a vector) is always changing direction. (b) True. From its definition, if the acceleration is zero, the velocity must be constant and so, therefore, must be the speed. 26 • r r Determine the Concept The average acceleration vector is defined by aav = ∆v / ∆t. r The direction of aav is that of r r r ∆v = vf − vi , as shown to the right. 27 • r r r Determine the Concept The velocity of B relative to A is v BA = v B − v A . r r r The direction of v BA = v B − v A is shown to the right. *28 •• r r (a) The vectors A(t ) and A(t + ∆t ) are of equal length but point in slightly different r r directions. ∆A is shown in the diagram below. Note that ∆A is nearly perpendicular r r r to A(t ) . For very small time intervals, ∆A and A(t ) are perpendicular to one another. r r Therefore, dA / dt is perpendicular to A. r (b) If A represents the position of a particle, the particle must be undergoing circular motion (i.e., it is at a constant distance from some origin). The velocity vector is tangent to the particle’s trajectory; in the case of a circle, it is perpendicular to the circle’s radius. (c) Yes, it could in the case of uniform circular motion. The speed of the particle is constant, but its heading is changing constantly. The acceleration vector in this case is Motion in One and Two Dimensions 131 always perpendicular to the velocity vector. 29 •• Determine the Concept The velocity vector is in the same direction as the change in the position vector while the acceleration vector is in the same direction as the change in the velocity vector. Choose a coordinate system in which the y direction is north and the x direction is east. (b) (a) Path AB BC CD DE EF (c) Direction of velocity vector north northeast east southeast south Path AB BC CD DE EF Direction of acceleration vector north southeast 0 southwest north The magnitudes are comparable, but larger for DE since the radius of the path is smaller there. *30 •• Determine the Concept We’ll assume that the cannons are identical and use a constantacceleration equation to express the displacement of each cannonball as a function of time. Having done so, we can then establish the condition under which they will have the same vertical position at a given time and, hence, collide. The modified diagram shown below shows the displacements of both cannonballs. Express the displacement of the cannonball from cannon A at any time t after being fired and before any collision: r r r ∆r = v0t + 1 gt 2 2 Express the displacement of the cannonball from cannon A at any time t′ after being fired and before any collision: r r r ′ ∆r ′ = v0t ′ + 1 gt ′2 2 132 Chapter 3 If the guns are fired simultaneously, t = t ' and the balls are the same distance 1 2 gt 2 below the line of sight at all times. Therefore, they should fire the guns simultaneously. Remarks: This is the ″monkey and hunter″ problem in disguise. If you imagine a monkey in the position shown below, and the two guns are fired simultaneously, and the monkey begins to fall when the guns are fired, then the monkey and the two cannonballs will all reach point P at the same time. 31 •• Determine the Concept The droplet leaving the bottle has the same horizontal velocity as the ship. During the time the droplet is in the air, it is also moving horizontally with the same velocity as the rest of the ship. Because of this, it falls into the vessel, which has the same horizontal velocity. Because you have the same horizontal velocity as the ship does, you see the same thing as if the ship were standing still. 32 • Determine the Concept r r Because A and D are tangent to the path of the stone, either of them could (a) represent the velocity of the stone. r r Let the vectors A(t ) and B (t + ∆t ) be of equal length but point in slightly different directions as the stone moves around the circle. These two r r vectors and ∆A are shown in the diagram above. Note that ∆A is nearly r r r (b) perpendicular to A(t ). For very small time intervals, ∆A and A(t ) are r r perpendicular to one another. Therefore, dA/dt is perpendicular to A and r only the vector E could represent the acceleration of the stone. Motion in One and Two Dimensions 133 33 • Determine the Concept True. An object accelerates when its velocity changes; that is, when either its speed or its direction changes. When an object moves in a circle the direction of its motion is continually changing. 34 •• Picture the Problem In the diagram, (a) shows the pendulum just before it reverses direction and (b) shows the pendulum just after it has reversed its direction. The acceleration of the bob is in the direction of r r r the change in the velocity ∆v = v f − v i and is tangent to the pendulum trajectory at the point of reversal of direction. This makes sense because, at an extremum of motion, v = 0, so there is no centripetal acceleration. However, because the velocity is reversing direction, the tangential acceleration is nonzero. 35 • Determine the Concept The principle reason is aerodynamic drag. When moving through a fluid, such as the atmosphere, the ball's acceleration will depend strongly on its velocity. Estimation and Approximation *36 •• Picture the Problem During the flight of the ball the acceleration is constant and equal to 9.81 m/s2 directed downward. We can find the flight time from the vertical part of the motion, and then use the horizontal part of the motion to find the horizontal distance. We’ll assume that the release point of the ball is 2 m above your feet. Make a sketch of the motion. Include coordinate axes, initial and final positions, and initial velocity components: Obviously, how far you throw the ball will depend on how fast you can throw it. A major league baseball pitcher can throw a fastball at 90 mi/h or so. Assume that you can throw a ball at two-thirds that speed to obtain: v0 = 60 mi/h × 0.447 m/s = 26.8 m/s 1 mi/h 134 Chapter 3 There is no acceleration in the x direction, so the horizontal motion is one of constant velocity. Express the horizontal position of the ball as a function of time: x = v0 xt Assuming that the release point of the ball is a distance h above the ground, express the vertical position of the ball as a function of time: y = h + v0 y t + 1 a y t 2 2 (a) For θ = 0 we have: v0 x = v0 cosθ 0 = (26.8 m/s )cos 0° (1) (2) = 26.8 m/s and v0 y = v0 sin θ 0 = (26.8 m/s )sin 0° = 0 Substitute in equations (1) and (2) to obtain: x = (26.8 m/s )t and ( ) y = 2 m + 1 − 9.81 m/s 2 t 2 2 Eliminate t between these equations to obtain: At impact, y = 0 and x = R: y = 2m − 4.91 m/s 2 2 x (26.8 m/s)2 0 = 2m − 4.91 m/s 2 2 R (26.8 m/s)2 Solve for R to obtain: R = 17.1 m (b) Using trigonometry, solve for v0x and v0y: v0 x = v0 cos θ 0 = (26.8 m/s ) cos 45° = 19.0 m/s and v0 y = v0 sin θ 0 = (26.8 m/s ) sin 45° = 19.0 m/s Substitute in equations (1) and (2) to obtain: x = (19.0 m/s )t and ( ) y = 2 m + (19.0 m/s )t + 1 − 9.81 m/s 2 t 2 2 Eliminate t between these equations to obtain: y = 2m + x − 4.905 m/s 2 2 x (19.0 m/s)2 Motion in One and Two Dimensions 135 At impact, y = 0 and x = R. Hence: 4.905 m/s 2 2 R 0 = 2m + R − (19.0 m/s)2 or R 2 − (73.60 m )R − 147.2 m 2 = 0 Solve for R (you can use the ″solver″ or ″graph″ functions of your calculator) to obtain: R = 75.6 m (c) Solve for v0x and v0y: v0 x = v0 = 26.8 m/s and v0 y = 0 Substitute in equations (1) and (2) to obtain: x = (26.8 m/s )t and ( ) y = 14 m + 1 − 9.81 m/s 2 t 2 2 Eliminate t between these equations to obtain: y = 14 m − 4.905 m/s 2 2 x (26.8 m/s)2 At impact, y = 0 and x = R: 4.905 m/s 2 2 R 0 = 14 m − (26.8 m/s)2 Solve for R to obtain: R = 45.3 m (d) Using trigonometry, solve for v0x and v0y: v0x = v0 y = 19.0 m / s Substitute in equations (1) and (2) to obtain: x = (19.0 m/s )t Eliminate t between these equations to obtain: At impact, y = 0 and x = R: Solve for R (you can use the ″solver″ or ″graph″ function of your calculator) to obtain: and y = 14 m + (19.0 m/s ) t + 1 (− 9.81 m/s 2 )t 2 2 4.905 m/s 2 2 y = 14 m + x − x (19.0 m/s)2 0 = 14 m + R − R = 85.6 m 4.905 m/s 2 2 R (19.0 m/s)2 136 Chapter 3 37 •• Picture the Problem We’ll ignore the height of Geoff’s release point above the ground and assume that he launched the brick at an angle of 45°. Because the velocity of the brick at the highest point of its flight is equal to the horizontal component of its initial velocity, we can use constant-acceleration equations to relate this velocity to the brick’s x and y coordinates at impact. The diagram shows an appropriate coordinate system and the brick when it is at point P with coordinates (x, y). Using a constant-acceleration equation, express the x coordinate of the brick as a function of time: Express the y coordinate of the brick as a function of time: x = x0 + v0 x t + 1 ax t 2 2 or, because x0 = 0 and ax = 0, x = v0 xt y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0 and ay = −g, y = v0 y t − 1 gt 2 2 Eliminate the parameter t to obtain: y = (tan θ 0 )x − g 2 x 2 2v0 x Use the brick’s coordinates when it strikes the ground to obtain: 0 = (tan θ 0 )R − g 2 R 2 2v0 x where R is the range of the brick. Solve for v0x to obtain: Substitute numerical values and evaluate v0x: v0 x = v0 x = gR 2 tan θ 0 (9.81m/s )(44.5 m ) = 2 2 tan 45° 14.8 m/s Note that, at the brick’s highest point, vy = 0. Vectors, Vector Addition, and Coordinate Systems 38 • Picture the Problem Let the positive y direction be straight up, the positive x direction r r be to the right, and A and B be the position vectors for the minute and hour hands. The Motion in One and Two Dimensions 137 pictorial representation below shows the orientation of the hands of the clock for parts (a) through (d). (a) The position vector for the minute hand at12:00 is: r A12:00 = (0.5 m ) ˆ j The position vector for the hour hand at 12:00 is: r B12:00 = (0.25 m ) ˆ j (b) At 3:30, the minute hand is positioned along the −y axis, while the hour hand is at an angle of (3.5 h)/12 h × 360° = 105°, measured clockwise from the top. The position vector for the minute hand is: r A3:30 = − (0.5 m ) ˆ j Find the x-component of the vector representing the hour hand: Bx = (0.25 m )sin 105° = 0.241m Find the y-component of the vector representing the hour hand: By = (0.25 m )cos105° = −0.0647 m The position vector for the hour hand is: r B3:30 = (0.241m )iˆ − (0.0647 m ) ˆ j (c) At 6:30, the minute hand is positioned along the −y axis, while the hour hand is at an angle of (6.5 h)/12 h × 360° = 195°, measured clockwise from the top. The position vector for the minute hand is: r A6:30 = − (0.5 m ) ˆ j Find the x-component of the vector representing the hour hand: Bx = (0.25 m )sin 195° = −0.0647 m Find the y-component of the vector representing the hour hand: By = (0.25 m )cos195° = −0.241m The position vector for the hour hand is: r ˆ B6:30 = − (0.0647 m )i − (0.241m ) ˆ j (d) At 7:15, the minute hand is positioned along the +x axis, while the hour hand is at an angle of (7.25 h)/12 h × 360° = 218°, measured clockwise from the top. 138 Chapter 3 The position vector for the minute hand is: r A7:15 = Find the x-component of the vector representing the hour hand: Bx = (0.25 m )sin 218° = −0.154 m Find the y-component of the vector representing the hour hand: By = (0.25 m )cos 218° = −0.197 m The position vector for the hour hand is: r ˆ B7:15 = − (0.154 m ) i − (0.197 m ) ˆ j r r (e) Find A − B at 12:00: (0.5 m ) iˆ r r A − B = (0.5 m ) ˆ − (0.25 m ) ˆ j j = r r Find A − B at 3:30: (0.25 m ) ˆ j r r A − B = −(0.5 m ) ˆ j [ ˆ j − (0.241 m ) i − (0.0647 m ) ˆ ˆ j = − (0.241m ) i − (0.435 m ) ˆ r r Find A − B at 6:30: r r A − B = −(0.5 m ) ˆ j [ ˆ j − (0.0647 m ) i − (0.241 m ) ˆ ˆ j = − (0.0647 m ) i − (0.259 m ) ˆ r r Find A − B at 7:15: r r A − B = (0.5 m ) ˆ j [ ˆ j − − (0.152 m ) i − (0.197 m ) ˆ = (0.152 m ) iˆ + (0.697 m ) ˆ j *39 • Picture the Problem The resultant displacement is the vector sum of the individual displacements. The two displacements of the bear and its resultant displacement are shown to the right: Motion in One and Two Dimensions 139 Using the law of cosines, solve for the resultant displacement: R 2 = (12 m ) + (12 m ) 2 2 − 2(12 m )(12 m )cos135° and R = 22.2 m Using the law of sines, solve for α: sin α sin 135° = 12 m 22.2 m ∴ α = 22.5° and the angle with the horizontal is 45° − 22.5° = 22.5° 40 • Picture the Problem The resultant displacement is the vector sum of the individual displacements. (a) Using the endpoint coordinates for her initial and final positions, draw the student’s initial and final position vectors and construct her displacement vector. Find the magnitude of her displacement and the angle this displacement makes with the positive x-axis: (b) Her displacement is 5 2 m @ 135°. His initial and final positions are the same as in (a), so his displacement is also 5 2 @ 135°. *41 • Picture the Problem Use the standard rules for vector addition. Remember that changing the sign of a vector reverses its direction. (a) (b) 140 Chapter 3 (c) (d) (e) 42 • Picture the Problem The figure shows the paths walked by the Scout. The length of path A is 2.4 km; the length of path B is 2.4 km; and the length of path C is 1.5 km: (a) Express the distance from the campsite to the end of path C: 2.4 km – 1.5 km = 0.9 km (b) Determine the angle θ subtended by the arc at the origin (campsite): θradians = arc length 2.4 km = radius 2.4 km = 1 rad = 57.3° His direction from camp is 1 rad north of east. (c) Express the total distance as the sum of the three parts of his walk: dtot = deast + darc + dtoward camp Substitute the given distances to find the total: dtot = 2.4 km + 2.4 km + 1.5 km = 6.3 km Motion in One and Two Dimensions 141 Express the ratio of the magnitude of his displacement to the total distance he walked and substitute to obtain a numerical value for this ratio: Magnitude of his displacement 0.9 km = Total distance walked 6.3 km = 1 7 43 • Picture the Problem The direction of a vector is determined by its components. The vector is in the fourth quadrant and ⎛ − 3.5 m/s ⎞ θ = tan −1 ⎜ ⎜ 5.5 m/s ⎟ = −32.5° ⎟ ⎝ ⎠ (b) is correct. 44 • Picture the Problem The components of the resultant vector can be obtained from the components of the vectors being added. The magnitude of the resultant vector can then be found by using the Pythagorean Theorem. A table such as the one shown to the right is useful in organizing the r information in this problem. Let D r r Vector x-component y-component r 6 −3 A r 4 −3 B r be the sum of vectors A, B, and C . r r r adding the components of A, B, r and C . Determine the components of D by Use the Pythagorean Theorem to r calculate the magnitude of D : r C r D 2 5 Dx = 5 and Dy = 6 2 D = Dx2 + D y = (5)2 + (6)2 = 7.81 and (d ) is correct. 45 • Picture the Problem The components of the given vector can be determined using righttriangle trigonometry. Use the trigonometric relationships between the magnitude of a vector and its components to calculate the x- and y-components of each vector. (a) (b) (c) (d) A 10 m 5m 7 km 5 km θ 30° 45° 60° 90° Ax 8.66 m 3.54 m 3.50 km 0 Ay 5m 3.54 m 6.06 km 5 km 142 Chapter 3 (e) 15 km/s 150° −13.0 km/s 7.50 km/s (f) 10 m/s 240° −5.00 m/s −8.66 m/s (g) 8 m/s2 270° 0 −8.00 m/s2 *46 • Picture the Problem Vectors can be added and subtracted by adding and subtracting their components. r Write A in component form: Ax = (8 m) cos 37° = 6.4 m Ay = (8 m) sin 37° = 4.8 m r ˆ ∴ A = (6.4 m ) i + (4.8 m ) ˆ j (a), (b), (c) Add (or subtract) x- and y-components: r D= r E= r F= r (d) Solve for G and add components to obtain: (0.4m ) iˆ + (7.8m ) ˆ j (− 3.4m ) iˆ − (9.8m ) ˆ j (− 17.6m ) iˆ + (23.8m ) ˆ j ( ) r r 1 r r G = − A + B + 2C 2 ˆ = (1.3 m ) i − (2.9 m ) ˆ j 47 •• Picture the Problem The magnitude of each vector can be found from the Pythagorean theorem and their directions found using the inverse tangent function. r ˆ j (a) A = 5 i + 3 ˆ r ˆ j (b) B = 10 i − 7 ˆ r ˆ ˆ (c) C = −2 i − 3 ˆ + 4k j A= 2 Ax2 + Ay = 5.83 r and, because A is in the 1st quadrant, ⎛A ⎞ θ = tan −1 ⎜ y ⎟ = 31.0° ⎜A ⎟ ⎝ x⎠ 2 B = Bx2 + B y = 12.2 r and, because B is in the 4th quadrant, ⎛B ⎞ θ = tan −1 ⎜ y ⎟ = − 35.0° ⎜B ⎟ ⎝ x⎠ 2 C = C x2 + C y + C z2 = 5.39 ⎛ Cz ⎞ ⎟ = 42.1° ⎝C⎠ θ = cos −1 ⎜ where θ is the polar angle measured from the positive z-axis and Motion in One and Two Dimensions 143 ⎛ Cx ⎞ −1 ⎛ − 2 ⎞ ⎟ = 112° ⎟ = cos ⎜ ⎝C ⎠ ⎝ 29 ⎠ φ = cos −1 ⎜ 48 • Picture the Problem The magnitude and direction of a two-dimensional vector can be found by using the Pythagorean Theorem and the definition of the tangent function. r ˆ j (a) A = −4i − 7 ˆ r ˆ B = 3i − 2 ˆ j r r r ˆ C = A + B = −i − 9 ˆ j (b) Follow the same steps as in (a). A= 2 Ax2 + Ay = 8.06 r and, because A is in the 3rd quadrant, ⎛A ⎞ θ = tan −1 ⎜ y ⎟ = 240° ⎜A ⎟ ⎝ x⎠ 2 B = Bx2 + B y = 3.61 r and, because B is in the 4th quadrant, ⎛B ⎞ θ = tan −1 ⎜ y ⎟ = − 33.7° ⎜B ⎟ ⎝ x⎠ 2 C = C x2 + C y = 9.06 r and, because C is in the 3rd quadrant, ⎛C ⎞ θ = tan −1 ⎜ y ⎟ = 264° ⎜C ⎟ ⎝ x⎠ A = 4.12 ; θ = − 76.0° B = 6.32 ; θ = 71.6° C = 3.61 ; θ = 33.7° 49 • Picture the Problem The components of these vectors are related to the magnitude of each vector through the Pythagorean Theorem and trigonometric functions. In parts (a) and (b), calculate the rectangular components of each vector and then express the vector in rectangular form. 144 Chapter 3 r (a) Express v in rectangular form: r ˆ v = vx i + v y ˆ j Evaluate vx and vy: vx = (10 m/s) cos 60° = 5 m/s and vy = (10 m/s) sin 60° = 8.66 m/s Substitute to obtain: r ˆ v = (5 m/s)i + (8.66 m/s) ˆ j r (b) Express v in rectangular form: r ˆ A = Ax i + Ay ˆ j Evaluate Ax and Ay: Ax = (5 m) cos 225° = −3.54 m and Ay = (5 m) sin 225° = −3.54 m Substitute to obtain: r ˆ A = (− 3.54 m )i + (− 3.54 m ) ˆ j r ˆ r = (14m )i − (6m ) ˆ j (c) There is nothing to calculate as we are given the rectangular components: 50 • Picture the Problem While there are infinitely many vectors B that can be constructed such that A = B, the simplest are those which lie along the coordinate axes. r Determine the magnitude of A : Write three vectors of the same r magnitude as A : The vectors are shown to the right: A= 2 Ax2 + Ay = 32 + 4 2 = 5 r r r ˆ ˆ B1 = 5i , B2 = −5i , and B3 = 5 ˆ j Motion in One and Two Dimensions 145 *51 •• Picture the Problem While there are several walking routes the fly could take to get from the origin to point C, its displacement will be the same for all of them. One possible route is shown in the figure. Express the fly’s r displacement D during its trip from the origin to point C and find its magnitude: r r r r D = A+ B +C ˆ ˆ = (3 m )i + (3 m ) ˆ + (3 m )k j and D= (3 m )2 + (3 m )2 + (3 m )2 = 5.20 m *52 • Picture the Problem The diagram shows the locations of the transmitters relative to the ship and defines the distances separating the transmitters from each other and from the ship. We can find the distance between the ship and transmitter B using trigonometry. Relate the distance between A and B to the distance from the ship to A and the angle θ: Solve for and evaluate the distance from the ship to transmitter B: tan θ = DSB = DAB DSB DAB 100 km = = 173 km tan θ tan 30° 146 Chapter 3 Velocity and Acceleration Vectors 53 • Picture the Problem For constant speed and direction, the instantaneous velocity is identical to the average velocity. Take the origin to be the location of the stationary radar and construct a pictorial representation. Express the average velocity: Determine the position vectors: r r ∆r vav = ∆t r r1 = (− 10 km ) ˆ j and r ˆ r2 = (14.1 km )i + (− 14.1 km ) ˆ j Find the displacement vector: r r r ∆r = r2 − r1 ˆ = (14.1 km )i + (− 4.1 km ) ˆ j r Substitute for ∆r and ∆t to find the average velocity. r (14.1km )iˆ + (− 4.1km ) ˆ j vav = 1h ˆ = (14.1 km/h )i + (− 4.1 km/h ) ˆ j 54 • Picture the Problem The average velocity is the change in position divided by the elapsed time. (a) The average velocity is: Find the position vectors and the displacement vector: vav = ∆r ∆t r ˆ r0 = (2m )i + (3m ) ˆ j r ˆ r2 = (6m )i + (7m ) ˆ j and r r r ˆ ∆r = r2 − r1 = (4 m )i + (4 m ) ˆ j Find the magnitude of the displacement vector for the interval between t = 0 and t = 2 s: ∆r02 = (4m )2 + (4m )2 = 5.66m Motion in One and Two Dimensions 147 Substitute to determine vav: vav = 5.66 m = 2.83 m/s 2s and ⎛ 4m ⎞ ⎟ = 45.0° measured 4m ⎟ ⎝ ⎠ θ = tan −1 ⎜ ⎜ from the positive x axis. (b) Repeat (a), this time using the displacement between t = 0 and t = 5 s to obtain: r ˆ r5 = (13 m )i + (14 m ) ˆ , j r r r ˆ ∆r05 = r5 − r0 = (11 m )i + (11 m ) ˆ , j ∆r05 = (11 m ) + (11 m ) = 15.6 m , 15.6 m vav = = 3.11 m/s , 5s 2 2 and ⎛ 11 m ⎞ ⎟ = 45.0° measured ⎟ ⎝ 11 m ⎠ θ = tan −1 ⎜ ⎜ from the positive x axis. *55 • Picture the Problem The magnitude of the velocity vector at the end of the 2 s of acceleration will give us its speed at that instant. This is a constant-acceleration problem. Find the final velocity vector of the particle: r Find the magnitude of v : r ˆ ˆ v = vx i + v y ˆ = vx 0 i + a y tˆ j j ( ) ˆ = (4.0 m/s ) i + 3.0 m/s 2 (2.0 s ) ˆ j ˆ = (4.0 m/s ) i + (6.0 m/s ) ˆ j v= (4.0 m/s)2 + (6.0 m/s)2 = 7.21 m/s and (b) is correct. 56 • Picture the Problem Choose a coordinate system in which north coincides with the positive y direction and east with the positive x direction. Expressing the west and north r r velocity vectors is the first step in determining ∆v and a av . (a) The magnitudes of r r v W and v N are 40 m/s and 30 m/s, respectively. The change in the magnitude of the particle’s velocity during this time is: ∆v = vN − vW = − 10 m/s 148 Chapter 3 (b) The change in the direction of the velocity is from west to north. The change in direction is 90° (c) The change in velocity is: r r r ˆ ∆v = v N − v W = (30 m/s ) ˆ − (− 40 m/s ) i j ˆ = (40 m/s ) i + (30 m/s ) ˆ j Calculate the magnitude and r direction of ∆v : ∆v = (40 m/s )2 + (30 m/s)2 and θ + x axis = tan −1 (d) Find the average acceleration during this interval: 30 m/s = 36.9° 40 m/s r r (40 m/s ) iˆ + (30 m/s) ˆ j aav ≡ ∆v ∆t = 5s 2 ˆ 2 ˆ = 8 m/s i + 6 m/s j ( The magnitude of this vector is: = 50 m/s aav = ) ( ) (8 m/s ) + (6 m/s ) 2 2 2 2 = 10 m/s 2 and its direction is ⎛ 6 m/s 2 ⎞ ⎟ = 36.9° measured 2 ⎟ ⎝ 8 m/s ⎠ θ = tan −1 ⎜ ⎜ from the positive x axis. 57 • Picture the Problem The initial and final positions and velocities of the particle are given. We can find the average velocity and average acceleration using their definitions ˆ by first calculating the given displacement and velocities using unit vectors i and ˆ. j (a) The average velocity is: r r vav ≡ ∆r ∆t The displacement of the particle during this interval of time is: r ˆ ∆r = (100 m )i + (80 m ) ˆ j Substitute to find the average velocity: r (100 m ) iˆ + (80 m ) ˆ j vav = 3s = (b) The average acceleration is: (33.3 m/s ) iˆ + (26.7 m/s) ˆ j r r aav = ∆v ∆t Motion in One and Two Dimensions 149 r r v Find v1 , v 2 , and ∆v : r ˆ v1 = (28.3 m/s ) i + (28.3 m/s ) ˆ j and r ˆ v 2 = (19.3 m/s ) i + (23.0 m/s ) ˆ j r ˆ ∴ ∆v = (− 9.00 m/s ) i + (− 5.30 m/s ) ˆ j Using ∆t = 3 s, find the average acceleration: (− 3.00 m/s )iˆ + (− 1.77 m/s ) ˆj r aav = 2 2 *58 •• Picture the Problem The acceleration is constant so we can use the constant-acceleration equations in vector form to find the velocity at t = 2 s and the position vector at t = 4 s. (a) The velocity of the particle, as a function of time, is given by: Substitute to find the velocity at t = 2 s: r r r v = v0 + at r ˆ v = (2 m/s) i + (−9 m/s) ˆ j ˆ + (4 m/s 2 ) i + (3 m/s 2 ) ˆ (2s ) j [ ˆ = (10 m/s) i + (−3 m/s) ˆ j (b) Express the position vector as a function of time: r r r r r = r0 + v0t + 1 at 2 2 Substitute and simplify: r ˆ r = (4 m) i + (3 m) ˆ j ˆ + (2 m/s) i + (-9 m/s) ˆ (4 s ) j 2 ˆ + 1 (4 m/s 2 ) i + (3 m/s 2 ) ˆ (4 s ) j [ 2 [ ˆ = (44 m) i + (−9 m) ˆ j Find the magnitude and direction of r r at t = 4 s: r (4 s) = (44 m )2 + (− 9 m )2 = 44.9 m r and, because r is in the 4th quadrant, ⎛ −9m ⎞ θ = tan −1 ⎜ ⎜ 44 m ⎟ = − 11.6° ⎟ ⎝ ⎠ 59 •• Picture the Problem The velocity vector is the time-derivative of the position vector and the acceleration vector is the time-derivative of the velocity vector. r Differentiate r with respect to time: r r dr d (30t )iˆ + 40t − 5t 2 ˆ v= = j dt dt ˆ = 30i + (40 − 10t ) ˆ j [ ( )] 150 Chapter 3 r where v has units of m/s if t is in seconds. r r r dv d ˆ a= = j 30i + (40 − 10t ) ˆ dt dt = − 10 m/s 2 ˆ j [ Differentiate v with respect to time: ( ) 60 •• Picture the Problem We can use the constant-acceleration equations in vector form to solve the first part of the problem. In the second part, we can eliminate the parameter t from the constant-acceleration equations and express y as a function of x. r r r [( r ) ( ) ] [( )] ) ] r ˆ v = 6m/s 2 i + 4m/s 2 ˆ t j (a) Use v = v0 + at with v0 = 0 to r find v : r r r r r r ˆ Use r = r0 + v0t + 1 at 2 with r0 = (10 m )i to find r : 2 [ ( r ˆ r = (10m ) + 3m/s 2 t 2 i + 2m/s 2 t 2 ˆ j (b) Obtain the x and y components of the path from the vector equation in (a): Eliminate the parameter t from these equations and solve for y to obtain: x = 10 m + (3 m/s 2 )t 2 and y = (2 m/s 2 )t 2 y= 20 2 x− m 3 3 Use this equation to plot the graph shown below. Note that the path in the xy plane is a straight line. 20 18 16 14 y (m) 12 10 8 6 4 2 0 0 10 20 x (m) 30 40 Motion in One and Two Dimensions 151 61 ••• Picture the Problem The displacements of the boat are shown in the figure. We need to determine each of the displacements in order to calculate the average velocity of the boat during the 30s trip. (a) Express the average velocity of the boat: r r ∆r vav = ∆t Express its total displacement: r ∆r = r ∆rN ( ) = 1 a N ∆t N 2 r ∆rW + 2 ( ) ˆ + v ∆t − i j W W ˆ v W = vN, f = aN ∆t N = 60 m/s To calculate the displacement we first have to find the speed after the first 20 s: so Substitute to find the average velocity: r ˆ j r ∆r (600m ) − i + ˆ vav = = 30s ∆t r 2 ˆ ∆r = 1 aN (∆t N ) ˆ − (60 m/s )∆t W i j 2 ˆ = (600m ) ˆ − (600m ) i j ( = (b) The average acceleration is given by: (c) The displacement of the boat from the dock at the end of the 30-s trip was one of the intermediate results we obtained in part (a). ) (20m/s )(− iˆ + ˆ ) j r r r r ∆r vf − vi aav = = ∆t ∆t (− 60 m/s) iˆ − 0 = = 30 s (− 2 m/s )iˆ r ˆ ∆r = (600m ) ˆ + (− 600m ) i j = (600m )(− iˆ + ˆ ) j 2 152 Chapter 3 *62 ••• Picture the Problem Choose a coordinate system with the origin at Petoskey, the positive x direction to the east, and the positive y direction to the north. Let t = 0 at 9:00 a.m. and θ be the angle between Robert’s velocity vector and the easterly direction and let ″M″ and ″R″ denote Mary and Robert, respectively. You can express the positions of Mary and Robert as functions of time and then equate their north (y) and east (x) coordinates at the time they rendezvous. Express Mary’s position as a function of time: r rM = vM t ˆ = (8t ) ˆ j j r where rM is in miles if t is in hours. Note that Robert’s initial position coordinates (xi, yi) are: (xi, yi) = (−13 mi, 22.5 mi) Express Robert’s position as a function of time: r r ˆ rR = [ xi + (vR cosθ )(t − 1)]) i + [ yi + (vR sinθ )(t − 1)] ˆ j ˆ = [−13 + {6(t − 1) cos θ }] i + [22.5 + {6(t − 1) sin θ }] ˆ j where rR is in miles if t is in hours. When Mary and Robert rendezvous, their coordinates will be the same. Equating their north and east coordinates yields: Solve equation (1) for cosθ : East: –13 + 6t cosθ – 6 cosθ = 0 North: 22.5 + 6t sinθ – 6 sinθ = 8t (2) cos θ = 13 6(t − 1) (3) sin θ = Solve equation (2) for sinθ : 8t − 22.5 6(t − 1) (4) Square and add equations (3) and (4) to obtain: 2 ⎡ 8t − 22.5 ⎤ ⎡ 13 ⎤ sin θ + cos θ = 1 = ⎢ ⎥ +⎢ ⎥ ⎣ 6(t − 1) ⎦ ⎣ 6(t − 1) ⎦ 2 (1) 2 2 Simplify to obtain a quadratic equation in t: 28t 2 − 288t + 639 = 0 Solve (you could use your calculator’s ″solver″ function) this t = 3.24 h = 3 h 15 min Motion in One and Two Dimensions 153 equation for the smallest value of t (both roots are positive) to obtain: Now you can find the distance traveled due north by Mary: rM = vM t = (8 mi/h )(3.24 h ) = 25.9 mi Finally, solving equation (3) for θ and substituting 3.24 h for t yields: ⎤ ⎡ 13 ⎤ 13 −1 ⎡ ⎥ = cos ⎢ 6(3.24 − 1)⎥ = 14.7° ⎣ 6(t − 1) ⎦ ⎣ ⎦ θ = cos −1 ⎢ and so Robert should head 14.7° north of east. Remarks: Another solution that does not depend on the components of the vectors utilizes the law of cosines to find the time t at which Mary and Robert meet and then uses the law of sines to find the direction that Robert must head in order to rendezvous with Mary. Relative Velocity 63 •• Picture the Problem Choose a coordinate system in which north is the positive y direction and east is the positive x direction. Let θ be the angle between north and the direction of the plane’s heading. The velocity of the plane relative r to the ground, v PG , is the sum of the velocity of the plane relative to the air, r v PA , and the velocity of the air relative to r the ground, vAG . i.e., r r r v PG = v PA + v AG The pilot must head in such a direction that r the east-west component of v PG is zero in order to make the plane fly due north. (a) From the diagram one can see that: Solve for and evaluate θ : vAG cos 45° = vPA sinθ ⎛ 56.6 km/h ⎞ ⎟ ⎟ ⎝ 250 km/h ⎠ θ = sin −1 ⎜ ⎜ = 13.1° west of north (b) Because the plane is headed due north, add the north components of r v PG = (250 km/h) cos 13.1° + (80 km/h) sin 45° 154 Chapter 3 r r v PA and v AG to determine the = 300 km/h plane’s ground speed: 64 •• r Picture the Problem Let vSB represent the velocity of the swimmer relative to the r bank; vSW the velocity of the swimmer r relative to the water; and v WB the velocity of the water relative to the shore; i.e., r r r vSB = vSW + v WB The current of the river causes the swimmer to drift downstream. (a) The triangles shown in the figure are similar right triangles. Set up a proportion between their sides and solve for the speed of the water relative to the bank: (b) Use the Pythagorean Theorem to solve for the swimmer’s speed relative to the shore: vWB 40 m = vSW 80 m and vWB = 1 2 (1.6 m/s ) = 0.800 m/s 2 2 vSB = vSW + v WS = (1.6 m/s)2 + (0.8 m/s)2 = 1.79 m/s (c) The swimmer should head in a direction such that the upstream component of her velocity is equal to the speed of the water relative to the shore: Use a trigonometric function to evaluate θ: ⎛ 0.8 m/s ⎞ ⎟ = 30.0° ⎟ ⎝ 1.6 m/s ⎠ θ = sin −1 ⎜ ⎜ Motion in One and Two Dimensions 155 *65 •• Picture the Problem Let the velocity of the plane relative to the ground be r represented by v PG ; the velocity of the r plane relative to the air by v PA , and the velocity of the air relative to the ground by r v AG . Then r r r v PG = v PA + v AG (1) Choose a coordinate system with the origin at point A, the positive x direction to the east, and the positive y direction to the north. θ is the angle between north and the direction of the plane’s heading. The pilot must head so that the east-west component r of vPG is zero in order to make the plane fly due north. Use the diagram to express the condition relating the eastward r component of v AG and the (50 km/h) cos 45° = (240 km/h) sinθ r westward component of v PA . This must be satisfied if the plane is to stay on its northerly course. [Note: this is equivalent to equating the xcomponents of equation (1).] ⎡ (50 km/h )cos45° ⎤ ⎥ = 8.47° ⎦ ⎣ 240 km/h Now solve for θ to obtain: θ = sin −1 ⎢ r Add the north components of v PA r and v AG to find the velocity of the plane relative to the ground: Finally, find the time of flight: vPG + vAGsin45° = vPAcos8.47° and vPG = (240 km/h)cos 8.47° − (50 km/h)sin 45° = 202 km/h t flight = = distance travelled vPG 520 km = 2.57 h 202 km/h 156 Chapter 3 66 •• r Picture the Problem Let v BS be the velocity of the boat relative to the shore; r v BW be the velocity of the boat relative to r the water; and v WS represent the velocity of the water relative to the shore. Independently of whether the boat is going upstream or downstream: r r r v BS = v BW + v WS Going upstream, the speed of the boat relative to the shore is reduced by the speed of the water relative to the shore. Going downstream, the speed of the boat relative to the shore is increased by the same amount. For the upstream leg of the trip: vBS = vBW − vWS For the downstream leg of the trip: vBS = vBW + vWS Express the total time for the trip in terms of the times for its upstream and downstream legs: ttotal = tupstream + tdownstream Multiply both sides of the equation by (v BW − v WS )(v BW + v WS ) (the product of the denominators) and rearrange the terms to obtain: Solve the quadratic equation for vBW. (Only the positive root is physically meaningful.) 67 •• r Picture the Problem Let v pg be the velocity of the plane relative to the ground; r v ag be the velocity of the air relative to the r ground; and v pa the velocity of the plane r r relative to the air. Then, v pg = v pa + r v ag . The wind will affect the flight times differently along these two paths. = L L + vBW − vWS vBW + vWS 2 vBW − 2L 2 vBW − vWS = 0 t total vBW = 5.18 km/h Motion in One and Two Dimensions 157 The velocity of the plane, relative to the ground, on its eastbound leg is equal to its velocity on its westbound leg. Using the diagram, find the velocity of the plane relative to the ground for both directions: 2 2 vpg = vpa − vag Express the time for the east-west roundtrip in terms of the distances and velocities for the two legs: troundtrip,EW = teastbound + t westbound = (15 m/s)2 − (5 m/s)2 = radius of the circle vpg,eastbound + = = 14.1m/s radius of the circle vpg,westbound 2 × 103 m = 141 s 14.1m/s Use the distances and velocities for the two legs to express and evaluate the time for the north-south roundtrip: troundtrip,NS = t northbound + tsouthbound = radius of the circle radius of the circle + vpg,northbound vpg,southbound 103 m 103 m = + = 150 s (15 m/s) − (5 m/s) (15 m/s) + (5 m/s) Because troundtrip,EW < troundtrip,NS , you should fly your plane across the wind. 68 • Picture the Problem This is a relative velocity problem. The given quantities are the direction of the velocity of the plane relative to the ground and the velocity (magnitude and direction) of the air relative to the ground. Asked for is the direction of the velocity of the air relative to the r r r ground. Using v PG = v PA + v AG , draw a vector addition diagram and solve for the unknown quantity. Calculate the heading the pilot must take: Because this is also the angle of the plane's heading clockwise from north, it is also its azimuth or the required true heading: θ = sin −1 30 kts = 11.5° 150 kts Az = (011.5°) 158 Chapter 3 *69 •• Picture the Problem The position of B relative to A is the vector from A to B; i.e., r r r rAB = rB − rA The velocity of B relative to A is r r v AB = drAB dt and the acceleration of B relative to A is r r a AB = dv AB dt Choose a coordinate system with the origin at the intersection, the positive x direction to the east, and the positive y direction to the north. r r r (a) Find rB , rA , and rAB : [ r rB = 40m − 1 (2m/s 2 )t 2 ˆ j 2 r ˆ rA = [(20m/s )t ]i and r r r rAB = rB − rA ˆ = [(− 20m/s ) t ] i [ ( ) ] + 40m − 1 2m/s 2 t 2 ˆ j 2 r Evaluate rAB at t = 6 s: r r (b) Find v AB = drAB dt : r ˆ rAB (6 s) = (120 m) i + (4 m) ˆ j r r r drAB d {(− 20 m/s)t}i v AB = = dt dt j + {40 m − 1 (2 m/s 2 )t 2 }ˆ 2 ˆ = (−20 m/s) i + (−2 m/s 2 ) t ˆ j [ r Evaluate v AB at t = 6 s: r r (c) Find a AB = dv AB dt : r v AB (6 s ) = (− 20 m/s)iˆ − (12 m/s ) ˆ j [ r d ˆ (−20 m/s) i + (−2 m/s 2 ) t ˆ aAB = j dt = − 2 m/s 2 ˆ j r Note that a AB is independent of time. ( ) *70 ••• Picture the Problem Let h and h′ represent the heights from which the ball is dropped and to which it rebounds, respectively. Let v and v′ represent the speeds with which the ball strikes the racket and rebounds from it. We can use a constant-acceleration equation to relate the pre- and post-collision speeds of the ball to its drop and rebound heights. Motion in One and Two Dimensions 159 (a) Using a constant-acceleration equation, relate the impact speed of the ball to the distance it has fallen: Relate the rebound speed of the ball to the height to which it rebounds: 2 v 2 = v0 + 2 gh or, because v0 = 0, v = 2 gh v 2 = v' 2 − 2 gh' or because v = 0, v' = 2 gh' Divide the second of these equations by the first to obtain: v' = v 2 gh' Substitute for h′ and evaluate the ratio of the speeds: v' = v 0.64h = 0.8 ⇒ v' = 0.8v h 2 gh = h' h (b) Call the speed of the racket V. In a reference frame where the racket is unmoving, the ball initially has speed V, moving toward the racket. After it "bounces" from the racket, it will have speed 0.8 V, moving away from the racket. In the reference frame where the racket is moving and the ball initially unmoving, we need to add the speed of the racket to the speed of the ball in the racket's rest frame. Therefore, the ball's speed is: (c) v' = V + 0.8V = 1.8V = 45 m/s ≈ 100 mi/h This speed is close to that of a tennis pro’s serve. Note that this result tells us that the ball is moving significantly faster than the racket. From the result in part (b), the ball can never move more than twice as fast as the racket. Circular Motion and Centripetal Acceleration 71 • Picture the Problem We can use the definition of centripetal acceleration to express ac in terms of the speed of the tip of the minute hand. We can find the tangential speed of the tip of the minute hand by using the distance it travels each revolution and the time it takes to complete each revolution. Express the acceleration of the tip of the minute hand of the clock as a function of the length of the hand and the speed of its tip: Use the distance the minute hand travels every hour to express its speed: ac = v= v2 R 2πR T 160 Chapter 3 Substitute to obtain: Substitute numerical values and evaluate ac: Express the ratio of ac to g: 4π 2 R ac = T2 ac = 4π 2 (0.5 m ) = 1.52 × 10− 6 m/s 2 2 (3600 s ) ac 1.52 × 10−6 m/s 2 = = 1.55 × 10− 7 2 9.81m/s g 72 • Picture the Problem The diagram shows the centripetal and tangential accelerations experienced by the test tube. The tangential acceleration will be zero when the centrifuge reaches its maximum speed. The centripetal acceleration increases as the tangential speed of the centrifuge increases. We can use the definition of centripetal acceleration to express ac in terms of the speed of the test tube. We can find the tangential speed of the test tube by using the distance it travels each revolution and the time it takes to complete each revolution. The tangential acceleration can be found from the change in the tangential speed as the centrifuge is spinning up. (a) Express the acceleration of the centrifuge arm as a function of the length of its arm and the speed of the test tube: Use the distance the test tube travels every revolution to express its speed: Substitute to obtain: Substitute numerical values and evaluate ac: ac = v= v2 R 2πR T ac = ac = 4π 2 R T2 4π 2 (0.15 m ) ⎛ 1 min 60 s ⎞ ⎜ ⎜ 15000 rev × min ⎟ ⎟ ⎝ ⎠ = 3.70 × 105 m/s 2 2 Motion in One and Two Dimensions 161 (b) Express the tangential acceleration in terms of the difference between the final and initial tangential speeds: 2πR −0 2πR vf − vi T = = at = ∆t ∆t T∆t Substitute numerical values and evaluate aT: at = 2π (0.15 m ) ⎛ 1 min 60 s ⎞ ⎜ ⎜ 15000 rev × min ⎟(75 s ) ⎟ ⎝ ⎠ = 3.14 m/s 2 73 • Picture the Problem The diagram includes a pictorial representation of the earth in its orbit about the sun and a force diagram showing the force on an object at the equator that is due to the earth’s rotation, r FR , and the force on the object due to the orbital motion of the earth about the sun, r Fo . Because these are centripetal forces, we can calculate the accelerations they require from the speeds and radii associated with the two circular motions. Express the radial acceleration due to the rotation of the earth: Express the speed of the object on the equator in terms of the radius of the earth R and the period of the earth’s rotation TR: 2 vR R 2πR vR = TR aR = Substitute for vR in the expression for aR to obtain: aR = Substitute numerical values and evaluate aR: aR = 4π 2 R TR2 ( 4π 2 6370 × 103 m ⎡ ⎛ 3600 s ⎞⎤ ⎢(24 h )⎜ ⎜ 1 h ⎟⎥ ⎟ ⎝ ⎠⎦ ⎣ ) 2 = 3.37 × 10−2 m/s 2 = 3.44 × 10−3 g Note that this effect gives rise to the wellknown latitude correction for g. 162 Chapter 3 Express the radial acceleration due to the orbital motion of the earth: 2 vo ao = r vo = 2π r To Substitute for vo in the expression for ao to obtain: ao = 4π 2 r To2 Substitute numerical values and evaluate ao: ao = Express the speed of the object on the equator in terms of the earth-sun distance r and the period of the earth’s motion about the sun To: ( 4π 2 1.5 × 1011 m ) ⎡ ⎛ 24 h ⎞ ⎛ 3600 s ⎞⎤ ⎢(365 d )⎜ ⎟ ⎟⎜ ⎜ 1d ⎟ ⎜ 1 h ⎟ ⎥ ⎠⎦ ⎠⎝ ⎝ ⎣ 2 = 5.95 × 10−3 m/s 2 = 6.07 × 10−4 g 74 •• Picture the Problem We can relate the acceleration of the moon toward the earth to its orbital speed and distance from the earth. Its orbital speed can be expressed in terms of its distance from the earth and its orbital period. From tables of astronomical data, we find that the sidereal period of the moon is 27.3 d and that its mean distance from the earth is 3.84×108 m. Express the centripetal acceleration of the moon: ac = Express the orbital speed of the moon: v= Substitute to obtain: Substitute numerical values and evaluate ac: v2 r 2πr T 4π 2 r ac = 2 T ac = ( 4π 2 3.84 × 108 m ) 24 h 3600 s ⎞ ⎛ × ⎜ 27.3 d × ⎟ d h ⎠ ⎝ = 2.72 × 10 −3 m/s 2 = 2.78 × 10 −4 g 2 Motion in One and Two Dimensions 163 Remarks: Note that ac radius of earth = (ac is just the acceleration g distance from earth to moon due to the earth’s gravity evaluated at the moon’s position). This is Newton’s famous ″falling apple″ observation. 75 • Picture the Problem We can find the number of revolutions the ball makes in a given period of time from its speed and the radius of the circle along which it moves. Because the ball’s centripetal acceleration is related to its speed, we can use this relationship to express its speed. Express the number of revolutions per minute made by the ball in terms of the circumference c of the circle and the distance x the ball travels in time t: n= x c (1) Relate the centripetal acceleration of the ball to its speed and the radius of its circular path: v2 ac = g = R Solve for the speed of the ball: v = Rg Express the distance x traveled in time t at speed v: x = vt Substitute to obtain: x = Rg t The distance traveled per revolution is the circumference c of the circle: c = 2π R Substitute in equation (1) to obtain: Substitute numerical values and evaluate n: n= Rg t 1 = 2π R 2π n= 1 2π g t R 9.81m/s 2 (60 s ) = 33.4 min −1 0.8 m Remarks: The ball will oscillate at the end of this string as a simple pendulum with a period equal to 1/n. Projectile Motion and Projectile Range 76 • Picture the Problem Neglecting air resistance, the accelerations of the ball are constant and the horizontal and vertical motions of the ball are independent of each other. We can use the horizontal motion to determine the time-of-flight and then use this information to determine the distance the ball drops. Choose a coordinate system in which the origin is at the point of release of the ball, downward is the positive y direction, and the horizontal 164 Chapter 3 direction is the positive x direction. Express the vertical displacement of the ball: ∆y = v0 y ∆t + 1 a y (∆t ) 2 2 or, because v0y = 0 and ay = g, ∆y = 1 g (∆t ) 2 2 Find the time of flight from vx = ∆x/∆t: ∆t = = Substitute to find the vertical displacement in 0.473 s: ∆y = ∆x vx (18.4 m )(3600 s/h ) = 0.473 s (140 km/h )(1000 m/km) 1 2 (9.81m/s )(0.473 s ) 2 2 = 1.10 m 77 • Picture the Problem In the absence of air resistance, the maximum height achieved by a projectile depends on the vertical component of its initial velocity. The vertical component of the projectile’s initial velocity is: v0y = v0 sinθ0 Use the constant-acceleration equation: 2 v 2 = v0 y + 2a y∆y y Set vy = 0, a = −g, and ∆y = h to obtain: h= (v0 sin θ 0 )2 2g *78 •• Picture the Problem Choose the coordinate system shown to the right. Because, in the absence of air resistance, the horizontal and vertical speeds are independent of each other, we can use constant-acceleration equations to relate the impact speed of the projectile to its components. The horizontal and vertical velocity components are: v0x = vx= v0cosθ and v0y = v0sinθ Using a constant-acceleration equation, relate the vertical 2 2 v y = v0 y + 2a y ∆y or, because ay = −g and ∆y = −h, Motion in One and Two Dimensions 165 component of the velocity to the vertical displacement of the projectile: 2 v y = (v0 sin θ ) + 2 gh Express the relationship between the magnitude of a velocity vector and its components, substitute for the components, and simplify to obtain: 2 2 2 v 2 = vx + v y = (v0 cos θ ) + v y Substitute for v: (1.2v0 )2 = v02 + 2 gh Set v = 1.2 v0, h = 40 m and solve for v0: v0 = 42.2 m/s 2 2 ( ) 2 = v0 sin 2 θ + cos 2 θ + 2 gh = v + 2 gh 2 0 Remarks: Note that v is independent of θ. This will be more obvious once conservation of energy has been studied. 79 •• Picture the Problem Example 3-12 shows that the dart will hit the monkey unless the dart hits the ground before reaching the monkey’s line of fall. What initial speed does the dart need in order to just reach the monkey’s line of fall? First, we will calculate the fall time of the monkey, and then we will calculate the horizontal component of the dart’s velocity. Using a constant-acceleration equation, relate the monkey’s fall distance to the fall time: h = 1 gt 2 2 Solve for the time for the monkey to fall to the ground: t= 2h g Substitute numerical values and evaluate t: t= 2(11.2 m ) = 1.51s 9.81 m/s 2 Let θ be the angle the barrel of the dart gun makes with the horizontal. Then: θ = tan −1 ⎜ ⎜ Use the fact that the horizontal velocity is constant to determine v0: v0 = ⎛ 10 m ⎞ ⎟ = 11.3° ⎟ ⎝ 50 m ⎠ (50 m 1.51s ) = 33.8 m/s vx = cos θ cos11.3° 166 Chapter 3 80 •• Picture the Problem Choose the coordinate system shown in the figure to the right. In the absence of air resistance, the projectile experiences constant acceleration in both the x and y directions. We can use the constant-acceleration equations to express the x and y coordinates of the projectile along its trajectory as functions of time. The elimination of the parameter t will yield an expression for y as a function of x that we can evaluate at (R, 0) and (R/2, h). Solving these equations simultaneously will yield an expression for θ. Express the position coordinates of the projectile along its flight path in terms of the parameter t: x = (v0 cos θ )t and y = (v0 sin θ )t − 1 gt 2 2 Eliminate the parameter t to obtain: y = (tan θ )x − Evaluate equation (1) at (R, 0) to obtain: R= g x2 2 2v cos θ (1) 2 0 2 2v0 sin θ cos θ g Evaluate equation (1) at (R/2, h) to obtain: (v0 sin θ )2 h= Equate R and h and solve the resulting equation for θ : θ = tan −1 (4) = 76.0° 2g Remarks: Note that this result is independent of v0. 81 •• Picture the Problem In the absence of air resistance, the motion of the ball is uniformly accelerated and its horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure to the right and use constant-acceleration equations to relate the x and y components of the ball’s initial velocity. Use the components of v0 to express θ in terms of v0x and v0y: θ = tan −1 v0 y v0 x (1) Motion in One and Two Dimensions 167 Use the Pythagorean relationship between the velocity and its components to express v0: 2 2 v0 = v0 x + v0 y (2) Using a constant-acceleration equation, express the vertical speed of the projectile as a function of its initial upward speed and time into the flight: vy= v0y+ ay t Because vy = 0 halfway through the flight (at maximum elevation): v0y = (9.81 m/s2)(1.22 s) = 12.0 m/s Determine v0x: Substitute in equation (2) and evaluate v0: v0x = ∆x 40 m = = 16.4 m/s ∆t 2.44 s v0 = (16.4 m/s)2 + (12.0 m/s)2 = 20.3 m/s Substitute in equation (1) and evaluate θ : ⎛ 12.0 m/s ⎞ ⎟ = 36.2° ⎟ ⎝ 16.4 m/s ⎠ θ = tan −1 ⎜ ⎜ *82 •• Picture the Problem In the absence of friction, the acceleration of the ball is constant and we can use the constantacceleration equations to describe its motion. The figure shows the launch conditions and an appropriate coordinate system. The speeds v, vx, and vy are related through the Pythagorean Theorem. The squares of the vertical and horizontal components of the object’s velocity are: 2 2 v y = v0 sin 2 θ − 2 gh The relationship between these variables is: 2 2 v 2 = vx + v y Substitute and simplify to obtain: 2 v 2 = v0 − 2 gh and 2 2 vx = v0 cos 2 θ Note that v is independent of θ ... as was to be shown. 168 Chapter 3 83 •• Picture the Problem In the absence of air resistance, the projectile experiences constant acceleration during its flight and we can use constant-acceleration equations to relate the speeds at half the maximum height and at the maximum height to the launch angle θ of the projectile. The angle the initial velocity makes with the horizontal is related to the initial velocity components. Write the equation 2 2 v y = v0 y + 2a∆y, for ∆y = h and tan θ = v0 y v0 x 2 ∆y = h ⇒ 0 = v0 y − 2 gh (1) vy = 0: Write the equation 2 2 v y = v0 y + 2a∆y, for ∆y = h/2: ∆y = We are given vy = (3/4)v0. Square both sides and express this using the components of the velocity. The x component of the velocity remains constant. ⎛3⎞ 2 2 v + v = ⎜ ⎟ v0 x + v0 y ⎝4⎠ where we have used v x = v 0x . h h 2 2 ⇒ v y = v0 y − 2 g 2 2 2 2 0x 2 y ( (2) ) (3) (Equations 1, 2, and 3 constitute three equations and four unknowns v0x, v0y, vy, and h. To solve for any of these unknowns, we first need a fourth equation. However, to solve for the ratio (v0y/v0x) of two of the unknowns, the three equations are sufficient. That is 2 because dividing both sides of each equation by v 0x gives three equations and three 2 unknowns vy/v0x, v0y/v0x, and h/ v 0 x . Solve equation 2 for gh and substitute in equation 1: 2 Substitute for v y in equation 3: 2 0y v = 2(v − v )⇒ v = 2 0y 2 y 2 h 2 ( 2 v0 y 2 1 2 ⎛3⎞ 2 2 v + v0 y = ⎜ ⎟ v0 x + v0 y 2 4⎠ ⎝ 2 0x ) Motion in One and Two Dimensions 169 2 Divide both sides by v 0 x and solve for v0y/v0x to obtain: 2 2 1 v0 y 9 ⎛ v0 y ⎞ = ⎜1 + 2 ⎟ 2 2 v0 x 16 ⎜ v0 x ⎟ ⎠ ⎝ and 1+ v0 y v0 x Using tan θ = v0y/v0x, solve for θ : = 7 ( ) ⎛ v0 y ⎞ ⎟ = tan −1 7 = 69.3° ⎟ ⎝ v0 x ⎠ θ = tan −1 ⎜ ⎜ 84 • Picture the Problem The horizontal speed of the crate, in the absence of air resistance, is constant and equal to the speed of the cargo plane. Choose a coordinate system in which the direction the plane is moving is the positive x direction and downward is the positive y direction and apply the constant-acceleration equations to describe the crate’s displacements at any time during its flight. (a) Using a constant-acceleration equation, relate the vertical displacement of the crate ∆y to the time of fall ∆t: Solve for ∆t: ∆y = v0 y ∆t + 1 g (∆t ) 2 2 or, because v0y = 0, ∆y = 1 g (∆t ) 2 2 ∆t = 2∆y g Substitute numerical values and evaluate ∆t: ∆t = 2(12 × 103 m ) = 49.5 s 9.81m/s 2 (b) The horizontal distance traveled in 49.5 s is: R = ∆x = v0 x ∆t ⎛ 1h ⎞ = (900 km/h )⎜ ⎜ 3600 s ⎟(49.5 s ) ⎟ ⎝ ⎠ = 12.4 km (c) Because the velocity of the plane is constant, it will be directly over the crate when it hits the ground; i.e., the distance to the aircraft will be the elevation of the aircraft. ∆y = 12.0 km 170 Chapter 3 *85 •• Picture the Problem In the absence of air resistance, the accelerations of both Wiley Coyote and the Roadrunner are constant and we can use constant-acceleration equations to express their coordinates at any time during their leaps across the gorge. By eliminating the parameter t between these equations, we can obtain an expression that relates their y coordinates to their x coordinates and that we can solve for their launch angles. (a) Using constant-acceleration equations, express the x coordinate of the Roadrunner while it is in flight across the gorge: Using constant-acceleration equations, express the y coordinate of the Roadrunner while it is in flight across the gorge: Eliminate the parameter t to obtain: Letting R represent the Roadrunner’s range and using the trigonometric identity sin2θ = 2sinθ cosθ, solve for and evaluate its launch speed: x = x0 + v0 x t + 1 a xt 2 2 or, because x0 = 0, ax = 0 and v0x = v0 cosθ0, x = (v0 cos θ 0 ) t y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, ay =−g and v0y = v0 sinθ0, y = (v0 sin θ 0 ) t − 1 gt 2 2 y = (tan θ 0 )x − g x2 2v cos 2 θ 0 Rg = v0 = sin 2θ 0 (1) 2 0 (16.5 m )(9.81m/s2 ) sin 30° = 18.0 m/s (b) Letting R represent Wiley’s range, solve equation (1) for his launch angle: θ 0 = sin −1 ⎜ 2 ⎟ ⎜v ⎟ 2 ⎝ 0 ⎠ Substitute numerical values and evaluate θ0: θ 0 = sin −1 ⎢ 1 ⎛ Rg ⎞ 1 2 ⎡ (14.5 m ) 9.81 m/s 2 ⎤ (18.0 m/s )2 ⎥ ⎣ ⎦ = 13.0° ( ) Motion in One and Two Dimensions 171 86 • Picture the Problem Because, in the absence of air resistance, the vertical and horizontal accelerations of the cannonball are constant, we can use constantacceleration equations to express the ball’s position and velocity as functions of time and acceleration. The maximum height of the ball and its time-of-flight are related to the components of its launch velocity. (a) Using a constant-acceleration equation, relate h to the initial and final speeds of the cannonball: or, because v = 0 and ay = −g, Find the vertical component of the firing speed: v0y = v0sinθ = (300 m/s)sin 45° = 212 m/s Solve for and evaluate h: (b) The total flight time is: 2 v 2 = v0 y + 2a y ∆y 2 0 = v0 y − 2 g∆y h= 2 v0 y 2g = (212 m/s)2 2(9.81m/s 2 ) = 2.29 km ∆t = tup + tdn = 2tup =2 v0 y g = 2(212 m/s ) = 43.2 s 9.81 m/s 2 (c) Express the x coordinate of the ball as a function of time: x = v0 x ∆t = (v0 cosθ )∆t Evaluate x (= R) when ∆t = 43.2 s: x = [(300 m/s )cos45°](43.2 s ) = 9.16 km 87 •• Picture the Problem Choose a coordinate system in which the origin is at the base of the tower and the x- and y-axes are as shown in the figure to the right. In the absence of air resistance, the horizontal speed of the stone will remain constant during its fall and a constant-acceleration equation can be used to determine the time of fall. The final velocity of the stone will be the vector sum of its x and y components. 172 Chapter 3 ∆y = v0 y ∆t + 1 a y (∆t ) 2 (a) Using a constant-acceleration equation, express the vertical displacement of the stone (the height of the tower) as a function of the fall time: or, because v0y = 0 and a = −g, Solve for and evaluate the time of fall: ∆t = − Use the definition of average velocity to find the velocity with which the stone was thrown from the tower: (b) Find the y component of the stone’s velocity after 2.21 s: 2 ∆y = − 1 g (∆t ) 2 2 2∆y 2(− 24 m ) = − = 2.21s g 9.81m/s 2 vx = v0 x ≡ ∆x 18 m = = 8.14 m / s ∆t 2.21s v y = v0 y − gt = 0 − (9.81 m/s2)(2.21 s) = −21.7 m/s Express v in terms of its components: 2 2 v = vx + v y Substitute numerical values and evaluate v: v= (8.14 m/s)2 + (− 21.7 m/s)2 = 23.2 m/s 88 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and its horizontal and vertical motions are independent of each other. We can use constant-acceleration equations to express the horizontal and vertical displacements of the projectile in terms of its time-of-flight. Using a constant-acceleration equation, express the horizontal displacement of the projectile as a function of time: ∆x = v0 x ∆t + 1 a x (∆t ) 2 or, because v0x = v0cosθ and ax = 0, ∆x = (v0 cosθ )∆t Using a constant-acceleration equation, express the vertical displacement of the projectile as a function of time: ∆y = v0 y ∆t + 1 a y (∆t ) 2 or, because v0y = v0sinθ and ay = −g, 2 ∆y = (v0 cos θ )∆t − 1 g (∆t ) 2 Substitute numerical values to obtain the quadratic equation: − 200 m = (60m/s )(sin 60°)∆t Solve for ∆t: ∆t = 13.6 s 2 2 − 1 (9.81 m/s 2 )(∆t ) 2 2 Motion in One and Two Dimensions 173 Substitute for ∆t and evaluate the horizontal distance traveled by the projectile: ∆x = (60 m/s)(cos60°)(13.6 s) = 408 m 89 •• Picture the Problem In the absence of air resistance, the acceleration of the cannonball is constant and its horizontal and vertical motions are independent of each other. Choose the origin of the coordinate system to be at the base of the cliff and the axes directed as shown and use constant- acceleration equations to describe both the horizontal and vertical displacements of the cannonball. Express the direction of the velocity vector when the projectile strikes the ground: Express the vertical displacement using a constant-acceleration equation: Set ∆x = −∆y (R = −h) to obtain: Solve for vx: Find the y component of the projectile as it hits the ground: Substitute and evaluate θ : 90 • Picture the Problem In the absence of air resistance, the vertical and horizontal motions of the projectile experience constant accelerations and are independent of each other. Use a coordinate system in which up is the positive y direction and horizontal is the positive x direction and use constant-acceleration equations to describe the horizontal and vertical displacements of the projectile as functions of the time into the flight. ⎛ vy ⎞ ⎟ ⎟ ⎝ vx ⎠ θ = tan −1 ⎜ ⎜ ∆y = v0 y ∆t + 1 a y (∆t ) 2 2 or, because v0y = 0 and ay = −g, ∆y = − 1 g (∆t ) 2 2 ∆x = vx ∆t = 1 g (∆t ) 2 2 vx = ∆x 1 = g∆t ∆t 2 v y = v0 y + a∆t = − g∆t = −2vx ⎛ vy ⎞ ⎟ = tan −1 (− 2 ) = − 63.4° ⎟ ⎝ vx ⎠ θ = tan −1 ⎜ ⎜ 174 Chapter 3 (a) Use a constant-acceleration equation to express the horizontal displacement of the projectile as a function of time: ∆x = v0 x ∆t Evaluate this expression when ∆t = 6 s: ∆x = (300 m/s )(cos60°)(6 s ) = 900 m (b) Use a constant-acceleration equation to express the vertical displacement of the projectile as a function of time: ∆y = (v0 sin θ )∆t − 1 g (∆t ) 2 = (v0 cosθ )∆t 2 Evaluate this expression when ∆t = 6 s: ( ) ∆y = (300 m/s )(sin60°)(6 s ) − 1 9.81 m/s 2 (6 s ) = 1.38 km 2 2 91 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and the horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure with the origin at the base of the cliff and the axes oriented as shown and use constant-acceleration equations to find the range of the cannonball. Using a constant-acceleration equation, express the horizontal displacement of the cannonball as a function of time: ∆x = v0 x ∆t + 1 a x (∆t ) 2 or, because v0x = v0cosθ and ax = 0, ∆x = (v0 cosθ )∆t Using a constant-acceleration equation, express the vertical displacement of the cannonball as a function of time: ∆y = v0 y ∆t + 1 a y (∆t ) 2 2 2 or, because y = −40 m, a = −g, and v0y = v0sinθ, − 40 m = (42.2 m/s )(sin 30°)∆t ( ) − 1 9.81 m/s 2 (∆t ) 2 2 Solve the quadratic equation for ∆t: ∆t = 5.73 s Calculate the range: R = ∆x = (42.2 m/s )(cos30°)(5.73 s ) = 209 m Motion in One and Two Dimensions 175 *92 •• Picture the Problem Choose a coordinate system in which the origin is at ground level. Let the positive x direction be to the right and the positive y direction be upward. We can apply constantacceleration equations to obtain parametric equations in time that relate the range to the initial horizontal speed and the height h to the initial upward speed. Eliminating the parameter will leave us with a quadratic equation in R, the solution to which will give us the range of the arrow. In (b), we’ll find the launch speed and angle as viewed by an observer who is at rest on the ground and then use these results to find the arrow’s range when the horse is moving at 12 m/s. (a) Use constant-acceleration equations to express the horizontal and vertical coordinates of the arrow’s motion: Solve the x-component equation for time: Eliminate time from the y-component equation: Solve for the range to obtain: R = ∆x = x − x0 = v0 xt and y = h + v0 y t + 1 (− g )t 2 2 where v0 x = v0 cosθ and v0 y = v0 sin θ t= R R = v0 x v0 cosθ 2 R 1 ⎛ R ⎞ ⎟ − g⎜ y = h + v0 y v0 x 2 ⎜ v0 x ⎟ ⎝ ⎠ and, at (R, 0), g R2 0 = h + (tan θ )R − 2 2 2v0 cos θ R= 2 ⎛ 2 gh v0 sin 2θ ⎜1 + 1 + 2 2 ⎜ 2g v0 sin θ ⎝ Substitute numerical values and evaluate R: R= ⎛ 2(9.81 m/s 2 )(2.25 m ) ⎞ ⎟ = 81.6 m sin 20°⎜1 + 1 + 2 2 ⎜ 2(9.81 m/s 2 ) (45 m/s) (sin 10°) ⎟ ⎝ ⎠ (45 m/s)2 ⎞ ⎟ ⎟ ⎠ 176 Chapter 3 (b) Express the speed of the arrow in the horizontal direction: v x = varrow + varcher Express the vertical speed of the arrow: v y = (45 m/s )sin10° = 7.81m/s = (45 m/s )cos10° + 12 m/s = 56.3 m/s Express the angle of elevation from the perspective of someone on the ground: θ = tan −1 ⎜ ⎜ Express the arrow’s speed relative to the ground: 2 2 v0 = v x + v y ⎛ vy ⎞ ⎛ 7.81 m/s ⎞ ⎟ = tan −1 ⎜ ⎜ 56.3 m/s ⎟ = 7.90° ⎟ ⎟ ⎝ ⎠ ⎝ vx ⎠ = (56.3 m/s)2 + (7.81 m/s)2 = 56.8 m/s Substitute numerical values and evaluate R: 2 ⎛ ⎞ ⎜1 + 1 + 2(9.81 m/s )(2.25 m ) ⎟ = 104 m R= sin15.8° ⎜ 2(9.81 m/s 2 ) (56.8 m/s)2 (sin 2 7.9°) ⎟ ⎝ ⎠ (56.8 m/s)2 Remarks: An alternative solution for part (b) is to solve for the range in the reference frame of the archer and then add to it the distance the frame travels, relative to the earth, during the time of flight. 93 • Picture the Problem In the absence of air resistance, the horizontal and vertical motions are independent of each other. Choose a coordinate system oriented as shown in the figure to the right and apply constant-acceleration equations to find the time-of-flight and the range of the spudplug. ∆y = v0 y ∆t + 1 a y (∆t ) 2 (a) Using a constant-acceleration equation, express the vertical displacement of the plug: or, because v0y = 0 and ay = −g, Solve for and evaluate the flight time ∆t: ∆t = − 2 ∆y = − 1 g (∆t ) 2 2 2∆y 2(− 1.00 m ) = − g 9.81m/s 2 = 0.452 s Motion in One and Two Dimensions 177 (b) Using a constant-acceleration equation, express the horizontal displacement of the plug: Substitute numerical values and evaluate R: ∆x = v0 x ∆t + 1 a x (∆t ) 2 2 or, because ax = 0 and v0x = v0, ∆x = v0 ∆t ∆x = R = (50 m/s )(0.452 s ) = 22.6 m 94 •• Picture the Problem An extreme value (i.e., a maximum or a minimum) of a function is determined by setting the appropriate derivative equal to zero. Whether the extremum is a maximum or a minimum can be determined by evaluating the second derivative at the point determined by the first derivative. Evaluate dR/dθ0: 2 2 dR v0 d [sin (2θ 0 )] = 2v0 cos(2θ 0 ) = dθ 0 g dθ 0 g Set dR/dθ0= 0 for extrema and solve for θ0: 2 2v0 cos(2θ 0 ) = 0 g and θ 0 = 1 cos −1 0 = 45° 2 Determine whether 45° is a maximum or a minimum: d 2R 2 dθ 0 [ 2 = − 4(v0 g )sin 2θ 0 θ0 =45° <0 ∴ R is a maximum at θ0 = 45° 95 • Picture the Problem We can use constantacceleration equations to express the x and y coordinates of a bullet in flight on the moon as a function of t. Eliminating this parameter will yield an expression for y as a function of x that we can use to find the range of the bullet. The necessity that the centripetal acceleration of an object in orbit at the surface of a body equal the acceleration due to gravity at the surface will allow us to determine the required muzzle velocity for orbital motion. (a) Using a constant-acceleration equation, express the x coordinate of a bullet in flight on the moon: x = x0 + v0 xt + 1 a xt 2 2 or, because x0 = 0, ax = 0 and v0x = v0cosθ0, x = (v0 cos θ 0 ) t θ0 =45° 178 Chapter 3 Using a constant-acceleration equation, express the y coordinate of a bullet in flight on the moon: y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, ay = −gmoon and v0y = v0sinθ0, y = (v0 sin θ 0 )t − 1 g moont 2 2 Eliminate the parameter t to obtain: When y = 0 and x = R: y = (tan θ 0 )x − g moon x2 2 2v cos θ 0 0 = (tan θ 0 )R − g moon R2 2 2v cos θ 0 2 0 2 0 and R= Substitute numerical values and evaluate R: 2 v0 g moon sin 2θ 0 (900 m/s)2 sin90° = 4.85 × 105 m R= 1.67 m/s 2 = 485 km This result is probably not very accurate because it is about 28% of the moon’s radius (1740 km). This being the case, we can no longer assume that the ground is ″flat″ because of the curvature of the moon. (b) Express the condition that the centripetal acceleration must satisfy for an object in orbit at the surface of the moon: ac = g moon Solve for and evaluate v: v = g moon r = v2 = r (1.67 m/s )(1.74 × 10 m ) 2 6 = 1.70 km/s 96 ••• Picture the Problem We can show that ∆R/R = –∆g/g by differentiating R with respect to g and then using a differential approximation. Differentiate the range equation with respect to g: 2 ⎞ dR d ⎛ v0 v2 ⎜ sin 2θ 0 ⎟ = − 02 sin 2θ 0 = ⎟ dg dg ⎜ g g ⎝ ⎠ =− R g Motion in One and Two Dimensions 179 Approximate dR/dg by ∆R/∆g: ∆R R =− g ∆g Separate the variables to obtain: ∆R ∆g =− R g i.e., for small changes in gravity ( g ≈ g ± ∆g ), the fractional change in R is linearly opposite to the fractional change in g. Remarks: This tells us that as gravity increases, the range will decrease, and vice versa. This is as it must be because R is inversely proportional to g. 97 ••• Picture the Problem We can show that ∆R/R = 2∆v0/ v0 by differentiating R with respect to v0 and then using a differential approximation. Differentiate the range equation with respect to v0: 2 ⎞ 2v dR d ⎛ v0 ⎜ sin 2θ 0 ⎟ = 0 sin 2θ 0 = ⎜g ⎟ dv0 dv0 ⎝ g ⎠ =2 R v0 Approximate dR/dv0 by ∆R/∆v0: ∆R R =2 v0 ∆v0 Separate the variables to obtain: ∆R ∆v =2 0 R v0 i.e., for small changes in the launch velocity ( v0 ≈ v0 ± ∆v0 ), the fractional change in R is twice the fractional change in v0. Remarks: This tells us that as launch velocity increases, the range will increase twice as fast, and vice versa. 98 ••• Picture the Problem Choose a coordinate system in which the origin is at the base of the surface from which the projectile is launched. Let the positive x direction be to the right and the positive y direction be upward. We can apply constant-acceleration equations to obtain parametric equations in time that relate the range to the initial horizontal speed and the height h to the initial upward speed. Eliminating the parameter will leave us with a quadratic equation in R, the solution to which is the result we are required to establish. Write the constant-acceleration equations for the horizontal and vertical parts of the projectile’s x = v0 xt and 180 Chapter 3 motion: y = h + v0 y t + 1 (− g ) t 2 2 where v0 x = v0 cosθ and v0 y = v0 sin θ Solve the x-component equation for time: t= x x = v0 x v0 cosθ Using the x-component equation, eliminate time from the y-component equation to obtain: y = h + (tan θ ) x − g x2 2v cos 2 θ When the projectile strikes the ground its coordinates are (R, 0) and our equation becomes: 0 = h + (tan θ )R − g R2 2 2v cos θ Using the plus sign in the quadratic formula to ensure a physically meaningful root (one that is positive), solve for the range to obtain: ⎛ 2 gh R = ⎜1 + 1 + 2 2 ⎜ v0 sin θ 0 ⎝ 2 0 2 0 2 ⎞ v0 ⎟ ⎟ 2 g sin 2θ 0 ⎠ *99 •• Picture the Problem We can use trigonometry to relate the maximum height of the projectile to its range and the sighting angle at maximum elevation and the range equation to express the range as a function of the launch speed and angle. We can use a constant-acceleration equation to express the maximum height reached by the projectile in terms of its launch angle and speed. Combining these relationships will allow us to conclude that tan φ = 1 tan θ . 2 Referring to the figure, relate the maximum height of the projectile to its range and the sighting angle φ: Express the range of the rocket and use the trigonometric identity sin 2θ = 2 sin θ cos θ to rewrite the expression as: Using a constant-acceleration equation, relate the maximum height of a projectile to the vertical component of its launch speed: Solve for the maximum height h: tan φ = R= h R2 v2 v2 sin( 2θ ) = 2 sin θ cosθ g g 2 2 v y = v0 y − 2 gh or, because vy = 0 and v0y = v0sinθ, 2 v0 sin 2 θ = 2 gh h= v2 sin 2 θ 2g Motion in One and Two Dimensions 181 Substitute for R and h and simplify to obtain: v2 2 sin 2 θ 2g tan φ = = 2 v 2 sin θ cos θ g 1 2 tan θ 100 • Picture the Problem In the absence of air resistance, the horizontal and vertical displacements of the projectile are independent of each other and describable by constant-acceleration equations. Choose the origin at the firing location and with the coordinate axes as shown in the figure and use constant-acceleration equations to relate the vertical displacement to vertical component of the initial velocity and the horizontal velocity to the horizontal displacement and the time of flight. (a) Using a constant-acceleration equation, express the vertical displacement of the projectile as a function of its time of flight: Solve for v0y: Substitute numerical values and evaluate v0y: ∆y = v0 y ∆t + 1 a y (∆t ) 2 2 or, because ay = −g, ∆y = v0 y ∆t − 1 g (∆t ) 2 2 ∆y + 1 g (∆t ) 2 ∆t 2 v0 y = 450 m + 1 (9.81 m/s 2 )(20 s ) 2 = 20 s 2 v0 y = 121 m/s (b) The horizontal velocity remains constant, so: *101 •• Picture the Problem In the absence of air resistance, the acceleration of the stone is constant and the horizontal and vertical motions are independent of each other. Choose a coordinate system with the origin at the throwing location and the axes oriented as shown in the figure and use constant- acceleration equations to express the x and y coordinates of the stone while it is in flight. v0 x = v x = ∆x 3000 m = = 150 m/s ∆t 20 s 182 Chapter 3 Using a constant-acceleration equation, express the x coordinate of the stone in flight: Using a constant-acceleration equation, express the y coordinate of the stone in flight: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0 and ax = 0, x = v0t y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = 0 and ay = g, y = 1 gt 2 2 Referring to the diagram, express the relationship between θ, y and x at impact: tan θ = y x Substitute for x and y and solve for the time to impact: tan θ = gt 2 g = t 2v0t 2v0 Solve for t to obtain: Referring to the diagram, express the relationship between θ, L, y and x at impact: Substitute for y to obtain: Substitute for t and solve for L to obtain: t= 2v0 tan θ g x = L cos θ = y tan θ gt 2 = L cosθ 2g L= 2 2v0 tan θ g cosθ 102 ••• Picture the Problem The equation of a particle’s trajectory is derived in the text so we’ll use it as our starting point in this derivation. We can relate the coordinates of the point of impact (x, y) to the angle φ and use this relationship to eliminate y from the equation for the cannonball’s trajectory. We can then solve the resulting equation for x and relate the horizontal component of the point of impact to the cannonball’s range. The equation of the cannonball’s trajectory is given in the text: ⎛ ⎞ 2 g y ( x) = (tan θ 0 ) x − ⎜ 2 ⎜ 2v cos 2 θ ⎟ x ⎟ 0 ⎠ ⎝ 0 Relate the x and y components of a point on the ground to the angle φ: y( x ) = (tan φ ) x Express the condition that the cannonball hits the ground: ⎛ ⎞ 2 g ⎟x 2 ⎟ ⎝ 2v cos θ 0 ⎠ (tan φ )x = (tan θ 0 ) x − ⎜ ⎜ 2 0 Motion in One and Two Dimensions 183 Solve for x to obtain: 2 2v0 cos 2 θ 0 (tan θ 0 − tan φ ) x= g Relate the range of the cannonball’s flight R to the horizontal distance x: x = R cos φ Substitute to obtain: Solve for R: R cos φ = R= 2 2v0 cos 2 θ 0 (tan θ 0 − tan φ ) g 2 2v0 cos 2 θ 0 (tan θ 0 − tan φ ) g cos φ 103 •• Picture the Problem In the absence of air resistance, the acceleration of the rock is constant and the horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure with the origin at the base of the building and the axes oriented as shown and apply constant-acceleration equations to relate the horizontal and vertical displacements of the rock to its time of flight. Find the horizontal and vertical components of v0: v0x = v0 cos53° = 0.602v0 v0y = v0 sin53° = 0.799v0 Using a constant-acceleration equation, express the horizontal displacement of the projectile: ∆x = 20 m = v0 x ∆t = (0.602v0 )∆t Using a constant-acceleration equation, express the vertical displacement of the projectile: ∆y = −20 m = v0 y ∆t − 1 g (∆t ) 2 2 = (0.799v0 )∆t − 1 g (∆t ) 2 2 Solve the x-displacement equation for ∆t: ∆t = Substitute ∆t into the expression for ∆y: − 20 m = (0.799v 0 )∆t − 4.91 m/s 2 (∆t ) Solve for v0 to obtain: v0 = 10.8 m/s Find ∆t at impact: 20 m 0.602v0 ( ∆t = 20 m = 3.08 s (10.8 m/s)cos53° ) 2 184 Chapter 3 Using constant-acceleration equations, find vy and vx at impact: vx = v0 x = 6.50 m/s and v y = v0 y − g∆t = −21 m/s Express the velocity at impact in vector form: r ˆ v = (6.50 m/s) i + (−21.6 m/s) ˆ j 104 •• Picture the Problem The ball experiences constant acceleration, except during its collision with the wall, so we can use the constant-acceleration equations in the analysis of its motion. Choose a coordinate system with the origin at the point of release, the positive x axis to the right, and the positive y axis upward. Using a constant-acceleration equation, express the vertical displacement of the ball as a function of ∆t: ∆y = v0 y ∆t − 1 g (∆t ) 2 When the ball hits the ground, ∆y = −2 m: − 2 m = (10 m/s ) ∆t Solve for the time of flight: t flight = ∆t = 2.22 s Find the horizontal distance traveled in this time: ∆x = (10 m/s) (2.22 s) = 22.2 m The distance from the wall is: ∆x – 4 m = 18.2 m 2 ( Hitting Targets and Related Problems 105 • Picture the Problem In the absence of air resistance, the acceleration of the pebble is constant. Choose the coordinate system shown in the diagram and use constantacceleration equations to express the coordinates of the pebble in terms of the time into its flight. We can eliminate the parameter t between these equations and solve for the launch velocity of the pebble. We can determine the launch angle from the sighting information and, once the range is known, the time of flight can be found using the horizontal component of the initial velocity. ) − 1 9.81 m/s 2 (∆t ) 2 2 Motion in One and Two Dimensions 185 Referring to the diagram, express θ in terms of the given distances: Use a constant-acceleration equation to express the horizontal position of the pebble as a function of time: Use a constant-acceleration equation to express the vertical position of the pebble as a function of time: Eliminate the parameter t to obtain: At impact, y = 0 and x = R: Solve for v0 to obtain: Substitute numerical values and evaluate v0: Substitute in equation (1) to relate R to tflight: Solve for and evaluate the time of flight: ⎛ 4.85 m ⎞ ⎟ = 6.91° ⎟ ⎝ 40 m ⎠ θ = tan −1 ⎜ ⎜ x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ, and ax = 0, x = (v0 cosθ )t (1) y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = v0sinθ, and ay = −g, y = (v0 sin θ )t − 1 gt 2 2 y = (tan θ )x − g x2 2 2v cos θ 0 = (tan θ )R − 2 0 g R2 2v cos 2 θ 2 0 v0 = Rg sin 2θ v0 = (40 m)(9.81 m/s 2 ) = 40.6 m/s sin 13.8° R = (v0 cos θ ) t flight tflight = 40 m = 0.992 s (40.6 m/s)cos6.91° *106 •• Picture the Problem The acceleration of the ball is constant (zero horizontally and – g vertically) and the vertical and horizontal components are independent of each other. Choose the coordinate system shown in the figure and assume that v and t are unchanged by throwing the ball slightly downward. Express the horizontal displacement of the ball as a function of time: ∆x = v0 x ∆t + 1 a x (∆t ) 2 2 186 Chapter 3 or, because ax = 0, ∆x = v0 x ∆t Solve for the time of flight if the ball were thrown horizontally: ∆t = 18.4 m ∆x = = 0.491 s v0 x 37.5 m/s ∆y = v0 y ∆t + 1 a y (∆t ) 2 Using a constant-acceleration equation, express the distance the ball would drop (vertical displacement) if it were thrown horizontally: or, because v0y = 0 and ay = −g, Substitute numerical values and evaluate ∆y: ∆y = − 1 9.81 m/s 2 (0.491 s ) = −1.18 m 2 The ball must drop an additional 0.62 m before it gets to home plate. y = (2.5 – 1.18) m = 1.32 m above ground Calculate the initial downward speed the ball must have to drop 0.62 m in 0.491 s: vy = 2 ∆y = − 1 g (∆t ) 2 2 ( Find the angle with horizontal: ) 2 − 0.62 m = −1.26 m 0.491s ⎛ vy ⎞ ⎛ − 1.26 m/s ⎞ ⎟ = tan −1 ⎜ ⎜ 37.5 m/s ⎟ ⎟ ⎟ ⎝ ⎠ ⎝ vx ⎠ θ = tan −1 ⎜ ⎜ = − 1.92° Remarks: One can readily show that 2 2 vx + v y = 37.5 m/s to within 1%; so the assumption that v and t are unchanged by throwing the ball downward at an angle of 1.93° is justified. 107 •• Picture the Problem The acceleration of the puck is constant (zero horizontally and –g vertically) and the vertical and horizontal components are independent of each other. Choose a coordinate system with the origin at the point of contact with the puck and the coordinate axes as shown in the figure and use constant-acceleration equations to relate the variables v0y, the time t to reach the wall, v0x, v0, and θ0. Using a constant-acceleration equation for the motion in the y direction, express v0y as a function of the puck’s displacement ∆y: 2 2 v y = v0 y + 2a y ∆y or, because vy= 0 and ay = −g, 2 0 = v0 y − 2 g∆y Motion in One and Two Dimensions 187 Solve for and evaluate v0y: v0 y = 2 g∆y = 2(2.80 m )(9.81m/s 2 ) = 7.41 m/s Find t from the initial velocity in the y direction: t= Use the definition of average velocity to find v0x: v0 x = vx = Substitute numerical values and evaluate v0: 2 2 v0 = v0 x + v0 y v0 y = g = 7.41 m/s = 0.756 s 9.81 m/s 2 ∆x 12.0 m = = 15.9 m/s 0.756 s t (15.9 m/s )2 + (7.41m/s )2 = 17.5 m/s Substitute numerical values and evaluate θ : ⎛ v0 y ⎞ ⎛ 7.41 m/s ⎞ ⎟ = tan −1 ⎜ ⎜ 15.9 m/s ⎟ ⎟ ⎟ ⎝ ⎠ ⎝ v0 x ⎠ θ = tan −1 ⎜ ⎜ = 25.0° 108 •• Picture the Problem In the absence of air resistance, the acceleration of Carlos and his bike is constant and we can use constant-acceleration equations to express his x and y coordinates as functions of time. Eliminating the parameter t between these equations will yield y as a function of x … an equation we can use to decide whether he can jump the creek bed as well as to find the minimum speed required to make the jump. (a) Use a constant-acceleration equation to express Carlos’ horizontal position as a function of time: Use a constant-acceleration equation to express Carlos’ vertical position as a function of time: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ, and ax = 0, x = (v0 cos θ ) t y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = v0sinθ, and ay = −g, y = (v0 sin θ )t − 1 gt 2 2 188 Chapter 3 Eliminate the parameter t to obtain: Substitute y = 0 and x = R to obtain: Solve for and evaluate R: y = (tan θ )x − g x2 2v cos 2 θ 0 = (tan θ )R − g R2 2 2v cos θ 2 0 2 0 2 (11.1m/s) sin 20° v0 sin (2θ 0 ) = 9.81 m/s 2 g = 4.30 m 2 R= He should apply the brakes! (b) Solve the equation we used in the previous step for v0,min: Letting R = 7 m, evaluate v0,min: v0, min = v0,min = Rg sin (2θ 0 ) (7m )(9.81m/s2 ) sin20° = 14.2m/s = 51.0 km/h 109 ••• Picture the Problem In the absence of air resistance, the bullet experiences constant acceleration along its parabolic trajectory. Choose a coordinate system with the origin at the end of the barrel and the coordinate axes oriented as shown in the figure and use constant-acceleration equations to express the x and y coordinates of the bullet as functions of time along its flight path. Use a constant-acceleration equation to express the bullet’s horizontal position as a function of time: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ, and ax = 0, x = (v0 cosθ )t Use a constant-acceleration equation to express the bullet’s vertical position as a function of time: Eliminate the parameter t to obtain: y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = v0sinθ, and ay = −g, y = (v0 sin θ )t − 1 gt 2 2 y = (tan θ )x − g x2 2 2v cos θ 2 0 Motion in One and Two Dimensions 189 Let y = 0 when x = R to obtain: Solve for the angle above the horizontal that the rifle must be fired to hit the target: Substitute numerical values and evaluate θ0: 0 = (tan θ )R − g R2 2v cos 2 θ 2 0 ⎛ Rg ⎞ θ 0 = 1 sin −1 ⎜ 2 ⎟ 2 ⎜v ⎟ ⎝ 0 ⎠ ( ) ⎡ (100 m ) 9.81 m/s 2 ⎤ (250 m/s)2 ⎥ ⎣ ⎦ = 0.450° Note: A second value for θ0, 89.6° is θ 0 = 1 sin −1 ⎢ 2 physically unreasonable. Referring to the diagram, relate h to θ0 and solve for and evaluate h: tan θ 0 = h 100 m and h = (100 m ) tan (0.450°) = 0.785 m General Problems 110 • Picture the Problem The sum and difference of two vectors can be found from the components of the two vectors. The magnitude and direction of a vector can be found from its components. (a) The table to the right r summarizes the components of A r and B . (b) The table to the right shows the r components of S . Vector r A r B Vector r A r B r S Determine the magnitude and r direction of S from its components: x component (m) 0.707 y component (m) 0.707 0.866 −0.500 x component (m) 0.707 y component (m) 0.707 0.866 −0.500 0.207 1.57 2 S = S x2 + S y = 1.59 m r and, because S is in the 1st ⎛S ⎞ θ S = tan −1 ⎜ y ⎟ = 7.50° ⎜S ⎟ ⎝ x⎠ 190 Chapter 3 (c) The table to the right shows the r Vector components of D : r A r B r D Determine the magnitude and r direction of D from its components: x component (m) 0.707 y component (m) 0.707 0.866 −0.500 1.21 −0.159 2 D = Dx2 + D y = 1.22 m r and, because D is in the 2nd quadrant, ⎛D ⎞ θ D = tan −1 ⎜ y ⎟ = 97.5° ⎜D ⎟ ⎝ x⎠ *111 • Picture the Problem A vector quantity can be resolved into its components relative to any coordinate system. In this example, the axes are orthogonal and the components of the vector can be found using trigonometric functions. r The x and y components of g are related to g through the sine and cosine functions: gx = gsin30° = 4.91 m/s 2 and gy = gcos30° = 8.50 m/s 2 112 • Picture the Problem The figure shows two arbitrary, co-planar vectors that (as drawn) do not satisfy the condition that A/B = Ax/Bx. Because Ax = A cos θ A and Bx = B cosθ B , cosθ A = 1 for the cosθ B condition to be satisfied. r r ∴ A/B = Ax/Bx if and only if A and B are parallel (θA = θB) or on opposite sides of the x-axis (θA = –θB). 113 • Picture the Problem We can plot the path of the particle by substituting values for t and r evaluating rx and ry coordinates of r . The velocity vector is the time derivative of the position vector. (a) We can assign values to t in the parametric equations x = (5 m/s)t and y = (10 m/s)t to obtain ordered pairs (x, y) that lie on the path of the particle. The path is shown in the following graph: Motion in One and Two Dimensions 191 25 y (m) 20 15 10 5 0 0 2 4 6 8 10 12 x (m) r (b) Evaluate dr dt : Use its components to find the r magnitude of v : r r dr d (5 m/s)t iˆ + (10m/s)t ˆ v= j = dt dt ˆ j = (5 m/s ) i + (10m/s ) ˆ [ 2 2 v = v x + v y = 11.2 m/s 114 •• Picture the Problem In the absence of air resistance, the hammer experiences constant acceleration as it falls. Choose a coordinate system with the origin and coordinate axes as shown in the figure and use constant-acceleration equations to describe the x and y coordinates of the hammer along its trajectory. We’ll use the equation describing the vertical motion to find the time of flight of the hammer and the equation describing the horizontal motion to determine its range. Using a constant-acceleration equation, express the x coordinate of the hammer as a function of time: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ0, and ax = 0, x = (v0 cosθ 0 )t Using a constant-acceleration equation, express the y coordinate of the hammer as a function of time: y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = h, v0y = v0sinθ, and ay = −g, y = h + (v0 sin θ ) t − 1 gt 2 2 192 Chapter 3 Substitute numerical values to obtain: y = 10 m + (4 m/s )(sin 30°) t Substitute the conditions that exist when the hammer hits the ground: 0 = 10 m − (4 m/s ) sin 30° t Solve for the time of fall to obtain: t = 1.24 s Use the x-coordinate equation to find the horizontal distance traveled by the hammer in 1.24 s: R = (4 m/s )(cos30°)(1.24 s ) ( ) − 1 9.81 m/s 2 t 2 2 − 1 (9.81 m/s 2 ) t 2 2 = 4.29 m 115 •• Picture the Problem We’ll model Zacchini’s flight as though there is no air resistance and, hence, the acceleration is constant. Then we can use constant- acceleration equations to express the x and y coordinates of Zacchini’s motion as functions of time. Eliminating the parameter t between these equations will leave us with an equation we can solve forθ. Because the maximum height along a parabolic trajectory occurs (assuming equal launch and landing elevations) occurs at half range, we can use this same expression for y as a function of x to find h. Use a constant-acceleration equation to express Zacchini’s horizontal position as a function of time: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ, and ax = 0, x = (v0 cos θ ) t Use a constant-acceleration equation to express Zacchini’s vertical position as a function of time: Eliminate the parameter t to obtain: Use Zacchini’s coordinates when he lands in a safety net to obtain: y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = v0sinθ, and ay = −g, y = (v0 sin θ ) t − 1 gt 2 2 y = (tan θ )x − g x2 2v cos 2 θ 0 = (tan θ )R − g R2 2 2v cos θ 2 0 2 0 Motion in One and Two Dimensions 193 Solve for his launch angle θ : ⎛ Rg ⎞ θ = 1 sin −1 ⎜ 2 ⎟ 2 ⎜v ⎟ ⎝ 0 ⎠ ( ) ⎡ (53 m ) 9.81 m/s 2 ⎤ ⎥ = 31.3° 2 ⎣ (24.2 m/s ) ⎦ Substitute numerical values and evaluate θ : θ = 1 sin −1 ⎢ 2 Use the fact that his maximum height was attained when he was halfway through his flight to obtain: R g ⎛R⎞ h = (tan θ ) − 2 ⎜ ⎟ 2 2 2v0 cos θ ⎝ 2 ⎠ 2 Substitute numerical values and evaluate h: 2 53 m 9.81 m/s 2 ⎛ 53 m ⎞ h = (tan 31.3°) − ⎜ ⎟ = 8.06 m 2 2 2 2(24.2 m/s ) cos 31.3° ⎝ 2 ⎠ 116 •• Picture the Problem Because the acceleration is constant; we can use the constantacceleration equations in vector form and the definitions of average velocity and average (instantaneous) acceleration to solve this problem. (a) The average velocity is given by: The average velocity can also be expressed as: r r r r ∆r r2 − r1 = v av = ∆t ∆t ˆ + (−2.5 m/s) ˆ = (3 m/s)i j r r r v1 + v 2 vav = 2 and r r r v1 = 2vav − v 2 Substitute numerical values to obtain: (b) The acceleration of the particle is given by: r ˆ v1 = (1 m/s) i + (1 m/s) ˆ j r r r r ∆v v 2 − v1 = a= ∆t ∆t ˆ = (2 m/s 2 ) i + (−3.5 m/s 2 ) ˆ j (c) The velocity of the particle as a function of time is: r r r ˆ v (t ) = v1 + at = [(1 m/s) + (2 m/s 2 )t ] i + [(1 m/s) + (−3.5 m/s 2 )t ] ˆ j (d) Express the position vector as a function of time: r r r r r (t ) = r1 + v1t + 1 at 2 2 194 Chapter 3 r Substitute numerical values and evaluate r (t ) : ( ) r ˆ r (t ) = [(4 m) + (1 m/s)t + (1 m/s 2 )t 2 ] i + [(3 m) + (1 m/s)t + − 1.75 m/s 2 t 2 ] ˆ j *117 •• Picture the Problem In the absence of air resistance, the steel ball will experience constant acceleration. Choose a coordinate system with its origin at the initial position of the ball, the x direction to the right, and the y direction downward. In this coordinate system y0 = 0 and a = g. Letting (x, y) be a point on the path of the ball, we can use constant-acceleration equations to express both x and y as functions of time and, using the geometry of the staircase, find an expression for the time of flight of the ball. Knowing its time of flight, we can find its range and identify the step it strikes first. The angle of the steps, with respect to the horizontal, is: Using a constant-acceleration equation, express the x coordinate of the steel ball in its flight: Using a constant-acceleration equation, express the y coordinate of the steel ball in its flight: The equation of the dashed line in the figure is: Solve for the flight time: ⎛ 0.18 m ⎞ ⎟ = 31.0° ⎟ ⎝ 0.3 m ⎠ θ = tan −1 ⎜ ⎜ x = x0 + v0t + 1 a y t 2 2 or, because x0 = 0 and ay = 0, x = v0t y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = 0, v0y = 0, and ay = g, y = 1 gt 2 2 y gt = tan θ = x 2v0 t= 2v0 tan θ g Find the x coordinate of the landing position: x= 2v 2 y = 0 tan θ tan θ g Substitute the angle determined in the first step: x= 2(3 m/s ) tan31° = 1.10 m 9.81 m/s 2 2 The first step with x > 1.10 m is the 4th step. Motion in One and Two Dimensions 195 118 •• Picture the Problem Ignoring the influence of air resistance, the acceleration of the ball is constant once it has left your hand and we can use constant-acceleration equations to express the x and y coordinates of the ball. Elimination of the parameter t will yield an equation from which we can determine v0. We can then use the y equation to express the time of flight of the ball and the x equation to express its range in terms of x0, v0,θ and the time of flight. Use a constant-acceleration equation to express the ball’s horizontal position as a function of time: Use a constant-acceleration equation to express the ball’s vertical position as a function of time: Eliminate the parameter t to obtain: For the throw while standing on level ground we have: x = x0 + v0 xt + 1 axt 2 2 or, because x0 = 0, v0x = v0cosθ, and ax = 0, x = (v0 cosθ )t (1) y = y0 + v0 y t + 1 a y t 2 2 or, because y0 = x0, v0y = v0sinθ, and ay = −g, y = x0 + (v0 sin θ )t − 1 gt 2 (2) 2 y = x0 + (tan θ )x − 0 = (tan θ )x0 − g x2 2 2v cos θ 2 0 g 2 x0 2v cos 2 θ 2 0 and x0 = 2 v0 v2 v2 sin 2θ = 0 sin 2(45°) = 0 g g g Solve for v0: v0 = gx0 At impact equation (2) becomes: 0 = x0 + ( tflight = x0 sin θ + sin 2 θ + 2 g Solve for the time of flight: ) 2 gx0 sin θ tflight − 1 gtflight 2 ( ) 196 Chapter 3 Substitute in equation (1) to express the range of the ball when thrown from an elevation x0 at an angle θ with the horizontal: ( =( R= ) x (sin θ + cosθ ) g gx0 cosθ tflight gx0 0 ( sin 2 θ + 2 = x0 cosθ sin θ + sin 2 θ + 2 Substitute θ = 0°, 30°, and 45°: ) ) x(0°) = 1.41x0 x(30°) = 1.73 x0 and x(45°) = 1.62 x0 119 ••• Picture the Problem Choose a coordinate system with its origin at the point where the motorcycle becomes airborne and with the positive x direction to the right and the positive y direction upward. With this choice of coordinate system we can relate the x and y coordinates of the motorcycle (which we’re treating as a particle) using Equation 3-21. (a) The path of the motorcycle is given by: For the jump to be successful, h < y(x). Solving for v0, we find: ⎛ ⎞ g y ( x ) = (tanθ ) x − ⎜ 2 2 ⎟ x 2 ⎜ 2v cos θ ⎟ ⎝ 0 ⎠ vmin > x cosθ g 2( x tan θ − h) (b) Use the values given to obtain: vmin > 26.0 m/s or 58.0 mph (c) In order for our expression for vmin to be real valued; i.e., to predict values for vmin that are physically meaningful, x tanθ − h > 0. ∴ hmax < x tanθ The interpretation is that the bike "falls away" from traveling on a straight-line path due to the free-fall acceleration downwards. No matter what the initial speed of the bike, it must fall a little bit before reaching the other side of the pit. Motion in One and Two Dimensions 197 120 ••• Picture the Problem Let the origin be at the position of the boat when it was engulfed by the fog. Take the x and y directions to be east and north, r respectively. Let v BW be the velocity of r the boat relative to the water, v BS be the velocity of the boat relative to the shore, r and v WS be the velocity of the water with respect to the shore. Then r r r v BS = v BW + v WS . r θ is the angle of v WS with respect to the x (east) direction. (a) Find the position vector for the boat at t = 3 h: r ˆ rboat = {(32 km )(cos 135°)t}i + {(32 km )(sin135°) t − 4 km} ˆ j ˆ = {(− 22.6 km )t}i + {(22.6 km ) t − 4 km} ˆ j Find the coordinates of the boat at t = 3 h: rx = [(10 km/h ) cos135° + vWS cos θ ](3 h ) and ry = [(10 km/h )sin 135° + vWS sin θ ](3 h ) Simplify the expressions involving rx and ry and equate these simplified expressions to the x and y components of the position vector of the boat: Divide the second of these equations by the first to obtain: 3vWS cosθ = −1.41 km/h and 3vWS sinθ = −2.586 km/h tan θ = − 2.586 km − 1.41 km or ⎛ − 2.586 km ⎞ ⎟ = 61.4° or 241.4° − 1.41 km ⎟ ⎝ ⎠ θ = tan −1 ⎜ ⎜ Because the boat has drifted south, use θ = 241.4° to obtain: vWS 1.41 km/h − vx 3 = = cos θ cos(241.4°) = 0.982 km/h at θ = 241.4° 198 Chapter 3 (b) Letting φ be the angle between east and the proper heading for the boat, express the components of the velocity of the boat with respect to the shore: vBS,x = (10 km/h) cosφ + (0.982 km/h) cos(241.3°) For the boat to travel northwest: vBS,x = –vBS,y Substitute the velocity components, square both sides of the equation, and simplify the expression to obtain the equations: sinφ + cosφ = 0.133, sin2φ + cos2φ + 2 sinφ cosφ = 0.0177, and 1 + sin(2φ) = 0.0177 Solve for φ: φ = 129.6° or 140.4° Because the current pushes south, the boat must head more northerly than 135°: Using 129.6°, the correct heading (c) Find vBS: vBS,x = –6.84 km/h and vBS = vBx /cos135° = 9.68 km/h To find the time to travel 32 km, divide the distance by the boat’s actual speed: t = (32 km)/(9.68 km/h) vBS,y = (10 km/h) sinφ + (0.982 km/h) sin(241.3°) is 39.6° west of north . = 3.31 h = 3 h 18 min *121 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and the equation of a projectile for equal initial and final elevations, which was derived from the constant-acceleration equations, is applicable. We can use the equation giving the range of a projectile for equal initial and final elevations to evaluate the ranges of launches that exceed or fall short of 45° by the same amount. Express the range of the projectile as a function of its initial speed and angle of launch: Let θ0= 45° ± θ: R= 2 v0 sin 2θ 0 g R= 2 v0 sin (90° ± 2θ ) g = Because cos(–θ) = cos(+θ) (the cosine function is an even function): 2 v0 cos(± 2θ ) g R(45° + θ ) = R(45° − θ ) Motion in One and Two Dimensions 199 122 •• Picture the Problem In the absence of air resistance, the acceleration of both balls is that due to gravity and the horizontal and vertical motions are independent of each other. Choose a coordinate system with the origin at the base of the cliff and the coordinate axes oriented as shown and use constant-acceleration equations to relate the x and y components of the ball’s speed. Independently of whether a ball is thrown upward at the angle α or downward at β, the vertical motion is described by: 2 2 v y = v0 y + 2a∆y The horizontal component of the motion is given by: vx = v0x Find v at impact from its components: 2 2 2 2 v = v x + v y = v0 x + v0 y − 2 gh 2 = v0 y − 2 gh = 2 v0 − 2 gh 200 Chapter 3 Chapter 4 Newton’s Laws Conceptual Problems *1 •• Determine the Concept A reference frame in which the law of inertia holds is called an inertial reference frame. If an object with no net force acting on it is at rest or is moving with a constant speed in a straight line (i.e., with constant velocity) relative to the reference frame, then the reference frame is an inertial reference frame. Consider sitting at rest in an accelerating train or plane. The train or plane is not an inertial reference frame even though you are at rest relative to it. In an inertial frame, a dropped ball lands at your feet. You are in a noninertial frame when the driver of the car in which you are riding steps on the gas and you are pushed back into your seat. 2 •• Determine the Concept A reference frame in which the law of inertia holds is called an inertial reference frame. A reference frame with acceleration a relative to the initial frame, and with any velocity relative to the initial frame, is inertial. 3 • Determine the Concept No. If the net force acting on an object is zero, its acceleration is zero. The only conclusion one can draw is that the net force acting on the object is zero. *4 • Determine the Concept An object accelerates when a net force acts on it. The fact that an object is accelerating tells us nothing about its velocity other than that it is always changing. Yes, the object must have an acceleration relative to the inertial frame of reference. According to Newton’s 1st and 2nd laws, an object must accelerate, relative to any inertial reference frame, in the direction of the net force. If there is ″only a single nonzero force,″ then this force is the net force. Yes, the object’s velocity may be momentarily zero. During the period in which the force is acting, the object may be momentarily at rest, but its velocity cannot remain zero because it must continue to accelerate. Thus, its velocity is always changing. 5 • Determine the Concept No. Predicting the direction of the subsequent motion correctly requires knowledge of the initial velocity as well as the acceleration. While the acceleration can be obtained from the net force through Newton’s 2nd law, the velocity can only be obtained by integrating the acceleration. 6 • Determine the Concept An object in an inertial reference frame accelerates if there is a net force acting on it. Because the object is moving at constant velocity, the net force acting on it is zero. (c ) is correct. 201 202 Chapter 4 7 • Determine the Concept The mass of an object is an intrinsic property of the object whereas the weight of an object depends directly on the local gravitational field. Therefore, the mass of the object would not change and wgrav = mg local . Note that if the gravitational field is zero then the gravitational force is also zero. *8 • Determine the Concept If there is a force on her in addition to the gravitational force, she will experience an additional acceleration relative to her space vehicle that is proportional to the net force required producing that acceleration and inversely proportional to her mass. She could do an experiment in which she uses her legs to push off from the wall of her space vehicle and measures her acceleration and the force exerted by the wall. She could calculate her mass from the ratio of the force exerted by the wall to the acceleration it produced. *9 • Determine the Concept One’s apparent weight is the reading of a scale in one’s reference frame. Imagine yourself standing on a scale that, in turn, is on a platform accelerating upward with an acceleration a. The freebody diagram shows the force the r gravitational field exerts on you, mg, and r the force the scale exerts on you, wapp . The scale reading (the force the scale exerts on you) is your apparent weight. Choose the coordinate system shown in the free-body diagram and apply r r ∑ F = ma to the scale: ∑F y = wapp − mg = ma y or wapp = mg + ma y So, your apparent weight would be greater than your true weight when observed from a reference frame that is accelerating upward. That is, when the surface on which you are standing has an acceleration a such that a y is positive: a y > 0 . 10 •• Determine the Concept Newton's 2nd law tells us that forces produce changes in the velocity of a body. If two observers pass each other, each traveling at a constant velocity, each will experience no net force acting on them, and so each will feel as if he or she is standing still. 11 • Determine the Concept Neither block is accelerating so the net force on each block is zero. Newton’s 3rd law states that objects exert equal and opposite forces on each other. Newton’s Laws 203 (a) and (b) Draw the free-body diagram for the forces acting on the block of mass m1: Apply r r ∑ F = ma to the block 1: ∑F y = Fn21 − m1 g = m1a1 or, because a1 = 0, Fn21 − m1 g = 0 Therefore, the magnitude of the force that block 2 exerts on block 1 is given by: Fn21 = m1 g From Newton’s 3rd law of motion we know that the force that block 1 exerts on block 2 is equal to, but opposite in direction, the force that block 2 exerts on block 1. r r Fn21 = − Fn12 ⇒ Fn12 = m1 g (c) and (d) Draw the free-body diagram for the forces acting on block 2: Apply r r ∑ F = ma to block 2: ∑F 2y = FnT2 − Fn12 − m2 g = m2 a2 or, because a2 = 0, FnT2 = FnT2 + m2 g = m1 g + m2 g = (m1 + m2 )g and the normal force that the table exerts on body 2 is FnT2 = From Newton’s 3rd law of motion we know that the force that block 2 exerts on the table is equal to, but opposite in direction, the force that the table exerts on block 2. (m1 + m 2 )g r r FnT2 = − Fn2T ⇒ Fn2T = (m1 + m2 ) g 204 Chapter 4 *12 • (a) True. By definition, action-reaction force pairs cannot act on the same object. (b) False. Action equals reaction independent of any motion of the two objects. 13 • Determine the Concept Newton’s 3rd law of motion describes the interaction between the man and his less massive son. According to the 3rd law description of the interaction of two objects, these are action-reaction forces and therefore must be equal in magnitude. (b) is correct. 14 • Determine the Concept According to Newton’s 3rd law the reaction force to a force exerted by object A on object B is the force exerted by object B on object A. The bird’s weight is a gravitational field force exerted by the earth on the bird. Its reaction force is the gravitational force the bird exerts on the earth. (b) is correct. 15 • Determine the Concept We know from Newton’s 3rd law of motion that the reaction to the force that the bat exerts on the ball is the force the ball exerts on the bat and is equal in magnitude but oppositely directed. The action-reaction pair consists of the force with which the bat hits the ball and the force the ball exerts on the bat. These forces are equal in magnitude, act in opposite directions. (c) is correct. 16 • Determine the Concept The statement of Newton’s 3rd law given in the problem is not complete. It is important to remember that the action and reaction forces act on different bodies. The reaction force does not cancel out because it does not act on the same body as the external force. *17 • Determine the Concept The force diagrams will need to include the ceiling, string, object, and earth if we are to show all of the reaction forces as well as the forces acting on the object. (a) The forces acting on the 2.5-kg r object are its weight W, and the r tension T1, in the string. The reaction r forces are W ' acting on the earth and r T1 ' acting on the string. Newton’s Laws 205 (b) The forces acting on the string are its weight, the weight of the object, r and F, the force exerted by the ceiling. The reaction forces are r r T1 acting on the string and F ' acting on the ceiling. 18 • Determine the Concept Identify the objects in the block’s environment that are exerting forces on the block and then decide in what directions those forces must be acting if the block is sliding down the inclined plane. Because the incline is frictionless, the force the incline exerts on the block must be normal to the surface. The second object capable of exerting a force on the block is the earth and its force; the weight of the block acts directly downward. The magnitude of the normal force is less than that of the weight because it supports only a portion of the weight. The forces shown in FBD (c) satisfy these conditions. 19 • Determine the Concept In considering these statements, one needs to decide whether they are consistent with Newton’s laws of motion. A good strategy is to try to think of a counterexample that would render the statement false. (a) True. If there are no forces acting on an object, the net force acting on it must be zero and, hence, the acceleration must be zero. (b) False. Consider an object moving with constant velocity on a frictionless horizontal surface. While the net force acting on it is zero (it is not accelerating), gravitational and normal forces are acting on it. (c) False. Consider an object that has been thrown vertically upward. While it is still rising, the direction of the gravitational force acting on it is downward. (d) False. The mass of an object is an intrinsic property that is independent of its location (the gravitational field in which it happens to be situated). 20 • Determine the Concept In considering these alternatives, one needs to decide which alternatives are consistent with Newton’s 3rd law of motion. According to Newton’s 3rd law, the magnitude of the gravitational force exerted by her body on the earth is equal and opposite to the force exerted by the earth on her. (a ) is correct. 206 Chapter 4 *21 • Determine the Concept In considering these statements, one needs to decide whether they are consistent with Newton’s laws of motion. In the absence of a net force, an object moves with constant velocity. (d ) is correct. 22 • Determine the Concept Draw the freebody diagram for the towel. Because the towel is hung at the center of the line, the r r magnitudes of T1 and T2 are the same. No. To support the towel, the tension in the line must have a vertical component equal to the towel’s weight. Thus θ > 0. 23 • Determine the Concept The free-body diagram shows the forces acting on a person in a descending elevator. The upward force exerted by the scale on the r person, wapp , is the person’s apparent weight. Apply ∑F y = ma y to the person and solve for wapp: wapp – mg = may or wapp = mg + may = m(g + ay) Because wapp is independent of v, the velocity of the elevator has no effect on the person' s apparent weight. Remarks: Note that a nonconstant velocity will alter the apparent weight. Estimation and Approximation 24 • Picture the Problem Assuming a stopping distance of 25 m and a mass of 80 kg, use Newton’s 2nd law to determine the force exerted by the seat belt. The force the seat belt exerts on the driver is given by: Fnet = ma, where m is the mass of the driver. Newton’s Laws 207 Using a constant-acceleration equation, relate the velocity of the car to its stopping distance and acceleration: Solve for a: 2 v 2 = v0 + 2a∆x or, because v = 0, 2 − v0 = 2a∆x a= Substitute numerical values and evaluate a: 2 − v0 2∆x ⎛ km 1h 103 m ⎞ ⎜ 90 ⎟ × × ⎜ h 3600 s km ⎟ ⎝ ⎠ a=− 2(25 m ) 2 = −12.5 m/s 2 Substitute for a and evaluate Fnet: ( Fnet = (80 kg ) − 12.5 m/s 2 ) = − 1.00 kN Fnet is negative because it is opposite the direction of motion. *25 ••• Picture the Problem The free-body diagram shows the forces acting on you and your bicycle as you are either ascending or descending the grade. The magnitude of the normal force acting on you and your bicycle is equal to the component of your weight in the y direction and the magnitude of the tangential force is the x component of your weight. Assume a combined mass (you plus your bicycle) of 80 kg. (a) Apply ∑F y = ma y to you and your bicycle and solve for Fn: Fn – mg cosθ = 0, because there is no acceleration in the y direction. ∴ Fn = mg cosθ Determine θ from the information concerning the grade: tanθ = 0.08 and θ = tan−1(0.08) = 4.57° Substitute to determine Fn: Fn = (80 kg)(9.81 m/s2) cos4.57° = 782 N Apply ∑F x = ma x to you and your bicycle and solve for Ft, the tangential force exerted by the road on the wheels: Ft – mg sinθ = 0, because there is no acceleration in the x direction. 208 Chapter 4 Evaluate Ft: Ft = (80 kg)(9.81 m/s2) sin4.57° = 62.6 N (b) Because there is no acceleration, the forces are the same going up and going down the incline. Newton’s First and Second Laws: Mass, Inertia, and Force 26 • Picture the Problem The acceleration of the particle can be found from the stopping distance by using a constant-acceleration equation. The mass of the particle and its acceleration are related to the net force through Newton’s second law of motion. Choose a coordinate system in which the direction the particle is moving is the v r positive x direction and apply Fnet = ma. Use Newton’s 2nd law to relate the mass of the particle to the net force acting on it and its acceleration: Because the force is constant, use a constant-acceleration equation with vx = 0 to determine a: m= Fnet ax 2 2 vx = v0 x + 2ax ∆x and 2 − v0 x ax = 2∆x Substitute to obtain: m= 2∆xFnet 2 v0 x Substitute numerical values and evaluate m: m= 2(62.5 m) (15.0 N) = 3.00 kg (25.0 m/s)2 and (b) is correct. 27 • Picture the Problem The acceleration of the object is related to its mass and the net force acting on it by Fnet = F0 = ma. (a) Use Newton’s 2nd law of motion to calculate the acceleration of the object: a= Fnet 2 F0 = m m ( ) = 2 3 m/s 2 = 6.00 m/s 2 Newton’s Laws 209 (b) Let the subscripts 1 and 2 distinguish the two objects. The ratio of the two masses is found from Newton’s 2nd law: 1 m2 F0 a2 a1 3 m/s 2 = = = = 2 3 m1 F0 a1 a2 9 m/s (c) The acceleration of the two-mass system is the net force divided by the total mass m = m1 + m2: a= = Fnet F0 = m m1 + m2 F0 m1 a1 = 1 + m2 m1 1 + 1 3 = 3 a1 = 2.25 m/s 2 4 28 • Picture the Problem The acceleration of an object is related to its mass and the net force acting on it by Fnet = ma . Let m be the mass of the ship, a1 be the acceleration of the ship when the net force acting on it is F1, and a2 be its acceleration when the net force is F1 + F2 . Using Newton’s 2nd law, express the net force acting on the ship when its acceleration is a1: F1 = ma1 Express the net force acting on the ship when its acceleration is a2: F1 + F2 = ma2 Divide the second of these equations by the first and solve for the ratio F2/F1: F1 + F2 ma1 = F1 ma2 and F2 a2 = −1 F1 a1 Substitute for the accelerations to determine the ratio of the accelerating forces and solve for F2: F2 (16 km/h ) (10 s ) = −1 = 3 (4 km/h ) (10 s ) F1 or F2 = 3F1 *29 •• Picture the Problem Because the deceleration of the bullet is constant, we can use a constant-acceleration equation to determine its acceleration and Newton’s 2nd law of motion to find the average resistive force that brings it to a stop. Apply r r ∑ F = ma to express the Fwood = ma force exerted on the bullet by the wood: Using a constant-acceleration 2 v 2 = v0 + 2a∆x 210 Chapter 4 equation, express the final velocity of the bullet in terms of its acceleration and solve for the acceleration: Substitute to obtain: Substitute numerical values and evaluate Fwood: and a= 2 2 v 2 − v0 − v0 = 2∆x 2∆x Fwood = − Fwood = − 2 mv0 2∆x (1.8 ×10 ) kg (500 m/s ) 2(0.06 m ) −3 2 = − 3.75 kN where the negative sign means that the direction of the force is opposite the velocity. *30 •• Picture the Problem The pictorial representation summarizes what we know about the motion. We can find the acceleration of the cart by using a constant-acceleration equation. The free-body diagram shows the forces acting on the cart as it accelerates along the air track. We can determine the net force acting on the cart using Newton’s 2nd law and our knowledge of its acceleration. (a) Apply ∑F x = ma x to the cart F = ma to obtain an expression for the net force F: Using a constant-acceleration equation, relate the displacement of the cart to its acceleration, initial speed, and travel time: ∆x = v0 ∆t + 1 a (∆t ) 2 2 or, because v0 = 0, Newton’s Laws 211 ∆x = 1 a (∆t ) 2 2 2∆x (∆t )2 Solve for a: a= Substitute for a in the force equation to obtain: F =m Substitute numerical values and evaluate F: F= (b) Using a constant-acceleration equation, relate the displacement of the cart to its acceleration, initial speed, and travel time: ∆x = v0 ∆t + 1 a ' (∆t ) 2 Solve for ∆t: 2(0.355 kg )(1.5 m ) = 0.0514 N (4.55 s )2 2 or, because v0 = 0, ∆x = 1 a ' (∆t ) 2 2 ∆t = If we assume that air resistance is negligible, the net force on the cart is still 0.0514 N and its acceleration is: Substitute numerical values and evaluate ∆t: 2∆x 2m∆x = (∆t )2 (∆t )2 a' = 2∆x a' 0.0514 N = 0.0713 m/s 2 0.722 kg ∆t = 2(1.5 m ) = 6.49 s 0.0713 m/s 2 31 • Picture the Problem The acceleration of an object is related to its mass and the net force r r acting on it according to Fnet = ma. Let m be the mass of the object and choose a coordinate system in which the direction of 2F0 in (b) is the positive x and the direction of the left-most F0 in (a) is the positive y direction. Because both force and acceleration are vector quantities, find the resultant force in each case and then find the resultant acceleration. (a) Calculate the acceleration of the object from Newton’s 2nd law of motion: Express the net force acting on the object: Find the magnitude and direction of this net force: r r Fnet a= m r ˆ ˆ Fnet = Fx i + Fy ˆ = F0 i + F0 ˆ j j and Fnet = Fx2 + Fy2 = 2 F0 and 212 Chapter 4 ⎛ Fy ⎝ Fx θ = tan −1 ⎜ ⎜ Use this result to calculate the magnitude and direction of the acceleration: ⎛F ⎞ ⎞ ⎟ = tan −1 ⎜ 0 ⎟ = 45° ⎜F ⎟ ⎟ ⎝ 0⎠ ⎠ Fnet 2 F0 = = 2 a0 m m = 2 (3 m/s 2 ) a = 4.24 m/s 2 @ 45.0° from each = force. (b) Calculate the acceleration of the object from Newton’s 2nd law of motion: Express the net force acting on the object: r r a = Fnet /m r ˆ Fnet = Fx i + Fy ˆ j ˆ = (− F0 sin 45°)i + (2 F0 + F0 cos 45°) ˆ j Find the magnitude and direction of this net force: (− F0 sin 45°)2 + (2 F0 + F0 cos 45°)2 Fnet = Fx2 + Fy2 = = 2.80 F0 and ⎛ Fy ⎝ Fx ⎞ ⎛ 2 F + F0 cos 45° ⎞ ⎟ = tan −1 ⎜ 0 ⎟ ⎜ − F sin 45° ⎟ = −75.4° ⎟ 0 ⎠ ⎝ ⎠ r = 14.6° from 2F0 θ = tan −1 ⎜ ⎜ Use this result to calculate the magnitude and direction of the acceleration: Fnet F = 2.80 0 = 2.80a0 m m 2 = 2.80(3 m/s ) a= r = 8.40 m/s 2 @ 14.6° from 2F0 32 • Picture the Problem The acceleration of an object is related to its mass and the net force r r acting on it according to a = Fnet m . r r Apply a = Fnet m to the object to obtain: r ˆ r Fnet (6 N ) i − (3 N ) ˆ j a= = m 1.5 kg = (4.00 m/s ) iˆ − (2.00 m/s ) ˆj 2 2 Newton’s Laws 213 r Find the magnitude of a : 2 2 a = ax + a y = (4.00 m/s ) + (2.00 m/s ) 2 2 2 2 = 4.47 m/s 2 33 • Picture the Problem The mass of the particle is related to its acceleration and the net force acting on it by Newton’s 2nd law of motion. Because the force is constant, we can use constant-acceleration formulas to calculate the acceleration. Choose a coordinate system in which the positive x direction is the direction of motion of the particle. The mass is related to the net force and the acceleration by Newton’s 2nd law: r ∑ F Fx m= r = a ax ∆x = v0 x t + 1 a x (∆t ) , where v0 x = 0, 2 Because the force is constant, the acceleration is constant. Use a constant-acceleration equation to find the acceleration: so Substitute this result into the first equation and solve for and evaluate the mass m of the particle: (12 N )(6 s ) F F (∆t ) = m= x = x 2∆x 2(18 m ) ax 2 ax = 2∆x (∆t )2 2 = 12.0 kg *34 • Picture the Problem The speed of either Al or Bert can be obtained from their accelerations; in turn, they can be obtained from Newtons 2nd law applied to each person. The free-body diagrams to the right show the forces acting on Al and Bert. The forces that Al and Bert exert on each other are action-and-reaction forces. (a) Apply ∑ Fx = ma x to Bert and solve for his acceleration: − FAl on Bert = mBert aBert aBert = − FAl on Bert mBert = − 20 N 100 kg = −0.200 m/s 2 Using a constant-acceleration equation, relate Bert’s speed to his initial speed, speed after 1.5 s, and acceleration and solve for his speed at the end of 1.5 s: v = v0 + a∆t = 0 + (−0.200 m/s2)(1.5 s) = − 0.300 m/s 2 214 Chapter 4 (b) From Newton's 3rd law, an equal but oppositely directed force acts on Al while he pushes Bert. Because the ice is frictionless, Al speeds off in the opposite direction. Apply Newton’s 2nd law to the forces acting on Al and solve for his acceleration: Using a constant-acceleration equation, relate Al’s speed to his initial speed, speed after 1.5 s, and acceleration; solve for his speed at the end of 1.5 s: ∑F x , Al = FBert on Al = mAl aAl and aAl = FBert on Al 20 N = mAl 80 kg = 0.250 m/s 2 v = v0 + a∆t = 0 + (0.250 m/s2)(1.5 s) = 0.375 m/s 35 • Picture the Problem The free-body diagrams show the forces acting on the two blocks. We can apply Newton’s second law to the forces acting on the blocks and eliminate F to obtain a relationship between the masses. Additional applications of Newton’s 2nd law to the sum and difference of the masses will lead us to values for the accelerations of these combinations of mass. (a) Apply ∑F x = ma x to the two blocks: ∑F x ,1 and ∑F x,2 Eliminate F between the two equations and solve for m2: = F = m1 a1 m2 = Express and evaluate the acceleration of an object whose mass is m2 – m1 when the net force acting on it is F: a= (b) Express and evaluate the acceleration of an object whose mass is m2 + m1 when the net force acting on it is F: a= = F = m2 a 2 a1 12 m/s 2 m1 = m1 = 4m1 a2 3 m/s 2 F F F = = m2 − m1 4m1 − m1 3m1 ( ) = 1 a1 = 1 12 m/s 2 = 4.00 m/s 2 3 3 = F F = m2 + m1 4m1 + m1 F = 1 a1 = 1 (12 m/s 2 ) 5 5 5m1 = 2.40 m/s 2 Newton’s Laws 215 36 • Picture the Problem Because the velocity is constant, the net force acting on the log must be zero. Choose a coordinate system in which the positive x direction is the direction of motion of the log. The freebody diagram shows the forces acting on the log when it is accelerating in the positive x direction. (a) Apply ∑F x = ma x to the log Fpull – Fres = max = 0 when it is moving at constant speed: Solve for and evaluate Fres: (b) Apply ∑F x = ma x to the log Fres = Fpull = 250 N Fpull – Fres = max when it is accelerating to the right: Solve for and evaluate Fpull: Fpull = Fres + max = 250 N + (75 kg) (2 m/s2) = 400 N 37 • Picture the Problem The acceleration can be found from Newton’s 2nd law. Because both forces are constant, the net force and the acceleration are constant; hence, we can use the constant-acceleration equations to answer questions concerning the motion of the object at various times. (a) Apply Newton’s 2nd law to the object to obtain: r r r r Fnet F1 + F2 a= = m m ˆ + (− 14 N ) ˆ (6 N ) i j = 4 kg ( ) ( ) ˆ = 1.50 m/s 2 i + − 3.50 m/s 2 ˆ j (b) Using a constant-acceleration equation, express the velocity of the object as a function of time and solve for its velocity when t = 3 s: (c) Express the position of the object in terms of its average velocity and evaluate this expression at t = 3 s: r r r v = v0 + at [( ) ( )] ˆ = 0 + 1.50 m/s 2 i + − 3.50 m/s 2 ˆ (3 s ) j = (4.50 m/s) iˆ + (− 10.5 m/s) ˆ j r r r = vavt r = 1 vt 2 = (6.75 m ) iˆ + (− 15.8 m ) ˆ j 216 Chapter 4 Mass and Weight *38 • Picture the Problem The mass of the astronaut is independent of gravitational fields and will be the same on the moon or, for that matter, out in deep space. Express the mass of the astronaut in terms of his weight on earth and the gravitational field at the surface of the earth: m= wearth 600 N = = 61.2 kg g earth 9.81 N/kg and (c) is correct. 39 • Picture the Problem The weight of an object is related to its mass and the gravitational field through w = mg. (a) The weight of the girl is: w = mg = (54 kg )(9.81 N/kg ) = 530 N (b) Convert newtons to pounds: w= 530 N = 119 lb 4.45 N/lb 40 • Picture the Problem The mass of an object is related to its weight and the gravitational field. Find the weight of the man in newtons: Calculate the mass of the man from his weight and the gravitational field: 165 lb = (165 lb )(4.45 N/lb ) = 734 N m= w 734 N = = 74.8 kg g 9.81 N/kg Contact Forces *41 • Picture the Problem Draw a free-body diagram showing the forces acting on the r block. Fk is the force exerted by the spring, r r r W = mg is the weight of the block, and Fn is the normal force exerted by the horizontal surface. Because the block is resting on a surface, Fk + Fn = W. (a) Calculate the force exerted by the spring on the block: Fx = kx = (600 N/m )(0.1 m ) = 60.0 N Newton’s Laws 217 (b) Choosing the upward direction to be positive, sum the forces acting on the block and solve for Fn: r F = 0 ⇒ Fk + Fn − W = 0 ∑ and Fn = W − Fk Substitute numerical values and evaluate Fn: Fn = (12 kg)(9.81 N/kg) − 60 N = 57.7 N 42 • Picture the Problem Let the positive x direction be the direction in which the spring is stretched. We can use Newton’s 2nd law and the expression for the force exerted by a stretched (or compressed) spring to express the acceleration of the box in terms of its mass m, the stiffness constant of the spring k, and the distance the spring is stretched x. Apply Newton’s 2nd law to the box to obtain: a= ∑F m Express the force exerted on the box by the spring: F = − kx Substitute to obtain: a= Substitute numerical values and evaluate a: a=− − kx m (800 N/m )(0.04 m ) 6 kg = − 5.33 m/s 2 where the minus sign tells us that the box’s acceleration is toward its equilibrium position. Free-Body Diagrams: Static Equilibrium 43 • Picture the Problem Because the traffic light is not accelerating, the net force acting on r r r it must be zero; i.e., T1 + T2 + mg = 0. Construct a free-body diagram showing the forces acting on the knot and choose the coordinate system shown: 218 Chapter 4 Apply ∑F x = ma x to the knot: Solve for T2 in terms of T1: T1cos30° − T2cos60° = max = 0 cos 30° T1 = 1.73T1 cos 60° ∴ T2 is greater than T1 T2 = 44 • Picture the Problem Draw a free-body diagram showing the forces acting on the lamp and apply Fy = 0 . ∑ From the FBD, it is clear that T1 supports the full weight mg = 418 N. Apply ∑F y = 0 to the lamp to T1 − w = 0 obtain: Solve for T1: Substitute numerical values and evaluate T1: T1 = w = mg ( ) T1 = (42.6 kg ) 9.81 m/s 2 = 418 N and (b) is correct. *45 •• Picture the Problem The free-body diagrams for parts (a), (b), and (c) are shown below. In both cases, the block is in equilibrium under the influence of the forces and we can use Newton’s 2nd law of motion and geometry and trigonometry to obtain relationships between θ and the tensions. (a) and (b) (c) (a) Referring to the FBD for part (a), use trigonometry to determine θ : θ = cos −1 0.5 m = 36.9° 0.625 m Newton’s Laws 219 2T sin θ − mg = 0 since a = 0 (b) Noting that T = T′, apply Fy = ma y to the 0.500-kg block ∑ and and solve for the tension T: T= Substitute numerical values and evaluate T: T= mg 2 sin θ (0.5 kg )(9.81m/s2 ) = 2sin36.9° (c) The length of each segment is: 1.25 m = 0.417 m 3 Find the distance d: 4.08 N d= Express θ in terms of d and solve for its value: 1 m − 0.417m 2 = 0.2915 m ⎛ ⎞ ⎛ 0.2915 m ⎞ d ⎟ = cos −1 ⎜ ⎟ ⎜ 0.417 m ⎟ ⎟ ⎝ 0.417 m ⎠ ⎝ ⎠ θ = cos −1 ⎜ ⎜ = 45.7° Apply ∑F y = ma y to the 0.250-kg block and solve for the tension T3: T3 sin θ − mg = 0 since a = 0. and T3 = Substitute numerical values and evaluate T3: Apply ∑F x = ma x to the 0.250-kg T3 = mg sin θ (0.25 kg )(9.81m/s2 ) = sin45.7° 3.43 N T3 cos θ − T2 = 0 since a = 0. block and solve for the tension T2: and Substitute numerical values and evaluate T2: T2 = (3.43 N ) cos 45.7° = 2.40 N By symmetry: T1 = T3 = 3.43 N T2 = T3 cosθ 220 Chapter 4 46 • Picture the Problem The suspended body is in equilibrium under the influence of the r r r forces Th , T45 , and mg; r r r i.e., Th + T45 + mg = 0 Draw the free-body diagram of the forces acting on the knot just above the 100-N body. Choose a coordinate system with the positive x direction to the right and the positive y direction upward. Apply the conditions for translational equilibrium to determine the tension in the horizontal cord. If the system is to remain in static equilibrium, the vertical component of T45 must be exactly balanced by, and therefore equal to, the tension in the string suspending the 100-N body: Tv = T45 sin45° = mg Express the horizontal component of T45: Th = T45 cos45° Because T45 sin45° = T45 cos45°: Th = mg = 100 N 47 • Picture the Problem The acceleration of any object is directly proportional to the net force acting on it. Choose a coordinate system in which the positive x direction is the r same as that of F1 and the positive y direction is to the right. Add the two forces to determine the net force and then use Newton’s 2nd law to find the acceleration of the r r r r object. If F3 brings the system into equilibrium, it must be true that F3 + F1 + F2 = 0. r (a) Find the components of F1 and r F2 : r ˆ F1 = (20 N) i r ˆ F2 = {(−30 N) sin 30°}i + {(30 N) cos 30°} ˆ j ˆ = (−15 N)i + (26 N) ˆ j r r r Add F1 and F2 to find Ftot : r ˆ Ftot = (5 N) i + (26 N) ˆ j Newton’s Laws 221 Apply r r F = ma to find the ∑ acceleration of the object: r r Ftot a= m ˆ = (0.500 m/s 2 ) i + (2.60 m/s 2 ) ˆ j (b) Because the object is in equilibrium under the influence of the three forces, it must be true that: r r r F3 + F1 + F2 = 0 and ( r r r F3 = − F1 + F2 ) (− 5.00 N ) iˆ + (− 26.0 N ) ˆ j = *48 • r r Picture the Problem The acceleration of the object equals the net force, T − mg , divided by the mass. Choose a coordinate system in which upward is the positive y direction. Apply Newton’s 2nd law to the forces acting on this body to find the acceleration of the object as a function of T. (a) Apply ∑F y = ma y to the T – w = T – mg = may object: Solve this equation for a as a function of T: ay = T −g m Substitute numerical values and evaluate ay: ay = 5N − 9.81 m/s 2 = − 8.81 m/s 2 5 kg (b) Proceed as in (a) with T = 10 N: a = − 7.81m/s 2 (c) Proceed as in (a) with T = 100 N: a = 10.2 m/s 2 49 •• Picture the Problem The picture is in equilibrium under the influence of the three forces r shown in the figure. Due to the symmetry of the support system, the vectors T and r T ' have the same magnitude T. Choose a coordinate system in which the positive x direction is to the right and the positive y direction is upward. Apply the condition for translational equilibrium to obtain an expression for T as a function of θ and w. (a) Referring to Figure 4-37, apply the condition for translational equilibrium in the vertical direction and solve for T: ∑F y =2T sin θ − w = 0 and T= w 2 sin θ 222 Chapter 4 Tmin occurs when sinθ is a maximum: Tmax occurs when sinθ is a minimum. Because the function is undefined when sinθ = 0, we can conclude that: (b) Substitute numerical values in the result in (a) and evaluate T: θ = sin −1 1 = 90° T → Tmax as θ → 0° (2 kg )(9.81m/s2 ) = T= 2sin30° 19.6 N Remarks: θ = 90° requires wires of infinite length; therefore it is not possible. As θ gets small, T gets large without limit. *50 ••• Picture the Problem In part (a) we can apply Newton’s 2nd law to obtain the given expression for F. In (b) we can use a symmetry argument to find an expression for tan θ0. In (c) we can use our results obtained in (a) and (b) to express xi and yi. (a) Apply ∑F y = 0 to the balloon: F + Ti sin θ i − Ti −1 sin θ i −1 = 0 Solve for F to obtain: F = Ti −1 sin θ i −1 − Ti sin θ i (b) By symmetry, each support must balance half of the force acting on the entire arch. Therefore, the vertical component of the force on the support must be NF/2. The horizontal component of the tension must be TH. Express tanθ0 in terms of NF/2 and TH: tan θ 0 = NF 2 NF = TH 2TH By symmetry, θN+1 = − θ0. Therefore, because the tangent function is odd: tan θ 0 = − tan θ N +1 = (c) Using TH = Ti cosθi = Ti−1cosθi−1, divide both sides of our result in (a) by TH and simplify to obtain: F Ti −1 sin θ i −1 Ti sin θ i = − TH Ti −1 cos θ i −1 Ti cos θ i Using this result, express tan θ1: NF 2TH = tan θ i −1 − tan θ i F tan θ1 = tan θ 0 − TH Substitute for tan θ0 from (a): tan θ1 = NF F F − = (N − 2) 2TH TH 2TH Newton’s Laws 223 Generalize this result to obtain: tan θ i = (N − 2i ) Express the length of rope between two balloons: l between balloons = Express the horizontal coordinate of the point on the rope where the ith balloon is attached, xi, in terms of xi−1 and the length of rope between two balloons: xi = xi −1 + Sum over all the coordinates to obtain: F 2TH L N +1 L cos θ i −1 N +1 xi = yi = Proceed similarly for the vertical coordinates to obtain: L i −1 ∑ cosθ j N + 1 j =0 L i −1 ∑ sin θ j N + 1 j =0 (d) A spreadsheet program is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell C9 Content/Formula ($B$2-2*B9)/(2*$B$4) D9 SIN(ATAN(C9)) E9 COS(ATAN(C9)) F10 F9+$B$1/($B$2+1)*E9 G10 G9+$B$1/($B$2+1)*D9 1 2 3 4 5 6 7 8 9 10 11 12 13 A L= N= F= TH= B C 10 m 10 1 N 3.72 N I 0 1 2 3 4 Algebraic Form (N − 2i ) F 2TH ( ) cos(tan θ ) sin tan −1 θ i −1 i L cos θ i −1 N +1 L yi −1 + cos θ i −1 N +1 xi −1 + D E F G tan(thetai) sin(thetai) cos(thetai) xi yi 1.344 0.802 0.597 0.000 0.000 1.075 0.732 0.681 0.543 0.729 0.806 0.628 0.778 1.162 1.395 0.538 0.474 0.881 1.869 1.966 0.269 0.260 0.966 2.670 2.396 224 Chapter 4 14 15 16 17 18 19 20 5 6 7 8 9 10 11 0.000 −0.269 −0.538 −0.806 −1.075 −1.344 0.000 −0.260 −0.474 −0.628 −0.732 −0.802 1.000 0.966 0.881 0.778 0.681 0.597 3.548 4.457 5.335 6.136 6.843 7.462 8.005 2.632 2.632 2.396 1.966 1.395 0.729 0.000 (e) A horizontal component of tension 3.72 N gives a spacing of 8 m. At this spacing, the arch is 2.63 m high, tall enough for someone to walk through. 3.0 2.5 yi 2.0 1.5 1.0 0.5 0.0 0 1 2 3 4 5 6 7 8 xi 51 •• Picture the Problem We know, because the speed of the load is changing in parts (a) and (c), that it is accelerating. We also know that, if the load is accelerating in a particular direction, there must be a net force in that direction. A free-body diagram for part (a) is shown to the right. We can apply Newton’s 2nd law of motion to each part of the problem to relate the tension in the cable to the acceleration of the load. Choose the upward direction to be the positive y direction. (a) Apply ∑F y = ma y to the load and solve for T: Substitute numerical values and evaluate T: (b) Because the crane is lifting the T – mg = ma and T = ma y + mg = m(a y + g ) (1) T = (1000 kg )(2 m/s 2 + 9.81 m/s 2 ) = 11.8 kN T = mg = 9.81 kN Newton’s Laws 225 load at constant speed, a = 0: (c) Because the acceleration of the load is downward, a is negative. Apply Fy = ma y to the load: T – mg = may Substitute numerical values in equation (1) and evaluate T: T = (1000 kg)(9.81 m/s2 − 2 m/s2) ∑ = 7.81kN 52 •• Picture the Problem Draw a free-body diagram for each of the depicted situations and use the conditions for translational equilibrium to find the unknown tensions. (a) ΣFx = T1cos60° − 30 N = 0 and T1 = (30 N)/cos60° = 60.0 N ΣFy = T1sin60° − T2 = 0 and T2 = T1sin60° = 52.0 N ∴ m = T2/g = 5.30 kg (b) ΣFx = (80 N)cos60° − T1sin60° = 0 and T1 = (80 N)cos60°/sin60° = 46.2 N ΣFy = (80 N)sin60° − T2 − T1cos60° = 0 T2 = (80 N)sin60° − (46.2 N)cos60° = 46.2 N m = T2/g = 4.71 kg (c) ΣFx = −T1cos60° + T3cos60° = 0 and T1 = T3 ΣFy = 2T1sin60° − mg = 0 and T1 = T3 = (58.9 N)/(2sin60°) = 34.0 N 226 Chapter 4 ∴ m = T1/g = 3.46 kg 53 •• Picture the Problem Construct the freebody diagram for that point in the rope at r which you exert the force F and choose the coordinate system shown on the FBD. We can apply Newton’s 2nd law to the rope to relate the tension to F. (a) Noting that T1 = T2 = T, apply Fy = ma y to the car: 2Tsinθ − F = may = 0 because the car’s acceleration is zero. Solve for and evaluate T: T= 400 N F = = 3.82 kN 2 sin θ 2 sin 3° (b) Proceed as in part (a): T= 600 N = 4.30 kN 2 sin 4° ∑ Free-Body Diagrams: Inclined Planes and the Normal Force *54 • Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it across the frictionless floor. We can apply Newton’s 2nd law of motion to the box to find its acceleration. Apply ∑F x = ma x to the box: F cos θ = ma x Solve for ax: ax = Substitute numerical values and evaluate ax: ax = F cos θ m (250 N )cos35° = 20 kg 10.2 m/s 2 Newton’s Laws 227 55 • Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it up the frictionless incline. We can apply Newton’s 2nd law of motion to the box to determine the smallest force that will move it up the incline at constant speed. Apply ∑F x = ma x to the box as it Fmin cos(40° − θ ) − mg sin θ = 0 moves up the incline with constant speed: Solve for Fmin: Fmin = Substitute numerical values and evaluate Fmin: Fmin = mg sin θ cos (40° − θ ) (20 kg )(9.81m/s2 ) = cos25° 56.0 N 56 • Picture the Problem Forces always occur in equal and opposite pairs. If object A exerts a r r r force, FB , A on object B, an equal but opposite force, F A, B = − FB , A is exerted by object B on object A. The forces acting on the box are its weight, r W, and the normal reaction force of the r inclined plane on the box, Fn . The reaction forces are the forces the box exerts on the inclined plane and the gravitational force the box exerts on the earth. The reaction forces are indicated with primes. 57 • Picture the Problem Because the block whose mass is m is in equilibrium, the sum r r r of the forces Fn , T, and mg must be zero. Construct the free-body diagram for this object, use the coordinate system shown on the free-body diagram, and apply Newton’s 2nd law of motion. Apply ∑F the incline: x = ma x to the block on T – mgsin40° = max = 0 because the system is in equilibrium. 228 Chapter 4 Solve for m: m= The tension must equal the weight of the 3.5-kg block because that block is also in equilibrium: T = (3.5 kg)g and T g sin 40° (3.5 kg) g 3.5 kg = g sin 40° sin 40° m= Because this expression is not included in the list of solution candidates, (d ) is correct. Remarks: Because the object whose mass is m does not hang vertically, its mass must be greater than 3.5 kg. *58 • Picture the Problem The balance(s) indicate the tension in the string(s). Draw free-body diagrams for each of these systems and apply the condition(s) for equilibrium. (a) ∑F y = T − mg = 0 and ( ) T = mg = (10 kg ) 9.81 m/s 2 = 98.1 N (b) ∑F x = T − T '= 0 or, because T ′= mg, T = T ' = mg ( ) = (10 kg ) 9.81m/s 2 = 98.1 N (c) ∑F y = 2T − mg = 0 and T = 1 mg 2 = (d) 1 2 ∑F x (10 kg )(9.81m/s2 ) = 49.1 N = T − mg sin 30° = 0 and T = mg sin 30° ( ) = (10 kg ) 9.81 m/s 2 sin 30° = 49.1 N Newton’s Laws 229 Remarks: Note that (a) and (b) give the same answers … a rather surprising result until one has learned to draw FBDs and apply the conditions for translational equilibrium. 59 •• Picture the Problem Because the box is held in place (is in equilibrium) by the forces acting on it, we know that r r r T + Fn + W = 0 Choose a coordinate system in which the positive x direction is in the direction of r T and the positive y direction is in the r direction of F n . Apply Newton’s 2nd law to the block to obtain expressions for r r T and Fn . (a) Apply ∑F x = ma x to the box: T − mg sin θ = 0 Solve for T: T = mg sin θ Substitute numerical values and evaluate T: T = (50 kg ) 9.81 m/s 2 sin 60° = 425 N Apply ∑F y = ma y to the box: Solve for Fn: Substitute numerical values and evaluate Fn: (b) Using the result for the tension from part (a) to obtain: ( ) Fn − mg cos θ = 0 Fn = mg cos θ Fn = (50 kg )(9.81m/s 2 )cos 60° = 245 N T90° = mg sin 90° = mg and T0° = mg sin 0° = 0 230 Chapter 4 60 •• Picture the Problem Draw a free-body diagram for the box. Choose a coordinate system in which the positive x-axis is parallel to the inclined plane and the positive y-axis is in the direction of the normal force the incline exerts on the block. Apply Newton’s 2nd law of motion to both the x and y directions. (a) Apply ∑F y = ma y to the block: Fn = mg cos 25° + (100 N )sin 25° Solve for Fn: Substitute numerical values and evaluate Fn: (b) Apply Fn − mg cos 25° − (100 N )sin 25° = 0 ∑F x = ma x to the block: Fn = (12 kg ) (9.81 m/s 2 )cos25° + (100 N )sin 25° = 149 N (100 N )cos 25° − mg sin 25° = ma a= Solve for a: Substitute numerical values and evaluate a: (100 N )cos 25° − g sin 25° a= (100 N )cos 25° − (9.81m/s2 )sin 25° m 12 kg = 3.41 m/s 2 *61 •• Picture the Problem The scale reading (the boy’s apparent weight) is the force the scale exerts on the boy. Draw a free-body diagram for the boy, choosing a coordinate system in which the positive x-axis is parallel to and down the inclined plane and the positive y-axis is in the direction of the normal force the incline exerts on the boy. Apply Newton’s 2nd law of motion in the y direction. Apply ∑F y = ma y to the boy to find Fn. Remember that there is no acceleration in the y direction: Fn − W cos 30° = 0 Newton’s Laws 231 Substitute for W to obtain: Fn − mg cos 30° = 0 Solve for Fn: Fn = mg cos 30° Substitute numerical values and evaluate Fn: ( ) Fn = (65 kg ) 9.81 m/s 2 cos 30° = 552 N 62 •• Picture the Problem The free-body diagram for the block sliding up the incline is shown to the right. Applying Newton’s 2nd law to the forces acting in the x direction will lead us to an expression for ax. Using this expression in a constantacceleration equation will allow us to express h as a function of v0 and g. The height h is related to the distance ∆x traveled up the incline: Using a constant-acceleration equation, relate the final speed of the block to its initial speed, acceleration, and distance traveled: Solve for ∆x to obtain: Apply ∑F x = ma x to the block and h = ∆xsin θ 2 v 2 = v0 + 2a x ∆x or, because v = 0, 2 0 = v0 + 2a x ∆x ∆x = 2 − v0 2a x − mg sin θ = ma x solve for its acceleration: and ax = −g sinθ Substitute these results in the equation for h and simplify: 2 ⎛ v0 ⎞ h = ∆x sin θ = ⎜ ⎟ ⎜ 2 g sin θ ⎟ sin θ ⎠ ⎝ = 2 v0 2g which is independent of the ramp’s angle θ. 232 Chapter 4 Free-Body Diagrams: Elevators 63 • Picture the Problem Because the elevator is descending at constant speed, the object r r is in equilibrium and T + mg = 0. Draw a free-body diagram of the object and let the upward direction be the positive y direction. Apply Newton’s 2nd law with a = 0. Because the downward speed is constant, the acceleration is zero. Apply Fy = ma y and solve for ∑ T – mg = 0 ⇒ T = mg and (a ) is correct. T: 64 • Picture the Problem The sketch to the right shows a person standing on a scale in a descending elevator. To its right is a freebody diagram showing the forces acting on the person. The force exerted by the scale r on the person, wapp , is the person’s apparent weight. Because the elevator is slowing down while descending, the acceleration is directed upward. Apply ∑F y = ma y to the person: Solve for wapp: wapp − mg = ma y wapp = mg + ma y > mg The apparent weight will be higher. Because an upward acceleration is required to "slow" a downward velocity, the normal force exerted on you by the scale (your apparent weight ) must be greater than your weight. *65 • Picture the Problem The sketch to the right shows a person standing on a scale in the elevator immediately after the cable breaks. To its right is the free-body diagram showing the forces acting on the person. The force exerted by the scale on r the person, wapp , is the person’s apparent weight. Newton’s Laws 233 r r r r From the free-body diagram we can see that wapp + mg = ma where g is the local r gravitational field and a is the acceleration of the reference frame (elevator). When the r r r r r r elevator goes into free fall ( a = g ), our equation becomes wapp + mg = ma = mg. This r tells us that wapp = 0. (e) is correct. 66 • Picture the Problem The free-body diagram shows the forces acting on the 10kg block as the elevator accelerates upward. Apply Newton’s 2nd law of motion to the block to find the minimum acceleration of the elevator required to break the cord. Apply ∑F y = ma y to the block: T – mg = may Solve for ay to determine the minimum breaking acceleration: ay = T − mg T = −g m m Substitute numerical values and evaluate ay: ay = 150 N − 9.81 m/s 2 = 5.19 m/s 2 10 kg 67 •• Picture the Problem The free-body diagram shows the forces acting on the 2-kg block as the elevator ascends at a constant velocity. Because the acceleration of the elevator is zero, the block is in equilibrium under the influence of r r T and mg. Apply Newton’s 2nd law of motion to the block to determine the scale reading. (a) Apply ∑F y = ma y to the block T − mg = ma y (1) to obtain: For motion with constant velocity, ay = 0: T − mg = 0 and T = mg Substitute numerical values and evaluate T: T = (2 kg ) 9.81m/s 2 = 19.6 N (b) As in part (a), for constant velocity, a = 0: T − mg = ma y ( and ) T = (2 kg ) (9.81 m/s 2 ) = 19.6 N 234 Chapter 4 (c) Solve equation (1) for T and simplify to obtain: Because the elevator is ascending and its speed is increasing, we have ay = 3 m/s2. Substitute numerical values and evaluate T: T = mg + ma y = m(g + a y ) (2) T = (2 kg ) (9.81 m/s 2 + 3m/s 2 ) = 25.6 N (d) For 0 < t < 5 s: ay = 0 and T0→5 s = 19.6 N Using its definition, calculate a for 5 s < t < 9 s: a= Substitute in equation (2) and evaluate T: ∆v 0 − 10 m/s = = −2.5 m/s 2 ∆t 4s ( T5 s→9 s = (2 kg ) 9.81 m/s 2 − 2.5m/s 2 ) = 14.6 N Free-Body Diagrams: Ropes, Tension, and Newton’s Third Law 68 • Picture the Problem Draw a free-body diagram for each object and apply Newton’s 2nd law of motion. Solve the resulting simultaneous equations for the ratio of T1 to T2. Draw the FBD for the box to the left and apply Fx = max : ∑ T1 = m1a1 Draw the FBD for the box to the Fx = ma x : right and apply ∑ T2 − T1 = m2a2 The two boxes have the same acceleration: a1 = a2 Divide the second equation by the first: T1 m = 1 T2 − T1 m2 Newton’s Laws 235 Solve for the ratio T1/T2 : T1 m1 = and ( d ) is correct. T2 m1 + m2 69 •• Picture the Problem Call the common acceleration of the boxes a. Assume that box 1 moves upward, box 2 to the right, and box 3 downward and take this direction to be the positive x direction. Draw free-body diagrams for each of the boxes, apply Newton’s 2nd law of motion, and solve the resulting equations simultaneously. (a) (c) (b) (a) Apply ∑F x = max to the box T1 – w1 = m1a whose mass is m1: Apply ∑F x = max to the box T2 – T1 = m2a whose mass is m2: Noting that T2 = T2' , apply ∑F x w3 – T2 = m3a = ma x to the box whose mass is m3: Add the three equations to obtain: w3 − w1 = (m1 + m2 + m3)a Solve for a: a= Substitute numerical values and evaluate a: a= (m3 − m1 )g m1 + m2 + m3 (2.5 kg − 1.5 kg )(9.81m/s2 ) 1.5 kg + 3.5 kg + 2.5 kg = 1.31m/s 2 (b) Substitute for the acceleration in the equations obtained above to find the tensions: T1 = 16.7 N and T2 = 21.3 N 236 Chapter 4 *70 •• Picture the Problem Choose a coordinate system in which the positive x direction is to the right and the positive y direction is r upward. Let F2,1 be the contact force r exerted by m2 on m1 and F1, 2 be the force exerted by m1 on m2. These forces are equal r r and opposite so F2,1 = − F1, 2 . The freebody diagrams for the blocks are shown to the right. Apply Newton’s 2nd law to each block separately and use the fact that their accelerations are equal. (a) Apply ∑F x = ma x to the first F − F2,1 = m1a1 = m1a block: Apply ∑F x = ma x to the second F1, 2 = m2 a2 = m2 a (1) block: Add these equations to eliminate F2,1 and F1,2 and solve for a = a1 = a2: a= Substitute your value for a into equation (1) and solve for F1,2: F1,2 = (b) Substitute numerical values in the equations derived in part (a) and evaluate a and F1,2: a= F m1 + m2 Fm2 m1 + m2 3.2 N = 0.400 m/s 2 2 kg + 6 kg and F1,2 = (3.2 N )(6 kg ) = 2 kg + 6 kg 2.40 N Remarks: Note that our results for the acceleration are the same as if the force F had acted on a single object whose mass is equal to the sum of the masses of the two blocks. In fact, because the two blocks have the same acceleration, we can consider them to be a single system with mass m1 + m2. Newton’s Laws 237 71 • Picture the Problem Choose a coordinate system in which the positive x direction is to the right and the positive y direction is r upward. Let F2,1 be the contact force r exerted by m2 on m1 and F1, 2 be the force exerted by m1 on m2. These forces are equal r r and opposite so F2,1 = − F1, 2 . The freebody diagrams for the blocks are shown. We can apply Newton’s 2nd law to each block separately and use the fact that their accelerations are equal. (a) Apply ∑F x = ma x to the first F − F1, 2 = m2 a2 = m2 a block: Apply ∑F x = ma x to the second F2,1 = m1a1 = m1a (1) block: Add these equations to eliminate F2,1 and F1,2 and solve for a = a1 = a2: Substitute your value for a into equation (1) and solve for F2,1: (b) Substitute numerical values in the equations derived in part (a) and evaluate a and F2,1: F m1 + m2 a= F2,1 = a= Fm1 m1 + m2 3.2 N = 0.400 m/s 2 2 kg + 6 kg and F2,1 = (3.2 N )(2 kg ) = 2 kg + 6 kg 0.800 N Remarks: Note that our results for the acceleration are the same as if the force F had acted on a single object whose mass is equal to the sum of the masses of the two blocks. In fact, because the two blocks have the same acceleration, we can consider them to be a single system with mass m1 + m2. 72 •• Picture the Problem The free-body diagrams for the boxes and the ropes are below. Because the vertical forces have no bearing on the problem they have not been included. Let the numeral 1 denote the 100-kg box to the left, the numeral 2 the rope connecting the boxes, the numeral 3 the box to the right and the numeral 4 the rope to which the force r r r F is applied. F3, 4 is the tension force exerted by m3 on m4, F4,3 is the tension force r r exerted by m4 on m3, F2,3 is the tension force exerted by m2 on m3, F3, 2 is the tension 238 Chapter 4 r r force exerted by m3 on m2, F1, 2 is the tension force exerted by m1 on m2, and F2,1 is the r tension force exerted by m2 on m1. The equal and opposite pairs of forces are F2,1 = r r r r r − F1, 2 , F3, 2 = − F2,3 , and F4,3 = − F3, 4 . We can apply Newton’s 2nd law to each box and rope separately and use the fact that their accelerations are equal. Apply r r F2,1 = m1a1 = m1a (1) r r F3, 2 − F1, 2 = m2 a2 = m2 a (2) (3) ∑ F = ma to the box whose mass is m1: Apply ∑ F = ma to the rope whose mass is m2: Apply r r F4,3 − F2,3 = m3a3 = m3a r r F − F3, 4 = m4 a4 = m4 a ∑ F = ma to the box whose mass is m3: Apply ∑ F = ma to the rope whose mass is m4: Add these equations to eliminate F2,1, F1,2, F3,2, F2,3, F4,3, and F3,4 and solve for F: F = (m1 + m2 + m3 + m4 )a = (202 kg )(1.0 m/s 2 ) = 202 N ( ) Use equation (1) to find the tension at point A: F2,1 = (100 kg ) 1.0 m/s 2 = 100 N Use equation (2) to find the tension at point B: F3, 2 = F1, 2 + m2 a ( = 100 N + (1 kg ) 1.0 m/s 2 ) = 101 N Use equation (3) to find the tension at point C: F4,3 = F2,3 + m3a ( = 101 N + (100 kg ) 1.0 m/s 2 = 201 N ) Newton’s Laws 239 73 •• Picture the Problem Because the distribution of mass in the rope is uniform, we can express the mass m′ of a length x of the rope in terms of the total mass of the rope M and its length L. We can then express the total mass that the rope must support at a distance x above the block and use Newton’s 2nd law to find the tension as a function of x. Set up a proportion expressing the mass m′ of a length x of the rope as a function of M and L and solve for m′: M m' M x = ⇒ m' = L x L Express the total mass that the rope must support at a distance x above the block: m + m' = m + Apply ∑F y = ma y to the block and a length x of the rope: Solve for T to obtain: M x L M ⎞ ⎛ T − w = T −⎜m + x ⎟g L ⎠ ⎝ M ⎞ ⎛ = ⎜m + x⎟ a L ⎠ ⎝ T= (a + g )⎛ m + M ⎜ ⎝ *74 •• Picture the Problem Choose a coordinate system with the positive y direction upward and denote the top link with the numeral 1, the second with the numeral 2, etc.. The free-body diagrams show the forces acting on links 1 and 2. We can apply Newton’s 2nd law to each link to obtain a system of simultaneous equations that we can solve for the force each link exerts on the link below it. Note that the net force on each link is the product of its mass and acceleration. (a) Apply ∑F y = ma y to the top link and solve for F: F − 5mg = 5ma and F = 5m ( g + a ) ⎞ x⎟ L ⎠ 240 Chapter 4 Substitute numerical values and evaluate F: (b) Apply ∑F y = ma y to a single link: ( F = 5(0.1kg ) 9.81 m/s 2 + 2.5 m/s 2 ) = 6.16 N ( F1 link = m1 link a = (0.1 kg ) 2.5 m/s 2 ) = 0.250 N ∑F y = ma y to the 1st through 5th links to obtain: Add equations (2) through (5) to obtain: Solve for F2 to obtain: Substitute numerical values and evaluate F2: F − F2 − mg = ma , F2 − F3 − mg = ma , (1) F3 − F4 − mg = ma , F4 − F5 − mg = ma , and F5 − mg = ma (c) Apply (3) (2) (4) (5) F2 − 4mg = 4ma F2 = 4mg + 4ma = 4m(a + g ) ( F2 = 4(0.1kg ) 9.81 m/s 2 + 2.5 m/s 2 = 4.92 N Substitute for F2 to find F3, and then substitute for F3 to find F4: F3 = 3.69 N and F4 = 2.46 N Solve equation (5) for F5: F5 = m( g + a ) Substitute numerical values and evaluate F5: ( F5 = (0.1 kg ) 9.81m/s 2 + 2.5 m/s 2 = 1.23 N 75 • Picture the Problem A net force is required to accelerate the object. In this problem the net force is the difference r r r between T and W (= mg ). The free-body diagram of the object is shown to the right. Choose a coordinate system in which the upward direction is positive. Apply r r ∑ F = ma to the object to Fnet = T – W = T – mg obtain: Solve for the tension in the lower portion of the rope: T = Fnet + mg = ma + mg = m(a + g) ) ) Newton’s Laws 241 Using its definition, find the acceleration of the object: a ≡ ∆v/∆t = (3.5 m/s)/(0.7 s) = 5.00 m/s2 Substitute numerical values and evaluate T: T = (40 kg)(5.00 m/s2 + 9.81 m/s2) = 592 N and (a ) is correct. 76 • Picture the Problem A net force in the downward direction is required to accelerate the truck downward. The net r r force is the difference between Wt and T . A free-body diagram showing these forces acting on the truck is shown to the right. Choose a coordinate system in which the downward direction is positive. Apply ∑F y = ma y to the truck to T − mt g = mt a y obtain: Solve for the tension in the lower portion of the cable: Substitute to find the tension in the rope: T = mt g + mt a y = mt (g + a y ) T = mt (g − 0.1g ) = 0.9mt g and (c) is correct. 77 •• Picture the Problem Because the string does not stretch or become slack, the two objects must have the same speed and therefore the magnitude of the acceleration is the same for each object. Choose a coordinate system in which up the incline is the positive x direction for the object of mass m1 and downward is the positive x direction for the object of mass m2. This idealized pulley acts like a piece of polished pipe; i.e., its only function is to change the direction the tension in the massless string acts. Draw a free-body diagram for each of the two objects, apply Newton’s 2nd law of motion to both objects, and solve the resulting equations simultaneously. (a) Draw the FBD for the object of mass m1: Apply ∑F x = max to the object whose mass is m1: T – m1gsinθ = m1a 242 Chapter 4 Draw the FBD for the object of mass m2: Apply ∑F x = max to the object m2g − T = m2a whose mass is m2: Add the two equations and solve for a: a= g (m2 − m1 sin θ ) m1 + m2 Substitute for a in either of the equations containing the tension and solve for T: T= gm1m2 (1 + sin θ ) m1 + m2 (b) Substitute the given values into the expression for a: a = 2.45 m/s 2 Substitute the given data into the expression for T: T = 36.8 N 78 • Picture the Problem The magnitude of the accelerations of Peter and the counterweight are the same. Choose a coordinate system in which up the incline is the positive x direction for the counterweight and downward is the positive x direction for Peter. The pulley changes the direction the tension in the rope acts. Let Peter’s mass be mP. Ignoring the mass of the rope, draw free-body diagrams for the counterweight and Peter, apply Newton’s 2nd law to each of them, and solve the resulting equations simultaneously. (a) Using a constant-acceleration equation, relate Peter’s displacement to her acceleration and descent time: ∆x = v0 ∆t + 1 a (∆t ) 2 2 or, because v0 = 0, ∆x = 1 a (∆t ) 2 2 Solve for the common acceleration of Peter and the counterweight: a= 2∆x (∆t )2 Substitute numerical values and evaluate a: a= 2(3.2 m ) = 1.32 m/s 2 (2.2 s )2 Newton’s Laws 243 Draw the FBD for the counterweight: Apply ∑F x = max to the T – mg sin50° = ma counterweight: Draw the FBD for Peter: Apply ∑F x = max to Peter: mPg – T = mPa Add the two equations and solve for m: m= Substitute numerical values and evaluate m: m= mP ( g − a ) a + g sin 50° (50 kg )(9.81m/s2 − 1.32 m/s2 ) 1.32 m/s 2 + (9.81m/s 2 ) sin50° = 48.0 kg (b) Substitute for m in the force equation for the counterweight and solve for T: T = m(a + g sin 50°) (b) Substitute numerical values and evaluate T: [ T = (48.0 kg ) 1.32 m/s 2 + (9.81m/s 2 ) sin50° = 424 N 79 •• Picture the Problem The magnitude of the accelerations of the two blocks are the same. Choose a coordinate system in which up the incline is the positive x direction for the 8-kg object and downward is the positive x direction for the 10-kg object. The peg changes the direction the tension in the rope acts. Draw free-body diagrams for each object, apply Newton’s 2nd law of motion to both of them, and solve the resulting equations simultaneously. 244 Chapter 4 (a) Draw the FBD for the 3-kg object: Apply ∑F x = ma x to the 3-kg block: T – m8g sin40° = m3a Draw the FBD for the 10-kg object: Apply ∑F x = ma x to the 10-kg block: Add the two equations and solve for and evaluate a: m10g sin50° − T = m10a a= g (m10 sin 50° − m8 sin 40°) m8 + m10 = 1.37 m/s 2 Substitute for a in the first of the two force equations and solve for T: T = m8 g sin 40° + m8 a Substitute numerical values and evaluate T: T = (8 kg ) 9.81 m/s 2 sin 40° [( + 1.37 m/s 2 ) = 61.4 N (b) Because the system is in equilibrium, set a = 0, express the force equations in terms of m1 and m2, add the two force equations, and solve for and evaluate the ratio m1/m2: T – m1g sin40° = 0 m2g sin50°− T = 0 ∴ m2g sin50°– m1g sin40° = 0 and m1 sin 50° = = 1.19 m2 sin 40° Newton’s Laws 245 80 •• Picture the Problem The pictorial representations shown to the right summarize the information given in this problem. While the mass of the rope is distributed over its length, the rope and the 6-kg block have a common acceleration. Choose a coordinate system in which the direction of the 100-N force is the positive x direction. Because the surface is horizontal and frictionless, the only force that influences our solution is the 100-N force. (a) Apply ∑F x = ma x to the 100 N = (m1 + m2)a objects shown for part (a): Solve for a to obtain: a= 100 N m1 + m2 Substitute numerical values and evaluate a: a= 100 N = 10.0 m/s 2 10 kg (b) Let m represent the mass of a length x of the rope. Assuming that the mass of the rope is uniformly distributed along its length: 4 kg m m2 = = x Lrope 5 m and ⎛ 4 kg ⎞ m=⎜ ⎜ 5m ⎟x ⎟ ⎝ ⎠ Let T represent the tension in the rope at a distance x from the point at which it is attached to the 6-kg Fx = ma x to the block. Apply ∑ system shown for part (b) and solve for T: T = (m1 + m)a ⎡ ⎛ 4 kg ⎞ ⎤ 2 = ⎢6 kg + ⎜ ⎜ 5 m ⎟ x ⎥ 10 m/s ⎟ ⎝ ⎠ ⎦ ⎣ ( = 60 N + (8 N/m )x *81 •• Picture the Problem Choose a coordinate system in which upward is the positive y direction and draw the free-body diagram for the frame-plus- painter. Noting that r r F = −T, apply Newton’s 2nd law of motion. (a) Letting mtot = mframe + mpainter, 2T – mtotg = mtota and ) 246 Chapter 4 apply ∑F y = ma y to the frame- T= plus-painter and solve T: Substitute numerical values and evaluate T: T= mtot (a + g ) 2 (75 kg )(0.8 m/s 2 + 9.81m/s 2 ) 2 = 398 N Because F = T: (b) Apply F = 398 N ∑F y = ma y with a = 0 2T – mtotg = 0 to obtain: Solve for T: T = 1 mtot g 2 Substitute numerical values and evaluate T: T= 1 2 (75 kg )(9.81m/s 2 ) = 368 N 82 ••• Picture the Problem Choose a coordinate system in which up the incline is the positive x r r direction and draw free-body diagrams for each block. Noting that a20 = −a10 , apply Newton’s 2nd law of motion to each block and solve the resulting equations simultaneously. Draw a FBD for the 20-kg block: Apply ∑F x = ma x to the block to T – m20gsin20° = m20a20 obtain: Draw a FBD for the 10-kg block. Because all the surfaces, including the surfaces between the blocks, are frictionless, the force the 20-kg block exerts on the 10-kg block must be normal to their surfaces as shown to the right. Apply obtain: ∑F x = ma x to the block to T – m10gsin20° = m10a10 Newton’s Laws 247 Because the blocks are connected by a taut string: a20 = −a10 Substitute for a20 and eliminate T between the two equations to obtain: a10 = 1.12 m/s 2 and a20 = − 1.12 m/s 2 Substitute for either of the accelerations in the force equations and solve for T: T = 44.8 N 83 ••• Picture the Problem Choose a coordinate system in which the positive x direction is to the right and draw free-body diagrams for each block. Because of the pulley, the string exerts a force of 2T. Apply Newton’s 2nd law of motion to both blocks and solve the resulting equations simultaneously. (a) Noting the effect of the pulley, express the distance the 20-kg block moves in a time ∆t: ∆x20 = 1 ∆x5 = 2 (b) Draw a FBD for the 20-kg block: Apply ∑F x = ma x to the block to 2T = m20a20 obtain: Draw a FBD for the 5-kg block: ∑F = ma x to the block to m5g − T = m5a5 Using a constant-acceleration equation, relate the displacement of the 5-kg block to its acceleration ∆x5 = 1 a5 (∆t ) 2 Apply x obtain: 2 1 2 (10 cm) = 5.00 cm 248 Chapter 4 and the time during which it is accelerated: Using a constant-acceleration equation, relate the displacement of the 20-kg block to its acceleration and the time during which it is accelerated: Divide the first of these equations by the second to obtain: ∆x20 = 1 a20 (∆t ) 2 2 1 a (∆t ) ∆x5 a = 2 5 = 5 2 ∆x20 1 a20 (∆t ) a20 2 2 Use the result of part (a) to obtain: a5 = 2a20 Let a20 = a. Then a5 = 2a and the force equations become: 2T = m20a and m5g – T = m5(2a) Eliminate T between the two equations to obtain: a = a20 = Substitute numerical values and evaluate a20 and a5: a20 = and m5 g 2m5 + 1 m20 2 (5 kg )(9.81m/s2 ) = 2(5 kg ) + 1 (20 kg ) 2 ( 2.45 m/s 2 ) a5 = 2 2.45 m/s 2 = 4.91 m/s 2 Substitute for either of the accelerations in either of the force equations and solve for T: T = 24.5 N Free-Body Diagrams: The Atwood’s Machine *84 •• Picture the Problem Assume that m1 > m2. Choose a coordinate system in which the positive y direction is downward for the block whose mass is m1 and upward for the block whose mass is m2 and draw free-body diagrams for each block. Apply Newton’s 2nd law of motion to both blocks and solve the resulting equations simultaneously. Draw a FBD for the block whose mass is m2: Newton’s Laws 249 Apply ∑F y = ma y to this block: T – m2g = m2a2 Draw a FBD for the block whose mass is m1: Apply ∑F y = ma y to this block: Because the blocks are connected by a taut string, let a represent their common acceleration: Add the two force equations to eliminate T and solve for a: m1g – T = m1a1 a = a1 = a2 m1 g − m2 g = m1 a + m2 a and a= Substitute for a in either of the force equations and solve for T: m1 − m2 g m1 + m2 T= 2m1m2 g m1 + m2 85 •• Picture the Problem The acceleration can be found from the given displacement during the first second. The ratio of the two masses can then be found from the acceleration using the first of the two equations derived in Problem 89 relating the acceleration of the Atwood’s machine to its masses. Using a constant-acceleration equation, relate the displacement of the masses to their acceleration and solve for the acceleration: ∆y = v0t + 1 a (∆t ) 2 2 or, because v0 = 0, ∆y = 1 a(∆t ) 2 2 2∆y 2(0.3 m ) = = 0.600 m/s 2 2 2 (1s ) (∆t ) Solve for and evaluate a: a= Solve for m1 in terms of m2 using the first of the two equations given in Problem 84: g + a 10.41 m/s 2 m1 = m2 = m2 = 1.13m2 g − a 9.21m/s 2 Find the second mass for m2 or m1 = 1.2 kg: m2nd mass = 1.36 kg or 1.06 kg 250 Chapter 4 86 •• Picture the Problem Let Fnm be the force the block of mass m2 exerts on the pebble of mass m. Because m2 < m1, the block of mass m2 accelerates upward. Draw a freebody diagram for the pebble and apply Newton’s 2nd law and the acceleration equation given in Problem 84. Apply ∑F y = ma y to the pebble: Solve for Fnm: Fnm – mg = ma Fnm = m(a + g ) m1 − m 2 g m1 + m 2 From Problem 84: a= Substitute for a and simplify to obtain: ⎛ m − m2 ⎞ 2m1m Fnm = m⎜ 1 ⎜ m +m g + g⎟ = m +m g ⎟ 2 1 2 ⎝ 1 ⎠ 87 •• Picture the Problem Note from the free-body diagrams for Problem 89 that the net force exerted by the accelerating blocks is 2T. Use this information, together with the expression for T given in Problem 84, to derive an expression for F = 2T. From Problem 84 we have: The net force, F, exerted by the Atwood’s machine on the hanger is: If m1 = m2 = m, then: If either m1 or m2 = 0, then: T= 2m1m2 g m1 + m2 F = 2T = F= 4m1m2 g m1 + m2 4m 2 g = 2mg … as expected. 2m F = 0 … also as expected. 88 ••• Picture the Problem Use a constant-acceleration equation to relate the displacement of the descending (or rising) mass as a function of its acceleration and then use one of the results from Problem 84 to relate a to g. Differentiation of our expression for g will allow us to relate uncertainty in the time measurement to uncertainty in the measured value for g … and to the values of m2 that would yield an experimental value for g that is good to within 5%. Newton’s Laws 251 m1 + m2 m1 − m2 (a) Use the result given in Problem 84 to express g in terms of a: g=a Using a constant-acceleration equation, express the displacement, L, as a function of t and solve for the acceleration: ∆y = L = v0 ∆t + 1 a (∆t ) 2 Substitute this expression for a: (b) Evaluate dg/dt to obtain: 2 or, because v0 = 0 and ∆t = t, a= 2L t2 (2) 2 L ⎛ m1 + m2 ⎞ ⎜ ⎟ t 2 ⎜ m1 − m2 ⎟ ⎝ ⎠ g= ⎛ m + m2 ⎞ dg = −4 Lt −3 ⎜ 1 ⎜m −m ⎟ ⎟ dt 2 ⎠ ⎝ 1 = Divide both sides of this expression by g and multiply both sides by dt: (1) − 2 ⎡ 2 L ⎤⎛ m1 + m2 ⎞ − 2 g ⎜ ⎟= t ⎢ t 2 ⎥⎜ m1 − m2 ⎟ t ⎣ ⎦⎝ ⎠ dg dt = −2 g t (c) We have: dg dt = ±0.05 and = ±0.025 g t Solve the second of these equations for t to obtain: t= dt 1s = = 4s 0.025 0.025 Substitute in equation (2) to obtain: a= 2(3 m ) = 0.375 m/s 2 2 (4 s ) Solve equation (1) for m2 to obtain: m2 = Evaluate m2 with m1 = 1 kg: 9.81m/s 2 − 0.375 m/s 2 (1kg ) m2 = 9.81 m/s 2 + 0.375 m/s 2 = 0.926 kg Solve equation (1) for m1 to obtain: m1 = m2 Substitute numerical values to obtain: 9.81 m/s2 + 0.375 m/s2 m1 = (0.926 kg ) 9.81 m/s2 − 0.375 m/s2 = 1.08 kg g −a m1 g+a g+a g −a 252 Chapter 4 Because the masses are interchangeable: m2 = 0.926 kg or 1.08 kg *89 •• Picture the Problem We can reason to this conclusion as follows: In the two extreme cases when the mass on one side or the other is zero, the tension is zero as well, because the mass is in free-fall. By symmetry, the maximum tension must occur when the masses on each side are equal. An alternative approach that is shown below is to treat the problem as an extreme-value problem. Express m2 in terms of M and m1: m2 = M − m1 Substitute in the equation from Problem 84 and simplify to obtain: T= Differentiate this expression with respect to m1 and set the derivative equal to zero for extreme values: dT ⎛ 2m ⎞ = 2 g ⎜1 − 1 ⎟ = 0 for extreme values dm1 M ⎠ ⎝ Solve for m1 to obtain: m1 = 1 M 2 Show that m1 = M/2 is a maximum value by evaluating the second derivative of T with respect to m1 at m1 = M/2: d 2T 4g =− < 0, independently of m1 2 dm1 M ⎛ m2 ⎞ 2 gm1 (M − m1 ) = 2 g ⎜ m1 − 1 ⎟ ⎜ m1 + M − m1 M ⎟ ⎝ ⎠ and we have shown that T is a maximum when m1 = m2 = 1 M . 2 Remarks: An alternative solution is to use a graphing calculator to show that T as a function of m1 is concave downward and has its maximum value when m1 = m2 = M/2. 90 ••• Picture the Problem The free-body diagrams show the forces acting on the objects whose masses are m1 and m2. The application of Newton’s 2nd law to these forces and the accelerations the net forces are responsible for will lead us to an expression for the tension in the string as a function of m1 and m2. Examination of this expression as for m2 >> m1 will yield the predicted result. (a) Apply ∑F y = ma y to the objects whose masses are m1 and m2 to obtain: T1 − m1 g = m1a1 and m2 g − T2 = m2 a2 Newton’s Laws 253 Assume that the role of the pulley is simply to change the direction the tension acts. Then T1 = T2 = T. Because the two objects have a common acceleration, let a = a1 = a2. Eliminate a between the two equations and solve for T to obtain: T= 2m1m2 g m1 + m2 Divide the numerator and denominator of this fraction by m2: T= 2m1 g m 1+ 1 m2 Take the limit of this fraction as m2 → ∞ to obtain: T = 2m1 g (b) Imagine the situation when m2 >> m1: Under these conditions, the object whose mass is m2 is essentially in freefall, so the object whose mass is m1 is accelerating upward with an acceleration of magnitude g. Under these conditions, the net force acting on the object whose mass is m1 is m1g and: T – m1g = m1g ⇒ T = 2m1g. Note that this result agrees with that obtained using more analytical methods. General Problems 91 • Picture the Problem Choose a coordinate system in which the force the tree exerts on the woodpecker’s head is in the negative-x direction and determine the acceleration of the woodpecker’s head from Newton’s 2nd law of motion. The depth of penetration, under the assumption of constant acceleration, can be determined using a constant- acceleration equation. Knowing the acceleration of the woodpecker’s head and the depth of penetration of the tree, we can calculate the time required to bring the head to rest. (a) Apply ∑F x = ma x to the woodpecker’s head to obtain: (b) Using a constant-acceleration equation, relate the depth-ofpenetration into the bark to the acceleration of the woodpecker’s head: Solve for and evaluate ∆x: ax = −6N ∑ Fx = = − 100 m/s 2 m 0.060 kg 2 v 2 = v0 + 2a∆x or, because v = 0, 2 0 = v0 + 2a∆x 2 − v0 − (3.5 m/s ) = = 6.13 cm 2a 2 − 100 m/s 2 2 ∆x = ( ) 254 Chapter 4 (c) Use the definition of acceleration to express the time required for the woodpecker’s head to come to rest: ∆t = v − v0 a or, because v = 0, ∆t = Substitute numerical values and evaluate ∆t: v − v0 a ∆t = − v0 − 3.5 m/s = = 35.0 ms a − 100 m/s 2 *92 •• Picture the Problem The free-body diagram shown to the right shows the forces acting on an object suspended from the ceiling of a car that is accelerating to the right. Choose the coordinate system shown and use Newton’s laws of motion and constant- acceleration equations in the determination of the influence of the forces on the behavior of the suspended object. The second free-body diagram shows the forces acting on an object suspended from the ceiling of a car that is braking while it moves to the right. (a) In accordance with Newton' s law of inertia, the object' s displacement will be in the direction opposite that of the acceleration. (b) Resolve the tension, T, into its r r components and apply ∑ F = ma to the object: Take the ratio of these two equations to eliminate T and m: (c) ΣFx = Tsinθ = ma and ΣFy = Tcosθ − mg = 0 T sin θ ma = T cos θ mg or a tanθ = ⇒ a = g tan θ g Because the acceleration is opposite the direction the car is moving, the accelerometer will swing forward. Newton’s Laws 255 Using a constant-acceleration equation, express the velocity of the car in terms of its acceleration and solve for the acceleration: 2 v 2 = v0 + 2a∆x or, because v = 0, 2 0 = v0 + 2a∆x a= 2 − v0 2∆x Substitute numerical values and evaluate a: a= − (50 km/h ) = − 1.61 m/s 2 2(60 m ) Solve the equation derived in (b) for θ: θ = tan −1 ⎜ ⎟ ⎜ ⎟ Substitute numerical values and evaluate θ : ⎛ 1.61m/s 2 ⎞ θ = tan ⎜ ⎜ 9.81 m/s 2 ⎟ = 9.32° ⎟ ⎝ ⎠ Solve for a: 2 ⎛a⎞ ⎝g⎠ −1 93 •• Picture the Problem The free-body diagram shows the forces acting at the top of the mast. Choose the coordinate system shown and use Newton’s 2nd and 3rd laws of motion to analyze the forces acting on the deck of the sailboat. Apply ∑F x = ma x to the top of the TFsinθF − TBsinθB = 0 mast: Find the angles that the forestay and backstay make with the vertical: ⎛ 3.6 m ⎞ ⎟ = 16.7° ⎟ ⎝ 12 m ⎠ θ F = tan −1 ⎜ ⎜ and ⎛ 6.4 m ⎞ ⎟ = 28.1° 12 m ⎟ ⎝ ⎠ θ B = tan −1 ⎜ ⎜ Solve the x-direction equation for TB: TB = TF sin θ F sin 16.7° = (500 N ) sin θ B sin 28.1° = 305 N Find the downward forces that TB and TF exert on the mast: ∑F Solve for Fmast to obtain: Fmast = TF cosθ F + TB cosθ B y = Fmast − TF cos θ F − TB cos θ B = 0 256 Chapter 4 Substitute numerical values and evaluate Fmast: Fmast = (500 N ) cos16.7° + (305 N )cos 28.1° = 748 N The force that the mast exerts on the deck is the sum of its weight and the downward forces exerted on it by the forestay and backstay: Fmast on the deck = 748 N + 800 N = 1.55 kN 94 •• Picture the Problem Let m be the mass of the block and M be the mass of the chain. The free-body diagrams shown below display the forces acting at the locations identified in the problem. We can apply Newton’s 2nd law with ay = 0 to each of the segments of the chain to determine the tensions. (a) (a) Apply (b) ∑F y = ma y to the block (c) Ta − mg = ma y and solve for Ta: or, because ay = 0, Substitute numerical values and evaluate Ta: Ta = (50 kg ) 9.81 m/s 2 = 491 N (b) Apply ∑F y Ta = mg = ma y to the block and half the chain and solve for Tb: ( ) M⎞ ⎛ Tb − ⎜ m + ⎟ g = ma y 2 ⎠ ⎝ or, because ay = 0, M⎞ ⎛ Tb = ⎜ m + ⎟ g 2 ⎠ ⎝ Substitute numerical values and evaluate Tb: (c) Apply ∑F y = ma y to the block and chain and solve for Tc: ( ) Tb = (50 kg + 10 kg ) 9.81 m/s 2 = 589 N Tc − (m + M )g = ma y or, because ay = 0, Tc = (m + M )g Newton’s Laws 257 Substitute numerical values and evaluate Tc: Tc = (50 kg + 20 kg ) (9.81 m/s 2 ) = 687 N *95 ••• Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it across a frictionless floor. Because the force is time-dependent, the acceleration will be, too. We can obtain the acceleration as a function of time from the application of Newton’s 2nd law and then find the velocity of the box as a function of time by integration. Finally, we can derive an expression for the displacement of the box as a function of time by integration of the velocity function. (a) The velocity is related to the acceleration according to: Apply ∑F x = ma x to the box and solve for its acceleration: dv = a(t ) dt F = ma and a= Because the box’s acceleration is a function of time, separate variables in equation (1) and integrate to find v as a function of time: Evaluate v at t = 3 s: (b) Integrate v = dx/dt between 0 and 3 s to find the displacement of the box during this time: (1) F (8 N/s )t = = (1 m/s3 )t 3 m 24 kg t v(t ) = ∫ a(t ')dt ' = ( ) 0 = ( t m/s3 ∫ t ' dt ' 1 3 0 ) t2 = ( ) 2 1 3 m/s3 1 6 m/s3 t 2 v(3 s ) = (1 m/s3 )(3 s ) = 1.50 m/s 6 2 3s ∆x = ∫ v(t ')dt ' = ( )∫ t ' 3s 1 6 m/s 3 2 dt ' 0 0 3s ⎡ t '3 ⎤ = ⎢ 1 m/s3 ⎥ = 1.50 m 6 3 ⎦0 ⎣ ( (c) The average velocity is given by: (d) Use Newton’s 2nd law to express the average force exerted on the box by the man: vave = ) ∆x 1.5 m = = 0.500 m/s ∆t 3s Fav = maav = m ∆v ∆t 258 Chapter 4 Substitute numerical values and evaluate Fav: Fav = (24 kg ) 1.5 m/s − 0 m/s = 12.0 N 3s 96 •• Picture the Problem The application of Newton’s 2nd law to the glider and the hanging weight will lead to simultaneous equations in their common acceleration a and the tension T in the cord that connects them. Once we know the acceleration of this system, we can use a constant-acceleration equation to predict how long it takes the cart to travel r r 1 m from rest. Note that the magnitudes of T and T ' are equal. (a) The free-body diagrams are shown to the right. m1 represents the mass of the cart and m2 the mass of the hanging weight. (b) Apply ∑F x = ma x to the cart and the suspended mass: T − m1 g sin θ = m1a1 and m2 g − T = m2 a2 Letting a represent the common accelerations of the two objects, eliminate T between the two equations and solve a: a= m2 − m1 sin θ g m1 + m2 Substitute numerical values and evaluate a: a= 0.075 kg − (0.270 kg )sin30° 0.075 kg + 0.270 kg ( × 9.81 m/s 2 ) = − 1.71 m/s 2 i.e., the acceleration is down the incline. Substitute for a in either of the force equations to obtain: T = 0.863 N (c) Using a constant-acceleration equation, relate the displacement of the cart down the incline to its initial speed and acceleration: ∆x = v0 ∆t + 1 a (∆t ) 2 Solve for ∆t: Substitute numerical values and evaluate ∆t: 2 or, because v0 = 0, ∆x = 1 a (∆t ) 2 2 ∆t = 2∆x a ∆t = 2(1 m ) = 1.08 s 1.71 m/s 2 Newton’s Laws 259 97 •• Picture the Problem Note that, while the mass of the rope is distributed over its length, the rope and the block have a common acceleration. Because the surface is horizontal r and smooth, the only force that influences our solution is F . The figure misrepresents the situation in that each segment of the rope experiences a gravitational force; the combined effect of which is that the rope must sag. r r (a) Apply a = Fnet / mtot to the ropeblock system to obtain: (b) Apply r r ∑ F = ma to the rope, substitute the acceleration of the system obtained in (a), and simplify to obtain: (c) Apply r r ∑ F = ma to the block, substitute the acceleration of the system obtained in (a), and simplify to obtain: a= F m1 + m2 ⎛ F ⎞ Fnet = m2 a = m2 ⎜ ⎜m +m ⎟ ⎟ 2 ⎠ ⎝ 1 m2 F = m1 + m2 ⎛ F ⎞ T = m1a = m1 ⎜ ⎜m +m ⎟ ⎟ 2 ⎠ ⎝ 1 m1 F = m1 + m2 r (d) The rope sags and so F has both vertical and horizontal components; with its horizontal component being r less than F . Consequently, a will be somewhat smaller. *98 •• Picture the Problem The free-body diagram shows the forces acting on the block. Choose the coordinate system shown on the diagram. Because the surface of the wedge is frictionless, the force it exerts on the block must be normal to its surface. (a) Apply to obtain: ∑F y = ma y to the block Fn sin 30° − w = ma y or, because ay = 0 and w = mg, Fn sin 30° − mg = 0 or 260 Chapter 4 Fn sin 30° = mg Apply ∑F x = ma x to the block: (1) Fn cos 30° = max (2) Divide equation (2) by equation (1) to obtain: ax = cot 30° g Solve for and evaluate ax: a x = g cot 30° = 9.81 m/s 2 cot 30° ( ) = 17.0 m/s 2 (b) An acceleration of the wedge greater than g cot30° would require that the normal force exerted on the body by the wedge be greater than that given in part (a); i.e., Fn > mg/sin30°. Under this condition, there would be a net force in the y direction and the block would accelerate up the wedge. 99 •• Picture the Problem Because the system is initially in equilibrium, it follows that T0 = 5mg. When one washer is removed on the left side, the remaining washers will accelerate upward (and those on the right side downward) in response to the net force that results. The free-body diagrams show the forces under this unbalanced condition. Applying Newton’s 2nd law to each collection of washers will allow us to determine both the acceleration of the system and the mass of a single washer. (a) Apply ∑F y = ma y to the rising T − 4mg = (4m )a (1) 5mg − T = (5m )a (2) masses: Apply ∑F y = ma y to the descending masses: Eliminate T between these equations to obtain: a=1g 9 Use this acceleration in equation (1) or equation (2) to obtain: T= Express the difference between T0 and T and solve for m: T0 − T = 5mg − and 40 mg 9 40 mg = 0.3 N 9 Newton’s Laws 261 m = 0.0550 kg = 55.0 g (b) Proceed as in (a) to obtain: T – 3mg = 3ma and 5mg – T = 5ma Eliminate T and solve for a: a=1g= 4 Eliminate a in either of the motion equations and solve for T to obtain: T= 15 mg 4 Substitute numerical values and evaluate T: T= 15 (0.0550 kg ) 9.81m/s2 4 1 4 (9.81m/s ) = 2 2.45 m/s 2 ( = 2.03 N 100 •• Picture the Problem The free-body diagram represents the Atwood’s machine with N washers moved from the left side to the right side. Application of Newton’s 2nd law to each collection of washers will result in two equations that can be solved simultaneously to relate N, a, and g. The acceleration can then be found from the given data. Apply ∑F y = ma y to the rising T – (5 – N)mg = (5 – N)ma = ma y to the (5 +N)mg – T = (5 + N)ma washers: Apply ∑F y descending washers: Add these equations to eliminate T: (5 + N )mg − (5 − N )mg = (5 − N )ma + (5 + N )ma Simplify to obtain: 2 Nmg = 10ma Solve for N: N = 5a/g Using a constant-acceleration equation, relate the distance the washers fell to their time of fall: ∆y = v0 ∆t + 1 a(∆t ) 2 2 or, because v0 = 0, ∆y = 1 a(∆t ) 2 2 ) 262 Chapter 4 Solve for the acceleration: a= 2∆y (∆t )2 Substitute numerical values and evaluate a: a= 2(0.471 m ) = 5.89 m/s 2 2 (0.40 s ) Substitute in the expression for N: ⎛ 5.89 m/s 2 ⎞ N = 5⎜ ⎜ 9.81 m/s 2 ⎟ = 3 ⎟ ⎝ ⎠ 101 •• Picture the Problem Draw the free-body diagram for the block of mass m and apply Newton’s 2nd law to obtain the acceleration of the system and then the tension in the rope connecting the two blocks. (a) Letting T be the tension in the connecting string, apply Fx = ma x to the block of T – F1 = ma ∑ mass m: Apply ∑F x = ma x to both blocks F2 – F1 = (m + 2m)a = (3m)a to determine the acceleration of the system: Substitute and solve for a: a = (F2 – F1)/3m Substitute for a in the first equation and solve for T: T= (b) Substitute for F1 and F2 in the equation derived in part (a): T = (2Ct +2Ct)/3 = 4Ct/3 Evaluate this expression for T = T0 and t = t0 and solve for t0: t0 = 1 3 (F2 + 2F1 ) 3T0 4C Newton’s Laws 263 *102 ••• Picture the Problem Because a constantupward acceleration has the same effect as an increase in the acceleration due to gravity, we can use the result of Problem 89 (for the tension) with a replaced by a + g. The application of Newton’s 2nd law to the object whose mass is m2 will connect the acceleration of this body to tension from Problem 84. In Problem 84 it is given that, when the support pulley is not accelerating, the tension in the rope and the acceleration of the masses are related according to: T= 2m1m2 g m1 + m2 Replace a with a + g: T= 2m1m2 (a + g ) m1 + m2 Apply ∑F y = ma y to the object whose mass is m2 and solve for a2: Substitute for T and simplify to obtain: The expression for a1 is the same as for a2 with all subscripts interchanged (note that a positive value for a1 represents acceleration upward): T – m2g = m2a2 and a2 = a2 = a1 = T − m2 g m2 (m1 − m2 )g + 2m1a m1 + m2 (m2 − m1 )g + 2m2a m1 + m2 264 Chapter 4 Chapter 5 Applications of Newton’s Laws Conceptual Problems 1 • Determine the Concept Because the objects are speeding up (accelerating), there must be a net force acting on them. The forces acting on an object are the normal force exerted by the floor of the truck, the weight of the object, and the friction force; also exerted by the floor of the truck. The force of friction between the object and the floor of the truck Of these forces, the only one that acts in the direction of the acceleration (chosen to be to the right in the free-body diagram) is the friction force. must be the force that causes the object to accelerate. *2 • Determine the Concept The forces acting on an object are the normal force exerted by the floor of the truck, the weight of the object, and the friction force; also exerted by the floor of the truck. Of these forces, the only one that acts in the direction of the acceleration (chosen to be to the right in the free-body diagram) is the friction force. Apply Newton’s 2nd law to the object to determine how the critical acceleration depends on its weight. Taking the positive x direction to be to the right, apply ΣFx = max and solve for ax: f = µsw = µsmg = max and ax = µsg Because ax is independent of m and w, the critical accelerations are the same. 265 266 Chapter 5 3 • Determine the Concept The forces acting r on the block are the normal force Fn exerted by the incline, the weight of the r block mg exerted by the earth, and the r static friction force f s exerted by an external agent. We can use the definition of µs and the conditions for equilibrium to determine the relationship between µs and θ. Apply ∑F = ma x to the block: fs − mgsinθ = 0 (1) Apply ∑F = ma y in the y Fn − mgcosθ = 0 (2) x y direction: Divide equation (1) by equation (2) to obtain: tan θ = Substitute for fs (≤ µsFn): tan θ ≤ fs Fn µs Fn Fn = µs and (d ) is correct. *4 • Determine the Concept The block is in r r equilibrium under the influence of Fn , mg, r and f s ; i.e., r r r Fn + mg + f s = 0 We can apply Newton’s 2nd law in the x direction to determine the relationship between fs and mg. Apply ∑F x Solve for fs: = 0 to the block: fs − mgsinθ = 0 fs = mgsinθ and (d ) is correct. Applications of Newton’s Laws 267 5 •• Picture the Problem The forces acting on the car as it rounds a curve of radius R at maximum speed are shown on the free-body diagram to the right. The centripetal force is the static friction force exerted by the roadway on the tires. We can apply Newton’s 2nd law to the car to derive an expression for its maximum speed and then compare the speeds under the two friction conditions described. Apply r r ∑ F = ma to the car: ∑ Fx = fs, max = m 2 vmax R and ∑F y = Fn − mg = 0 From the y equation we have: Fn = mg Express fs,max in terms of Fn in the x equation and solve for vmax: vmax = µs gR or vmax = constant µs ' Express v'max for µs = 1 µs : 2 v'max = constant µs 2 and (b) is correct. *6 •• Picture the Problem The normal reaction force Fn provides the centripetal force and the force of static friction, µsFn, keeps the cycle from sliding down the wall. We can apply Newton’s 2nd law and the definition of fs,max to derive an expression for vmin. Apply r r ∑ F = ma to the motorcycle: ∑ Fx = Fn = m and v2 R = .707vmax ≈ 71%vmax 268 Chapter 5 ∑F For the minimum speed: Substitute for fs, eliminate Fn between the force equations, and solve for vmin: Assume that R = 6 m and µs = 0.8 and solve for vmin: y = f s − mg = 0 fs = fs,max = µsFn vmin = vmin = Rg µs (6 m )(9.81m/s2 ) 0.8 = 8.58 m/s = 30.9 km/h 7 •• Determine the Concept As the spring is extended, the force exerted by the spring on the block increases. Once that force is greater than the maximum value of the force of static friction on the block, the block will begin to move. However, as it accelerates, it will shorten the length of the spring, decreasing the force that the spring exerts on the block. As this happens, the force of kinetic friction can then slow the block to a stop, which starts the cycle over again. One interesting application of this to the real world is the bowing of a violin string: The string under tension acts like the spring, while the bow acts as the block, so as the bow is dragged across the string, the string periodically sticks and frees itself from the bow. 8 • True. The velocity of an object moving in a circle is continually changing independently of whether the object’s speed is changing. The change in the velocity vector and the acceleration vector and the net force acting on the object all point toward the center of circle. This center-pointing force is called a centripetal force. 9 • Determine the Concept A particle traveling in a vertical circle experiences a downward gravitational force plus an additional force that constrains it to move along a circular path. Because the net force acting on the particle will vary with location along its trajectory, neither (b), (c), nor (d) can be correct. Because the velocity of a particle moving along a circular path is continually changing, (a) cannot be correct. (e) is correct. *10 • Determine the Concept We can analyze these demonstrations by drawing force diagrams for each situation. In both diagrams, h denotes ″hand″, g denotes ″gravitational″, m denotes ″magnetic″, and n denotes ″normal″. Applications of Newton’s Laws 269 Demonstration 2: (a) Demonstration 1: (b) Because the magnet doesn’t lift the iron in the first demonstration, the force exerted on the iron must be less than its (the iron’s) weight. This is still true when the two are falling, but the motion of the iron is not restrained by the table, and the motion of the magnet is not restrained by the hand. Looking at the second diagram, the net force pulling the magnet down is greater than its weight, implying that its acceleration is greater than g. The opposite is true for the iron: the magnetic force acts upwards, slowing it down, so its acceleration will be less than g. Because of this, the magnet will catch up to the iron piece as they fall. *11 ••• Picture the Problem The free-body diagrams show the forces acting on the two objects some time after block 2 is dropped. r r Note that, while T1 ≠ T2 , T1 = T2. The only force pulling block 2 to the left is the horizontal component of the tension. Because this force is smaller than the magnitude of the tension, the acceleration of block 1, which is identical to block 2, to the right (T1 = T2) will always be greater than the acceleration of block 2 to the left. Because the initial distance from block 1 to the pulley is the same as the initial distance of block 2 to the wall, block 1 will hit the pulley before block 2 hits the wall. 12 • 1n True. The terminal speed of an object is given by vt = (mg b ) , where b depends on the shape and area of the falling object as well as upon the properties of the medium in which the object is falling. 13 • 1n Determine the Concept The terminal speed of a sky diver is given by vt = (mg b ) , where b depends on the shape and area of the falling object as well as upon the properties of the medium in which the object is falling. The sky diver’s orientation as she falls 270 Chapter 5 determines the surface area she presents to the air molecules that must be pushed aside. (d ) is correct. 14 •• Determine the Concept In your frame of reference (the accelerating reference frame of the car), the direction of the force must point toward the center of the circular path along which you are traveling; that is, in the direction of the centripetal force that keeps you moving in a circle. The friction between you and the seat you are sitting on supplies this force. The reason you seem to be "pushed" to the outside of the curve is that your body’s inertia "wants" , in accordance with Newton’s law of inertia, to keep it moving in a straight line–that is, tangent to the curve. *15 • Determine the Concept The centripetal force that keeps the moon in its orbit around the earth is provided by the gravitational force the earth exerts on the moon. As described by Newton’s 3rd law, this force is equal in magnitude to the force the moon exerts on the earth. (d ) is correct. 16 • Determine the Concept The only forces acting on the block are its weight and the force the surface exerts on it. Because the loop-the-loop surface is frictionless, the force it exerts on the block must be perpendicular to its surface. Point A: the weight is downward and the normal force is to the right. Free-body diagram 3 Point B: the weight is downward, the normal force is upward, and the normal force is greater than the weight so that their difference is the centripetal force. Free-body diagram 4 Point C: the weight is downward and the normal force is to the left. Free-body diagram 5 Point D: both the weight and the normal forces are downward. Free-body diagram 2 Applications of Newton’s Laws 271 17 •• Picture the Problem Assume that the drag force on an object is given by the Newtonian formula FD = 1 CAρv 2 , where A is the projected surface area, v is the object’s speed, ρ 2 is the density of air, and C a dimensionless coefficient. Express the net force acting on the falling object: Fnet = mg − FD = ma Substitute for FD under terminal speed conditions and solve for the terminal speed: 2 mg − 1 CAρvT = 0 2 or vT = 2mg CAρ Thus, the terminal velocity depends on the ratio of the mass of the object to its surface area. For a rock, which has a relatively small surface area compared to its mass, the terminal speed will be relatively high; for a lightweight, spread-out object like a feather, the opposite is true. Another issue is that the higher the terminal velocity is, the longer it takes for a falling object to reach terminal velocity. From this, the feather will reach its terminal velocity quickly, and fall at an almost constant speed very soon after being dropped; a rock, if not dropped from a great height, will have almost the same acceleration as if it were in freefall for the duration of its fall, and thus be continually speeding up as it falls. An interesting point is that the average drag force acting on the rock will be larger than that acting on the feather precisely because the rock’s average speed is larger than the feather's, as the drag force increases as v2. This is another reminder that force is not the same thing as acceleration. Estimation and Approximation *18 • Picture the Problem The free-body diagram shows the forces on the Tercel as it slows from 60 to 55 mph. We can use Newton’s 2nd law to calculate the average force from the rate at which the car’s speed decreases and the rolling force from its definition. The drag force can be inferred from the average and rolling friction forces and the drag coefficient from the defining equation for the drag force. (a) Apply ∑F x = max to the car to relate the average force acting on it to its average velocity: Fav = maav = m ∆v ∆t 272 Chapter 5 Substitute numerical values and evaluate Fav: Fav = (1020 kg ) 5 mi km 1h 1000 m × 1.609 × × h mi 3600 s km = 581 N 3.92 s (b) Using its definition, express and evaluate the force of rolling friction: f rolling = µ rolling Fn = µ rolling mg ( = (0.02 )(1020 kg ) 9.81 m/s 2 ) = 200 N Assuming that only two forces are acting on the car in the direction of its motion, express their relationship and solve for and evaluate the drag force: (c) Convert 57.5 mi/h to m/s: Fav = Fdrag + Frolling and Fdrag = Fav − Frolling = 581 N − 200 N = 381 N 57.5 mi mi 1.609 km = 57.5 × h h mi 1h 103 m × × 3600 s km = 25.7 m/s Using the definition of the drag force and its calculated value from (b) and the average speed of the car during this 5 mph interval, solve for C: Substitute numerical values and evaluate C: Fdrag = 1 Cρ Av 2 ⇒ C = 2 C= 2 Fdrag ρ Av 2 2(381 N ) 2 1.21 kg/m 1.91 m 2 (25.7 m/s ) ( 3 )( ) = 0.499 19 • Picture the Problem We can use the dimensions of force and velocity to determine the dimensions of the constant b and the dimensions of ρ, r, and v to show that, for n = 2, Newton’s expression is consistent dimensionally with our result from part (b). In parts (d) and (e), we can apply Newton’s 2nd law under terminal velocity conditions to find the terminal velocity of the sky diver near the surface of the earth and at a height of 8 km. (a) Solve the drag force equation for b with n = 1: b= Fd v Applications of Newton’s Laws 273 Substitute the dimensions of Fd and v and simplify to obtain: ML 2 [b] = TL = M T T and the units of b are kg/s Fd v2 (b) Solve the drag force equation for b with n = 2: b= Substitute the dimensions of Fd and v and simplify to obtain: ML 2 [b] = T 2 = M L ⎛L⎞ ⎜ ⎟ ⎝T ⎠ and the units of b are kg/m (c) Express the dimensions of Newton’s expression: [Fd ] = [1 ρπr 2v 2 ] = ⎛ M ⎞(L )2 ⎛ L ⎞ ⎜ 3⎟ ⎜ ⎟ 2 ⎝L ⎠ = From part (b) we have: 2 ⎝T ⎠ ML T2 [Fd ] = [bv 2 ] = ⎛ M ⎞⎛ L ⎞ ⎜ ⎟⎜ ⎟ 2 ⎝ L ⎠⎝ T ⎠ = (d) Letting the downward direction be the positive y direction, apply Fy = ma y to the sky diver: ML T2 mg − 1 ρπr 2 vt2 = 0 2 ∑ Solve for and evaluate vt: vt = 2mg = ρπ r 2 ( ( ) = 56.9 m/s (e) Evaluate vt at a height of 8 km: vt = ( ) 2(56 kg ) 9.81m/s 2 2 π 0.514 kg/m 3 (0.3 m ) ( = 86.9 m/s ) 2(56 kg ) 9.81 m/s 2 2 π 1.2 kg/m 3 (0.3 m ) ) 274 Chapter 5 20 •• Picture the Problem From Newton’s 2nd law, the equation describing the motion of falling raindrops and large hailstones is mg – Fd = ma where Fd = 1 ρπ r 2v 2 = bv 2 is the 2 drag force. Under terminal speed conditions (a = 0), the drag force is equal to the weight of the falling object. Take the radius of a raindrop rr to be 0.5 mm and the radius of a golf-ball sized hailstone rh to be 2 cm. Using b = 1 πρ r 2 , evaluate br and bh: 2 ( )( br = 1 π 1.2 kg/m 3 0.5 × 10 −3 m 2 ) 2 = 4.71 × 10 −7 kg/m and ( )( bh = 1 π 1.2 kg/m 3 2 × 10 −2 m 2 ) 2 = 7.54 × 10 −4 kg/m 4π r 3 ρ 3 Express the mass of a sphere in terms of its volume and density: m = ρV = Using ρr = 103 kg/m3 and ρh = 920 kg/m3, evaluate mr and mh: 4π (0.5 × 10 −3 m ) (103 kg/m 3 ) mr = 3 −7 = 5.24 × 10 kg 3 and 4π (2 × 10 −2 m ) (920 kg/m 3 ) 3 −2 = 3.08 × 10 kg 3 mh = Express the relationship between vt and the weight of a falling object under terminal speed conditions and solve for vt: Use numerical values to evaluate vt,r and vt,h: mg b bvt2 = mg ⇒ vt = vt,r = (5.24 ×10 −7 )( ) )( ) kg 9.81 m/s 2 4.71× 10 −7 kg/m = 3.30 m/s and vt,h = (3.08 ×10 −2 kg 9.81 m/s 2 7.54 × 10 −4 kg/m = 20.0 m/s Applications of Newton’s Laws 275 Friction *21 • Picture the Problem The block is in r equilibrium under the influence of Fn , r r mg, and f k ; i.e., r r r Fn + mg + f k = 0 We can apply Newton’s 2nd law to determine the relationship between fk, θ, and mg. Using its definition, express the coefficient of kinetic friction: Apply ∑F x µk = = max to the block: ∑F y (1) fk − mgsinθ = max = 0 because ax = 0 fk = mgsinθ Solve for fk: Apply fk Fn = ma y to the block: Fn − mgcosθ = may = 0 because ay = 0 Solve for Fn: Fn = mgcosθ Substitute in equation (1) to obtain: µk = mg sin θ = tan θ mg cos θ and (b) is correct. 22 • Picture the Problem The block is in r equilibrium under the influence of Fn , r r r mg, Fapp , and f k ; i.e., r r r r Fn + mg + Fapp + f k = 0 We can apply Newton’s 2nd law to determine fk. Apply ∑F Solve for fk: x = max to the block: Fapp − fk = max = 0 because ax = 0 fk = Fapp = 20 N and (e) is correct. 276 Chapter 5 *23 • Picture the Problem Whether the friction force is that due to static friction or kinetic friction depends on whether the applied tension is greater than the maximum static friction force. We can apply the definition of the maximum static friction to decide whether fs,max or T is greater. Calculate the maximum static friction force: fs,max = µsFn = µsw = (0.8)(20 N) = 16 N (a) Because fs,max > T: f = fs = T = 15.0 N (b) Because T > fs,max: f = fk = µkw = (0.6)(20 N) = 12.0 N 24 • Picture the Problem The block is in equilibrium under the influence of the r r r forces T, f k , and mg; i.e., r r r T + f k + mg = 0 We can apply Newton’s 2nd law to determine the relationship between T and fk . Apply ∑F x = max to the block: Solve for fk: T cosθ − fk = max = 0 because ax = 0 fk = T cosθ and (b) is correct. 25 • Picture the Problem Whether the friction force is that due to static friction or kinetic friction depends on whether the applied tension is greater than the maximum static friction force. Calculate the maximum static fs,max = µsFn = µsw Applications of Newton’s Laws 277 = (0.6)(100 kg)(9.81 m/s2) = 589 N friction force: Because fs,max > Fapp, the box does not move and : Fapp = f s = 500 N 26 • Picture the Problem Because the box is moving with constant velocity, its acceleration is zero and it is in equilibrium r r r under the influence of Fapp , Fn , w , and r f ; i.e., r r r r Fapp + Fn + w + f = 0 We can apply Newton’s 2nd law to determine the relationship between f and mg. The definition of µk is: Apply ∑F y = ma y to the box: Solve for Fn: Apply ∑F x µk = fk Fn Fn – w = may = 0 because ay = 0 Fn = w = 600 N = max to the box: ΣFx = Fapp – f = max = 0 because ax = 0 Solve for fk: Fapp = fk = 250 N Substitute to obtain µk: µk = (250 N)/(600 N) = 0.417 27 • Picture the Problem Assume that the car is traveling to the right and let the positive x direction also be to the right. We can use Newton’s 2nd law of motion and the definition of µs to determine the maximum acceleration of the car. Once we know the car’s maximum acceleration, we can use a constant-acceleration equation to determine the least stopping distance. 278 Chapter 5 (a) Apply Apply ∑F x ∑F y = max to the car: = ma y to the car and solve for Fn: Substitute (2) in (1) and solve for ax,max: (b) Using a constant-acceleration equation, relate the stopping distance of the car to its initial velocity and its acceleration and solve for its displacement: Substitute numerical values and evaluate ∆x: −fs,max = −µsFn = max (1) Fn − w = may = 0 or, because ay = 0, Fn = mg (2) ax,max = µs g = (0.6)(9.81 m/s 2 ) = − 5.89 m/s 2 2 v 2 = v0 + 2a∆x or, because v = 0, ∆x = 2 − v0 2a ∆x = − (30 m/s ) = 76.4 m 2 − 5.89 m/s 2 2 ( ) *28 • Picture the Problem The free-body diagram shows the forces acting on the drive wheels, the ones we’re assuming support half the weight of the car. We can use the definition of acceleration and apply Newton’s 2nd law to the horizontal and vertical components of the forces to determine the minimum coefficient of friction between the road and the tires. (a) Because µs > µ k , f will be greater if the wheels do not slip. (b) Apply Apply ∑F x ∑F y = max to the car: = ma y to the car and fs = µsFn = max (1) Fn − 1 mg = ma y 2 solve for Fn: Because ay = 0, Fn − 1 mg = 0 ⇒ Fn = 1 mg 2 2 Find the acceleration of the car: ax = ∆v (90 km/h )(1000 m/km ) = 12 s ∆t = 2.08 m/s 2 Applications of Newton’s Laws 279 Solve equation (1) for µs: µs = ma x 2a x = 1 g 2 mg Substitute numerical values and evaluate ax: µs = 2(2.08 m/s 2 ) = 0.424 9.81 m/s 2 29 • Picture the Problem The block is in equilibrium under the influence of the forces shown on the free-body diagram. We can use Newton’s 2nd law and the definition of µs to solve for fs and Fn. (a) Apply ∑F y = ma y to the block and solve for fs: Solve for and evaluate fs: f s − mg = ma y or, because ay = 0, f s − mg = 0 f s = mg = (5 kg )(9.81 m/s 2 ) = 49.1 N (b) Use the definition of µs to express Fn: Fn = Substitute numerical values and evaluate Fn: Fn = 30 • Picture the Problem The free-body diagram shows the forces acting on the book. The normal force is the net force the student exerts in squeezing the book. Let the horizontal direction be the x direction and upward the y direction. Note that the normal force is the same on either side of the book because it is not accelerating in the horizontal direction. The book could be accelerating downward. We can apply Newton’s 2nd law to relate the minimum force required to hold the book in place to its mass and to the coefficients of static friction. In part (b), we can proceed similarly to relate the acceleration of the f s,max µs 49.1 N = 123 N 0.4 280 Chapter 5 book to the coefficients of kinetic friction. (a) Apply r r ∑ F = ma to the book: ∑F = F2,min − F1,min = 0 x and ∑F = µ s ,1 F1',min + µ s,2 F2',min − mg = 0 Fmin = mg µ,1 + µs,2 y Noting that F1',min = F2' ,min , solve the y equation for Fmin: Substitute numerical values and evaluate Fmin: (b) Apply ∑F y = ma y with the Fmin = (10.2 kg )(9.81m/s2 ) = ∑F = µ k ,1 F + µ k,2 F − mg = ma y 0.32 + 0.16 208 N book accelerating downward, to obtain: Solve for a to obtain: Substitute numerical values and evaluate a: a= µk , + µk ,2 m a = F−g 0.2 + 0.09 (195 N ) − 9.81 m/s 2 10.2 kg = − 4.27 m/s 2 31 • Picture the Problem A free-body diagram showing the forces acting on the car is shown to the right. The friction force that the ground exerts on the tires is the force fs shown acting up the incline. We can use the definition of the coefficient of static friction and Newton’s 2nd law to relate the angle of the incline to the forces acting on the car. Apply r r ∑ F = ma to the car: ∑F x = f s − mg sin θ = 0 (1) = Fn − mg cos θ = 0 (2) and ∑F y Solve equation (1) for fs and equation (2) for Fn: f s = mg sin θ and Applications of Newton’s Laws 281 Fn = mg cos θ f s mg sin θ = = tan θ Fn mg cos θ Use the definition of µs to relate fs and Fn: µs = Solve for and evaluate θ : θ = tan −1 (µs ) = tan −1 (0.08) = 4.57° *32 • Picture the Problem The free-body diagrams for the two methods are shown to the right. Method 1 results in the box being pushed into the floor, increasing the normal force and the static friction force. Method 2 partially lifts the box,, reducing the normal force and the static friction force. We can apply Newton’s 2nd law to obtain expressions that relate the maximum static r friction force to the applied force F . (a) Method 2 is preferable as it reduces Fn and, therefore, f s. (b) Apply ∑F x = max to the box: ∑F = ma y to F cosθ − fs = Fcosθ − µsFn = 0 the block and solve for Fn: Fn – mg − Fsinθ = 0 ∴ Fn = mg + Fsinθ Relate fs,max to Fn: fs,max = µsFn = µs(mg + Fsinθ) Method 1: Apply y ∑F = ma y to (1) the forces in the y direction and solve for Fn: Fn – mg + Fsinθ = 0 and Fn = mg − Fsinθ Relate fs,max to Fn: fs,max = µsFn = µs(mg − Fsinθ) (2) Express the condition that must be satisfied to move the box by either method: fs,max = Fcosθ (3) Method 1: Substitute (1) in (3) and solve for F: F1 = Method 2: Apply y µ s mg cosθ − µ s sin θ (4) 282 Chapter 5 µ s mg cosθ + µ s sin θ Method 2: Substitute (2) in (3) and solve for F: F2 = Evaluate equations (4) and (5) with θ = 30°: F1 (30°) = 520 N Evaluate (4) and (5) with θ = 0°: F1 (0°) = F2 (0°) = µ s mg = 294 N (5) F2 (30°) = 252 N 33 • Picture the Problem Draw a free-body diagram for each object. In the absence of friction, the 3-kg box will move to the right, and the 2-kg box will move down. r The friction force is indicated by f without r r subscript; it is f s for (a) and f k for (b). For values of µs less than the value found in part (a) required for equilibrium, the system will accelerate and the fall time for a given distance can be found using a constantacceleration equation. (a) Apply ∑F x = max to the 3-kg T – fs = 0 because ax = 0 (1) Fn,3 – m3g = 0 because ay = 0 and T – µs m3g = 0 (2) m2g – T = 0 because ax = 0 (3) box: Apply ∑F y = ma y to the 3-kg box, solve for Fn,3, and substitute in (1): Apply ∑F x = max to the 2-kg box: Solve (2) and (3) simultaneously and solve for µs: (b) The time of fall is related to the acceleration, which is constant: µs = m2 = 0.667 m3 ∆x = v0 ∆t + 1 a(∆t ) 2 2 or, because v0 =0, ∆x = 1 a(∆t ) 2 2 Solve for ∆t: ∆t = 2 ∆x a (4) Applications of Newton’s Laws 283 Apply ∑F x = max to each box: Add equations (5) and (6) and solve for a: Substitute numerical values and evaluate a: T – µk m3g = m3a and m2g – T = m2a a= (5) (6) (m2 − µk m3 )g m2 + m3 [2 kg − 0.3(3 kg )](9.81m/s2 ) a= 2 kg + 3 kg = 2.16 m/s 2 Substitute numerical values in equation (4) and evaluate ∆t: ∆t = 2(2 m ) = 1.36 s 2.16 m/s 2 34 •• Picture the Problem The application of Newton’s 2nd law to the block will allow us to express the coefficient of kinetic friction in terms of the acceleration of the block. We can then use a constant-acceleration equation to determine the block’s acceleration. The pictorial representation summarizes what we know about the motion. A free-body diagram showing the forces acting on the block is shown to the right. Apply ∑F = max to the block: – fk = −µkFn = ma (1) Apply ∑F = ma y to the block and Fn – mg = 0 because ay = 0 and Fn = mg (2) x y solve for Fn: 284 Chapter 5 Substitute (2) in (1) and solve for µk: µk = −a/g Using a constant-acceleration equation, relate the initial and final velocities of the block to its displacement and acceleration: 2 v12 = v0 + 2a∆x Solve for a to obtain: Substitute for a in equation (3) to obtain: (3) or, because v1 = 0, v0 = v, and ∆x = d, 0 = v 2 + 2ad a= − v2 2d µk = v2 2 gd *35 •• Picture the Problem We can find the speed of the system when it has moved a given distance by using a constantacceleration equation. Under the influence of the forces shown in the free-body diagrams, the blocks will have a common acceleration a. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fk, will allow us to determine the acceleration of the system. Using a constant-acceleration equation, relate the speed of the system to its acceleration and displacement; solve for its speed: r r Apply Fnet = ma to the block whose mass is m1: Using fk = µkFn, substitute (3) in (2) to obtain: Apply ∑F x = max to the block 2 v 2 = v0 + 2a∆x and, because v0 = 0, v = 2a∆x (1) ΣFx = T – fk – m1gsin30° = m1a and ΣFy = Fn,1 – m1gcos30° = 0 (2) (3) T – µk m1g cos30° – m1gsin30° = m1a m2g – T = m2a whose mass is m2: Add the last two equations to eliminate T and solve for a to a= (m2 − µk m1 cos 30° − m1 sin 30°)g m1 + m2 Applications of Newton’s Laws 285 obtain: Substitute numerical values and evaluate a: a = 1.16 m/s 2 Substitute numerical values in equation (1) and evaluate v: v = 2 1.16 m/s 2 (0.3 m ) = 0.835 m/s ( ) and (a ) is correct. 36 •• Picture the Problem Under the influence of the forces shown in the free-body diagrams, the blocks are in static equilibrium. While fs can be either up or down the incline, the free-body diagram shows the situation in which motion is impending up the incline. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fs, will allow us to determine the range of values for m2. (a) Apply r r ∑ F = ma to the block whose mass is m1: Using fs,max = µsFn, substitute (2) in (1) to obtain: Apply ∑F x = max to the block ΣFx = T ± fs,max – m1gsin30° = 0 and ΣFy = Fn,1 – m1gcos30° = 0 T ± µ s m1 g cos 30° − m1 g sin 30° = m1a m2g – T = 0 (1) (2) (3) (4) whose mass is m2: Add equations (3) and (4) to eliminate T and solve for m2: Evaluate (5) denoting the value of m2 with the plus sign as m2,+ and the value of m2 with the minus sign as m2,- to determine the range of values of m2 for which the system is in static equilibrium: m2 = m1 (± µs cos 30° + sin 30°) = (4 kg )[± (0.4) cos 30° + sin 30°] m2, + = 3.39 kg and m 2,- = 0.614 kg ∴ 0.614 kg ≤ m2 ≤ 3.39 kg (5) 286 Chapter 5 (b) With m2 = 1 kg, the impending motion is down the incline and the static friction force is up the incline. Apply Fx = max to the block T + fs – m1gsin30° = 0 (6) ∑ whose mass is m1: Apply ∑F x = max to the block m2g – T = 0 (7) whose mass is m2: Add equations (6) and (7) and solve for and evaluate fs: fs = (m1sin30° – m2)g = [(4 kg)sin30° – 1 kg](9.81 m/s2) = 9.81 N 37 •• Picture the Problem Under the influence of the forces shown in the free-body diagrams, the blocks will have a common acceleration a. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fk, will allow us to determine the acceleration of the system. Finally, we can substitute for the tension in either of the motion equations to determine the acceleration of the masses. Apply r r ∑ F = ma to the block whose mass is m1: Using fk = µkFn, substitute (2) in (1) to obtain: Apply ∑F x = max to the block ΣFx = T – fk – m1gsin30° = m1a and ΣFy = Fn,1 – m1gcos30° = 0 T − µ k m1 g cos 30° − m1 g sin 30° = m1a m2g – T = m2a (1) (2) (3) (4) whose mass is m2: (m2 − µk m1 cos 30° − m1 sin 30°)g Add equations (3) and (4) to eliminate T and solve for a to obtain: a= Substituting numerical values and evaluating a yields: a = 2.36 m/s 2 m1 + m2 Applications of Newton’s Laws 287 T = 37.3 N Substitute for a in equation (3) to obtain: *38 •• Picture the Problem The truck will stop in the shortest possible distance when its acceleration is a maximum. The maximum acceleration is, in turn, determined by the maximum value of the static friction force. The free-body diagram shows the forces acting on the box as the truck brakes to a stop. Assume that the truck is moving in the positive x direction and apply Newton’s 2nd law and the definition of fs,max to find the shortest stopping distance. 2 v 2 = v0 + 2a∆x Using a constant-acceleration equation, relate the truck’s stopping distance to its acceleration and initial velocity; solve for the stopping distance: r or, since v = 0, ∆xmin = r Apply Fnet = ma to the block: 2 − v0 2amax ΣFx = – fs,max = mamax and ΣFy = Fn – mg = 0 (2) fs,max ≡ µsFn and amax = −µsg = − (0.3)(9.81 m/s2) = −2.943 m/s2 Using the definition of fs,max, solve equations (1) and (2) simultaneously for a: Substitute numerical values and evaluate ∆xmin: − (80 km/h ) (1000 km/m ) (1 h/3600 s ) = 9.16 m 2 − 2.943 m/s 2 2 ∆xmin = (1) 2 ( 2 ) 288 Chapter 5 39 •• Picture the Problem We can find the coefficient of friction by applying Newton’s 2nd law and determining the acceleration from the given values of displacement and initial velocity. We can find the displacement and speed of the block by using constant-acceleration equations. During its motion up the incline, the sum of the kinetic friction force and a component of the object’s weight will combine to bring the object to rest. When it is moving down the incline, the difference between the weight component and the friction force will be the net force. (a) Draw a free-body diagram for the block as it travels up the incline: Apply r r ∑ F = ma to the block: Substitute fk = µkFn and Fn from (2) in (1) and solve for µk: ΣFx = – fk – mgsin37°= ma and ΣFy = Fn – mg cos37° = 0 µk = − g sin 37° − a g cos 37° a = − tan 37° − g cos 37° Using a constant-acceleration equation, relate the final velocity of the block to its initial velocity, acceleration, and displacement: Solving for a yields: Substitute numerical values and evaluate a: (1) (2) (3) 2 v12 = v0 + 2a∆x a= 2 v12 − v0 2∆x (5.2 m/s)2 − (14 m/s)2 a= 2(8 m ) = −10.6 m/s 2 Applications of Newton’s Laws 289 Substitute for a in (3) to obtain: − 10.6 m/s 2 µ k = − tan 37° − 9.81m/s 2 cos37° ( ) = 0.599 (b) Use the same constantacceleration equation used above but with v1 = 0 to obtain: 2 0 = v0 + 2a∆x ∆x = 2 − v0 2a Substitute numerical values and evaluate ∆x: ∆x = − (14 m/s ) = 9.25 m 2 − 10.6 m/s 2 (c) When the block slides down the incline, fk is in the positive x direction: ΣFx = fk – mgsin37°= ma and ΣFy = Fn – mgcos37° = 0 Solve for a as in part (a): a = g (µ k cos 37° − sin 37°) = −1.21 m/s 2 Solve for ∆x to obtain: 2 ( ) Use the same constant-acceleration equation used in part (b) to obtain: 2 v 2 = v0 + 2a∆x Set v0 = 0 and solve for v: v = 2a∆x Substitute numerical values and evaluate v: v = 2 − 1.21 m/s 2 (− 9.25 m ) ( ) = 4.73 m/s 40 •• Picture the Problem We can find the stopping distances by applying Newton’s 2nd law to the automobile and then using a constant-acceleration equation. The friction force the road exerts on the tires and the component of the car’s weight along the incline combine to provide the net force that stops the car. The pictorial representation summarizes what we know about the motion of the car. We can use Newton’s 2nd law to determine the acceleration of the car and a constant-acceleration equation to obtain its stopping distance. 290 Chapter 5 (a) Using a constant-acceleration equation, relate the final speed of the car to its initial speed, acceleration, and displacement; solve for its displacement: 2 v12 = v0 + 2amax ∆xmin or, because v1 = 0, ∆xmin = 2 − v0 2amax Draw the free-body diagram for the car going up the incline: Apply r r ∑ F = ma to the car: Substitute fs,max = µsFn and Fn from (2) in (1) and solve for a: Substitute numerical values in the expression for ∆xmin to obtain: (b) Draw the free-body diagram for the car going down the incline: ΣFx = −fs,max – mgsin15° = ma and ΣFy = Fn – mgcos15° = 0 amax = − g (µ s cos 15° + sin 15°) = −9.17 m/s 2 − (30 m/s ) = 49.1 m 2 − 9.17 m/s 2 2 ∆xmin = ( ) (1) (2) Applications of Newton’s Laws 291 Apply r r F = ma to the car: ∑ ΣFx = fs,max – mgsin15° = ma and ΣFy = Fn – mgcos15° = 0 Proceed as in (a) to obtain amax: amax = g (µs cos15° − sin 15°) = 4.09 m/s 2 Again, proceed as in (a) to obtain the displacement of the car: ∆xmin = 2 (30 m/s ) = 110 m − v0 = 2amax 2 4.09 m/s 2 2 ( ) 41 •• Picture the Problem The friction force the road exerts on the tires provides the net force that accelerates the car. The pictorial representation summarizes what we know about the motion of the car. We can use Newton’s 2nd law to determine the acceleration of the car and a constant-acceleration equation to calculate how long it takes it to reach 100 km/h. (a) Because 40% of the car’s weight is on its two drive wheels and the accelerating friction forces act just on these wheels, the free-body diagram shows just the forces acting on the drive wheels. Apply r r F = ma to the car: ∑ Use the definition of fs,max in equation (1) and eliminate Fn between the two equations to obtain: (b) Using a constant-acceleration equation, relate the initial and final ΣFx = fs,max = ma and ΣFy = Fn – 0.4mg = 0 (1) (2) ( a = 0.4µs g = 0.4(0.7 ) 9.81 m/s 2 = 2.75 m/s 2 v1 = v0 + a∆t ) 292 Chapter 5 or, because v0 = 0 and ∆t = t1, velocities of the car to its acceleration and the elapsed time; solve for the time: t1 = v1 a Substitute numerical values and evaluate t1: t1 = (100 km/h )(1h/3600 s )(1000 m/km) = 2.75 m/s 2 10.1s *42 •• Picture the Problem To hold the box in place, the acceleration of the cart and box must be great enough so that the static friction force acting on the box will equal the weight of the box. We can use Newton’s 2nd law to determine the minimum acceleration required. (a) Apply r r ∑ F = ma to the box: ΣFx = Fn = mamin and ΣFy = fs,max – mg = 0 (1) (2) Substitute µFn for fs,max in equation µFn − mg = 0 , µ(ma min ) − mg = 0 (2), eliminate Fn between the two equations and solve for and evaluate amin: and (b) Solve equation (2) for fs,max, and substitute numerical values and evaluate fs,max: fs,max = mg (c) If a is twice that required to hold the box in place, fs will still have its maximum value given by: fs,max = 19.6 N amin = g µs = 9.81 m/s 2 = 16.4 m/s 2 0.6 = (2 kg)(9.81 m/s2) = 19.6 N (d) Because g µs is amin , the box will not fall if a ≥ g µs . 43 •• Picture the Problem Note that the blocks have a common acceleration and that the tension in the string acts on both blocks in accordance with Newton’s third law of motion. Let down the incline be the positive x direction. Draw the freebody diagrams for each block and apply Newton’s second law of motion and the definition of the kinetic friction force to each block to obtain simultaneous Applications of Newton’s Laws 293 equations in ax and T. Draw the free-body diagram for the block whose mass is m1: r r ∑ F = ma to the upper block: ΣFx = −fk,1 + T1 + m1gsinθ = m1ax and ΣFy = Fn,1 – m1gcosθ = 0 (2) The relationship between fk,1 and Fn,1 is: fk,1 = µk,1Fn,1 (3) Eliminate fk,1 and Fn,1 between (1), (2), and (3) to obtain: −µk,1m1gcosθ + T1 + m1gsinθ = m1ax (4) ΣFx = − fk,2 – T2 + m2gsinθ = m2ax and ΣFy = Fn,2 – m2gcosθ = 0 (5) The relationship between fk,2 and Fn,2 is: fk,2 = µk,2Fn,2 (7) Eliminate fk,2 and Fn,2 between (5), (6), and (7) to obtain: −µk,2m2gcosθ – T2 + m2gsinθ = m2ax (8) Apply (1) Draw the free-body diagram for the block whose mass is m2: Apply r r ∑ F = ma to the block: (6) 294 Chapter 5 Noting that T2 = T1 = T, add equations (4) and (8) to eliminate T and solve for ax: µ m + µ k,2 m2 ⎡ ⎤ a x = ⎢sin θ − k,1 1 cosθ ⎥ g m1 + m2 ⎣ ⎦ Substitute numerical values and evaluate ax to obtain: a x = − 0.965 m/s 2 where the minus sign tells us that the acceleration is directed up the incline. m1m2 (µ k,2 − µ k,1 )g cosθ m1 + m2 (b) Eliminate ax between equations (4) and (8) and solve for T = T1 = T2 to obtain: T= Substitute numerical values and evaluate T: T = 0.184 N *44 •• Picture the Problem The free-body diagram shows the forces acting on the two blocks as they slide down the incline. Down the incline has been chosen as the positive x direction. T is the force transmitted by the stick; it can be either tensile (T > 0) or compressive (T < 0). By applying Newton’s 2nd law to these blocks, we can obtain equations in T and a from which we can eliminate either by solving them simultaneously. Once we have expressed T, the role of the stick will become apparent. (a) Apply r r ∑ F = ma to block 1: ∑F x = T1 + m1 g sin θ − f k,1 = m1a and ∑F = Fn,1 − m1 g cos θ = 0 ∑F = m2 g sin θ − T2 − f k,2 = m2 a y Apply r r ∑ F = ma to block 2: x and ∑F y Letting T1 = T2 = T, use the definition of the kinetic friction force to eliminate fk,1 and Fn,1 between the equations for block 1 and fk,2 and Fn,1 between the equations for block 2 to obtain: = Fn,2 − m2 g cos θ = 0 m1 a = m1 g sin θ + T − µ1 m1 g cos θ (1) and m 2 a = m2 g sin θ − T − µ 2 m 2 g cos θ (2) Applications of Newton’s Laws 295 Add equations (1) and (2) to eliminate T and solve for a: (b) Rewrite equations (1) and (2) by dividing both sides of (1) by m1 and both sides of (2) by m2 to obtain. ⎛ ⎞ µ m + µ 2 m2 a = g ⎜ sin θ − 1 1 cosθ ⎟ ⎜ ⎟ m1 + m2 ⎝ ⎠ a = g sin θ + (3) T − µ 2 g cos θ m2 (4) and a = g sin θ − Subtracting (4) from (3) and rearranging yields: T − µ1 g cosθ m1 ⎛ mm T= ⎜ 1 2 ⎜m −m 2 ⎝ 1 ⎞ ⎟(µ1 − µ 2 )g cosθ ⎟ ⎠ If µ1 = µ 2 , T = 0 and the blocks move down the incline with the same acceleration of g (sinθ − µ cosθ ). Inserting a stick between them can' t change this; therefore, the stick must exert no force on either block. 45 •• Picture the Problem The pictorial representation shows the orientation of the two blocks on the inclined surface. Draw the free-body diagrams for each block and apply Newton’s 2nd law of motion and the definition of the static friction force to each block to obtain simultaneous equations in θc and T. (a) Draw the free-body diagram for the lower block: Apply r r ∑ F = ma to the block: The relationship between fs,1 and Fn,1 is: ΣFx = m1gsinθc – fs,1 − T = 0 and ΣFy = Fn,1 – m1gcosθc = 0 (2) fs,1 = µs,1Fn,1 (3) (1) 296 Chapter 5 m1gsinθc − µs,1m1gcosθc − T = 0 (4) ΣFx =T + m2gsinθc – fs,2 = 0 and ΣFy = Fn,2 – m2gcosθc = 0 (5) The relationship between fs,2 and Fn,2 is: fs,2 = µs,2Fn,2 (7) Eliminate fs,2 and Fn,2 between (5), (6), and (7) to obtain: T + m2gsinθc – µs,2m2gcosθc = 0 (8) Eliminate fs,1 and Fn,1 between (1), (2), and (3) to obtain: Draw the free-body diagram for the upper block: Apply r r F = ma to the block: ∑ ⎡ µs,1m1 + µs,2 m2 ⎤ ⎥ ⎣ m1 + m2 ⎦ Add equations (4) and (8) to eliminate T and solve for θc: (6) θ c = tan −1 ⎢ ⎡ (0.4)(0.2 kg ) + (0.6)(0.1 kg )⎤ = tan −1 ⎢ ⎥ 0.1 kg + 0.2 kg ⎣ ⎦ = 25.0° T = m1 g (sin θ C − µs,1 cos θ C ) (b) Because θc is greater than the angle of repose (tan−1(µs,1) = tan−1(0.4) = 21.8°) for the lower block, it would slide if T = 0. Solve equation (4) for T: Substitute numerical values and evaluate T: ( ) T = (0.2 kg ) 9.81 m/s 2 [sin25° − (0.4 )cos25°] = 0.118 N Applications of Newton’s Laws 297 46 •• Picture the Problem The pictorial representation shows the orientation of the two blocks with a common acceleration on the inclined surface. Draw the free-body diagrams for each block and apply Newton’s 2nd law and the definition of the kinetic friction force to each block to obtain simultaneous equations in a and T. (a) Draw the free-body diagram for the lower block: r r F = ma to the lower ∑ ΣFx = m1gsin20° − fk,1 – T = m1a and ΣFy = Fn,1 − m1gcos20° = 0 (2) Express the relationship between fk,1 and Fn,1: fk,1 = µk,1Fn,1 (3) Eliminate fk,1 and Fn,1 between (1), (2), and (3) to obtain: m1 g sin 20° − µ k,1m1 g cos 20° Apply block: − T = m1a (1) (4) Draw the free-body diagram for the upper block: Apply r r ∑ F = ma to the upper block: Express the relationship between fk,2 and Fn,2 : ΣFx = T + m2gsin20° − fk,2 = m2a and ΣFy = Fn,2 – m2gcos20° = 0 (6) fk,2 = µk,2Fn,2 (7) (5) 298 Chapter 5 Eliminate fk,2 and Fn,2 between (5), (2), and (7) to obtain: T + m2 g sin 20° − µ k,2 m2 g cos 20° Add equations (4) and (8) to eliminate T and solve for a: ⎛ ⎞ µ m + µ 2 m2 a = g ⎜ sin 20° − 1 1 cos 20° ⎟ ⎜ ⎟ m1 + m2 ⎝ ⎠ Substitute the given values and evaluate a: a = 0.944 m/s 2 (b) Substitute for a in either equation (4) or equation (8) to obtain: T = − 0.426 N ; i.e., the rod is under = m2 a (8) compression. *47 •• Picture the Problem The vertical r component of F reduces the normal force; hence, the static friction force between the surface and the block. The horizontal component is responsible for any tendency to move and equals the static friction force until it exceeds its maximum value. We can apply Newton’s 2nd law to the box, under equilibrium conditions, to relate F to θ. (a) The static-frictional force opposes the motion of the object, and the maximum value of the static-frictional force is proportional to the normal force FN. The normal force is equal to the weight minus the vertical component FV of the force F. Keeping the magnitude F constant while increasing θ from zero results in a decrease in FV and thus a corresponding decrease in the maximum static-frictional force fmax. The object will begin to move if the horizontal component FH of the force F exceeds fmax. An increase in θ results in a decrease in FH. As θ increases from 0, the decrease in FN is larger than the decrease in FH, so the object is more and more likely to slip. However, as θ approaches 90°, FH approaches zero and no movement will be initiated. If F is large enough and if θ increases from 0, then at some value of θ the block will start to move. (b) Apply r r ∑ F = ma to the block: Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F: ΣFx =Fcosθ – fs = 0 and ΣFy = Fn + Fsinθ – mg = 0 F= µ s mg cosθ + µ s sin θ (1) (2) Applications of Newton’s Laws 299 Use this function with mg = 240 N to generate the table shown below: θ F (deg) (N) 0 240 10 220 20 210 30 206 40 208 50 218 60 235 The following graph of F(θ) was plotted using a spreadsheet program. 240 235 F (N) 230 225 220 215 210 205 0 10 20 30 40 50 60 theta (degrees) From the graph, we can see that the minimum value for F occurs when θ ≈ 32°. Remarks: An alternative to manually plotting F as a function of θ or using a spreadsheet program is to use a graphing calculator to enter and graph the function. 48 ••• Picture the Problem The free-body diagram shows the forces acting on the block. We can apply Newton’s 2nd law, under equilibrium conditions, to relate F to θ and then set its derivative with respect to θ equal to zero to find the value of θ that minimizes F. (a) Apply r r F = ma to the block: ∑ ΣFx =Fcosθ – fs = 0 and ΣFy = Fn + Fsinθ – mg = 0 (1) (2) 300 Chapter 5 Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F: F= µ s mg cosθ + µ s sin θ (3) To find θmin, differentiate F with respect to θ and set the derivative equal to zero for extrema of the function: (cosθ + µs sin θ ) d (µs mg ) µs mg d (cosθ + µs sin θ ) dF dθ dθ = − dθ (cosθ + µs sin θ )2 (cosθ + µs sin θ )2 µ mg (− sin θ + µs cosθ ) = s = 0 for extrema (cosθ + µs sin θ )2 Solve for θmin to obtain: θ min = tan −1 µ s (b) Use the reference triangle shown below to substitute for cosθ and sinθ in equation (3): Fmin = µ s mg 1 1+ µ = 2 s + µs µs 1 + µ s2 µ s mg 1 + µ s2 1 + µ s2 = µs 1 + µ s2 mg (c) The coefficient of kinetic friction is less than the coefficient of static friction. An analysis identical to the one above shows that the minimum force one should apply to keep the block moving should be applied at an angle given by θ min = tan −1 µ k . Therefore, once the block is moving the coefficient of friction will decrease, so the angle can be decreased. 49 •• r Picture the Problem The vertical component of F increases the normal force and the static friction force between the surface and the block. The horizontal component is responsible for any tendency to move and equals the static friction force until it exceeds its maximum value. We can apply Newton’s 2nd law to the box, under equilibrium conditions, to relate F to θ. Applications of Newton’s Laws 301 (a) As θ increases from zero, F increases the normal force exerted by the surface and the static friction force. As the horizontal component of F decreases with increasing θ, one would expect F to continue to increase. (b) Apply r r ∑ F = ma to the block: ΣFx =Fcosθ – fs = 0 and ΣFy = Fn – Fsinθ – mg = 0 Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F: F= (1) (2) µ s mg cosθ − µ s sin θ (3) Use this function with mg = 240 N to generate the table shown below. θ F (deg) (N) 0 240 10 273 20 327 30 424 40 631 50 1310 60 very large The graph of F as a function of θ, plotted using a spreadsheet program, confirms our prediction that F continues to increase with θ. 1400 1200 F (N) 1000 800 600 400 200 0 0 10 20 30 theta (degrees) (a) From the graph we see that: θ min = 0° 40 50 302 Chapter 5 (b) Evaluate equation (3) for θ = 0° to obtain: F= µs mg = µs mg cos 0° − µs sin 0° You should keep the angle at 0°. (c) Remarks: An alternative to the use of a spreadsheet program is to use a graphing calculator to enter and graph the function. 50 •• Picture the Problem The forces acting on each of these masses are shown in the freebody diagrams below. m1 represents the mass of the 20-kg mass and m2 that of the 100-kg mass. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fk,1 (= fk,2) act on both masses but in opposite directions. Newton’s 2nd law and the definition of kinetic friction forces can be used to determine the various forces and the acceleration called for in this problem. (a) Draw a free-body diagram showing the forces acting on the 20-kg mass: Apply r r ∑ F = ma to this mass: Solve equation (1) for fk,1: ΣFx = fk,1 = m1a1 and ΣFy = Fn,1 – m1g = 0 (1) (2) fk,1 = m1a1 = (20 kg)(4 m/s2) = 80.0 N (b) Draw a free-body diagram showing the forces acting on the 100-kg mass: Apply ∑F x = max to the 100-kg object and evaluate Fnet: Fnet = m2 a2 ( ) = (100 kg ) 6 m/s 2 = 600 N Applications of Newton’s Laws 303 Express F in terms of Fnet and fk,2: F = Fnet + fk,2 = 600 N + 80 N = 680 N (c) When the 20-kg mass falls off, the 680-N force acts just on the 100-kg mass and its acceleration is given by Newton’s 2nd law: a= Fnet 680 N = = 6.80 m/s 2 m 100 kg 51 •• Picture the Problem The forces acting on each of these blocks are shown in the freebody diagrams to the right. m1 represents the mass of the 60-kg block and m2 that of the 100-kg block. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fk,1 (= fk,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of kinetic friction forces can be used to determine the coefficient of kinetic friction and acceleration of the 100-kg block. (a) Apply r r ∑ F = ma to the 60-kg block: Apply ∑F x = max to the 100-kg ΣFx = F − fk,1 = m1a1 and ΣFy = Fn,1 – m1g = 0 (2) fk,2 = m2a2 (3) (1) block: Using equation (2), express the relationship between the kinetic r r friction forces f k ,1 and f k , 2 : fk,1 = fk,2 = fk = µ kFn,1 = µ km1g (4) Substitute equation (4) into equation (1) and solve for µ k: µk = F − m1a1 m1 g Substitute numerical values and evaluate µ k: µk = 320 N − (60 kg ) 3 m/s 2 = 0.238 (60 kg ) 9.81m/s2 (b) Substitute equation (4) into equation (3) and solve for a2: a2 = ( µ k m1 g m2 ( ) ) 304 Chapter 5 Substitute numerical values and evaluate a2: a2 = (0.238)(60 kg )(9.81m/s 2 ) 100 kg = 1.40 m/s 2 *52 •• Picture the Problem The accelerations of the truck can be found by applying Newton’s 2nd law of motion. The free-body diagram for the truck climbing the incline with maximum acceleration is shown to the right. (a) Apply r r ∑ F = ma to the truck when it is climbing the incline: Solve equation (2) for Fn and use the definition of fs,max to obtain: Substitute equation (3) into equation (1) and solve for a: Substitute numerical values and evaluate a: (b) When the truck is descending the incline with maximum acceleration, the static friction force points down the incline; i.e., its direction is reversed on the FBD. Apply Fx = max to the truck under ΣFx = fs,max – mgsin12° = ma and ΣFy = Fn – mgcos12° = 0 (1) fs,max = µsmgcos12° (3) (2) a = g (µs cos12° − sin 12°) a = (9.81 m/s 2 )[(0.85) cos12° − sin 12°] = 6.12 m/s 2 – fs,max – mgsin12° = ma (4) ∑ these conditions: Substitute equation (3) into equation (4) and solve for a: Substitute numerical values and evaluate a: a = − g (µs cos12° + sin 12°) ( a = − 9.81 m/s 2 )[(0.85)cos12° + sin 12°] = − 10.2 m/s 2 Applications of Newton’s Laws 305 53 •• Picture the Problem The forces acting on each of the blocks are shown in the freebody diagrams to the right. m1 represents the mass of the 2-kg block and m2 that of the 4-kg block. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fs,1 (= fs,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of the maximum static friction force can be used to determine the maximum force acting on the 4-kg block for which the 2-kg block does not slide. (a) Apply r r ∑ F = ma to the 2-kg block: Apply r r ∑ F = ma to the 4-kg block: Using equation (2), express the relationship between the static r ΣFx = fs,1,max = m1amax and ΣFy = Fn,1 – m1g = 0 (1) ΣFx = F – fs,2,max = m2amax and ΣFy = Fn,2 – Fn,1 - m2g = 0 (3) (4) fs,1,max = fs,2,max = µs m1g (5) (2) r friction forces f s ,1,max and f s , 2,max : Substitute (5) in (1) and solve for amax: amax = µsg = (0.3)g = 2.94 m/s2 Solve equation (3) for F = Fmax: Fmax = m2 amax + µs m1 g Substitute numerical values and evaluate Fmax: ( ) Fmax = (4 kg ) 2.94 m/s 2 + (0.3)(2 kg ) ( × 9.81 m/s 2 ) = 17.7 N (b) Use Newton’s 2nd law to express the acceleration of the blocks moving as a unit: a= Substitute numerical values and evaluate a: a= F m1 + m2 1 2 (17.7 N ) 2 kg + 4 kg = 1.47 m/s 2 306 Chapter 5 Because the friction forces are an action-reaction pair, the friction force acting on each block is given by: fs = m1a = (2 kg)(1.47 m/s2) (c) If F = 2Fmax, then m1 slips on m2 and the friction force (now kinetic) is given by: f = fk = µkm1g Use ∑F x = max to relate the acceleration of the 2-kg block to the net force acting on it and solve for a 1: Use ∑F x = max to relate the = 2.94 N fk = µkm1g = m1a1 and a1 = µkg = (0.2)g = 1.96 m/s 2 F − µkm1g = m2a2 acceleration of the 4-kg block to the net force acting on it: Solve for a2: a2 = F − µ k m1 g m2 Substitute numerical values and evaluate a2: a2 = 2(17.7 N ) − (0.2 )(2 kg ) 9.81 m/s 2 4 kg ( ) = 7.87 m/s 2 54 •• Picture the Problem Let the positive x direction be the direction of motion of these blocks. The forces acting on each of the blocks are shown, for the static friction case, on the free-body diagrams to the right. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fs,1 (= fs,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of the maximum static friction force can be used to determine the maximum acceleration of the block whose mass is m1. (a) Apply block: r r ∑ F = ma to the 2-kg ΣFx = fs,1,max = m1amax and (1) Applications of Newton’s Laws 307 ΣFy = Fn,1 – m1g = 0 (2) ΣFx = T – fs,2,max = m2amax and ΣFy = Fn,2 – Fn,1 – m2g = 0 (3) (4) Using equation (2), express the relationship between the static r r friction forces f s ,1,max and f s , 2,max : fs,1,max = fs,2,max = µs m1g (5) Substitute (5) in (1) and solve for amax: amax = µsg = (0.6)g = 5.89 m/s 2 Apply r r ∑ F = ma to the 4-kg block: (b) Use ∑F x = ma x to express the T = (m1 + m2) amax (6) m3g – T = m3 amax (7) acceleration of the blocks moving as a unit: Apply ∑F x = max to the object whose mass is m3: µs (m1 + m2 ) (0.6)(10 kg + 5 kg ) = 1 − µs 1 − 0.6 Add equations (6) and (7) to eliminate T and then solve for and evaluate m3: m3 = (c) If m3 = 30 kg, then m1 will slide on m2 and the friction force (now kinetic) is given by: f = fk = µkm1g Use ∑F x = max to relate the = 22.5 kg m3g – T = m3a3 (8) acceleration of the 30-kg block to the net force acting on it: g (m3 − µ k m1 ) m2 + m3 Noting that a2 = a3 and that the friction force on the body whose mass is m2 is due to kinetic friction, add equations (3) and (8) and solve for and evaluate the common acceleration: a2 = a3 = With block 1 sliding on block 2, the fk = µkm1g = m1a1 = (9.81m/s )[30 kg − (0.4)(5 kg )] 2 10 kg + 30 kg = 6.87 m/s 2 (1′) 308 Chapter 5 friction force acting on each is kinetic and equations (1) and (3) become: Solve equation (1′) for and evaluate a 1: T – fk = T – µkm1g = m2a2 ( a1 = µ k g = (0.4 ) 9.81 m/s 2 (3′) ) = 3.92 m/s 2 T = m2 a2 + µ k m1 g Solve equation (3′) for T: Substitute numerical values and evaluate T: T = (10 kg ) (6.87 m/s 2 ) + (0.4 )(5 kg ) (9.81 m/s 2 ) = 88.3 N 55 • Picture the Problem Let the direction of motion be the positive x direction. The free-body diagrams show the forces acting on both the block (M) and the r r counterweight (m). While T1 ≠ T2 , T1 = T2. By applying Newton’s 2nd law to these blocks, we can obtain equations in T and a from which we can eliminate the tension. Once we know the acceleration of the block, we can use constant-acceleration equations to determine how far it moves in coming to a momentary stop. (a) Apply r r ∑ F = ma to the block on the incline: ∑F x = T1 − Mg sin θ − f k = Ma and ∑F = Fn − Mg cosθ = 0 ∑F = mg − T2 = ma y Apply r r ∑ F = ma to the x (1) counterweight: Letting T1 = T2 = T and using the definition of the kinetic friction force, eliminate fk and Fn between the equations for the block on the incline to obtain: Eliminate T from equations (1) and (2) by adding them and solve for a: T − Mg sin θ − µ k Mg cos θ = Ma a= m − M (sin θ + µ k cos θ ) g m+M (2) Applications of Newton’s Laws 309 Substitute numerical values and evaluate a: a= 550 kg − (1600 kg ) (sin 10° + 0.15 cos10°) 9.81m/s 2 = 0.163 m/s2 550 kg + 1600 kg ( (b) Using a constant-acceleration equation, relate the speed of the block at the instant the rope breaks to its acceleration and displacement as it slides to a stop. Solve for its displacement: ) vf2 = vi2 + 2a∆x or, because vf = 0, ∆x = − vi2 2a (3) The block had been accelerating up the incline for 3 s before the rope broke, so it has an initial speed of : (0.163 m/s2)(3 s) = 0.489 m/s From equation (2) we can see that, when the rope breaks (T = 0) and: a = − g (sin θ + µ k cos θ ) ( ) = − 9.81 m/s 2 [sin 10° + (0.15)cos10°] = −3.15 m/s 2 where the minus sign indicates that the block is being accelerated down the incline, although it is still sliding up the incline. Substitute in equation (3) and evaluate ∆x: − (0.489 m/s ) = 0.0380 m ∆x = 2 − 3.15 m/s 2 (c) When the block is sliding down the incline, the kinetic friction force will be up the incline. Express the block’s acceleration: a = − g (sin θ − µ k cos θ ) 2 ( ( ) ) = − 9.81 m/s 2 [sin 10° − (0.15) cos10°] = − 0.254 m/s 2 56 ••• Picture the Problem If the 10-kg block is not to slide on the bracket, the maximum r value for F must be equal to the maximum value of fs and will produce the maximum acceleration of this block and the bracket. We can apply Newton’s 2nd law and the definition of fs,max to first calculate the maximum acceleration and then the maximum value of F. (a) and (b) Apply r r ∑ F = ma to the 10-kg block when it is experiencing its maximum acceleration: ΣFx = fs,max – F = m2a2,max and ΣFy = Fn,2 – m2g = 0 (1) (2) 310 Chapter 5 Express the static friction force acting on the 10-kg block: fs,max = µsFn,2 (3) Eliminate fs,max and Fn,2 from equations (1), (2) and (3) to obtain: µsm2g – F = m2a2,max (4) 2F – µsm2g = m1a1,max (5) Apply ∑F x = max to the bracket to obtain: Because a1,max = a2,max, denote this acceleration by amax. Eliminate F from equations (4) and (5) and solve for amax: amax = Substitute numerical values and evaluate amax: amax = µs m2 g m1 + 2m2 (0.4)(10 kg )(9.81m/s2 ) 5 kg + 2(10 kg ) = 1.57 m/s 2 Solve equation (4) for F = Fmax: Substitute numerical values and evaluate F: F = µs m2 g − m2 amax = m2 (µs g − amax ) [ ( ) F = (10 kg ) (0.4) 9.81 m/s 2 − 1.57 m/s 2 = 23.5 N *57 •• Picture the Problem The free-body diagram shows the forces acting on the block as it is moving up the incline. By applying Newton’s 2nd law, we can obtain expressions for the accelerations of the block up and down the incline. Adding and subtracting these equations, together with the data found in the notebook, will lead to values for gV and µk. Apply r r Fi = ma to the block when ∑i it is moving up the incline: ∑F x and ∑F Using the definition of fk, eliminate Fn between the two equations to obtain: = − f k − mg V sin θ = maup y = Fn − mg V cosθ = 0 aup = − µ k g V cos θ − g V sin θ (1) Applications of Newton’s Laws 311 When the block is moving down the incline, fk is in the positive x direction, and its acceleration is: adown = µ k g V cos θ − g V sin θ (2) Add equations (1) and (2) to obtain: aup + adown = −2 g V sin θ (3) Solve equation (3) for gV: gV = Determine θ from the figure: aup + adown − 2 sin θ ⎡ 0.73 glapp ⎤ ⎥ = 10.8° ⎣ 3.82 glapp ⎦ θ = tan −1 ⎢ Substitute the data from the notebook in equation (4) to obtain: 1.73 glapp/plipp 2 + 1.42 glapp/plipp 2 gV = = − 8.41glapp/plipp 2 − 2 sin 10.8° Subtract equation (1) from equation (2) to obtain: Solve for µk: adown − aup = 2µ k g V cos θ µk = adown − aup 2 g V cosθ Substitute numerical values and evaluate µk: µk = − 1.42 glapp/plipp 2 − 1.73 glapp/plipp 2 = 0.191 2 − 8.41glapp/plipp 2 cos10.8° ( ) *58 •• Picture the Problem The free-body diagram shows the block sliding down the incline under the influence of a friction force, its weight, and the normal force exerted on it by the inclined surface. We can find the range of values for m for the two situations described in the problem statement by applying Newton’s 2nd law of motion to, first, the conditions under which the block will not move or slide if pushed, and secondly, if pushed, the block will move up the incline. (a) Assume that the block is sliding down the incline with a constant velocity and with no hanging weight r r (m = 0) and apply ∑ F = ma to ∑F x = − f k + Mg sin θ = 0 and ∑F y = Fn − Mg cosθ = 0 (4) 312 Chapter 5 the block: Using fk = µkFn, eliminate Fn between the two equations and solve for the net force acting on the block: Fnet = − µ k Mg cosθ + Mg sin θ If the block is moving, this net force must be nonnegative and: (− µ k cosθ + sin θ )Mg ≥ 0 This condition requires that: µ k ≤ tan θ = tan 18° = 0.325 Because µk = 0.2, this condition is satisfied and: mmin = 0 To find the maximum value, note that the maximum possible value for the tension in the rope is mg. For the block to move down the incline, the component of the block’s weight parallel to the incline minus the frictional force must be greater than or equal to the tension in the rope: Mgsinθ – µkMgcosθ ≥ mg Solve for mmax: mmax ≤ M (sin θ − µ k cos θ ) Substitute numerical values and evaluate mmax: mmax ≤ (100 kg )[sin 18° − (0.2)cos18°] The range of values for m is: = 11.9 kg 0 ≤ m ≤ 11.9 kg (b) If the block is being dragged up the incline, the frictional force will point down the incline, and: Mg sinθ + µkMg cosθ < mg Solve for and evaluate mmin: mmin > M (sinθ + µk cosθ) = (100 kg)[sin18° + (0.2)cos18°] = 49.9 kg If the block is not to move unless pushed: Mg sinθ + µs Mg cosθ > mg Solve for and evaluate mmax: mmax < M (sinθ + µs cosθ) = (100 kg)[sin18° + (0.4)cos18°] = 68.9 kg The range of values for m is: 49.9 kg ≤ m ≤ 68.9 kg Applications of Newton’s Laws 313 59 ••• Picture the Problem The free-body diagram shows the forces acting on the 0.5 kg block when the acceleration is a minimum. Note the choice of coordinate system is consistent with the direction of r F . Apply Newton’s 2nd law to the block and solve the resulting equations for amin and amax. r r F = ma to the 0.5-kg ∑ ΣFx = Fnsinθ – fscosθ = ma and ΣFy = Fncosθ + fssinθ – mg = 0 (2) Under minimum acceleration, fs = fs,max. Express the relationship between fs,max and Fn: fs,max = µsFn (3) Substitute fs,max for fs in equation (2) and solve for Fn: Fn = Substitute for Fn in equation (1) and solve for a = amin: amin = g Substitute numerical values and evaluate amin: amin = 9.81 m/s 2 (a) Apply block: (1) mg cosθ + µs sin θ sin θ − µs cos θ cosθ + µs sin θ ( ) sin35°°− ((0.8))cos35° cos35 + 0.8 sin35° = −0.627 m/s 2 Treat the block and incline as a single object to determine Fmin: Fmin = mtotamin = (2.5 kg)( –0.627 m/s2) To find the maximum acceleration, r reverse the direction of f s and apply ΣFx = Fnsinθ + fscosθ = ma and ΣFy = Fncosθ – fssinθ – mg = 0 r r F = ma to the block: ∑ = − 1.57 N (4) (5) sin θ + µs cosθ cosθ − µs sin θ Proceed as above to obtain: amax = g Substitute numerical values and evaluate amax: amax = 9.81m/s 2 ( = 33.5 m/s 2 ° ) sin35°°+ ((0.8))cos35° cos35 − 0.8 sin35 314 Chapter 5 Treat the block and incline as a single object to determine Fmax: Fmax = mtotamax = (2.5 kg)(33.5 m/s2) (b) Repeat (a) with µs = 0.4 to obtain: Fmin = 5.75 N and Fmax = 37.5 N = 83.8 N 60 • Picture the Problem The kinetic friction force fk is the product of the coefficient of sliding friction µk and the normal force Fn the surface exerts on the sliding object. By applying Newton’s 2nd law in the vertical direction, we can see that, on a horizontal surface, the normal force is the weight of the sliding object. Note that the acceleration of the block is opposite its direction of motion. (a) Relate the force of kinetic friction to µk and the normal force acting on the sliding wooden object: Substitute v = 10 m/s and evaluate fk : (b) Substitute v = 20 m/s and evaluate fk: f k = µ k Fn = fk = fk = 0.11 (1 + 2.3 ×10 −4 ( 0.11(100 kg ) 9.81 m/s 2 (1 + 2.3 ×10 −4 (10 m/s) ( (1 + 2.3 ×10 2 (20 m/s) mg )= ) 2 2 0.11(100 kg ) 9.81 m/s 2 −4 ) v2 103 N ) ) 2 2 = 90.5 N 61 •• Picture the Problem The pictorial representation shows the block sliding from left to right and coming to rest when it has traveled a distance ∆x. Note that the direction of the motion is opposite that of the block’s acceleration. The acceleration and stopping distance of the blocks can be found from constant-acceleration equations. Let the direction of motion of the sliding blocks be the positive x direction. Because the surface is horizontal, the normal force acting on the sliding block is the block’s weight. Applications of Newton’s Laws 315 (a) Using a constant-acceleration equation, relate the block’s stopping distance to its initial speed and acceleration; solve for the stopping distance: Apply ∑F x = max to the sliding block, introduce Konecny’s empirical expression, and solve for the block’s acceleration: Evaluate a with m = 10 kg: 2 v 2 = v0 + 2a∆x or, because v = 0, 2 − v0 ∆x = 2a (1) Fnet,x − f k 0.4 Fn0.91 = =− m m m 0.91 0.4(mg ) =− m a= (0.4)[(10 kg )(9.81m/s 2 )]0.91 a=− 10 kg = − 2.60 m/s 2 Substitute in equation (1) and evaluate the stopping distance when v0 = 10 m/s: ∆x = (b) Proceed as in (a) with m = 100 kg to obtain: a=− − (10 m/s ) = 19.2 m 2 − 2.60 m/s 2 2 ( ) (0.4)[(100 kg )(9.81m/s2 )] 0.91 100 kg = − 2.11 m/s 2 Find the stopping distance as in (a): − (10 m/s ) ∆x = = 23.7 m 2 − 2.11 m/s 2 2 ( ) *62 ••• Picture the Problem The kinetic friction force fk is the product of the coefficient of sliding friction µk and the normal force Fn the surface exerts on the sliding object. By applying Newton’s 2nd law in the vertical direction, we can see that, on a horizontal surface, the normal force is the weight of the sliding object. We can apply Newton’s 2nd law in the horizontal (x) direction to relate the block’s acceleration to the net force acting on it. In the spreadsheet program, we’ll find the acceleration of the block from this net force (which is velocity dependent), calculate the increase in the block’s speed from its acceleration and the elapsed time and add this increase to its speed at end of the previous time interval, determine how far it has moved in this time interval, and add this distance to its previous position to find its current position. We’ll also calculate the position of the block x2, under the assumption that µk = 0.11, using a constant-acceleration equation. 316 Chapter 5 The spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell Formula/Content C9 C8+$B$6 D9 D8+F9*$B$6 E9 $B$5−($B$3)*($B$2)*$B$5/ (1+$B$4*D9^2)^2 Algebraic Form F− t + ∆t v + a∆t µ k mg (1 + 2.34 ×10 F9 E10/$B$5 G9 K9 G9+D10*$B$6 0.5*5.922*I10^2 J10-K10 v2 ) 2 Fnet / m x + v∆t 2 1 2 at L9 −4 x − x2 A 1 2 3 4 5 6 B g= 9.81 Coeff1= 0.11 Coeff2= 2.30E04 Mass= 10 Applied 70 Force= Time 0.05 step= 7 8 9 C m/s^2 D E F G H I J t x x2 x−x2 kg N s Net force 10 11 12 13 14 15 t 0.00 0.05 0.10 0.15 0.20 0.25 v 0.00 0.30 0.59 0.89 1.18 1.48 a 59.22 59.22 59.22 59.22 59.23 205 206 9.75 9.80 61.06 61.40 66.84 6.68 292.37 66.88 6.69 295.44 5.92 5.92 5.92 5.92 5.92 x 0.00 0.01 0.04 0.09 0.15 0.22 0.00 0.05 0.10 0.15 0.20 0.25 9.75 9.80 mu=variable mu=constant 0.00 0.00 0.00 0.01 0.01 0.01 0.04 0.03 0.01 0.09 0.07 0.02 0.15 0.12 0.03 0.22 0.19 0.04 292.37 295.44 281.48 284.37 10.89 11.07 Applications of Newton’s Laws 317 207 208 209 210 9.85 9.90 9.95 10.00 61.73 62.07 62.40 62.74 66.91 66.94 66.97 67.00 6.69 6.69 6.70 6.70 298.53 301.63 304.75 307.89 9.85 9.90 9.95 10.00 298.53 301.63 304.75 307.89 287.28 290.21 293.15 296.10 11.25 11.42 11.61 11.79 The displacement of the block as a function of time, for a constant coefficient of friction (µk = 0.11) is shown as a solid line on the graph and for a variable coefficient of friction, is shown as a dotted line. Because the coefficient of friction decreases with increasing particle speed, the particle travels slightly farther when the coefficient of friction is variable. 300 mu = variable 250 mu = constant x (m) 200 150 100 50 0 0 2 4 6 8 10 t (s) The velocity of the block, with variable coefficient of kinetic friction, is shown below. 70 60 v (m/s) 50 40 30 20 10 0 0 2 4 6 t (s) 8 10 318 Chapter 5 63 •• Picture the Problem The free-body diagram shows the forces acting on the block as it moves to the right. The kinetic friction force will slow the block and, eventually, bring it to rest. We can relate the coefficient of kinetic friction to the stopping time and distance by applying Newton’s 2nd law and then using constantacceleration equations. (a) Apply r r ∑ F = ma to the block of wood: ∑F x = − f k = ma and ∑F y = Fn − mg = 0 Using the definition of fk, eliminate Fn between the two equations to obtain: a = −µ k g Use a constant-acceleration equation to relate the acceleration of the block to its displacement and its stopping time: ∆x = v0 ∆t + 1 a(∆t ) 2 (1) 2 v0 + v ∆t 2 = 1 v0 ∆t since v = 0. 2 (2) Relate the initial speed of the block, v0, to its displacement and stopping distance: ∆x = vav ∆t = Use this result to eliminate v0 in equation (2): ∆x = − 1 a(∆t ) 2 Substitute equation (1) in equation (4) and solve for µk: µk = 2∆x 2 g (∆t ) Substitute for ∆x = 1.37 m and ∆t = 0.97 s to obtain: µk = 2(1.37 m ) = 0.297 2 9.81 m/s 2 (0.97 s ) (b) Use equation (3) to find v0: 2 v0 = ( (3) (4) ) 2∆x 2(1.37 m ) = = 2.82 m/s ∆t 0.97 s Applications of Newton’s Laws 319 *64 •• Picture the Problem The free-body diagram shows the forces acting on the block as it slides down an incline. We can apply Newton’s 2nd law to these forces to obtain the acceleration of the block and then manipulate this expression algebraically to show that a graph of a/cosθ versus tanθ will be linear with a slope equal to the acceleration due to gravity and an intercept whose absolute value is the coefficient of kinetic friction. (a) Apply r r ∑ F = ma to the block as it slides down the incline: ∑F x = mg sin θ − f k = ma and ∑F y = Fn − mg cosθ = 0 Substitute µkFn for fk and eliminate Fn between the two equations to obtain: a = g (sin θ − µ k cos θ ) Divide both sides of this equation by cosθ to obtain: a = g tan θ − gµ k cos θ Note that this equation is of the form y = mx + b: Thus, if we graph a/cosθ versus tanθ, we should get a straight line with slope g and y-intercept −gµk. (b) A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell C7 D7 E7 a TAN(C7*PI()/180) F7 D7/COS(C7*PI()/180) 6 7 8 9 Formula/Content Algebraic Form θ C theta 25 27 29 D a 1.691 2.104 2.406 π ⎞ ⎛ tan⎜θ × ⎟ ⎝ 180 ⎠ a π ⎞ ⎛ cos⎜θ × ⎟ ⎝ 180 ⎠ E tan(theta) 0.466 0.510 0.554 F a/cos(theta) 1.866 2.362 2.751 320 Chapter 5 10 11 12 13 14 15 16 17 31 33 35 37 39 41 43 45 2.888 3.175 3.489 3.781 4.149 4.326 4.718 5.106 0.601 0.649 0.700 0.754 0.810 0.869 0.933 1.000 3.370 3.786 4.259 4.735 5.338 5.732 6.451 7.220 A graph of a/cosθ versus tanθ is shown below. From the curve fit (Excel’s Trendline was used), g = 9.77 m/s2 and µ k = 2.62 m/s 2 = 0.268. 9.77 m/s 2 The percentage error in g from the commonly accepted value of 9.81 m/s2 is ⎛ 9.81 m/s 2 − 9.77 m/s 2 ⎞ ⎟ = 0.408% 100⎜ ⎜ ⎟ 9.81 m/s 2 ⎝ ⎠ 8 y = 9.7681x - 2.6154 R2 = 0.9981 7 a /cos(theta) 6 5 4 3 2 1 0 0.4 0.5 0.6 0.7 tan(theta) 0.8 0.9 1.0 Applications of Newton’s Laws 321 Motion Along a Curved Path 65 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the horizontal is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. r r ∑ F = ma to the stone: ΣFx = Tcosθ = mac = mv2/r and ΣFy= Tsinθ – mg = 0 (2) Use the right triangle in the diagram to relate r, L, and θ : r = Lcosθ (3) Eliminate T and r between equations (1), (2) and (3) and solve for v2: v 2 = gL cot θ cosθ (4) Express the velocity of the stone in terms of its period: v= Eliminate v between equations (4) and (5) and solve for θ : θ = sin −1 ⎜ ⎜ Substitute numerical values and evaluate θ : θ = sin −1 ⎢ Apply 2πr t1 rev (1) (5) ⎛ gt12rev ⎞ ⎟ 4π 2 L ⎟ ⎠ ⎝ ( ) ⎡ 9.81 m/s 2 (1.22 s )2 ⎤ ⎥ = 25.8° 2 ⎣ 4π (0.85 m ) ⎦ and (c) is correct. 322 Chapter 5 66 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the horizontal is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. r r F = ma to the stone: ∑ ΣFx = Tcosθ = mac = mv2/r and ΣFy= Tsinθ – mg = 0 (2) Use the right triangle in the diagram to relate r, L, and θ: r = Lcosθ (3) Eliminate T and r between equations (1), (2), and (3) and solve for v: v = gL cot θ cosθ Substitute numerical values and evaluate v: v= Apply (1) (9.81m/s )(0.8 m )cot 20° cos 20° 2 = 4.50 m/s 67 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the vertical is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. (a) Apply r r ∑ F = ma to the stone: ΣFx = Tsinθ = mac = mv2/r and (1) Applications of Newton’s Laws 323 ΣFy= Tcosθ – mg = 0 Eliminate T between equations (1) and (2) and solve for v: v = rg tan θ Substitute numerical values and evaluate v: v= (b) Solve equation (2) for T: T= Substitute numerical values and evaluate T: T= (2) (0.35 m )(9.81m/s 2 )tan30° = 1.41 m/s mg cosθ (0.75 kg )(9.81m/s2 ) = cos30° *68 •• Picture the Problem The sketch shows the forces acting on the pilot when her plane is r at the lowest point of its dive. Fn is the force the airplane seat exerts on her. We’ll apply Newton’s 2nd law for circular motion to determine Fn and the radius of the circular path followed by the airplane. (a) Apply ∑F y = ma y to the pilot: Fn − mg = mac Solve for and evaluate Fn: Fn = mg + mac = m(g + ac) = m(g + 8.5g) = 9.5mg = (9.5) (50 kg) (9.81 m/s2) = 4.66 kN (b) Relate her acceleration to her velocity and the radius of the circular arc and solve for the radius: ac = v2 v2 ⇒ r= r ac Substitute numerical values and evaluate r : [(345 km/h )(1h/3600 s )(1000 m/km)] 2 r= ( 8.5 9.81 m/s 2 ) = 110 m 8.50 N 324 Chapter 5 69 •• Picture the Problem The diagram shows the forces acting on the pilot when her plane is at the lowest point of its dive. r Fn is the force the airplane seat exerts on her. We’ll use the definitions of centripetal acceleration and centripetal force and apply Newton’s 2nd law to calculate these quantities and the normal force acting on her. (a) Her acceleration is centripetal and given by: Substitute numerical values and evaluate ac: ac = ac v2 , upward r [(180 km/h )(1h/3600 s)(10 = 3 /km 300 m = 8.33 m/s , upward 2 (b) The net force acting on her at the bottom of the circle is the force responsible for her centripetal acceleration: (c) Apply ∑F y = ma y to the pilot: ( Fnet = mac = (65 kg ) 8.33 m/s 2 ) = 541 N, upward Fn – mg = mac Solve for Fn: Fn = mg + mac = m(g + ac) Substitute numerical values and evaluate Fn: Fn = (65 kg)(9.81 m/s2 + 8.33 m/s2) 70 •• Picture the Problem The free-body diagrams for the two objects are shown to the right. The hole in the table changes the direction the tension in the string (which provides the centripetal force required to keep the object moving in a circular path) acts. The application of Newton’s 2nd law and the definition of centripetal force will lead us to an expression for r as a function of m1, m2, and the time T for one revolution. = 1.18 kN, upward )] 2 Applications of Newton’s Laws 325 Apply ∑F x = max to both objects and use the definition of centripetal acceleration to obtain: Because F1 = F2 we can eliminate both of them between these equations to obtain: Express the speed v of the object in terms of the distance it travels each revolution and the time T for one revolution: Substitute to obtain: m2g – F2 = 0 and F1 = m1ac = m1v2/r m2 g − m1 v= v2 =0 r 2πr T m2 g − m1 4π 2 r 2 =0 rT 2 or 4π 2 r m2 g − m1 2 = 0 T Solve for r: r= m2 gT 2 4π 2 m1 *71 •• Picture the Problem The free-body diagrams show the forces acting on each block. We can use Newton’s 2nd law to relate these forces to each other and to the masses and accelerations of the blocks. Apply ∑F x = max to the block whose mass is m1: Apply ∑F x = max to the block whose mass is m2: Relate the speeds of each block to their common period and their distance from the center of the v12 T1 − T2 = m1 L1 T2 = m2 v1 = 2 v2 L1 + L2 2πL1 2π (L1 + L2 ) and v2 = T T 326 Chapter 5 circle: 2 Solve the first force equation for T2, substitute for v2, and simplify to obtain: ⎛ 2π ⎞ T2 = [m2 (L1 + L2 )] ⎜ ⎟ ⎝ T ⎠ Substitute for T2 and v1 in the first force equation to obtain: ⎛ 2π ⎞ T1 = [m2 (L1 + L2 ) + m1 L1 ] ⎜ ⎟ ⎝ T ⎠ 2 *72 •• Picture the Problem The path of the particle and its position at 1-s intervals are shown. The displacement vectors are also shown. The velocity vectors for the average velocities in the first and second r r intervals are along r01 and r12 , respectively, and are shown in the lower diagram. r ∆ v points toward the center of the circle. Use the diagram to the right to find ∆r: ∆r = 2rsin22.5°= 2(4 cm) sin22.5° = 3.06 cm Find the average velocity of the particle along the chords: vav = ∆r/∆t = (3.06 cm)/(1 s) = 3.06 cm/s Using the lower diagram and the fact that the angle between r r v1 and v 2 is 45°, express ∆v in ∆v = 2v1sin22.5° terms of v1 (= v2): Evaluate ∆v using vav as v1: ∆v = 2(3.06 cm/s)sin22.5° = 2.34 cm/s Now we can determine a = ∆v/∆t: a= 2.34 cm/s = 2.34 cm/s 2 1s Applications of Newton’s Laws 327 2πr 2π(4 cm ) = = 3.14 cm/s T 8s Find the speed v (= v1 = v2 …) of the particle along its circular path: v= Calculate the radial acceleration of the particle: ac = Compare ac and a by taking their ratio: ac 2.46 cm/s 2 = = 1.05 a 2.34 cm/s 2 v 2 (3.14 cm/s) = = 2.46 cm/s 2 r 4 cm 2 or ac = 1.05a 73 •• Picture the Problem The diagram to the right has the free-body diagram for the child superimposed on a pictorial representation of her motion. The force her r father exerts is F and the angle it makes with respect to the direction we’ve chosen as the positive y direction is θ. We can infer her speed from the given information concerning the radius of her path and the period of her motion. Applying Newton’s 2nd law as it describes circular motion will allow us to find both the direction and r magnitude of F . Apply r r ∑ F = ma to the child: ΣFx = Fsinθ = mv2/r and ΣFy = Fcosθ − mg = 0 ⎡ v2 ⎤ ⎥ ⎣ rg ⎦ Eliminate F between these equations and solve for θ : θ = tan −1 ⎢ Express v in terms of the radius and period of the child’s motion: v= Substitute for v in the expression for θ to obtain: θ = tan −1 ⎢ 2πr T ⎡ 4π 2 r ⎤ 2 ⎥ ⎣ gT ⎦ 328 Chapter 5 4π 2 (0.75 m ) ⎤ = 53.3° 2⎥ 2 ⎣ 9.81 m/s (1.5 s ) ⎦ ⎡ Substitute numerical values and evaluate θ : θ = tan −1 ⎢ Solve the y equation for F: F= Substitute numerical values and evaluate F: F= ( ) mg cos θ (25 kg )(9.81m/s2 ) = cos53.3° 410 N 74 •• Picture the Problem The diagram to the right has the free-body diagram for the bob of the conical pendulum superimposed on a pictorial representation of its motion. The r tension in the string is F and the angle it makes with respect to the direction we’ve chosen as the positive x direction isθ. We can findθ from the y equation and the information provided about the tension. Then, by using the definition of the speed of the bob in its orbit and applying Newton’s 2nd law as it describes circular motion, we can find the period T of the motion. Apply r r ∑ F = ma to the pendulum bob: ΣFx = Fcosθ = mv2/r and ΣFy = Fsinθ − mg = 0 ⎛ mg ⎞ −1 ⎛ mg ⎞ ⎟ = sin ⎜ ⎟ ⎜ 6mg ⎟ = 9.59° ⎝ F ⎠ ⎠ ⎝ Using the given information that F = 6mg, solve the y equation for θ: θ = sin −1 ⎜ With F = 6mg, solve the x equation for v: v = 6rg cosθ Relate the period T of the motion to the speed of the bob and the radius of the circle in which it moves: T= From the diagram, one can see that: r =Lcosθ 2π r = v 2π r 6rg cosθ Applications of Newton’s Laws 329 Substitute for r in the expression for the period to obtain: T = 2π L 6g Substitute numerical values and evaluate T: T = 2π 0.5 m = 0.579 s 6 9.81 m/s 2 ( ) 75 •• Picture the Problem The static friction force fs is responsible for keeping the coin from sliding on the turntable. Using Newton’s 2nd law of motion, the definition of the period of the coin’s motion, and the definition of the maximum static friction force, we can find the magnitude of the friction force and the value of the coefficient of static friction for the two surfaces. (a) Apply r r F = ma to the coin: ∑ ∑ Fx = f s = m v2 r and ∑F y = Fn − mg = 0 2πr T If T is the period of the coin’s motion, its speed is given by: v= Substitute for v in the force equation and simplify to obtain: fs = Substitute numerical values and evaluate fs: 4π 2 ((0.1kg )(0.1 m ) fs = = 0.395 N (1s )2 (b) Determine Fn from the y equation: If the coin is about to slide at r = 16 cm, fs = fs,max. Solve for µs in terms of fs,max and Fn: 4π 2 mr T2 Fn = mg µs = f s ,max Fn 4π 2 mr 2 4π 2 r = T = mg gT 2 330 Chapter 5 Substitute numerical values and evaluate µs: 4π 2 (0.16 m ) µs = = 0.644 (9.81m/s2 )(1s)2 76 •• Picture the Problem The forces acting on the tetherball are shown superimposed on a pictorial representation of the motion. The r horizontal component of T is the centripetal force. Applying Newton’s 2nd law of motion and solving the resulting equations will yield both the tension in the cord and the speed of the ball. (a) Apply r r ∑ F = ma to the tetherball: ∑ Fx = T sin 20° = m v2 r and ∑F y Solve the y equation for T: T= Substitute numerical values and evaluate T: T= = T cos 20° − mg = 0 mg cos 20° (0.25 kg )(9.81m/s2 ) = cos 20° (b) Eliminate T between the force equations and solve for v: v = rg tan 20° Note from the diagram that: r = Lsin20° Substitute for r in the expression for v to obtain: v = gL sin 20° tan 20° Substitute numerical values and evaluate v: v = 2.61 N (9.81m/s )(1.2 m )sin 20° tan 20° = 1.21 m/s 2 Applications of Newton’s Laws 331 *77 •• Picture the Problem The diagram includes a pictorial representation of the earth in its orbit about the sun and a force diagram showing the force on an object at the equator that is due to the earth’s r rotation, FR , and the force on the object due to the orbital motion of the earth about r the sun, Fo . Because these are centripetal forces, we can calculate the accelerations they require from the speeds and radii associated with the two circular motions. aR = 2 vR R vR = 2πR TR Substitute for vR in the expression for aR to obtain: aR = 4π 2 R TR2 Substitute numerical values and evaluate aR: aR = Express the radial acceleration due to the rotation of the earth: Express the speed of the object on the equator in terms of the radius of the earth R and the period of the earth’s rotation TR: 4π 2 (6370 km )(1000 m/km ) ⎡ ⎛ 3600 s ⎞⎤ ⎢(24 h )⎜ ⎜ 1 h ⎟⎥ ⎟ ⎝ ⎠⎦ ⎣ −2 2 = 3.37 × 10 m/s = 3.44 × 10 −3 g Express the radial acceleration due to the orbital motion of the earth: Express the speed of the object on the equator in terms of the earth-sun distance r and the period of the earth’s motion about the sun To: ao = 2 vo r vo = 2π r To 2 332 Chapter 5 Substitute for vo in the expression for ao to obtain: 4π 2 r ao = 2 To Substitute numerical values and evaluate ac: ao = ( 4π 2 1.5 × 1011 m ) ⎡ ⎛ 24 h ⎞ ⎛ 3600 s ⎞⎤ ⎢(365 d )⎜ ⎜ 1d ⎟ ⎜ 1 h ⎟ ⎥ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎦ ⎣ 2 = 5.95 × 10−3 m/s 2 = 6.07 × 10−4 g 78 • Picture the Problem The most significant force acting on the earth is the gravitational force exerted by the sun. More distant or less massive objects exert forces on the earth as well, but we can calculate the net force by considering the radial acceleration of the earth in its orbit. Similarly, we can calculate the net force acting on the moon by considering its radial acceleration in its orbit about the earth. (a) Apply ∑F r = mar to the earth: Fon earth = m Express the orbital speed of the earth in terms of the time it takes to make one trip around the sun (i.e., its period) and its average distance from the sun: v= Substitute for v to obtain: v2 r 2π r T Fon earth = 4π 2 mr T2 Substitute numerical values and evaluate Fon earth: Fon earth = ( )( 4π 2 5.98 × 10 24 kg 1.496 × 1011 m 24 h 3600 s ⎞ ⎛ × ⎜ 365.24 d × ⎟ d h ⎠ ⎝ 2 )= 3.55 × 10 22 N )= 2.00 × 10 20 N (b) Proceed as in (a) to obtain: Fon moon = ( )( 4π 2 7.35 × 10 22 kg 3.844 × 108 m 24 h 3600 s ⎞ ⎛ × ⎜ 27.32 d × ⎟ d h ⎠ ⎝ 2 Applications of Newton’s Laws 333 79 •• Picture the Problem The semicircular wire of radius 10 cm limits the motion of the bead in the same manner as would a 10-cm string attached to the bead and fixed at the center of the semicircle. The horizontal component of the normal force the wire exerts on the bead is the centripetal force. The application of Newton’s 2nd law, the definition of the speed of the bead in its orbit, and the relationship of the frequency of a circular motion to its period will yield the angle at which the bead will remain stationary relative to the rotating wire. Apply r r ∑ F = ma to the bead: ∑ Fx = Fn sin θ = m v2 r and ∑F y = Fn cosθ − mg = 0 v2 rg Eliminate Fn from the force equations to obtain: tan θ = The frequency of the motion is the reciprocal of its period T. Express the speed of the bead as a function of the radius of its path and its period: v= Using the diagram, relate r to L and θ: r = L sin θ 2πr T ⎡ gT 2 ⎤ 2 ⎥ ⎣ 4π L ⎦ Substitute for r and v in the expression for tanθ and solve for θ : θ = cos −1 ⎢ Substitute numerical values and evaluate θ : θ = cos −1 ⎢ ( ) ⎡ 9.81 m/s 2 (0.5 s )2 ⎤ ⎥ = 51.6° 4π 2 (0.1 m ) ⎦ ⎣ 334 Chapter 5 80 ••• Picture the Problem Note that the acceleration of the bead has two components, the radial component r perpendicular to v , and a tangential component due to friction that is opposite r to v . The application of Newton’s 2nd law will result in a differential equation with separable variables. Its integration will lead to an expression for the speed of the bead as a function of time. Apply r r ∑ F = ma to the bead in the radial and tangential directions: ∑ Fr = Fn = m v2 r and ∑F = − f t k = mat = m Express fk in terms of µk and Fn: fk = µkFn Substitute for Fn and fk in the tangential equation to obtain the differential equation: µ dv = − k v2 dt r Separate the variables to obtain: µ dv = − k dt 2 v r Express the integral of this equation with the limits of integration being from v0 to v on the left-hand side and from 0 to t on the right-hand side: 1 µ ∫ v' 2 dv' = − rk ∫ dt' 0 v0 Evaluate these integrals to obtain: ⎛1 1 ⎞ ⎛µ ⎞ − ⎜ − ⎟ = −⎜ k ⎟t ⎜v v ⎟ ⎝ r ⎠ 0 ⎠ ⎝ Solve this equation for v: v t ⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ v = v0 ⎜ ⎛ µ k v0 ⎞ ⎟ ⎟t ⎟ ⎜1+ ⎜ ⎝ ⎝ r ⎠ ⎠ dv dt Applications of Newton’s Laws 335 81 ••• Picture the Problem Note that the acceleration of the bead has two components−the radial component r perpendicular to v , and a tangential component due to friction that is opposite r to v . The application of Newton’s 2nd law will result in a differential equation with separable variables. Its integration will lead to an expression for the speed of the bead as a function of time. (a) In Problem 81 it was shown that: ⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ v = v0 ⎜ ⎛ µ k v0 ⎞ ⎟ ⎟t ⎟ ⎜1+ ⎜ ⎝ ⎝ r ⎠ ⎠ Express the centripetal acceleration of the bead: ⎛ ⎞ ⎟ 2 2 ⎜ v v0 ⎜ 1 ⎟ ac = = r r ⎜ ⎛ µ k v0 ⎞ ⎟ ⎟t ⎟ ⎜1+ ⎜ ⎝ ⎝ r ⎠ ⎠ (b) Apply Newton’s 2nd law to the bead: v2 ∑ Fr = Fn = m r 2 and ∑F = − f t k = mat = m dv dt v2 = − µ k ac r Eliminate Fn and fk to rewrite the radial force equation and solve for at: at = − µ k (c) Express the resultant acceleration in terms of its radial and tangential components: a = at2 + ac2 = 2 = ac 1 + µ k (− µ k ac )2 + ac2 336 Chapter 5 Concepts of Centripetal Force *82 • Picture the Problem The diagram depicts a seat at its highest and lowest points. Let ″t″ denote the top of the loop and ″b″ the bottom of the loop. Applying Newton’s 2nd law to the seat at the top of the loop will establish the value of mv2/r; this can then be used at the bottom of the loop to determine Fn,b. Apply ∑F r = mar to the seat at the mg +Fn,t = 2mg = mar = mv2/r top of the loop: Apply ∑F r = mar to the seat at the Fn,b – mg = mv2/r bottom of the loop: Solve for Fn,b and substitute for mv2/r to obtain: Fn,b = 3mg and (d ) is correct. 83 • Picture the Problem The speed of the roller coaster is imbedded in the expression for its radial acceleration. The radial acceleration is determined by the net radial force acting on the passenger. We can use Newton’s 2nd law to relate the net force on the passenger to the speed of the roller coaster. Apply ∑F radial = maradial to the mg + 0.4mg = mv2/r passenger: Solve for v: v = 1.4 gr Substitute numerical values and evaluate v: v = 1.4 9.81 m/s 2 (12.0 m ) ( = 12.8 m/s ) Applications of Newton’s Laws 337 84 • Picture the Problem The force F the passenger exerts on the armrest of the car door is the radial force required to maintain the passenger’s speed around the curve and is related to that speed through Newton’s 2nd law of motion. Apply ∑F x = max to the forces acting on the passenger: Solve this equation for v: Substitute numerical values and evaluate v: F =m v2 r v= rF m v= (80 m )(220 N ) = 15.9 m/s 70 kg and (a ) is correct. *85 ••• Picture the Problem The forces acting on the bicycle are shown in the force diagram. The static friction force is the centripetal force exerted by the surface on the bicycle that allows it to move in a circular path. r r Fn + f s makes an angle θ with the vertical direction. The application of Newton’s 2nd law will allow us to relate this angle to the speed of the bicycle and the coefficient of static friction. (a) Apply r r ∑ F = ma to the bicycle: ∑ Fx = fs = mv 2 r and ∑F y Relate Fn and fs to θ : = Fn − mg = 0 mv 2 f v2 tan θ = s = r = Fn mg rg 338 Chapter 5 Solve for v: v = rg tan θ Substitute numerical values and evaluate v: v= (b) Relate fs to µs and Fn: fs = Solve for µs and substitute for fs to obtain: 2 f s 2v 2 = µs = mg rg Substitute numerical values and evaluate µs 2(7.25 m/s ) µs = = 0.536 (20 m ) 9.81m/s2 (20 m )(9.81m/s2 )tan15° = 7.25 m/s 1 2 f s,max = 1 µs mg 2 2 ( ) 86 •• Picture the Problem The diagram shows the forces acting on the plane as it flies in a horizontal circle of radius R. We can apply Newton’s 2nd law to the plane and eliminate the lift force in order to obtain an expression for R as a function of v and θ. Apply r r ∑ F = ma to the plane: ∑ Fx = Flift sin θ = m v2 R and ∑F y Eliminate Flift between these equations to obtain: Solve for R: Substitute numerical values and evaluate R: = Flift cosθ − mg = 0 tan θ = R= v2 Rg v2 g tan θ 2 ⎛ km 1h ⎞ ⎜ 480 ⎟ × ⎜ h 3600 s ⎟ ⎝ ⎠ = 2.16 km R= 2 (9.81m/s )tan40° Applications of Newton’s Laws 339 87 • Picture the Problem Under the conditions described in the problem statement, the only forces acting on the car are the normal force exerted by the road and the gravitational force exerted by the earth. The horizontal component of the normal force is the centripetal force. The application of Newton’s 2nd law will allow us to express θ in terms of v, r, and g. Apply r r ∑ F = ma to the car: ∑ Fx = Fn sin θ = m v2 r and ∑F y Eliminate Fn from the force equations to obtain: = Fn cosθ − mg = 0 tan θ = Solve for θ : v2 rg ⎡ v2 ⎤ ⎥ ⎣ rg ⎦ θ = tan −1 ⎢ Substitute numerical values and evaluate θ: ⎧ [(90 km/h )(1 h 3600 s )(1000 m/km )] 2 ⎫ ⎬ = 21.7° (160 m ) 9.81m/s2 ⎩ ⎭ θ = tan −1 ⎨ ( *88 •• Picture the Problem Both the normal force and the static friction force contribute to the centripetal force in the situation described in this problem. We can apply Newton’s 2nd law to relate fs and Fn and then solve these equations simultaneously to determine each of these quantities. ) 340 Chapter 5 (a) Apply r r ∑ F = ma to the car: ∑ Fx = Fn sin θ + f s cosθ = m v2 r and ∑F y Multiply the x equation by sinθ and the y equation by cosθ to obtain: = Fn cosθ − f s sin θ − mg = 0 f s sin θ cosθ + Fn sin 2 θ = m v2 sin θ r and Fn cos 2 θ − f s sin θ cosθ − mg cosθ = 0 Add these equations to eliminate fs: Fn − mg cosθ = m Solve for Fn: v2 sin θ r Fn = mg cosθ + m v2 sin θ r ⎛ ⎞ v2 ⎜ g cosθ + sin θ ⎟ = m⎜ ⎟ r ⎝ ⎠ Substitute numerical values and evaluate Fn: ⎡ (85 km/h )2 (1000 m/km)2 (1h/3600 s )2 sin10°⎤ 2 Fn = (800 kg ) ⎢ 9.81 m/s cos10° + ⎥ 150 m ⎣ ⎦ ( ) = 8.25 kN (b) Solve the y equation for fs: fs = Fn cos θ − mg sin θ Substitute numerical values and evaluate fs: fs (8.25 kN )cos10° − (800 kg )(9.81m/s2 ) = = sin10° 1.59 kN (c) Express µs,min in terms of fs and Fn : µs,min = fs Fn Substitute numerical values and evaluate µs,min: µs,min = 1.59 kN = 0.193 8.25 kN Applications of Newton’s Laws 341 89 •• Picture the Problem Both the normal force and the static friction force contribute to the centripetal force in the situation described in this problem. We can apply Newton’s 2nd law to relate fs and Fn and then solve these equations simultaneously to determine each of these quantities. (a) Apply r r ∑ F = ma to the car: v2 r ∑ Fy = Fn cosθ − f s sinθ − mg = 0 ∑ Fx = Fn sin θ + f s cosθ = m Multiply the x equation by sinθ and the y equation by cosθ : v2 sin θ r Fn cos 2 θ − f s sin θ cosθ − mg cosθ = 0 Add these equations to eliminate fs: Fn − mg cosθ = m Solve for Fn: Fn = mg cosθ + m f s sin θ cosθ + Fn sin 2 θ = m v2 sin θ r v2 sin θ r ⎛ ⎞ v2 = m⎜ g cosθ + sin θ ⎟ ⎜ ⎟ r ⎝ ⎠ Substitute numerical values and evaluate Fn: ⎡ (38 km/h )2 (1000 m/km)2 (1h/3600 s )2 sin10°⎤ 2 Fn = (800 kg )⎢ 9.81 m/s cos10° + ⎥ 150 m ⎣ ⎦ ( ) = 7.832 kN (b) Solve the y equation for fs: Substitute numerical values and evaluate fs: Fn cos θ − mg sin θ mg = Fn cot θ − sin θ fs = 342 Chapter 5 (800 kg ) (9.81 m/s 2 ) = f s = (7.832 kN ) cot 10° − sin10° − 777 N The negative sign tells us that fs points upward along the inclined plane rather than as shown in the force diagram. *90 ••• Picture the Problem The free-body diagram to the left is for the car at rest. The static friction force up the incline balances the downward component of the car’s weight and prevents it from sliding. In the free-body diagram to the right, the static friction force points in the opposite direction as the tendency of the moving car is to slide toward the outside of the curve. r r ∑ F = ma to the car that is Apply at rest: ∑F y and ∑F x Substitute fs = fs,max = µsFn in equation (2) and solve for and evaluate the maximum allowable value of θ: Apply r r F = ma to the car that is ∑ moving with speed v: Substitute numerical values into (5) = Fn sin θ − f s cosθ = 0 (2) θ = tan −1 (µs ) = tan −1 (0.08) = 4.57° ∑F y ∑F x Substitute fs = µsFn in equations (3) and (4) and simplify to obtain: = Fn cosθ + f s sin θ − mg = 0 (1) = Fn cosθ − f s sin θ − mg = 0 (3) = Fn sin θ + f s cosθ = m Fn (cosθ − µs sin θ ) = mg Fn (µs cosθ + sin θ ) = m 0.9904Fn = mg v2 r (4) (5) 2 v r (6) Applications of Newton’s Laws 343 and (6) to obtain: and v2 0.1595Fn = m r Eliminate Fn and solve for r: Substitute numerical values and evaluate r: v2 r= 0.1610 g r= (60 km/h ×1h/3600 s ×1000 m/km)2 ( 0.1610 9.81 m/s 2 ) = 176 m 91 ••• Picture the Problem The free-body diagram to the left is for the car rounding the curve at the minimum (not sliding down the incline) speed. The static friction force up the incline balances the downward component of the car’s weight and prevents it from sliding. In the free-body diagram to the right, the static friction force points in the opposite direction as the tendency of the car moving with the maximum safe speed is to slide toward the outside of the curve. Application of Newton’s 2nd law and the simultaneous solution of the force equations will yield vmin and vmax. Apply r r ∑ F = ma to a car traveling around the curve when the coefficient of static friction is zero: ∑ Fx = Fn sin θ = m and ∑F y Divide the first of these equations by the second to obtain: 2 vmin r = Fn cosθ − mg =0 2 v2 −1 ⎛ v ⎞ tan θ = or θ = tan ⎜ ⎟ ⎜ rg ⎟ rg ⎝ ⎠ Substitute numerical values and evaluate the banking angle: 344 Chapter 5 ( ) ⎡ (40 km/h )2 (1000 m/km )2 1h/3600 s 2 ⎤ ⎥ = 22.8° (30 m ) 9.81m/s2 ⎣ ⎦ θ = tan −1 ⎢ Apply r r F = ma to the car ∑ traveling around the curve at minimum speed: ( 2 vmin ∑ Fx = Fn sin θ − fs cosθ = m r and ∑F y Substitute fs = fs,max = µsFn in the force equations and simplify to obtain: Evaluate these equations for θ = 22.8° and µs = 0.3: ) = Fn cos θ + f s sin θ − mg = 0 2 vmin Fn (µs cosθ − sin θ ) = m r and Fn (cosθ + µs sin θ ) = mg 0.1102Fn= m 2 vmin r and 1.038Fn = mg Eliminate Fn between these two equations and solve for vmin: vmin = 0.106rg Substitute numerical values and evaluate vmin: vmin = 0.106(30 m ) 9.81m/s 2 Apply r r ∑ F = ma to the car traveling around the curve at maximum speed: ( = 5.59 m/s = 20.1 km/h ∑ Fx = Fn sin θ + fs cosθ = m Evaluate these equations for θ = 22.8° and µs = 0.3: 2 vmax r and ∑F y Substitute fs = fs,max = µsFn in the force equations and simplify to obtain: ) = Fn cosθ − f s sin θ − mg =0 Fn (µs cosθ + sin θ ) = m and 2 vmax r Fn (cosθ − µ s sin θ ) = mg 0.6641Fn= m 2 vmax r and 0.8056Fn = mg Applications of Newton’s Laws 345 Eliminate Fn between these two equations and solve for vmax: vmax = 0.8243rg Substitute numerical values and evaluate vmax: vmax = (0.8243)(30 m )(9.81m/s2 ) = 15.6 m/s = 56.1 km/h Drag Forces 92 • Picture the Problem We can apply Newton’s 2nd law to the particle to obtain its equation of motion. Applying terminal speed conditions will yield an expression for b that we can evaluate using the given numerical values. Apply ∑F y = ma y to the particle: mg − bv = ma y When the particle reaches its terminal speed v = vt and ay = 0: mg − bvt = 0 Solve for b to obtain: b= Substitute numerical values and evaluate b: b= mg vt (10 −13 )( kg 9.81 m/s 2 3 × 10 −4 m/s ) = 3.27 ×10 −9 kg/s 93 • Picture the Problem We can apply Newton’s 2nd law to the Ping-Pong ball to obtain its equation of motion. Applying terminal speed conditions will yield an expression for b that we can evaluate using the given numerical values. Apply ∑F y = ma y to the Ping- mg − bv 2 = ma y Pong ball: When the Ping-Pong ball reaches its terminal speed v = vt and ay = 0: mg − bvt2 = 0 Solve for b to obtain: b= mg vt2 346 Chapter 5 Substitute numerical values and evaluate b: b= (2.3 ×10 −3 )( kg 9.81 m/s 2 (9 m/s)2 ) = 2.79 × 10−4 kg/m *94 • Picture the Problem Let the upward direction be the positive y direction and apply Newton’s 2nd law to the sky diver. (a) Apply ∑F y = ma y to the sky Fd − mg = ma y diver: or, because ay = 0, Substitute numerical values and evaluate Fd: Fd = (60 kg ) 9.81 m/s 2 = 589 N (b) Substitute Fd = b vt2 in equation bvt2 = mg Fd = mg (1) ( ) (1) to obtain: Solve for b: b= mg Fd = 2 vt2 vt Substitute numerical values and evaluate b: b= 589 N = 0.942 kg/m (25 m/s)2 95 •• Picture the Problem The free-body diagram shows the forces acting on the car as it descends the grade with its terminal velocity. The application of Newton’s 2nd law with a = 0 and Fd equal to the given function will allow us to solve for the terminal velocity of the car. Apply ∑F x = max to the car: mg sin θ − Fd = ma x or, because v = vt and ax = 0, mg sin θ − Fd = 0 Substitute for Fd to obtain: ( ) mg sin θ − 100 N − 1.2 N ⋅ s 2 / m 2 vt2 = 0 Applications of Newton’s Laws 347 Solve for vt: vt = Substitute numerical values and evaluate vt: vt = mg sin θ − 100 N 1.2 N ⋅ s 2 / m 2 (800 kg )(9.81m/s 2 )sin 6° − 100 N 1.2 N ⋅ s 2 / m 2 = 24.5 m/s = 88.2 km/h 96 ••• Picture the Problem Let the upward direction be the positive y direction and apply Newton’s 2nd law to the particle to obtain an equation from which we can find the particle’s terminal speed. (a) Apply ∑F y = ma y to a mg − 6πηrv = ma y pollution particle: or, because ay = 0, Solve for vt to obtain: vt = Express the mass of a sphere in terms of its volume: ⎛ 4π r 3 ⎞ m = ρV = ρ ⎜ ⎜ 3 ⎟ ⎟ ⎝ ⎠ Substitute for m to obtain: Substitute numerical values and evaluate vt: mg − 6πηrvt = 0 vt = mg 6πηr 2r 2 ρg 9η ( )( ( 2 )( 2 10 −5 m 2000 kg/m 3 9.81 m/s 2 vt = 9 1.8 ×10 −5 N ⋅ s/m 2 ) ) = 2.42 cm/s (b) Use distance equals average speed times the fall time to find the time to fall 100 m at 2.42 cm/s: t= 10 4 cm = 4.13 × 103 s = 1.15 h 2.42 cm/s *97 ••• Picture the Problem The motion of the centrifuge will cause the pollution particles to migrate to the end of the test tube. We can apply Newton’s 2nd law and Stokes’ law to derive an expression for the terminal speed of the sedimentation particles. We can then use this terminal speed to calculate the sedimentation time. We’ll use the 12 cm distance 348 Chapter 5 from the center of the centrifuge as the average radius of the pollution particles as they settle in the test tube. Let R represent the radius of a particle and r the radius of the particle’s circular path in the centrifuge. Express the sedimentation time in terms of the sedimentation speed vt: Apply ∑F radial = maradial to a ∆t sediment = ∆x vt 6πηRvt = mac pollution particle: Express the mass of the particle in terms of its radius R and density ρ: m = ρV = 4 π R 3 ρ 3 Express the acceleration of the pollution particles due to the motion of the centrifuge in terms of their orbital radius r and period T: ⎛ 2π r ⎞ ⎜ ⎟ 2 2 v ⎝ T ⎠ = 4π r = ac = r r T2 Substitute for m and ac and simplify to obtain: ⎛ 4π 2 r ⎞ 16π 3 ρ rR 3 6πηRvt = π R ρ ⎜ 2 ⎟ = ⎜ T ⎟ 3T 2 ⎝ ⎠ Solve for vt: Find the period T of the motion from the number of revolutions the centrifuge makes in 1 second: 2 4 3 3 vt = 8π 2 ρ rR 2 9ηT 2 T= 1 = 1.25 × 10 −3 min/rev 800 rev / min = 1.25 × 10−3 min/rev × 60 s/min = 75.0 × 10-3 s/rev Substitute numerical values and evaluate vt: Find the time it takes the particles to move 8 cm as they settle in the test tube: vt = ( ) ( 8π 2 2000 kg/m 3 (0.12 m ) 10−5 m ( )( 9 1.8 × 10−5 N ⋅ s/m 2 75 × 10 −3 s = 2.08 m/s ∆tsediment = 8 cm ∆x = 208 cm/s v = 38.5 ms ) ) 2 2 Applications of Newton’s Laws 349 8 cm ∆x = 2.42 cm/s v In Problem 96 it was shown that the rate of fall of the particles in air is 2.42 cm/s. Find the time required to fall 8 cm in air under the influence of gravity: ∆tair = Find the ratio of the two times: ∆tair/∆tsediment ≈ 100 = 3.31s Euler’s Method 98 •• Picture the Problem The free-body diagram shows the forces acting on the baseball sometime after it has been thrown downward but before it has reached its terminal speed. In order to use Euler’s method, we’ll need to determine how the acceleration of the ball varies with its speed. We can do this by applying Newton’s 2nd law to the ball and using its terminal speed to express the constant in the acceleration equation in terms of the ball’s terminal speed. We can then use vn+1 = vn + an ∆t to find the speed of the ball at any given time. Apply Newton’s 2nd law to the ball to obtain: Solve for dv/dt to obtain: When the ball reaches its terminal speed: Substitute to obtain: Express the position of the ball to obtain: Letting an be the acceleration of the ball at time tn, express its speed when t = tn + 1: mg − bv 2 = m dv dt dv b = g − v2 dt m 0=g− b g b 2 = 2 vt ⇒ m vt m ⎛ v2 ⎞ dv = g ⎜1 − 2 ⎟ ⎜ v ⎟ dt t ⎠ ⎝ xn+1 = xn + vn+1 + vn ∆t 2 vn+1 = vn + an ∆t where ⎛ v2 ⎞ a n = g ⎜1 − n ⎟ ⎜ v2 ⎟ t ⎠ ⎝ 350 Chapter 5 and ∆t is an arbitrarily small interval of time. A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell A10 B10 Formula/Content B9+$B$1 B9+0.5*(C9+C10)*$B$1 C10 C9+D9*$B$1 D10 $B$4*(1−C10^2/$B$5^2) A ∆t= x0= v0= a0= vt= Algebraic Form t + ∆t xn+1 = xn + vn+1 + vn ∆t 2 vn+1 = vn+ an∆t ⎛ v2 ⎞ a n = g ⎜1 − n ⎟ ⎜ v2 ⎟ t ⎠ ⎝ B C 0.5 0 9.722 9.81 41.67 D s m m/s m/s^2 m/s 1 2 3 4 5 6 7 8 9 10 11 12 t (s) 0.0 0.5 1.0 1.5 x (m) 0 6 14 25 v (m/s) 9.7 14.4 18.7 22.6 a (m/s^2) 9.28 8.64 7.84 6.92 28 29 30 9.5 10.0 10.5 317 337 358 41.3 41.4 41.5 0.17 0.13 0.10 38 39 40 41 42 14.5 15.0 15.5 16.0 16.5 524 545 566 587 608 41.6 41.7 41.7 41.7 41.7 0.01 0.01 0.01 0.01 0.00 From the table we can see that the speed of the ball after 10 s is approximately 41.4 m/s. We can estimate the uncertainty in this result by halving ∆t and recalculating the speed of the ball at t = 10 s. Doing so yields v(10 s) ≈ 41.3 m/s, a difference of about 0.02%. The graph shows the velocity of the ball thrown straight down as a function of time. Applications of Newton’s Laws 351 v (m/s) Ball Throw n Straight Dow n 45 40 35 30 25 20 15 10 5 0 0 5 10 15 20 t (s) Reset ∆t to 0.5 s and set v0 = 0. Ninety-nine percent of 41.67 m/s is approximately 41.3 m/s. Note that the ball will reach this speed in about 10.5 s and that the distance it travels in this time is about 322 m. The following graph shows the distance traveled by the ball dropped from rest as a function of time. Ball Dropped From Rest 400 350 300 x (m) 250 200 150 100 50 0 0 2 4 6 t (s) *99 •• Picture the Problem The free-body diagram shows the forces acting on the baseball after it has left your hand. In order to use Euler’s method, we’ll need to determine how the acceleration of the ball varies with its speed. We can do this by applying Newton’s 2nd law to the baseball. We can then use vn+1 = vn + an ∆t and xn +1 = xn + vn ∆t to find the speed and 8 10 12 352 Chapter 5 position of the ball. Apply ∑F y = ma y to the baseball: − bv v − mg = m dv dt where v = v for the upward part of the flight of the ball and v = −v for the downward part of the flight. Solve for dv/dt: Under terminal speed conditions ( v = −vt ): dv b = −g − v v dt m 0 = −g + b 2 vt m and b g = 2 m vt Substitute to obtain: ⎛ vv dv g = − g − 2 v v = − g ⎜1 + 2 ⎜ dt vt vt ⎝ Letting an be the acceleration of the ball at time tn, express its position and speed when t = tn + 1: yn+1 = yn + 1 (vn + vn−1 )∆t 2 ⎞ ⎟ ⎟ ⎠ and vn+1 = vn + an ∆t where ⎛ v v a n = − g ⎜1 + n 2 n ⎜ vt ⎝ ⎞ ⎟ ⎟ ⎠ and ∆t is an arbitrarily small interval of time. A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell D11 E10 E11 F10 F11 Formula/Content D10+$B$6 41.7 E10−$B$4* (1+E10*ABS(E10)/($B$5^2))*$B$6 0 F10+0.5*(E10+E11)*$B$6 G10 G11 0 $E$10*D11−0.5*$B$4*D11^2 Algebraic Form t + ∆t v0 vn+1 = vn + an ∆t y0 yn+1 = yn + 1 (vn + vn−1 )∆t 2 y0 v0t − 1 gt 2 2 Applications of Newton’s Laws 353 D E F G t 0.0 0.1 0.2 v 41.70 39.74 37.87 y 0.00 4.07 7.95 y no drag 0.00 4.12 8.14 40 41 42 43 44 45 46 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.01 2.03 1.05 0.07 −0.91 −1.89 −2.87 60.13 60.39 60.54 60.60 60.55 60.41 60.17 81.00 82.18 83.26 84.25 85.14 85.93 86.62 78 79 80 81 6.8 6.9 7.0 7.1 −28.34 −28.86 −29.37 −29.87 6.26 3.41 0.49 −2.47 56.98 54.44 51.80 49.06 4 5 6 7 8 9 10 11 12 A B C g= 9.81 m/s^2 vt= 41.7 m/s ∆t= 0.1 s From the table we can see that, after 3.5 s, the ball reaches a height of about 60.4 m. It reaches its peak a little earlier−at about 3.3 s, and its height at t = 3.3 s is 60.6 m. The ball hits the ground at about t = 7 s −so it spends a little longer coming down than going up. The solid curve on the following graph shows y(t) when there is no drag on the baseball and the dotted curve shows y(t) under the conditions modeled in this problem. 354 Chapter 5 90 80 70 y (m) 60 50 40 x with drag 30 x with no drag 20 10 0 0 1 2 3 4 5 6 7 t (s) 100 •• Picture the Problem The pictorial representation shows the block in its initial position against the compressed spring, later as the spring accelerates it to the right, and finally when it has reached its maximum speed at xf = 0. In order to use Euler’s method, we’ll need to determine how the acceleration of the block varies with its position. We can do this by applying Newton’s 2nd law to the box. We can then use vn+1 = vn + an ∆t and xn +1 = xn + vn ∆t to find the speed and position of the block. Apply ∑F x = max to the block: Solve for an: Express the position and speed of the block when t = tn + 1: k (0.3 m − xn ) = man an = k (0.3 m − xn ) m xn +1 = xn + vn ∆t and vn+1 = vn + an ∆t where an = k (0.3 m − xn ) m and ∆t is an arbitrarily small interval of time. A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Applications of Newton’s Laws 355 Cell A10 B10 Formula/Content A9+$B$1 B9+C10*$B$1 Algebraic Form t + ∆t C10 C9+D9*$B$1 vn + an ∆t k (0.3 − xn ) m D10 ($B$4/$B$5)*(0.3−B10) A ∆t= x0= v0= k= m= B 0.005 0 0 50 0.8 xn + vn ∆t C D 1 2 3 4 5 6 7 8 9 10 11 12 t (s) 0.000 0.005 0.010 0.015 x (m) 0.00 0.00 0.00 0.00 v (m/s) 0.00 0.09 0.19 0.28 a (m/s^2) 18.75 18.72 18.69 18.63 45 46 47 48 49 0.180 0.185 0.190 0.195 0.200 0.25 0.27 0.28 0.29 0.30 2.41 2.42 2.43 2.44 2.44 2.85 2.10 1.34 0.58 −0.19 s m m/s N/m kg From the table we can see that it took about 0.200 s for the spring to push the block 30 cm and that it was traveling about 2.44 m/s at that time. We can estimate the uncertainty in this result by halving ∆t and recalculating the speed of the ball at t = 10 s. Doing so yields v(0.200 s) ≈ 2.41 m/s, a difference of about 1.2%. 356 Chapter 5 General Problems 101 • Picture the Problem The forces that act on the block as it slides down the incline are shown on the free-body diagram to the right. The acceleration of the block can be determined from the distance-and-time information given in the problem. The application of Newton’s 2nd law to the block will lead to an expression for the coefficient of kinetic friction as a function of the block’s acceleration and the angle of the incline. Apply r r ∑ F = ma to the block: ΣFx = mgsinθ − fk = ma and ΣFy = Fn − mg = 0 g sin θ − a g cosθ Set fk = µkFn, Fn between the two equations, and solve for µk: µk = Using a constant-acceleration equation, relate the distance the block slides to its sliding time: ∆x = v0 ∆t + 1 a (∆t ) where v0 = 0 2 Solve for a: a= 2∆x (∆t )2 Substitute numerical values and evaluate a: a= 2(2.4 m ) = 0.1775 m/s 2 2 (5.2 s ) Find µk for a = 0.1775 m/s2 and θ = 28°: 2 µk (9.81m/s ) sin28° − 0.1775 m/s = (9.81m/s ) cos28° 2 2 = 0.511 2 Applications of Newton’s Laws 357 102 • Picture the Problem The free-body diagram shows the forces acting on the model airplane. The speed of the plane can be calculated from the data concerning the radius of its path and the time it takes to make one revolution. The application of Newton’s 2nd law will give us the tension F in the string. (a) Express the speed of the airplane in terms of the circumference of the circle in which it is flying and its period: v= Substitute numerical values and evaluate v: v= (b) Apply ∑F x = max to the model airplane: Substitute numerical values and evaluate F: 2πr T 2π(5.7 m ) = 10.7 m/s 4 s 1.2 F =m v2 r F = (0.4 kg ) (10.7 m/s)2 5.7 m = 8.03 N *103 •• Picture the Problem The free-body diagram shows the forces acting on the box. If the student is pushing with a force of 200 N and the box is on the verge of moving, the static friction force must be at its maximum value. In part (b), the motion is impending up the incline; therefore the direction of fs,max is down the incline. (a) Apply r r ∑ F = ma to the box: ∑F x = f s + F − mg sin θ = 0 and ∑F y = Fn − mg cosθ = 0 358 Chapter 5 F mg cosθ Substitute fs = fs,max = µsFn, eliminate Fn between the two equations, and solve for µs: µs = tan θ − Substitute numerical values and evaluate µs: µs = tan 30° − 200 N (800 N )cos30° = 0.289 (b) Find fs,max from the x-direction force equation: Substitute numerical values and evaluate fs,max: f s,max = mg sin θ − F f s,max = (800 N )sin30° − 200 N = 200 N If the block is on the verge of sliding up the incline, fs,max must act down the incline. The x-direction force equation becomes: − f s, max + F − mg sin θ = 0 Solve the x-direction force equation for F: F = mg sin θ + f s,max Substitute numerical values and evaluate F: F = (800 N )sin30° + 200 N = 600 N 104 • Picture the Problem The path of the particle is a circle if r is a constant. Once we have shown that it is, we can calculate its value from its components. The direction of the particle’s motion can be determined by examining two positions of the particle at times that are close to each other. (a) and (b) Express the magnitude of r r in terms of its components: r = rx2 + ry2 Evaluate r with rx = −10 m cos ωt and ry = 10 m sinωt: r= [(− 10 m )cos ωt ] 2 + [(10 m )sin ωt ] 2 ( ) = 100 cos 2ωt + sin 2 ωt m = 10.0 m Applications of Newton’s Laws 359 (c) Evaluate rx and ry at t = 0 s: Evaluate rx and ry at t = ∆t, where ∆t is small: rx = −(10 m ) cos 0° = −10 m ry = (10 m )sin 0° = 0 rx = −(10 m )cos ω∆t ≈ −(10 m )cos 0° = −10 m ry = (10 m )sin ω∆t = ∆y where ∆y is positive the motion is clockwise and r (d) Differentiate r with respect to r time to obtain v : r Use the components of v to find its speed: r r v = dr / dt ˆ = [(10ω sin ωt ) m] i + [(10ω cos ωt )m] ˆ j 2 2 v = vx + v y [(10ω sin ωt )m] 2 + [(10ω cos ωt )m] 2 = (10 m )ω = (10 m )(2 s −1 ) = = 20.0 m/s (e) Relate the period of the particle’s motion to the radius of its path and its speed: T= 2πr 2π (10 m ) = = πs v 20 m/s 105 •• Picture the Problem The free-body diagram shows the forces acting on the crate of books. The kinetic friction force opposes the motion of the crate up the incline. Because the crate is moving at constant speed in a straight line, its acceleration is zero. We can determine F by applying Newton’s 2nd law to the crate, substituting for fk, eliminating the normal force, and solving for the required force. Apply r r ∑ F = ma to the crate, with both ax and ay equal to zero, to the crate: ∑F x = F cosθ − f k − mg sin θ = 0 and ∑F y = Fn − F sin θ − mg cosθ = 0 360 Chapter 5 Substitute µsFn for fk and eliminate Fn to obtain: F= mg (sin θ + µ k cos θ ) cos θ − µ k sin θ Substitute numerical values and evaluate F: F= (100 kg )(9.81m/s2 )(sin30° + (0.5)cos30°) cos30° − (0.5)sin30° = 1.49 kN 106 •• Picture the Problem The free-body diagram shows the forces acting on the object as it slides down the inclined plane. We can calculate its speed at the bottom of the incline from its acceleration and displacement and find its acceleration from Newton’s 2nd law. Using a constant-acceleration equation, relate the initial and final velocities of the object to its acceleration and displacement: solve for the final velocity: Apply r r ∑ F = ma to the sliding 2 v 2 = v0 + 2a∆x Because v0 = 0, v = ∑F x 2a∆x (1) = − f k + mg sin θ = ma and object: ∑F Solve the y equation for Fn and using fk = µkFn, eliminate both Fn and fk from the x equation and solve for a: Substitute equation (2) in equation (1) and solve for v: y = Fn − mg cosθ = 0 a = g (sin θ − µ k cosθ ) (2) v = 2 g (sin θ − µ k cosθ )∆x Substitute numerical values and evaluate v: ( ) v = 2 9.81m/s 2 (sin 30° − (0.35)cos 30°)(72 m ) = 16.7 m/s and (d ) is correct. Applications of Newton’s Laws 361 *107 •• Picture the Problem The free-body diagram shows the forces acting on the brick as it slides down the inclined plane. We’ll apply Newton’s 2nd law to the brick when it is sliding down the incline with constant speed to derive an expression for µk in terms of θ0. We’ll apply Newton’s 2nd law a second time for θ = θ1 and solve the equations simultaneously to obtain an expression for a as a function of θ0 and θ1. Apply r r ∑ F = ma to the brick ∑F x = − f k + mg sin θ 0 = 0 when it is sliding with constant speed: and Solve the y equation for Fn and using fk = µkFn, eliminate both Fn and fk from the x equation and solve for µk: µ k = tan θ 0 Apply r r ∑ F = ma to the brick when θ = θ1: Solve the y equation for Fn, use fk = µkFn to eliminate both Fn and fk from the x equation, and use the expression for µk obtained above to obtain: ∑F y ∑F x = Fn − mg cosθ 0 = 0 = − f k + mg sin θ 1 = ma and ∑F y = Fn − mg cosθ 1 = 0 a = g (sin θ 1 − tan θ 0 cosθ 1 ) 108 •• Picture the Problem The fact that the object is in static equilibrium under the influence r r r of the three forces means that F1 + F2 + F3 = 0. Drawing the corresponding force triangle will allow us to relate the forces to the angles between them through the law of sines and the law of cosines. 362 Chapter 5 (a) Using the fact that the object is in static equilibrium, redraw the force diagram connecting the forces head-to-tail: Apply the law of sines to the triangle: Use the trigonometric identity sin(π − α) = sinα to obtain: (b) Apply the law of cosines to the triangle: Use the trigonometric identity cos(π − α) = −cosα to obtain: F3 F1 F2 = = sin (π − θ 23 ) sin (π − θ13 ) sin (π − θ12 ) F1 F2 F3 = = sin θ 23 sin θ13 sin θ12 F12 = F22 + F32 − 2 F2 F3 cos(π − θ 23 ) F12 = F22 + F32 + 2 F2 F3 cos θ 23 109 •• Picture the Problem We can calculate the acceleration of the passenger from his/her speed that, in turn, is a function of the period of the motion. To determine the longest period of the motion, we focus our attention on the situation at the very top of the ride when the seat belt is exerting no force on the rider. We can use Newton’s 2nd law to relate the period of the motion to the acceleration and speed of the rider. (a) Because the motion is at constant speed, the acceleration is entirely radial and is given by: v2 ac = r Express the speed of the motion of the ride as a function of the radius of the circle and the period of its motion: v= 2π r T Applications of Newton’s Laws 363 Substitute in the expression for ac to obtain: 4π 2 r ac = 2 T Substitute numerical values and evaluate ac: ac = (b) Apply r r F = ma to the ∑ 4π 2 (5 m ) = 49.3 m/s 2 2 (2 s ) ∑F r = mg = ma c passenger when he/she is at the top of the circular path and solve for ac: and ac = g Relate the acceleration of the motion to its radius and speed and solve for v: v2 g = ⇒ v = gr r Express the period of the motion as a function of the radius of the circle and the speed of the passenger and solve for Tm: Substitute numerical values and evaluate Tm: Tm = 2π r r = 2π v g Tm = 2π 5m = 4.49 s 9.81 m/s 2 Remarks: The rider is ″weightless″ under the conditions described in part (b). *110 •• Picture the Problem The pictorial representation to the right shows the cart and its load on the inclined plane. The load will not slip provided its maximum acceleration is not exceeded. We can find that maximum acceleration by applying Newton’s 2nd law to the load. We can then apply Newton’s 2nd law to the cart-plusload system to determine the tension in the rope when the system is experiencing its maximum acceleration. 364 Chapter 5 Draw the free-body diagram for the cart and its load: Apply ∑F x = max to the cart plus T − (m1 + m2 )g sin θ = (m1 + m2 )amax (1) its load: Draw the free-body diagram for the load of mass m2 on top of the cart: Apply r r F = ma to the load on ∑ top of the cart: ∑F x = f s,max − m2 g sin θ = m2 a max and ∑F y = Fn , 2 − m2 g cosθ = 0 Using fs,max = µsFn,2, eliminate Fn,2 between the two equations and solve for the maximum acceleration of the load: a max = g (µ s cosθ − sin θ ) Substitute equation (2) in equation (1) and solve for T : T= 111 •• Picture the Problem The free-body diagram for the sled while it is held stationary by the static friction force is shown to the right. We can solve this problem by repeatedly applying Newton’s 2nd law under the conditions specified in each part of the problem. (m1 + m2 )gµ s cosθ (2) Applications of Newton’s Laws 365 (a) Apply ∑F y = ma y to the sled: Fn,1 − m1 g cosθ = 0 Solve for Fn,1: Fn,1 = m1 g cosθ Substitute numerical values and evaluate Fn,1: Fn,1 = (200 N ) cos15° = 193 N (b) Apply ∑F x = max to the sled: f s − m1 g sin θ = 0 Solve for fs: f s = m1 g sin θ Substitute numerical values and evaluate fs: f s = (200 N )sin 15° = 51.8 N (c) Draw the free-body diagram for the sled when the child is pulling on the rope: Apply r r ∑ F = ma to the sled to ∑F x = Fnet = F cos 30° − m1 g sin θ − f s,max determine whether it moves: and ∑F Solve the y-direction equation for Fn,1: Substitute numerical values and evaluate Fn,1: y = Fn,1 + F sin 30° − m1 g cosθ = 0 Fn,1 = − F sin 30° + m1 g cosθ Fn,1 = −(100 N )sin 30° + (200 N )cos15° = 143 N Express fs,max: fs,max = µsFn,1 = (0.5)(143 N) = 71.5 N Use the x-direction force equation to evaluate Fnet: Fnet = (100 N)cos30° − (200 N)sin15° − 71.5 N = −36.7 N 366 Chapter 5 Because the net force is negative, the sled does not move: f k is undetermined (d) Because the sled does not move: µ k is undetermined (e) Draw the FBD for the child: Express the net force Fc exerted on the child by the incline: Noting that the child is stationary, r r apply F = ma to the child: ∑ 2 Fc = Fn2 + f s,2max ∑F x (1) = f s,max − F cos 30° − m2 g sin 15° =0 and ∑F y Solve the x equation for fs,max and the y equation for Fn2: = Fn2 − m2 g sin 15° − F sin 30° = 0 f s,max = F cos 30° + m2 g sin 15° and Fn2 = m2 g sin 15° + F sin 30° Substitute numerical values and evaluate Fx and Fn2: f s,max = (500 N ) cos 30° + (100 N ) sin 15° = 459 N and Fn2 = (100 N )sin 15° + (500 N )sin 30° = 276 N Substitute numerical values in equation (1) and evaluate F: Fc = (276 N )2 + (459 N )2 = 536 N 112 • Picture the Problem Let v represent the speed of rotation of the station, and r the distance from the center of the station. Because the O’Neill colony is, presumably, in deep space, the only acceleration one would experience in it would be that due to its rotation. Applications of Newton’s Laws 367 (a) Express the acceleration of anyone who is standing inside the station: a = v2/r This acceleration is directed toward the axis of rotation. If someone inside the station drops an apple, the apple will not have any forces acting on it once released, but will move along a straight line at constant speed. However, from the point of view of our observer inside the station, if he views himself as unmoving, the apple is perceived to have an acceleration of mv2/r directed away from the axis of rotation (a "centrifugal" force). (b) Each deck must rotate the central axis with the same period T. Relate the speed of a person on a particular deck to his/her distance r from the center: Express the "acceleration of gravity" perceived by someone a distance r from the center: (c) Relate the desired acceleration to the radius of Babylon 5 and its period: Solve for T: v= 2π r T v 2 4π 2 r = r T2 i.e., the " acceleration due to gravity" decreases as r decreases. a= T= Substitute numerical values and evaluate T: T= 4π 2 r T2 4π 2 r a 1.609 km ⎞ ⎛ 4π 2 ⎜ 0.3 mi × ⎟ mi ⎠ ⎝ 9.8 m/s 2 = 44.1s = 0.735 min Take the reciprocal of this time to find the number of revolutions per minute Babylon 5 has to make in order to provide this ″earth-like″ acceleration: T −1 = 1.36 rev / min 368 Chapter 5 113 •• Picture the Problem The free-body diagram shows the forces acting on the child as she slides down the incline. We’ll first use Newton’s 2nd law to derive an expression for µk in terms of her acceleration and then use Newton’s 2nd law to find her acceleration when riding the frictionless cart. Using a constantacceleration equation, we’ll relate these two accelerations to her descent times and solve for her acceleration when sliding. Finally, we can use this acceleration in the expression for µk. Apply r r ∑ F = ma to the child as she slides down the incline: Using fk = µkFn, eliminate fk and Fn between the two equations and solve for µk: Apply ∑F x = max to the child as ∑F x = mg sin 30° − f k = ma1 and ∑F y = Fn − mg cos 30° = 0 µ k = tan 30° − a1 g cos 30° mg sin 30° = ma2 she rides the frictionless cart down the incline and solve for her acceleration a2: and Letting s represent the distance she slides down the incline, use a constant-acceleration equation to relate her sliding times to her accelerations and distance traveled down the slide : s = v0t1 + 1 a1t12 where v0 = 0 2 Equate these expressions, substitute t2 = 1 t1 and solve for a1: 2 a 2 = g sin 30° = 4.91 m/s 2 and 2 s = v0t2 + 1 a2t2 where v0 = 0 2 a1 = 1 a2 = 1 g sin 30° = 1.23 m/s2 4 4 (1) Applications of Newton’s Laws 369 1.23 m/s 2 µ k = tan 30° − 9.81 m/s 2 cos 30° Evaluate equation (1) with a1 = 1.23 m/s2: ( ) = 0.433 *114 •• Picture the Problem The path of the particle is a circle if r is a constant. Once we have shown that it is, we can calculate its value from its components and determine the particle’s velocity and acceleration by differentiation. The direction of the net force acting on the particle can be determined from the direction of its acceleration. r (a) Express the magnitude of r in terms of its components: r = rx2 + ry2 Evaluate r with rx = Rsinωt and ry = Rcosωt: r= [R sin ωt ] 2 + [R cos ωt ] 2 ( ) = R 2 sin 2ωt + cos 2 ωt = R = 4.0 m ∴the path of the particle is a circle centered at the origin. r (b) Differentiate r with respect to r time to obtain v : r r ˆ v = dr / dt = [Rω cos ω t ] i + [− Rω sin ω t ] ˆ j = Express the ratio vx : vy Express the ratio − y : x vx 8π cos ω t = = − cot ω t v y − 8π sin ω t − ∴ r (c) Differentiate v with respect to r time to obtain a : [(8π cos 2π t ) m/s] iˆ − [(8π sin 2π t ) m/s] ˆ j R cos ω t y =− = − cot ω t x R sin ω t vx y =− vy x r r a = dv / dt [(− 16π m/s )sin ω t ] iˆ + [(− 16π m/s )cos ω t ] ˆ j 2 = 2 2 2 370 Chapter 5 r Factor −4π2/s2 from a to obtain: ( )[ ) ( circle in which the particle is traveling. Find the ratio (d) Apply v 2 (8π m/s ) = = 16π 2 m/s 2 = a r 4m v2 : r r 2 r ∑ F = ma to the particle: Fnet = ma = (0.8 kg )(16π 2 m/s 2 ) = 12.8π 2 N r Because the direction of Fnet is the r r Fnet is toward the center of the circle. same as that of a : 115 •• Picture the Problem The free-body diagram showing the forces acting on a rider being held in place by the maximum static friction force is shown to the right. The application of Newton’s 2nd law and the definition of the maximum static friction force will be used to determine the period T of the motion. The reciprocal of the period will give us the minimum number of revolutions required per unit time to hold the riders in place. Apply r r ∑ F = ma to the riders while they are held in place by friction: Using f s,max = µ s Fn and v = 2π r , T eliminate Fn between the force equations and solve for the period of the motion: r ˆ a = − 4π 2 / s 2 (4 sin ω t ) i + (4 cos ω t ) ˆ j r = − 4π 2 / s 2 r r Because a is in the opposite direction from r r , it is directed toward the center of the ∑ Fx = Fn = m v2 r and ∑F y = f s,max − mg = 0 T = 2π µs r g Applications of Newton’s Laws 371 Substitute numerical values and evaluate T: T = 2π (0.4)(4 m ) 9.81 m/s 2 = 2.54 s = 0.00423 min The number of revolutions per minute is the reciprocal of the period in minutes: 23.6 rev/min 116 •• Picture the Problem The free-body diagrams to the right show the forces acting on the blocks whose masses are m1 and m2. The application of Newton’s 2nd law and the use of a constant-acceleration equation will allow us to find a relationship between the coefficient of kinetic friction and m1. The repetition of this procedure with the additional object on top of the object whose mass is m1 will lead us to a second equation that, when solved simultaneously with the former equation, leads to a quadratic equation in m1. Finally, its solution will allow us to substitute in an expression for µk and determine its value. Using a constant-acceleration equation, relate the displacement of the system in its first configuration as a function of its acceleration and fall time: ∆x = v0 ∆t + 1 a1 (∆t ) 2 2 or, because v0 = 0, ∆x = 1 a1 (∆t ) 2 2 Solve for a1: a1 = 2∆x (∆t )2 Substitute numerical values and evaluate a1: a1 = 2(1.5 m ) = 4.46 m/s 2 2 (0.82 s ) Apply ∑F x = max to the object whose mass is m2 and solve for T1: m2 g − T1 = m2 a1 and T1 = m2 (g − a ) 372 Chapter 5 Substitute numerical values and evaluate T1: Apply r r ∑ F = ma to the object whose mass is m1: ( T1 = (2.5 kg ) 9.81 m/s 2 − 4.46 m/s 2 = 13.375 N ∑F = T1 − f k = m1 a1 x and ∑F y = Fn,1 − m1 g = 0 Using fk = µkFn, eliminate Fn between the two equations to obtain: T1 − µ k m1 g = m1 a1 Find the acceleration a2 for the second run: a2 = Evaluate T2: T2 = m2 ( g − a ) Apply ∑F x ) (1) 2∆x 2(1.5 m ) = = 1.775 m/s 2 2 2 (∆t ) (1.3 s ) ( = (2.5 kg ) 9.81 m/s 2 − 1.775 m/s 2 = 20.1 N = max to the 1.2-kg T2 − µ k (m1 + 1.2 kg )g (2) = (m1 + 1.2 kg )a2 object in place: ) T1 − m1a1 m1 g Solve equation (1) for µk: µk = Substitute for µk in equation (2) and simplify to obtain the quadratic equation in m1: 2.685m12 + 9.947 m1 − 16.05 = 0 Solve the quadratic equation to obtain: m1 = (− 1.85 ± 3.07 )kg ⇒ m1 = 1.22 kg Substitute numerical values in equation (3) and evaluate µk: µk = (3) ( 13.375 N − (1.22 kg ) 4.66 m/s 2 (1.22 kg ) 9.81m/s2 = 0.643 ( ) ) Applications of Newton’s Laws 373 *117 ••• Picture the Problem The diagram shows a point on the surface of the earth at latitude θ. The distance R to the axis of rotation is given by R = rcosθ. We can use the definition of centripetal acceleration to express the centripetal acceleration of a point on the surface of the earth due to the rotation of the earth. (a) Referring to the figure, express ac for a point on the surface of the earth at latitude θ : v2 ac = where R = rcosθ R Express the speed of the point due to the rotation of the earth: v= Substitute for v in the expression for ac and simplify to obtain: ac = Substitute numerical values and evaluate ac: 4π 2 (6370 km )cosθ ac = [(24 h )(3600 s/h )] 2 2πR T where T is the time for one revolution. = 4π 2 r cosθ T2 (3.37 cm/s )cosθ , toward the 2 earth' s axis. (b) A stone dropped from a hand at a location on earth. The effective weight of r r the stone is equal to mast, surf , where ast, surf is the acceleration of the falling stone (neglecting air resistance) relative to the local surface of the earth. The r r gravitational force on the stone is equal to mast,iner , where ast, iner is the acceleration of the local surface of the earth relative to the inertial frame (the acceleration of the surface due to the rotation of the earth). Multiplying r r r through this equation by m and rearranging gives mast, surf = mast, iner − masurf, iner , which relates the apparent weight to the acceleration due to gravity and the acceleration due to the earth' s rotation. A vector addition diagram can be used r r to show that the magnitude of mast, surf is slightly less than that of mast, iner . 374 Chapter 5 (c) At the equator, the gravitational acceleration and the radial acceleration are both directed toward the center of the earth. Therefore: g = g eff + ac ( ) = 978 cm/s2 + 3.37 cm/s 2 cos0° = 981.4 cm/s 2 At latitude θ the gravitational acceleration points toward the center of the earth whereas the centripetal acceleration points toward the axis of rotation. Use the law of cosines to relate geff, g, and a c: 2 g eff = g 2 + a c2 − 2 ga c cosθ Substitute for θ, geff, and ac and simplify to obtain the quadratic equation: g 2 − 4.75 cm/s 2 g − 962350 cm 2 /s 4 = 0 Solve for the physically meaningful (i.e., positive) root to obtain: g = 983 cm/s 2 ( ) *118 ••• Picture the Problem The diagram shows the block in its initial position, an intermediate position, and as it is separating from the sphere. Because the sphere is frictionless, the only forces acting on the block are the normal and gravitational forces. We’ll apply Newton’s 2nd law and set Fn equal to zero to determine the angle θc at which the block leaves the surface. Taking the inward direction to be positive, apply Fr = mar to the ∑ v2 mg cosθ − Fn = m R block: Apply the separation condition to obtain: Solve for cosθc: Apply ∑F t = mat to the block: mg cos θ c = m v2 cosθ c = gR mg sin θ = mat or v2 R (1) Applications of Newton’s Laws 375 at = dv = g sin θ dt Note that a is not constant and, hence, we cannot use constant-acceleration equations. Multiply the left-hand side of the equation by one in the form of dθ/dθ and rearrange to obtain: dv dθ = g sin θ dt dθ and dθ dv = g sin θ dt dθ Relate the arc distance s the block travels to the angle θ and the radius R of the sphere: θ= dθ 1 ds v s = = and R dt R dt R where v is the block’s instantaneous speed. Substitute to obtain: v dv = g sin θ R dθ Separate the variables and integrate from v′ = 0 to v and θ = 0 to θc: v θc 0 0 ∫ v'dv' = gR ∫ sin θdθ or v 2 = 2 gR(1 − cosθ c ) Substitute in equation (1) to obtain: Solve for and evaluate θc: 2 gR(1 − cos θ c ) gR = 2(1 − cos θ c ) cos θ c = ⎛2⎞ ⎝3⎠ θ c = cos −1 ⎜ ⎟ = 48.2° 376 Chapter 5 Chapter 6 Work and Energy Conceptual Problems *1 • Determine the Concept A force does work on an object when its point of application moves through some distance and there is a component of the force along the line of motion. (a) False. The net force acting on an object is the vector sum of all the forces acting on the object and is responsible for displacing the object. Any or all of the forces contributing to the net force may do work. (b) True. The object could be at rest in one reference frame and moving in another. If we consider only the frame in which the object is at rest, then, because it must undergo a displacement in order for work to be done on it, we would conclude that the statement is true. (c) True. A force that is always perpendicular to the velocity of a particle changes neither it’s kinetic nor potential energy and, hence, does no work on the particle. 2 • Determine the Concept If we ignore the work that you do in initiating the horizontal motion of the box and the work that you do in bringing it to rest when you reach the second table, then neither the kinetic nor the potential energy of the system changed as you moved the box across the room. Neither did any forces acting on the box produce displacements. Hence, we must conclude that the minimum work you did on the box is zero. 3 • False. While it is true that the person’s kinetic energy is not changing due to the fact that she is moving at a constant speed, her gravitational potential energy is continuously changing and so we must conclude that the force exerted by the seat on which she is sitting is doing work on her. *4 • Determine the Concept The kinetic energy of any object is proportional to the square of its speed. Because K = 1 mv 2 , replacing v by 2v yields 2 ( ) K' = 1 m(2v ) = 4 1 mv 2 = 4 K . Thus doubling the speed of a car quadruples its kinetic 2 2 2 energy. 371 372 Chapter 6 5 • r Determine the Concept No. The work done on any object by any force F is defined as r r r dW = F ⋅ dr . The direction of Fnet is toward the center of the circle in which the object r is traveling and dr is tangent to the circle. No work is done by the net force because r r Fnet and dr are perpendicular so the dot product is zero. 6 • Determine the Concept The kinetic energy of any object is proportional to the square of its speed and is always positive. Because K = 1 mv 2 , replacing v by 3v yields 2 ( ) K' = 1 m(3v ) = 9 1 mv 2 = 9 K . Hence tripling the speed of an object increases its 2 2 2 kinetic energy by a factor of 9 and (d ) is correct. *7 • Determine the Concept The work required to stretch or compress a spring a distance x is given by W = 1 kx 2 where k is the spring’s stiffness constant. Because W ∝ x2, doubling 2 the distance the spring is stretched will require four times as much work. 8 • Determine the Concept No. We know that if a net force is acting on a particle, the particle must be accelerated. If the net force does no work on the particle, then we must conclude that the kinetic energy of the particle is constant and that the net force is acting perpendicular to the direction of the motion and will cause a departure from straight-line motion. 9 • Determine the Concept We can use the definition of power as the scalar product of force and velocity to express the dimension of power. r r Power is defined as: P ≡ F ⋅v Express the dimension of force: [M][L/T 2] Express the dimension of velocity: [L/T] Express the dimension of power in terms of those of force and velocity: [M][L/T 2][L/T] = [M][L]2/[T]3 and (d ) is correct. Work and Energy 373 10 • Determine the Concept The change in gravitational potential energy, over elevation changes that are small enough so that the gravitational field can be considered constant, is mg∆h, where ∆h is the elevation change. Because ∆h is the same for both Sal and Joe, their gains in gravitational potential energy are the same. (c) is correct. 11 • (a) False. The definition of work is not limited to displacements caused by conservative forces. (b) False. Consider the work done by the gravitational force on an object in freefall. (c) True. This is the definition of work done by a conservative force. *12 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x; i.e., Fx = − dU dx . (a) Examine the slopes of the curve at each of the lettered points, remembering that Fx is the negative of the slope of the potential energy graph, to complete the table: (b) Find the point where the slope is steepest: Point dU/dx Fx + A − 0 0 B + C − 0 0 D + E − 0 0 F At point C Fx is greatest. (c) If d2U/dx2 < 0, then the curve is concave downward and the equilibrium is unstable. At point B the equilibrium is unstable. If d2U/dx2 > 0, then the curve is concave upward and the equilibrium is stable. At point D the equilibrium is stable. Remarks: At point F, d2U/dx2 = 0 and the equilibrium is neither stable nor unstable; it is said to be neutral. 374 Chapter 6 13 • (a) False. Any force acting on an object may do work depending on whether the force produces a displacement … or is displaced as a consequence of the object’s motion. (b) False. Consider an element of area under a force-versus-time graph. Its units are N⋅s whereas the units of work are N⋅m. 14 • r r r Determine the Concept Work dW = F ⋅ ds is done when a force F produces a ( ) r r r displacement ds . Because F ⋅ ds ≡ Fds cos θ = (F cos θ ) ds, W will be negative if the value of θ is such that Fcosθ is negative. (d ) is correct. Estimation and Approximation *15 •• Picture the Problem The diagram depicts the situation when the tightrope walker is at the center of rope. M represents her mass and the vertical components of tensions r r T1 and T2 , equal in magnitude, support her weight. We can apply a condition for static equilibrium in the vertical direction to relate the tension in the rope to the angle θ and use trigonometry to find s as a function of θ. (a) Use trigonometry to relate the sag s in the rope to its length L and θ: Apply ∑F y = 0 to the tightrope walker when she is at the center of the rope to obtain: Solve for θ to obtain: Substitute numerical values and evaluate θ : tan θ = s L and s = tan θ 1 2 2 L 2T sin θ − Mg = 0 where T is the r r magnitude of T1 and T2 . ⎛ Mg ⎞ ⎟ ⎝ 2T ⎠ θ = sin −1 ⎜ ( ) ⎡ (50 kg ) 9.81 m/s 2 ⎤ ⎥ = 2.81° 2(5000 N ) ⎣ ⎦ θ = sin −1 ⎢ Work and Energy 375 Substitute to obtain: s= (b) Express the change in the tightrope walker’s gravitational potential energy as the rope sags: ∆U = U at center − U end = Mg∆y Substitute numerical values and evaluate ∆U: 10 m tan 2.81° = 0.245 m 2 ( ) ∆U = (50 kg ) 9.81 m/s 2 (− 0.245 m ) = − 120 J 16 • Picture the Problem You can estimate your change in potential energy due to this change in elevation from the definition of ∆U. You’ll also need to estimate the height of one story of the Empire State building. We’ll assume your mass is 70 kg and the height of one story to be 3.5 m. This approximation gives us a height of 1170 ft (357 m), a height that agrees to within 7% with the actual height of 1250 ft from the ground floor to the observation deck. We’ll also assume that it takes 3 min to ride non-stop to the top floor in one of the high-speed elevators. (a) Express the change in your gravitational potential energy as you ride the elevator to the 102nd floor: Substitute numerical values and evaluate ∆U: ∆U = mg∆h ( ) ∆U = (70 kg ) 9.81m/s 2 (357 m ) = 245 kJ (b) Ignoring the acceleration intervals at the beginning and the end of your ride, express the work done on you by the elevator in terms of the change in your gravitational potential energy: W = Fh = ∆U Solve for and evaluate F: F= ∆U 245 kJ = = 686 N h 357 m (c) Assuming a 3 minute ride to the top, express and evaluate the average power delivered to the elevator: P= ∆U 245 kJ = (3 min )(60 s/min ) ∆t = 1.36 kW 376 Chapter 6 17 • Picture the Problem We can find the kinetic energy K of the spacecraft from its definition and compare its energy to the annual consumption in the U.S. W by examining the ratio K/W. (10000 kg )(3 ×107 m/s) 2 Using its definition, express and evaluate the kinetic energy of the spacecraft: K = 1 mv 2 = 2 Express this amount of energy as a percentage of the annual consumption in the United States: K 4.50 × 1018 J ≈ ≈ 1% E 5 × 10 20 J 1 2 = 4.50 × 1018 J *18 •• Picture the Problem We can find the orbital speed of the Shuttle from the radius of its orbit and its period and its kinetic energy from K = 1 mv 2 . We’ll ignore the variation in 2 the acceleration due to gravity to estimate the change in the potential energy of the orbiter between its value at the surface of the earth and its orbital value. (a) Express the kinetic energy of the orbiter: K = 1 mv 2 2 Relate the orbital speed of the orbiter to its radius r and period T: v= Substitute and simplify to obtain: 2π 2 mr 2 ⎛ 2π r ⎞ K = m ⎟ = T2 ⎝ T ⎠ 2π r T 2 1 2 Substitute numerical values and evaluate K: K= ( ) 2π 2 8 × 10 4 kg [(200 mi + 3960 mi )(1.609 km/mi)] = 2.43 TJ [(90 min )(60 s / min )] 2 (b) Assuming the acceleration due to gravity to be constant over the 200 mi and equal to its value at the surface of the earth (actually, it is closer to 9 m/s2 at an elevation of 200 mi), express the change in gravitational potential energy of the orbiter, relative to the surface of the earth, as the Shuttle goes into orbit: 2 ∆U = mgh Work and Energy 377 Substitute numerical values and evaluate ∆U: ( )( ) ∆U = 8 ×10 4 kg 9.81 m/s 2 × (200 mi )(1.609 km/mi) = 0.253 TJ No, they shouldn' t be equal because there is more than just the force of gravity to consider here. When the shuttle is resting on the surface of (c) the earth, it is supported against the force of gravity by the normal force the earth exerts upward on it. We would need to take into consideration the change in potential energy of the surface of earth in its deformation under the weight of the shuttle to find the actual change in potential energy. 19 • Picture the Problem Let’s assume that the width of the driveway is 18 ft. We’ll also assume that you lift each shovel full of snow to a height of 1 m, carry it to the edge of the driveway, and drop it. We’ll ignore the fact that you must slightly accelerate each shovel full as you pick it up and as you carry it to the edge of the driveway. While the density of snow depends on the extent to which it has been compacted, one liter of freshly fallen snow is approximately equivalent to 100 mL of water. Express the work you do in lifting the snow a distance h: W = ∆U = mgh = ρ snowVsnow gh where ρ is the density of the snow. Using its definition, express the densities of water and snow: ρ snow = Divide the first of these equations by the second to obtain: ρ snow Vwater V or ρ snow = ρ water water = ρ water Vsnow Vsnow Substitute and evaluate the ρsnow: ρ snow = (103 kg/m 3 ) Calculate the volume of snow covering the driveway: ⎛ 10 ⎞ Vsnow = (50 ft )(18 ft )⎜ ft ⎟ ⎝ 12 ⎠ 28.32 L 10−3 m 3 = 750 ft 3 × × ft 3 L 3 = 21.2 m Substitute numerical values in the expression for W to obtain an estimate (a lower bound) for the work you would do on the snow in removing it: msnow m and ρ water = water Vsnow Vwater ( )( 100 mL = 100 kg/m 3 L )( ) W = 100 kg/m 3 21.2 m 3 9.81 m/s 2 (1 m ) = 20.8 kJ 378 Chapter 6 Work and Kinetic Energy *20 • Picture the Problem We can use 1 2 mv 2 to find the kinetic energy of the bullet. K = 1 mv 2 2 (a) Use the definition of K: = 1 2 (0.015 kg )(1.2 ×103 m/s) 2 = 10.8 kJ (b) Because K ∝ v2: K' = 1 K = 2.70 kJ 4 (c) Because K ∝ v2: K' = 4 K = 43.2 kJ 21 • Picture the Problem We can use 1 2 mv 2 to find the kinetic energy of the baseball and the jogger. (a) Use the definition of K: K = 1 mv 2 = 2 1 2 (0.145 kg )(45 m/s)2 = 147 J (b) Convert the jogger’s pace of 9 min/mi into a speed: ⎛ 1 mi ⎞ ⎛ 1 min ⎞ ⎛ 1609 m ⎞ v=⎜ ⎟ ⎟⎜ ⎟⎜ ⎜ 9 min ⎟ ⎜ 60 s ⎟ ⎜ 1 mi ⎟ ⎠ ⎠⎝ ⎠⎝ ⎝ = 2.98 m/s Use the definition of K: K = 1 mv 2 = 2 1 2 (60 kg )(2.98 m/s)2 = 266 J 22 • Picture the Problem The work done in raising an object a given distance is the product of the force producing the displacement and the displacement of the object. Because the weight of an object is the gravitational force acting on it and this force acts downward, the work done by gravity is the negative of the weight of the object multiplied by its displacement. The change in kinetic energy of an object is equal to the work done by the net force acting on it. (a) Use the definition of W: r r W = F ⋅ ∆y = F∆y Work and Energy 379 = (80 N)(3 m) = 240 J (b) Use the definition of W: r r r W = F ⋅ ∆y = −mg∆y, because F and r ∆y are in opposite directions. ∴ W = − (6 kg)(9.81 m/s2)(3 m) = − 177 J (c) According to the work-kinetic energy theorem: K = W + Wg = 240 J + (−177 J) = 63.0 J 23 • Picture the Problem The constant force of 80 N is the net force acting on the box and the work it does is equal to the change in the kinetic energy of the box. ( Using the work-kinetic energy theorem, relate the work done by the constant force to the change in the kinetic energy of the box: W = K f − K i = 1 m vf2 − vi2 2 Substitute numerical values and evaluate W: W = 1 2 ) (5 kg )[(68 m/s )2 − (20 m/s )2 ] = 10.6 kJ *24 •• Picture the Problem We can use the definition of kinetic energy to find the mass of your friend. Using the definition of kinetic energy and letting ″1″ denote your mass and speed and ″2″ your girlfriend’s, express the equality of your kinetic energies and solve for your girlfriend’s mass as a function of both your masses and speeds: Express the condition on your speed that enables you to run at the same speed as your girlfriend: 1 2 2 m1v12 = 1 m2 v2 2 and ⎛v ⎞ m2 = m1 ⎜ 1 ⎟ ⎜v ⎟ ⎝ 2⎠ v2 = 1.25v1 2 (1) (2) 380 Chapter 6 Substitute equation (2) in equation (1) to obtain: 2 ⎛v ⎞ ⎛ 1 ⎞ m2 = m1 ⎜ 1 ⎟ = (85 kg )⎜ ⎟ ⎜v ⎟ ⎝ 1.25 ⎠ ⎝ 2⎠ 2 = 54.4 kg Work Done by a Variable Force 25 •• Picture the Problem The pictorial representation shows the particle as it moves along the positive x axis. The particle’s kinetic energy increases because work is done on it. We can calculate the work done on it from the graph of Fx vs. x and relate its kinetic energy when it is at x = 4 m to its kinetic energy when it was at the origin and the work done on it by using the work-kinetic energy theorem. (a) Calculate the kinetic energy of the particle when it is at x = 0: (b) Because the force and displacement are parallel, the work done is the area under the curve. Use the formula for the area of a triangle to calculate the area under the F as a function of x graph: K 0 = 1 mv 2 = 2 1 2 (3 kg )(2 m/s) 2 = 6.00 J (base)(altitude) = 1 (4 m )(6 N ) 2 W0→4 = 1 2 = 12.0 J (c) Express the kinetic energy of the particle at x = 4 m in terms of its speed and mass and solve for its speed: v4 = Using the work-kinetic energy theorem, relate the work done on the particle to its change in kinetic energy and solve for the particle’s kinetic energy at x = 4 m: W0→4 = K4 – K0 2K 4 m (1) K 4 = K 0 + W0→4 = 6.00 J + 12.0 J = 18.0 J Work and Energy 381 Substitute numerical values in equation (1) and evaluate v4: 2(18.0 J ) = 3.46 m/s 3 kg v4 = *26 •• Picture the Problem The work done by this force as it displaces the particle is the area under the curve of F as a function of x. Note that the constant C has units of N/m3. Because F varies with position nonlinearly, express the work it does as an integral and evaluate the integral between the limits x = 1.5 m and x = 3 m: ( ) ∫ x' 3m W = C N/m 3 3 dx' 1.5 m ( ) [ x' ] (C N/m ) [(3 m) − (1.5 m) ] = = C N/m 3 3m 4 1 4 1.5 m 3 4 4 4 = 19C J 27 •• Picture the Problem The work done on the dog by the leash as it stretches is the area under the curve of F as a function of x. We can find this area (the work Lou does holding the leash) by integrating the force function. Because F varies with position nonlinearly, express the work it does as an integral and evaluate the integral between the limits x = 0 and x = x1: x1 ( ) W = ∫ − kx'−ax'2 dx' 0 [ = − 1 kx'2 − 1 ax'3 2 3 x1 0 = − 1 kx12 − 1 ax13 2 3 28 •• Picture the Problem The work done on an object can be determined by finding the area bounded by its graph of Fx as a function of x and the x axis. We can find the kinetic energy and the speed of the particle at any point by using the work-kinetic energy theorem. (a) Express W, the area under the curve, in terms of the area of one square, Asquare, and the number of squares n: W = n Asquare Determine the work equivalent of one square: W = (0.5 N)(0.25 m) = 0.125 J 382 Chapter 6 Estimate the number of squares under the curve between x = 0 and x = 2 m: n ≈ 22 Substitute to determine W: W = 22(0.125 J) = 2.75 J (b) Relate the kinetic energy of the object at x = 2 m, K2, to its initial kinetic energy, K0, and the work that was done on it between x = 0 and x = 2 m: K 2 = K 0 + W0→2 = 1 2 (3 kg )(2.40 m/s)2 + 2.75 J = 11.4 J (c) Calculate the speed of the object at x = 2 m from its kinetic energy at the same location: v= (d) Estimate the number of squares under the curve between x = 0 and x = 4 m: 2(11.4 J ) = 2.76 m/s 3 kg n ≈ 26 Substitute to determine W: (e) Relate the kinetic energy of the object at x = 4 m, K4, to its initial kinetic energy, K0, and the work that was done on it between x = 0 and x = 4 m: Calculate the speed of the object at x = 4 m from its kinetic energy at the same location: 2K 2 = m W = 26(0.125 J ) = 3.25 J K 4 = K 0 + W0→4 = 1 2 (3 kg )(2.40 m/s )2 + 3.25 J = 11.9 J v= 2K 4 = m 2(11.9 J ) = 2.82 m/s 3 kg *29 •• Picture the Problem We can express the mass of the water in Margaret’s bucket as the difference between its initial mass and the product of the rate at which it loses water and her position during her climb. Because Margaret must do work against gravity in lifting and carrying the bucket, the work she does is the integral of the product of the gravitational field and the mass of the bucket as a function of its position. (a) Express the mass of the bucket and the water in it as a function of m( y ) = 40 kg − ry Work and Energy 383 its initial mass, the rate at which it is losing water, and Margaret’s position, y, during her climb: Find the rate, r = ∆m , at which ∆y r= ∆m 20 kg = = 1 kg/m ∆y 20 m Margaret’s bucket loses water: m( y ) = 40 kg − ry = 40 kg − Substitute to obtain: 1 kg y m (b) Integrate the force Margaret exerts on the bucket, m(y)g, between the limits of y = 0 and y = 20 m: 20 m W=g ∫ 0 ( )[ 1 kg ⎞ ⎛ y ' ⎟dy ' = 9.81 m/s 2 (40 kg ) y '− 1 (1 kg/m ) y '2 ⎜ 40 kg − 2 m ⎠ ⎝ 20 m 0 = 5.89 kJ Remarks: We could also find the work Margaret did on the bucket, at least approximately, by plotting a graph of m(y)g and finding the area under this curve between y = 0 and y = 20 m. Work, Energy, and Simple Machines 30 • Picture the Problem The free-body diagram shows the forces that act on the block as it slides down the frictionless incline. We can find the work done by these forces as the block slides 2 m by finding their components in the direction of, or opposite to, the motion. When we have determined the work done on the block, we can use the work-kinetic energy theorem or a constant-acceleration equation to calculate its kinetic energy and its speed at any given location. 384 Chapter 6 From the free - body diagram, we see that the forces acting on the block are (a) a gravitational force that acts downward and the normal force that the incline exerts perpendicularly to the incline. Identify the component of mg that acts down the incline and calculate the work done by it: Express the work done by this force: Substitute numerical values and evaluate W: Fx = mg sin 60° W = Fx ∆x = mg∆x sin 60° ( ) W = (6 kg ) 9.81 m/s 2 (2 m )sin 60° = 102 J Remarks: Fn and mgcos60°, being perpendicular to the motion, do no work on the block (b) The total work done on the block is the work done by the net force: W = Fnet ∆x = mg∆x sin 60° ( ) = (6 kg ) 9.81 m/s 2 (2 m )sin 60° = 102 J (c) Express the change in the kinetic energy of the block in terms of the distance, ∆x, it has moved down the incline: ∆K = Kf – Ki = W = (mgsin60°)∆x or, because Ki = 0, Kf = W = (mgsin60°)∆x Relate the speed of the block when it has moved a distance ∆x down the incline to its kinetic energy at that location: v= Determine this speed when ∆x = 1.5 m: (d) As in part (c), express the change in the kinetic energy of the block in terms of the distance, ∆x, it has moved down the incline and 2K = m 2mg∆x sin 60° m = 2 g∆x sin 60° ( ) v = 2 9.81 m/s 2 (1.5 m )sin 60° = 5.05 m/s ∆K = Kf – Ki =W = (mg sin 60°)∆x and Work and Energy 385 solve for Kf: Kf = (mg sin 60°)∆x + Ki Substitute for the kinetic energy terms and solve for vf to obtain: vf = 2 g sin 60°∆x + vi2 Substitute numerical values and evaluate vf: ( ) vf = 2 9.81 m/s 2 (1.5 m ) sin60° + (2 m/s ) = 5.43 m/s 2 31 • Picture the Problem The free-body diagram shows the forces acting on the object as in moves along its circular path on a frictionless horizontal surface. We can use Newton’s 2nd law to obtain an expression for the tension in the string and the definition of work to determine the amount of work done by each force during one revolution. (a) Apply ∑F r = mar to the 2-kg object and solve for the tension: (2.5 m/s) v2 T = m = (2 kg ) r 3m 2 = 4.17 N (b) From the FBD we can see that the forces acting on the object are: r r r T , Fg , and Fn Because all of these forces act perpendicularly to the direction of motion of the object, none of them do any work. 386 Chapter 6 *32 • Picture the Problem The free-body r diagram, with F representing the force required to move the block at constant speed, shows the forces acting on the block. We can apply Newton’s 2nd law to the block to relate F to its weight w and then use the definition of the mechanical advantage of an inclined plane. In the second part of the problem we’ll use the definition of work. (a) Express the mechanical advantage M of the inclined plane: Apply ∑F x = ma x to the block: M = w F F − w sin θ = 0 because ax = 0. Solve for F and substitute to obtain: M = Refer to the figure to obtain: sin θ = Substitute to obtain: M = w 1 = w sin θ sin θ H L 1 L = sin θ H (b) Express the work done pushing the block up the ramp: Wramp = FL = mgL sin θ Express the work done lifting the block into the truck: Wlifting = mgH = mgL sin θ and Wramp = Wlifting 33 • Picture the Problem We can find the work done per revolution in lifting the weight and the work done in each revolution of the handle and then use the definition of mechanical advantage. Express the mechanical advantage of the jack: M = Express the work done by the jack in one complete revolution (the weight W is raised a distance p): Wlifting = Wp Express the work done by the force F in one complete revolution: Wturning = 2π RF W F Work and Energy 387 Equate these expressions to obtain: Solve for the ratio of W to F: Wp = 2π RF M = 2π R W = F p Remarks: One does the same amount of work turning as lifting; exerting a smaller force over a greater distance. 34 • r Picture the Problem The object whose weight is w is supported by two portions of the rope resulting in what is known as a mechanical advantage of 2. The work that is done in each instance is the product of the force doing the work and the displacement of the object on which it does the work. (a) If w moves through a distance h: F moves a distance 2h (b) Assuming that the kinetic energy of the weight does not change, relate the work done on the object to the change in its potential energy to obtain: W = ∆U = wh cos θ = wh (c) Because the force you exert on the rope and its displacement are in the same direction: Determine the tension in the ropes supporting the object: W = F (2h )cosθ = F (2h ) ∑F vertical = 2F − w = 0 and F=1w 2 Substitute for F: W = F (2h ) = 1 w (2h ) = wh 2 (d) The mechanical advantage of the inclined plane is the ratio of the weight that is lifted to the force required to lift it, i.e.: M = w w = 1 = 2 F 2w Remarks: Note that the mechanical advantage is also equal to the number of ropes supporting the load. 388 Chapter 6 Dot Products *35 • r r Picture the Problem Because A ⋅ B ≡ AB cos θ we can solve for cosθ and use the fact r r that A ⋅ B = − AB to find θ. r r A⋅ B θ = cos AB Solve for θ : −1 r r Substitute for A ⋅ B and evaluate θ : θ = cos −1 (− 1) = 180° 36 • r r Picture the Problem We can use its definition to evaluate A ⋅ B . r r Express the definition of A ⋅ B : Substitute numerical values and r r evaluate A ⋅ B : r r A ⋅ B = AB cos θ r r A ⋅ B = (6 m )(6 m )cos 60° = 18.0 m 2 37 • r r Picture the Problem The scalar product of two-dimensional vectors A and B is AxBx + A yB y. r ˆ (a) For A = 3 i − 6 ˆ and j r ˆ B = −4 i + 2 ˆ : j r ˆ j (b) For A = 5 i + 5 ˆ and r ˆ j B = 2 i −4 ˆ : r r ˆ ˆ j j (c) For A = 6 i + 4 ˆ and B = 4 i − 6 ˆ : r r A ⋅ B = (3)( −4) + (−6)(2) = − 24 r r A ⋅ B = (5)(2) + (5)( −4) = − 10 r r A ⋅ B = (6)(4) + (4)( −6) = 0 Work and Energy 389 38 • r r Picture the Problem The scalar product of two-dimensional vectors A and B is AB cos θ r r = AxBx + AyBy. Hence the angle between vectors A and B is given by θ = cos −1 Ax B x + Ay B y AB . r r A ⋅ B = (3)( −4) + (−6)(2) = −24 r ˆ j (a) For A = 3 i − 6 ˆ and r ˆ j B = −4 i + 2 ˆ : A= B= (3)2 + (− 6)2 (− 4)2 + (2)2 and θ = cos −1 = 20 − 24 = 143° 45 20 r r A ⋅ B = (5)(2) + (5)(−4) = -10 r ˆ (b) For A = 5 i + 5 ˆ and j r ˆ B = 2i − 4 ˆ : j A= B= (5)2 + (5)2 = 50 (2)2 + (− 4)2 = 20 and θ = cos −1 r = 45 − 10 = 108° 50 20 r r A ⋅ B = (6)(4) + (4)( −6) = 0 r ˆ ˆ j j (c) For A = 6 i + 4 ˆ and B = 4 i − 6 ˆ : A= B= (6)2 + (4)2 = 52 (4)2 + (− 6)2 = 52 and θ = cos −1 0 = 90.0° 52 52 39 • r r Picture the Problem The work W done by a force F during a displacement ∆ s for r r which it is responsible is given by F ⋅∆ s . 390 Chapter 6 r r W = F ⋅ ∆s ˆ ˆ = 2 N i −1N ˆ +1N k j (a) Using the definitions of work and the scalar product, calculate the work done by the given force during the specified displacement: ( ( ) ) ˆ ˆ ⋅ 3m i + 3m ˆ − 2 m k j = [(2)(3) + (− 1)(3) + (1) (− 2)] N ⋅ m = 1.00 J W = F∆s cos θ = (F cos θ )∆s (b) Using the definition of work that includes the angle between the force and displacement vectors, solve for r the component of F in the direction r of ∆ s : and F cosθ = Substitute numerical values and evaluate Fcosθ : F cos θ = W ∆s 1J (3 m )2 + (3 m )2 + (− 2 m )2 = 0.213 N 40 •• Picture the Problem The component of a vector that is along another vector is the scalar product of the former vector and a unit vector that is parallel to the latter vector. r ˆ ˆ j A Ax i + Ay ˆ + Az k ˆ = uA = 2 A Ax2 + Ay + Az2 (a) By definition, the unit vector r that is parallel to the vector A is: r r B ˆ uB = = B (b) Find the unit vector parallel to B : r r ˆ 3i + 4 ˆ j (3) ( 2 + (4 ) 2 = 3ˆ 4 ˆ i+ j 5 5 ) r ˆ j ˆ ⎛ 3 ˆ 4 j⎞ ˆ A ⋅ uB = 2i − ˆ − k ⋅ ⎜ i + ˆ ⎟ 5 ⎠ ⎝5 ⎛3⎞ ⎛4⎞ = (2)⎜ ⎟ + (− 1)⎜ ⎟ + (− 1)(0) ⎝5⎠ ⎝5⎠ The component of A along B is: = 0.400 *41 •• Picture the Problem We can use the definitions of the magnitude of a vector and the dot r v r v r r product to show that if A + B = A − B , then A ⊥ B . Work and Energy 391 ( ) 2 ( ) 2 r r2 r r2 r r A+ B = A+ B Express A − B : r v r r2 r r A− B = A− B Equate these expressions to obtain: r v r v (A + B ) = (A − B ) Express A + B : Expand both sides of the equation to obtain: 2 2 r r r r A2 + 2 A ⋅ B + B 2 = A2 − 2 A ⋅ B + B 2 r r 4A ⋅ B = 0 Simplify to obtain: or r r A⋅ B = 0 From the definition of the dot product we have: r v Because neither A nor B is the zero vector: r r A ⋅ B = AB cos θ r v where θ is the angle between A and B. r r cos θ = 0 ⇒ θ = 90° and A ⊥ B. 42 •• Picture the Problem The diagram shows ˆ ˆ the unit vectors A and B arbitrarily st located in the 1 quadrant. We can express these vectors in terms of the unit vectors ˆ i and ˆ and their x and y components. We j can then form the dot product of ˆ ˆ A and B to show that cos(θ1 − θ2) = cosθ1cosθ2 + sinθ1sinθ2. ˆ (a) Express A in terms of the unit ˆ ˆ A = Axi + Ay ˆ j ˆ j vectors i and ˆ : where Ax = cos θ1 and Ay = sin θ1 Proceed as above to obtain: ˆ ˆ B = Bxi + By ˆ j where Bx = cos θ 2 ˆ ˆ (b) Evaluate A ⋅ B : ( and B y = sin θ 2 ) ˆ ˆ ˆ A ⋅ B = cos θ1i + sin θ1 ˆ j ˆ ⋅ cos θ 2 i + sin θ 2 ˆ j ( ) = cos θ1 cos θ 2 + sin θ1 sin θ 2 From the diagram we note that: ˆ ˆ A ⋅ B = cos(θ1 − θ 2 ) 392 Chapter 6 Substitute to obtain: cos(θ1 − θ 2 ) = cos θ1 cos θ 2 + sin θ1 sin θ 2 43 • r r Picture the Problem In (a) we’ll show that it does not follow that B = C by giving a counterexample. r r ˆ ˆ Let A = i , B = 3i + 4 ˆ and j r r r r r ˆ C = 3i − 4 ˆ. Form A ⋅ B and A ⋅ C : j ( ) and r r ˆ ˆ A ⋅ C = i ⋅ (3i − 4 ˆ ) = 3 j r r ˆ ˆ A ⋅ B = i ⋅ 3i + 4 ˆ = 3 j No. We' ve shown by a counter r example that B is not necessarily r equal to C . 44 •• r r Picture the Problem We can form the dot product of A and r and require that r r r A ⋅ r = 1 to show that the points at the head of all such vectors r lie on a straight line. r We can use the equation of this line and the components of A to find the slope and intercept of the line. r ˆ (a) Let A = a x i + a y ˆ . Then: j ( )( r r ˆ ˆ A ⋅ r = ax i + a y ˆ ⋅ x i + y ˆ j j ) = ax x + a y y = 1 Solve for y to obtain: y= − ax 1 x+ ay ay which is of the form y = mx + b and hence is the equation of a line. r ˆ j (b) Given that A = 2 i − 3 ˆ : m=− ax 2 2 =− = ay 3 −3 and b= 1 1 1 = = − ay − 3 3 Work and Energy 393 (c) The equation we obtained in (a) specifies all vectors whose component r parallel to A has constant magnitude; therefore, we can write such a vector as r r r r A r = r 2 + B , where B is any vector A r perpendicular to A. This is shown graphically to the right. r Because all possiblervectors B lie in a plane, the resultant r must lie in a plane as well, as is shown above. *45 •• Picture the Problem The rules for the differentiation of vectors are the same as those for the differentiation of scalars and scalar multiplication is commutative. r r (a) Differentiate r ⋅ r = r2 = constant: r r Because v ⋅ r = 0 : r r (b) Differentiate v ⋅ v = v2 = constant with respect to time: r r Because a ⋅ v = 0 : r r r r d r r r dr dr r (r ⋅ r ) = r ⋅ + ⋅ r = 2v ⋅ r dt dt dt d = (constant ) = 0 dt r r v ⊥r r r r r d r r r dv dv r (v ⋅ v ) = v ⋅ + ⋅ v = 2a ⋅ v dt dt dt d = (constant ) = 0 dt r r a ⊥v The results of (a) and (b) tell us that r r a is perpendicular to r and and r parallel (or antiparallel) to r . r r (c) Differentiate v ⋅ r = 0 with respect to time: r r d r r r dr r dv (v ⋅ r ) = v ⋅ + r ⋅ dt dt dt r r d = v 2 + r ⋅ a = (0) = 0 dt 394 Chapter 6 r r r r r ⋅ a = −v 2 Because v 2 + r ⋅ a = 0 : Express ar in terms of θ, where θ is r r the angle between r and a : (1) ar = a cos θ Express r ⋅ a : r r r ⋅ a = ra cos θ = rar Substitute in equation (1) to obtain: rar = −v 2 Solve for ar: ar = − r r v2 r Power 46 •• Picture the Problem The power delivered by a force is defined as the rate at which the force does work; i.e., P = dW . dt Calculate the rate at which force A does work: PA = 5J = 0.5 W 10 s Calculate the rate at which force B does work: PB = 3J = 0.6 W and PB > PA 5s 47 • Picture the Problem The power delivered by a force is defined as the rate at which the force does work; i.e., P = r r dW = F ⋅ v. dt (a) If the box moves upward with a constant velocity, the net force acting it must be zero and the force that is doing work on the box is: The power input of the force is: Substitute numerical values and evaluate P: F = mg P = Fv = mgv ( ) P = (5 kg ) 9.81 m/s 2 (2 m/s ) = 98.1 W Work and Energy 395 (b) Express the work done by the force in terms of the rate at which energy is delivered: W = Pt = (98.1 W) (4 s) = 392 J 48 • Picture the Problem The power delivered by a force is defined as the rate at which the force does work; i.e., P = r r dW = F ⋅ v. dt P 6W = = 2 m/s F 3N (a) Using the definition of power, express Fluffy’s speed in terms of the rate at which he does work and the force he exerts in doing the work: v= (b) Express the work done by the force in terms of the rate at which energy is delivered: W = Pt = (6 W) (4 s) = 24.0 J 49 Picture the Problem We can use Newton’s 2nd law and the definition of acceleration to express the velocity of this object as a function of time. The power input of the force accelerating the object is defined to be the rate at which it does work; i.e., r r P = dW dt = F ⋅ v . (a) Express the velocity of the object as a function of its acceleration and time: Apply r r ∑ F = ma to the object: Substitute for a in the expression for v: (b) Express the power input as a function of F and v and evaluate P: (c) Substitute t = 3 s: v = at a = F/m v= F 5N t= t= m 8 kg ( ( 5 8 ) m/s 2 t ) 5 P = Fv = (5 N ) 8 m/s 2 t = 3.13t W/s P = (3.13 W/s )(3 s ) = 9.38 W 396 Chapter 6 50 • Picture the Problem The power delivered by a force is defined as the rate at which the force does work; i.e., P = r r dW = F ⋅ v. dt r ˆ ˆ (a) For F = 4 N i + 3 N k and r ˆ v = 6 m/s i : ( )( r r ˆ ˆ ˆ P = F ⋅ v = 4 N i + 3 N k ⋅ 6 m/s i ) = 24.0 W r ˆ (b) For F = 6 N i − 5 N ˆ and j r ˆ v = − 5 m/s i + 4 m/s ˆ : j r r P = F ⋅v ( )( ˆ ˆ = 6 N i − 5 N ˆ ⋅ − 5 m/s i + 4 m/s ˆ j j ) = − 50.0 W r ˆ j (c) For F = 3 N i + 6 N ˆ r ˆ j and v = 2 m/s i + 3 m/s ˆ : r r P = F ⋅v ˆ ˆ = 3 N i + 6 N ˆ ⋅ 2 m/s i + 3 m/s ˆ j j ( )( ) = 24.0 W *51 • Picture the Problem Choose a coordinate system in which upward is the positive y direction. We can find Pin from the given information that Pout = 0.27 Pin . We can express Pout as the product of the tension in the cable T and the constant speed v of the dumbwaiter. We can apply Newton’s 2nd law to the dumbwaiter to express T in terms of its mass m and the gravitational field g. Express the relationship between the motor’s input and output power: Pout = 0.27 Pin or Pin = 3.7 Pout Express the power required to move the dumbwaiter at a constant speed v: Apply ∑F y = ma y to the Pout = Tv T − mg = ma y dumbwaiter: or, because ay = 0, Substitute to obtain: Pin = 3.7Tv = 3.7 mgv Substitute numerical values and evaluate Pin: T = mg ( ) Pin = 3.7(35 kg ) 9.81 m/s 2 (0.35 m/s ) = 445 W Work and Energy 397 52 •• Picture the Problem Choose a coordinate system in which upward is the positive y direction. We can express Pdrag as the product of the drag force Fdrag acting on the skydiver and her terminal velocity vt. We can apply Newton’s 2nd law to the skydiver to express Fdrag in terms of her mass m and the gravitational field g. r r r Pdrag = Fdrag ⋅ v t r r or, because Fdrag and v t are antiparallel, (a) Express the power due to drag force acting on the skydiver as she falls at her terminal velocity vt: Pdrag = − Fdrag vt Apply ∑F y Fdrag − mg = ma y = ma y to the skydiver: or, because ay = 0, Fdrag = mg Pdrag = − mgvt Substitute to obtain, for the magnitude of Pdrag: (1) Substitute numerical values and evaluate P: Pdrag = − (55 kg) (9.81 m/s 2 ) (120 mi 1h 1.609 km × × ) = 2.89 × 10 4 W h 3600 s mi (b) Evaluate equation (1) with v = 15 mi/h: 1h 1.609 km ⎛ mi ⎞ Pdrag = − (55 kg )(9.81 m/s 2 ) ⎜15 ⎟ × × ) = 3.62 kW mi ⎝ h ⎠ 3600 s *53 •• Picture the Problem Because, in the absence of air resistance, the acceleration of the cannonball is constant, we can use a constant-acceleration equation to relate its velocity to the time it has been in flight. We can apply Newton’s 2nd law to the cannonball to find r r the net force acting on it and then form the dot product of F and v to express the rate at which the gravitational field does work on the cannonball. Integrating this expression over the time-of-flight T of the ball will yield the desired result. Express the velocity of the cannonball as a function of time while it is in the air: Apply r r ∑ F = ma to the r ˆ v (t ) = 0i + (v0 − gt ) ˆ j r F = − mg ˆ j cannonball to express the force acting on it while it is in the air: r r Evaluate F ⋅ v : r r F ⋅ v = −mg ˆ ⋅ (v0 − gt ) ˆ j j = −mgv0 + mg 2t 398 Chapter 6 r r r r dW = F ⋅ v = − mgv0 + mg 2t dt Relate F ⋅ v to the rate at which work is being done on the cannonball: Separate the variables and integrate over the time T that the cannonball is in the air: ( T ) W = ∫ − mgv0 + mg 2 t dt 0 (1) = 1 mg 2T 2 − mgv0T 2 2 v 2 = v0 + 2a∆y Using a constant-acceleration equation, relate the speed v of the cannonball when it lands at the bottom of the cliff to its initial speed v0 and the height of the cliff H: or, because a = g and ∆y = H, 2 v 2 = v0 + 2 gH Solve for v to obtain: v= Using a constant-acceleration equation, relate the time-of-flight T to the initial and impact speeds of the cannonball: v = v0 − gT Solve for T to obtain: T= Substitute for T in equation (1) and simplify to evaluate W: v0 + 2 gH 2 v0 − v g W = 1 mg 2 2 v 02 − 2vv0 + v 2 g2 ⎛v −v⎞ − mgv0 ⎜ 0 ⎜ g ⎟ ⎟ ⎝ ⎠ 2 = 1 mv 2 − 1 mv0 = ∆K 2 2 54 •• Picture the Problem If the particle is acted on by a single force, that force is the net force acting on the particle and is responsible for its acceleration. The rate at which r r energy is delivered by the force is P = F ⋅ v . Express the rate at which this force r r r r P = F ⋅v does work in terms of F and v : The velocity of the particle, in terms of its acceleration and the time that the force has acted is: r r v = at Work and Energy 399 Using Newton’s 2nd law, substitute r for a : r Substitute for v in the expression for P and simplify to obtain: r r F v= t m r r r r F F2 F ⋅F P=F⋅ t= t= t m m m Potential Energy 55 • Picture the Problem The change in the gravitational potential energy of the earth-man system, near the surface of the earth, is given by ∆U = mg∆h, where ∆h is measured relative to an arbitrarily chosen reference position. Express the change in the man’s gravitational potential energy in terms of his change in elevation: Substitute for m, g and ∆h and evaluate ∆U: ∆U = mg∆h ∆U = (80 kg ) (9.81 m/s 2 ) (6 m ) = 4.71 kJ 56 • Picture the Problem The water going over the falls has gravitational potential energy relative to the base of the falls. As the water falls, the falling water acquires kinetic energy until, at the base of the falls; its energy is entirely kinetic. The rate at which energy is delivered to the base of the falls is given by P = dW / dt = − dU / dt. Express the rate at which energy is being delivered to the base of the falls; remembering that half the potential energy of the water is converted to electric energy: Substitute numerical values and evaluate P: dW dU =− dt dt d dm = − 1 (mgh ) = − 1 gh 2 2 dt dt P= ( ) × (1.4 × 10 kg/s ) P = − 1 9.81 m/s 2 (− 128 m ) 2 6 = 879 MW 400 Chapter 6 57 • Picture the Problem In the absence of friction, the sum of the potential and kinetic energies of the box remains constant as it slides down the incline. We can use the conservation of the mechanical energy of the system to calculate where the box will be and how fast it will be moving at any given time. We can also use Newton’s 2nd law to show that the acceleration of the box is constant and constant-acceleration equations to calculate where the box will be and how fast it will be moving at any given time. (a) Express and evaluate the gravitational potential energy of the box, relative to the ground, at the top of the incline: (b) Using a constant-acceleration equation, relate the displacement of the box to its initial speed, acceleration and time-of-travel: Apply ∑F x = max to the box as it Ui = mgh = (2 kg) (9.81 m/s2) (20 m) = 392 J ∆x = v0 ∆t + 1 a (∆t ) 2 2 or, because v0 = 0, ∆x = 1 a (∆t ) 2 2 mg sin θ = ma ⇒ a = g sin θ slides down the incline and solve for its acceleration: Substitute for a and evaluate ∆x(t = 1 s): ∆x(1s ) = (g sin θ )(∆t )2 2 = 1 (9.81 m/s 2 )(sin30°)(1s ) 2 1 2 = 2.45 m Using a constant-acceleration equation, relate the speed of the box at any time to its initial speed and acceleration and solve for its speed when t = 1 s: v = v0 + at where v0 = 0 and v(1s ) = a∆t = (g sin θ )∆t ( ) = 9.81m/s 2 (sin 30°)(1s ) = 4.91 m/s Work and Energy 401 (c) Calculate the kinetic energy of the box when it has traveled for 1 s: Express the potential energy of the box after it has traveled for 1 s in terms of its initial potential energy and its kinetic energy: (d) Express the kinetic energy of the box at the bottom of the incline in terms of its initial potential energy and solve for its speed at the bottom of the incline: Substitute numerical values and evaluate v: K = 1 mv 2 = 2 1 2 (2 kg )(4.91m/s )2 = 24.1 J U = U i − K = 392 J − 24.1 J = 368 J K = U i = 1 mv 2 = 392 J 2 and v= 2U i m v= 2(392 J ) = 19.8 m/s 2 kg 58 • Picture the Problem The potential energy function U (x) is defined by the equation x U ( x ) − U ( x0 ) = − ∫ Fdx. We can use the given force function to determine U(x) and then x0 the conditions on U to determine the potential functions that satisfy the given conditions. (a) Use the definition of the potential energy function to find the potential energy function associated with Fx: x U (x ) = U (x0 ) − ∫ Fx dx x0 x = U (x0 ) − ∫ (6 N )dx' x0 = − (6 N )(x − x0 ) because U(x0) = 0. (b) Use the result obtained in (a) to find U (x) that satisfies the condition that U(4 m) = 0: U (4 m ) = −(6 N )(4 m − x0 ) = 0 ⇒ x0 = 4 m and U (x ) = −(6 N )(x − 4 m ) = 24 J − (6 N )x 402 Chapter 6 (c) Use the result obtained in (a) to find U that satisfies the condition that U(6 m) = 14 J: U (6 m ) = −(6 N )(6 m − x0 ) = 14 J ⇒ x0 = 50 m and 25 ⎞ ⎛ U (x ) = −(6 N )⎜ x − m ⎟ 3 ⎠ ⎝ = 50 J − (6 N )x 59 • Picture the Problem The potential energy of a stretched or compressed ideal spring Us is related to its force (stiffness) constant k and stretch or compression ∆x by U s = 1 kx 2 . 2 (a) Relate the potential energy stored in the spring to the distance it has been stretched: U s = 1 kx 2 2 Solve for x: x= 2U s k Substitute numerical values and evaluate x: x= 2(50 J ) = 0.100 m 10 4 N/m (b) Proceed as in (a) with Us = 100 J: x= 2(100 J ) = 0.141 m 10 4 N/m *60 •• Picture the Problem In a simple Atwood’s machine, the only effect of the pulley is to connect the motions of the two objects on either side of it; i.e., it could be replaced by a piece of polished pipe. We can relate the kinetic energy of the rising and falling objects to the mass of the system and to their common speed and relate their accelerations to the sum and difference of their masses … leading to simultaneous equations in m1 and m2. K= Use the definition of the kinetic energy of the system to determine the total mass being accelerated: a= (m1 + m2 )v 2 and In Chapter 4, the acceleration of the masses was shown to be: 1 2 m1 + m2 = 2K 2(80 J ) = = 10.0 kg (1) 2 v (4 m/s)2 m1 − m2 g m1 + m2 Work and Energy 403 Because v(t) = at, we can eliminate a in the previous equation to obtain: Solve for m1 − m2 : v(t ) = m1 − m2 gt m1 + m2 m1 − m2 = (m1 + m2 )v(t ) gt (10 kg )(4 m/s) = 1.36 kg (9.81m/s2 )(3 s ) Substitute numerical values and evaluate m1 − m2 : m1 − m2 = Solve equations (1) and (2) simultaneously to obtain: m1 = 5.68 kg and m2 = 4.32 kg (2) 61 •• Picture the Problem The gravitational potential energy of this system of two objects is the sum of their individual potential energies and is dependent on an arbitrary choice of where, or under what condition(s), the gravitational potential energy is zero. The best choice is one that simplifies the mathematical details of the expression of U. In this problem let’s choose U = 0 where θ = 0. (a) Express U for the 2-object system as the sum of their gravitational potential energies; noting that because the object whose mass is m2 is above the position we have chosen for U = 0, its potential energy is positive while that of the object whose mass is m1 is negative: (b) Differentiate U with respect toθ and set this derivative equal to zero to identify extreme values: To be physically meaningful, U (θ ) = U1 + U 2 = m2 gl 2 sin θ − m1 gl 1 sin θ = (m2l 2 − m1l 1 )g sin θ dU = (m2 l 2 − m1l 1 )g cosθ = 0 dθ from which we can conclude that cosθ = 0 and θ = cos−10. ∴θ = ± π 2 −π 2 ≤ θ ≤ π 2 : Express the 2nd derivative of U with respect to θ and evaluate this derivative at θ = ± π 2 : d 2U = −(m2l 2 − m1l 1 )g sin θ dθ 2 404 Chapter 6 If we assume, in the expression for U that we derived in (a), that m2l2 – m1l1 >0, then U(θ) is a sine function and, in the interval of interest, − π 2 ≤ θ ≤ π 2 , takes on its minimum value when θ = −π/2: d 2U dθ 2 >0 −π 2 and U is a minimum at θ = − π 2 d 2U dθ 2 <0 π 2 and U is a maximum at θ = π 2 (c) If m1l1 = m2l2, then (m2l2 − m1l1) = 0 and U = 0 independently of θ . Remarks: An alternative approach to establishing the U is a maximum at θ = π/2 is to plot its graph and note that, in the interval of interest, U is concave downward with its maximum value at θ = π/2. Force, Potential Energy, and Equilibrium 62 • Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is, Fx = − dU dx . Consequently, given U as a function of x, we can find Fx by differentiating U with respect to x. (a) Evaluate Fx = − dU : dx (b) Set Fx = 0 and solve for x: Fx = − ( ) d Ax 4 = − 4 Ax 3 dx Fx = 0 ⇒ x = 0 63 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is Fx = − dU dx . Consequently, given U as a function of x, we can find Fx by differentiating U with respect to x. (a) Evaluate Fx = − (b) Because C > 0: dU : dx Fx = − d ⎛C ⎞ C ⎜ ⎟= 2 dx ⎝ x ⎠ x Fx is positive for x ≠ 0 and therefore r F is directed away from the origin. Work and Energy 405 (c) Because U is inversely proportional to x and C > 0: (d) With C < 0: Because U is inversely proportional to x and C < 0, U(x) becomes less negative as x increases: U ( x ) decreases with increasing x. Fx is negative for x ≠ 0 and therefore r F is directed toward from the origin. U ( x ) increases with increasing x. *64 •• Picture the Problem Fy is defined to be the negative of the derivative of the potential function with respect to y, i.e. Fy = − dU dy . Consequently, we can obtain Fy by examining the slopes of the graph of U as a function of y. The table to the right summarizes the information we can obtain from Figure 6-40: Slope Fy Interval (N) (N) 2 −2 A→B B→C transitional −2 → 1.4 1.4 −1.4 C→D The graph of F as a function of y is shown to the right: 2.5 2.0 1.5 F (N) 1.0 0.5 0.0 0 1 2 3 4 5 6 -0.5 -1.0 -1.5 y (m) 65 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. Fx = − dU dx . Consequently, given F as a function of x, we can find U by integrating Fx with respect to x. Evaluate the integral of Fx with respect to x: U ( x ) = − ∫ F ( x ) dx = − ∫ = a dx x2 a +U0 x where U0 is a constant determined by whatever conditions apply to U. 406 Chapter 6 66 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is, Fx = − dU dx . Consequently, given U as a function of x, we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we’ll evaluate d 2U dx 2 at the point of interest. (a) Evaluate Fx = − dU : dx (b) We know that, at equilibrium, Fx = 0: Fx = − ( ) d 3x 2 − 2 x 3 = 6 x( x − 1) dx When Fx =0, 6x(x – 1) = 0. Therefore, the object is in equilibrium at x = 0 and x = 1 m. (c) To decide whether the equilibrium at a particular point is stable or unstable, evaluate the 2nd derivative of the potential energy function at the point of interest: Evaluate d 2U at x = 0: dx 2 ( ) dU d = 3x 2 − 2 x 3 = 6 x − 6 x 2 dx dx and d 2U = 6 − 12 x dx 2 d 2U dx 2 =6>0 x =0 ⇒ stable equilibrium at x = 0 Evaluate d 2U at x = 1 m: dx 2 d 2U dx 2 = 6 − 12 < 0 x =1 m ⇒ unstable equilibrium at x = 1 m 67 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. Fx = − dU dx . Consequently, given U as a function of x, we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we’ll evaluate d 2U dx 2 at the point of interest. Work and Energy 407 (a) Evaluate the negative of the derivative of U with respect to x: dU dx d = − (8 x 2 − x 4 ) = 4 x 3 − 16 x dx Fx = − = 4 x( x + 2)( x − 2) (b) The object is in equilibrium wherever Fnet = Fx = 0: 4 x( x + 2)( x − 2) = 0 ⇒ the equilibrium points are x = −2 m, 0, and 2 m. ( ) (c) To decide whether the equilibrium at a particular point is stable or unstable, evaluate the 2nd derivative of the potential energy function at the point of interest: d 2U d = 16 x − 4 x 3 = 16 − 12 x 2 2 dx dx d 2U at x = −2 m: Evaluate dx 2 d 2U dx 2 = −32 < 0 x =−2 m ⇒ Evaluate d 2U at x = 0: dx 2 d 2U dx 2 unstable equilibrium at x = −2 m = 16 > 0 x =0 ⇒ stable equilibrium at x = 0 Evaluate d 2U at x = 2 m: dx 2 d 2U dx 2 = −32 < 0 x =2 m ⇒ unstable equilibrium at x = 2 m Remarks: You could also decide whether the equilibrium positions are stable or unstable by plotting F(x) and examining the curve at the equilibrium positions. 68 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. Fx = − dU dx . Consequently, given F as a function of x, we can find U by integrating Fx with respect to x. Examination of d 2U dx 2 at extreme points will determine the nature of the stability at these locations. 408 Chapter 6 Determine the equilibrium locations by setting Fnet = F(x) = 0: F(x) = x3 – 4x = x(x2 – 4) = 0 ∴ the positions of stable and unstable equilibrium are at Evaluate the negative of the integral of F(x) with respect to x: U (x ) = − ∫ F (x ) ( x = −2, 0 and 2 . ) = − ∫ x 3 − 4 x dx =− x4 + 2x2 + U 0 4 where U0 is a constant whose value is determined by conditions on U(x). Differentiate U(x) twice: dU = − Fx = − x 3 + 4 x dx and d 2U = −3 x 2 + 4 2 dx d 2U Evaluate at x = −2: dx 2 d 2U dx 2 = −8 < 0 x =−2 ∴ the equilibrium is unstable at x = − 2 Evaluate d 2U at x = 0: dx 2 d 2U dx 2 =4>0 x =0 ∴ the equilibrium is stable at x = 0 Evaluate d 2U at x = 2: dx 2 d 2U dx 2 = −8 < 0 x =2 ∴ the equilibrium is unstable at x = 2 Thus U(x) has a local minimum at x = 0 and local maxima at x = ±2. 69 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. Fx = − dU dx . Consequently, given U as a function of x, we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we can examine the graph of U. Work and Energy 409 (a) Evaluate Fx = − dU for x ≤ 3 m: dx Set Fx = 0 to identify those values of x for which the 4-kg object is in equilibrium: Evaluate Fx = − dU for x > 3 m: dx Fx = − ( ) d 3x 2 − x 3 = 3x(2 − x ) dx When Fx = 0, 3x(2 – x) = 0. Therefore, the object is in equilibrium at x = 0 and x = 2 m. Fx = 0 because U = 0. Therefore, the object is in neutral equilibrium for x > 3 m. (b) A graph of U(x) in the interval –1 m ≤ x ≤ 3 m is shown to the right: 4.0 3.5 3.0 U (J) 2.5 2.0 1.5 1.0 0.5 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 x (m) (c) From the graph, U(x) is a minimum at x = 0: ∴ stable equilibrium at x = 0 From the graph, U(x) is a maximum at x = 2 m: ∴ unstable equilibrium at x = 2 m (d) Relate the kinetic energy of the object to its total energy and its potential energy: K = 1 mv 2 = E − U 2 Solve for v: v= 2( E − U ) m Evaluate U(x = 2 m): U ( x = 2 m ) = 3(2) − (2) = 4 J Substitute in the equation for v to obtain: v= 2 3 2(12 J − 4 J ) = 2.00 m/s 4 kg 3.0 410 Chapter 6 70 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is Fx = − dU dx . Consequently, given F as a function of x, we can find U by integrating Fx with respect to x. (a) Evaluate the negative of the integral of F(x) with respect to x: U ( x ) = − ∫ F ( x ) = − ∫ Ax −3 dx = 1 A + U0 2 x2 where U0 is a constant whose value is determined by conditions on U(x). For x > 0: (b) As x → ∞, U decreases as x increases 1 A → 0: 2 x2 ∴ U0 = 0 and U (x ) = (c) The graph of U(x) is shown to the right: 1 A 1 8 N ⋅ m3 4 = = 2 N ⋅ m3 2 2 2x 2 x x 400 350 300 250 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 x (m) *71 ••• Picture the Problem Let L be the total length of one cable and the zero of gravitational potential energy be at the top of the pulleys. We can find the value of y for which the potential energy of the system is an extremum by differentiating U(y) with respect to y and setting this derivative equal to zero. We can establish that this value corresponds to a minimum by evaluating the second derivative of U(y) at the point identified by the first derivative. We can apply Newton’s 2nd law to the clock to confirm the result we obtain by examining the derivatives of U(y). (a) Express the potential energy of the system as the sum of the potential energies of the clock and counterweights: Substitute to obtain: U ( y ) = U clock ( y ) + U weights ( y ) ( U ( y ) = − mgy − 2 Mg L − y 2 + d 2 ) Work and Energy 411 (b) Differentiate U(y) with respect to y: [ ( dU ( y ) d =− mgy + 2Mg L − y 2 + d 2 dy dy ⎡ = − ⎢mg − 2Mg ⎢ ⎣ ⎤ ⎥ y2 + d 2 ⎥ ⎦ y or y' mg − 2Mg Solve for y′ to obtain: Find d 2U ( y ) : dy 2 y' = d m2 4M 2 − m 2 d 2U ( y ) d ⎡ = − ⎢mg − 2Mg dy 2 dy ⎢ ⎣ 2Mgd 2 = 32 y2 + d 2 ( d 2U ( y ) Evaluate at y = y′: dy 2 = 0 for extrema y' 2 + d 2 ) 2Mgd 2 d 2U ( y ) = 32 dy 2 y' y2 + d 2 ( = ⎤ ⎥ y2 + d 2 ⎥ ⎦ y ) y' 2Mgd ⎞ ⎛ m2 ⎜ ⎟ ⎜ 4M 2 − m 2 + 1⎟ ⎠ ⎝ >0 32 and the potential energy is a minimum at y= d m2 4M 2 − m 2 (c) The FBD for the clock is shown to the right: Apply ∑F y = 0 to the clock: 2Mg sin θ − mg = 0 and sin θ = m 2M )] 412 Chapter 6 Express sinθ in terms of y and d: Substitute to obtain: sin θ = m = 2M y y2 + d 2 y y2 + d 2 which is equivalent to the first equation in part (b). This is a point of stable equilibrium. Ifthe clock is displaced downward, θ increases, leading to a larger upward force on the clock. Similarly, if the clock is displaced upward, the net force from the cables decreases. Because of this, the clock will be pulled back toward the equilibrium point if it is displaced away from it. Remarks: Because we’ve shown that the potential energy of the system is a minimum at y = y′ (i.e., U(y) is concave upward at that point), we can conclude that this point is one of stable equilibrium. General Problems *72 • Picture the Problem 25 percent of the electrical energy generated is to be diverted to do the work required to change the potential energy of the American people. We can calculate the height to which they can be lifted by equating the change in potential energy to the available energy. Express the change in potential energy of the population of the United States in this process: ∆U = Nmgh Letting E represent the total energy generated in February 2002, relate the change in potential to the energy available to operate the elevator: Nmgh = 0.25E Solve for h: h= 0.25 E Nmg Work and Energy 413 Substitute numerical values and evaluate h: h= ⎟ ⎜ (0.25)(60.7 ×109 kW ⋅ h ) ⎛ 3600 s ⎞ ⎟ ⎜ ⎝ 1h ⎠ 287 × 10 (60 kg ) 9.81 m/s 2 ( 6 ) ( ) = 323 km 73 • Picture the Problem We can use the definition of the work done in changing the potential energy of a system and the definition of power to solve this problem. (a) Find the work done by the crane in changing the potential energy of its load: W = mgh = (6×106 kg) (9.81 m/s2) (12 m) (b) Use the definition of power to find the power developed by the crane: P≡ = 706 MJ dW 706 MJ = = 11.8 MW dt 60 s 74 • Picture the Problem The power P of the engine needed to operate this ski lift is related to the rate at which it changes the potential energy U of the cargo of the gondolas according to P = ∆U/∆t. Because as many empty gondolas are descending as are ascending, we do not need to know their mass. ∆U ∆t Express the rate at which work is done as the cars are lifted: P= Letting N represent the number of gondola cars and M the mass of each, express the change in U as they are lifted a vertical displacement ∆h: ∆U = NMg∆h Substitute to obtain: P≡ Relate ∆h to the angle of ascent θ and the length L of the ski lift: ∆h = Lsinθ Substitute for ∆h in the expression for P: P= ∆U NMg∆h = ∆t ∆t NMgL sin θ ∆t 414 Chapter 6 Substitute numerical values and evaluate P: P= ( ) 12(550 kg ) 9.81 m/s 2 (5.6 km )sin30° = 50.4 kW (60 min )(60 s/min ) 75 • Picture the Problem The application of Newton’s 2nd law to the forces shown in the free-body diagram will allow us to relate R to T. The unknown mass and speed of the object can be eliminated by introducing its kinetic energy. Apply ∑F radial = maradial the object and solve for R: mv 2 mv 2 and R = T= T R Express the kinetic energy of the object: K = 1 mv 2 2 Eliminate mv2 between the two equations to obtain: R= 2K T Substitute numerical values and evaluate R: R= 2(90 J ) = 0.500 m 360 N *76 • Picture the Problem We can solve this problem by equating the expression for the gravitational potential energy of the elevated car and its kinetic energy when it hits the ground. Express the gravitational potential energy of the car when it is at a distance h above the ground: U = mgh Express the kinetic energy of the car when it is about to hit the ground: K = 1 mv 2 2 Equate these two expressions (because at impact, all the potential energy has been converted to kinetic energy) and solve for h: h= v2 2g Work and Energy 415 Substitute numerical values and evaluate h: [(100 km/h )(1h/3600 s )] 2 h= ( 2 9.81 m/s 2 ) = 39.3 m 77 ••• Picture the Problem The free-body diagram shows the forces acting on one of the strings at the bridge. The force whose magnitude is F is one-fourth of the force (103 N) the bridge exerts on the strings. We can apply the condition for equilibrium in the y direction to find the tension in each string. Repeating this procedure at the site of the plucking will yield the restoring force acting on the string. We can find the work done on the string as it returns to equilibrium from the product of the average force acting on it and its displacement. (a) Noting that, due to symmetry, T′ = T, apply Fy = 0 to the string ∑ F − 2T sin 18° = 0 at the point of contact with the bridge: Solve for and evaluate T: T= 1 (103 N ) = 41.7 N F = 4 2 sin 18° 2 sin 18° (b) A free-body diagram showing the forces restoring the string to its equilibrium position just after it has been plucked is shown to the right: Express the net force acting on the string immediately after it is released: Use trigonometry to find θ: Substitute and evaluate Fnet: Fnet = 2T cos θ ⎛ 16.3 cm 10 mm ⎞ ⎟ = 88.6° × cm ⎟ ⎠ ⎝ 4 mm θ = tan −1 ⎜ ⎜ Fnet = 2(34.4 N )cos88.6° = 1.68 N 416 Chapter 6 (c) Express the work done on the string in displacing it a distance dx′: dW = Fdx' If we pull the string out a distance x′, the magnitude of the force pulling it down is approximately: F = (2T ) Substitute to obtain: dW = Integrate to obtain: 4T 2T 2 W= ∫ x'dx' = L x L 0 x' 4T = x' L2 L 4T x' dx' L x where x is the final displacement of the string. Substitute numerical values to obtain: W= 2(41.7 N ) 4 × 10−3 m −2 32.6 × 10 m ( ) 2 = 4.09 mJ 78 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is Fx = − dU dx . Consequently, given F as a function of x, we can find U by integrating Fx with respect to x. Evaluate the integral of Fx with respect to x: Apply the condition that U(0) = 0 to determine U0: ( ) U (x ) = − ∫ F (x ) dx = − ∫ − ax 2 dx = 1 ax 3 + U 0 3 U(0) = 0 + U0 = 0 ⇒ U0 = 0 ∴U (x ) = 1 3 ax 3 The graph of U(x) is shown to the right: 3 2 U (J) 1 0 -2.0 -1.5 -1.0 -0.5 0.0 -1 -2 -3 x (m) 0.5 1.0 1.5 2.0 Work and Energy 417 *79 •• Picture the Problem We can use the definition of work to obtain an expression for the position-dependent force acting on the cart. The work done on the cart can be calculated from its change in kinetic energy. (a) Express the force acting on the cart in terms of the work done on it: F (x ) = dW dx Because U is constant: F (x ) = d 1 2 d (2 mv ) = dx dx [ m(Cx ) ] 2 1 2 = mC 2 x (b) The work done by this force changes the kinetic energy of the cart: 2 W = ∆K = 1 mv12 − 1 mv0 2 2 = 1 mv12 − 0 = 1 m(Cx1 ) 2 2 2 = 1 2 mC 2 x12 80 •• r Picture the Problem The work done by F depends on whether it causes a displacement in the direction it acts. r r r (a) Because F is along x-axis and the displacement is along y-axis: W = ∫ F ⋅ ds = 0 (b) Calculate the work done by r F during the displacement from x = 2 m to 5 m: r r W = ∫ F ⋅ ds = ( = 2 N/m 2 ) ∫ (2 N/m ) x dx 5m 2 2 2m 5m ⎡ x3 ⎤ ⎢ 3 ⎥ = 78.0 J ⎣ ⎦2 m 81 •• Picture the Problem The velocity and acceleration of the particle can be found by differentiation. The power delivered to the particle can be expressed as the product of its velocity and the net force acting on it, and the work done by the force and can be found from the change in kinetic energy this work causes. In the following, if t is in seconds and m is in kilograms, then v is in m/s, a is in m/s2, P is in W, and W is in J. 418 Chapter 6 (a) The velocity of the particle is given by: v= ( (6t ) dv d a= = (6t dt dt = The acceleration of the particle is given by: ) dx d = 2t 3 − 4t 2 dt dt 2 − 8t 2 − 8t ) = (12t − 8) (b) Express and evaluate the rate at which energy is delivered to this particle as it accelerates: (c) Because the particle is moving in such a way that its potential energy is not changing, the work done by the force acting on the particle equals the change in its kinetic energy: P = Fv = mav ( ) 8mt (9t − 18t + 8) = m(12t − 8) 6t 2 − 8t = 2 W = ∆K = K1 − K 0 [ = 1 m (v(t1 )) − (v(0 )) 2 2 2 [( = 1 m 6t12 − 8t1 2 )] 2 −0 = 2mt12 (3t1 − 4) 2 Remarks: We could also find W by integrating P(t) with respect to time. 82 •• Picture the Problem We can calculate the work done by the given force from its r r definition. The power can be determined from P = F ⋅ v and v from the change in kinetic energy of the particle produced by the work done on it. (a) Calculate the work done from its definition: r r 3m W = ∫ F ⋅ ds = ∫ 6 + 4 x − 3x 2 dx ( ) 0 3m ⎡ 4 x 3x3 ⎤ = ⎢6 x + − = 9.00 J 2 3 ⎥0 ⎣ ⎦ 2 (b) Express the power delivered to the particle in terms of Fx=3 m and its velocity: Relate the work done on the particle to its kinetic energy and solve for its velocity: r r P = F ⋅ v = Fx =3 m v W = ∆K = K final = 1 mv 2 since v0 = 0 2 Work and Energy 419 Solve for and evaluate v: v= 2(9 J ) = 2.45 m/s 3 kg 2K = m Evaluate Fx=3 m: Fx=3 m = 6 + 4(3) − 3(3) = −9 N Substitute for Fx=3 m and v: P = (− 9 N )(2.45 m/s ) = − 22.1 W 2 *83 •• Picture the Problem We’ll assume that the firing height is negligible and that the bullet lands at the same elevation from which it was fired. We can use the equation 2 R = v0 g sin 2θ to find the range of the bullet and constant-acceleration equations to ( ) find its maximum height. The bullet’s initial speed can be determined from its initial kinetic energy. 2 v0 sin 2θ g Express the range of the bullet as a function of its firing speed and angle of firing: R= Rewrite the range equation using the trigonometric identity sin2θ = 2sinθ cosθ: 2 2 v0 sin 2θ 2v0 sin θ cos θ = R= g g Express the position coordinates of the projectile along its flight path in terms of the parameter t: Eliminate the parameter t and make use of the fact that the maximum height occurs when the projectile is at half the range to obtain: x = (v0 cos θ )t and y = (v0 sin θ )t − 1 gt 2 2 h= (v0 sin θ )2 2g Equate R and h and solve the resulting equation for θ: tan θ = 4 ⇒ θ = tan −1 4 = 76.0° Relate the bullet’s kinetic energy to its mass and speed and solve for the square of its speed: 2 2 K = 1 mv0 and v0 = 2 2 Substitute for v0 and θ and evaluate R: R= 2K m 2(1200 J ) sin2(76°) (0.02 kg ) 9.81m/s2 = 5.74 km ( ) 420 Chapter 6 84 •• Picture the Problem The work done on the particle is the area under the force-versusdisplacement curve. Note that for negative displacements, F is positive, so W is negative for x < 0. (a) Use either the formulas for the areas of simple geometric figures or counting squares and multiplying by the work represented by one square to complete the table to the right: (b) Choosing U(0) = 0, and using the definition of ∆U = −W, complete the third column of the table to the right: x W (m) (J) −4 −11 −3 −10 −2 −7 −1 −3 0 0 1 1 2 0 3 −2 4 −3 ∆U (J) (J) −11 11 −10 10 −7 7 −3 3 0 0 1 −1 0 0 −2 2 −3 3 x W (m) −4 −3 −2 −1 0 1 2 3 4 12 10 8 6 U (J) The graph of U as a function of x is shown to the right: 4 2 0 -4 -3 -2 -1 0 -2 x (m) 1 2 3 4 Work and Energy 421 85 •• Picture the Problem The work done on the particle is the area under the force-versusdisplacement curve. Note that for negative displacements, F is negative, so W is positive for x < 0. (a) Use either the formulas for the areas of simple geometric figures or counting squares and multiplying by the work represented by one square to complete the table to the right: x (m) −4 −3 −2 −1 0 1 2 3 4 W (J) 6 4 2 0.5 0 0.5 1.5 2.5 3 (b) Choosing U(0) = 0, and using the definition of ∆U = −W, complete the third column of the table to the right: x W (m) −4 −3 −2 −1 0 1 2 3 4 (J) 6 4 2 0.5 0 0.5 1.5 2.5 3 ∆U (J) −6 −4 −2 −0.5 0 −0.5 −1.5 −2.5 −3 0 -4 -3 -2 -1 0 -1 -2 U (J) The graph of U as a function of x is shown to the right: -3 -4 -5 -6 x (m) 1 2 3 4 422 Chapter 6 86 •• Picture the Problem The pictorial representation shows the box at its initial position 0 at the bottom of the inclined plane and later at position 1. We’ll assume that the block is at position 0. Because the surface is frictionless, the work done by the tension will change both the potential and kinetic energy of the block. We’ll use Newton’s 2nd law to find the acceleration of the block up the incline and a constantacceleration equation to express v in terms of T, x, M, and θ. Finally, we can express the power produced by the tension in terms of the tension and the speed of the box. (a) Use the definition of work to express the work the tension T does moving the box a distance x up the incline: (b) Apply ∑F x = Max to the box: Solve for ax: Using a constant-acceleration equation, express the speed of the box in terms of its acceleration and the distance x it has moved up the incline: Substitute for ax to obtain: (c) The power produced by the tension in the string is given by: W = Tx T − Mg sin θ = Max ax = T − Mg sin θ T = − g sin θ M M 2 v 2 = v0 + 2a x x or, because v0 = 0, v = 2a x x v= ⎛T ⎞ 2⎜ − g sin θ ⎟ x ⎝M ⎠ ⎛T ⎞ P = Tv = T 2⎜ − g sin θ ⎟ x ⎝M ⎠ Work and Energy 423 87 ••• Picture the Problem We can use the definition of the magnitude of vector to show that r the magnitude of F is F0 and the definition of the scalar product to show that its direction r is perpendicular to r . The work done as the particle moves in a circular path can be found from its definition. r r F = Fx2 + Fy2 (a) Express the magnitude of F : 2 ⎛F ⎞ ⎛ F ⎞ = ⎜ 0 y⎟ + ⎜− 0 x⎟ ⎝ r ⎠ ⎝ r ⎠ F = 0 x2 + y2 r Because r = r F F = 0 r x2 + y2 : r r Form the scalar product of F and r : F0 r = F0 r x2 + y2 = ( 2 )( r r ⎛F ⎞ ˆ ˆ F ⋅r = ⎜ 0 ⎟ yi − x ˆ ⋅ xi + y ˆ j j ⎝ r ⎠ ⎛F ⎞ = ⎜ 0 ⎟(xy − xy ) = 0 ⎝ r ⎠ r r r ) r Because F ⋅ r = 0, F ⊥ r r r r (b) Because F ⊥ r , F is tangential to the circle and constant. At (5 m, r j 0), F points in the − ˆ direction. If r ds is in the − ˆ direction, dW > 0. j The work it does in one revolution is: W = F0 (2π r ) = 2π (5 m )F0 = (10π m )F0 if the rotation is clockwise and W = (− 10π m )F0 if the rotation is counterclockwise. W = (10π m) F0 if the rotation is clockwise, − (10π m) F0if the rotation is r counterclockwise. Because W ≠ 0 for a complete circuit, F is not conservative. *88 ••• r ˆ j Picture the Problem We can substitute for r and xi + yˆ in F to show that the magnitude of the force varies as the inverse of the square of the distance to the origin, and that its direction is opposite to the radius vector. We can find the work done by this force by evaluating the integral of F with respect to x from an initial position x = 2 m, y = 0 m to a final position x = 5 m, y = 0 m. Finally, we can apply Newton’s 2nd law to the particle to relate its speed to its radius, mass, and the constant b. 424 Chapter 6 r ⎛ ⎞ 2 b ⎟ x + y2 r ˆ F = −⎜ 32 ⎟ ⎜ x2 + y2 ⎝ ⎠ ˆ where r is a unit vector pointing from the (a) Substitute for r and r ˆ x i + y ˆ in F to obtain: j ( ) origin toward the point of application of r F. r ⎛ 1 ⎞ b ˆ F = −b⎜ 2 ⎜ x + y 2 ⎟r = − r 2 r ⎟ˆ ⎝ ⎠ Simplify to obtain: i.e., the magnitude of the force varies as the inverse of the square of the distance to the origin, and its direction is antiparallel r ˆ j (opposite) to the radius vector r = xi + yˆ. (b) Find the work done by this force by evaluating the integral of F with respect to x from an initial position x = 2 m, y = 0 m to a final position x = 5 m, y = 0 m: 5m 5m b ⎡1⎤ W = − ∫ 2 dx' = b ⎢ ⎥ x' ⎣ x' ⎦ 2 m 2m 1 ⎞ ⎛ 1 = 3 N ⋅ m2 ⎜ − ⎟ = − 0.900 J ⎝5m 2m⎠ (c) No work is done as the force is perpendicular to the velocity. (d) Because the particle is moving in a circle, the force on the particle must be supplying the centripetal acceleration keeping it moving in the circle. Apply Fr = mac to b v2 =m r2 r ∑ the particle: Solve for v: v= Substitute numerical values and evaluate v: b mr v= 3 N ⋅ m2 = 0.463 m/s (2 kg )(7 m ) 89 ••• Picture the Problem A spreadsheet program to calculate the potential is shown below. The constants used in the potential function and the formula used to calculate the ″6-12″ potential are as follows: Cell B2 B3 Content/Formula 1.09×10−7 6.84×10−5 Algebraic Form a b Work and Energy 425 D8 $B$2/C8^12−$B$3/C8^6 C9 C8+0.1 a b − 6 12 r r r + ∆r (a) A B C D 1 2 a = 1.09E-07 3 b = 6.84E-05 4 5 6 7 r U 8 3.00E-01 1.11E-01 9 3.10E-01 6.13E-02 10 3.20E-01 3.08E-02 11 3.30E-01 1.24E-02 12 3.40E-01 1.40E-03 13 3.50E-01 −4.95E-03 45 46 47 48 6.70E-01 6.80E-01 6.90E-01 7.00E-01 −7.43E-04 −6.81E-04 −6.24E-04 −5.74E-04 The graph shown below was generated from the data in the table shown above. Because the force between the atomic nuclei is given by F = −(dU dr ) , we can conclude that the shape of the potential energy function supports Feynman’s claim. "6-12" Potential 0.12 0.10 U (eV) 0.08 0.06 0.04 0.02 0.00 0.30 -0.02 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 r (nm) (b) The minimum value is about −0.0107 eV, occurring at a separation of approximately 0.380 nm. Because the function is concave upward (a potential ″well″) at this separation, 426 Chapter 6 this separation is one of stable equilibrium, although very shallow. dU d ⎡ a b⎤ = − ⎢ 12 − 6 ⎥ dr dr ⎣ r r ⎦ 12a 6b = 13 − 7 r r (c) Relate the force of attraction between two argon atoms to the slope of the potential energy function: F =− Substitute numerical values and evaluate F(5 Å): F= ( ) ( ) 12 1.09 × 10−7 6 6.84 × 10 −5 eV 1.6 × 10−19 J 1 nm − = −4.18 × 10 −2 × × −9 nm eV 10 m (0.5 nm )13 (0.5 nm )7 = − 6.69 × 10 −12 N where the minus sign means that the force is attractive. Substitute numerical values and evaluate F(3.5 Å): F= ( ) ( ) 12 1.09 × 10−7 6 6.84 × 10 −5 eV 1.6 × 10−19 J 1 nm − = 4.68 × 10 −1 × × −9 nm eV 10 m (0.35 nm )13 (0.35 nm )7 = 7.49 × 10−11 N where the plus sign means that the force is repulsive. *90 ••• Picture the Problem A spreadsheet program to plot the Yukawa potential is shown below. The constants used in the potential function and the formula used to calculate the Yukawa potential are as follows: Cell Content/Formula Algebraic Form B1 4 U0 B2 2.5 a D8 −$B$1*($B$2/C9)*EXP(−C9/$B$2) ⎛ a ⎞ −r / a C10 − U 0 ⎜ ⎟e ⎝r⎠ r + ∆r C9+0.1 (a) A 1 2 3 7 8 9 10 B U0= 4 a= 2.5 C D r 0.5 0.6 U −16.37 −13.11 pJ fm Work and Energy 427 11 12 13 14 0.7 0.8 0.9 1 −10.80 −9.08 −7.75 −6.70 64 65 66 67 68 69 70 6 6.1 6.2 6.3 6.4 6.5 6.6 −0.15 −0.14 −0.14 −0.13 −0.12 −0.11 −0.11 U as a function of r is shown below. 0 -2 0 1 2 3 4 5 6 7 -4 U (pJ) -6 -8 -10 -12 -14 -16 -18 r (fm) (b) Relate the force between the nucleons to the slope of the potential energy function: dU (r ) dr ⎤ d ⎡ ⎛a⎞ = − ⎢ − U 0 ⎜ ⎟e − r a ⎥ dr ⎣ ⎝r⎠ ⎦ F (r ) = − ⎛ a 1⎞ = − U 0 e −r / a ⎜ 2 + ⎟ r⎠ ⎝r (c) Evaluate F(2a): ⎛ a 1 ⎞ F (2a ) = −U 0 e −2 a / a ⎜ + ⎟ 2 ⎜ (2a ) 2a ⎟ ⎠ ⎝ ⎛ 3 ⎞ = −U 0 e −2 ⎜ ⎟ ⎝ 4a ⎠ 428 Chapter 6 Evaluate F(a): ⎛ a 1⎞ F (a ) = −U 0 e −a / a ⎜ 2 + ⎟ ⎜ (a ) a⎟ ⎝ ⎠ ⎛1 1⎞ ⎛2⎞ = −U 0 e −1 ⎜ + ⎟ = −U 0 e −1 ⎜ ⎟ ⎝a a⎠ ⎝a⎠ Express the ratio F(2a)/F(a): ⎛ 3 ⎞ − U 0 e −2 ⎜ ⎟ F (2a ) ⎝ 4a ⎠ = 3 e −1 = F (a ) ⎛2⎞ 8 − U 0 e −1 ⎜ ⎟ ⎝a⎠ = 0.138 (d) Evaluate F(5a): ⎛ a 1 ⎞ F (5a ) = −U 0 e −5 a / a ⎜ + ⎟ 2 ⎜ (5a ) 5a ⎟ ⎝ ⎠ ⎛ 6 ⎞ = −U 0 e −5 ⎜ ⎟ ⎝ 25a ⎠ Express the ratio F(5a)/F(a): ⎛ 6 ⎞ − U 0 e −5 ⎜ ⎟ F (5a ) ⎝ 25a ⎠ = 3 e −4 = 25 F (a ) ⎛2⎞ − U 0 e −1 ⎜ ⎟ ⎝a⎠ = 2.20 × 10 −3 Chapter 7 Conservation of Energy Conceptual Problems *1 • Determine the Concept Because the peg is frictionless, mechanical energy is conserved as this system evolves from one state to another. The system moves and so we know that ∆K > 0. Because ∆K + ∆U = constant, ∆U < 0. (a ) is correct. 2 • Determine the Concept Choose the zero of gravitational potential energy to be at ground level. The two stones have the same initial energy because they are thrown from the same height with the same initial speeds. Therefore, they will have the same total energy at all times during their fall. When they strike the ground, their gravitational potential energies will be zero and their kinetic energies will be equal. Thus, their speeds at impact will be equal. The stone that is thrown at an angle of 30° above the horizontal has a longer flight time due to its initial upward velocity and so they do not strike the ground at the same time. (c) is correct. 3 • (a) False. Forces that are external to a system can do work on the system to change its energy. (b) False. In order for some object to do work, it must exert a force over some distance. The chemical energy stored in the muscles of your legs allows your muscles to do the work that launches you into the air. 4 • Determine the Concept Your kinetic energy increases at the expense of chemical energy. *5 • Determine the Concept As she starts pedaling, chemical energy inside her body is converted into kinetic energy as the bike picks up speed. As she rides it up the hill, chemical energy is converted into gravitational potential and thermal energy. While freewheeling down the hill, potential energy is converted to kinetic energy, and while braking to a stop, kinetic energy is converted into thermal energy (a more random form of kinetic energy) by the frictional forces acting on the bike. *6 • Determine the Concept If we define the system to include the falling body and the earth, then no work is done by an external agent and ∆K + ∆Ug + ∆Etherm= 0. Solving for the change in the gravitational potential energy we find ∆Ug = −(∆K + friction energy). 437 438 Chapter 7 (b) is correct. 7 •• Picture the Problem Because the constant friction force is responsible for a constant acceleration, we can apply the constant-acceleration equations to the analysis of these statements. We can also apply the work-energy theorem with friction to obtain expressions for the kinetic energy of the car and the rate at which it is changing. Choose the system to include the earth and car and assume that the car is moving on a horizontal surface so that ∆U = 0. (a) A constant frictional force causes a constant acceleration. The stopping distance of the car is related to its speed before the brakes were applied through a constantacceleration equation. (b) Apply the work-energy theorem with friction to obtain: Express the rate at which K is dissipated: 2 v 2 = v0 + 2a∆s where v = 0. 2 − v0 where a < 0. 2a 2 Thus, ∆s ∝ v0 and statement (a) is false. ∴∆s = ∆K = −Wf = − µ k mg∆s ∆K ∆s = − µ k mg ∆t ∆t Thus, ∆K ∝ v and therefore not constant. ∆t Statement (b) is false. (c) In part (b) we saw that: K ∝ ∆s Because ∆s ∝ ∆t: K ∝ ∆t and statement (c) is false. Because none of the above are correct: (d ) is correct. 8 • Picture the Problem We’ll let the zero of potential energy be at the bottom of each ramp and the mass of the block be m. We can use conservation of energy to predict the speed of the block at the foot of each ramp. We’ll consider the distance the block travels on each ramp, as well as its speed at the foot of the ramp, in deciding its descent times. Use conservation of energy to find the speed of the blocks at the bottom of each ramp: ∆K + ∆U = 0 or K bot − K top + U bot − U top = 0 Conservation of Energy 439 Because Ktop = Ubot = 0: K bot − U top = 0 Substitute to obtain: 1 2 Solve for vbot: vbot = 2 gH independently of the shape of 2 mvbot − mgH = 0 the ramp. Because the block sliding down the circular arc travels a greater distance (an arc length is greater than the length of the chord it defines) but arrives at the bottom of the ramp with the same speed that it had at the bottom of the inclined plane, it will require more time to arrive at the bottom of the arc. (b) is correct. 9 •• Determine the Concept No. From the work-kinetic energy theorem, no total work is being done on the rock, as its kinetic energy is constant. However, the rod must exert a tangential force on the rock to keep the speed constant. The effect of this force is to cancel the component of the force of gravity that is tangential to the trajectory of the rock. Estimation and Approximation *10 •• Picture the Problem We’ll use the data for the "typical male" described above and assume that he spends 8 hours per day sleeping, 2 hours walking, 8 hours sitting, 1 hour in aerobic exercise, and 5 hours doing moderate physical activity. We can approximate his energy utilization using Eactivity = APactivity ∆tactivity , where A is the surface area of his body, Pactivity is the rate of energy consumption in a given activity, and ∆tactivity is the time spent in the given activity. His total energy consumption will be the sum of the five terms corresponding to his daily activities. (a) Express the energy consumption of the hypothetical male: E = Esleeping + Ewalking + Esitting Evaluate Esleeping: Esleeping = APsleeping ∆tsleeping + Emod. act. + Eaerobic act. ( )( ) = 2 m 2 40 W/m 2 (8 h )(3600 s/h ) = 2.30 × 10 J 6 Evaluate Ewalking: Ewalking = APwalking ∆t walking ( )( ) = 2 m 2 160 W/m 2 (2 h )(3600 s/h ) = 2.30 × 106 J Evaluate Esitting: Esitting = APsitting ∆tsitting ( )( ) = 2 m 2 60 W/m 2 (8 h )(3600 s/h ) = 3.46 × 106 J 440 Chapter 7 Evaluate Emod. act.: Emod. act. = APmod. act.∆tmod. act. ( )( ) = 2 m 2 175 W/m 2 (5 h )(3600 s/h ) = 6.30 × 106 J Evaluate Eaerobic act.: Eaerobic act. = APaerobic act.∆taerobic act. ( )( ) = 2 m 2 300 W/m 2 (1 h )(3600 s/h ) = 2.16 × 106 J Substitute to obtain: E = 2.30 × 106 J + 2.30 × 106 J + 3.46 × 106 J + 6.30 × 106 J + 2.16 × 106 J = 16.5 × 106 J Express the average metabolic rate represented by this energy consumption: Pav = E 16.5 × 106 J = = 191 W ∆t (24 h )(3600 s/h ) or about twice that of a 100 W light bulb. (b) Express his average energy consumption in terms of kcal/day: (c) E= 16.5 × 106 J/day = 3940 kcal/day 4190 J/kcal 3940 kcal = 22.5 kcal/lb is higher than the estimate given in the statement of the 175 lb problem. However, by adjusting the day's activities, the metabolic rate can vary by more than a factor of 2. 11 • Picture the Problem The rate at which you expend energy, i.e., do work, is defined as power and is the ratio of the work done to the time required to do the work. Relate the rate at which you can expend energy to the work done in running up the four flights of stairs and solve for your running time: Express the work done in climbing the stairs: Substitute for ∆W to obtain: P= ∆W ∆W ⇒ ∆t = P ∆t ∆W = mgh ∆t = mgh P Conservation of Energy 441 Assuming that your weight is 600 N, evaluate ∆t: ∆t = (600 N )(4 × 3.5 m ) = 250 W 33.6 s 12 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation E0 = mc 2 . (a) Relate the rest mass consumed to the energy produced and solve for and evaluate m: (b) Express the energy required as a function of the power of the light bulb and evaluate E: Substitute in equation (1) to obtain: E0 = mc 2 ⇒ m = m= E0 c2 1J (2.998 ×10 8 m/s ) (1) 2 = 1.11× 10−17 kg E = 3Pt = 3(100 W )(10 y ) ⎛ 365.24 d ⎞ ⎛ 24 h ⎞ ⎛ 3600 s ⎞ ⎟⎜ ×⎜ ⎜ ⎟ d ⎟⎜ h ⎟ y ⎠⎝ ⎠ ⎝ ⎠⎝ 10 = 9.47 × 10 J m= 9.47 × 1010 J (2.998 ×10 8 m/s ) 2 = 1.05 µg *13 • Picture the Problem There are about 3×108 people in the United States. On the assumption that the average family has 4 people in it and that they own two cars, we have a total of 1.5×108 automobiles on the road (excluding those used for industry). We’ll assume that each car uses about 15 gal of fuel per week. Calculate, based on the assumptions identified above, the total annual consumption of energy derived from gasoline: (1.5 ×10 8 gal J ⎞ ⎛ ⎞ ⎛ weeks ⎞ ⎛ ⎟ ⎜ 2.6 × 108 ⎟ = 3.04 × 1019 J/y auto ⎜15 ⎟ ⎜ 52 ⎜ ⎟⎜ y ⎠⎝ gal ⎟ ⎝ auto ⋅ week ⎠ ⎝ ⎠ ) Express this rate of energy use as a fraction of the total annual energy use by the US: 3.04 × 1019 J/y ≈ 6% 5 × 1020 J/y Remarks: This is an average power expenditure of roughly 9x1011 watt, and a total cost (assuming $1.15 per gallon) of about 140 billion dollars per year. 14 • Picture the Problem The energy consumption of the U.S. works out to an average power consumption of about 1.6×1013 watt. The solar constant is roughly 103 W/m2 (reaching 442 Chapter 7 the ground), or about 120 W/m2 of useful power with a 12% conversion efficiency. Letting P represent the daily rate of energy consumption, we can relate the power available at the surface of the earth to the required area of the solar panels using P = IA . Relate the required area to the electrical energy to be generated by the solar panels: Solve for and evaluate A: P = IA where I is the solar intensity that reaches the surface of the Earth. A= ( P 2 1.6 × 1013 W = I 120 W/m 2 ) = 2.67 × 1011 m 2 where the factor of 2 comes from the fact that the sun is only up for roughly half the day. Find the side of a square with this area: s = 2.67 × 1011 m 2 = 516 km Remarks: A more realistic estimate that would include the variation of sunlight over the day and account for latitude and weather variations might very well increase the area required by an order of magnitude. 15 Picture the Problem We can relate the energy available from the water in terms of its mass, the vertical distance it has fallen, and the efficiency of the process. Differentiation of this expression with respect to time will yield the rate at which water must pass through its turbines to generate Hoover Dam’s annual energy output. Assuming a total efficiencyη, use the expression for the gravitational potential energy near the earth’s surface to express the energy available from the water when it has fallen a distance h: E = ηmgh Differentiate this expression with respect to time to obtain: P= Solve for dV/dt: dV P = dt ηρgh Using its definition, relate the dam’s annual power output to the energy produced: P= Substitute numerical values to obtain: 4 × 109 kW ⋅ h P= = 4.57 × 108 W (365.24 d )(24 h/d ) d [ηmgh] = ηgh dm = ηρgh dV dt dt dt (1) ∆E ∆t Conservation of Energy 443 Substitute in equation (1) and evaluate dV/dt: 4.57 × 108 W dV = dt 0.2(1kg/L ) 9.81 m/s 2 (211m ) ( ) = 1.10 × 106 L/s The Conservation of Mechanical Energy 16 • Picture the Problem The work done in compressing the spring is stored in the spring as potential energy. When the block is released, the energy stored in the spring is transformed into the kinetic energy of the block. Equating these energies will give us a relationship between the compressions of the spring and the speeds of the blocks. Let the numeral 1 refer to the first case and the numeral 2 to the second case. Relate the compression of the spring in the second case to its potential energy, which equals its initial kinetic energy when released: 1 2 Relate the compression of the spring in the first case to its potential energy, which equals its initial kinetic energy when released: 1 2 2 2 kx2 = 1 m2 v2 2 = 1 2 (4m1 )(3v1 )2 = 18m1v12 kx12 = 1 m1v12 2 or m1v12 = kx12 2 kx2 = 18kx12 Substitute to obtain: 1 2 Solve for x2: x 2 = 6x1 17 • Picture the Problem Choose the zero of gravitational potential energy to be at the foot of the hill. Then the kinetic energy of the woman on her bicycle at the foot of the hill is equal to her gravitational potential energy when she has reached her highest point on the hill. Equate the kinetic energy of the rider at the foot of the incline and her gravitational potential energy when she has reached her highest point on the hill and solve for h: Relate her displacement along the 1 2 mv 2 = mgh ⇒ h = d = h/sinθ v2 2g 444 Chapter 7 incline d to h and the angle of the incline: Substitute for h to obtain: d sin θ = Solve for d: d= Substitute numerical values and evaluate d: d= v2 2g v2 2 g sin θ (10 m/s) 2 ( ) 2 9.81 m/s 2 sin3° and = 97.4 m (c) is correct. *18 • Picture the Problem The diagram shows the pendulum bob in its initial position. Let the zero of gravitational potential energy be at the low point of the pendulum’s swing, the equilibrium position. We can find the speed of the bob at it passes through the equilibrium position by equating its initial potential energy to its kinetic energy as it passes through its lowest point. Equate the initial gravitational potential energy and the kinetic energy of the bob as it passes through its lowest point and solve for v: Express ∆h in terms of the length L of the pendulum: Substitute and simplify: mg∆h = 1 mv 2 2 and v = 2 g∆h ∆h = v= L 4 gL 2 19 • Picture the Problem Choose the zero of gravitational potential energy to be at the foot of the ramp. Let the system consist of the block, the earth, and the ramp. Then there are Conservation of Energy 445 no external forces acting on the system to change its energy and the kinetic energy of the block at the foot of the ramp is equal to its gravitational potential energy when it has reached its highest point. Relate the gravitational potential energy of the block when it has reached h, its highest point on the ramp, to its kinetic energy at the foot of the ramp: mgh = 1 mv 2 2 Solve for h: v2 h= 2g Relate the displacement d of the block along the ramp to h and the angle the ramp makes with the horizontal: d = h/sinθ Substitute for h: v2 d sin θ = 2g Solve for d: d= Substitute numerical values and evaluate d: d= v2 2 g sin θ ( (7 m/s) 2 ) 2 9.81 m/s 2 sin40° = 3.89 m 20 • Picture the Problem Let the system consist of the earth, the block, and the spring. With this choice there are no external forces doing work to change the energy of the system. Let Ug = 0 at the elevation of the spring. Then the initial gravitational potential energy of the 3-kg object is transformed into kinetic energy as it slides down the ramp and then, as it compresses the spring, into potential energy stored in the spring. (a) Apply conservation of energy to relate the distance the spring is compressed to the initial potential energy of the block: Solve for x: Wext = ∆K + ∆U = 0 and, because ∆K = 0, − mgh + 1 kx 2 = 0 2 x= 2mgh k 446 Chapter 7 Substitute numerical values and evaluate x: x= ( ) 2(3 kg ) 9.81 m/s 2 (5 m ) 400 N/m = 0.858 m (b) The energy stored in the compressed spring will accelerate the block, launching it back up the incline: The block will retrace its path, rising to a height of 5 m. 21 • Picture the Problem With Ug chosen to be zero at the uncompressed level of the spring, the ball’s initial gravitational potential energy is negative. The difference between the initial potential energy of the spring and the gravitational potential energy of the ball is first converted into the kinetic energy of the ball and then into gravitational potential energy as the ball rises and slows … eventually coming momentarily to rest. Apply the conservation of energy to the system as it evolves from its initial to its final state: − mgx + 1 kx 2 = mgh 2 Solve for h: h= Substitute numerical values and evaluate h: kx 2 −x 2mg (600 N/m )(0.05 m )2 − 0.05 m h= 2(0.015 kg ) (9.81 m/s 2 ) = 5.05 m 22 • Picture the Problem Let the system include the earth and the container. Then the work done by the crane is done by an external force and this work changes the energy of the system. Because the initial and final speeds of the container are zero, the initial and final kinetic energies are zero and the work done by the crane equals the change in the gravitational potential energy of the container. Choose Ug = 0 to be at the level of the deck of the freighter. Apply conservation of energy to the system: Because ∆K = 0: Wext = ∆E sys = ∆K + ∆U Wext = ∆U = mg∆h Conservation of Energy 447 Evaluate the work done by the crane: Wext = mg∆h ( ) = (4000 kg ) 9.81 m/s 2 (− 8 m ) = − 314 kJ 23 • Picture the Problem Let the system consist of the earth and the child. Then Wext = 0. Choose Ug,i = 0 at the child’s lowest point as shown in the diagram to the right. Then the child’s initial energy is entirely kinetic and its energy when it is at its highest point is entirely gravitational potential. We can determine h from energy conservation and then use trigonometry to determine θ. Using the diagram, relate θ to h and L: h⎞ ⎛ L−h⎞ −1 ⎛ ⎟ = cos ⎜1 − ⎟ ⎝ L⎠ ⎝ L ⎠ θ = cos −1 ⎜ mvi2 − mgh = 0 Apply conservation of energy to the system to obtain: 1 2 Solve for h: h= Substitute to obtain: θ = cos −1 ⎜1 − ⎜ vi2 2g ⎛ ⎝ Substitute numerical values and evaluate θ : vi2 ⎞ ⎟ 2 gL ⎟ ⎠ ⎛ (3.4 m/s) 2 ⎞ ⎜1 − ⎟ θ = cos ⎜ 2 9.81 m/s 2 (6 m ) ⎟ ⎝ ⎠ −1 ( ) = 25.6° *24 •• Picture the Problem Let the system include the two objects and the earth. Then Wext = 0. Choose Ug = 0 at the elevation at which the two objects meet. With this choice, the initial potential energy of the 3-kg object is positive and that of the 2-kg object is negative. Their sum, however, is positive. Given our choice for Ug = 0, this initial potential energy is transformed entirely into kinetic energy. Apply conservation of energy: Wext = ∆K + ∆U g = 0 or, because Wext = 0, 448 Chapter 7 ∆K = −∆Ug mvf2 − 1 mvi2 = − ∆U g 2 Substitute for ∆K and solve for vf; noting that m represents the sum of the masses of the objects as they are both moving in the final state: or, because vi = 0, Express and evaluate ∆Ug: ∆U g = U g,f − U g,i 1 2 − 2∆U g vf = m = 0 − (3 kg − 2 kg )(0.5 m ) ( × 9.81 m/s 2 ) = −4.91 J Substitute and evaluate vf: − 2(− 4.91 J ) = 1.40 m/s 5 kg vf = 25 •• Picture the Problem The free-body diagram shows the forces acting on the block when it is about to move. Fsp is the force exerted by the spring and, because the block is on the verge of sliding, fs = fs,max. We can use Newton’s 2nd law, under equilibrium conditions, to express the elongation of the spring as a function of m, k and θ and then substitute in the expression for the potential energy stored in a stretched or compressed spring. Express the potential energy of the spring when the block is about to move: Apply r r ∑ F = ma, under equilibrium conditions, to the block: U = 1 kx 2 2 ∑F x and ∑F y Using fs,max = µsFn and Fsp = kx, eliminate fs,max and Fsp from the x equation and solve for x: = Fsp − f s,max − mg sin θ = 0 x= = Fn − mg cosθ = 0 mg (sin θ + µs cos θ ) k Conservation of Energy 449 Substitute for x in the expression for U: ⎡ mg (sin θ + µs cosθ ) ⎤ U = k⎢ ⎥ k ⎣ ⎦ 2 1 2 = [mg (sin θ + µs cosθ )] 2 2k 26 •• Picture the Problem The mechanical energy of the system, consisting of the block, the spring, and the earth, is initially entirely gravitational potential energy. Let Ug = 0 where the spring is compressed 15 cm. Then the mechanical energy when the compression of the spring is 15 cm will be partially kinetic and partially stored in the spring. We can use conservation of energy to relate the initial potential energy of the system to the energy stored in the spring and the kinetic energy of block when it has compressed the spring 15 cm. Apply conservation of energy to the system: ∆U + ∆K = 0 or U g,f − U g,i + U s,f − U s,i + K f − K i = 0 Because Ug,f = Us,I = Ki = 0: − U g,i + U s,f + K f = 0 Substitute to obtain: − mg (h + x ) + 1 kx 2 + 1 mv 2 = 0 2 2 Solve for v: v = 2 g (h + x ) − kx 2 m Substitute numerical values and evaluate v: ( ) v = 2 9.81 m/s 2 (5 m + 0.15 m ) − (3955 N/m )(0.15 m )2 2.4 kg = 8.00 m/s 450 Chapter 7 *27 •• Picture the Problem The diagram represents the ball traveling in a circular path with constant energy. Ug has been chosen to be zero at the lowest point on the circle and the superimposed free-body diagrams show the forces acting on the ball at the top and bottom of the circular path. We’ll apply Newton’s 2nd law to the ball at the top and bottom of its path to obtain a relationship between TT and TB and the conservation of mechanical energy to relate the speeds of the ball at these two locations. Apply ∑F radial = maradial to the ball at the bottom of the circle and solve for TB: 2 vB TB − mg = m R and 2 vB TB = mg + m R Apply ∑F radial = maradial to the ball at the top of the circle and solve for TT: TT + mg = m 2 vT R and TT = −mg + m Subtract equation (2) from equation (1) to obtain: (1) 2 vT R TB − TT = mg + m (2) 2 vB R ⎛ v2 ⎞ ⎜ − mg + m T ⎟ −⎜ R⎟ ⎝ ⎠ 2 2 v v = m B − m T + 2mg R R Using conservation of energy, relate the mechanical energy of the ball at the bottom of its path to its mechanical energy at the top of the circle and solve for m 1 2 2 2 mvB = 1 mvT + mg (2 R ) 2 m 2 vB v2 − m T = 4mg R R 2 vB v2 −m T : R R Substitute in equation (3) to obtain: TB − TT = 6mg (3) Conservation of Energy 451 28 •• Picture the Problem Let Ug = 0 at the lowest point in the girl’s swing. Then we can equate her initial potential energy to her kinetic energy as she passes through the low point on her swing to relate her speed v to R. The FBD show the forces acting on the girl at the low point of her swing. Applying Newton’s 2nd law to her will allow us to establish the relationship between the tension T and her speed. Apply ∑F radial = maradial to the girl T − mg = m at her lowest point and solve for T: v2 R and T = mg + m Equate the girl’s initial potential energy to her final kinetic energy and solve for mg v2 R R 1 2 v2 = 2 mv ⇒ =g 2 R v2 : R Substitute for v2/R2 and simplify to obtain: T = mg + mg = 2mg 29 •• Picture the Problem The free-body diagram shows the forces acting on the car when it is upside down at the top of the loop. Choose Ug = 0 at the bottom of the loop. We can express Fn in terms of v and R by apply Newton’s 2nd law to the car and then obtain a second expression in these same variables by applying the conservation of mechanical energy. The simultaneous solution of these equations will yield an expression for Fn in terms of known quantities. Apply ∑F radial = maradial to the car at the top of the circle and solve for Fn : Fn + mg = m v2 R and Fn = m v2 − mg R (1) 452 Chapter 7 Using conservation of energy, relate the energy of the car at the beginning of its motion to its energy when it is at the top of the loop: Solve for m v2 : R mgH = 1 mv 2 + mg (2 R ) 2 m Substitute equation (2) in equation (1) to obtain: v2 ⎛H ⎞ = 2mg ⎜ − 2 ⎟ R ⎝R ⎠ ⎛H ⎞ Fn = 2mg ⎜ − 2 ⎟ − mg ⎝R ⎠ ⎛ 2H ⎞ = mg ⎜ − 5⎟ ⎝ R ⎠ Substitute numerical values and evaluate Fn: ⎡ 2(23 m ) ⎤ − 5⎥ = 1.67 × 10 4 N ⇒ (c) is correct. Fn = (1500 kg ) 9.81 m/s 2 ⎢ ⎣ 7.5 m ⎦ ( ) 30 • Picture the Problem Let the system include the roller coaster, the track, and the earth and denote the starting position with the numeral 0 and the top of the second hill with the numeral 1. We can use the workenergy theorem to relate the energies of the coaster at its initial and final positions. (a) Use conservation of energy to relate the work done by external forces to the change in the energy of the system: Because the track is frictionless, Wext = 0: Wext = ∆Esys = ∆K + ∆U ∆K + ∆U = 0 and K1 − K 0 + U1 − U 0 = 0 2 mv12 − 1 mv0 + mgh1 − mgh0 = 0 2 Substitute to obtain: 1 2 Solve for v0: v0 = v12 + 2 g (h1 − h0 ) If the coaster just makes it to the top of the second hill, v1 = 0 and: v0 = 2 g (h1 − h0 ) (2) Conservation of Energy 453 Substitute numerical values and evaluate v0: (b) ( ) v0 = 2 9.81 m/s 2 (9.5 m − 5 m ) = 9.40 m/s No. Note that the required speed depends only on the difference in the heights of the two hills. 31 •• Picture the Problem Let the radius of the loop be R and the mass of one of the riders be m. At the top of the loop, the centripetal force on her is her weight (the force of gravity). The two forces acting on her at the bottom of the loop are the normal force exerted by the seat of the car, pushing up, and the force of gravity, pulling down. We can apply Newton’s 2nd law to her at both the top and bottom of the loop to relate the speeds at those locations to m and R and, at b, to F, and then use conservation of energy to relate vt and vb. Apply ∑F radial = ma radial to the rider at the bottom of the circular arc: Solve for F to obtain: Apply ∑F radial = ma radial to the rider at the top of the circular arc: F − mg = m 2 vb R F = mg + m 2 vb R mg = m (1) vt2 R Solve for vt2 : vt2 = gR Use conservation of energy to relate the energies of the rider at the top and bottom of the arc: Kb − Kt + U b − U t = 0 or, because Ub = 0, Kb − Kt − U t = 0 2 mvb − 1 mvt2 − 2mgR = 0 2 Substitute to obtain: 1 2 2 Solve for vb : 2 vb = 5 gR Substitute in equation (1) to obtain: F = mg + m 5 gR = 6mg R i.e., the rider will feel six times heavier than her normal weight. 454 Chapter 7 *32 •• Picture the Problem Let the system consist of the stone and the earth and ignore the influence of air resistance. Then Wext = 0. Choose Ug = 0 as shown in the figure. Apply the law of the conservation of mechanical energy to describe the energy transformations as the stone rises to the highest point of its trajectory. Apply conservation of energy: Wext = ∆K + ∆U = 0 and K1 − K 0 + U1 − U 0 = 0 Because U0 = 0: K1 − K 0 + U1 = 0 Substitute to obtain: 1 2 2 mvx − 1 mv 2 + mgH = 0 2 In the absence of air resistance, the r horizontal component of v is constant and equal to vx = vcosθ. Hence: 1 2 m(v cosθ ) − 1 mv 2 + mgH = 0 2 Solve for v: Substitute numerical values and evaluate v: 2 v= 2 gH 1 − cos 2 θ v= 2 9.81 m/s 2 (24 m ) = 27.2 m/s 1 − cos 2 53° ( ) 33 •• Picture the Problem Let the system consist of the ball and the earth. Then Wext = 0. The figure shows the ball being thrown from the roof of a building. Choose Ug = 0 at ground level. We can use the conservation of mechanical energy to determine the maximum height of the ball and its speed at impact with the ground. We can use the definition of the work done by gravity to calculate how much work was done by gravity as the ball rose to its maximum height. (a) Apply conservation of energy: Wext = ∆K + ∆U = 0 Conservation of Energy 455 or K 2 − K1 + U 2 − U 1 = 0 Substitute for the energies to obtain: Note that, at point 2, the ball is moving horizontally and: Substitute for v2 and h2: 1 2 2 mv 2 − 1 mv12 + mgh2 − mgh1 = 0 2 v 2 = v1 cosθ 1 2 m(v1 cosθ ) − 1 mv12 + mgH 2 2 − mgh1 = 0 ( ) v2 cos 2 θ − 1 2g Solve for H: H = h1 − Substitute numerical values and evaluate H: H = 12 m − (30 m/s)2 (cos 2(9.81 m/s ) 2 2 ) 40° − 1 = 31.0 m (b) Using its definition, express the work done by gravity: Substitute numerical values and evaluate Wg: ( Wg = − ∆U = − U H − U hi ) = −(mgH − mghi ) = −mg (H − hi ) ( ) Wg = −(0.17 kg ) 9.81 m/s 2 (31 m − 12 m ) = − 31.7 J mvi2 + mghi = 1 mvf2 2 (c) Relate the initial mechanical energy of the ball to its just-beforeimpact energy: 1 2 Solve for vf: vf = vi2 + 2ghi Substitute numerical values and evaluate vf vf = (30 m/s) 2 + 2(9.81m/s2 )(12 m ) = 33.7 m/s 456 Chapter 7 34 •• Picture the Problem The figure shows the pendulum bob in its release position and in the two positions in which it is in motion with the given speeds. Choose Ug = 0 at the low point of the swing. We can apply the conservation of mechanical energy to relate the two angles of interest to the speeds of the bob at the intermediate and low points of its trajectory. (a) Apply conservation of energy: Wext = ∆K + ∆U = 0 or Kf − Ki + U f − U i = 0 where U f and K i equal zero. ∴Kf − U i = 0 Express Ui: U i = mgh = mgL(1 − cos θ 0 ) Substitute for Kf and Ui: 1 2 Solve for θ0: ⎛ v2 ⎞ θ 0 = cos ⎜1 − ⎜ 2 gL ⎟ ⎟ ⎝ ⎠ Substitute numerical values and evaluate θ0: θ 0 = cos −1 ⎢1 − mvf2 − mgL(1 − cos θ 0 ) = 0 −1 ⎡ ⎣ ⎤ (2.8 m/s)2 ⎥ 2 2(9.81m/s )(0.8 m ) ⎦ = 60.0° (b) Letting primed quantities describe the indicated location, use the law of the conservation of mechanical energy to relate the speed of the bob at this point to θ : Express U f' : Substitute for K f' , U f' and U i : K f' − K i + U f' − U i = 0 where K i = 0. ∴K f ' + U f ' − U i = 0 U f' = mgh' = mgL(1 − cosθ ) 1 2 m(vf' ) + mgL(1 − cos θ ) 2 − mgL(1 − cos θ 0 ) = 0 Conservation of Energy 457 Solve for θ : Substitute numerical values and evaluate θ : ⎡ (vf ')2 ⎤ + cosθ 0 ⎥ ⎣ 2 gL ⎦ θ = cos −1 ⎢ ⎤ (1.4 m/s) 2 + cos 60°⎥ 2 ⎣ 2(9.81m/s )(0.8 m ) ⎦ ⎡ θ = cos −1 ⎢ = 51.3° *35 •• Picture the Problem Choose Ug = 0 at the bridge, and let the system be the earth, the jumper and the bungee cord. Then Wext = 0. Use the conservation of mechanical energy to relate to relate her initial and final gravitational potential energies to the energy stored in the stretched bungee, Us cord. In part (b), we’ll use a similar strategy but include a kinetic energy term because we are interested in finding her maximum speed. (a) Express her final height h above the water in terms of L, d and the distance x the bungee cord has stretched: Use the conservation of mechanical energy to relate her gravitational potential energy as she just touches the water to the energy stored in the stretched bungee cord: h=L–d−x (1) Wext = ∆K + ∆U = 0 Because ∆K = 0 and ∆U = ∆Ug + ∆Us, − mgL + 1 kx 2 = 0, 2 where x is the maximum distance the bungee cord has stretched. 2mgL x2 Solve for k: k= Find the maximum distance the bungee cord stretches: x = 310 m – 50 m = 260 m. Evaluate k: k= 2(60 kg ) (9.81m/s 2 )(310 m ) (260 m )2 = 5.40 N/m 458 Chapter 7 Fnet = kx − mg = 0 Express the relationship between the forces acting on her when she has finally come to rest and solve for x: and Evaluate x: (60 kg )(9.81m/s 2 ) = 109 m x= x= mg k 5.40 N/m Substitute in equation (1) and evaluate h: (b) Using conservation of energy, express her total energy E: Because v is a maximum when K is a maximum, solve for K:: h = 310 m − 50 m − 109 m = 151 m E = K + U g + U s = Ei = 0 K = −U g − U s (1) = mg (d + x ) − 1 kx 2 2 Use the condition for an extreme value to obtain: dK = mg − kx = 0 for extreme values dx Solve for and evaluate x: x= From equation (1) we have: 1 2 Solve for v to obtain: v = 2 g (d + x ) − ( ) mg (60 kg ) 9.81m/s 2 = = 109 m k 5.40 N/m mv 2 = mg (d + x ) − 1 kx 2 2 kx 2 m Substitute numerical values and evaluate v for x = 109 m: ( v = 2 9.81 m/s Because 2 (5.4 N/m )(109 m )2 )(50 m + 109 m) − 60 kg = 45.3 m/s d 2K = −k < 0, x = 109 m corresponds to Kmax and so v is a maximum. dx 2 Conservation of Energy 459 36 •• Picture the Problem Let the system be the earth and pendulum bob. Then Wext = 0. Choose Ug = 0 at the low point of the bob’s swing and apply the law of the conservation of mechanical energy to its motion. When the bob reaches the 30° position its energy will be partially kinetic and partially potential. When it reaches its maximum height, its energy will be entirely potential. Applying Newton’s 2nd law will allow us to express the tension in the string as a function of the bob’s speed and its angular position. (a) Apply conservation of energy to relate the energies of the bob at points 1 and 2: Wext = ∆K + ∆U = 0 or K 2 − K1 + U 2 − U 1 = 0 1 2 Because U1 = 0: 2 mv2 − 1 mv12 + U 2 = 0 2 U 2 = mgL(1 − cos θ ) Express U2: Substitute for U2 to obtain: 1 2 2 mv2 − 1 mv12 + mgL(1 − cosθ ) = 0 2 v2 = v12 − 2 gL(1 − cos θ ) Solve for v2: Substitute numerical values and evaluate v2: v2 = (4.5 m/s )2 − 2(9.81m/s 2 )(3 m )(1 − cos30°) = (b) From (a) we have: Substitute numerical values and evaluate U2: (c) Apply ∑F obtain: Solve for T: radial = maradial to the bob to 3.52 m/s U 2 = mgL(1 − cos θ ) ( ) U 2 = (2 kg ) 9.81 m/s 2 (3 m )(1 − cos30°) = 7.89 J 2 v2 T − mg cos θ = m L 2 ⎛ v2 ⎞ T = m⎜ g cos θ + ⎟ ⎜ L⎟ ⎝ ⎠ 460 Chapter 7 Substitute numerical values and evaluate T: ⎡ (3.52 m/s)2 ⎤ = 25.3 N 2 T = (2 kg )⎢ 9.81 m/s cos30° + ⎥ 3m ⎣ ⎦ ( ) (d) When the bob reaches its greatest height: Substitute for K1 and Umax Solve for θmax: U = U max = mgL(1 − cos θ max ) and K1 + U max = 0 − 1 mv12 + mgL(1 − cos θ max ) = 0 2 θ max Substitute numerical values and evaluate θmax: ⎛ v2 ⎞ ⎜1 − 1 ⎟ = cos ⎜ ⎟ ⎝ 2 gL ⎠ −1 ⎡ θ max = cos −1 ⎢1 − ⎣ = 49.0° 37 •• Picture the Problem Let the system consist of the earth and pendulum bob. Then Wext = 0. Choose Ug = 0 at the bottom of the circle and let points 1, 2 and 3 represent the bob’s initial point, lowest point and highest point, respectively. The bob will gain speed and kinetic energy until it reaches point 2 and slow down until it reaches point 3; so it has its maximum kinetic energy when it is at point 2. We can use Newton’s 2nd law at points 2 and 3 in conjunction with the law of the conservation of mechanical energy to find the maximum kinetic energy of the bob and the tension in the string when the bob has its maximum kinetic energy. (a) Apply ∑F radial = maradial to the bob at the top of the circle and solve 2 for v3 : mg = m and 2 v3 = gL 2 v3 L (4.5 m/s) 2 ⎤ ⎥ 2(9.81 m/s 2 )(3 m ) ⎦ Conservation of Energy 461 Use conservation of energy to express the relationship between K2, K3 and U3 and solve for K2: K 3 − K 2 + U 3 − U 2 = 0 where U 2 = 0 Therefore, K 2 = K max = K 3 + U 3 2 = 1 mv3 + mg (2 L ) 2 2 Substitute for v3 and simplify to K max = 1 m(gL ) + 2mgL = 2 5 2 mgL obtain: (b) Apply ∑F radial = maradial to the bob at the bottom of the circle and solve for T2: Fnet = T2 − mg = m 2 v2 L and T2 = mg + m 2 v2 L K 3 − K 2 + U 3 − U 2 = 0 where U 2 = 0 Use conservation of energy to relate the energies of the bob at points 2 and 3 and solve for K2: K 2 = K3 + U 3 2 Substitute for v3 and K2 and solve 1 2 2 = 1 mv3 + mg (2 L ) 2 2 mv2 = 1 m( gL ) + mg (2 L ) 2 2 for v2 : and Substitute in equation (1) to obtain: T2 = 6mg 38 •• Picture the Problem Let the system consist of the earth and child. Then Wext = 0. In the figure, the child’s initial position is designated with the numeral 1; the point at which the child releases the rope and begins to fall with a 2, and its point of impact with the water is identified with a 3. Choose Ug = 0 at the water level. While one could use the law of the conservation of energy between points 1 and 2 and then between points 2 and 3, it is more direct to consider the energy transformations between points 1 and 3. Given our choice of the zero of gravitational potential energy, the initial potential energy at point 1 is transformed into kinetic energy at point 3. (1) 2 v2 = 5 gL 462 Chapter 7 Apply conservation of energy to the energy transformations between points 1 and 3: Wext = ∆K + ∆U = 0 K 3 − K1 + U 3 − U1 = 0 where U 3 and K1are zero. 2 mv3 − mg [h + L(1 − cos θ )] = 0 Substitute for K3 and U1; 1 2 Solve for v3: v3 = 2 g [h + L(1 − cos θ )] Substitute numerical values and evaluate v3: ( ) v3 = 2 9.81m/s 2 [3.2 m + (10.6 m )(1 − cos23°)] = 8.91m/s *39 •• Picture the Problem Let the system consist of you and the earth. Then there are no external forces to do work on the system and Wext = 0. In the figure, your initial position is designated with the numeral 1, the point at which you release the rope and begin to fall with a 2, and your point of impact with the water is identified with a 3. Choose Ug = 0 at the water level. We can apply Newton’s 2nd law to the forces acting on you at point 2 and apply conservation of energy between points 1 and 2 to determine the maximum angle at which you can begin your swing and then between points 1 and 3 to determine the speed with which you will hit the water. (a) Use conservation of energy to relate your speed at point 2 to your potential energy there and at point 1: Wext = ∆K + ∆U = 0 or K 2 − K1 + U 2 − U 1 = 0 1 2 Because K1 = 0: Solve this equation for θ : 2 mv2 + mgh − [mgL(1 − cos θ ) + mgh] = 0 ⎡ θ = cos −1 ⎢1 − ⎣ Apply ∑F radial = maradial to T − mg = m 2 v2 ⎤ ⎥ 2 gL ⎦ 2 v2 L (1) Conservation of Energy 463 yourself at point 2 and solve for T: and 2 v2 T = mg + m L Because you’ve estimated that the rope might break if the tension in it exceeds your weight by 80 N, it must be that: 2 v2 m = 80 N L or (80 N )L 2 v2 = m Let’s assume your weight is 650 N. Then your mass is 66.3 kg and: 2 v2 = Substitute numerical values in equation (1) to obtain: θ = cos −1 ⎢1 − (80 N )(4.6 m ) = 5.55 m 2 /s 2 66.3kg ⎡ ⎣ ⎤ 5.55 m 2 /s 2 2 2 9.81 m/s (4.6 m )⎥ ⎦ ( ) = 20.2° (b) Apply conservation of energy to the energy transformations between points 1 and 3: Wext = ∆K + ∆U = 0 K 3 − K1 + U 3 − U 1 = 0 where U 3 and K1are zero 2 mv3 − mg [h + L(1 − cos θ )] = 0 Substitute for K3 and U1 to obtain: 1 2 Solve for v3: v3 = 2 g [h + L(1 − cos θ )] Substitute numerical values and evaluate v3: ( ) v3 = 2 9.81 m/s 2 [1.8 m + (4.6 m )(1 − cos20.2°)] = 6.39 m/s 464 Chapter 7 40 •• Picture the Problem Choose Ug = 0 at point 2, the lowest point of the bob’s trajectory and let the system consist of the bob and the earth. Given this choice, there are no external forces doing work on the system. Because θ << 1, we can use the trigonometric series for the sine and cosine functions to approximate these functions. The bob’s initial energy is partially gravitational potential and partially potential energy stored in the stretched spring. As the bob swings down to point 2 this energy is transformed into kinetic energy. By equating these energies, we can derive an expression for the speed of the bob at point 2. Apply conservation of energy to the system as the pendulum bob swings from point 1 to point 2: Note, from the figure, that x ≈ Lsinθ when θ << 1: Also, when θ << 1: Substitute, simplify and solve for v2: 1 2 2 mv 2 = 1 kx 2 + mgL(1 − cosθ ) 2 1 2 2 mv 2 = 1 k (L sin θ ) + mgL(1 − cosθ ) 2 2 sin θ ≈ θ and cosθ ≈ 1 − 1 θ 2 2 v 2 = Lθ k g + m L Conservation of Energy 465 41 ••• Picture the Problem Choose Ug = 0 at point 2, the lowest point of the bob’s trajectory and let the system consist of the earth, ceiling, spring, and pendulum bob. Given this choice, there are no external forces doing work to change the energy of the system. The bob’s initial energy is partially gravitational potential and partially potential energy stored in the stretched spring. As the bob swings down to point 2 this energy is transformed into kinetic energy. By equating these energies, we can derive an expression for the speed of the bob at point 2. 1 2 Apply conservation of energy to the system as the pendulum bob swings from point 1 to point 2: 2 mv 2 = 1 kx 2 + mgL(1 − cosθ ) (1) 2 Apply the Pythagorean theorem to the lower triangle in the diagram to obtain: (x + 1 L )2 = L2 [sin 2 θ + ( 3 cosθ )2 ] = L2 [sin 2 θ + 9 − 3 cosθ + cos 2 θ ] = L2 (13 − 3 cosθ ) 2 2 4 4 Take the square root of both sides of the equation to obtain: x+ 1 L= L 2 Solve for x: x=L [( 13 4 (13 − 3 cosθ ) 4 − 3 cos θ ) − 1 2 Substitute for x in equation (1): 1 2 2 mv2 = 1 kL2 2 [( 13 4 − 3 cos θ ) − 1 2 2 + mgL(1 − cosθ ) 2 Solve for v2 to obtain: 2 v2 = 2 gL(1 − cos θ ) + [ ( k 2 L m k ⎡ g = L2 ⎢2 (1 − cos θ ) + m ⎣ L 13 4 − 3 cosθ − 1 2 13 4 − 3 cosθ − 1 2 2 ) 2 ⎤ ⎥ ⎦ 466 Chapter 7 Finally, solve for v2: v2 = L 2 g (1 − cosθ ) + k L m ( 13 4 − 3 cos θ − 1 2 ) 2 The Conservation of Energy 42 • Picture the Problem The energy of the eruption is initially in the form of the kinetic energy of the material it thrusts into the air. This energy is then transformed into gravitational potential energy as the material rises. E = mg∆h (a) Express the energy of the eruption in terms of the height ∆h to which the debris rises: m V Relate the density of the material to its mass and volume: ρ= Substitute for m to obtain: E = ρVg∆h Substitute numerical values and evaluate E: ( )( )( ) E = 1600 kg/m 3 4 km 3 9.81 m/s 2 (500 m ) = 3.14 × 1016 J (b) Convert 3.13×1016 J to megatons of TNT: 3.14 × 1016 J = 3.14 × 1016 J × 1Mton TNT = 7.48 Mton TNT 4.2 × 1015 J 43 •• Picture the Problem The work done by the student equals the change in his/her gravitational potential energy and is done as a result of the transformation of metabolic energy in the climber’s muscles. (a) The increase in gravitational potential energy is: ∆U = mg∆h ( ) = (80 kg ) 9.81 m/s 2 (120 m ) = 94.2 kJ Conservation of Energy 467 (b) The energy required to do this work comes from chemical energy stored in the body. (c) Relate the chemical energy expended by the student to the change in his/her potential energy and solve for E: 0.2 E = ∆U and E = 5∆U = 5(94.2 kJ ) = 471 kJ Kinetic Friction 44 • Picture the Problem Let the car and the earth be the system. As the car skids to a stop on a horizontal road, its kinetic energy is transformed into internal (i.e., thermal) energy. Knowing that energy is transformed into heat by friction, we can use the definition of the coefficient of kinetic friction to calculate its value. (a) The energy dissipated by friction is given by: Apply the work-energy theorem for problems with kinetic friction: f∆s = ∆Etherm Wext = ∆Emech + ∆Etherm = ∆Emech + f∆s or, because ∆Emech = ∆K = − K i and Wext = 0, 0 = − 1 mvi2 + f∆s 2 Solve for f∆s to obtain: f∆s = 1 mvi2 2 Substitute numerical values and evaluate f∆s: f∆s = (b) Relate the kinetic friction force to the coefficient of kinetic friction and the weight of the car and solve for the coefficient of kinetic friction: f k = µ k mg ⇒ µ k = Express the relationship between the energy dissipated by friction and the kinetic friction force and solve fk: ∆Etherm = f k ∆s ⇒ f k = Substitute to obtain: µk = 1 2 (2000 kg )(25 m/s)2 = ∆Etherm mg∆s fk mg ∆Etherm ∆s 625 kJ 468 Chapter 7 Substitute numerical values and evaluate µk: µk = 625 kJ (2000 kg ) 9.81 m/s 2 (60 m ) ( ) = 0.531 45 • Picture the Problem Let the system be the sled and the earth. Then the 40-N force is external to the system. The free-body diagram shows the forces acting on the sled as it is pulled along a horizontal road. The work done by the applied force can be found using the definition of work. To find the energy dissipated by friction, we’ll use Newton’s 2nd law to determine fk and then use it in the definition of work. The change in the kinetic energy of the sled is equal to the net work done on it. Finally, knowing the kinetic energy of the sled after it has traveled 3 m will allow us to solve for its speed at that location. (a) Use the definition of work to calculate the work done by the applied force: (b) Express the energy dissipated by friction as the sled is dragged along the surface: Apply ∑F y = ma y to the sled and r r Wext ≡ F ⋅ s = Fs cosθ = (40 N )(3 m )cos 30° = 104 J ∆Etherm = f∆x = µ k Fn ∆x Fn + F sin θ − mg = 0 solve for Fn: and Substitute to obtain: ∆Etherm = µ k ∆x(mg − F sin θ ) Substitute numerical values and evaluate ∆Etherm: Fn = mg − F sin θ [ ( ∆Etherm = (0.4)(3 m ) (8 kg ) 9.81m/s 2 − (40 N )sin30°] ) = 70.2 J (c) Apply the work-energy theorem Wext = ∆Emech + ∆Etherm = ∆Emech + f∆s Conservation of Energy 469 for problems with kinetic friction: or, because ∆Emech = ∆K + ∆U and ∆U = 0, Wext = ∆K + ∆Etherm Solve for and evaluate ∆K to obtain: ∆K = Wext − ∆Etherm = 104 J − 70.2 J = 33.8 J (d) Because Ki = 0: K f = ∆K = 1 mvf2 2 Solve for vf: vf = 2∆K m Substitute numerical values and evaluate vf: vf = 2(33.8 J ) = 2.91 m/s 8 kg *46 • Picture the Problem Choose Ug = 0 at the foot of the ramp and let the system consist of the block, ramp, and the earth. Then the kinetic energy of the block at the foot of the ramp is equal to its initial kinetic energy less the energy dissipated by friction. The block’s kinetic energy at the foot of the incline is partially converted to gravitational potential energy and partially dissipated by friction as the block slides up the incline. The free-body diagram shows the forces acting on the block as it slides up the incline. Applying Newton’s 2nd law to the block will allow us to determine fk and express the energy dissipated by friction. (a) Apply conservation of energy to the system while the block is moving horizontally: Wext = ∆Emech + ∆Etherm = ∆K + ∆U + f∆s or, because ∆U = Wext = 0, 0 = ∆K + f∆s = K f − K i + f∆s Solve for Kf: K f = K i − f∆s 470 Chapter 7 mvf2 = 1 mvi2 − µ k mg∆x 2 Substitute for Kf, Ki, and f∆s to obtain: 1 2 Solving for vf yields: vf = vi2 − 2µ k g∆x Substitute numerical values and evaluate vf: vf = (b) Apply conservation of energy to the system while the block is on the incline: Wext = ∆Emech + ∆Etherm (7 m/s)2 − 2(0.3)(9.81m/s2 )(2 m ) = 6.10 m/s = ∆K + ∆U + f∆s or, because Kf = Wext = 0, 0 = − K i + ∆U + f∆s Apply ∑F y = ma y to the block Fn − mg cos θ = 0 ⇒ Fn = mg cos θ when it is on the incline: Express f∆s: The final potential energy of the block is: Substitute for Uf, Ui, and f∆s to obtain: f∆s = f k L = µ k Fn L = µ k mgL cosθ U f = mgL sin θ 0 = − K i + mgL sin θ + µ k mgL cos θ Solving for L yields: L= vi2 g (sin θ + µ k cos θ ) Substitute numerical values and evaluate L: L= (6.10 m/s)2 (9.81 m/s 2 )(sin40° + (0.3)cos40°) 1 2 1 2 = 2.17 m 47 • Picture the Problem Let the system include the block, the ramp and horizontal surface, and the earth. Given this choice, there are no external forces acting that will change the energy of the system. Because the curved ramp is frictionless, mechanical energy is conserved as the block slides down it. We can calculate its speed at the bottom of the ramp by using the law of the conservation of energy. The potential energy of the block at the top of the ramp or, equivalently, its kinetic energy at the bottom of the ramp is Conservation of Energy 471 converted into thermal energy during its slide along the horizontal surface. (a) Choosing Ug = 0 at point 2 and letting the numeral 1 designate the initial position of the block and the numeral 2 its position at the foot of the ramp, use conservation of energy to relate the block’s potential energy at the top of the ramp to its kinetic energy at the bottom: Solve for v2 to obtain: Substitute numerical values and evaluate v2: (b) The energy dissipated by friction is responsible for changing the thermal energy of the system: Because ∆K = 0 for the slide: Substitute numerical values and evaluate Wf: (c) The energy dissipated by friction is given by: Wext = ∆Emech + ∆Etherm or, because Wext = Ki = Uf = ∆Etherm = 0, 2 0 = 1 mv2 − mg∆h = 0 2 v2 = 2 g∆h ( ) v2 = 2 9.81 m/s 2 (3 m ) = 7.67 m/s Wf + ∆K + ∆U = ∆E therm + ∆K + ∆U = 0 Wf = −∆U = −(U 2 − U1 ) = U1 ( ) Wf = mg∆h = (2 kg ) 9.81 m/s 2 (3 m ) = 58.9 J ∆Etherm = f∆s = µ k mg∆x Solve for µk: µk = ∆Etherm mg∆x Substitute numerical values and evaluate µk: µk = 58.9 J = 0.333 (2 kg ) 9.81 m/s 2 (9 m ) ( ) 48 •• Picture the Problem Let the system consist of the earth, the girl, and the slide. Given this choice, there are no external forces doing work to change the energy of the system. By the time she reaches the bottom of the slide, her potential energy at the top of the slide has been converted into kinetic and thermal energy. Choose Ug = 0 at the bottom of the slide and denote the top and bottom of the slide as shown in 472 Chapter 7 the figure. We’ll use the work-energy theorem with friction to relate these quantities and the forces acting on her during her slide to determine the friction force that transforms some of her initial potential energy into thermal energy. Wext = ∆K + ∆U + ∆Etherm = 0 (a) Express the work-energy theorem: 0 = K 2 − U1 + ∆Etherm = 0 Because U2 = K1 = Wext = 0: or 2 ∆Etherm = U1 − K 2 = mg∆h − 1 mv2 2 Substitute numerical values and evaluate ∆Etherm: ( ) ∆Etherm = (20 kg ) 9.81 m/s 2 (3.2 m ) − 1 (20 kg )(1.3 m/s ) = 611 J 2 (b) Relate the energy dissipated by friction to the kinetic friction force and the distance over which this force acts and solve for µk: Apply ∑F y = ma y to the girl and 2 ∆Etherm = f∆s = µ k Fn ∆s and µk = ∆Etherm Fn ∆s Fn − mg cos θ = 0 ⇒ Fn = mg cosθ solve for Fn: Referring to the figure, relate ∆h to ∆s and θ: ∆s = Substitute for ∆s and Fn to obtain: µk = Substitute numerical values and evaluate µk: ∆h sin θ ∆Etherm ∆Etherm tan θ = ∆h mg∆h cos θ mg sin θ Conservation of Energy 473 µk = (611 J )tan20° = (20 kg )(9.81 m/s 2 )(3.2 m ) 0.354 49 •• Picture the Problem Let the system consist of the two blocks, the shelf, and the earth. Given this choice, there are no external forces doing work to change the energy of the system. Due to the friction between the 4-kg block and the surface on which it slides, not all of the energy transformed during the fall of the 2-kg block is realized in the form of kinetic energy. We can find the energy dissipated by friction and then use the workenergy theorem with kinetic friction to find the speed of either block when they have moved the given distance. ∆Etherm = f∆s = µ k m1 gy (a) The energy dissipated by friction when the 2-kg block falls a distance y is given by: ( ) ∆Etherm = (0.35)(4 kg ) 9.81 m/s 2 y Substitute numerical values and evaluate ∆Etherm: = (13.7 N ) y Wext = ∆Emech + ∆Etherm (b) From the work-energy theorem with kinetic friction we have: or, because Wext = 0, ∆Emech = −∆Etherm = − (13.7 N ) y (m1 + m2 )v 2 − m2 gy = −∆Etherm (c) Express the total mechanical energy of the system: 1 2 Solve for v to obtain: v= 2(m2 gy − ∆E therm ) m1 + m2 (1) Substitute numerical values and evaluate v: v= [ ( ) 2 (2 kg ) 9.81 m/s 2 (2 m ) − (13.73 N )(2 m ) = 1.98 m/s 4 kg + 2 kg *50 •• Picture the Problem Let the system consist of the particle, the table, and the earth. Then Wext = 0 and the energy dissipated by friction during one revolution is the change in the thermal energy of the system. (a) Apply the work-energy theorem Wext = ∆K + ∆U + ∆Etherm 474 Chapter 7 with kinetic friction to obtain: or, because ∆U = Wext = 0, 0 = ∆K + ∆Etherm Substitute for ∆Kf and simplify to obtain: ( ∆Etherm = − 1 mvf2 − 1 mvi2 2 2 [ = − 1 m( 1 v0 ) − 1 m(v0 ) 2 2 2 = (b) Relate the energy dissipated by friction to the distance traveled and the coefficient of kinetic friction: Substitute for ∆E and solve for µk to obtain: (c) ) 3 8 2 2 2 mv0 ∆Etherm = f∆s = µ k mg∆s = µ k mg (2πr ) µk = 2 3 mv 2 ∆Etherm 3v0 = 8 0 = 2πmgr 2πmgr 16πgr Because it lost 3 K i in one revolution, it will only require another 1/3 4 revolution to lose the remaining 1 K i . 4 51 •• Picture the Problem The box will slow down and stop due to the dissipation of thermal energy. Let the system be the earth, the box, and the inclined plane and apply the work-energy theorem with friction. With this choice of the system, there are no external forces doing work to change the energy of the system. The freebody diagram shows the forces acting on the box when it is moving up the incline. Apply the work-energy theorem with friction to the system: Substitute for ∆K, ∆U, and ∆Etherm to obtain: Referring to the FBD, relate the normal force to the weight of the box and the angle of the incline: Relate ∆h to the distance L along the Wext = ∆Emech + ∆Etherm = ∆K + ∆U + ∆Etherm 2 0 = 1 mv12 − 1 mv0 + mg∆h + µ k Fn L 2 2 Fn = mg cos θ ∆h = L sin θ (1) Conservation of Energy 475 incline: Substitute in equation (1) to obtain: 2 µ k mgL cos θ + 1 mv12 − 1 mv0 2 2 + mgL sin θ = 0 (2) Solving equation (2) for L yields: L= 2 v0 2 g (µ k cos θ + sin θ ) Substitute numerical values and evaluate L: L= (3.8 m/s)2 2(9.81 m/s 2 )[(0.3)cos37° + sin37°] = 0.875 m Let vf represent the box’s speed as it passes its starting point on the way down the incline. For the block’s descent, equation (2) becomes: Set v1 = 0 (the block starts from rest at the top of the incline) and solve for vf : µ k mgL cos θ + 1 mvf2 − 1 mv12 2 2 − mgL sin θ = 0 vf = 2 gL(sin θ − µk cosθ ) Substitute numerical values and evaluate vf: ( ) vf = 2 9.81m/s2 (0.875 m)[sin37° − (0.3)cos37°] = 2.49 m/s 52 ••• Picture the Problem Let the system consist of the earth, the block, the incline, and the spring. With this choice of the system, there are no external forces doing work to change the energy of the system. The free-body diagram shows the forces acting on the block just before it begins to move. We can apply Newton’s 2nd law to the block to obtain an expression for the extension of the spring at this instant. We’ll apply the work-energy theorem with friction to the second part of the problem. (a) Apply r r ∑ F = ma to the block ∑F x = Fspring − f s,max − mg sin θ = 0 476 Chapter 7 when it is on the verge of sliding: Eliminate Fn, fs,max, and Fspring between the two equations to obtain: Solve for and evaluate d: (b) Begin with the work-energy theorem with friction and no work being done by an external force: Because the block is at rest in both its initial and final states, ∆K = 0 and: Let Ug = 0 at the initial position of the block. Then: Express the change in the energy stored in the spring as it relaxes to its unstretched length: The energy dissipated by friction is: and ∑F y = Fn − mg cos θ = 0 kd − µ s mg cos θ − mg sin θ = 0 mg (sin θ + µ s cos θ ) k d= Wext = ∆Emech + ∆Etherm = ∆K + ∆U g + ∆U s + ∆Etherm ∆U g + ∆U s + ∆Etherm = 0 (1) ∆U g = U g,final − U g,initial = mgh − 0 = mgd sin θ ∆U s = U s,final − U s,initial = 0 − 1 kd 2 2 = − 1 kd 2 2 ∆Etherm = f∆s = − f k d = − µ k Fn d = − µ k mgd cosθ Substitute in equation (1) to obtain: mgd sin θ − 1 kd 2 − µ k mgd cos θ = 0 2 Finally, solve for µk: µk = 1 2 (tan θ − µs ) Mass and Energy 53 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation E0 = mc 2 . Conservation of Energy 477 (a) Relate the rest mass consumed to the energy produced and solve for and evaluate m: (b) Express kW⋅h in joules: E0 = mc 2 ( )( = 1× 10 −3 kg 3 × 108 m/s ) 2 = 9.00 × 1013 J ( ) 1 kW ⋅ h = 1× 103 J/s (1 h )(3600 s/h ) = 3.60 × 10 J 6 Convert 9×1013 J to kW⋅h: ⎛ 1 kW ⋅ h ⎞ 9 × 1013 J = 9 × 1013 J ⎜ ⎜ 3.60 × 106 J ⎟ ⎟ ⎝ ⎠ 7 = 2.50 × 10 kW ⋅ h Determine the price of the electrical energy: ⎛ $0.10 ⎞ Price = 2.50 × 107 kW ⋅ h ⎜ ⎟ ⎝ kW ⋅ h ⎠ ( ) ( ) = $2.5 × 106 (c) Relate the energy consumed to its rate of consumption and the time and solve for the latter: E = Pt and t= E 9 × 1013 J = 100 W P = 9 × 1011 s = 28,500 y 54 • Picture the Problem We can use the equation expressing the equivalence of energy and matter, E = mc2, to find the mass equivalent of the energy from the explosion. Solve E = mc2 for m: Substitute numerical values and evaluate m: m= m= E c2 5 × 1012 J (2.998 ×10 8 m/s ) 2 = 5.56 × 10 −5 kg 55 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation E0 = mc 2 . Relate the rest mass of a muon to its rest energy: m0 = E c2 478 Chapter 7 Express 1 MeV in joules: 1 MeV = 1.6×10−13 J Substitute numerical values and evaluate m0: m0 = (105.7 MeV )(1.6 ×10−13 J/MeV ) (3 ×10 8 m/s ) 2 = 1.88 × 10 −28 kg *56 • Picture the Problem We can differentiate the mass-energy equation to obtain an expression for the rate at which the black hole gains energy. Using the mass-energy relationship, express the energy radiated by the black hole: E = 0.01mc 2 Differentiate this expression to obtain an expression for the rate at which the black hole is radiating energy: dm dE d = 0.01mc 2 = 0.01c 2 dt dt dt Solve for dm/dt: dm dE dt = dt 0.01c 2 Substitute numerical values and evaluate dm/dt: dm 4 × 1031 watt = dt (0.01) 2.998 × 108 m/s [ ( ) 2 = 4.45 × 1016 kg/s 57 • Picture the Problem The number of reactions per second is given by the ratio of the power generated to the energy released per reaction. The number of reactions that must take place to produce a given amount of energy is the ratio of the energy per second (power) to the energy released per second. In Example 7-15 it is shown that the energy per reaction is 17.59 MeV. Convert this energy to joules: The number of reactions per second is: 17.59 MeV = (17.59 MeV ) ( × 1.6 × 10 −19 J/eV = 28.1 × 10 −13 J 1000 J/s 28.1 × 10 −13 J/reaction = 3.56 × 1014 reactions/s ) Conservation of Energy 479 58 • Picture the Problem The energy required for this reaction is the difference between the rest energy of 4He and the sum of the rest energies of 3He and a neutron. He→3 He + n Express the reaction: 4 The rest energy of a neutron (Table 7-1) is: 939.573 MeV The rest energy of 4He (Example 7-15) is: 3727.409 MeV The rest energy of 3He is: 2808.432 MeV Substitute numerical values to find the difference in the rest energy of 4He and the sum of the rest energies of 3He and n: E = [3727.409 − (2808.41 + 939.573)] MeV = 20.574 MeV 59 • Picture the Problem The energy required for this reaction is the difference between the rest energy of a neutron and the sum of the rest energies of a proton and an electron. The rest energy of a proton (Table 7-1) is: 938.280 MeV The rest energy of an electron (Table 7-1) is: 0.511 MeV The rest energy of a neutron (Table 7-1) is: 939.573 MeV Substitute numerical values to find the difference in the rest energy of a neutron and the sum of the rest energies of a positron and an electron: E = [939.573 − (938.280 + 0.511)] MeV = 0.782 MeV 60 •• Picture the Problem The reaction is 2 H + 2 H→ 4 He + E . The energy released in this reaction is the difference between twice the rest energy of 2H and the rest energy of 4He. 480 Chapter 7 The number of reactions that must take place to produce a given amount of energy is the ratio of the energy per second (power) to the energy released per reaction. (a) The rest energy of 4He (Example 7-14) is: 3727.409 MeV The rest energy of a deuteron, 2H, (Table 7-1) is: 1875.628 MeV The energy released in the reaction is: (b) The number of reactions per second is: E = [2(1875.628) − 3727.409] MeV = 23.847 MeV = 3.816 × 10 −12 J 1000 J/s 3.816 × 10 −12 J/reaction = 2.62 × 1014 reactions/s 61 •• Picture the Problem The annual consumption of matter by the fission plant is the ratio of its annual energy output to the square of the speed of light. The annual consumption of coal in a coal-burning power plant is the ratio of its annual energy output to energy per unit mass of the coal. (a) Express m in terms of E: Assuming an efficiency of 33 percent, find the energy produced annually: m= E c2 ( ) E = 3P∆t = 3 3 × 109 J/s (1 y ) ( ) = 3 3 × 109 J/s (3600 s/h ) × (24 h/d )(365.24 d ) = 2.84 × 1017 J Substitute to obtain: (b) Assuming an efficiency of 38 percent, express the mass of coal required in terms of the annual energy production and the energy released per kilogram: m= mcoal 2.84 × 1017 J (3 ×10 8 m/s ) 2 = 3.16 kg Eannual 9.47 × 1016 J = = 0.38(E / m ) 0.38 3.1× 107 J/kg ( = 8.04 × 109 kg ) Conservation of Energy 481 General Problems *62 •• Picture the Problem Let the system consist of the block, the earth, and the incline. Then the tension in the string is an external force that will do work to change the energy of the system. Because the incline is frictionless; the work done by the tension in the string as it displaces the block on the incline is equal to the sum of the changes in the kinetic and gravitational potential energies. Relate the work done by the tension force to the changes in the kinetic and gravitational potential energies of the block: Referring to the figure, express the change in the potential energy of the block as it moves from position 1 to position 2: Wtension force = Wext = ∆U + ∆K ∆U = mg∆h = mgL sin θ Because the block starts from rest: ∆K = K 2 = 1 mv 2 2 Substitute to obtain: Wtension force = mgL sin θ + 1 mv 2 2 and (c) is correct. 63 •• Picture the Problem Let the system include the earth, the block, and the inclined plane. Then there are no external forces to do work on the system and Wext = 0. Apply the work-energy theorem with friction to find an expression for the energy dissipated by friction. Express the work-energy theorem with friction: Wext = ∆K + ∆U + ∆Etherm = 0 482 Chapter 7 Because the velocity of the block is constant, ∆K = 0 and: In time ∆t the block slides a distance v∆t . From the figure: Substitute to obtain: ∆Etherm = − ∆U = −mg∆h ∆h = v∆t sin θ ∆Etherm = − mgv∆t sin θ and (b) is correct. 64 • Picture the Problem Let the system include the earth and the box. Then the applied force is external to the system and does work on the system in compressing the spring. This work is stored in the spring as potential energy. Express the work-energy theorem: Wext = ∆K + ∆U g + ∆U s + ∆Etherm Because ∆K = ∆U g = ∆Etherm = 0 : Wext = ∆U s Substitute for Wext and ∆Us: Fx = 1 kx 2 2 Solve for x: x= 2F k Substitute numerical values and evaluate x: x= 2(70 N ) = 2.06 cm 6800 N/m *65 • Picture the Problem The solar constant is the average energy per unit area and per unit time reaching the upper atmosphere. This physical quantity can be thought of as the power per unit area and is known as intensity. P ∆E / ∆t = A A Letting Isurface represent the intensity of the solar radiation at the surface of the earth, express Isurface as a function of power and the area on which this energy is incident: I surface = Solve for ∆E: ∆E = I surface A∆t Conservation of Energy 483 Substitute numerical values and evaluate ∆E: ( )( ) ∆E = 1 kW/m 2 2 m 2 (8 h )(3600 s/h ) = 57.6 MJ 66 •• Picture the Problem The luminosity of the sun (or of any other object) is the product of the power it radiates per unit area and its surface area. If we let L represent the sun’s luminosity, I the power it radiates per unit area (also known as the solar constant or the intensity of its radiation), and A its surface area, then L = IA. We can estimate the solar lifetime by dividing the number of hydrogen nuclei in the sun by the rate at which they are being transformed into energy. (a) Express the total energy the sun radiates every second in terms of the solar constant: L = IA Letting R represent its radius, express the surface area of the sun: A = 4πR 2 Substitute to obtain: L = 4πR 2 I Substitute numerical values and evaluate L: L = 4π (1.5 ×1011 m ) (1.35 kW/m 2 ) 2 = 3.82 ×10 26 watt Note that this result is in good agreement with the value given in the text of 3.9×1026 watt. (b) Express the solar lifetime in terms of the mass of the sun and the rate at which its mass is being converted to energy: tsolar = where M is the mass of the sun, m the mass of a hydrogen nucleus, and n is the number of nuclei used up. Substitute numerical values to obtain: tsolar 1.99 × 1030 kg 1.67 ×10 −27 kg/H nucleus = ∆n ∆t = For each reaction, 4 hydrogen nuclei are "used up"; so: N H nuclei M m = ∆n ∆t ∆n ∆t 1.19 ×1057 H nuclei ∆n ∆t ( ∆n 4 3.82 × 1026 J/s = ∆t 4.27 × 10−12 J = 3.57 × 1038 s −1 ) 484 Chapter 7 ⎛ 1.19 × 1057 H nuclei ⎞ tsolar = 0.1⎜ ⎜ 3.57 × 1038 s −1 ⎟ ⎟ ⎝ ⎠ Because we’ve assumed that the sun will continue burning until roughly 10% of its hydrogen fuel is used up, the total solar lifetime should be: = 3.33 × 1017 s = 1.06 × 1010 y 67 • Picture the Problem Let the system include the earth and the Spirit of America. Then there are no external forces to do work on the car and Wext = 0. We can use the workenergy theorem to relate the coefficient of kinetic friction to the given information. A constant-acceleration equation will yield the car’s velocity when 60 s have elapsed. (a) Apply the work-energy theorem with friction to relate the coefficient of kinetic friction µk to the initial and final kinetic energies of the car: Solve for µk: Substitute numerical values and evaluate µk: (b) Express the kinetic energy of the car: Using a constant-acceleration equation, relate the speed of the car to its acceleration, initial speed, and the elapsed time: Express the braking force acting on the car: 1 2 2 mv 2 − 1 mv0 + µ k mg∆s = 0 2 or, because v = 0, 2 − 1 mv0 + µ k mg∆s = 0 2 µk = v2 2 g∆s [(708 km/h )(1h/3600 s )] 2 µk = 2(9.81 m/s 2 )(9.5 km ) K = 1 mv 2 2 (1) v = v0 + a∆t Fnet = − f k = − µ k mg = ma Solve for a: a = −µk g Substitute for a to obtain: v = v0 − µ k g∆t Substitute in equation (1) to obtain: K = 1 m(v0 − µ k g∆t ) 2 Substitute numerical values and evaluate K: = 0.208 2 Conservation of Energy 485 K= 1 2 (1250 kg )[708 ×103 m/h − (0.208)(9.81m/s2 )(60 s )] 2 = 3.45 MJ 68 •• Picture the Problem The free-body diagram shows the forces acting on the skiers as they are towed up the slope at constant speed. Because the power r r required to move them is F ⋅ v , we need to find F as a function of mtot, θ, and µk. We can apply Newton’s 2nd law to obtain such a function. Express the power required as a function of force on the skiers and their speed: Apply r r F = ma to the skiers: ∑ P = Fv ∑F x (1) = F − f k − mtot g sin θ = 0 and ∑F y Eliminate fk = µkFn and Fn between the two equations and solve for F: Substitute in equation (1) to obtain: = Fn − mtot g cos θ = 0 F = mtot g sin θ + µ k mtot g cos θ P = (mtot g sin θ + µ k mtot g cos θ )v = mtot gv(sin θ + µ k cos θ ) Substitute numerical values and evaluate P: ( ) P = 80(75 kg ) 9.81 m/s 2 (2.5 m/s )[sin 15° + (0.06 ) cos15°] = 46.6 kW 486 Chapter 7 69 •• Picture the Problem The free-body diagram for the box is superimposed on the pictorial representation shown to the right. The work done by friction slows and momentarily stops the box as it slides up the incline. The box’s speed when it returns to bottom of the incline will be less than its speed when it started up the incline due to the energy dissipated by friction while it was in motion. Let the system include the box, the earth, and the incline. Then Wext = 0. We can use the work-energy theorem with friction to solve the several parts of this problem. From the FBD we can see that the forces acting on the box are the (a) normal force exerted by the inclined plane, a kinetic friction force, and the gravitational force (the weight of the box) exerted by the earth. (b) Apply the work-energy theorem with friction to relate the distance ∆x the box slides up the incline to its initial kinetic energy, its final potential energy, and the work done against friction: − 1 mv12 + mg∆h + µ k mg∆x cos θ = 0 2 Referring to the figure, relate ∆h to ∆x to obtain: ∆h = ∆x sin θ Substitute for ∆h to obtain: − 1 mv12 + mg∆x sin θ 2 + µ k mg∆x cos θ = 0 Solve for ∆x: Substitute numerical values and evaluate ∆x: ∆x = ∆x = v12 2 g (sin θ + µ k cos θ ) ( ) (3 m/s)2 2 9.81m/s 2 [sin60° + (0.3)cos60°] = 0.451 m Conservation of Energy 487 (c) Express and evaluate the energy dissipated by friction: ∆Etherm = f k ∆x = µ k mg∆x cosθ ( ) = (0.3)(2 kg ) 9.81 m/s 2 (0.451 m )cos60° = 1.33 J (d) Use the work-energy theorem with friction to obtain: Wext = ∆K + ∆U + ∆Etherm = 0 or K1 − K 2 + U1 − U 2 + ∆Etherm = 0 Because K2 = U1 = 0 we have: K1 − U 2 + ∆Etherm = 0 or 1 2 mv12 − mg∆x sin θ + µ k mg∆x cos θ = 0 Solve for v1: v1 = 2 g∆x(sin θ − µ k cos θ ) Substitute numerical values and evaluate v1: v1 = 2(9.81 m/s 2 )(0.451 m )[sin60° − (0.3)cos60°] = 2.52 m/s *70 • Picture the Problem The power provided by a motor that is delivering sufficient energy to exert a force F on a load which it is moving at a speed v is Fv. The power provided by the motor is given by: Because the elevator is ascending with constant speed, the tension in the support cable(s) is: Substitute for F to obtain: Substitute numerical values and evaluate P: P = Fv F = (melev + mload )g P = (melev + mload )gv P = (2000 kg ) (9.81 m/s 2 )(2.3 m/s ) = 45.1 kW 488 Chapter 7 71 •• Picture the Problem The power a motor must provide to exert a force F on a load that it is moving at a speed v is Fv. The counterweight does negative work and the power of the motor is reduced from that required with no counterbalance. The power provided by the motor is given by: Because the elevator is counterbalanced and ascending with constant speed, the tension in the support cable(s) is: Substitute and evaluate P: Substitute numerical values and evaluate P: Without a load: P = Fv F = (melev + mload − mcw )g P = (melev + mload − mcw )gv ( ) P = (500 kg ) 9.81 m/s 2 (2.3 m/s ) = 11.3 kW F = (melev − mcw )g and P = (melev − mcw )gv ( ) = (− 300 kg ) 9.81 m/s 2 (2.3 m/s ) = − 6.77 kW 72 •• Picture the Problem We can use the work-energy theorem with friction to describe the energy transformation within the dart-spring-air-earth system. With this choice of the system, there are no external forces to do work on the system; i.e., Wext = 0. Choose Ug = 0 at the elevation of the dart on the compressed spring. The energy initially stored in the spring is transformed into gravitational potential energy and thermal energy. During the dart’s descent, its gravitational potential energy is transformed into kinetic energy and thermal energy. Apply conservation of energy during the dart’s ascent: Wext = ∆K + ∆U + ∆Etherm = 0 or, because ∆K = 0, U g,f − U g,i + U s,f − U s,i + ∆Etherm = 0 Because U g,i = U s,f = 0 : U g,f − U s,i + ∆Etherm = 0 Conservation of Energy 489 Substitute for Ug,i and Us,f and solve for ∆Etherm: ∆Etherm = U s,i − U g,f = 1 kx 2 − mgh 2 Substitute numerical values and evaluate ∆Etherm: ∆Etherm = (5000 N/m )(0.03 m )2 − (0.007 kg )(9.81 m/s 2 )(24 m ) 1 2 = 0.602 J Wext = ∆K + ∆U + ∆Etherm = 0 Apply conservation of energy during the dart’s descent: or, because Ki = Ug,f = 0, K f − U g,i + ∆Etherm = 0 1 2 Substitute for Kf and Ug,i to obtain: Solve for vf: mvf2 − mgh + ∆Etherm = 0 vf = 2(mgh − ∆Etherm ) m Substitute numerical values and evaluate vf: vf = [ ( ) 2 (0.007 kg ) 9.81 m/s 2 (24 m ) − 0.602 J = 17.3 m/s 0.007 kg *73 •• Picture the Problem Let the system consist of the earth, rock and air. Given this choice, there are no external forces to do work on the system and Wext = 0. Choose Ug = 0 to be where the rock begins its upward motion. The initial kinetic energy of the rock is partially transformed into potential energy and partially dissipated by air resistance as the rock ascends. During its descent, its potential energy is partially transformed into kinetic energy and partially dissipated by air resistance. (a) Using the definition of kinetic energy, calculate the initial kinetic energy of the rock: (b) Apply the work-energy theorem with friction to relate the energies of the system as the rock ascends: Because Kf = 0: K i = 1 mvi2 = 2 1 2 (2 kg )(40 m/s )2 = 1.60 kJ ∆K + ∆U + ∆Etherm = 0 − K i + ∆U + ∆Etherm = 0 and ∆Etherm = K i − ∆U 490 Chapter 7 Substitute numerical values and evaluate ∆Etherm: (c) Apply the work-energy theorem with friction to relate the energies of the system as the rock descends: ( ) ∆Etherm = 1600 J − (2 kg ) 9.81 m/s 2 (50 m ) = 619 J ∆K + ∆U + 0.7∆Etherm = 0 Because Ki = Uf = 0: K f − U i + 0.7∆Etherm = 0 Substitute for the energies to obtain: 1 2 mvf2 − mgh + 0.7∆Etherm = 0 1.4∆Etherm m Solve for vf: vf = 2 gh − Substitute numerical values and evaluate vf: vf = 2 9.81 m/s 2 (50 m ) − ( ) 1.4(619 J ) 2 kg = 23.4 m/s 74 •• Picture the Problem Let the distance the block slides before striking the spring be L. The pictorial representation shows the block at the top of the incline (1), just as it strikes the spring (2), and the block against the fully compressed spring (3). Let the block, spring, and the earth comprise the system. Then Wext = 0. Let Ug = 0 where the spring is at maximum compression. We can apply the work-energy theorem to relate the energies of the system as it evolves from state 1 to state 3. Express the work-energy theorem: ∆K + ∆U g + ∆U s = 0 or ∆K + U g,3 − U g,1 + U s,3 − U s,1 = 0 Conservation of Energy 491 Because ∆K = Ug,3 = Us,1 = 0: − U g,1 + U s,3 = 0 Substitute for each of these energy terms to obtain: − mgh1 + 1 kx 2 = 0 2 Substitute for h3 and h1: − mg (L + x )sin θ + 1 kx 2 = 0 2 Rewrite this equation explicitly as a quadratic equation: x2 − 2mg sin θ 2mgL sin θ x− =0 k k Solve this quadratic equation to obtain: 2 2mgL mg ⎛ mg ⎞ 2 sin θ + ⎜ sin θ x= ⎟ sin θ + k k ⎝ k ⎠ Note that the negative sign between the two terms leads to a non-physical solution. *75 • Picture the Problem We can find the work done by the girder on the slab by calculating the change in the potential energy of the slab. (a) Relate the work the girder does on the slab to the change in potential energy of the slab: Substitute numerical values and evaluate W: W = ∆U = mg∆h ( )( ) W = 1.5 × 104 kg 9.81 m/s 2 (0.001 m ) = 147 J The energy is transferred to the girder from its surroundings, which are (b) warmer than the girder. As the temperature of the girder rises, the atoms in the girder vibrate with a greater average kinetic energy, leading to a larger average separation, which causes the girder' s expansion. 76 •• Picture the Problem The average power delivered by the car’s engine is the rate at which it changes the car’s energy. Because the car is slowing down as it climbs the hill, its potential energy increases and its kinetic energy decreases. Express the average power delivered by the car’s engine: Pav = ∆E ∆t 492 Chapter 7 Express the increase in the car’s mechanical energy: ∆E = ∆K + ∆U = K top − K bot + U top − U bot 2 2 = 1 mvtop − 1 mvbot + mg∆h 2 2 ( 2 2 = 1 m vtop − vbot + 2 g∆h 2 ) Substitute numerical values and evaluate ∆E: ∆E = 1 2 (1500 kg )[(10 m/s)2 − (24 m/s )2 + 2(9.81 m/s 2 )(120 m )]= 1.41 MJ v top + v bot Assuming that the acceleration of the car is constant, find its average speed during this climb: vav = Using the vav, find the time it takes the car to climb the hill: ∆t = ∆s 2000 m = = 118 s vav 17 m/s Substitute to determine Pav: Pav = 1.41 MJ = 11.9 kW 118 s 2 = 17 m/s *77 •• Picture the Problem Given the potential energy function as a function of y, we can find the net force acting on a given system from F = −dU / dy . The maximum extension of the spring; i.e., the lowest position of the mass on its end, can be found by applying the work-energy theorem. The equilibrium position of the system can be found by applying the work-energy theorem with friction … as can the amount of thermal energy produced as the system oscillates to its equilibrium position. (a) The graph of U as a function of y is shown to the right. Because k and m are not specified, k has been set equal to 2 and mg to 1. The spring is unstretched when y = y0 = 0. Note that the minimum value of U (a position of stable equilibrium) occurs near y = 5 m. Conservation of Energy 493 1.0 0.8 U (J) 0.6 0.4 0.2 0.0 -0.2 -0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 y (m) (b) Evaluate the negative of the derivative of U with respect to y: F =− dU d =− dy dy ( 1 2 ky 2 − mgy ) = − ky + mg (c) Apply conservation of energy to the movement of the mass from y = 0 to y = ymax: ∆K + ∆U + ∆Etherm = 0 Because ∆K = 0 (the object starts from rest and is momentarily at rest at y = ymax) and ∆Etherm = 0 (no friction), it follows that: ∆U = U(ymax) – U(0) = 0 Because U(0) = 0: U(ymax) = 0 ⇒ Solve for ymax: y max = (d) Express the condition of F at equilibrium and solve for yeq: 2 kymax − mgymax = 0 2mg k Feq = 0 ⇒ − ky eq + mg = 0 and yeq = (e) Apply the conservation of energy to the movement of the mass from y = 0 to y = yeq and solve for ∆Etherm: 1 2 mg k ∆K + ∆U + ∆Etherm = 0 or, because ∆K = 0. ∆Etherm = −∆U = U i − U f 494 Chapter 7 ( Because U i = U (0 ) = 0 : 2 ∆Etherm = −U f = − 1 kyeq − mgyeq 2 Substitute for yeq and simplify to obtain: ∆Etherm = ) m2 g 2 2k 78 •• Picture the Problem The energy stored in the compressed spring is initially transformed into the kinetic energy of the signal flare and then into gravitational potential energy and thermal energy as the flare climbs to its maximum height. Let the system contain the earth, the air, and the flare so that Wext = 0. We can use the work-energy theorem with friction in the analysis of the energy transformations during the motion of the flare. (a) The work done on the spring in compressing it is equal to the kinetic energy of the flare at launch. Therefore: (b) Ignoring changes in gravitational potential energy (i.e., assume that the compression of the spring is small compared to the maximum elevation of the flare), apply the conservation of energy to the transformation that takes place as the spring decompresses and gives the flare its launch speed: Ws = K i, flare = 1 2 2 mv0 ∆K + ∆U s = 0 or K f − K i + U s,f − U s,i = 0 Because Ki = ∆Ug = Us,f: K f − U s,i = 0 Substitute for K f and U s,i : 1 2 Solve for k to obtain: (c) Apply the work-energy theorem with friction to the upward trajectory of the flare: 2 mv0 − 1 kd 2 = 0 2 k= 2 mv0 d2 ∆K + ∆U g + ∆Etherm = 0 Conservation of Energy 495 Solve for ∆Etherm: ∆Etherm = −∆K − ∆U g = Ki − Kf + U i − U f Because Kf = Ui = 0: ∆Etherm = 1 2 2 mv0 − mgh 79 •• Picture the Problem Let UD = 0. Choose the system to include the earth, the track, and the car. Then there are no external forces to do work on the system and change its energy and we can use Newton’s 2nd law and the work-energy theorem to describe the system’s energy transformations to point G … and then the work-energy theorem with friction to determine the braking force that brings the car to a stop. The free-body diagram for point C is shown to the right. The free-body diagram for point D is shown to the right. The free-body diagram for point F is shown to the right. (a) Apply the work-energy theorem to the system’s energy transformations between A and B: If we assume that the car arrives at point B with vB = 0, then: ∆K + ∆U = 0 or KB − KA +UB −UA = 0 2 − 1 mvA + mg∆h = 0 2 where ∆h is the difference in elevation between A and B. 496 Chapter 7 Solve for and evaluate ∆h: (12 m/s) = 7.34 m v2 ∆h = A = 2 g 2 9.81 m/s 2 The height above the ground is: h + ∆h = 10 m + 7.34 m = 17.3 m (b) If the car just makes it to point B; i.e., if it gets there with vB = 0, then the force exerted by the track on the car will be the normal force: (c) Apply ∑F x = max to the car at point C (see the FBD) and solve for a: 2 ( ) Ftrack on car = Fn = mg ( = (500 kg ) 9.81 m/s 2 = 4.91 kN mg sin θ = ma and a = g sin θ = (9.81 m/s 2 )sin30° = 4.91 m/s 2 (d) Apply ∑F y = ma y to the car at point D (see the FBD) and solve for Fn : Fn − mg = m 2 vD R and Fn = mg + m 2 vD R ∆K + ∆U = 0 Apply the work-energy theorem to the system’s energy transformations between B and D: or Because KB = UD = 0: KD −UB = 0 Substitute to obtain: 1 2 2 Solve for vD : 2 vD = 2 g (h + ∆h ) Substitute to find Fn: KD − KB +UD −UB = 0 2 mv D − mg (h + ∆h ) = 0 2 vD R 2 g (h + ∆h ) = mg + m R ⎡ 2(h + ∆h ) ⎤ = mg ⎢1 + ⎥ R ⎣ ⎦ Fn = mg + m ) Conservation of Energy 497 Substitute numerical values and evaluate Fn: ⎡ 2(17.3 m )⎤ Fn = (500 kg ) 9.81 m/s 2 ⎢1 + 20 m ⎥ ⎦ ⎣ ( ) = 13.4 kN, directed upward. (e) F has two components at point F; one horizontal (the inward force that the track exerts) and the other vertical (the normal force). Apply r r F = ma to the car at point F: ∑ Express the resultant of these two forces: ∑F = Fn − mg = 0 ⇒ Fn = mg y and ∑ Fx = Fc = m F = Fc2 + Fn2 2 ⎛ v2 ⎞ 2 = ⎜ m F ⎟ + (mg ) ⎜ R⎟ ⎝ ⎠ =m Substitute numerical values and evaluate F: 2 vF R 4 vF + g2 R2 F = (500 kg ) (12 m/s)4 + (9.81m/s2 ) 2 (30 m )2 = 5.46 kN Find the angle the resultant makes with the x axis: ⎛ Fn ⎞ ⎛ gR ⎞ ⎟ = tan −1 ⎜ 2 ⎟ ⎜v ⎟ ⎟ ⎝ F ⎠ ⎝ Fc ⎠ θ = tan −1 ⎜ ⎜ ( ) ⎡ 9.81 m/s 2 (30 m ) ⎤ = 63.9° = tan ⎢ (12 m/s)2 ⎥ ⎣ ⎦ −1 (f) Apply the work-energy theorem with friction to the system’s energy transformations between F and the car’s stopping position: The work done by friction is also given by: Equate the two expressions for ∆Etherm and solve for Fbrake: − K G + ∆Etherm = 0 and 2 ∆Etherm = K G = 1 mvG 2 ∆Etherm = f∆s = Fbrake d where d is the stopping distance. Fbrake 2 mvF = 2d 498 Chapter 7 Substitute numerical values and evaluate Fbrake: (500 kg )(12 m/s)2 Fbrake = 2(25 m ) = 1.44 kN *80 • Picture the Problem The rate of conversion of mechanical energy can be r r determined from P = F ⋅ v . The pictorial representation shows the elevator moving downward just as it goes into freefall as state 1. In state 2 the elevator is moving faster and is about to strike the relaxed spring. The momentarily at rest elevator on the compressed spring is shown as state 3. Let Ug = 0 where the spring has its maximum compression and the system consist of the earth, the elevator, and the spring. Then Wext = 0 and we can apply the conservation of mechanical energy to the analysis of the falling elevator and compressing spring. (a) Express the rate of conversion of mechanical energy to thermal energy as a function of the speed of the elevator and braking force acting on it: Because the elevator is moving with constant speed, the net force acting on it is zero and: Substitute for Fbraking and evaluate P: P = Fbraking v0 Fbraking = Mg P = Mgv0 ( ) = (2000 kg ) 9.81 m/s 2 (1.5 m/s ) = 29.4 kW ∆K + ∆U g + ∆U s = 0 (b) Apply the conservation of energy to the falling elevator and compressing spring: or Because K3 = Ug,3 = Us,1 = 0: 2 − 1 Mv0 − Mg (d + ∆y ) + 1 k (∆y ) = 0 2 2 K 3 − K 1 + U g,3 − U g,1 + U s,3 − U s,1 = 0 2 Conservation of Energy 499 Rewrite this equation as a quadratic equation in ∆y, the maximum compression of the spring: Solve for ∆y to obtain: (∆y )2 − ⎛ 2Mg ⎞∆y − M (2 gd + v02 ) = 0 ⎜ ⎟ ⎝ k ⎠ ∆y = k ( Mg M 2g2 M 2 ± + 2 gd + v0 2 k k k ) Substitute numerical values and evaluate ∆y: ∆y = (2000 kg )(9.81m/s 2 ) 1.5 × 10 4 N/m + (2000 kg )2 (9.81m/s 2 )2 + (1.5 ×10 4 N/m ) 2 [( ) 2000 kg 2 2 9.81 m/s 2 (5 m ) + (1.5 m/s ) 4 1.5 × 10 N/m = 5.19 m 81 • Picture the Problem We can use Newton’s 2nd law to determine the force of friction as a function of the angle of the hill for a given constant speed. The power output of the r r engine is given by P = Ff ⋅ v . FBD for (a): (a) Apply FBD for (b): ∑F x = max to the car: Evaluate Ff for the two speeds: mg sin θ − Ff = 0 ⇒ Ff = mg sin θ ( ) ( ) F20 = (1000 kg ) 9.81 m/s 2 sin2.87° = 491 N and F30 = (1000 kg ) 9.81 m/s 2 sin5.74° = 981 N (b) Express the power an engine must deliver on a level road in order P = Ff v P20 = (491 N )(20 m/s ) = 9.82 kW 500 Chapter 7 to overcome friction loss and evaluate this expression for v = 20 m/s and 30 m/s: (c) Apply ∑F x = max to the car: and P30 = (981 N )(30 m/s ) = 29.4 kW ∑F x = F − mg sin θ − F f = 0 P v Relate F to the power output of the engine and the speed of the car: Since P = Fv, F = Substitute for F and solve for θ : ⎤ ⎡P ⎢ v − F20 ⎥ θ = sin −1 ⎢ ⎥ ⎢ mg ⎥ ⎦ ⎣ Substitute numerical values and evaluate θ : ⎡ 40 kW ⎢ 20 m/s − 491 N −1 θ = sin ⎢ 2 ⎢ (1000 kg ) 9.81 m/s ⎢ ⎣ ( ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ) = 8.85° (d) Express the equivalence of the work done by the engine in driving the car at the two speeds: Let ∆V represent the volume of fuel consumed by the engine driving the car on a level road and divide both sides of the work equation by ∆V to obtain: Solve for (∆s )30 : ∆V Substitute numerical values and evaluate (∆s )30 ∆V : Wengine = F20 (∆s )20 = F30 (∆s )30 F20 (∆s )20 ∆V (∆s )30 ∆V (∆s )30 ∆V = F30 (∆s )30 ∆V = F20 (∆s )20 F30 ∆V = 491 N (12.7 km/L) 981 N = 6.36 km/L 82 •• Picture the Problem Let the system include the earth, block, spring, and incline. Then Wext = 0. The pictorial representation to the left shows the block sliding down the incline Conservation of Energy 501 and compressing the spring. Choose Ug = 0 at the elevation at which the spring is fully compressed. We can use the conservation of mechanical energy to determine the maximum compression of the spring. The pictorial representation to the right shows the block sliding up the rough incline after being accelerated by the fully compressed spring. We can use the work-energy theorem with friction to determine how far up the incline the block slides before stopping. (a) Apply conservation of mechanical energy to the system as it evolves from state 1 to state 3: ∆K + ∆U g + ∆U s = 0 or K 3 − K1 + U g,3 − U g,1 + U s,3 − U s,1 = 0 Because K 3 = K 1 = U g,3 = U s,1 = 0 : − U g,1 + U s,3 = 0 or − mg∆h + 1 kx 2 = 0 2 Relate ∆h to L + x and θ and substitute to obtain: ∆h = (L + x )sin θ ∴ 1 kx 2 − mg (L + x )sin θ = 0 2 kx 2 − (mg sin θ )x − mgL sin θ = 0 Rewrite this equation in the form of an explicit quadratic equation: 1 2 Substitute for k, m, g, θ and L to obtain: ⎛ N⎞ 2 ⎜ 50 ⎟ x − (9.81 N )x − 39.24 J = 0 ⎝ m⎠ Solve for the physically meaningful (i.e., positive) root: x = 0.989 m (b) Proceed as in (a) but include energy dissipated by friction: The mechanical energy transformed to thermal energy is given by: − U g,1 + U s,3 + ∆Etherm = 0 ∆Etherm = Ff (L + x ) = µ k Fn (L + x ) = µ k mg cos θ (L + x ) 502 Chapter 7 Substitute for ∆h and ∆Etherm to obtain: − mg (L + x ) sin θ + 1 kx 2 2 Substitute for k, m, g, θ, µk and L to obtain: ⎛ N⎞ 2 ⎜ 50 ⎟ x − (6.41 N )x − 25.65 J = 0 ⎝ m⎠ Solve for the positive root: x = 0.783 m (c) Apply the work-energy theorem with friction to the system as it evolves from state 3 to state 4: Because K 4 = K 1 = U g,3 = U s,4 = 0 : + µ k mg cos θ (L + x ) = 0 K 4 − K 3 + U g,4 − U g,3 + U s,4 − U s,3 + ∆Etherm = 0 U g,4 − U s,3 + ∆Etherm = 0 or − mg∆h'+ 1 kx 2 + ∆Etherm = 0 2 Substitute for ∆h′ and ∆Etherm to obtain: − mg (L'+ x ) sin θ + 1 kx 2 2 Solve for L′ with x = 0.783 m: L' = 1.54 m + µ k mg cos θ (L'+ x ) = 0 83 •• Picture the Problem The work done by the engines maintains the kinetic energy of the cars and overcomes the work done by frictional forces. Let the system include the earth, track, and the cars but not the engines. Then the engines will do external work on the system and we can use this work to find the power output of the train’s engines. (a) Use the definition of kinetic energy: K = 1 mv 2 2 = 1 2 ⎛ km 1 h ⎞ ⎟ ⋅ 2 × 10 kg ⎜15 ⎜ h 3600 s ⎟ ⎝ ⎠ ( 6 ) 2 = 17.4 MJ (b) The change in potential energy of the train is: ∆U = mg∆h ( )( = 1.39 × 1010 J (c) Express the energy dissipated by kinetic friction: ) = 2 × 10 6 kg 9.81 m/s 2 (707 m ) ∆Etherm = f∆s Conservation of Energy 503 f = 0.008mg Express the frictional force: Substitute for f and evaluate ∆Etherm: ( )( ) ∆Etherm = 0.008mg∆s = 0.008 2 × 106 kg 9.81 m/s 2 (62 km ) = 9.73 × 109 J (d) Express the power output of the train’s engines in terms of the work done by them: ∆W ∆t P= Wext = ∆K + ∆U + ∆Etherm Use the work-energy theorem with friction to find the work done by the train’s engines: or, because ∆K = 0, Find the time during which the engines do this work: ∆t = Substitute in the expression for P to obtain: P= Wext = ∆U + ∆Etherm ∆s v (∆U + ∆Etherm )v ∆s Substitute numerical values and evaluate P: ⎛ km 1 h ⎞ ⎜15 ⎟ ⋅ ⎜ h 3600 s ⎟ 10 9 ⎝ ⎠ = 1.59 MW P = 1.39 × 10 J + 9.73 × 10 J 62 km ( ) *84 •• Picture the Problem While on a horizontal surface, the work done by an automobile engine changes the kinetic energy of the car and does work against friction. These energy transformations are described by the work-energy theorem with friction. Let the system include the earth, the roadway, and the car but not the car’s engine. (a) The required energy equals the change in the kinetic energy of the car: ∆K = 1 mv 2 2 ⎛ km 1 h ⎞ ⎟ = (1200 kg )⎜ 50 ⋅ ⎜ h 3600 s ⎟ ⎝ ⎠ 1 2 = 116 kJ (b) The required energy equals the ∆Etherm = f∆s 2 504 Chapter 7 work done against friction: Substitute numerical values and evaluate ∆Etherm: (c) Apply the work-energy theorem with friction to express the required energy: Divide both sides of the equation by E to express the ratio of the two energies: Substitute numerical values and evaluate E′/E: ∆Etherm = (300 N )(300 m ) = 90.0 kJ E ' = Wext = ∆K + ∆Etherm = ∆K + 0.75E E ' ∆K = + 0.75 E E E ' 116 kJ = + 0.75 = 2.04 90 kJ E *85 •• Picture the Problem Assume that the bob is moving with speed v as it passes the top vertical point when looping around the peg. There are two forces acting on the bob: the tension in the string (if any) and the force of gravity, Mg; both point downward when the ball is in the topmost position. The minimum possible speed for the bob to pass the vertical occurs when the tension is 0; from this, gravity must supply the centripetal force required to keep the ball moving in a circle. We can use conservation of energy to relate v to L and R. Express the condition that the bob swings around the peg in a full circle: Simplify to obtain: Use conservation of energy to relate the kinetic energy of the bob at the bottom of the loop to its potential energy at the top of its swing: Solve for v2: Substitute to obtain: M v2 > Mg R v2 >g R 1 Mv 2 = Mg (L − 2 R ) 2 v 2 = 2 g (L − 2 R ) 2 g (L − 2 R ) >g R Conservation of Energy 505 Solve for R: R< 2 L 5 86 •• Picture the Problem If the wood exerts an average force F on the bullet, the work it does has magnitude FD. This must be equal to the change in the kinetic energy of the bullet, or because the final kinetic energy of the bullet is zero, to the negative of the initial kinetic energy. We’ll let m be the mass of the bullet and v its initial speed and apply the work-kinetic energy theorem to relate the penetration depth to v. Wtotal = ∆K = K f − K i Apply the work-kinetic energy theorem to relate the penetration depth to the change in the kinetic energy of the bullet: or, because Kf = 0, Substitute for Wtotal and Ki to obtain: FD = − 1 mv 2 2 Solve for D to obtain: Wtotal = − K i D=− mv 2 2F For an identical bullet with twice the speed we have: FD' = − 1 m(2v ) 2 Solve for D′ to obtain: ⎛ mv 2 ⎞ D' = 4⎜ − ⎜ 2F ⎟ = 4D ⎟ ⎝ ⎠ 2 and (c) is correct. 87 •• Picture the Problem For part (a), we’ll let the system include the glider, track, weight, and the earth. The speeds of the glider and the falling weight will be the same while they are in motion. Let their common speed when they have moved a distance Y be v and let the zero of potential energy be at the elevation of the weight when it has fallen the distance Y. We can use conservation of energy to relate the speed of the glider (and the weight) to the distance the weight has fallen. In part (b), we’ll let the direction of motion be the x direction, the tension in the connecting string be T, and apply Newton’s 2nd law to the glider and the weight to find their common acceleration. Because this acceleration is constant, we can use a constant-acceleration equation to find their common speed when they have moved a distance Y. (a) Use conservation of energy to relate the kinetic and potential energies of the system: ∆K + ∆U = 0 or Kf − Ki + U f − U i = 0 Because the system starts from rest and Uf = 0: Kf − U i = 0 Substitute to obtain: 1 2 mv 2 + 1 Mv 2 − mgY = 0 2 506 Chapter 7 Solve for v: v= 2mgY M +m (b) The free-body diagrams for the glider and the weight are shown to the right: Apply Newton’s 3rd law to obtain: r r T1 = T2 = T Apply ∑F = ma to the glider: T = Ma Apply ∑F = ma to the weight: mg − T = ma x x Add these equations to eliminate T and obtain: mg = Ma + ma Solve for a to obtain: a=g m m+M 2 v 2 = v0 + 2aY Using a constant-acceleration equation, relate the speed of the glider to its initial speed and to the distance that the weight has fallen: or, because v0 = 0, Substitute for a and solve for v to obtain: v= v 2 = 2aY 2mgY , the same result we M +m obtained in part (a). *88 •• Picture the Problem We’re given P = dW / dt and are asked to evaluate it under the assumed conditions. Express the rate of energy expenditure by the man: Express the rate of energy expenditure P′ assuming that his P = 3mv 2 = 3(10 kg )(3 m/s ) = 270 W 2 P = 1 P' 5 Conservation of Energy 507 muscles have an efficiency of 20%: Solve for and evaluate P′: P' = 5 P = 5(270 W ) = 1.35 kW 89 •• Picture the Problem The pictorial representation shows the bob swinging through an angle θ before the thread is cut and it is launched horizontally. Let its speed at position 1 be v. We can use conservation of energy to relate v to the change in the potential energy of the bob as it swings through the angle θ . We can find its flight time ∆t from a constantacceleration equation and then express D as the product of v and ∆t. Relate the distance D traveled horizontally by the bob to its launch speed v and time of flight ∆t: D = v∆t Use conservation of energy to relate its launch speed v to the length of the pendulum L and the angle θ : K1 − K 0 + U1 − U 0 = 0 (1) or, because U1 = K0 = 0, K1 − U 0 = 0 mv 2 − mgL(1 − cosθ ) = 0 Substitute to obtain: 1 2 Solving for v yields: v = 2 gL(1 − cosθ ) In the absence of air resistance, the horizontal and vertical motions of the bob are independent of each other and we can use a constantacceleration equation to express the time of flight (the time to fall a distance H): ∆y = v0 y ∆t + 1 a y (∆t ) 2 2 or, because ∆y = −H, ay = −g, and v0y = 0, − H = − 1 g (∆t ) 2 2 Solve for ∆t to obtain: ∆t = 2 H / g Substitute in equation (1) and simplify to obtain: D = 2 gL(1 − cosθ ) 2H g = 2 HL(1 − cosθ ) which shows that, while D depends on θ, it is independent of g. 508 Chapter 7 90 •• Picture the Problem The pictorial representation depicts the block in its initial position against the compressed spring (1), as it separates from the spring with its maximum kinetic energy (2), and when it has come to rest after moving a distance x + d. Let the system consist of the earth, the block, and the surface on which the block slides. With this choice, Wext = 0. We can use the work-energy theorem with friction to determine how far the block will slide before coming to rest. (a) The work done by the spring on the block is given by: Wspring = ∆U spring = 1 kx 2 2 Substitute numerical values and evaluate Wspring: Wspring = (b) The energy dissipated by friction is given by: Substitute numerical values and evaluate ∆Etherm: (c) Apply the conservation of energy between points 1 and 2: 1 2 (20 N/cm)(3 cm)2 = 0.900 J ∆Etherm = f∆s = µ k Fn ∆x = µ k mg∆x ( ) ∆Etherm = (0.2)(5 kg ) 9.81m/s 2 (0.03 m ) = 0.294 J K 2 − K1 + U s,2 − U s,1 + ∆Etherm = 0 Because K1 = Us,2 = 0: K 2 − U s,1 + ∆Etherm = 0 Substitute to obtain: 1 2 Solve for v2: Substitute numerical values and evaluate v2: 2 mv2 − 1 kx 2 + ∆Etherm = 0 2 v2 = v2 = kx 2 − 2∆Etherm m (20 N/cm)(3 cm)2 − 2(0.294 J ) = 0.492 m/s 5 kg Conservation of Energy 509 (d) Apply the conservation of energy between points 1 and 3: Because ∆K = Us,3 = 0: ∆K + U s,3 − U s,1 + ∆Etherm = 0 − U s,1 + ∆Etherm = 0 or − 1 kx 2 + µ k mg ( x + d ) = 0 2 Solve for d: Substitute numerical values and evaluate d: kx 2 d= −x 2µ k mg (20 N/cm)(3 cm)2 − 0.03 m d= 2(0.2)(5 kg )(9.81m/s 2 ) = 6.17 cm 91 •• Picture the Problem The pictorial representation shows the block initially at rest at point 1, falling under the influence of gravity to point 2, partially compressing the spring as it continues to gain kinetic energy at point 3, and finally coming to rest at point 4 with the spring fully compressed. Let the system consist of the earth, the block, and the spring so that Wext = 0. Let Ug = 0 at point 3 for part (a) and at point 4 for part (b). We can use the work-energy theorem to express the kinetic energy of the system as a function of the block’s position and then use this function to maximize K as well as determine the maximum compression of the spring and the location of the block when the system has half its maximum kinetic energy. ∆K + ∆U g + ∆U s = 0 (a) Apply conservation of mechanical energy to describe the energy transformations between state 1 and state 3: or Because K1 = Ug,3 = Us,1 = 0: K 3 − U g,1 + U s,3 = 0 K 3 − K 1 + U g,3 − U g,1 + U s,3 − U s,1 = 0 510 Chapter 7 and K 3 = K = mg (h + x ) − 1 kx 2 2 Differentiate K with respect to x and set this derivative equal to zero to identify extreme values: dK = mg − kx = 0 for extreme values. dx Solve for x: x= Evaluate the second derivative of K with respect to x: d 2K = −k < 0 dt 2 mg ⇒ x= maximizes K . k Evaluate K for x = mg/k: mg k K max ⎛ mg ⎞ 1 ⎛ mg ⎞ = mgh + mg ⎜ ⎟ − 2 k⎜ ⎟ ⎝ k ⎠ ⎝ k ⎠ 2 m2 g 2 = mgh + 2k (b) The spring will have its maximum compression at point 4 where K = 0: Solve for x and keep the physically meaningful root: 2 mg (h + x max ) − 1 kx max = 0 2 or 2 x max − x max 2mg 2mgh x max − =0 k k mg m 2 g 2 2mgh = + + k k k2 ∆K + ∆U g + ∆U s = 0 (c) Apply conservation of mechanical energy to the system as it evolves from state 1 to the state in which K = 1 K max : 2 or Because K1 = Ug,3 = Us,1 = 0: K − U g,1 + U s,3 = 0 K − K 1 + U g,3 − U g,1 + U s,3 − U s,1 = 0 and K = mg (h + x ) − 1 kx 2 2 Conservation of Energy 511 Substitute for K to obtain: Express this equation in quadratic form: Solve for the positive value of x: 1 2 ⎛ m2 g 2 ⎞ ⎜ mgh + ⎟ = mg (h + x ) − 1 kx 2 2 ⎜ 2k ⎟ ⎝ ⎠ x2 − ⎛ m 2 g 2 mgh ⎞ 2mg x+⎜ ⎜ 2k 2 − k ⎟ = 0 ⎟ k ⎝ ⎠ x= mg 2m 2 g 2 4mgh + + k k k2 92 ••• Picture the Problem The free-body diagram shows the forces acting on the pendulum bob. The application of Newton’s 2nd law leads directly to the required expression for the tangential acceleration. Recall that, provided θ is in radian measure, s = Lθ. Differentiation with respect to time produces the result called for in part (b). The remaining parts of the problem simply require following the directions for each part. (a) Apply ∑F x = max to the bob: Solve for atan: (b) Relate the arc distance s to the length of the pendulum L and the angle θ : Ftan = − mg sin θ = matan a tan = dv / dt = − g sin θ s = Lθ Differentiate with respect to time: ds / dt = v = Ldθ / dt dv dθ and by dt dθ dθ substitute for from part (b): dt dv dv dθ dv dθ = = dt dt dθ dθ dt (c) Multiply = dv ⎛ v ⎞ ⎜ ⎟ dθ ⎝ L ⎠ 512 Chapter 7 (d) Equate the expressions for dv/dt from (a) and (c): dv ⎛ v ⎞ ⎜ ⎟ = − g sin θ dθ ⎝ L ⎠ vdv = − gL sin θ dθ Separate the variables to obtain: v 0 0 θ0 (e) Integrate the left side of the equation in part (d) from v = 0 to the final speed v and the right side from θ = θ0 to θ = 0: ∫ v' dv' = ∫ − gL sin θ ' dθ ' Evaluate the limits of integration to obtain: 1 2 Note, from the figure, that h = L(1 − cosθ0). Substitute and solve for v: v= v 2 = gL(1 − cosθ 0 ) 2 gh 93 ••• Picture the Problem The potential energy of the climber is the sum of his gravitational potential energy and the potential energy stored in the spring-like bungee cord. Let θ be the angle which the position of the rock climber on the cliff face makes with a vertical axis and choose the zero of gravitational potential energy to be at the bottom of the cliff. We can use the definitions of Ug and Uspring to express the climber’s total potential energy. (a) Express the total potential energy of the climber: U (s ) = U bungee cord + U g U ( s ) = 1 k (s − L ) + Mgy 2 Substitute to obtain: 2 = 1 k (s − L ) + MgH cos θ 2 2 ⎛ s ⎞ 2 = 1 k (s − L ) + MgH cos ⎜ ⎟ 2 ⎝H⎠ A spreadsheet solution is shown below. The constants used in the potential energy function and the formulas used to calculate the potential energy are as follows: Cell B3 B4 B5 B6 B7 D11 D12 Content/Formula 300 5 60 85 9.81 60 D11+1 Algebraic Form H k L M g s s+1 Conservation of Energy 513 E11 0.5*$B$4*(D11−$B$5)^2 +$B$6*$B$7*$B$3*(cos(D11/$B$3)) G11 E11−E61 A ⎛ s ⎞ 2 k (s − L ) + MgH cos ⎜ ⎟ ⎝H⎠ U (60 m ) − U (110 m ) 300 5 60 85 9.81 C E U(s) 2.45E+05 2.45E+05 2.45E+05 2.45E+05 2.45E+05 196 197 198 199 200 H= k= L= m= g= D s 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 B 1 2 2.45E+05 2.45E+05 2.45E+05 2.45E+05 2.46E+05 m N/m m kg m/s^2 147 148 149 150 151 The following graph was plotted using the data from columns D (s) and E (U(s)). 246 245 244 U (kJ) 243 242 241 240 239 238 50 70 90 110 130 s (m) 150 170 190 210 514 Chapter 7 *94 ••• Picture the Problem The diagram shows the forces each of the springs exerts on the block. The change in the potential energy stored in the springs is due to the elongation of both springs when the block is displaced a distance x from its equilibrium position and we can find ∆U using 1 k (∆L ) . We can find the magnitude of the force pulling the block 2 back toward its equilibrium position by finding the sum of the magnitudes of the y components of the forces exerted by the springs. In Part (d) we can use conservation of energy to find the speed of the block as it passes through its equilibrium position. 2 (a) Express the change in the potential energy stored in the springs when the block is displaced a distance x: [ ∆U = 2 1 k (∆L ) = k (∆L ) 2 2 2 where ∆L is the change in length of a spring. Referring to the force diagram, express ∆L: ∆L = L2 + x 2 − L Substitute to obtain: ∆U = k (b) Sum the forces acting on the block to express Frestoring: Frestoring = 2 F cos θ = 2k∆L cos θ ( L + x − L) 2 2 x = 2k∆L Substitute for ∆L to obtain: Frestoring = 2k 2 L + x2 2 ( L + x − L) 2 x 2 ⎛ L = 2kx ⎜1 − ⎜ 2 L + x2 ⎝ L + x2 2 ⎞ ⎟ ⎟ ⎠ (c) A spreadsheet program to calculate U(x) is shown below. The constants used in the potential energy function and the formulas used to calculate the potential energy are as follows: Cell B1 B2 B3 C8 D7 Content/Formula 1 1 1 C7+0.01 $B$2*((C7^2+$B$1^2)^0.5−$B$1)^2 Algebraic Form L k M x U(x) Conservation of Energy 515 1 2 3 4 5 6 7 8 9 10 11 12 A B L = 0.1 k= 1 M= 1 C m N/m kg D x 0 0.01 0.02 0.03 0.04 0.05 0.16 0.17 0.18 0.19 0.20 23 24 25 26 27 U(x) 0 2.49E−07 3.92E−06 1.94E−05 5.93E−05 1.39E−04 7.86E−03 9.45E−03 1.12E−02 1.32E−02 1.53E−02 The following graph was plotted using the data from columns C (x) and D (U(x)). 16 14 12 U (mJ) 10 8 6 4 2 0 0.00 0.05 0.10 0.15 x (m) (d) Use conservation of energy to relate the kinetic energy of the block as it passes through the equilibrium position to the change in its potential energy as it returns to its equilibrium position: K equilibrium = ∆U or 1 2 Mv 2 = ∆U 0.20 516 Chapter 7 Solve for v to obtain: v= = 2∆U = M 2k ( L + x − L) 2 2 ( L + x − L) 2 2 M 2k M Substitute numerical values and evaluate v: v=⎛ ⎜ ⎝ (0.1 m )2 + (0.1 m )2 − 0.1 m ⎞ ⎟ ⎠ 2(1 N/m ) = 5.86 cm/s 1 kg 2 Chapter 8 Systems of Particles and Conservation of Momentum Conceptual Problems 1 • Determine the Concept A doughnut. The definition of the center of mass of an object does not require that there be any matter at its location. Any hollow sphere (such as a basketball) or an empty container with any geometry are additional examples of threedimensional objects that have no mass at their center of mass. *2 • Determine the Concept The center of mass is midway between the two balls and is in free-fall along with them (all forces can be thought to be concentrated at the center of mass.) The center of mass will initially rise, then fall. Because the initial velocity of the center of mass is half of the initial velocity of the ball thrown upwards, the mass thrown upwards will rise for twice the time that the center of mass rises. Also, the center of mass will rise until the velocities of the two balls are equal but opposite. (b) is correct. 3 • Determine the Concept The acceleration of the center of mass of a system of particles is r described by Fnet,ext = r r Fi,ext = Macm , where M is the total mass of the system. ∑ i Express the acceleration of the center of mass of the two pucks: acm = Fnet,ext F1 = M m1 + m2 and (b) is correct. 4 • Determine the Concept The acceleration of the center of mass of a system of particles r is described by Fnet,ext = r ∑F i,ext r = Macm , where M is the total mass of the system. i Express the acceleration of the center of mass of the two pucks: acm = Fnet,ext F1 = M m1 + m2 because the spring force is an internal force. (b) is correct. 517 518 Chapter 8 *5 • Determine the Concept No. Consider a 1-kg block with a speed of 1 m/s and a 2- kg block with a speed of 0.707 m/s. The blocks have equal kinetic energies but momenta of magnitude 1 kg·m /s and 1.414 kg·m/s, respectively. 6 • (a) True. The momentum of an object is the product of its mass and velocity. Therefore, if we are considering just the magnitudes of the momenta, the momentum of a heavy object is greater than that of a light object moving at the same speed. (b) True. Consider the collision of two objects of equal mass traveling in opposite directions with the same speed. Assume that they collide inelastically. The mechanical energy of the system is not conserved (it is transformed into other forms of energy), but the momentum of the system is the same after the collision as before the collision, i.e., zero. Therefore, for any inelastic collision, the momentum of a system may be conserved even when mechanical energy is not. (c) True. This is a restatement of the expression for the total momentum of a system of particles. 7 • Determine the Concept To the extent that the system in which the rifle is being fired is an isolated system, i.e., the net external force is zero, momentum is conserved during its firing. Apply conservation of momentum to the firing of the rifle: r r prifle + pbullet = 0 or r r prifle = − pbullet *8 • Determine the Concept When she jumps from a boat to a dock, she must, in order for momentum to be conserved, give the boat a recoil momentum, i.e., her forward momentum must be the same as the boat’s backward momentum. The energy she 2 imparts to the boat is Eboat = pboat 2mboat . When she jumps from one dock to another, the mass of the dock plus the earth is so large that the energy she imparts to them is essentially zero. *9 •• Determine the Concept Conservation of momentum requires only that the net external force acting on the system be zero. It does not require the presence of a medium such as air. Systems of Particles and Conservation of Momentum 519 10 • 2 Determine the Concept The kinetic energy of the sliding ball is 1 mvcm . The kinetic 2 2 energy of the rolling ball is 1 mvcm + K rel , where its kinetic energy relative to its center 2 of mass is K rel . Because the bowling balls are identical and have the same velocity, the rolling ball has more energy. 11 • Determine the Concept Think of someone pushing a box across a floor. Her push on the box is equal but opposite to the push of the box on her, but the action and reaction forces act on different objects. You can only add forces when they act on the same object. 12 • Determine the Concept It’s not possible for both to remain at rest after the collision, as that wouldn't satisfy the requirement that momentum is conserved. It is possible for one to remain at rest: This is what happens for a one-dimensional collision of two identical particles colliding elastically. 13 • Determine the Concept It violates the conservation of momentum! To move forward requires pushing something backwards, which Superman doesn’t appear to be doing when flying around. In a similar manner, if Superman picks up a train and throws it at Lex Luthor, he (Superman) ought to be tossed backwards at a pretty high speed to satisfy the conservation of momentum. *14 •• Determine the Concept There is only one force which can cause the car to move forward−the friction of the road! The car’s engine causes the tires to rotate, but if the road were frictionless (as is closely approximated by icy conditions) the wheels would simply spin without the car moving anywhere. Because of friction, the car’s tire pushes backwards against the road−from Newton’s third law, the frictional force acting on the tire must then push it forward. This may seem odd, as we tend to think of friction as being a retarding force only, but true. 15 •• Determine the Concept The friction of the tire against the road causes the car to slow down. This is rather subtle, as the tire is in contact with the ground without slipping at all times, and so as you push on the brakes harder, the force of static friction of the road against the tires must increase. Also, of course, the brakes heat up, and not the tires. 16 • Determine the Concept Because ∆p = F∆t is constant, a safety net reduces the force acting on the performer by increasing the time ∆t during which the slowing force acts. 17 • Determine the Concept Assume that the ball travels at 80 mi/h ≈ 36 m/s. The ball stops in a distance of about 1 cm. So the distance traveled is about 2 cm at an average speed of 520 Chapter 8 about 18 m/s. The collision time is 0.02 m ≈ 1 ms . 18 m/s 18 • Determine the Concept The average force on the glass is less when falling on a carpet because ∆t is longer. 19 • (a) False. In a perfectly inelastic collision, the colliding bodies stick together but may or may not continue moving, depending on the momentum each brings to the collision. (b) True. In a head-on elastic collision both kinetic energy and momentum are conserved and the relative speeds of approach and recession are equal. (c) True. This is the definition of an elastic collision. *20 •• Determine the Concept All the initial kinetic energy of the isolated system is lost in a perfectly inelastic collision in which the velocity of the center of mass is zero. 21 •• Determine the Concept We can find the loss of kinetic energy in these two collisions by finding the initial and final kinetic energies. We’ll use conservation of momentum to find the final velocities of the two masses in each perfectly elastic collision. pbefore = pafter (a) Letting V represent the velocity of the masses after their perfectly inelastic collision, use conservation of momentum to determine V: or Express the loss of kinetic energy for the case in which the two objects have oppositely directed velocities of magnitude v/2: ⎛ ⎛ v ⎞2 ⎞ ∆K = K f − K i = 0 − 2⎜ 1 m⎜ ⎟ ⎟ ⎜2 ⎝2⎠ ⎟ ⎝ ⎠ 2 mv =− 4 Letting V represent the velocity of the masses after their perfectly inelastic collision, use conservation of momentum to determine V: mv − mv = 2mV ⇒ V = 0 pbefore = pafter or mv = 2mV ⇒ V = 1 v 2 Systems of Particles and Conservation of Momentum 521 Express the loss of kinetic energy for the case in which the one object is initially at rest and the other has an initial velocity v: ∆K = K f − K i 2 mv ⎛v⎞ = (2m )⎜ ⎟ − 1 mv 2 = − 2 4 ⎝2⎠ 2 1 2 The loss of kinetic energy is the same in both cases. (b) Express the percentage loss for the case in which the two objects have oppositely directed velocities of magnitude v/2: 2 1 ∆K 4 mv = 1 2 = 100% K before 4 mv Express the percentage loss for the case in which the one object is initially at rest and the other has an initial velocity v: 2 1 ∆K 4 mv = = 50% K before 1 mv 2 2 The percentage loss is greatest for the case in which the two objects have oppositely directed velocities of magnitude v/2. *22 •• Determine the Concept A will travel farther. Both peas are acted on by the same force, but pea A is acted on by that force for a longer time. By the impulse-momentum theorem, its momentum (and, hence, speed) will be higher than pea B’s speed on leaving the shooter. 23 •• Determine the Concept Refer to the particles as particle 1 and particle 2. Let the direction particle 1 is moving before the collision be the positive x direction. We’ll use both conservation of momentum and conservation of mechanical energy to obtain an expression for the velocity of particle 2 after the collision. Finally, we’ll examine the ratio of the final kinetic energy of particle 2 to that of particle 1 to determine the condition under which there is maximum energy transfer from particle 1 to particle 2. Use conservation of momentum to obtain one relation for the final velocities: Use conservation of mechanical energy to set the velocity of m1v1,i = m1v1,f + m2 v 2,f (1) v 2,f − v1,f = −(v 2,i − v1,i ) = v1,i (2) 522 Chapter 8 recession equal to the negative of the velocity of approach: To eliminate v1,f, solve equation (2) for v1,f, and substitute the result in equation (1): Solve for v2,f: Express the ratio R of K2,f to K1,i in terms of m1 and m2: v1,f = v 2,f + v1,i m1v1, i = m1 (v2, f − v1,i ) + m2v2, f v 2,f = 2 R= = Differentiate this ratio with respect to m2, set the derivative equal to zero, and obtain the quadratic equation: Solve this equation for m2 to determine its value for maximum energy transfer: 2m1 v1,i m1 + m2 − K 2,f K1,i ⎛ 2m1 ⎞ 2 1 ⎜ ⎟ v1,i 2 m2 ⎜ m1 + m2 ⎟ ⎝ ⎠ = 1 m1v12,i 2 m2 4m12 m1 (m1 + m2 )2 2 m2 +1 = 0 m12 m2 = m1 (b) is correct because all of 1' s kinetic energy is transferred to 2 when m2 = m1. 24 • Determine the Concept In the center-of-mass reference frame the two objects approach with equal but opposite momenta and remain at rest after the collision. 25 • Determine the Concept The water is changing direction when it rounds the corner in the nozzle. Therefore, the nozzle must exert a force on the stream of water to change its direction, and, from Newton’s 3rd law, the water exerts an equal but opposite force on the nozzle. 26 • Determine the Concept The collision usually takes place in such a short period of time that the impulse delivered by gravity or friction is negligible. Systems of Particles and Conservation of Momentum 523 27 • r r Determine the Concept No. Fext,net = dp dt defines the relationship between the net force acting on a system and the rate at which its momentum changes. The net external force acting on the pendulum bob is the sum of the force of gravity and the tension in the string and these forces do not add to zero. *28 •• Determine the Concept We can apply conservation of momentum and Newton’s laws of motion to the analysis of these questions. (a) Yes, the car should slow down. An easy way of seeing this is to imagine a "packet" of grain being dumped into the car all at once: This is a completely inelastic collision, with the packet having an initial horizontal velocity of 0. After the collision, it is moving with the same horizontal velocity that the car does, so the car must slow down. (b) When the packet of grain lands in the car, it initially has a horizontal velocity of 0, so it must be accelerated to come to the same speed as the car of the train. Therefore, the train must exert a force on it to accelerate it. By Newton’s 3rd law, the grain exerts an equal but opposite force on the car, slowing it down. In general, this is a frictional force which causes the grain to come to the same speed as the car. (c) No it doesn’t speed up. Imagine a packet of grain being "dumped" out of the railroad car. This can be treated as a collision, too. It has the same horizontal speed as the railroad car when it leaks out, so the train car doesn’t have to speed up or slow down to conserve momentum. *29 •• Determine the Concept Think of the stream of air molecules hitting the sail. Imagine that they bounce off the sail elastically−their net change in momentum is then roughly twice the change in momentum that they experienced going through the fan. Another way of looking at it: Initially, the air is at rest, but after passing through the fan and bouncing off the sail, it is moving backward−therefore, the boat must exert a net force on the air pushing it backward, and there must be a force on the boat pushing it forward. Estimation and Approximation 30 •• Picture the Problem We can estimate the time of collision from the average speed of the car and the distance traveled by the center of the car during the collision. We’ll assume a car length of 6 m. We can calculate the average force exerted by the wall on the car from the car’s change in momentum and it’s stopping time. (a) Relate the stopping time to the assumption that the center of the car travels halfway to the wall with constant deceleration: ∆t = d stopping vav = 1 2 ( 1 Lcar ) = 2 vav 1 4 Lcar vav 524 Chapter 8 Express and evaluate vav: vav = vi + vf 2 km 1h 1000 m × × h 3600 s km = 2 = 12.5 m/s 0 + 90 Substitute for vav and evaluate ∆t: ∆t = 1 4 (6 m ) 12.5 m/s = 0.120 s (b) Relate the average force exerted by the wall on the car to the car’s change in momentum: ⎛ Fav = ∆p = ∆t (2000 kg ) ⎜ 90 km × ⎜ ⎝ h 1h 1000 m ⎞ ⎟ × 3600 s km ⎟ ⎠ 0.120 s = 417 kN 31 •• Picture the Problem Let the direction the railcar is moving be the positive x direction and the system include the earth, the pumpers, and the railcar. We’ll also denote the railcar with the letter c and the pumpers with the letter p. We’ll use conservation of momentum to relate the center of mass frame velocities of the car and the pumpers and then transform to the earth frame of reference to find the time of fall of the car. (a) Relate the time of fall of the railcar to the distance it falls and its velocity as it leaves the bank: Use conservation of momentum to find the speed of the car relative to the velocity of its center of mass: Relate uc to up and solve for uc: ∆t = ∆y vc r r pi = pf or mc uc + mp u p = 0 u c − u p = 4 m/s ∴ u p = u c − 4 m/s Substitute for up to obtain: mc uc + mp (uc − 4 m/s ) = 0 Systems of Particles and Conservation of Momentum 525 Solve for and evaluate uc: Relate the speed of the car to its speed relative to the center of mass of the system: Substitute and evaluate ∆t: (b) Find the speed with which the pumpers hit the ground: uc = 4 m/s 4 m/s = = 1.85 m/s m 350 kg 1+ c 1+ 4(75 kg ) mp vc = uc + vcm m km 1h 1000 m + 32 × × s h 3600 s km = 10.74 m/s = 1.85 ∆t = 25 m = 2.33 s 10.74 m/s vp = vc − up = 10.74 m/s − 4 m/s = 6.74 m/s Hitting the ground at this speed, they may be injured. *32 •• Picture the Problem The diagram depicts the bullet just before its collision with the melon and the motion of the melon-and-bullet-less-jet and the jet just after the collision. We’ll assume that the bullet stays in the watermelon after the collision and use conservation of momentum to relate the mass of the bullet and its initial velocity to the momenta of the melon jet and the melon less the plug after the collision. Apply conservation of momentum to the collision to obtain: Solve for v2f: Express the kinetic energy of the jet of melon in terms of the initial kinetic energy of the bullet: m1v1i = (m2 − m3 + m1 ) v2f + 2m3 K 3 v2f = m1v1i − 2m3 K 3 m2 − m3 + m1 1 1 K 3 = 10 K1 = 10 ( 1 2 ) 2 m1v1i = 1 20 2 m1v1i 526 Chapter 8 Substitute and simplify to obtain: v2f = = 2 1 m1v1i − 2m3 ( 20 m1v1i ) ( m2 − m3 + m1 v1i m1 − 0.1m1m3 m2 − m3 + m1 ) Substitute numerical values and evaluate v2f: ( ) ⎛ ft 1 m ⎞ 0.0104 kg − 0.1(0.0104 kg )(0.14 kg ) ⎟ = −0.386 m/s v2f = ⎜1800 × ⎜ s 3.281 ft ⎟ 2.50 kg − 0.14 kg + 0.0104 kg ⎠ ⎝ = − 1.27 ft/s Note that this result is in reasonably good agreement with experimental results. Finding the Center of Mass 33 • Picture the Problem We can use its definition to find the center of mass of this system. Apply its definition to find xcm: xcm = m1 x1 + m2 x2 + m3 x3 (2 kg )(0) + (2 kg )(0.2 m ) + (2 kg )(0.5 m ) = = 0.233 m 2 kg + 2 kg + 2 kg m1 + m2 + m3 Because the point masses all lie along the x axis: y cm = 0 and the center of mass of this system of particles is at (0.233 m, 0) . *34 • Picture the Problem Let the left end of the handle be the origin of our coordinate system. We can disassemble the club-ax, find the center of mass of each piece, and then use these coordinates and the masses of the handle and stone to find the center of mass of the club-ax. Express the center of mass of the handle plus stone system: Assume that the stone is drilled and the stick passes through it. Use symmetry considerations to locate the center of mass of the stick: xcm = mstick xcm, stick + mstone xcm, stone mstick + mstone xcm,stick = 45.0 cm Systems of Particles and Conservation of Momentum 527 Use symmetry considerations to locate the center of mass of the stone: Substitute numerical values and evaluate xcm: xcm,stone = 89.0 cm xcm = (2.5 kg )(45 cm) + (8 kg )(89 cm) 2.5 kg + 8 kg = 78.5cm 35 • Picture the Problem We can treat each of balls as though they are point objects and apply the definition of the center of mass to find (xcm, ycm). Use the definition of xcm: xcm = = m A x A + m B x B + mC x C m A + m B + mC (3 kg )(2 m ) + (1 kg )(1 m ) + (1 kg )(3 m ) 3 kg + 1 kg + 1 kg = 2.00 m Use the definition of ycm: ycm = = m A y A + mB y B + mC yC m A + mB + mC (3 kg )(2 m ) + (1 kg )(1 m ) + (1 kg )(0) 3 kg + 1 kg + 1 kg = 1.40 m The center of mass of this system of particles is at: (2.00 m,1.40 m ) 36 • Picture the Problem The figure shows an equilateral triangle with its y-axis vertex above the x axis. The bisectors of the vertex angles are also shown. We can find x coordinate of the center-of-mass by inspection and the y coordinate using trigonometry. From symmetry considerations: xcm = 0 528 Chapter 8 Express the trigonometric relationship between a/2, 30°, and ycm: Solve for ycm: tan 30° = ycm a2 ycm = 1 a tan 30° = 0.289a 2 The center of mass of an equilateral triangle oriented as shown above is at (0, 0.289a ) . *37 •• Picture the Problem Let the subscript 1 refer to the 3-m by 3-m sheet of plywood before the 2-m by 1-m piece has been cut from it. Let the subscript 2 refer to 2-m by 1-m piece that has been removed and let σ be the area density of the sheet. We can find the centerof-mass of these two regions; treating the missing region as though it had negative mass, and then finding the center-of-mass of the U-shaped region by applying its definition. Express the coordinates of the center of mass of the sheet of plywood: Use symmetry to find xcm,1, ycm,1, xcm,2, and ycm,2: xcm = ycm = m1 xcm,1 − m2 xcm, 2 m1 − m2 m1 ycm,1 − m2 ycm , 2 m1 − m2 xcm,1 = 1.5 m, ycm,1 = 1.5 m and xcm,2 = 1.5 m, ycm,2 = 2.0 m Determine m1 and m2: m1 = σA1 = 9σ kg and m2 = σA2 = 2σ kg Substitute numerical values and evaluate xcm: xcm = (9σ kg )(1.5 m ) − (2σ kg )(1.5 kg ) 9σ kg − 2σ kg = 1.50 m Substitute numerical values and evaluate ycm: ycm = (9σ kg )(1.5 m ) − (2σ kg )(2 m ) 9σ kg − 2σ kg = 1.36 m The center of mass of the U-shaped sheet of plywood is at (1.50 m,1.36 m ) . Systems of Particles and Conservation of Momentum 529 38 •• Picture the Problem We can use its definition to find the center of mass of the can plus water. By setting the derivative of this function equal to zero, we can find the value of x that corresponds to the minimum height of the center of mass of the water as it drains out and then use this extreme value to express the minimum height of the center of mass. (a) Using its definition, express the location of the center of mass of the can + water: Let the cross-sectional area of the cup be A and use the definition of density to relate the mass m of water remaining in the can at any given time to its depth x: Solve for m to obtain: ⎛H⎞ ⎛ x⎞ M ⎜ ⎟ + m⎜ ⎟ 2 ⎝2⎠ xcm = ⎝ ⎠ M +m M m ρ= = AH Ax m= Substitute to obtain: xcm x M H ⎛H⎞ ⎛ x ⎞⎛ x ⎞ M ⎜ ⎟ + ⎜ M ⎟⎜ ⎟ 2 H ⎠⎝ 2 ⎠ = ⎝ ⎠ ⎝ x M+ M H ⎛ ⎛ x ⎞2 ⎞ ⎜1+ ⎜ ⎟ ⎟ H ⎜ ⎝H⎠ ⎟ = 2 ⎜ 1+ x ⎟ ⎜ ⎟ ⎜ H ⎟ ⎝ ⎠ (b) Differentiate xcm with respect to x and set the derivative equal to zero for extrema: 2⎤ ⎫ ⎧ ⎡ ⎛ x ⎞2 ⎤ d ⎛ ⎡ x ⎛ ⎛ x ⎞2 ⎞ ⎪ ⎛1 + x ⎞ d ⎢1 + ⎛ x ⎞ ⎥ ⎢1 + ⎜ ⎟ ⎥ ⎜1 + ⎞ ⎪ ⎜1+ ⎜ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ H dx H ⎠⎪ dxcm H d ⎜ ⎝ H ⎠ ⎟ H ⎪ ⎝ H ⎠ dx ⎢ ⎝ H ⎠ ⎥ ⎪ ⎪ ⎣ ⎦ ⎣ ⎦ −⎢ ⎝ ⎠ ⎥ ⎝ = ⎬ ⎜ ⎟= ⎨ 2 2 x dx 2 dx ⎜ x⎞ x⎞ ⎪ ⎛ ⎛ ⎟ 2⎪ 1+ ⎜1 + ⎟ ⎜1 + ⎟ H ⎟ ⎜ ⎪ ⎪ ⎝ H⎠ ⎝ H⎠ ⎝ ⎠ ⎪ ⎪ ⎭ ⎩ ⎡ ⎛ x ⎞ 2 ⎤⎛ 1 ⎞ ⎫ ⎧⎛ x x 1 ⎪ ⎪ ⎜1 + ⎞2⎛ ⎞⎛ ⎞ ⎢1 + ⎜ H ⎟ ⎥⎜ H ⎟ ⎪ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎝ ⎠ ⎥⎝ H⎪ ⎪ H ⎠ ⎝ H ⎠⎝ H ⎠ ⎣ ⎦ = ⎨⎝ − ⎬=0 2 2 2 ⎪ x⎞ x⎞ ⎪ ⎛ ⎛ ⎜1 + ⎟ ⎜1 + ⎟ ⎪ ⎪ ⎝ H⎠ ⎝ H⎠ ⎩ ⎪ ⎭ Simplify this expression to obtain: 2 ⎛ x⎞ ⎛ x⎞ ⎜ ⎟ + 2⎜ ⎟ − 1 = 0 ⎝H ⎠ ⎝H ⎠ 530 Chapter 8 x=H Solve for x/H to obtain: ( ) 2 − 1 ≈ 0.414 H where we’ve kept the positive solution because a negative value for x/H would make no sense. Use your graphing calculator to convince yourself that the graph of xcm as a function of x is concave upward at x ≈ 0.414 H and that, therefore, the minimum value of xcm occurs at x ≈ 0.414 H . ( ) 2 − 1 to obtain: ( ) ⎛ ⎛ H 2 − 1 ⎞2 ⎞ ⎜1+ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ H H ⎠ ⎟ ⎝ xcm x= H ( 2 −1) = ⎜ 2⎜ H 2 −1 ⎟ ⎟ ⎜ 1+ H ⎟ ⎜ ⎠ ⎝ = H ( *39 •• Picture the Problem A semicircular disk and a surface element of area dA is shown in the diagram. Because the disk is a continuous object, we’ll use r r Mrcm = ∫ r dm and symmetry to find its center of mass. xcm = 0 by symmetry. ycm = ) 2 −1 Finding the Center of Mass by Integration Express the coordinates of the center of mass of the semicircular disk: ) ( Evaluate xcm at x = H ∫ yσ dA M Express y as a function of r and θ : y = r sin θ Express dA in terms of r and θ : dA = r dθ dr Express M as a function of r and θ : M = σAhalf disk = 1 σπR 2 2 Systems of Particles and Conservation of Momentum 531 Rπ Substitute and evaluate ycm: ycm = = σ ∫ ∫ r 2 sin θ dθ dr 0 0 M 2σ 2 = r dr M ∫ 0 R 2σ 3 4 R = R 3M 3π 40 ••• Picture the Problem Because a solid hemisphere is a continuous object, we’ll use r r Mrcm = ∫ r dm to find its center of mass. The volume element for a sphere is dV = r2 sinθ dθ dφ dr, where θ is the polar angle and φ the azimuthal angle. Let the base of the hemisphere be the xy plane and ρ be the mass density. Then: Express the z coordinate of the center of mass: Evaluate M = ∫ ρdV : z = r cos θ zcm = ∫ rρdV ∫ ρdV M = ∫ ρdV = 1 ρVsphere 2 ( ) = 1 ρ 4 π R 3 = 2 πρR 3 3 2 3 Evaluate ∫ rρdV : R π / 2 2π ∫ rρdV = ∫ 0 = Substitute and simplify to find zcm: zcm = 1 4 2 3 ∫ ∫r 0 πρR 3 sin θ cosθdθdφdr 0 4 2 πρR 4 = πρR 3 [ sin θ ] 2 1 2 3 8 π /2 0 = πρR 4 4 R 41 ••• Picture the Problem Because a thin hemisphere shell is a continuous object, we’ll use r r Mrcm = r dm to find its center of mass. The element of area on the shell is dA = 2πR2 ∫ sinθ dθ, where R is the radius of the hemisphere. Let σ be the surface mass density and express the z coordinate of the center of mass: zcm = ∫ zσ dA ∫ σ dA 532 Chapter 8 ∫ Evaluate M = σ dA : M = ∫ σ dA = 1 σAspherical shell 2 ( ) = 1 σ 4π R 2 = 2πσR 2 2 Evaluate ∫ zσ dA : π /2 ∫ zσ dA = 2πR σ ∫ sin θ cos θ dθ 3 0 = πR 3σ π /2 ∫ sin 2θ dθ 0 = πR σ 3 Substitute and simplify to find zcm: zcm = πR 3σ = 2πσR 2 1 2 R 42 ••• Picture the Problem The parabolic sheet is shown to the right. Because the area of the sheet is distributed symmetrically with respect to the y axis, xcm = 0. We’ll integrate the element of area dA (= xdy) to obtain the total area of the sheet and yxdy to obtain the numerator of the definition of the center of mass. b Express ycm: ycm = ∫ xydy 0 b ∫ xdy 0 b Evaluate ∫ xydy : 0 b b 0 0 ∫ xydy = ∫ = b Evaluate ∫ xdy : 0 2 5 a b b 0 0 ∫ xdy = ∫ = b y1 2 1 32 ydy = ∫ y dy a a0 b5 2 b y1 2 1 12 dy = ∫ y dy a a0 2 3 a b3 2 Systems of Particles and Conservation of Momentum 533 2 Substitute and simplify to determine ycm: b5 2 = 3b ycm = 5 a 2 32 5 b 3 a Note that, by symmetry: xcm = 0 (0, 3 b ) 5 The center of mass of the parabolic sheet is at: Motion of the Center of Mass 43 • Picture the Problem The velocity of the center of mass of a system of particles is related r to the total momentum of the system through P = r ∑mv i i r = Mv cm . i r Use the expression for the total momentum of a system to relate the velocity of the center of mass of the two-particle system to the momenta of the individual particles: r v cm = ∑m v i i i M r r m1v1 + m2 v 2 = m1 + m2 r r r r r (3 kg )(v1 + v 2 ) = 1 (v + v ) vcm = 1 2 2 6 kg ˆ = 1 (6 m/s ) i − (3 m/s ) ˆ j Substitute numerical values and r evaluate v cm : 2 = [ (3 m/s ) iˆ − (1.5 m/s ) ˆ j *44 • Picture the Problem Choose a coordinate system in which east is the positive x r direction and use the relationship P = r ∑mv i i r = Mv cm to determine the velocity of the i center of mass of the system. Use the expression for the total momentum of a system to relate the velocity of the center of mass of the two-vehicle system to the momenta of the individual vehicles: Express the velocity of the truck: r r v cm = ∑m v i i i M r r mt v t + mc v c = mt + mc r ˆ v t = (16 m/s ) i 534 Chapter 8 r ˆ v c = (− 20 m/s ) i Express the velocity of the car: r Substitute numerical values and evaluate v cm : r (3000 kg )(16 m/s) iˆ + (1500 kg )(− 20 m/s) iˆ = v cm = 3000 kg + 1500 kg (4.00 m/s) iˆ 45 • Picture the Problem The acceleration of the center of mass of the ball is related to the r r net external force through Newton’s 2nd law: Fnet,ext = Ma cm . Use Newton’s 2nd law to express the acceleration of the ball: r Fnet,ext r acm = M Substitute numerical values and r evaluate a cm : r acm = (12 N ) iˆ 3 kg + 1 kg + 1 kg = (2.4 m/s )iˆ 2 46 •• Picture the Problem Choose a coordinate system in which upward is the positive y r r direction. We can use Newton’s 2nd law Fnet,ext = Ma cm to find the acceleration of the center of mass of this two-body system. (a) Yes; initially the scale reads ( M + m) g ; while m is in free fall, the reading is Mg. (b) Using Newton’s 2nd law, express the acceleration of the center of mass of the system: Substitute to obtain: (c) Use Newton’s 2nd law to express the net force acting on the scale while the object of mass m is falling: Substitute and simplify to obtain: r Fnet,ext r acm = mtot r mg ˆ a cm = − j M +m Fnet,ext = (M + m )g − ( M + m)acm ⎛ mg ⎞ Fnet,ext = (M + m )g − ( M + m) ⎜ ⎟ ⎝M +m⎠ = Mg Systems of Particles and Conservation of Momentum 535 as expected, given our answer to part (a). *47 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a) and the work-energy theorem in conjunction with Newton’s 2nd law in parts (b) and (c). (a) Apply ∑F y = ma y to the ∑F y = Fn − mp g − Fball on spring = 0 spring when it is compressed a distance d: Solve for Fn: Fn = mp g + Fball on spring ⎛m g⎞ = mp g + kd = mp g + k ⎜ b ⎟ ⎝ k ⎠ = mp g + mb g = (mp + mb )g (b) Use conservation of mechanical energy, with Ug = 0 at the position at which the spring is fully compressed, to relate the gravitational potential energy of the system to the energy stored in the fully compressed spring: ∆K + ∆U g + ∆U s = 0 Because ∆K = Ug,f = Us,i = 0, U g,i − U s,f = 0 or mb gd − 1 kd 2 = 0 2 2m b g k Solve for d: d= Evaluate our force equation in (a) Fn = mp g + Fball on spring with d = 2m b g : k ⎛ 2m g ⎞ = mp g + kd = mp g + k ⎜ b ⎟ ⎝ k ⎠ = mp g + 2mb g = (mp + 2mb )g 536 Chapter 8 (c) When the ball is in its original position, the spring is relaxed and exerts no force on the ball. Therefore: Fn = scale reading = mp g *48 •• Picture the Problem Assume that the object whose mass is m1 is moving downward and take that direction to be the positive direction. We’ll use Newton’s 2nd law for a system of particles to relate the acceleration of the center of mass to the acceleration of the individual particles. (a) Relate the acceleration of the center of mass to m1, m2, mc and their accelerations: r r r r Ma cm = m1a1 + m2 a 2 + mc a c m1 − m2 m1 + m2 + mc Because m1 and m2 have a common acceleration a and ac = 0: a cm = a From Problem 4-81 we have: a=g Substitute to obtain: ⎛ m − m2 ⎞ ⎛ m1 − m2 ⎞ acm = ⎜ 1 ⎜ m + m g ⎟⎜ m + m + m ⎟ ⎟⎜ ⎟ 2 2 c ⎠ ⎝ 1 ⎠⎝ 1 = (b) Use Newton’s 2nd law for a system of particles to obtain: Solve for F and substitute for acm from part (a): m1 − m2 m1 + m2 (m1 − m2 )2 g (m1 + m2 )(m1 + m2 + mc ) F − Mg = − Macm where M = m1 + m2 + mc and F is positive upwards. F = Mg − Macm (m1 − m2 )2 g = Mg − m1 + m2 ⎡ 4m1 m2 ⎤ = ⎢ + mc ⎥ g ⎣ m1 + m2 ⎦ (c) From Problem 4-81: T= 2m1 m2 g m1 + m2 Systems of Particles and Conservation of Momentum 537 Substitute in our result from part (b) to obtain: ⎡ 2m1m2 ⎤ F = ⎢2 + mc ⎥ g ⎣ m1 + m2 ⎦ ⎡ T ⎤ = ⎢2 + mc ⎥ g = 2T + mc g ⎣ g ⎦ 49 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a) and the result of Problem 7-96 part (b) to obtain the scale reading when the ball is dropped from a height h above the cup. (a) Apply ∑F y = ma y to the spring ∑F y = Fn − mp g − Fball on spring = 0 when it is compressed a distance d: Solve for Fn: Fn = mp g + Fball on spring ⎛m g⎞ = mp g + kd = mp g + k ⎜ b ⎟ ⎝ k ⎠ = mp g + mb g = (mp + mb )g (b) From Problem 7-96, part (b): From part (a): x max = mb g ⎛ ⎜1 + 1 + 2kh k ⎜ mb g ⎝ ⎞ ⎟ ⎟ ⎠ Fn = mp g + Fball on spring = mp g + kxmax ⎛ 2kh = mp g + mb g ⎜1 + 1 + ⎜ mb g ⎝ ⎞ ⎟ ⎟ ⎠ The Conservation of Momentum 50 • Picture the Problem Let the system include the woman, the canoe, and the earth. Then the net external force is zero and linear momentum is conserved as she jumps off the 538 Chapter 8 canoe. Let the direction she jumps be the positive x direction. Apply conservation of momentum to the system: Substitute to obtain: r Solve for v canoe : r ∑m v i i r r = mgirl v girl + mcanoe v canoe = 0 r (55 kg )(2.5 m/s ) iˆ + (75 kg ) vcanoe = 0 r v canoe = (− 1.83 m/s) iˆ 51 • Picture the Problem If we include the earth in our system, then the net external force is zero and linear momentum is conserved as the spring delivers its energy to the two objects. Apply conservation of momentum to the system: Substitute numerical values to obtain: r Solve for v10 : r ∑m v i i r r = m5 v 5 + m10 v10 = 0 r (5 kg )(− 8 m/s ) iˆ + (10 kg ) v10 = 0 r v10 = (4 m/s) iˆ *52 • Picture the Problem This is an explosion-like event in which linear momentum is conserved. Thus we can equate the initial and final momenta in the x direction and the initial and final momenta in the y direction. Choose a coordinate system in the positive x direction is to the right and the positive y direction is upward. Equate the momenta in the y direction before and after the explosion: ∑p y,i = ∑ py,f = mv2 − 2mv1 = m(2v1 ) − 2mv1 = 0 We can conclude that the momentum was entirely in the x direction before the particle exploded. ∑p = ∑ p x,f Equate the momenta in the x direction before and after the explosion: ∴4mvi = mv3 Solve for v3: vi = 1 v3 and (c) is correct. 4 x,i Systems of Particles and Conservation of Momentum 539 53 • Picture the Problem Choose the direction the shell is moving just before the explosion to be the positive x direction and apply conservation of momentum. Use conservation of momentum to relate the masses of the fragments to their velocities: r r r pi = pf or r ˆ mvi = 1 mvˆ + 1 mv ' j 2 2 r ˆ j v ' = 2vi − vˆ Solve for v ' : *54 •• Picture the Problem Let the system include the earth and the platform, gun and block. r Then Fnet,ext = 0 and momentum is conserved within the system. (a) Apply conservation of momentum to the system just before and just after the bullet leaves the gun: r r r r pbefore = pafter or r r 0 = pbullet + pplatform Substitute for pbullet and pplatform and r ˆ 0 = mb vb i + mp v platform solve for v platform : and r r m ˆ v platform = − b vb i mp (b) Apply conservation of momentum to the system just before the bullet leaves the gun and just after it comes to rest in the block: (c) Express the distance ∆s traveled by the platform: Express the velocity of the bullet relative to the platform: r r p before = pafter or r r 0 = p platform ⇒ v platform = 0 ∆s = vplatform ∆t vrel = vb − vplatform = vb + mb vb mp ⎛ m ⎞ m + mb vb = ⎜1 + b ⎟v b = p ⎜ m ⎟ mp p ⎠ ⎝ Relate the time of flight ∆t to L and vrel: ∆t = L v rel 540 Chapter 8 Substitute to find the distance ∆s moved by the platform in time ∆t: ⎛m ⎞⎛ L ⎞ ⎟ ∆s = vplatform ∆t = ⎜ b vb ⎟⎜ ⎜m ⎟⎜ v ⎟ ⎝ p ⎠⎝ rel ⎠ ⎞ ⎛ ⎟ ⎜ ⎛ mb ⎞ ⎜ L ⎟ vb ⎟ ⎜ =⎜ ⎜m ⎟ m + mb ⎟ ⎝ p ⎠⎜ p vb ⎟ ⎟ ⎜ m p ⎠ ⎝ = mb L mp + mb 55 •• Picture the Problem The pictorial representation shows the wedge and small object, initially at rest, to the left, and, to the right, both in motion as the small object leaves the wedge. Choose the direction the small object is moving when it leaves the wedge be the positive x direction and the zero of potential energy to be at the surface of the table. Let the speed of the small object be v and that of the wedge V. We can use conservation of momentum to express v in terms of V and conservation of energy to express v in terms of h. Apply conservation of momentum to the small object and the wedge: r r pi, x = pf, x or r ˆ 0 = mvi + 2mV r Solve for V : r ˆ V = − 1 vi 2 (1) and V = 1v 2 Use conservation of energy to determine the speed of the small object when it exits the wedge: Because Uf = Ki = 0: ∆K + ∆U = 0 or Kf − Ki + U f − U i = 0 1 2 mv 2 + 1 (2m )V 2 − mgh = 0 2 Systems of Particles and Conservation of Momentum 541 Substitute for V to obtain: 1 2 mv 2 + 1 (2m )( 1 v ) − mgh = 0 2 2 2 Solve for v to obtain: v=2 Substitute in equation (1) to r determine V : r ⎛ gh ⎞ ˆ ⎟i = − V = − 1 ⎜2 2⎜ 3 ⎟ ⎝ ⎠ gh 3 gh ˆ i 3 i.e., the wedge moves in the direction opposite to that of the small object with a gh . 3 speed of *56 •• Picture the Problem Because no external forces act on either cart, the center of mass of the two-cart system can’t move. We can use the data concerning the masses and separation of the gliders initially to calculate its location and then apply the definition of the center of mass a second time to relate the positions X1 and X2 of the centers of the carts when they first touch. We can also use the separation of the centers of the gliders when they touch to obtain a second equation in X1 and X2 that we can solve simultaneously with the equation obtained from the location of the center of mass. (a) Apply its definition to find the center of mass of the 2-glider system: xcm = = m1 x1 + m2 x2 m1 + m2 (0.1kg )(0.1m ) + (0.2 kg )(1.6 m ) 0.1 kg + 0.2 kg = 1.10 m from the left end of the air track. Use the definition of the center of mass to relate the coordinates of the centers of the two gliders when they first touch to the location of the center of mass: 1.10 m = = m1 X 1 + m2 X 2 m1 + m2 (0.1kg )X 1 + (0.2 kg )X 2 0.1 kg + 0.2 kg = X1 + 2 X 2 3 1 3 (10 cm + 20 cm) = 0.15 m Also, when they first touch, their centers are separated by half their combined lengths: X 2 − X1 = Thus we have: 0.333 X 1 + 0.667 X 2 = 1.10 m 1 2 and X 2 − X 1 = 0.15 m 542 Chapter 8 Solve these equations simultaneously to obtain: (b) X 1 = 1.00 m and X 2 = 1.15 m No. The initial momentum of the system is zero, so it must be zero after the collision. Kinetic Energy of a System of Particles *57 • Picture the Problem Choose a coordinate system in which the positive x direction is to the right. Use the expression for the total momentum of a system to find the velocity of the center of mass and the definition of relative velocity to express the sum of the kinetic energies relative to the center of mass. (a) Find the sum of the kinetic energies: K = K1 + K 2 2 = 1 m1v12 + 1 m2v2 2 2 = 1 2 (3 kg )(5 m/s )2 + 1 (3 kg )(2 m/s )2 2 = 43.5 J (b) Relate the velocity of the center of mass of the system to its total momentum: r r r Mv cm = m1v1 + m2 v 2 Solve for vcm : r r r m v + m2 v 2 v cm = 1 1 m1 + m2 Substitute numerical values and r evaluate vcm : r (3 kg )(5 m/s ) iˆ − (3 kg )(2 m/s ) iˆ vcm = 3 kg + 3 kg r ˆ = (1.50 m/s ) i (c) The velocity of an object relative to the center of mass is given by: r r r v rel = v − v cm Systems of Particles and Conservation of Momentum 543 Substitute numerical values to obtain: r ˆ ˆ v1,rel = (5 m/s ) i − (1.5 m/s ) i (3.50 m/s) iˆ r ˆ ˆ v 2,rel = (− 2 m/s ) i − (1.5 m/s ) i = (− 3.50 m/s) iˆ = (d) Express the sum of the kinetic energies relative to the center of mass: Substitute numerical values and evaluate Krel: 2 K rel = K1,rel + K 2,rel = 1 m1v12,rel + 1 m2v2,rel 2 2 K rel = 1 2 (3 kg )(3.5 m/s )2 2 + 1 (3 kg )(− 3.5 m/s ) 2 = 36.75 J (e) Find Kcm: 2 K cm = 1 mtot vcm = 2 1 2 (6 kg )(1.5 m/s )2 = 6.75 J = 43.5 J − 36.75 J = K − K rel 58 • Picture the Problem Choose a coordinate system in which the positive x direction is to the right. Use the expression for the total momentum of a system to find the velocity of the center of mass and the definition of relative velocity to express the sum of the kinetic energies relative to the center of mass. (a) Express the sum of the kinetic energies: Substitute numerical values and evaluate K: (b) Relate the velocity of the center of mass of the system to its total momentum: r Solve for v cm : 2 K = K1 + K 2 = 1 m1v12 + 1 m2 v2 2 2 K= 1 2 (3 kg )(5 m/s )2 + 1 (5 kg )(3 m/s )2 2 = 6 0 .0 J r r r Mv cm = m1v1 + m2 v 2 r r r m1v1 + m2 v 2 v cm = m1 + m2 544 Chapter 8 Substitute numerical values and r evaluate v cm : r (3 kg )(5 m/s) iˆ + (5 kg )(3 m/s) iˆ vcm = 3 kg + 5 kg (3.75 m/s) iˆ = (c) The velocity of an object relative to the center of mass is given by: Substitute numerical values and evaluate the relative velocities: r r r v rel = v − v cm r ˆ ˆ v1,rel = (5 m/s ) i − (3.75 m/s ) i ˆ = (1.25 m/s ) i and r ˆ ˆ v 2,rel = (3 m/s ) i − (3.75 m/s ) i (− 0.750 m/s) iˆ = (d) Express the sum of the kinetic energies relative to the center of mass: Substitute numerical values and evaluate Krel: K rel = K1,rel + K 2,rel 2 = 1 m1v12,rel + 1 m2 v2,rel 2 2 K rel = (3 kg )(1.25 m/s)2 2 + 1 (5 kg )(− 0.75 m/s ) 2 1 2 = 3.75 J (e) Find Kcm: 2 K cm = 1 mtot vcm = 2 1 2 (8 kg )(3.75 m/s )2 = 56.3 J = K − K rel Impulse and Average Force 59 • Picture the Problem The impulse imparted to the ball by the kicker equals the change in the ball’s momentum. The impulse is also the product of the average force exerted on the ball by the kicker and the time during which the average force acts. I = ∆p = p f − pi (a) Relate the impulse delivered to the ball to its change in momentum: = mvf since vi = 0 Substitute numerical values and evaluate I: I = (0.43 kg )(25 m/s ) = 10.8 N ⋅ s Systems of Particles and Conservation of Momentum 545 (b) Express the impulse delivered to the ball as a function of the average force acting on it and solve for and evaluate Fav : I = Fav ∆t and Fav = I 10.8 N ⋅ s = = 1.34 kN ∆t 0.008 s 60 • Picture the Problem The impulse exerted by the ground on the brick equals the change in momentum of the brick and is also the product of the average force exerted by the ground on the brick and the time during which the average force acts. (a) Express the impulse exerted by the ground on the brick: Because pf,brick = 0: I = ∆pbrick = pf,brick − pi,brick I = pi,brick = mbrick v (1) ∆K + ∆U = 0 Use conservation of energy to determine the speed of the brick at impact: or Kf − Ki + U f − U i = 0 Because Uf = Ki = 0: Kf −Ui = 0 or 1 2 mbrick v 2 − mbrick gh = 0 Solve for v: v = 2 gh Substitute in equation (1) to obtain: I = mbrick 2 gh Substitute numerical values and evaluate I: (c) Express the impulse delivered to the brick as a function of the average force acting on it and solve for and evaluate Fav : ( ) I = (0.3 kg ) 2 9.81 m/s 2 (8 m ) = 3.76 N ⋅ s I = Fav ∆t and Fav = I 3.76 N ⋅ s = = 2.89 kN ∆t 0.0013 s *61 • Picture the Problem The impulse exerted by the ground on the meteorite equals the change in momentum of the meteorite and is also the product of the average force exerted by the ground on the meteorite and the time during which the average force acts. 546 Chapter 8 I = ∆pmeteorite = pf − pi Express the impulse exerted by the ground on the meteorite: Relate the kinetic energy of the meteorite to its initial momentum and solve for its initial momentum: pi2 Ki = ⇒ pi = 2mK i 2m Express the ratio of the initial and final kinetic energies of the meteorite: pi2 p2 Ki = 2m = i2 = 2 Kf p f2 pf 2m Solve for pf: pf = pi 2 pi ⎛ 1 ⎞ − pi = pi ⎜ − 1⎟ 2 ⎝ 2 ⎠ ⎛ 1 ⎞ = 2mK i ⎜ − 1⎟ ⎝ 2 ⎠ Substitute in our expression for I and simplify: I= Because our interest is in its magnitude, evaluate I : I = ( )( ) ⎛ 1 ⎞ 2 30.8 × 103 kg 617 ×106 J ⎜ − 1⎟ = 1.81 MN ⋅ s ⎝ 2 ⎠ Express the impulse delivered to the meteorite as a function of the average force acting on it and solve for and evaluate Fav : I = Fav ∆t and Fav = I 1.81MN ⋅ s = = 0.602 MN ∆t 3s 62 •• Picture the Problem The impulse exerted by the bat on the ball equals the change in momentum of the ball and is also the product of the average force exerted by the bat on the ball and the time during which the bat and ball were in contact. (a) Express the impulse exerted by the bat on the ball in terms of the change in momentum of the ball: r r r r I = ∆pball = pf − pi ˆ ˆ ˆ = mv i − − mv i = 2mv i f ( where v = vf = vi i ) Systems of Particles and Conservation of Momentum 547 Substitute for m and v and evaluate I: (b) Express the impulse delivered to the ball as a function of the average force acting on it and solve for and evaluate Fav : I = 2(0.15 kg )(20 m/s ) = 6.00 N ⋅ s I = Fav ∆t and Fav = I 6.00 N ⋅ s = = 4.62 kN ∆t 1.3 ms *63 •• Picture the Problem The figure shows the handball just before and immediately after its collision with the wall. Choose a coordinate system in which the positive x direction is to the right. The wall changes the momentum of the ball by exerting a force on it during the ball’s collision with it. The reaction to this force is the force the ball exerts on the wall. Because these action and reaction forces are equal in magnitude, we can find the average force exerted on the ball by finding the change in momentum of the ball. Using Newton’s 3rd law, relate the average force exerted by the ball on the wall to the average force exerted by the wall on the ball: Relate the average force exerted by the wall on the ball to its change in momentum: r Express ∆v x for the ball: r r Fav on wall = − Fav on ball and Fav on wall = Fav on ball (1) r r r ∆p m∆v Fav on ball = = ∆t ∆t r ˆ ˆ ∆v x = vf , x i − vi , x i or, because vi,x = vcosθ and vf,x = −vcosθ, r ˆ ˆ ˆ ∆v x = −v cos θ i − v cos θ i = −2v cos θ i Substitute in our expression for r Fav on ball : r r m∆v 2mv cos θ ˆ Fav on ball = i =− ∆t ∆t 548 Chapter 8 r 2mv cosθ ∆t 2(0.06 kg )(5 m/s )cos40° = 2 ms = 230 N Evaluate the magnitude of Fav on ball : Fav on ball = Substitute in equation (1) to obtain: Fav on wall = 230 N 64 •• Picture the Problem The pictorial representation shows the ball during the interval of time you are exerting a force on it to accelerate it upward. The average force you exert can be determined from the change in momentum of the ball. The change in the velocity of the ball can be found by applying conservation of mechanical energy to its rise in the air once it has left your hand. (a) Relate the average force exerted by your hand on the ball to the change in momentum of the ball: Letting Ug = 0 at the initial elevation of your hand, use conservation of mechanical energy to relate the initial kinetic energy of the ball to its potential energy when it is at its highest point: Substitute for Kf and Ui and solve for v2: Fav = ∆p p2 − p1 mv2 = = ∆t ∆t ∆t because v1 and, hence, p1 = 0. ∆K + ∆U = 0 or − Ki + U f = 0 since K f = U i = 0 2 − 1 mv 2 + mgh = 0 2 and v 2 = 2 gh Relate ∆t to the average speed of the ball while you are throwing it upward: ∆t = d d 2d = = vav v 2 v2 2 Systems of Particles and Conservation of Momentum 549 Substitute for ∆t and v2 in the expression for Fav to obtain: Fav = Substitute numerical values and evaluate Fav: Fav = mgh d (0.15 kg )(9.81m/s 2 )(40 m ) 0.7 m = 84.1 N (b) Express the ratio of the weight of the ball to the average force acting on it: w mg (0.15 kg ) (9.81 m/s 2 ) = = < 2% Fav Fav 84.1 N Because the weight of the ball is less than 2% of the average force exerted on the ball, it is reasonable to have neglected its weight. 65 •• Picture the Problem Choose a coordinate system in which the direction the ball is moving after its collision with the wall is the positive x direction. The impulse delivered to the wall or received by the player equals the change in the momentum of the ball. We can find the average forces from the rate of change in the momentum of the ball. (a) Relate the impulse delivered to the wall to the change in momentum of the handball: r r r r I = ∆p = mvf − mvi ˆ = (0.06 kg )(8 m/s ) i [ ˆ − − (0.06 kg )(10 m/s ) i ˆ = (1.08 N ⋅ s ) i directed into wall. (b) Find Fav from the change in the ball’s momentum: ∆p 1.08 N ⋅ s = 0.003 s ∆t Fav = = 360 N, into wall. (c) Find the impulse received by the player from the change in momentum of the ball: I = ∆pball = m∆v = (0.06 kg )(8 m/s ) = 0.480 N ⋅ s, away from wall. (d) Relate Fav to the change in the ball’s momentum: Fav = Express the stopping time in terms of the average speed vav of the ball ∆t = ∆p ball ∆t d vav 550 Chapter 8 and its stopping distance d: Substitute to obtain: Fav = Substitute numerical values and evaluate Fav: Fav = vav ∆pball d (4 m/s)(0.480 N ⋅ s ) 0.5 m = 3.84 N, away from wall. 66 ••• Picture the Problem The average force exerted on the limestone by the droplets of water equals the rate at which momentum is being delivered to the floor. We’re given the number of droplets that arrive per minute and can use conservation of mechanical energy to determine their velocity as they reach the floor. (a) Letting N represent the rate at which droplets fall, relate Fav to the change in the droplet’s momentum: Find the mass of the droplets: Fav = ∆pdroplets ∆t =N m∆v ∆t m = ρV = (1 kg/L )(0.03 mL ) = 3 × 10 −5 kg Letting Ug = 0 at the point of impact of the droplets, use conservation of mechanical energy to relate their speed at impact to their fall distance: ∆K + ∆U = 0 or Kf − Ki + U f − U i = 0 mvf2 − mgh = 0 Because Ki = Uf = 0: 1 2 Solve for and evaluate v = vf: v = 2 gh = 2(9.81 m/s 2 )(5 m ) = 9.90 m/s Substitute numerical values and evaluate Fav: ⎛N⎞ Fav = ⎜ ⎟m∆v ⎝ ∆t ⎠ ⎛ droplets 1 min ⎞ ⎟ = ⎜10 × ⎜ min 60 s ⎟ ⎝ ⎠ × 3 × 10 −5 kg (9.90 m/s ) ( ) = 4.95 × 10 −5 N Systems of Particles and Conservation of Momentum 551 (b) Calculate the ratio of the weight of a droplet to Fav: w mg = Fav Fav = (3 × 10 −5 )( ) kg 9.81 m/s 2 ≈ 6 4.95 × 10 −5 N Collisions in One Dimension *67 • Picture the Problem We can apply conservation of momentum to this perfectly inelastic collision to find the after-collision speed of the two cars. The ratio of the transformed kinetic energy to kinetic energy before the collision is the fraction of kinetic energy lost in the collision. (a) Letting V be the velocity of the two cars after their collision, apply conservation of momentum to their perfectly inelastic collision: Solve for and evaluate V: pinitial = pfinal or mv1 + mv2 = (m + m )V V= v1 + v2 30 m/s + 10 m/s = 2 2 = 20.0 m/s (b) Express the ratio of the kinetic energy that is lost to the kinetic energy of the two cars before the collision and simplify: ∆K K − K initial = final K initial K initial = = = Substitute numerical values to obtain: K final −1 K initial 1 2 1 2 (2m )V 2 2 mv12 + 1 mv2 2 −1 2V 2 −1 2 v12 + v2 2(20 m/s ) ∆K = −1 K initial (30 m/s )2 + (10 m/s )2 = −0.200 2 20% of the initial kinetic energy is transformed into heat, sound, and the deformation of metal. 552 Chapter 8 68 • Picture the Problem We can apply conservation of momentum to this perfectly inelastic collision to find the after-collision speed of the two players. Letting the subscript 1 refer to the running back and the subscript 2 refer to the linebacker, apply conservation of momentum to their perfectly inelastic collision: pi = pf or m1v1 = (m1 + m2 )V Solve for V: V= m1 v1 m1 + m2 Substitute numerical values and evaluate V: V= 85 kg (7 m/s) = 3.13 m/s 85 kg + 105 kg 69 • Picture the Problem We can apply conservation of momentum to this collision to find the after-collision speed of the 5-kg object. Let the direction the 5-kg object is moving before the collision be the positive direction. We can decide whether the collision was elastic by examining the initial and final kinetic energies of the system. pi = pf (a) Letting the subscript 5 refer to the 5-kg object and the subscript 2 refer to the 10-kg object, apply conservation of momentum to obtain: or m5vi,5 − m10 vi,10 = m5vf,5 Solve for vf,5: vf,5 = Substitute numerical values and evaluate vf,5: vf,5 = m5vi,5 − m10 vi,10 m5 (5 kg )(4 m/s) − (10 kg )(3 m/s) 5 kg = − 2.00 m/s where the minus sign means that the 5-kg object is moving to the left after the collision. Systems of Particles and Conservation of Momentum 553 (b) Evaluate ∆K for the collision: ∆K = K f − K i = 1 2 (5 kg )(2 m/s )2 − [1 (5 kg )(4 m/s )2 + 1 (10 kg )(3 m/s )2 ] = −75.0 J 2 2 Because ∆K ≠ 0, the collision was inelastic. 70 • Picture the Problem The pictorial representation shows the ball and bat just before and just after their collision. Take the direction the bat is moving to be the positive direction. Because the collision is elastic, we can equate the speeds of recession and approach, with the approximation that vi,bat ≈ vf,bat to find vf,ball. Express the speed of approach of the bat and ball: Because the mass of the bat is much greater than that of the ball: vf, bat − vf, ball = −(vi, bat − vi, ball ) vi,bat ≈ vf,bat Substitute to obtain: vf, bat − vf, ball = −(vf, bat − vi, ball ) Solve for and evaluate vf,ball: vf,ball = vf,bat + (vf,bat − vi,ball ) = −vi,ball + 2vf,bat = v + 2v = 3v *71 •• Picture the Problem Let the direction the proton is moving before the collision be the positive x direction. We can use both conservation of momentum and conservation of mechanical energy to obtain an expression for velocity of the proton after the collision. r r r (a) Use the expression for the total P = mi vi = Mvcm i momentum of a system to find vcm: ∑ and r vcm = = r mv p,i m + 12m 1 ˆ = 13 (300 m/s ) i (23.1m/s) iˆ 554 Chapter 8 (b) Use conservation of momentum to obtain one relation for the final velocities: Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach: To eliminate vnuc,f, solve equation (2) for vnuc,f, and substitute the result in equation (1): Solve for and evaluate vp,f: mp v p,i = mp v p,f + mnuc v nuc,f (1) v nuc,f − v p,f = −(v nuc,i − v p,i ) = v p,i (2) v nuc,f = v p,i + v p,f mp v p,i = mp v p,f + mnuc (v p,i + v p,f ) vp,f = = mp − mnuc mp + mnuc vp,i m − 12m (300 m/s) = − 254 m/s 13m 72 •• Picture the Problem We can use conservation of momentum and the definition of an elastic collision to obtain two equations in v2f and v3f that we can solve simultaneously. Use conservation of momentum to obtain one relation for the final velocities: Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach: Solve equation (2) for v3f , substitute in equation (1) to eliminate v3f, and solve for and evaluate v2f: Use equation (2) to find v3f: m3 v3i = m3 v3f + m2 v 2f (1) v 2f − v3f = −(v 2i − v3i ) = v3i v2f = (2) 2m3v3i 2(3 kg )(4 m/s ) = m2 + m3 2 kg + 3 kg = 4.80 m/s v3f = v2f − v3i = 4.80 m/s − 4.00 m/s = 0.800 m/s Evaluate Ki and Kf: 2 K i = K 3i = 1 m3v3i = 2 = 24.0 J 1 2 (3 kg )(4 m/s )2 Systems of Particles and Conservation of Momentum 555 and 2 2 K f = K 3f + K 2f = 1 m3v3f + 1 m2v2f 2 2 (3 kg )(0.8 m/s)2 2 + 1 (2 kg )(4.8 m/s ) 2 = 1 2 = 24.0 J Because K i = K f , we can conclude that the values obtained for v2f and v3f are consistent with the collision having been elastic. 73 •• Picture the Problem We can find the velocity of the center of mass from the definition of the total momentum of the system. We’ll use conservation of energy to find the maximum compression of the spring and express the initial (i.e., before collision) and final (i.e., at separation) velocities. Finally, we’ll transform the velocities from the center of mass frame of reference to the table frame of reference. r r r P = ∑ mi v i = Mv cm (a) Use the definition of the total momentum of a system to relate the initial momenta to the velocity of the center of mass: or Solve for vcm: vcm = Substitute numerical values and evaluate vcm: vcm = i m1v1i = (m1 + m2 ) vcm m1v1i + m2 v2i m1 + m2 (2 kg )(10 m/s) + (5 kg )(3 m/s) 2 kg + 5 kg = 5.00 m/s (b) Find the kinetic energy of the system at maximum compression (u1 = u2 = 0): 2 K = K cm = 1 Mvcm 2 = 1 2 (7 kg )(5 m/s)2 = 87.5 J ∆K + ∆U s = 0 Use conservation of energy to relate the kinetic energy of the system to the potential energy stored in the spring at maximum compression: or Because Kf = Kcm and Usi = 0: K cm − K i + 1 k (∆x ) = 0 2 K f − K i + U sf − U si = 0 2 556 Chapter 8 Solve for ∆x: ∆x = 2(K i − K cm ) k [ = 2 2 1 m1v12i + 1 m2v2i − K cm 2 2 k = 2 m1v12i + m2v2i − 2 K cm k Substitute numerical values and evaluate ∆x: ∆x = (2 kg )(10 m/s)2 + (5 kg )(3 m/s)2 − 2(87.5 J ) ⎤ = 1120 N/m ⎥ ⎦ 1120 N/m (c) Find u1i, u2i, and u1f for this elastic collision: 0.250 m u1i = v1i − vcm = 10 m/s − 5 m/s = 5 m/s, u 2i = v2i − vcm = 3 m/s − 5 m/s = −2 m/s, and u1f = v1f − vcm = 0 − 5 m/s = −5 m/s Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach and solve for u2f: Transform u1f and u2f to the table frame of reference: u2f − u1f = −(u2i − u1i ) and u2f = −u2i + u1i + u1f = −(− 2 m/s ) + 5 m/s − 5 m/s = 2 m/s v1f = u1f + vcm = −5 m/s + 5 m/s = 0 and v 2f = u 2f + vcm = 2 m/s + 5 m/s = 7.00 m/s *74 •• Picture the Problem Let the system include the earth, the bullet, and the sheet of plywood. Then Wext = 0. Choose the zero of gravitational potential energy to be where the bullet enters the plywood. We can apply both conservation of energy and conservation of momentum to obtain the various physical quantities called for in this problem. (a) Use conservation of mechanical energy after the bullet exits the sheet of plywood to relate its exit speed to the height to which it rises: ∆K + ∆U = 0 or, because Kf = Ui = 0, 2 − 1 mvm + mgh = 0 2 Systems of Particles and Conservation of Momentum 557 Solve for vm: vm = 2 gh Proceed similarly to relate the initial velocity of the plywood to the height to which it rises: vM = 2 gH r r pi = pf (b) Apply conservation of momentum to the collision of the bullet and the sheet of plywood: or Substitute for vm and vM and solve for vmi: vmi = (c) Express the initial mechanical energy of the system (i.e., just before the collision): Express the final mechanical energy of the system (i.e., when the bullet and block have reached their maximum heights): (d) Use the work-energy theorem with Wext = 0 to find the energy dissipated by friction in the inelastic collision: mvmi = mvm + MvM 2 gh + M m 2 gH 2 Ei = 1 mvmi 2 ⎡ 2M = mg ⎢h + m ⎢ ⎣ 2 ⎛M ⎞ ⎤ hH + ⎜ ⎟ H ⎥ ⎝m⎠ ⎥ ⎦ Ef = mgh + MgH = g (mh + MH ) Ef − Ei + Wfriction = 0 and Wfriction = Ei − Ef ⎡ h M ⎤ = gMH ⎢2 + − 1⎥ ⎣ H m ⎦ 75 •• Picture the Problem We can find the velocity of the center of mass from the definition of the total momentum of the system. We’ll use conservation of energy to find the speeds of the particles when their separation is least and when they are far apart. (a) Noting that when the distance between the two particles is least, both move at the same speed, namely vcm, use the definition of the total momentum of a system to relate the initial momenta to the velocity of r r r P = ∑ mi v i = Mv cm i or mp vpi = (mp + mα )vcm . 558 Chapter 8 the center of mass: Solve for and evaluate vcm: vcm = v' = mp vpi + mα vαi m1 + m2 = mv0 + 0 m + 4m = 0.200 v0 (b) Use conservation of momentum to obtain one relation for the final velocities: Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach: Solve equation (2) for vpf , substitute in equation (1) to eliminate vpf, and solve for vαf: mp v0 = mpvpf + mα vαf (1) vpf − vαf = −(vpi − vαi ) = −vpi (2) vαf = 2 m p v0 mp + mα = 2mv0 = 0.400 v0 m + 4m 76 • Picture the Problem Let the numeral 1 denote the electron and the numeral 2 the hydrogen atom. We can find the final velocity of the electron and, hence, the fraction of its initial kinetic energy that is transferred to the atom, by transforming to the center-ofmass reference frame, calculating the post-collision velocity of the electron, and then transforming back to the laboratory frame of reference. Express f, the fraction of the electron’s initial kinetic energy that is transferred to the atom: f = Ki − Kf K =1− f Ki Ki ⎛v m1v12f =1− = 1 − ⎜ 1f 2 ⎜v m1v1i ⎝ 1i 1 2 1 2 Find the velocity of the center of mass: vcm = ⎞ ⎟ ⎟ ⎠ 2 m1v1i m1 + m2 or, because m2 = 1840m1, vcm = Find the initial velocity of the electron in the center-of-mass reference frame: m1v1i 1 = v1i m1 + 1840m1 1841 u1i = v1i − vcm = v1i − 1 ⎞ ⎛ = ⎜1 − ⎟v1i ⎝ 1841 ⎠ 1 v1i 1841 (1) Systems of Particles and Conservation of Momentum 559 Find the post-collision velocity of the electron in the center-of-mass reference frame by reversing its velocity: ⎛ 1 ⎞ − 1⎟v1i u1f = −u1i = ⎜ ⎝ 1841 ⎠ To find the final velocity of the electron in the original frame, add vcm to its final velocity in the centerof-mass reference frame: ⎛ 2 ⎞ − 1⎟v1i v1f = u1f + vcm = ⎜ ⎝ 1841 ⎠ Substitute in equation (1) to obtain: ⎛⎛ 2 ⎞ ⎞ − 1⎟v1i ⎟ ⎜⎜ 2 1841 ⎠ ⎟ ⎛ 2 ⎞ = 1− ⎜ − 1⎟ f = 1− ⎜ ⎝ ⎟ ⎜ v1i ⎝ 1841 ⎠ ⎟ ⎜ ⎠ ⎝ 2 = 2.17 × 10 −3 = 0.217% 77 •• Picture the Problem The pictorial representation shows the bullet about to imbed itself in the bob of the ballistic pendulum and then, later, when the bob plus bullet have risen to their maximum height. We can use conservation of momentum during the collision to relate the speed of the bullet to the initial speed of the bob plus bullet (V). The initial kinetic energy of the bob plus bullet is transformed into gravitational potential energy when they reach their maximum height. Hence we apply conservation of mechanical energy to relate V to the angle through which the bullet plus bob swings and then solve the momentum and energy equations simultaneously for the speed of the bullet. Use conservation of momentum to relate the speed of the bullet just before impact to the initial speed of the bob plus bullet: mvb = (m + M )V Solve for the speed of the bullet: ⎛ M⎞ vb = ⎜ 1 + ⎟ V m⎠ ⎝ Use conservation of energy to relate ∆K + ∆U = 0 (1) 560 Chapter 8 or, because Kf = Ui = 0, the initial kinetic energy of the bullet to the final potential energy of the system: − Ki + U f = 0 − 1 (m + M )V 2 2 Substitute for Ki and Uf and solve for V: + (m + M )gL(1 − cos θ ) = 0 and V = 2 gL(1 − cos θ ) M⎞ ⎛ vb = ⎜1 + ⎟ 2 gL(1 − cos θ ) m⎠ ⎝ Substitute for V in equation (1) to obtain: Substitute numerical values and evaluate vb: ⎛ 1.5 kg ⎞ 2 vb = ⎜ 1 + ⎜ 0.016 kg ⎟ 2 9.81 m/s (2.3 m )(1 − cos60°) = 450 m/s ⎟ ⎝ ⎠ ( ) *78 •• Picture the Problem We can apply conservation of momentum and the definition of an elastic collision to obtain equations relating the initial and final velocities of the colliding objects that we can solve for v1f and v2f. Apply conservation of momentum to the elastic collision of the particles to obtain: m1v1f + m2 v2f = m1v1i + m2 v2i Relate the initial and final kinetic energies of the particles in an elastic collision: 1 2 Rearrange this equation and factor to obtain: 2 2 m2 v2f − v2i = m1 v12i − v12f 2 2 m1v12f + 1 m2 v2 f = 1 m1v12i + 1 m2 v2i 2 2 2 ( ) ( ) or m2 (v2 f − v2i )(v2f + v2i ) = m1 (v1i − v1f )(v1i + v1f ) Rearrange equation (1) to obtain: (1) m2 (v2f − v2i ) = m1 (v1i − v1f ) Divide equation (2) by equation (3) to obtain: v1f − v2f = v2i − v1i (3) v2 f + v2i = v1i + v1f Rearrange this equation to obtain equation (4): (2) Multiply equation (4) by m2 and add it to equation (1) to obtain: (4) (m1 + m2 )v1f = (m1 − m2 )v1i + 2m2v2i Systems of Particles and Conservation of Momentum 561 Solve for v1f to obtain: Multiply equation (4) by m1 and subtract it from equation (1) to obtain: Solve for v2f to obtain: v1 f = 2m2 m1 − m2 v1i + v2 i m1 + m2 m1 + m2 (m1 + m2 )v2f = (m2 − m1 )v2i + 2m1v1i v2 f = 2m1 m − m1 v1i + 2 v 2i m1 + m2 m1 + m2 Remarks: Note that the velocities satisfy the condition that v 2f − v1f = −(v 2i − v1i ) . This verifies that the speed of recession equals the speed of approach. 79 •• Picture the Problem As in this problem, Problem 78 involves an elastic, onedimensional collision between two objects. Both solutions involve using the conservation of momentum equation m1v1f + m2 v2 f = m1v1i + m2 v2i and the elastic collision equation v1f − v2 f = v2i − v1i . In part (a) we can simply set the masses equal to each other and substitute in the equations in Problem 78 to show that the particles "swap" velocities. In part (b) we can divide the numerator and denominator of the equations in Problem 78 by m2 and use the condition that m2 >> m1 to show that v1f ≈ −v1i+2v2i and v2f ≈ v2i. m1 − m2 2m2 v1i + v2i m1 + m2 m1 + m2 (1) v2f = (a) From Problem 78 we have: m − m1 2m1 v1i + 2 v 2i m1 + m2 m1 + m2 (2) v1f = 2m v2 i = v2 i m+m v1f = and Set m1 = m2 = m to obtain: and v2 f = 2m v1i = v1i m+m (b) Divide the numerator and denominator of both terms in equation (1) by m2 to obtain: m1 −1 2 m2 v1f = v1i + v2i m1 m1 +1 +1 m2 m2 If m2 >> m1: v1f ≈ −v1i +2v2i 562 Chapter 8 Divide the numerator and denominator of both terms in equation (2) by m2 to obtain: If m2 >> m1: 2 v2f = m1 m2 m1 +1 m2 1− v1i + m1 m2 m1 +1 m2 v 2i v2f ≈ v 2i Remarks: Note that, in both parts of this problem, the velocities satisfy the condition that v 2f − v1f = −(v 2i − v1i ) . This verifies that the speed of recession equals the speed of approach. Perfectly Inelastic Collisions and the Ballistic Pendulum 80 •• Picture the Problem Choose Ug = 0 at the bob’s equilibrium position. Momentum is conserved in the collision of the bullet with bob and the initial kinetic energy of the bob plus bullet is transformed into gravitational potential energy as it swings up to the top of the circle. If the bullet plus bob just makes it to the top of the circle with zero speed, it will swing through a complete circle. Use conservation of momentum to relate the speed of the bullet just before impact to the initial speed of the bob plus bullet: m1v = (m1 + m2 )V Solve for the speed of the bullet: ⎛ m ⎞ v = ⎜1 + 2 ⎟ V ⎜ m ⎟ 1 ⎠ ⎝ Use conservation of energy to relate the initial kinetic energy of the bob plus bullet to their potential energy at the top of the circle: ∆K + ∆U = 0 (1) or, because Kf = Ui = 0, − Ki + U f = 0 Substitute for Ki and Uf: − 1 (m1 + m2 )V 2 + (m1 + m2 )g (2 L ) = 0 2 Solve for V: V = gL Substitute for V in equation (1) and simplify to obtain: ⎛ m v = ⎜1 + 2 ⎜ m1 ⎝ ⎞ ⎟ gL ⎟ ⎠ Systems of Particles and Conservation of Momentum 563 *81 •• Picture the Problem Choose Ug = 0 at the equilibrium position of the ballistic pendulum. Momentum is conserved in the collision of the bullet with the bob and kinetic energy is transformed into gravitational potential energy as the bob swings up to its maximum height. Letting V represent the initial speed of the bob as it begins its upward swing, use conservation of momentum to relate this speed to the speeds of the bullet just before and after its collision with the bob: m1v = m1 ( 1 v ) + m2V 2 m1 v 2m 2 Solve for the speed of the bob: V = Use conservation of energy to relate the initial kinetic energy of the bob to its potential energy at its maximum height: ∆K + ∆U = 0 (1) or, because Kf = Ui = 0, − Ki + U f = 0 Substitute for Ki and Uf: − 1 m2V 2 + m2 gh = 0 2 Solve for h: h= Substitute V from equation (1) in equation (2) and simplify to obtain: ⎛ m1 ⎞ ⎜ 2 ⎜ 2m v ⎟ ⎟ 2 ⎝ 2 ⎠ = v ⎛ m1 ⎞ ⎜ ⎟ h= 2g 8 g ⎜ m2 ⎟ ⎝ ⎠ V2 2g (2) 2 82 • Picture the Problem Let the mass of the bullet be m, that of the wooden block M, the pre-collision velocity of the bullet v, and the post-collision velocity of the block+bullet be V. We can use conservation of momentum to find the velocity of the block with the bullet imbedded in it just after their perfectly inelastic collision. We can use Newton’s 2nd law to find the acceleration of the sliding block and a constant-acceleration equation to find the distance the block slides. 564 Chapter 8 0 = V 2 + 2a∆x Using a constant-acceleration equation, relate the velocity of the block+bullet just after their collision to their acceleration and displacement before stopping: because the final velocity of the block+bullet is zero. Solve for the distance the block slides before coming to rest: ∆x = − Use conservation of momentum to relate the pre-collision velocity of the bullet to the post-collision velocity of the block+bullet: V2 2a mv = (m + M )V Solve for V: V = Substitute in equation (1) to obtain: 1 ⎛ m ⎞ ∆x = − ⎜ v⎟ 2a ⎝ m + M ⎠ Apply r r F = ma to the ∑ block+bullet (see the FBD in the diagram): Use the definition of the coefficient of kinetic friction and equation (4) to obtain: (1) m v m+M ∑F x and ∑F y 2 (2) = − f k = (m + M ) a (3) =Fn − (m + M )g = 0 (4) f k = µ k Fn = µ k (m + M )g Substitute in equation (3): − µ k (m + M )g = (m + M ) a Solve for a to obtain: a = −µk g Substitute in equation (2) to obtain: ∆x = 1 ⎛ m ⎞ v⎟ ⎜ 2µ k g ⎝ m + M ⎠ 2 Systems of Particles and Conservation of Momentum 565 Substitute numerical values and evaluate ∆x: 1 ∆x = 2(0.22 ) 9.81 m/s 2 ( 2 ) ⎛ ⎞ 0.0105 kg ⎜ (750 m/s)⎟ = 0.130 m ⎜ 0.0105 kg + 10.5 kg ⎟ ⎝ ⎠ 83 •• Picture the Problem The collision of the ball with the box is perfectly inelastic and we can find the speed of the box-and-ball immediately after their collision by applying conservation of momentum. If we assume that the kinetic friction force is constant, we can use a constant-acceleration equation to find the acceleration of the box and ball combination and the definition of µk to find its value. Using its definition, express the coefficient of kinetic friction of the table: Use conservation of momentum to relate the speed of the ball just before the collision to the speed of the ball+box immediately after the collision: µk = f k (M + m ) a a = = Fn (M + m )g g MV = (m + M ) v Solve for v: v= Use a constant-acceleration equation to relate the sliding distance of the ball+box to its initial and final velocities and its acceleration: vf2 = vi2 + 2a∆x Solve for a: Substitute in equation (1) to obtain: Use equation (2) to eliminate v: (1) MV m+M (2) or, because vf = 0 and vi = v, 0 = v 2 + 2a∆x a=− v2 2∆x µk = v2 2 g∆x µk = 1 ⎛ MV ⎞ ⎜ ⎟ 2 g∆x ⎝ m + M ⎠ ⎛ ⎞ 1 ⎜ V ⎟ ⎜ ⎟ = 2 g∆x ⎜ m + 1 ⎟ ⎜ ⎟ ⎝M ⎠ 2 2 566 Chapter 8 Substitute numerical values and evaluate µk: 2 ⎛ ⎞ ⎜ ⎟ 1 ⎜ 1.3 m/s ⎟ = 0.0529 µk = 2 9.81 m/s 2 (0.52 m ) ⎜ 0.327 kg + 1 ⎟ ⎜ 0.425 kg ⎟ ⎝ ⎠ ( ) *84 •• Picture the Problem Jane’s collision with Tarzan is a perfectly inelastic collision. We can find her speed v1 just before she grabs Tarzan from conservation of energy and their speed V just after she grabs him from conservation of momentum. Their kinetic energy just after their collision will be transformed into gravitational potential energy when they have reached their greatest height h. Use conservation of energy to relate the potential energy of Jane and Tarzan at their highest point (2) to their kinetic energy immediately after Jane grabbed Tarzan: Solve for h to obtain: U 2 = K1 or mJ+T gh = 1 mJ+TV 2 2 V2 h= 2g Use conservation of momentum to relate Jane’s velocity just before she collides with Tarzan to their velocity just after their perfectly inelastic collision: mJ v1 = mJ+TV Solve for V: V = Apply conservation of energy to relate Jane’s kinetic energy at 1 to her potential energy at 0: K1 = U 0 (1) mJ v1 mJ+T or 1 2 mJ v12 = mJ gL (2) Systems of Particles and Conservation of Momentum 567 Solve for v1: v1 = 2 gL Substitute in equation (2) to obtain: V = Substitute in equation (1) and simplify: ⎛ mJ ⎞ 1 ⎛ mJ ⎞ ⎜ h= ⎜ m ⎟ 2 gL = ⎜ m ⎟ L ⎟ ⎜ ⎟ 2 g ⎝ J +T ⎠ ⎝ J +T ⎠ Substitute numerical values and evaluate h: ⎛ ⎞ 54 kg h=⎜ ⎜ 54 kg + 82 kg ⎟ (25 m ) = 3.94 m ⎟ ⎝ ⎠ mJ 2 gL mJ+T 2 2 2 Exploding Objects and Radioactive Decay 85 •• Picture the Problem This nuclear reaction is 4Be → 2α + 1.5×10−14 J. In order to conserve momentum, the alpha particles will have move in opposite directions with the same velocities. We’ll use conservation of energy to find their speeds. Letting E represent the energy released in the reaction, express conservation of energy for this process: ( ) 2 2 K α = 2 1 mα vα = E 2 Solve for vα: vα = E mα Substitute numerical values and evaluate vα: vα = 1.5 × 10 −14 J = 1.50 × 10 6 m/s − 27 6.68 × 10 kg 86 •• Picture the Problem This nuclear reaction is 5Li → α + p + 3.15 × 10−13 J. To conserve momentum, the alpha particle and proton must move in opposite directions. We’ll apply both conservation of energy and conservation of momentum to find the speeds of the proton and alpha particle. Use conservation of momentum in this process to express the alpha particle’s velocity in terms of the proton’s: pi = p f = 0 and 0 = mp vp − mα vα 568 Chapter 8 Solve for vα and substitute for mα to obtain: Letting E represent the energy released in the reaction, apply conservation of energy to the process: vα = mp mα vp = mp 4mp vp = 1 v p 4 K p + Kα = E or 1 2 2 2 mp vp + 1 mα vα = E 2 Substitute for vα: 1 2 2 mp vp + 1 mα (1 vp ) = E 2 4 Solve for vp and substitute for mα to obtain: vp = 32 E 32 E = 16mp + mα 16mp + 4mp Substitute numerical values and evaluate vp: vp = 32 3.15 × 10 −13 J 20 1.67 × 10 − 27 kg 2 ( ( ) ) = 1.74 × 10 7 m/s Use the relationship between vp and vα to obtain vα: vα = 1 vp = 4 1 4 (1.74 ×10 7 m/s ) = 4.34 × 106 m/s 87 ••• Picture the Problem The pictorial representation shows the projectile at its maximum elevation and is moving horizontally. It also shows the two fragments resulting from the explosion. We chose the system to include the projectile and the earth so that no external forces act to change the momentum of the system during the explosion. With this choice of system we can also use conservation of energy to determine the elevation of the projectile when it explodes. We’ll also find it useful to use constant-acceleration equations in our description of the motion of the projectile and its fragments. Systems of Particles and Conservation of Momentum 569 (a) Use conservation of momentum to relate the velocity of the projectile before its explosion to the velocities of its two parts after the explosion: The only way this equality can hold is if: Express v3 in terms of v0 and substitute for the masses to obtain: r r pi = pf r r r m3v 3 = m1v1 + m2 v 2 ˆ ˆ m3v3 i = m1v x1i + m1v y1 ˆ − m2 v y 2 ˆ j j m3v3 = m1v x1 and m1v y1 = m2 v y 2 vx1 = 3v3 = 3v0 cosθ = 3(120 m/s )cos30° = 312 m/s and v y1 = 2v y 2 Using a constant-acceleration equation with the downward direction positive, relate vy2 to the time it takes the 2-kg fragment to hit the ground: With Ug = 0 at the launch site, apply conservation of energy to the climb of the projectile to its maximum elevation: Solve for ∆y: Substitute numerical values and evaluate ∆y: Substitute in equation (2) and evaluate vy2: Substitute in equation (1) and evaluate vy1: (1) ∆y = v y 2 ∆t + 1 g (∆t ) 2 2 ∆y − 1 g (∆t ) 2 ∆t 2 vy2 = (2) ∆K + ∆U = 0 Because Kf = Ui = 0, − K i + U f = 0 or 2 − 1 m3v y 0 + m3 g∆y = 0 2 ∆y = ∆y = vy2 2 vy0 2g = (v0 sin 30°)2 2g [(120 m/s)sin30°] 2 ( 2 9.81 m/s 2 ( ) = 183.5 m ) 183.5 m − 1 9.81 m/s 2 (3.6 s ) 2 = 3.6 s = 33.3 m/s v y1 = 2(33.3 m/s ) = 66.6 m/s 2 570 Chapter 8 r Express v1 in vector form: r ˆ v1 = vx1i + v y1 ˆ j = (b) Express the total distance d traveled by the 1-kg fragment: Relate ∆x to v0 and the time-toexplosion: (312 m/s) iˆ + (66.6 m/s) ˆ j d = ∆x + ∆x' (3) ∆x = (v0 cos θ )(∆t exp ) (4) vy0 v0 sin θ g Using a constant-acceleration equation, express ∆texp: ∆texp = Substitute numerical values and evaluate ∆texp: ∆texp = Substitute in equation (4) and evaluate ∆x: ∆x = (120 m/s )(cos30°)(6.12 s ) = 636.5 m Relate the distance traveled by the 1-kg fragment after the explosion to the time it takes it to reach the ground: Using a constant-acceleration equation, relate the time ∆t′ for the 1-kg fragment to reach the ground to its initial speed in the y direction and the distance to the ground: Substitute to obtain the quadratic equation: Solve the quadratic equation to find ∆t′: Substitute in equation (3) and evaluate d: = g (120 m/s)sin30° = 6.12 s 9.81 m/s 2 ∆x' = vx1∆t' ∆y = v y1∆t' − 1 g (∆t' ) 2 2 (∆t' )2 − (13.6 s )∆t' − 37.4 s 2 = 0 ∆t′ = 15.9 s d = ∆x + ∆x' = ∆x + v x1∆t' = 636.5 m + (312 m/s )(15.9 s ) = 5.61 km Systems of Particles and Conservation of Momentum 571 (c) Express the energy released in the explosion: Find the kinetic energy of the fragments after the explosion: Eexp = ∆K = K f − K i (5) 2 K f = K1 + K 2 = 1 m1v12 + 1 m2v2 2 2 (1kg )(312 m/s)2 + (66.6 m/s)2 ] [ 2 + 1 (2 kg )(33.3 m/s ) 2 = 1 2 = 52.0 kJ Find the kinetic energy of the projectile before the explosion: 2 K i = 1 m3v3 = 1 m3 (v0 cos θ ) 2 2 2 = 1 2 (3 kg )[(120 m/s ) cos 30°] 2 = 16.2 kJ Substitute in equation (5) to determine the energy released in the explosion: Eexp = K f − K i = 52.0 kJ − 16.2 kJ = 35.8 kJ *88 ••• Picture the Problem This nuclear reaction is 9B → 2α + p + 4.4×10−14 J. Assume that the proton moves in the –x direction as shown in the figure. The sum of the kinetic energies of the decay products equals the energy released in the decay. We’ll use conservation of momentum to find the angle between the velocities of the proton and the alpha particles. Note that vα = vα ' . Express the energy released to the kinetic energies of the decay products: Solve for vα: K p + 2 Kα = Erel or 1 2 ( ) 2 2 mp vp + 2 1 mα vα = Erel 2 vα = 2 Erel − 1 mp vp 2 mα 572 Chapter 8 Substitute numerical values and evaluate vα: 1 (1.67 ×10−27 kg )(6 ×106 m/s) = 1.44 ×106 m/s 4.4 × 10 −14 J −2 6.68 × 10 −27 kg 6.68 × 10 −27 kg 2 vα = Given that the boron isotope was at rest prior to the decay, use conservation of momentum to relate the momenta of the decay products: Solve for θ : r r pf = pi = 0 ⇒ p xf = 0 ∴ 2(mα vα cos θ ) − mp vp = 0 or 2(4mp vα cos θ ) − mp vp = 0 ⎡ vp ⎤ ⎥ ⎣ 8vα ⎦ θ = cos −1 ⎢ ⎡ 6 × 106 m/s ⎤ = cos −1 ⎢ ⎥ = ±58.7° 6 ⎣ 8 1.44 ×10 m/s ⎦ ( Let θ′ equal the angle the velocities of the alpha particles make with that of the proton: ) θ' = ±(180° − 58.7°) = ± 121° Coefficient of Restitution 89 • Picture the Problem The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which the ball was dropped and the height to which it rebounded by using conservation of mechanical energy. vrec vapp Use its definition to relate the coefficient of restitution to the velocities of approach and recession: e= Letting Ug = 0 at the surface of the steel plate, apply conservation of energy to express the velocity of approach: ∆K + ∆U = 0 Because Ki = Uf = 0, Kf − Ui = 0 or 1 2 Solve for vapp: 2 mvapp − mghapp = 0 vapp = 2 ghapp Systems of Particles and Conservation of Momentum 573 In like manner, show that: vrec = 2 ghrec Substitute in the equation for e to obtain: e= Substitute numerical values and evaluate e: e= 2 ghrec 2 ghapp = hrec happ 2.5 m = 0.913 3m *90 • Picture the Problem The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which an object was dropped and the height to which it rebounded by using conservation of mechanical energy. vrec vapp Use its definition to relate the coefficient of restitution to the velocities of approach and recession: e= Letting Ug = 0 at the surface of the steel plate, apply conservation of energy to express the velocity of approach: ∆K + ∆U = 0 Because Ki = Uf = 0, Kf − Ui = 0 or 1 2 2 mvapp − mghapp = 0 Solve for vapp: vapp = 2 ghapp In like manner, show that: vrec = 2 ghrec Substitute in the equation for e to obtain: e= Find emin: emin = 173 cm = 0.825 254 cm Find emax: emax = 183 cm = 0.849 254 cm 2 ghrec 2 ghapp = hrec happ 574 Chapter 8 and 0.825 ≤ e ≤ 0.849 91 • Picture the Problem Because the rebound kinetic energy is proportional to the rebound height, the percentage of mechanical energy lost in one bounce can be inferred from knowledge of the rebound height. The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which an object was dropped and the height to which it rebounded by using conservation of mechanical energy. (a) We know, from conservation of energy, that the kinetic energy of an object dropped from a given height h is proportional to h: K α h. If, for each bounce of the ball, hrec = 0.8happ: 20% of its mechanical energy is lost. vrec vapp (b) Use its definition to relate the coefficient of restitution to the velocities of approach and recession: e= Letting Ug = 0 at the surface from which the ball is rebounding, apply conservation of energy to express the velocity of approach: ∆K + ∆U = 0 Because Ki = Uf = 0, Kf − Ui = 0 or 1 2 2 mvapp − mghapp = 0 Solve for vapp: vapp = 2 ghapp In like manner, show that: vrec = 2 ghrec Substitute in the equation for e to obtain: e= Substitute for hrec to obtain: happ 2 ghrec 2 ghapp = hrec happ e = 0.8 = 0.894 Systems of Particles and Conservation of Momentum 575 92 •• Picture the Problem Let the numeral 2 refer to the 2-kg object and the numeral 4 to the 4-kg object. Choose a coordinate system in which the direction the 2-kg object is moving before the collision is the positive x direction and let the system consist of the earth, the surface on which the objects slide, and the objects. Then we can use conservation of momentum to find the velocity of the recoiling 4-kg object. We can find the energy transformed in the collision by calculating the difference between the kinetic energies before and after the collision and the coefficient of restitution from its definition. r r pi = p f (a) Use conservation of momentum in one dimension to relate the initial and final momenta of the participants in the collision: or m2 v 2i = m4 v 4f − m2 v 2f Solve for and evaluate the final velocity of the 4-kg object: v4 f = = m2v2i + m2v2 f m4 (2 kg )(6 m/s + 1 m/s) = 4 kg Elost = K i − K f (b) Express the energy lost in terms of the kinetic energies before and after the collision: 2 = 1 m2 v2i − 2 = 1 2 [m (v 2 2 2i ( 1 2 3.50 m/s 2 2 m2 v2f + 1 m4 v4 f 2 ) 2 2 − v2 f − m4 v4 f ) Substitute numerical values and evaluate Elost: Elost = 1 2 [((2 kg ){(6 m/s) − (1m/s) })− (4 kg )(3.5 m/s) ] = 2 2 2 10.5 J (c) Use the definition of the coefficient of restitution: e= vrec v4f − v2f 3.5 m/s − (− 1 m/s ) = = = 0.750 vapp v2 i 6 m/s 93 •• Picture the Problem Let the numeral 2 refer to the 2-kg block and the numeral 3 to the 3-kg block. Choose a coordinate system in which the direction the blocks are moving before the collision is the positive x direction and let the system consist of the earth, the surface on which the blocks move, and the blocks. Then we can use conservation of momentum find the velocity of the 2-kg block after the collision. We can find the coefficient of restitution from its definition. 576 Chapter 8 r r pi = p f (a) Use conservation of momentum in one dimension to relate the initial and final momenta of the participants in the collision: or m2 v 2i + m3 v3i = m2 v 2 f + m3 v3f Solve for the final velocity of the 2-kg object: v2 f = m2v2i + m3v3i − m3v3f m2 Substitute numerical values and evaluate v2f: v2 f = (2 kg )(5 m/s) + (3 kg )(2 m/s − 4.2 m/s) = 2 kg (b) Use the definition of the coefficient of restitution: e= 1.70 m/s vrec v3f − v2 f 4.2 m/s − 1.7 m/s = = 5 m/s − 2 m/s vapp v2i − v3i = 0.833 Collisions in Three Dimensions *94 •• Picture the Problem We can use the definition of the magnitude of a vector and the definition of the dot product to establish the result called for in (a). In part (b) we can use the result of part (a), the conservation of momentum, and the definition of an elastic collision (kinetic energy is conserved) to show that the particles separate at right angles. r r (a) Find the dot product of B + C with itself: r r r Because A = B + C : Substitute to obtain: (b) Apply conservation of momentum to the collision of the particles: Form the dot product of each side of this equation with itself to obtain: r r r r (B + C ) ⋅ (B + C ) r r = B2 + C 2 + 2B ⋅ C ( )( r r2 r r r r A2 = B + C = B + C ⋅ B + C r r A2 = B 2 + C 2 + 2 B ⋅ C r r r p1 + p2 = P r r r r r r ( p1 + p2 ) ⋅ ( p1 + p2 ) = P ⋅ P or r r 2 p12 + p2 + 2 p1 ⋅ p2 = P 2 Apply the definition of an elastic collision to obtain: ) 2 p12 p2 P2 + = 2m 2m 2m (1) Systems of Particles and Conservation of Momentum 577 or 2 p12 + p2 = P 2 Subtract equation (1) from equation (2) to obtain: r r 2 p1 ⋅ p2 = 0 or (2) r r p1 ⋅ p2 = 0 i.e., the particles move apart along paths that are at right angles to each other. 95 • Picture the Problem Let the initial direction of motion of the cue ball be the positive x direction. We can apply conservation of energy to determine the angle the cue ball makes with the positive x direction and the conservation of momentum to find the final velocities of the cue ball and the eight ball. (a) Use conservation of energy to relate the velocities of the collision participants before and after the collision: This Pythagorean relationship tells r r r us that v ci , v cf , and v8 form a right triangle. Hence: (b) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision: Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: Solve these equations simultaneously to obtain: 1 2 2 2 2 mvci = 1 mvcf + 1 mv8 2 2 or 2 2 2 vci = vcf + v8 θ cf + θ 8 = 90° and θ cf = 60° r r pxi = pxf or mvci = mvcf cos θ cf + mv8 cos θ 8 r r pyi = pyf or 0 = mvcf sin θ cf + mv8 sin θ 8 vcf = 2.50 m/s and v8 = 4.33 m/s 578 Chapter 8 96 •• Picture the Problem We can find the final velocity of the object whose mass is M1 by using the conservation of momentum. Whether the collision was elastic can be decided by examining the difference between the initial and final kinetic energy of the interacting objects. r r pi = pf (a) Use conservation of momentum to relate the initial and final velocities of the two objects: or Simplify to obtain: r ˆ v0 i + v0 ˆ = 1 v0 i + v1f j 2 ˆ ( ) ( ) r ˆ ˆ mv0 i + 2m 1 v0 ˆ = 2m 1 v0 i + mv1f j 2 4 r r v1f = Solve for v1f : 1 2 ˆ v0 i + v0 ˆ j (b) Express the difference between the kinetic energy of the system before the collision and its kinetic energy after the collision: [M v + M v − M v − 2mv ] = m[v + 2v − v − 2v ] − 2( v )] = mv ∆E = K i − K f = K1i + K 2i − (K1f + K 2f ) = = 1 2 = 1 2 [mv m[v 2 1i + 2mv − mv 2 0 2 2 + 2 1 v0 − 5 v0 4 4 2 2i ( ) 2 1f 2 2f 1 16 2 0 2 1 1i 1 2 2 1i 1 2 1 16 2 2 2i 2 2i 2 1f 2 1 1f 2 − M 2 v2 f 2 2f 2 0 Because ∆E ≠ 0, the collision is inelastic. *97 •• Picture the Problem Let the direction of motion of the puck that is moving before the collision be the positive x direction. Applying conservation of momentum to the collision in both the x and y directions will lead us to two equations in the unknowns v1 and v2 that we can solve simultaneously. We can decide whether the collision was elastic by either calculating the system’s kinetic energy before and after the collision or by determining whether the angle between the final velocities is 90°. (a) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision: pxi = pxf or mv = mv1 cos 30° + mv2 cos 60° or v = v1 cos 30° + v2 cos 60° Systems of Particles and Conservation of Momentum 579 pyi = pyf Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: or 0 = mv1 sin 30° − mv2 sin 60° or 0 = v1 sin 30° − v2 sin 60° v1 = 1.73 m/s and v2 = 1.00 m/s Solve these equations simultaneously to obtain: r r (b) Because the angle between v1 and v 2 is 90°, the collision was elastic. 98 •• Picture the Problem Let the direction of motion of the object that is moving before the collision be the positive x direction. Applying conservation of momentum to the motion in both the x and y directions will lead us to two equations in the unknowns v2 and θ2 that we can solve simultaneously. We can show that the collision was elastic by showing that the system’s kinetic energy before and after the collision is the same. (a) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision: Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: pxi = pxf or 3mv0 = 5mv0 cos θ1 + 2mv2 cos θ 2 or 3v0 = 5v0 cos θ1 + 2v2 cos θ 2 pyi = pyf or 0 = 5mv0 sin θ1 − 2mv2 sin θ 2 or 0 = 5v0 sin θ1 − 2v2 sin θ 2 Note that if tanθ1 = 2, then: Substitute in the momentum equations to obtain: cosθ 1 = 1 and sin θ 1 = 5 3v0 = 5v0 1 + 2v2 cos θ 2 5 or v0 = v2 cos θ 2 and 2 5 580 Chapter 8 2 − 2v2 sin θ 2 5 0 = 5v0 or 0 = v0 − v2 sin θ 2 Solve these equations simultaneously for θ2 : θ 2 = tan −1 1 = 45.0° Substitute to find v2: v2 = (b) To show that the collision was elastic, find the before-collision and after-collision kinetic energies: v0 v0 = = cos θ 2 cos 45° 2v0 2 K i = 1 m(3v0 ) = 4.5mv0 2 2 and ( ) ( K f = 1 m 5v0 + 1 (2m ) 2v0 2 2 2 ) 2 2 = 4.5mv0 Because K i = K f , the collision is elastic. *99 •• Picture the Problem Let the direction of motion of the ball that is moving before the collision be the positive x direction. Let v represent the velocity of the ball that is moving before the collision, v1 its velocity after the collision and v2 the velocity of the initially-atrest ball after the collision. We know that because the collision is elastic and the balls have the same mass, v1 and v2 are 90° apart. Applying conservation of momentum to the collision in both the x and y directions will lead us to two equations in the unknowns v1 and v2 that we can solve simultaneously. Noting that the angle of deflection for the recoiling ball is 60°, use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision: Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: pxi = pxf or mv = mv1 cos 30° + mv2 cos 60° or v = v1 cos 30° + v2 cos 60° pyi = pyf or 0 = mv1 sin 30° − mv2 sin 60° or 0 = v1 sin 30° − v2 sin 60° Systems of Particles and Conservation of Momentum 581 Solve these equations simultaneously to obtain: v1 = 8.66 m/s and v 2 = 5.00 m/s 100 •• Picture the Problem Choose the coordinate system shown in the diagram below with the x-axis the axis of initial approach of the first particle. Call V the speed of the target particle after the collision. In part (a) we can apply conservation of momentum in the x and y directions to obtain two equations that we can solve simultaneously for tanθ. In part (b) we can use conservation of momentum in vector form and the elastic-collision equation to show that v = v0cosφ. (a) Apply conservation of momentum in the x direction to obtain: v0 = v cos φ + V cos θ (1) Apply conservation of momentum in the y direction to obtain: v sin φ = V sin θ (2) Solve equation (1) for Vcosθ : V cos θ = v0 − v cos φ (3) Divide equation (2) by equation (3) to obtain: V sin θ v sin φ = V cos θ v0 − v cos φ or tan θ = (b) Apply conservation of momentum to obtain: v sin φ v0 − v cos φ r r r v0 = v + V Draw the vector diagram representing this equation: Use the definition of an elastic 2 v0 = v 2 + V 2 582 Chapter 8 collision to obtain: If this Pythagorean condition is to hold, the third angle of the triangle must be a right angle and, using the definition of the cosine function: v = v0 cos φ Center-of-Mass Frame 101 •• Picture the Problem The total kinetic energy of a system of particles is the sum of the kinetic energy of the center of mass and the kinetic energy relative to the center of mass. The kinetic energy of a particle of mass m is related to momentum according to K = p 2 2m . Express the total kinetic energy of the system: Relate the kinetic energy relative to the center of mass to the momenta of the two particles: K = K rel + K cm K rel (1) p12 p12 p12 (m1 + m2 ) = + = 2m1 2m2 2m1m2 Express the kinetic energy of the center of mass of the two particles: (2 p1 )2 = 2 p12 K cm = 2(m1 + m2 ) m1 + m2 Substitute in equation (1) and simplify to obtain: p12 (m1 + m2 ) 2 p12 K= + 2m1 m2 m1 + m2 = In an elastic collision: 2 p12 ⎡ m12 + 6m1 m2 + m2 ⎤ ⎥ ⎢ 2 2 ⎣ m12 m2 + m1 m2 ⎦ Ki = Kf = = Simplify to obtain: 2 p12 ⎡ m12 + 6m1m2 + m2 ⎤ 2 2 ⎢ m12 m2 + m1m2 ⎥ ⎣ ⎦ 2 p'12 ⎡ m12 + 6m1m2 + m2 ⎤ 2 2 ⎢ m12 m2 + m1m2 ⎥ ⎣ ⎦ (p ) = ( p ) ' 2 1 1 2 ' ⇒ p1 = ± p1 and ' If p1 = + p1 , the particles do not collide. Systems of Particles and Conservation of Momentum 583 *102 •• Picture the Problem Let the numerals 3 and 1 denote the blocks whose masses are 3 kg r r mi v i = Mv cm to find the velocity of the center-ofand 1 kg respectively. We can use ∑ i mass of the system and simply follow the directions in the problem step by step. (a) Express the total momentum of this two-particle system in terms of the velocity of its center of mass: r r r r P = ∑ mi vi = m1v1 + m3v3 i r r = Mvcm = (m1 + m3 )vcm Solve for vcm : r r r m v + m1v1 vcm = 3 3 m3 + m1 Substitute numerical values and r evaluate vcm : r (3 kg )(− 5 m/s ) iˆ + (1 kg )(3 m/s) iˆ v cm = 3 kg + 1 kg r = (b) Find the velocity of the 3-kg block in the center of mass reference frame: Find the velocity of the 1-kg block in the center of mass reference frame: (c) Express the after-collision velocities of both blocks in the center of mass reference frame: (d) Transform the after-collision velocity of the 3-kg block from the center of mass reference frame to the original reference frame: Transform the after-collision velocity of the 1-kg block from the center of mass reference frame to the original reference frame: (e) Express Ki in the original frame of (− 3.00 m/s ) iˆ r r r ˆ ˆ u3 = v3 − vcm = (− 5 m/s ) i − (− 3 m/s ) i = (− 2.00 m/s) iˆ r r r ˆ ˆ u1 = v1 − vcm = (3 m/s ) i − (− 3 m/s ) i = (6.00 m/s) iˆ r' u3 = (2.00 m/s) iˆ and r' u1 = (− 6.00 m/s) iˆ r' r' r ˆ ˆ v3 = u3 + vcm = (2 m/s ) i + (− 3 m/s ) i = (− 1.00 m/s ) iˆ r r' r ˆ ˆ v1' = u1 + v cm = (− 6 m/s ) i + (− 3 m/s ) i = (− 9.00 m/s ) iˆ 2 K i = 1 m3v3 + 1 m1v12 2 2 584 Chapter 8 reference: [(3 kg )(5 m/s) + (1kg )(3 m/s) ] Substitute numerical values and evaluate Ki: Ki = Express Kf in the original frame of reference: K f = 1 m3v'32 + 1 m1v'12 2 2 Substitute numerical values and evaluate Kf: Kf = 1 2 2 2 = 42.0 J 1 2 [(3 kg )(1m/s) + (1kg )(9 m/s) ] 2 2 = 42.0 J 103 •• Picture the Problem Let the numerals 3 and 1 denote the blocks whose masses are 3 kg r r mi v i = Mv cm to find the velocity of the center-ofand 1 kg respectively. We can use ∑ i mass of the system and simply follow the directions in the problem step by step. (a) Express the total momentum of this two-particle system in terms of the velocity of its center of mass: r r r r P = ∑ mi vi = m3v3 + m5v5 i r r = Mvcm = (m3 + m5 ) vcm Solve for vcm : r r r m3v3 + m5v5 vcm = m3 + m5 Substitute numerical values and r evaluate vcm : r (3 kg )(− 5 m/s) iˆ + (5 kg )(3 m/s) iˆ v cm = 3 kg + 5 kg r = 0 (b) Find the velocity of the 3-kg block in the center of mass reference frame: Find the velocity of the 5-kg block in the center of mass reference frame: (c) Express the after-collision velocities of both blocks in the center of mass reference frame: r r r ˆ u3 = v3 − vcm = (− 5 m/s ) i − 0 = (− 5 m/s ) iˆ r r r ˆ u5 = v5 − vcm = (3 m/s ) i − 0 = (3 m/s ) iˆ r' u3 = (5 m/s) iˆ and Systems of Particles and Conservation of Momentum 585 ' u5 = 0.75 m/s r' r' r ˆ v3 = u3 + vcm = (5 m/s ) i + 0 (d) Transform the after-collision velocity of the 3-kg block from the center of mass reference frame to the original reference frame: (5 m/s) iˆ = r' r' r ˆ v5 = u5 + vcm = (− 3 m/s ) i + 0 Transform the after-collision velocity of the 5-kg block from the center of mass reference frame to the original reference frame: (− 3 m/s) iˆ = 2 2 K i = 1 m3v3 + 1 m5v5 2 2 (e) Express Ki in the original frame of reference: [(3 kg )(5 m/s) + (5 kg )(3 m/s) ] Substitute numerical values and evaluate Ki: Ki = Express Kf in the original frame of reference: K f = 1 m3v'32 + 1 m5v'52 2 2 1 2 2 2 = 60.0 J Substitute numerical values and evaluate Kf: Kf = 1 2 [(3 kg )(5 m/s) 2 + (5 kg )(3 m/s ) = 60.0 J 2 Systems With Continuously Varying Mass: Rocket Propulsion 104 •• Picture the Problem The thrust of a rocket Fth depends on the burn rate of its fuel dm/dt and the relative speed of its exhaust gases uex according to Fth = dm dt uex . Using its definition, relate the rocket’s thrust to the relative speed of its exhaust gases: Substitute numerical values and evaluate Fth: Fth = dm uex dt Fth = (200 kg/s )(6 km/s ) = 1.20 MN 586 Chapter 8 105 •• Picture the Problem The thrust of a rocket Fth depends on the burn rate of its fuel dm/dt and the relative speed of its exhaust gases uex according to Fth = dm dt uex . The final velocity vf of a rocket depends on the relative speed of its exhaust gases uex, its payload to initial mass ratio mf/m0 and its burn time according to vf = −uex ln (mf m0 ) − gt b . (a) Using its definition, relate the rocket’s thrust to the relative speed of its exhaust gases: Fth = dm uex dt Fth = (200 kg/s )(1.8 km/s ) = 360 kN Substitute numerical values and evaluate Fth: (b) Relate the time to burnout to the mass of the fuel and its burn rate: tb = mfuel 0.8m0 = dm / dt dm / dt Substitute numerical values and evaluate tb: tb = 0.8(30,000 kg ) = 120 s 200 kg/s (c) Relate the final velocity of a rocket to its initial mass, exhaust velocity, and burn time: ⎛m ⎞ vf = −uex ln⎜ f ⎟ − gtb ⎜m ⎟ ⎝ 0⎠ Substitute numerical values and evaluate vf: ( ) ⎛1⎞ vf = −(1.8 km/s ) ln⎜ ⎟ − 9.81 m/s 2 (120 s ) = 1.72 km/s ⎝5⎠ *106 •• Picture the Problem We can use the dimensions of thrust, burn rate, and acceleration to show that the dimension of specific impulse is time. Combining the definitions of rocket thrust and specific impulse will lead us to uex = gI sp . (a) Express the dimension of specific impulse in terms of the dimensions of Fth, R, and g: [I ] sp M⋅L 2 [F ] = th = T = T [R][g ] M ⋅ L T T2 (b) From the definition of rocket thrust we have: Fth = Ruex Solve for uex: uex = Fth R Systems of Particles and Conservation of Momentum 587 Substitute for Fth to obtain: uex = RgI sp (c) Solve equation (1) for Isp and substitute for uex to obtain: I sp = Fth Rg From Example 8-21 we have: R = 1.384×104 kg/s and Fth = 3.4×106 N Substitute numerical values and evaluate Isp: 3.4 × 106 N I sp = 1.384 × 104 kg/s 9.81m/s 2 R = gI sp ( (1) )( ) = 25.0 s *107 ••• Picture the Problem We can use the rocket equation and the definition of rocket thrust to show that τ 0 = 1+ a0 g . In part (b) we can express the burn time tb in terms of the initial and final masses of the rocket and the rate at which the fuel burns, and then use this equation to express the rocket’s final velocity in terms of Isp, τ0, and the mass ratio m0/mf. In part (d) we’ll need to use trial-and-error methods or a graphing calculator to solve the transcendental equation giving vf as a function of m0/mf. (a) Express the rocket equation: − mg + Ruex = ma From the definition of rocket thrust we have: Fth = Ruex Substitute to obtain: − mg + Fth = ma Solve for Fth at takeoff: Fth = m0 g + m0 a0 Divide both sides of this equation by m0g to obtain: Fth a = 1+ 0 m0 g g Because τ 0 = Fth /( m0 g ) : (b) Use equation 8-42 to express the final speed of a rocket that starts from rest with mass m0: Express the burn time in terms of the burn rate R (assumed constant): τ 0 = 1+ a0 g vf = uex ln m0 − gtb , mf where tb is the burn time. tb = m0 − mf m0 ⎛ mf ⎞ ⎜1 − ⎟ = R R ⎜ m0 ⎟ ⎝ ⎠ (1) 588 Chapter 8 Multiply tb by one in the form gT/gT and simplify to obtain: tb = gFth m0 ⎛ mf ⎞ ⎜1 − ⎟ gFth R ⎜ m0 ⎟ ⎝ ⎠ gm0 Fth ⎛ mf ⎞ ⎜1 − ⎟ Fth gR ⎜ m0 ⎟ ⎝ ⎠ I ⎛ m ⎞ = sp ⎜1 − f ⎟ τ 0 ⎜ m0 ⎟ ⎝ ⎠ = Substitute in equation (1): m0 gI sp ⎛ mf ⎞ ⎜1 − ⎟ − τ 0 ⎜ m0 ⎟ mf ⎝ ⎠ vf = uex ln uex = gI sp , From Problem 32 we have: where uex is the exhaust velocity of the propellant. Substitute and factor to obtain: vf = gI sp ln m0 gI sp ⎛ mf ⎞ ⎜1 − ⎟ − τ 0 ⎜ m0 ⎟ mf ⎝ ⎠ ⎡ ⎛ m ⎞ 1 ⎛ m ⎞⎤ = gI sp ⎢ln⎜ 0 ⎟ − ⎜1 − f ⎟⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎝ mf ⎠ τ 0 ⎝ m0 ⎠⎦ (c) A spreadsheet program to calculate the final velocity of the rocket as a function of the mass ratio m0/mf is shown below. The constants used in the velocity function and the formulas used to calculate the final velocity are as follows: Cell B1 B2 B3 D9 E8 1 2 3 4 5 6 7 8 9 Content/Formula 250 9.81 2 D8 + 0.25 $B$2*$B$1*(LOG(D8) − (1/$B$3)*(1/D8)) A B C Isp = 250 s g = 9.81 m/s^2 tau = 2 D Algebraic Form Isp g τ m0/mf ⎡ ⎛ m ⎞ 1 ⎛ m ⎞⎤ gI sp ⎢ln⎜ 0 ⎟ − ⎜1 − f ⎟⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎝ mf ⎠ τ 0 ⎝ m0 ⎠⎦ E mass ratio vf 2.00 1.252E+02 2.25 3.187E+02 Systems of Particles and Conservation of Momentum 589 10 11 12 2.50 2.75 3.00 4.854E+02 6.316E+02 7.614E+02 36 9.00 2.204E+03 37 9.25 2.237E+03 38 9.50 2.269E+03 39 9.75 2.300E+03 40 10.00 2.330E+03 41 725.00 7.013E+03 A graph of final velocity as a function of mass ratio is shown below. v f (km/s) 2 1 0 2 4 6 8 10 m 0/m f (d) Substitute the data given in part (c) in the equation derived in part (b) to obtain: ⎛ m 1 ⎛ m ⎞⎞ 7 km/s = 9.81 m/s 2 (250 s )⎜ ln 0 − ⎜1 − f ⎟ ⎟ ⎜ m 2 ⎜ m ⎟⎟ f 0 ⎠⎠ ⎝ ⎝ ( ) or 2.854 = ln x − 0.5 + Use trial-and-error methods or a graphing calculator to solve this transcendental equation for the root greater than 1: 0.5 where x = m0/mf. x x = 28.1 , a value considerably larger than the practical limit of 10 for single-stage rockets. 108 •• Picture the Problem We can use the velocity-at-burnout equation from Problem 106 to find vf and constant-acceleration equations to approximate the maximum height the rocket will reach and its total flight time. (a) Assuming constant acceleration, relate the maximum height reached 2 h = 1 gt top 2 (1) 590 Chapter 8 by the model rocket to its time-totop-of-trajectory: From Problem 106 we have: ⎛ ⎛ m ⎞ 1 ⎛ m ⎞⎞ vf = gI sp ⎜ ln⎜ 0 ⎟ − ⎜1 − f ⎟ ⎟ ⎜ ⎜ m ⎟ τ ⎜ m ⎟⎟ 0 ⎠⎠ ⎝ ⎝ ⎝ f⎠ Evaluate the velocity at burnout vf for Isp = 100 s, m0/mf = 1.2, and τ = 5: vf = 9.81 m/s 2 (100 s ) Assuming that the time for the fuel to burn up is short compared to the total flight time, find the time to the top of the trajectory: t top = vf 146 m/s = = 14.9 s g 9.81 m/s 2 Substitute in equation (1) and evaluate h: h= (9.81m/s )(14.9 s) (b) Find the total flight time from the time it took the rocket to reach its maximum height: tflight = 2t top = 2(14.9 s ) = 29.8 s (c) Express and evaluate the fuel burn time tb: ( ) ⎡ 1⎛ 1 ⎞⎤ × ⎢ln (1.2) − ⎜1 − ⎟ 5 ⎝ 1.2 ⎠⎥ ⎣ ⎦ = 146 m/s 1 2 2 I sp ⎛ m f ⎜1 − τ ⎜ m0 ⎝ = 3.33 s tb = 2 = 1.09 km ⎞ 100 s ⎛ 1 ⎞ ⎟= ⎜1 − ⎟ ⎟ 5 ⎝ 1.2 ⎠ ⎠ Because this burn time is approximately 1/5 of the total flight time, we can' t expect the answer we obtained in Part (b) to be very accurate. It should, however, be good to about 30% accuracy, as the maximum distance the model rocket could possibly move in this time is 1 vtb = 243 m, assuming 2 constant acceleration until burnout. General Problems 109 • Picture the Problem Let the direction of motion of the 250-g car before the collision be the positive x direction. Let the numeral 1 refer to the 250-kg car, the numeral 2 refer to the 400-kg car, and V represent the velocity of the linked cars. Let the system include the earth and the cars. We can use conservation of momentum to find their speed after they have linked together and the definition of kinetic energy to find their initial and final kinetic energies. Systems of Particles and Conservation of Momentum 591 Use conservation of momentum to relate the speeds of the cars immediately before and immediately after their collision: pix = pfx or m1v1 = (m1 + m2 )V Solve for V: V= m1v1 m1 + m2 Substitute numerical values and evaluate V: V = (0.250 kg )(0.50 m/s ) = Find the initial kinetic energy of the cars: K i = 1 m1v12 = 2 Find the final kinetic energy of the coupled cars: Kf = 0.250 kg + 0.400 kg 1 2 0.192 m/s (0.250 kg )(0.50 m/s )2 = 31.3 mJ (m1 + m2 )V 2 2 = 1 (0.250 kg + 0.400 kg )(0.192 m/s ) 2 1 2 = 12.0 mJ 110 • Picture the Problem Let the direction of motion of the 250-g car before the collision be the positive x direction. Let the numeral 1 refer to the 250-kg car and the numeral 2 refer to the 400-g car and the system include the earth and the cars. We can use conservation of momentum to find their speed after they have linked together and the definition of kinetic energy to find their initial and final kinetic energies. (a) Express and evaluate the initial kinetic energy of the cars: (b) Relate the velocity of the center of mass to the total momentum of the system: K i = 1 m1v12 = 2 1 2 (0.250 kg )(0.50 m/s)2 = 31.3 mJ r r r P = ∑ mi v i = mv cm i Solve for vcm: vcm = Substitute numerical values and evaluate vcm: vcm = m1v1 + m2 v2 m1 + m2 (0.250 kg )(0.50 m/s) = 0.192 m/s 0.250 kg + 0.400 kg 592 Chapter 8 Find the initial velocity of the 250-g car relative to the velocity of the center of mass: Find the initial velocity of the 400-g car relative to the velocity of the center of mass: Express the initial kinetic energy of the system relative to the center of mass: Substitute numerical values and evaluate Ki,rel: u1 = v1 − vcm = 0.50 m/s − 0.192 m/s = 0.308 m/s u 2 = v2 − vcm = 0 m/s − 0.192 m/s = − 0.192 m/s 2 K i,rel = 1 m1u12 + 1 m2u 2 2 2 K i,rel = (0.250 kg )(0.308 m/s )2 2 + 1 (0.400 kg )(− 0.192 m/s ) 2 1 2 = 19.2 mJ (c) Express the kinetic energy of the center of mass: Substitute numerical values and evaluate Kcm: (d) Relate the initial kinetic energy of the system to its initial kinetic energy relative to the center of mass and the kinetic energy of the center of mass: 2 K cm = 1 Mvcm 2 K cm = 1 2 (0.650 kg )(0.192 m/s )2 = 12.0 mJ K i = K i,rel + K cm = 19.2 mJ + 12.0 mJ = 31.2 mJ ∴ K i = K i,rel + K cm *111 • Picture the Problem Let the direction the 4-kg fish is swimming be the positive x direction and the system include the fish, the water, and the earth. The velocity of the larger fish immediately after its lunch is the velocity of the center of mass in this perfectly inelastic collision. Relate the velocity of the center of mass to the total momentum of the system: r r r P = ∑ mi v i = mv cm i Systems of Particles and Conservation of Momentum 593 Solve for vcm: Substitute numerical values and evaluate vcm: m4 v4 − m1.2 v1.2 m4 + m1.2 vcm = vcm = (4 kg )(1.5 m/s) − (1.2kg) (3 m/s) 4 kg + 1.2 kg = 0.462 m/s 112 • Picture the Problem Let the direction the 3-kg block is moving be the positive x direction and include both blocks and the earth in the system. The total kinetic energy of the two-block system is the sum of the kinetic energies of the blocks. We can relate the momentum of the system to the velocity of its center of mass and use this relationship to find vcm. Finally, we can use the definition of kinetic energy to find the kinetic energy relative to the center of mass. (a) Express the total kinetic energy of the system in terms of the kinetic energy of the blocks: Substitute numerical values and evaluate Ktot: (b) Relate the velocity of the center of mass to the total momentum of the system: 2 2 K tot = 1 m3v3 + 1 m6 v6 2 2 K tot = 1 2 (3 kg )(6 m/s )2 + 1 (6 kg )(3 m/s)2 2 = 81.0 J r r r P = ∑ mi v i = mv cm i Solve for vcm: vcm = Substitute numerical values and evaluate vcm: vcm = m3v3 + m6 v6 m1 + m2 (3 kg )(6 m/s) + (6 kg )(3 m/s) 3 kg + 6 kg = 4.00 m/s (c) Find the center of mass kinetic energy from the velocity of the center of mass: 2 K cm = 1 Mvcm = 2 = 72.0 J 1 2 (9 kg )(4 m/s)2 594 Chapter 8 K rel = K tot − K cm (d) Relate the initial kinetic energy of the system to its initial kinetic energy relative to the center of mass and the kinetic energy of the center of mass: = 81.0 J − 72.0 J = 9.00 J 113 • Picture the Problem Let east be the positive x direction and north the positive y direction. Include both cars and the earth in the system and let the numeral 1 denote the 1500-kg car and the numeral 2 the 2000-kg car. Because the net external force acting on the system is zero, momentum is conserved in this perfectly inelastic collision. r r r r r p = p1 + p2 = m1v1 + m2 v 2 ˆ =mv ˆ−m v i j (a) Express the total momentum of the system: 1 1 2 2 r Substitute numerical values and evaluate p : r ˆ p = (1500 kg )(70 km/h ) ˆ − (2000 kg )(55 km/h ) i j ( ) ( ) ˆ = − 1.10 ×105 kg ⋅ km/h i + 1.05 × 105 kg ⋅ km/h ˆ j r r r p v f = v cm = M (b) Express the velocity of the wreckage in terms of the total momentum of the system: r Substitute numerical values and evaluate v f : ( ) ( ) ˆ r − 1.10 × 105 kg ⋅ km/h i 1.05 × 105 kg ⋅ km/h ˆ j vf = + 1500 kg + 2000 kg 1500 kg + 2000 kg ˆ = −(31.4 km/h ) i + (30.0 km/h ) ˆ j (31.4 km/h )2 + (30.0 km/h )2 Find the magnitude of the velocity of the wreckage: vf = Find the direction of the velocity of the wreckage: θ = tan −1 ⎢ = 43.4 km/h ⎡ 30.0 km/h ⎤ ⎥ = −43.7° ⎣ − 31.4 km/h ⎦ The direction of the wreckage is 46.3° west of north. Systems of Particles and Conservation of Momentum 595 *114 •• Picture the Problem Take the origin to be at the initial position of the right-hand end of raft and let the positive x direction be to the left. Let ″w″ denote the woman and ″r″ the raft, d be the distance of the end of the raft from the pier after the woman has walked to its front. The raft moves to the left as the woman moves to the right; with the center of mass of the woman-raft system remaining fixed (because Fext,net = 0). The diagram shows the initial (xw,i) and final (xw,f) positions of the woman as well as the initial (xr_cm,i) and final (xr_cm,f) positions of the center of mass of the raft both before and after the woman has walked to the front of the raft. CM x × xr_cm,i x w i =6 m , xC M 0 0.5 m CM x × xr_cm,f xr_cm,i 0 xw f , (a) Express the distance of the raft from the pier after the woman has walked to the front of the raft: d d = 0.5 m + xf,w Express xcm before the woman has walked to the front of the raft: xcm = Express xcm after the woman has walked to the front of the raft: xcm = Because Fext,net = 0, the center of mass remains fixed and we can equate these two expressions for xcm to obtain: P I E R (1) mw xw,i + mr xr_cm, i m w + mr mw xw,f + mr xr_cm,f m w + mr mw xw ,i + mr xr_cm,i = mw xw,f + mr xr_cm,f Solve for xw,f: xw,f = xw,i − From the figure it can be seen that xr_cm,f – xr_cm,i = xw,f. Substitute xw,f xw,f = mr (xr_cm,f − xr_cm,i ) mw mw xw,i m w + mr 596 Chapter 8 for xr_cm,f – xr_cm,i and to obtain: (60 kg )(6 m ) Substitute numerical values and evaluate xw,f: xw,f = Substitute in equation (1) to obtain: d = 2.00 m + 0.5 m = 2.50 m (b) Express the total kinetic energy of the system: Noting that the elapsed time is 2 s, find vw and vr: 60 kg + 120 kg = 2.00 m 2 K tot = 1 mw vw + 1 mr vr2 2 2 vw = x w,f − x w,i ∆t = 2m − 6m = −2 m/s 2s relative to the dock, and vr = xr,f − xr,i ∆t = 2.50 m − 0.5 m = 1 m/s , 2s also relative to the dock. Substitute numerical values and evaluate Ktot: K tot = (60 kg )(− 2 m/s)2 2 + 1 (120 kg )(1 m/s ) 2 1 2 = 180 J Evaluate K with the raft tied to the pier: 2 K tot = 1 mw vw = 2 1 2 (60 kg )(3 m/s )2 = 270 J All the kinetic energy derives from the chemical energy of the woman and, (c) assuming she stops via static friction, the kinetic energy is transformed into her internal energy. After the shot leaves the woman' s hand, the raft - woman system constitutes an inertial reference frame. In that frame the shot has the same initial (d) velocity as did the shot that had a range of 6 m in the reference frame of the land. Thus, in the raft - woman frame, the shot also has a range of 6 m and lands at the front of the raft. Systems of Particles and Conservation of Momentum 597 115 •• Picture the Problem Let the zero of gravitational potential energy be at the elevation of the 1-kg block. We can use conservation of energy to find the speed of the bob just before its perfectly elastic collision with the block and conservation of momentum to find the speed of the block immediately after the collision. We’ll apply Newton’s 2nd law to find the acceleration of the sliding block and use a constant-acceleration equation to find how far it slides before coming to rest. ∆K + ∆U = 0 (a) Use conservation of energy to find the speed of the bob just before or its collision with the block: K − K +U −U = 0 f Because Ki = Uf = 0: 1 2 i f i 2 mball vball + mball g∆h = 0 and vball = 2 g∆h Substitute numerical values and evaluate vball: ( ) vball = 2 9.81 m/s 2 (2 m ) = 6.26 m/s Because the collision is perfectly elastic and the ball and block have the same mass: vblock = vball = 6.26 m/s (b) Using a constant-acceleration equation, relate the displacement of the block to its acceleration and initial speed and solve for its displacement: vf2 = vi2 + 2ablock ∆x Apply r r ∑ F = ma to the sliding block: Since vf = 0, ∆x = − vi2 − v2 = block 2ablock 2ablock ∑F = − f k = mablock x and ∑F y = Fn − mblock g = 0 Using the definition of fk (µkFn) eliminate fk and Fn between the two equations and solve for ablock: ablock = − µ k g Substitute for ablock to obtain: ∆x = 2 − vblock v2 = block − 2µ k g 2µ k g 598 Chapter 8 Substitute numerical values and evaluate ∆x: ∆x = (6.26 m/s)2 = 2(0.1) (9.81 m/s 2 ) 20.0 m *116 •• Picture the Problem We can use conservation of momentum in the horizontal direction to find the recoil velocity of the car along the track after the firing. Because the shell will neither rise as high nor be moving as fast at the top of its trajectory as it would be in the absence of air friction, we can apply the work-energy theorem to find the amount of thermal energy produced by the air friction. (a) No. The vertical reaction force of the rails is an external force and so the momentum of the system will not be conserved. ∆p x = 0 (b) Use conservation of momentum in the horizontal (x) direction to obtain: or Solve for and evaluate vrecoil: vrecoil = Substitute numerical values and evaluate vrecoil: vrecoil = mv cos 30° − Mvrecoil = 0 mv cos 30° M (200 kg )(125 m/s)cos30° 5000 kg = 4.33 m/s (c) Using the work-energy theorem, relate the thermal energy produced by air friction to the change in the energy of the system: Substitute for ∆U and ∆K to obtain: Wext = Wf = ∆Esys = ∆U + ∆K Wext = mgyf − mgyi + 1 mvf2 − 1 mvi2 2 2 ( = mg ( yf − yi ) + 1 m vf2 − vi2 2 ) Substitute numerical values and evaluate Wext: ( ) [ Wext = (200 kg ) 9.81 m/s 2 (180 m ) + 1 (200 kg ) (80 m/s ) − (125 m/s ) = − 569 kJ 2 2 2 Systems of Particles and Conservation of Momentum 599 117 •• Picture the Problem Because this is a perfectly inelastic collision, the velocity of the block after the collision is the same as the velocity of the center of mass before the collision. The distance the block travels before hitting the floor is the product of its velocity and the time required to fall 0.8 m; which we can find using a constantacceleration equation. Relate the distance D to the velocity of the center of mass and the time for the block to fall to the floor: Relate the velocity of the center of mass to the total momentum of the system and solve for vcm: D = vcm ∆t r r r P = ∑ mi v i = Mv cm i and vcm = Substitute numerical values and evaluate vcm: Using a constant-acceleration equation, find the time for the block to fall to the floor: vcm = mbullet vbullet + mblock vblock mbullet + mblock (0.015 kg )(500 m/s) = 9.20 m/s 0.015 kg + 0.8 kg ∆y = v0 ∆t + 1 a(∆t ) 2 2 Because v0 = 0, ∆t = 2∆y g 2∆y g Substitute to obtain: D = vcm Substitute numerical values and evaluate D: D = (9.20 m/s ) 2(0.8 m ) = 3.72 m 9.81 m/s 2 118 •• Picture the Problem Let the direction the particle whose mass is m is moving initially be the positive x direction and the direction the particle whose mass is 4m is moving r initially be the negative y direction. We can determine the impulse delivered by F and, hence, the change in the momentum of the system from the change in the momentum of r the particle whose mass is m. Knowing ∆p , we can express the final momentum of the particle whose mass is 4m and solve for its final velocity. Express the impulse delivered by the r force F : r r r r r I = FT = ∆p = pf − pi ˆ ˆ ˆ = m(4v ) i − mv i = 3mv i 600 Chapter 8 r Express p' 4 m : r Solve for v ' : r r r r p' 4m = 4mv ' = p4 m (0 ) + ∆p ˆ = −4mv ˆ + 3mv i j r v' = 3 4 ˆ vi v ˆ j 119 •• Picture the Problem Let the numeral 1 refer to the basketball and the numeral 2 to the baseball. The left-hand side of the diagram shows the balls after the basketball’s elastic collision with the floor and just before they collide. The right-hand side of the diagram shows the balls just after their collision. We can apply conservation of momentum and the definition of an elastic collision to obtain equations relating the initial and final velocities of the masses of the colliding objects that we can solve for v1f and v2f. (a) Because both balls are in freefall, and both are in the air for the same amount of time, they have the same velocity just before the basketball rebounds. After the basketball rebounds elastically, its velocity will have the same magnitude, but the opposite direction than just before it hit the ground. The velocity of the basketball will be equal in magnitude but opposite in direction to the velocity of the baseball. (b) Apply conservation of momentum to the collision of the balls to obtain: m1v1f + m2 v2f = m1v1i + m2 v2i Relate the initial and final kinetic energies of the balls in their elastic collision: 1 2 Rearrange this equation and factor to obtain: 2 2 m2 v2f − v2i = m1 v12i − v12f 2 2 m1v12f + 1 m2 v2 f = 1 m1v12i + 1 m2 v2i 2 2 2 ( ) ( ) or m2 (v2 f − v2i )(v2 f + v2i ) = m1 (v1i − v1f )(v1i + v1f ) Rearrange equation (1) to obtain: Divide equation (2) by equation (3) to obtain: (1) m2 (v2f − v2i ) = m1 (v1i − v1f ) v2 f + v2i = v1i + v1f (2) (3) Systems of Particles and Conservation of Momentum 601 Rearrange this equation to obtain equation (4): Multiply equation (4) by m2 and add it to equation (1) to obtain: Solve for v1f to obtain: v1f − v2f = v2i − v1i (4) (m1 + m2 )v1f = (m1 − m2 )v1i + 2m2v2i v1f = m1 − m2 2m2 v1i + v2 i m1 + m2 m1 + m2 or, because v2i = −v1i, v1f = = For m1 = 3m2 and v1i = v: (c) Multiply equation (4) by m1 and subtract it from equation (1) to obtain: Solve for v2f to obtain: v1f = 2m2 m1 − m2 v1i − v1i m1 + m2 m1 + m2 m1 − 3m2 v1i m1 + m2 3m2 − 3m2 v= 0 3m2 + m2 (m1 + m2 )v2f = (m2 − m1 )v2i + 2m1v1i v2 f = 2m1 m − m1 v 2i v1i + 2 m1 + m2 m1 + m2 or, because v2i = −v1i, v2 f = = For m1 = 3m2 and v1i = v: v2 f = 2m1 m − m1 v1i − 2 v1i m1 + m2 m1 + m2 3m1 − m2 v1i m1 + m2 3(3m2 ) − m2 v = 2v 3m2 + m2 602 Chapter 8 120 ••• Picture the Problem In Problem 119 only two balls are dropped. They collide head on, each moving at speed v, and the collision is elastic. In this problem, as it did in Problem 119, the solution involves using the conservation of momentum equation m1v1f + m2 v2f = m1v1i + m2 v2i and the elastic collision equation v1f − v2 f = v2i − v1i , where the numeral 1 refers to the baseball, and the numeral 2 to the top ball. The diagram shows the balls just before and just after their collision. From Problem 119 we know that that v1i = 2v and v2i = −v. m1 − m2 2m2 v1i + v2 i m1 + m2 m1 + m2 (a) Express the final speed v1f of the baseball as a function of its initial speed v1i and the initial speed of the top ball v2i (see Problem 78): v1f = Substitute for v1i and , v2i to obtain: m1 − m2 (2v ) + 2m2 (− v ) m1 + m2 m1 + m2 m1 −1 m2 (2v ) + m 2 (− v ) v1f = m1 1 +1 +1 m2 m2 Divide the numerator and denominator of each term by m2 to introduce the mass ratio of the upper ball to the lower ball: Set the final speed of the baseball v1f equal to zero, let x represent the mass ratio m1/m2, and solve for x: v1f = 0= x −1 (2v ) + 2 (− v ) x +1 x +1 and x= m1 1 = m2 2 (b) Apply the second of the two equations in Problem 78 to the collision between the top ball and the baseball: v2 f = 2m1 m − m1 v1i + 2 v 2i m1 + m2 m1 + m2 Substitute v1i = 2v and are given that v2i = −v to obtain: v2 f = 2m1 (2v ) + m2 − m1 (− v ) m1 + m2 m1 + m2 Systems of Particles and Conservation of Momentum 603 v3f = 2(2m1 ) (2v ) − 2m1 − m1 v m1 + 2m1 m1 + 2m1 = In part (a) we showed that m2 = 2m1. Substitute and simplify: 4m1 (2v ) − m1 v = 8 v − 1 v 3 3 3m1 3m1 = 7 3 v *121 •• Picture the Problem Let the direction the probe is moving after its elastic collision with Saturn be the positive direction. The probe gains kinetic energy at the expense of the kinetic energy of Saturn. We’ll relate the velocity of approach relative to the center of mass to urec and then to v. (a) Relate the velocity of recession to the velocity of recession relative to the center of mass: Find the velocity of approach: v = u rec + vcm uapp = −9.6 km/s − 10.4 km/s = −20.0 km/s Relate the relative velocity of approach to the relative velocity of recession for an elastic collision: Because Saturn is so much more massive than the space probe: Substitute and evaluate v: u rec = −uapp = 20.0 km/s vcm = vSaturn = 9.6 km/s v = urec + vcm = 20 km/s + 9.6 km/s = 29.6 km/s (b) Express the ratio of the final kinetic energy to the initial kinetic energy: 2 1 ⎛v ⎞ Kf 2 Mvrec = 1 = ⎜ rec ⎟ 2 ⎜ v ⎟ Ki ⎝ i ⎠ 2 Mvi 2 2 ⎛ 29.6 km/s ⎞ =⎜ ⎟ ⎜ 10.4 km/s ⎟ = 8.10 ⎠ ⎝ The energy comes from an immeasurably small slowing of Saturn. 604 Chapter 8 *122 •• Picture the Problem We can use the relationships P = c∆m and ∆E = ∆mc 2 to show that P = ∆E c . We can then equate this expression with the change in momentum of the flashlight to find the latter’s final velocity. (a) Express the momentum of the mass lost (i.e., carried away by the light) by the flashlight: P = c∆m Relate the energy carried away by the light to the mass lost by the flashlight: ∆m = ∆E c2 P=c ∆E ∆E = 2 c c Substitute to obtain: (b) Relate the final momentum of the flashlight to ∆E: ∆E = ∆p = mv c because the flashlight is initially at rest. ∆E mc Solve for v: v= Substitute numerical values and evaluate v: 1.5 × 103 J v= (1.5 kg ) 2.998 × 108 m/s ( ) = 3.33 × 10 −6 m/s = 3.33 µm/s 123 • Picture the Problem We can equate the change in momentum of the block to the momentum of the beam of light and relate the momentum of the beam of light to the mass converted to produce the beam. Combining these expressions will allow us to find the speed attained by the block. Relate the change in momentum of the block to the momentum of the beam: Express the momentum of the mass converted into a well-collimated beam of light: Substitute to obtain: Solve for v: (M − m )v = Pbeam because the block is initially at rest. Pbeam = mc (M − m )v = mc v= mc M −m Systems of Particles and Conservation of Momentum 605 Substitute numerical values and evaluate v: v= (0.001kg )(2.998 × 108 m/s) 1kg − 0.001 kg = 3.00 × 105 m/s 124 •• Picture the Problem Let the origin of the coordinate system be at the end of the boat at which your friend is sitting prior to changing places. If we let the system include you and your friend, the boat, the water and the earth, then Fext,net = 0 and the center of mass is at the same location after you change places as it was before you shifted. Express the center of mass of the system prior to changing places: xcm = = Substitute numerical values and simplify to obtain an expression for xcm in terms of m: xcm = = mboat xboat + myou xyou + mxfriend mboat + myou + mfriend xyou (mboat + myou ) + mxfriend mboat + myou + m (2 m )(60 kg + 80 kg ) + (0) m 60 kg + 80 kg + m 280 kg ⋅ m 140 kg + m Find the center of mass of the system after changing places: x'cm = mboat xboat + myou xyou + mxfriend mboat + myou + mfriend = (mboat + m )(2 m ± 0.2 m ) mboat + myou + m + myou (± 0.2 m ) mboat + myou + m Substitute numerical values and simplify to obtain: x'cm = (60 kg + m )(2 m ± 0.2 m ) 60 kg + 80 kg + m + + (80 kg )(± 0.2 m ) 60 kg + 80 kg + m (2 m )m ± 0.2m m ± 16 kg ⋅ m 140 kg + m Because Fext,net = 0, x'cm = xcm . 120 kg ⋅ m ± 12 kg ⋅ m 140 kg + m m= (160 ± 28) kg (2 ± 0.2) m= (160 + 28) kg = (2 − 0.2) Equate the two expressions and solve for m to obtain: Calculate the largest possible mass for your friend: = 104 kg 606 Chapter 8 Calculate the smallest possible mass for your friend: m= (160 − 28) kg = (2 + 0.2) 60.0 kg 125 •• Picture the Problem Let the system include the woman, both vehicles, and the earth. Then Fext,net = 0 and acm = 0. Include the mass of the man in the mass of the truck. We can use Newton’s 2nd and 3rd laws to find the acceleration of the truck and net force acting on both the car and the truck. (a) Relate the action and reaction forces acting on the car and truck: Fcar = Ftruck or mcar acar = mtruck+ woman atruck Solve for the acceleration of the truck: Substitute numerical values and evaluate atruck: (b) Apply Newton’s 2nd law to either vehicle to obtain: Substitute numerical values and evaluate Fnet: atruck = atruck mcar acar mtruck + woman (800 kg )(1.2 m/s 2 ) = = 1600 kg 0.600 m/s 2 Fnet = mcar acar ( ) Fnet = (800 kg ) 1.2 m/s 2 = 960 N 126 •• Picture the Problem Let the system include the block, the putty, and the earth. Then Fext,net = 0 and momentum is conserved in this perfectly inelastic collision. We’ll use conservation of momentum to relate the after-collision velocity of the block plus blob and conservation of energy to find their after-collision velocity. Noting that, because this is a perfectly elastic collision, the final velocity of the block plus blob is the velocity of the center of mass, use conservation of momentum to relate the velocity of the center of mass to the velocity of the glob before the collision: pi = p f or mgl vgl = Mvcm where M = mgl + mbl. Systems of Particles and Conservation of Momentum 607 Solve for vgl to obtain: Use conservation of energy to find the initial energy of the block plus glob: Use fk = µkMg to eliminate fk and solve for vcm: Substitute numerical values and evaluate vcm: Substitute numerical values in equation (1) and evaluate vgl: vgl = M vcm mgl (1) ∆K + ∆U + Wf = 0 Because ∆U = Kf = 0, 2 − 1 Mvcm + f k ∆x = 0 2 vcm = 2 µ k g∆x ( ) vcm = 2(0.4) 9.81 m/s 2 (0.15 m ) = 1.08 m/s vgl = 13 kg + 0.4 kg (1.08 m/s) 0.4 kg = 36.2 m/s *127 •• Picture the Problem Let the direction the moving car was traveling before the collision be the positive x direction. Let the numeral 1 denote this car and the numeral 2 the car that is stopped at the stop sign and the system include both cars and the earth. We can use conservation of momentum to relate the speed of the initially-moving car to the speed of the meshed cars immediately after their perfectly inelastic collision and conservation of energy to find the initial speed of the meshed cars. Using conservation of momentum, relate the before-collision velocity to the after-collision velocity of the meshed cars: Solve for v1: Using conservation of energy, relate the initial kinetic energy of the meshed cars to the work done by friction in bringing them to a stop: Substitute for Ki and, using fk = µkFn = µkMg, eliminate fk to pi = pf or m1v1 = (m1 + m2 )V v1 = ⎛ m ⎞ m1 + m2 V = ⎜1 + 2 ⎟ V ⎜ m ⎟ m1 1 ⎠ ⎝ ∆K + ∆Ethermal = 0 or, because Kf = 0 and ∆Ethermal = f∆s, − K i + f k ∆s = 0 − 1 MV 2 + µ k Mg∆x = 0 2 608 Chapter 8 obtain: Solve for V: V = 2 µ k g∆x Substitute to obtain: ⎛ m ⎞ v1 = ⎜1 + 2 ⎟ 2 µ k g∆x ⎜ m1 ⎟ ⎝ ⎠ Substitute numerical values and evaluate v1: ⎛ 900 kg ⎞ 2 v1 = ⎜1 + ⎟ ⎜ 1200 kg ⎟ 2(0.92 ) (9.81 m/s )(0.76 m ) = 6.48 m/s = 23.3 km/h ⎠ ⎝ The driver was not telling the truth. He was traveling at 23.3 km/h. 128 •• Picture the Problem Let the zero of gravitational potential energy be at the lowest point of the bob’s swing and note that the bob can swing either forward or backward after the collision. We’ll use both conservation of momentum and conservation of energy to relate the velocities of the bob and the block before and after their collision. 2 pm 2m Express the kinetic energy of the block in terms of its after-collision momentum: Km = Solve for m to obtain: m= Use conservation of energy to relate Km to the change in the potential energy of the bob: ∆K + ∆U = 0 or, because Ki = 0, Solve for Km: K m = −U f + U i 2 pm 2K m (1) Km + Uf − Ui = 0 = mbob g [L(1 − cos θ i ) − L(1 − cos θ f )] = mbob gL[cos θ f − cos θ i ] Substitute numerical values and evaluate Km: ( ) K m = (0.4 kg ) 9.81 m/s 2 (1.6 m )[cos5.73° − cos53°] = 2.47 J Systems of Particles and Conservation of Momentum 609 Use conservation of energy to find the velocity of the bob just before its collision with the block: ∆K + ∆U = 0 or, because Ki = Uf = 0, Kf − Ui = 0 ∴ 1 mbob v 2 − mbob gL(1 − cos θ i ) = 0 2 or v = 2 gL(1 − cos θ i ) Substitute numerical values and evaluate v: ( ) v = 2 9.81 m/s 2 (1.6 m )(1 − cos53°) = 3.544 m/s ∆K + ∆U = 0 Use conservation of energy to find the velocity of the bob just after its collision with the block: or, because Kf = Ui = 0, Substitute for Ki and Uf to obtain: − 1 mbob v'2 + mbob gL(1 − cos θ f ) = 0 2 Solve for v′: v' = 2 gL(1 − cos θ f ) Substitute numerical values and evaluate v′: − Ki + U f = 0 ( ) v' = 2 9.81 m/s 2 (1.6 m )(1 − cos5.73°) = 0.396 m/s pi = p f Use conservation of momentum to relate pm after the collision to the momentum of the bob just before and just after the collision: or mbob v = mbob v'± pm Solve for and evaluate pm: pm = mbob v ± mbob v' = (0.4 kg )(3.544 m/s ± 0.396 m/s ) = 1.418 kg ⋅ m/s ± 0.158 kg ⋅ m/s Find the larger value for pm: pm = 1.418 kg ⋅ m/s + 0.158 kg ⋅ m/s = 1.576 kg ⋅ m/s Find the smaller value for pm: pm = 1.418 kg ⋅ m/s − 0.158 kg ⋅ m/s = 1.260 kg ⋅ m/s Substitute in equation (1) to determine the two values for m: m= (1.576 kg ⋅ m/s) 2 2(2.47 J ) = 0.503 kg 610 Chapter 8 or m= (1.260 kg ⋅ m/s) 2 2(2.47 J ) = 0.321kg 129 •• Picture the Problem Choose the zero of gravitational potential energy at the location of the spring’s maximum compression. Let the system include the spring, the blocks, and the earth. Then the net external force is zero as is work done against friction. We can use conservation of energy to relate the energy transformations taking place during the evolution of this system. Apply conservation of energy: ∆K + ∆U g + ∆U s = 0 Because ∆K = 0: ∆U g + ∆U s = 0 ∆U g = − mg∆h − Mgx sin θ Express the change in the gravitational potential energy: Express the change in the potential energy of the spring: ∆U s = 1 kx 2 2 Substitute to obtain: − mg∆h − Mgx sin θ + 1 kx 2 = 0 2 Solve for M: M = Relate ∆h to the initial and rebound positions of the block whose mass is m: 1 2 kx 2 − mg∆h kx 2m∆h = − gx sin 30° g x ∆h = (4 m − 2.56 m ) sin 30° = 0.720 m Substitute numerical values and evaluate M: M = (11×10 ) N/m (0.04 m ) 2(1 kg )(0.72 m ) − = 8.85 kg 2 9.81 m/s 0.04 m 3 *130 •• Picture the Problem By symmetry, xcm = 0. Let σ be the mass per unit area of the disk. The mass of the modified disk is the difference between the mass of the whole disk and the mass that has been removed. Systems of Particles and Conservation of Momentum 611 Start with the definition of ycm: ycm = = Express the mass of the complete disk: Express the mass of the material removed: Substitute and simplify to obtain: ∑m y i i i M − mhole mdisk ydisk − mhole y hole M − mhole M = σA = σπ r 2 2 mhole ⎛r⎞ = σπ ⎜ ⎟ = 1 σπ r 2 = 1 M 4 4 ⎝2⎠ ycm = M (0) − ( 1 M )(− 1 r ) 4 2 = M −1M 4 1 6 r 131 •• Picture the Problem Let the horizontal axis by the y axis and the vertical axis the z axis. By symmetry, xcm = ycm = 0. Let ρ be the mass per unit volume of the sphere. The mass of the modified sphere is the difference between the mass of the whole sphere and the mass that has been removed. Start with the definition of ycm: zcm = = ∑m y i i i M − mhole msphere ysphere − mhole y hole M − mhole Express the mass of the complete sphere: M = ρV = 4 ρπ r 3 3 Express the mass of the material removed: ⎛r⎞ mhole = 4 ρπ ⎜ ⎟ = 3 ⎝2⎠ Substitute and simplify to obtain: zcm = 3 ( 1 4 8 3 ρπ r 3 ) = 1 M 8 M (0) − ( 1 M )(− 1 r ) 8 2 = 1 M −8M 1 14 r *132 •• Picture the Problem In this elastic head-on collision, the kinetic energy of recoiling nucleus is the difference between the initial and final kinetic energies of the neutron. We can derive the indicated results by using both conservation of energy and conservation of momentum and writing the kinetic energies in terms of the momenta of the particles before and after the collision. 612 Chapter 8 (a) Use conservation of energy to relate the kinetic energies of the particles before and after the collision: Apply conservation of momentum to obtain a second relationship between the initial and final momenta: Eliminate pnf in equation (1) using equation (2): 2 Use equation (3) to write pni 2m in terms of pnucleus: Use equation (4) to express 2 K nucleus = pnucleus 2M in terms of 2 2 2 pni pnf pnucleus = + 2m 2m 2M (1) pni = pnf + pnucleus (2) pnucleus pnucleus pni + − =0 2M 2m m (3) 2 (M + m ) p2 pni = K n = nucleus 2 2m 8M m 2 (4) ⎡ 4Mm ⎤ K nucleus = K n ⎢ 2⎥ ⎣ (M + m ) ⎦ (5) Kn: (b) Relate the change in the kinetic energy of the neutron to the aftercollision kinetic energy of the nucleus: Using equation (5), express the fraction of the energy lost in the collision: ∆K n = − K nucleus − ∆K n = Kn m 4 4 Mm M = 2 (M + m ) ⎛ m ⎞ 2 ⎜1 + ⎟ ⎝ M⎠ 133 •• Picture the Problem Problem 132 (b) provides an expression for the fractional loss of energy per collision. (a) Using the result of Problem 132 (b), express the fractional loss of energy per collision: Evaluate this fraction to obtain: Express the kinetic energy of one K nf K ni − ∆K n (M − m ) = = K ni E0 (M + m )2 2 K nf (12m − m ) = = 0.716 E0 (12m + m )2 2 K nf = 0.716 N E0 Systems of Particles and Conservation of Momentum 613 neutron after N collisions: (b) Substitute for Knf and E0 to obtain: 0.716 N = 10 −8 Take the logarithm of both sides of the equation and solve for N: N= −8 ≈ 55 log 0.716 134 •• Picture the Problem We can relate the number of collisions needed to reduce the energy of a neutron from 2 MeV to 0.02 eV to the fractional energy loss per collision and solve the resulting exponential equation for N. (a) Using the result of Problem 132 (b), express the fractional loss of energy per collision: K nf K ni − ∆K n K ni − 0.63K ni = = K ni E0 K ni = 0.37 Express the kinetic energy of one neutron after N collisions: K nf = 0.37 N E0 Substitute for Knf and E0 to obtain: 0.37 N = 10 −8 Take the logarithm of both sides of the equation and solve for N: N= (b) Proceed as in (a) to obtain: −8 ≈ 19 log 0.37 K nf K ni − ∆K n K ni − 0.11K ni = = K ni E0 K ni = 0.89 Express the kinetic energy of one neutron after N collisions: K nf = 0.89 N E0 Substitute for Knf and E0 to obtain: 0.89 N = 10 −8 Take the logarithm of both sides of the equation and solve for N: N= −8 ≈ 158 log 0.89 614 Chapter 8 135 •• Picture the Problem Let λ = M/L be the mass per unit length of the rope, the subscript 1 refer to the portion of the rope that is being supported by the force F at any given time, and the subscript 2 refer to the rope that is still on the table at any given time. We can find the height hcm of the center of mass as a function of time and then differentiate this expression twice to find the acceleration of the center of mass. m1h1,cm + m2 h2,cm M (a) Apply the definition of the center of mass to obtain: hcm = From the definition of λ we have: M m1 M = ⇒ m1 = vt L vt L h1,cm and h2,cm are given by : h1,cm = Substitute for m1, h1,cm, and h2,cm in equation (1) and simplify to obtain: (b) Differentiate hcm twice to obtain acm: hcm (1) 1 vt and h2,cm = 0 2 ⎛M ⎞ ⎜ vt ⎟h1,cm + m2 (0 ) v2 2 L ⎠ =⎝ = t M 2L ⎛ v2 ⎞ v2 dhcm = 2⎜ ⎟t = t ⎜ 2L ⎟ dt L ⎝ ⎠ and d 2 hcm v2 = acm = dt 2 L (c) Letting N represent the normal force that the table exerts on the rope, apply Fy = macm to the F + N − Mg = Macm ∑ rope to obtain: Solve for F, substitute for acm and N to obtain: Use the definition of λ again to obtain: F = Mg + Macm − N v2 = Mg + M − m2 g L m2 M ⎛ vt ⎞ ⇒ m2 = M ⎜1 − ⎟ = L − vt L ⎝ L⎠ Systems of Particles and Conservation of Momentum 615 Substitute for m2 and simplify: F = Mg + M = ⎛ v2 v2 vt ⎛ vt ⎞ − M ⎜1 − ⎟ g = M ⎜ g + − g + ⎜ L L L ⎝ L⎠ ⎝ ⎞ ⎛ v 2 vt ⎞ g ⎟ = Mg ⎜ ⎟ ⎜ gL + L ⎟ ⎟ ⎠ ⎝ ⎠ ⎞ vt ⎛ v ⎜ + 1⎟ Mg ⎜ gt ⎟ L⎝ ⎠ 136 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a). We’ll use both conservation of energy and momentum to obtain the scale reading when the ball collides inelastically with the cup. (a) Apply ∑F y = ma y to the Fn − mp g − Fball on spring = 0 spring when it is compressed a distance d: Solve for Fn: Fn = mp g + Fball on spring = mp g + kd ⎛m g⎞ = mp g + k ⎜ b ⎟ ⎝ k ⎠ = mp g + mb g = (mp + mb )g (b) Letting the zero of gravitational energy be at the initial elevation of the cup and vbi represent the velocity of the ball just before it hits the cup, use conservation of energy to find this velocity: Use conservation of momentum to ∆K + ∆U g = 0 where K i = U gf = 0 2 ∴ 1 mb vbi − mgh = 0 2 and vbi = 2 gh r r pi = pf 616 Chapter 8 find the velocity of the center of mass: ∴ vcm = ⎡ mb ⎤ mb vbi = 2 gh ⎢ ⎥ mb + mc ⎣ mb + mc ⎦ ∆K cm + ∆U s = 0 Apply conservation of energy to the collision to obtain: or, with Kf = Usi = 0, Substitute for vcm and solve for kx2: 2 kx 2 = (mb + mc ) vcm 2 − 1 (mb + mc )vcm + 1 kx 2 = 0 2 2 ⎡ mb ⎤ = 2 gh(mb + mc ) ⎢ ⎥ ⎣ mb + mc ⎦ 2 2 ghmb = mb + mc Solve for x: From part (a): x = mb 2 2 gh k (mb + mc ) Fn = mp g + kx = mp g + kmb 2 gh k (mb + mc ) ⎞ ⎛ 2kh ⎟ = g ⎜ mp + mb ⎜ g (mb + mc ) ⎟ ⎠ ⎝ (c) Because the collision is inelastic, the ball never returns to its original height. 137 •• Picture the Problem Let the direction that astronaut 1 first throws the ball be the positive direction and let vb be the initial speed of the ball in the laboratory frame. Note that each collision is perfectly inelastic. We can apply conservation of momentum and the definition of the speed of the ball relative to the thrower to each of the perfectly inelastic collisions to express the final speeds of each astronaut after one throw and one catch. Use conservation of momentum to relate the speeds of astronaut 1 and the ball after the first throw: Relate the speed of the ball in the laboratory frame to its speed relative m1v1 + mb v b = 0 (1) v = v b − v1 (2) Systems of Particles and Conservation of Momentum 617 to astronaut 1: Eliminate vb between equations (1) and (2) and solve for v1: v1 = − Substitute equation (3) in equation (2) and solve for vb: vb = Apply conservation of momentum to express the speed of astronaut 2 and the ball after the first catch: mb v m1 + mb m1 v m1 + mb 0 = mb v b = (m2 + mb )v 2 Solve for v2: v2 = mb vb m2 + mb Express v2 in terms of v by substituting equation (4) in equation (6): v2 = mb m1 v m2 + mb m1 + mb Use conservation of momentum to express the speed of astronaut 2 and the ball after she throws the ball: Relate the speed of the ball in the laboratory frame to its speed relative to astronaut 2: ⎡ ⎤ mb m1 =⎢ ⎥v ⎣ (m2 + mb )(m1 + mb ) ⎦ (3) (4) (5) (6) (7) (m2 + mb )v2 = mb vbf + m2v2f (8) v = v 2f − v bf (9) Eliminate vbf between equations (8) and (9) and solve for v2f: ⎛ mb ⎞ ⎡ m1 ⎤ ⎟ ⎢1 + v2 f = ⎜ ⎥v ⎜m +m ⎟ m1 + mb ⎦ b ⎠⎣ ⎝ 2 (10) Substitute equation (10) in equation (9) and solve for vbf: ⎡ mb ⎤ vbf = − ⎢1 − ⎥ ⎣ m2 + mb ⎦ ⎡ m1 ⎤ × ⎢1 + ⎥v ⎣ m1 + mb ⎦ (11) (m1 + mb )v1f (12) Apply conservation of momentum to express the speed of astronaut 1 and the ball after she catches the ball: = mb vbf + m1v1 618 Chapter 8 Using equations (3) and (11), eliminate vbf and v1 in equation (12) and solve for v1f: m2 mb (2m1 + mb ) v1f = − (m1 + mb )2 (m2 + mb ) v *138 •• Picture the Problem We can use the definition of the center of mass of a system containing multiple objects to locate the center of mass of the earth−moon system. Any object external to the system will exert accelerating forces on the system. (a) Express the center of mass of the earth−moon system relative to the center of the earth: r r Mrcm = ∑ mi ri i or rcm = = Substitute numerical values and evaluate rcm: M e (0) + mm rem mm rem = M e + mm M e + mm rem Me +1 mm 3.84 × 105 km rcm = = 4670 km 81.3 + 1 Because this distance is less than the radius of the earth, the position of the center of mass of the earth − moon system is below the surface of the earth. (b) (c) Any object not in the earth − moon system exerts forces on the system, e.g., the sun and other planets. Because the sun exerts the dominant external force on the earth − moon system, the acceleration of the system is toward the sun. (d) Because the center of mass is at a fixed distance from the sun, the distance d moved by the earth in this time interval is: d = 2rem = 2(4670 km ) = 9340 km 139 •• Picture the Problem Let the numeral 2 refer to you and the numeral 1 to the water leaving the hose. Apply conservation of momentum to the system consisting of yourself, the water, and the earth and then differentiate this expression to relate your recoil acceleration to your mass, the speed of the water, and the rate at which the water is Systems of Particles and Conservation of Momentum 619 leaving the hose. Use conservation of momentum to relate your recoil velocity to the velocity of the water leaving the hose: Differentiate this expression with respect to t: r r p1 + p 2 = 0 or m1v1 + m2 v 2 = 0 m1 dv1 dm dv dm2 + v1 1 + m2 2 + v 2 =0 dt dt dt dt or m1 a1 + v1 Because the acceleration of the water leaving the hose, a1, is zero … as is dm2 , the rate at which you are dt losing mass: Substitute numerical values and evaluate a2: dm1 dm2 + ma 2 + v 2 =0 dt dt dm1 + m2 a 2 = 0 dt and v dm1 a2 = − 1 m 2 dt v1 a2 = − 30 m/s (2.4 kg/s) 75 kg = − 0.960 m/s 2 *140 ••• Picture the Problem Take the zero of gravitational potential energy to be at the elevation of the pan and let the system include the balance, the beads, and the earth. We can use conservation of energy to find the vertical component of the velocity of the beads as they hit the pan and then calculate the net downward force on the pan from Newton’s 2nd law. Use conservation of energy to relate the y component of the bead’s velocity as it hits the pan to its height of fall: Solve for vy: Substitute numerical values and evaluate vy: Express the change in momentum in the y direction per bead: ∆K + ∆U = 0 or, because Ki = Uf = 0, 1 2 2 mv y − mgh = 0 v y = 2 gh ( ) v y = 2 9.81 m/s 2 (0.5 m ) = 3.13 m/s ∆p y = p yf − p yi = mv y − (− mv y ) = 2mv y 620 Chapter 8 Use Newton’s 2nd law to express the net force in the y direction exerted on the pan by the beads: Fnet, y = N ∆p y ∆t ∆p y Letting M represent the mass to be placed on the other pan, equate its weight to the net force exerted by the beads, substitute for ∆py, and solve for M: and Substitute numerical values and evaluate M: M = (100 / s ) Mg = N M = ∆t N ⎛ 2mv y ⎞ ⎜ ⎟ ∆t ⎜ g ⎟ ⎝ ⎠ [2(0.0005 kg )(3.13 m/s)] 9.81m/s 2 = 31.9 g 141 ••• Picture the Problem Assume that the connecting rod goes halfway through both balls, i.e., the centers of mass of the balls are separated by L. Let the system include the dumbbell, the wall and floor, and the earth. Let the zero of gravitational potential be at the center of mass of the lower ball and use conservation of energy to relate the speeds of the balls to the potential energy of the system. By symmetry, the speeds will be equal when the angle with the vertical is 45°. Use conservation of energy to express the relationship between the initial and final energies of the system: Express the initial energy of the system: Express the energy of the system when the angle with the vertical is 45°: Ei = E f Ei = mgL Ef = mgL sin 45° + 1 (2m ) v 2 2 Substitute to obtain: ⎛ 1 ⎞ 2 gL = gL⎜ ⎟+v ⎝ 2⎠ Solve for v: 1 ⎞ ⎛ v = gL⎜1 − ⎟ 2⎠ ⎝ Systems of Particles and Conservation of Momentum 621 Substitute numerical values and evaluate v: v= (9.81m/s )L⎛1 − ⎜ 2 ( ) ⎝ = 1.70 m 2 /s L 1 1 ⎞ ⎟ 2⎠ 622 Chapter 8 Chapter 9 Rotation Conceptual Problems *1 • Determine the Concept Because r is greater for the point on the rim, it moves the greater distance. Both turn through the same angle. Because r is greater for the point on the rim, it has the greater speed. Both have the same angular velocity. Both have zero tangential acceleration. Both have zero angular acceleration. Because r is greater for the point on the rim, it has the greater centripetal acceleration. 2 • ⎡1⎤ (a) False. Angular velocity has the dimensions ⎢ ⎥ whereas linear velocity has ⎣T ⎦ ⎡L⎤ ⎣ ⎦ dimensions ⎢ ⎥ . T (b) True. The angular velocity of all points on the wheel is dθ/dt. (c) True. The angular acceleration of all points on the wheel is dω/dt. 3 •• Picture the Problem The constant-acceleration equation that relates the given variables 2 is ω 2 = ω0 + 2α∆θ . We can set up a proportion to determine the number of revolutions required to double ω and then subtract to find the number of additional revolutions to accelerate the disk to an angular speed of 2ω. Using a constant-acceleration equation, relate the initial and final angular velocities to the angular acceleration: 2 ω 2 = ω0 + 2α∆θ 2 or, because ω0 = 0, ω 2 = 2α∆θ Let ∆θ10 represent the number of revolutions required to reach an angular velocity ω: ω 2 = 2α∆θ10 (1) Let ∆θ2ω represent the number of revolutions required to reach an angular velocity ω: (2ω)2 = 2α∆θ2ω (2) Divide equation (2) by equation (1) and solve for ∆θ2ω: ∆θ2ω = 623 (2ω)2 ∆θ ω2 10 = 4∆θ10 624 Chapter 9 The number of additional revolutions is: 4∆θ10 − ∆θ10 = 3∆θ10 = 3(10 rev ) = 30 rev and (c) is correct. *4 • ⎡ ML2 ⎤ . 2 ⎥ ⎣T ⎦ Determine the Concept Torque has the dimension ⎢ ⎡ ML ⎤ . ⎣ T ⎥ ⎦ ⎡ ML2 ⎤ (b) Energy has the dimension ⎢ 2 ⎥ . ⎣T ⎦ (a) Impulse has the dimension ⎢ (b) is correct. ⎡ ML ⎤ . ⎣ T ⎥ ⎦ (c) Momentum has the dimension ⎢ 5 • Determine the Concept The moment of inertia of an object is the product of a constant that is characteristic of the object’s distribution of matter, the mass of the object, and the square of the distance from the object’s center of mass to the axis about which the object is rotating. Because both (b) and (c) are correct (d ) is correct. *6 • Determine the Concept Yes. A net torque is required to change the rotational state of an object. In the absence of a net torque an object continues in whatever state of rotational motion it was at the instant the net torque became zero. 7 • Determine the Concept No. A net torque is required to change the rotational state of an object. A net torque may decrease the angular speed of an object. All we can say for sure is that a net torque will change the angular speed of an object. 8 • (a) False. The net torque acting on an object determines the angular acceleration of the object. At any given instant, the angular velocity may have any value including zero. (b) True. The moment of inertia of a body is always dependent on one’s choice of an axis of rotation. (c) False. The moment of inertia of an object is the product of a constant that is characteristic of the object’s distribution of matter, the mass of the object, and the square of the distance from the object’s center of mass to the axis about which the object is Rotation 625 rotating. 9 • Determine the Concept The angular acceleration of a rotating object is proportional to the net torque acting on it. The net torque is the product of the tangential force and its lever arm. Express the angular acceleration of the disk as a function of the net torque acting on it: Because α ∝ d , doubling d will double the angular acceleration. α= τ net = I i.e., α ∝ d Fd F = d I I (b) is correct. *10 • Determine the Concept From the parallel-axis theorem we know that I = I cm + Mh 2 , where Icm is the moment of inertia of the object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Therefore, I is always greater than Icm by Mh2. (d ) is correct. 11 • Determine the Concept The power delivered by the constant torque is the product of the torque and the angular velocity of the merry-go-round. Because the constant torque causes the merry-go-round to accelerate, neither the power input nor the angular velocity of the merry-go-round is constant. (b) is correct. 12 • Determine the Concept Let’s make the simplifying assumption that the object and the surface do not deform when they come into contact, i.e., we’ll assume that the system is rigid. A force does no work if and only if it is perpendicular to the velocity of an object, and exerts no torque on an extended object if and only if it’s directed toward the center of the object. Because neither of these conditions is satisfied, the statement is false. 13 • Determine the Concept For a given applied force, this increases the torque about the hinges of the door, which increases the door’s angular acceleration, leading to the door being opened more quickly. It is clear that putting the knob far from the hinges means that the door can be opened with less effort (force). However, it also means that the hand on the knob must move through the greatest distance to open the door, so it may not be the quickest way to open the door. Also, if the knob were at the center of the door, you would have to walk around the door after opening it, assuming the door is opening toward you. 626 Chapter 9 *14 • Determine the Concept If the wheel is rolling without slipping, a point at the top of the wheel moves with a speed twice that of the center of mass of the wheel, but the bottom of the wheel is momentarily at rest. (c) is correct. 15 •• Picture the Problem The kinetic energies of both objects is the sum of their translational and rotational kinetic energies. Their speed dependence will differ due to the differences in their moments of inertia. We can express the total kinetic of both objects and equate them to decide which of their translational speeds is greater. Express the kinetic energy of the cylinder: 2 2 K cyl = 1 I cylω cyl + 1 mvcyl 2 2 = ( 1 1 2 2 mr 2 2 vcyl )r 2 2 + 1 mvcyl 2 2 = 3 mvcyl 4 Express the kinetic energy of the sphere: 2 2 K sph = 1 I sphω sphl + 1 mvsph 2 2 = ( 1 2 2 5 mr 2 2 vsph )r 2 2 + 1 mvsph 2 2 7 = 10 mvsph Equate the kinetic energies and simplify to obtain: vcyl = v 14 15 sph < vsph and (b) is correct. *16 • Determine the Concept You could spin the pipes about their center. The one which is easier to spin has its mass concentrated closer to the center of mass and, hence, has a smaller moment of inertia. 17 •• Picture the Problem Because the coin and the ring begin from the same elevation, they will have the same kinetic energy at the bottom of the incline. The kinetic energies of both objects is the sum of their translational and rotational kinetic energies. Their speed dependence will differ due to the differences in their moments of inertia. We can express the total kinetic of both objects and equate them to their common potential energy loss to decide which of their translational speeds is greater at the bottom of the incline. Rotation 627 Express the kinetic energy of the coin at the bottom of the incline: 2 2 K coin = 1 I cylω coin + 1 mcoin vcoin 2 2 = ( 1 1 2 2 mcoin r 2 ) vr 2 coin 2 2 + 1 mcoin vcoin 2 2 = 3 mcoin vcoin 4 Express the kinetic energy of the ring at the bottom of the incline: 2 2 K ring = 1 I ringω ring + 1 mring vring 2 2 = 1 2 (m ring r 2 2 vring )r 2 2 + 1 mring vring 2 2 = mring vring Equate the kinetic of the coin to its change in potential energy as it rolled down the incline and solve for vcoin: Equate the kinetic of the ring to its change in potential energy as it rolled down the incline and solve for vring: 3 4 2 mcoin v coin = mcoin gh and 2 v coin = 4 gh 3 2 mring v ring = mring gh and 2 v ring = gh Therefore, vcoin > vring and (b) is correct. 18 •• Picture the Problem We can use the definitions of the translational and rotational kinetic energies of the hoop and the moment of inertia of a hoop (ring) to express and compare the kinetic energies. Express the translational kinetic energy of the hoop: K trans = 1 mv 2 2 Express the rotational kinetic energy of the hoop: K rot = 1 I hoopω 2 = 2 1 2 (mr ) v r 2 2 2 = 1 mv 2 2 Therefore, the translational and rotational kinetic energies are the same and (c) is correct. 628 Chapter 9 19 •• Picture the Problem We can use the definitions of the translational and rotational kinetic energies of the disk and the moment of inertia of a disk (cylinder) to express and compare the kinetic energies. Express the translational kinetic energy of the disk: K trans = 1 mv 2 2 Express the rotational kinetic energy of the disk: K rot = 1 I hoopω 2 = 2 ( 1 1 2 2 mr 2 )v r 2 2 = 1 mv 2 4 Therefore, the translational kinetic energy is greater and (a ) is correct. 20 •• Picture the Problem Let us assume that f ≠ 0 and acts along the direction of motion. Now consider the acceleration of the center of mass and the angular acceleration about the point of contact with the plane. Because Fnet ≠ 0, acm ≠ 0. However, τ = 0 because l = 0, so α = 0. But α = 0 is not consistent with acm ≠ 0. Consequently, f = 0. 21 • Determine the Concept True. If the sphere is slipping, then there is kinetic friction which dissipates the mechanical energy of the sphere. 22 • Determine the Concept Because the ball is struck high enough to have topspin, the frictional force is forward; reducing ω until the nonslip condition is satisfied. (a ) is correct. Estimation and Approximation 23 •• Picture the Problem Assume the wheels are hoops, i.e., neglect the mass of the spokes, and express the total kinetic energy of the bicycle and rider. Let M represent the mass of the rider, m the mass of the bicycle, mw the mass of each bicycle wheel, and r the radius of the wheels. Express the ratio of the kinetic energy associated with the rotation of the wheels to that associated with the total kinetic energy of the bicycle and rider: K rot K rot = K tot K trans + K rot (1) Rotation 629 Express the translational kinetic energy of the bicycle and rider: K trans = K bicycle + K rider Express the rotational kinetic energy of the bicycle wheels: K rot = 2 K rot, 1 wheel = 2 1 I wω 2 2 = 1 mv 2 + 1 Mv 2 2 2 ( ( ) ) v2 = mw r 2 = mw v 2 r 2 Substitute in equation (1) to obtain: K rot m v2 mw 2 = 1 2 1 w 2 = 1 = 2 1 m+M K tot 2 mv + 2 Mv + mw v 2 m + 2 M + mw 2+ mw K rot 2 Substitute numerical values and = 10.3 % = K tot 2 + 14 kg + 38 kg evaluate Krot/Ktot: 3 kg 24 •• Picture the Problem We can apply the definition of angular velocity to find the angular orientation of the slice of toast when it has fallen a distance of 0.5 m from the edge of the table. We can then interpret the orientation of the toast to decide whether it lands jellyside up or down. Relate the angular orientation θ of the toast to its initial angular orientation, its angular velocity ω, and time of fall ∆t: θ = θ 0 + ω∆t Use the equation given in the problem statement to find the angular velocity corresponding to this length of toast: 9.81 m/s 2 ω = 0.956 = 9.47 rad/s 0.1m Using a constant-acceleration equation, relate the distance the toast falls ∆y to its time of fall ∆t: ∆y = v0 y ∆t + 1 a y (∆t ) 2 Solve for ∆t: Substitute numerical values and evaluate ∆t: (1) 2 or, because v0y = 0 and ay = g, ∆y = 1 g (∆t ) 2 2 ∆t = 2∆y g ∆t = 2(0.5 m ) = 0.319 s 9.81 m/s 2 630 Chapter 9 (vf ')2 + cos θ 2 gL find θ : 0 Substitute in equation (1) to θ= π 6 + (9.47 rad/s )(0.319 s ) = 3.54 rad × 180° = 203° π rad The orientation of the slice of toast will therefore be at an angle of 203° with respect to the ground, i.e. with the jelly - side down. *25 •• Picture the Problem Assume that the mass of an average adult male is about 80 kg, and that we can model his body when he is standing straight up with his arms at his sides as a cylinder. From experience in men’s clothing stores, a man’s average waist circumference seems to be about 34 inches, and the average chest circumference about 42 inches. We’ll also assume that about 20% of the body’s mass is in the two arms, and each has a length L = 1 m, so that each arm has a mass of about m = 8 kg. Letting Iout represent his moment of inertia with his arms straight out and Iin his moment of inertia with his arms at his side, the ratio of these two moments of inertia is: I out I body + I arms = I in I in Express the moment of inertia of the ″man as a cylinder″: I in = 1 MR 2 2 Express the moment of inertia of his arms: I arms = 2( 1 )mL2 3 Express the moment of inertia of his body-less-arms: I body = 1 2 (M − m )R 2 Substitute in equation (1) to obtain: I out = I in 1 2 (M − m )R 2 + 2( 1 )mL2 3 Assume the circumference of the cylinder to be the average of the average waist circumference and the average chest circumference: Find the radius of a circle whose circumference is 38 in: cav = 1 2 MR 2 34 in + 42 in = 38 in 2 38 in × cav = 2π = 0.154 m R= Substitute numerical values and evaluate Iout/ Iin: (1) 2.54 cm 1m × in 100 cm 2π Rotation 631 I out = I in 1 2 (80 kg − 16 kg )(0.154 m )2 + 2 (8 kg )(1m )2 3 1 (80 kg )(0.154 m )2 2 = 6.42 Angular Velocity and Angular Acceleration 26 • Picture the Problem The tangential and angular velocities of a particle moving in a circle are directly proportional. The number of revolutions made by the particle in a given time interval is proportional to both the time interval and its angular speed. (a) Relate the angular velocity of the particle to its speed along the circumference of the circle: v = rω Solve for and evaluate ω: ω= (b) Using a constant-acceleration equation, relate the number of revolutions made by the particle in a given time interval to its angular velocity: ⎛ 1 rev ⎞ rad ⎞ ⎛ ∆θ = ω ∆t = ⎜ 0.278 ⎟ (30 s )⎜ ⎟ ⎜ 2π rad ⎟ s ⎠ ⎝ ⎠ ⎝ v 25 m/s = = 0.278 rad/s r 90 m = 1.33 rev 27 • Picture the Problem Because the angular acceleration is constant; we can find the various physical quantities called for in this problem by using constant-acceleration equations. (a) Using a constant-acceleration equation, relate the angular velocity of the wheel to its angular acceleration and the time it has been accelerating: ω = ω 0 + α∆t or, when ω0 = 0, ω = α∆t Evaluate ω when ∆t = 6 s: ω = ⎛ 2.6 rad/s2 ⎞ (6 s ) = 15.6 rad/s ⎜ ⎟ (b) Using another constantacceleration equation, relate the angular displacement to the wheel’s angular acceleration and the time it ∆θ = ω 0 ∆t + 1 α (∆t ) 2 ⎝ ⎠ or, when ω0 = 0, ∆θ = 1 α (∆t ) 2 2 2 632 Chapter 9 has been accelerating: Evaluate ∆θ when ∆t = 6 s: (c) Convert ∆θ (6 s ) from rad to revolutions: ∆θ (6 s ) = 1 2 (2.6 rad/s )(6 s ) 2 2 ∆θ (6 s ) = 46.8 rad × = 46.8 rad 1 rev = 7.45 rev 2π rad (d) Relate the angular velocity of the particle to its tangential speed and evaluate the latter when ∆t = 6 s: v = rω = (0.3 m )(15.6 rad/s ) = 4.68 m/s Relate the resultant acceleration of the point to its tangential and centripetal accelerations when ∆t = 6 s: a = at2 + ac2 = Substitute numerical values and evaluate a: a = (0.3 m ) (2.6 rad/s 2 ) + (15.6 rad/s ) (rα )2 + (rω 2 )2 = r α 2 + ω4 2 4 = 73.0 m/s 2 *28 • Picture the Problem Because we’re assuming constant angular acceleration; we can find the various physical quantities called for in this problem by using constant-acceleration equations. (a) Using its definition, express the angular acceleration of the turntable: Substitute numerical values and evaluate α: α= α= ∆ω ω − ω0 = ∆t ∆t 0 − 33 1 3 rev 2π rad 1 min × × min rev 60 s 26 s = 0.134 rad/s 2 Rotation 633 (b) Because the angular acceleration is constant, the average angular velocity is the average of its initial and final values: ωav = = ω0 + ω 2 33 1 3 rev 2π rad 1 min × × min rev 60 s 2 = 1.75 rad/s (c) Using the definition of ωav, find the number or revolutions the turntable makes before stopping: ∆θ = ωav ∆t = (1.75 rad/s )(26 s ) = 45.5 rad × 1 rev = 7.24 rev 2π rad 29 • Picture the Problem Because the angular acceleration of the disk is constant, we can use a constant-acceleration equation to relate its angular velocity to its acceleration and the time it has been accelerating. We can find the tangential and centripetal accelerations from their relationships to the angular velocity and angular acceleration of the disk. (a) Using a constant-acceleration equation, relate the angular velocity of the disk to its angular acceleration and time during which it has been accelerating: ω = ω 0 + α ∆t or, because ω0 = 0, ω = α ∆t Evaluate ω when t = 5 s: ω (5 s ) = (8 rad/s 2 )(5 s ) = 40.0 rad/s (b) Express at in terms of α: a t = rα Evaluate at when t = 5 s: at (5 s ) = (0.12 m ) 8 rad/s 2 ( ) = 0.960 m/s 2 Express ac in terms of ω: a c = rω 2 Evaluate ac when t = 5 s: ac (5 s ) = (0.12 m )(40.0 rad/s ) 2 = 192 m/s 2 30 • Picture the Problem We can find the angular velocity of the Ferris wheel from its definition and the linear speed and centripetal acceleration of the passenger from the relationships between those quantities and the angular velocity of the Ferris wheel. 634 Chapter 9 (a) Find ω from its definition: (b) Find the linear speed of the passenger from his/her angular speed: Find the passenger’s centripetal acceleration from his/her angular velocity: ω= ∆θ 2π rad = = 0.233 rad/s ∆t 27 s v = rω = (12 m )(0.233 rad/s ) = 2.79 m/s ac = rω 2 = (12 m )(0.233 rad/s ) 2 = 0.651 m/s 2 31 • Picture the Problem Because the angular acceleration of the wheels is constant, we can use constant-acceleration equations in rotational form to find their angular acceleration and their angular velocity at any given time. (a) Using a constant-acceleration equation, relate the angular displacement of the wheel to its angular acceleration and the time it has been accelerating: ∆θ = ω 0 ∆t + 1 α (∆t ) 2 2 or, because ω0 = 0, ∆θ = 1 α (∆t ) 2 2 2∆θ (∆t )2 Solve for α: α= Substitute numerical values and evaluate α: ⎛ 2π rad ⎞ 2(3 rev )⎜ ⎟ ⎝ rev ⎠ = 0.589 rad/s 2 α= (8 s )2 (b) Using a constant-acceleration equation, relate the angular velocity of the wheel to its angular acceleration and the time it has been accelerating: ω = ω 0 + α∆t Evaluate ω when ∆t = 8 s: or, when ω0 = 0, ω = α∆t ω (8 s ) = (0.589 rad/s 2 )(8 s ) = 4.71 rad/s Rotation 635 32 • Picture the Problem The earth rotates through 2π radians every 24 hours. Find ω using its definition: ω≡ ∆θ 2π rad = ∆t 24 h × 3600 s h = 7.27 × 10 −5 rad/s 33 • Picture the Problem When the angular acceleration of a wheel is constant, its average angular velocity is the average of its initial and final angular velocities. We can combine this relationship with the always applicable definition of angular velocity to find the initial angular velocity of the wheel. Express the average angular velocity of the wheel in terms of its initial and final angular speeds: ω0 + ω ω av = 2 or, because ω = 0, ω av = 1 ω 0 2 Express the definition of the average angular velocity of the wheel: Equate these two expressions and solve for ω0: ω0 = ∆θ ∆t ω av ≡ 2∆θ 2(5 rad ) = = 3.57 s and ∆t 2.8 s (d ) is correct. 34 • Picture the Problem The tangential and angular accelerations of the wheel are directly proportional to each other with the radius of the wheel as the proportionality constant. Provided there is no slippage, the acceleration of a point on the rim of the wheel is the same as the acceleration of the bicycle. We can use its defining equation to determine the acceleration of the bicycle. Relate the tangential acceleration of a point on the wheel (equal to the acceleration of the bicycle) to the wheel’s angular acceleration and solve for its angular acceleration: a = a t = rα and α= a r 636 Chapter 9 Use its definition to express the acceleration of the wheel: a= ∆v v − v0 = ∆t ∆t or, because v0 = 0, a= v ∆t Substitute in the expression for α to obtain: α= v r∆t Substitute numerical values and evaluate α: ⎛ km ⎞ ⎛ 1 h ⎞ ⎛ 1000 m ⎞ ⎟⎜ ⎜ 24 ⎟⎜ ⎟ h ⎠ ⎜ 3600 s ⎟ ⎝ km ⎠ ⎝ ⎝ ⎠ α= (0.6 m )(14.0 s ) = 0.794 rad/s 2 *35 •• Picture the Problem The two tapes will have the same tangential and angular velocities when the two reels are the same size, i.e., have the same area. We can calculate the tangential speed of the tape from its length and running time and relate the angular velocity to the constant tangential speed and the radius of the reels when they are turning with the same angular velocity. Relate the angular velocity of the tape to its tangential speed: Letting Rf represent the outer radius of the reel when the reels have the same area, express the condition that they have the same speed: Solve for Rf: Substitute numerical values and evaluate Rf: Find the tangential speed of the tape from its length and running time: ω= v r (1) π Rf2 − π r 2 = 1 (π R 2 − π r 2 ) 2 Rf = Rf = R2 + r 2 2 (45 mm)2 + (12 mm)2 L v= = ∆t 2 = 32.9 mm 100 cm m = 3.42 cm/s 3600 s 2h × h 246 m × Rotation 637 Substitute in equation (1) and evaluate ω: ω= 3.42 cm/s 1 cm 32.9 mm × 10 mm v = Rf = 1.04 rad/s Convert 1.04 rad/s to rev/min: 1.04 rad/s = 1.04 rad 1 rev 60 s × × s 2π rad min = 9.93 rev/min Torque, Moment of Inertia, and Newton’s Second Law for Rotation 36 • Picture the Problem The force that the woman exerts through her axe, because it does not act at the axis of rotation, produces a net torque that changes (decreases) the angular velocity of the grindstone. (a) From the definition of angular acceleration we have: Substitute numerical values and evaluate α: ∆ω ω − ω0 = ∆t ∆t or, because ω = 0, − ω0 α= ∆t α= 730 α =− rev 2π rad 1 min × × min rev 60 s 9s = − 8.49 rad/s 2 where the minus sign means that the grindstone is slowing down. (b) Use Newton’s 2nd law in rotational form to relate the angular acceleration of the grindstone to the net torque slowing it: τ net = Iα Express the moment of inertia of disk with respect to its axis of rotation: I = 1 MR 2 2 638 Chapter 9 Substitute to obtain: Substitute numerical values and evaluate τnet: τ net = 1 MRα 2 τ net = 1 (1.7 kg )(0.08 m )2 (8.49 rad/s 2 ) 2 = 0.0462 N ⋅ m *37 • Picture the Problem We can find the torque exerted by the 17-N force from the definition of torque. The angular acceleration resulting from this torque is related to the torque through Newton’s 2nd law in rotational form. Once we know the angular acceleration, we can find the angular velocity of the cylinder as a function of time. (a) Calculate the torque from its definition: τ = Fl = (17 N )(0.11 m ) = 1.87 N ⋅ m (b) Use Newton’s 2nd law in rotational form to relate the acceleration resulting from this torque to the torque: α= Express the moment of inertia of the cylinder with respect to its axis of rotation: I = 1 MR 2 2 Substitute to obtain: α= 2τ MR 2 Substitute numerical values and evaluate α: α= 2(1.87 N ⋅ m ) = 124 rad/s 2 (2.5 kg )(0.11m )2 (c) Using a constant-acceleration equation, express the angular velocity of the cylinder as a function of time: ω = ω0 + α t Evaluate ω (5 s): τ I or, because ω0 = 0, ω =αt ω (5 s ) = (124 rad/s 2 )(5 s ) = 620 rad/s 38 •• Picture the Problem We can find the angular acceleration of the wheel from its definition and the moment of inertia of the wheel from Newton’s 2nd law. Rotation 639 (a) Express the moment of inertia of the wheel in terms of the angular acceleration produced by the applied torque: Find the angular acceleration of the wheel: I= τ α ∆ω = ∆t α= 600 rev 2π rad 1 min × × min rev 60 s 20 s = 3.14 rad/s 2 50 N ⋅ m = 15.9 kg ⋅ m 2 3.14 rad/s 2 Substitute and evaluate I: I= (b) Because the wheel takes 120 s to slow to a stop (it took 20 s to acquire an angular velocity of 600 rev/min) and its angular acceleration is directly proportional to the accelerating torque: τ fr = 1 τ = 6 1 6 (50 N ⋅ m ) = 39 •• Picture the Problem The pendulum and the forces acting on it are shown in the free-body diagram. Note that the tension in the string is radial, and so exerts no tangential force on the ball. We can use Newton’s 2nd law in both translational and rotational form to find the tangential component of the acceleration of the bob. (a) Referring to the FBD, express r the component of mg that is tangent Ft = mg sin θ to the circular path of the bob: Use Newton’s 2nd law to express the tangential acceleration of the bob: at = (b) Noting that, because the line-ofaction of the tension passes through the pendulum’s pivot point, its lever arm is zero and the net torque is due ∑τ Ft = g sin θ m pivot point = mgL sin θ 8.33 N ⋅ m 640 Chapter 9 to the weight of the bob, sum the torques about the pivot point to obtain: (c) Use Newton’s 2nd law in rotational form to relate the angular acceleration of the pendulum to the net torque acting on it: τ net = mgL sin θ = Iα Solve for α to obtain: α= Express the moment of inertia of the bob with respect to the pivot point: I = mL2 Substitute to obtain: α= Relate α to at: ⎛ g sin θ ⎞ a t = rα = L ⎜ ⎟ = g sin θ ⎝ L ⎠ mgL sin θ I mgL sin θ g sin θ = mL2 L *40 ••• Picture the Problem We can express the velocity of the center of mass of the rod in terms of its distance from the pivot point and the angular velocity of the rod. We can find the angular velocity of the rod by using Newton’s 2nd law to find its angular acceleration and then a constant-acceleration equation that relates ω to α. We’ll use the impulsemomentum relationship to derive the expression for the force delivered to the rod by the pivot. Finally, the location of the center of percussion of the rod will be verified by setting the force exerted by the pivot to zero. L ω 2 (a) Relate the velocity of the center of mass to its distance from the pivot point: vcm = Express the torque due to F0: τ = F0 x = I pivotα Solve for α: α= Express the moment of inertia of the rod with respect to an axis through I pivot = 1 ML2 3 F0 x I pivot (1) Rotation 641 its pivot point: 3F0 x ML2 Substitute to obtain: α= Express the angular velocity of the rod in terms of its angular acceleration: ω = α ∆t = Substitute in equation (1) to obtain: (b) Let IP be the impulse exerted by the pivot on the rod. Then the total impulse (equal to the change in momentum of the rod) exerted on the rod is: vcm = 3F0 x∆t ML2 3F0 x∆t 2ML I P + F0 ∆t = Mvcm and I P = Mvcm − F0 ∆t 3F0 x∆t ⎛ 3x ⎞ − F0 ∆t = F0 ∆t ⎜ − 1⎟ 2L ⎝ 2L ⎠ Substitute our result from (a) to obtain: IP = Because I P = FP ∆t : ⎛ 3x ⎞ − 1⎟ FP = F0 ⎜ ⎝ 2L ⎠ In order for FP to be zero: 2L 3x −1 = 0 ⇒ x = 3 2L 41 ••• Picture the Problem We’ll first express the torque exerted by the force of friction on the elemental disk and then integrate this expression to find the torque on the entire disk. We’ll use Newton’s 2nd law to relate this torque to the angular acceleration of the disk and then to the stopping time for the disk. (a) Express the torque exerted on the elemental disk in terms of the friction force and the distance to the elemental disk: dτ f = rdf k (1) Using the definition of the coefficient of friction, relate the df k = µ k gdm (2) 642 Chapter 9 force of friction to µk and the weight of the circular element: Letting σ represent the mass per unit area of the disk, express the mass of the circular element: dm = 2π rσ dr (3) Substitute equations (2) and (3) in (1) to obtain: dτ f = 2π µ kσ g r 2 dr (4) Because σ = M : π R2 (b) Integrate dτ f to obtain the total torque on the elemental disk: (c) Relate the disk’s stopping time to its angular velocity and acceleration: Using Newton’s 2nd law, express α in terms of the net torque acting on the disk: τf = 2 µk M g 2 ∫ r dr = R2 0 ∆t = ω α α= R 2 3 MRµ k g τf I I = 1 MR 2 2 The moment of inertia of the disk, with respect to its axis of rotation, is: Substitute and simplify to obtain: 2 µk M g 2 r dr R2 dτ f = ∆t = 3 Rω 4µ k g Calculating the Moment of Inertia 42 • Picture the Problem One can find the formula for the moment of inertia of a thin spherical shell in Table 9-1. The moment of inertia of a thin spherical shell about its diameter is: I = 2 MR 2 3 Rotation 643 Substitute numerical values and evaluate I: I= (0.057 kg )(0.035 m )2 2 3 = 4.66 × 10 −5 kg ⋅ m 2 *43 • Picture the Problem The moment of inertia of a system of particles with respect to a given axis is the sum of the products of the mass of each particle and the square of its distance from the given axis. Use the definition of the moment of inertia of a system of particles to obtain: Substitute numerical values and evaluate I: I = ∑ mi ri 2 i = m1r12 + m2 r22 + m3 r32 + m4 r42 ( I = (3 kg )(2 m ) + (4 kg ) 2 2 m 2 ) 2 + (4 kg )(0) + (3 kg )(2 m ) 2 2 = 56.0 kg ⋅ m 2 44 • Picture the Problem Note, from symmetry considerations, that the center of mass of the system is at the intersection of the diagonals connecting the four masses. Thus the distance of each particle from the axis through the center of mass is 2 m . According to the parallel-axis theorem, I = I cm + Mh 2 , where Icm is the moment of inertia of the object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Express the parallel axis theorem: I = I cm + Mh 2 Solve for Icm and substitute from Problem 44: I cm = I − Mh 2 ( = 56.0 kg ⋅ m 2 − (14 kg ) 2 m ) 2 = 28.0 kg ⋅ m 2 Use the definition of the moment of inertia of a system of particles to express Icm: Substitute numerical values and evaluate Icm: I cm = ∑ mi ri 2 i = m1r12 + m2 r22 + m3 r32 + m4 r42 ( ) ( I cm = (3 kg ) 2 m + (4 kg ) 2 m ( 2 ) ( ) 2 + (4 kg ) 2 m + (3 kg ) 2 m = 28.0 kg ⋅ m 2 2 ) 2 644 Chapter 9 45 • Picture the Problem The moment of inertia of a system of particles with respect to a given axis is the sum of the products of the mass of each particle and the square of its distance from the given axis. (a) Apply the definition of the moment of inertia of a system of particles to express Ix: Substitute numerical values and evaluate Ix: I x = ∑ mi ri 2 i = m1r12 + m2 r22 + m3 r32 + m4 r42 I x = (3 kg )(2 m ) + (4 kg )(2 m ) 2 2 + (4 kg )(0 ) + (3 kg )(0) = 28.0 kg ⋅ m 2 (b) Apply the definition of the moment of inertia of a system of particles to express Iy: Substitute numerical values and evaluate Iy: I y = ∑ mi ri 2 i = m1r12 + m2 r22 + m3 r32 + m4 r42 I y = (3 kg )(0) + (4 kg )(2 m ) 2 + (4 kg )(0) + (3 kg )(2 m ) 2 = 28.0 kg ⋅ m 2 Remarks: We could also use a symmetry argument to conclude that Iy = Ix . 46 • Picture the Problem According to the parallel-axis theorem, I = I cm + Mh 2 , where Icm is the moment of inertia of the object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Use Table 9-1 to find the moment of inertia of a sphere with respect to an axis through its center of mass: 2 I cm = 5 MR 2 Express the parallel axis theorem: I = I cm + Mh 2 Substitute for Icm and simplify to obtain: 2 I = 5 MR 2 + MR 2 = 7 5 MR 2 Rotation 645 47 •• Picture the Problem The moment of inertia of the wagon wheel is the sum of the moments of inertia of the rim and the six spokes. Express the moment of inertia of the wagon wheel as the sum of the moments of inertia of the rim and the spokes: I wheel = I rim + I spokes Using Table 9-1, find formulas for the moments of inertia of the rim and spokes: I rim = M rim R 2 and Substitute to obtain: I wheel = M rim R 2 + 6 1 M spoke L2 3 I spoke = 1 M spoke L2 3 ( ) = M rim R 2 + 2M spoke L2 Substitute numerical values and evaluate Iwheel: I wheel = (8 kg )(0.5 m ) + 2(1.2 kg )(0.5 m ) 2 2 = 2.60 kg ⋅ m 2 *48 •• Picture the Problem The moment of inertia of a system of particles depends on the axis with respect to which it is calculated. Once this choice is made, the moment of inertia is the sum of the products of the mass of each particle and the square of its distance from the chosen axis. (a) Apply the definition of the moment of inertia of a system of particles: I = ∑ mi ri 2 = m1 x 2 + m2 (L − x ) (b) Set the derivative of I with respect to x equal to zero in order to identify values for x that correspond to either maxima or minima: dI = 2m1 x + 2m2 (L − x )(− 1) dx = 2(m1 x + m2 x − m2 L ) If dI = 0 , then: dx 2 i = 0 for extrema m1 x + m2 x − m2 L = 0 646 Chapter 9 Solve for x: x= Convince yourself that you’ve found 2 a minimum by showing that d I is dx 2 m2 L m1 + m2 x= m2 L is, by definition, the m1 + m2 distance of the center of mass from m. positive at this point. 49 •• Picture the Problem Let σ be the mass per unit area of the uniform rectangular plate. Then the elemental unit has mass dm = σ dxdy. Let the corner of the plate through which the axis runs be the origin. The distance of the element whose mass is dm from the corner r is related to the coordinates of dm through the Pythagorean relationship r2 = x2 + y2. (a) Express the moment of inertia of the element whose mass is dm with respect to an axis perpendicular to it and passing through one of the corners of the uniform rectangular plate: Integrate this expression to find I: dI = σ (x 2 + y 2 )dxdy a b ( ) I = σ ∫ ∫ x 2 + y 2 dxdy 0 0 ( ) = 1 σ a 3b + ab 3 = 3 (b) Letting d represent the distance from the origin to the center of mass of the plate, use the parallel axis theorem to relate the moment of inertia found in (a) to the moment of inertia with respect to an axis through the center of mass: Using the Pythagorean theorem, relate the distance d to the center of 1 3 ( m a 2 + b3 ) I = I cm + md 2 or ( ) I cm = I − md 2 = 1 m a 2 + b 2 − md 2 3 d 2 = ( 1 a ) + ( 1 b) = 2 2 2 2 1 4 (a 2 + b2 ) Rotation 647 mass to the lengths of the sides of the plate: Substitute for d2 in the expression for Icm and simplify to obtain: ( ) ( I cm = 1 m a 2 + b 2 − 1 m a 2 + b 2 4 3 = 1 12 ( m a2 + b2 ) ) 2 *50 •• Picture the Problem Corey will use the point-particle relationship I app = mi ri 2 = m1r12 + m2 r22 for his calculation whereas Tracey’s calculation will take ∑ i into account not only the rod but also the fact that the spheres are not point particles. (a) Using the point-mass approximation and the definition of the moment of inertia of a system of particles, express Iapp: I app = ∑ mi ri 2 = m1r12 + m2 r22 i Substitute numerical values and evaluate Iapp: I app = (0.5 kg )(0.2 m ) + (0.5 kg )(0.2 m ) Express the moment of inertia of the two spheres and connecting rod system: I = I spheres + I rod Use Table 9-1 to find the moments of inertia of a sphere (with respect to its center of mass) and a rod (with respect to an axis through its center of mass): 2 I sphere = 5 M sphere R 2 Because the spheres are not on the axis of rotation, use the parallel axis theorem to express their moment of inertia with respect to the axis of rotation: Substitute to obtain: Substitute numerical values and evaluate I: 2 2 = 0.0400 kg ⋅ m 2 and 1 I rod = 12 M rod L2 2 I sphere = 5 M sphere R 2 + M sphere h 2 where h is the distance from the center of mass of a sphere to the axis of rotation. { } 2 1 I = 2 5 M sphere R 2 + M sphere h 2 + 12 M rod L2 648 Chapter 9 { (0.5 kg )(0.05 m) I =2 2 5 2 } 1 + (0.5 kg )(0.2 m ) + 12 (0.06 kg )(0.3 m ) 2 2 = 0.0415 kg ⋅ m 2 Compare I and Iapp by taking their ratio: (b) I app 0.0400 kg ⋅ m 2 = = 0.964 I 0.0415 kg ⋅ m 2 The rotational inertia would increase because I cm of a hollow sphere is greater than I cm of a solid sphere. 51 •• Picture the Problem The axis of rotation passes through the center of the base of the tetrahedron. The carbon atom and the hydrogen atom at the apex of the tetrahedron do not contribute to I because the distance of their nuclei from the axis of rotation is zero. From the geometry, the distance of the three H nuclei from the rotation axis is a / 3 , where a is the length of a side of the tetrahedron. Apply the definition of the moment of inertia for a system of particles to obtain: I = ∑ mi ri 2 = mH r12 + mH r22 + mH r32 Substitute numerical values and evaluate I: I = 1.67 × 10−27 kg 0.18 × 10 −9 m 52 •• Picture the Problem Let the mass of the element of volume dV be dm = ρdV = 2πρhrdr where h is the height of the cylinder. We’ll begin by expressing the moment of inertia dI for the element of volume and then integrating it between R1 and R2. i 2 ⎛ a ⎞ = 3mH ⎜ ⎟ = mH a 2 ⎝ 3⎠ ( )( = 5.41× 10−47 kg ⋅ m 2 ) 2 Rotation 649 Express the moment of inertia of the element of mass dm: Integrate dI from R1 to R2 to obtain: dI = r 2 dm = 2πρ hr 3 dr R2 4 I = 2πρ h ∫ r 3dr = 1 πρ h(R2 − R14 ) 2 R1 2 2 = 1 πρ h(R2 − R12 )(R2 + R12 ) 2 The mass of the hollow cylinder 2 is m = π ρ h R2 − R12 , so: ( ρ= ) m π h R22 − R12 ( ) Substitute for ρ and simplify to obtain: ⎛ m I = 1π ⎜ 2 ⎜ π h R2 − R2 2 1 ⎝ ( ⎞ 2 2 ⎟ h R2 − R12 R2 + R12 = ⎟ ⎠ ) ( )( ) 1 2 ( 2 m R2 + R12 ) 53 ••• Picture the Problem We can derive the given expression for the moment of inertia of a spherical shell by following the procedure outlined in the problem statement. Find the moment of inertia of a sphere, with respect to an axis through a diameter, in Table 9-1: 2 I = 5 mR 2 Express the mass of the sphere as a function of its density and radius: m = 4 π ρ R3 3 Substitute to obtain: 8 I = 15 π ρ R 5 Express the differential of this expression: dI = 8 π ρ R 4 dR 3 (1) Express the increase in mass dm as the radius of the sphere increases by dR: dm = 4π ρ R 2 dR (2) Eliminate dR between equations (1) and (2) to obtain: dI = 2 R 2 dm 3 Therefore, the moment of inertia of the spherical shell of mass m is 2 mR 2 . 3 650 Chapter 9 *54 ••• Picture the Problem We can find C in terms of M and R by integrating a spherical shell of mass dm with the given density function to find the mass of the earth as a function of M and then solving for C. In part (b), we’ll start with the moment of inertia of the same spherical shell, substitute the earth’s density function, and integrate from 0 to R. (a) Express the mass of the earth using the given density function: R M = ∫ dm = ∫ 4π ρ r 2 dr 0 4π C 3 r dr = 4π C ∫ 1.22r dr − R ∫ 0 0 R R 2 = 4π 1.22CR 3 − π CR 3 3 M R3 Solve for C as a function of M and R to obtain: C = 0.508 (b) From Problem 9-40 we have: dI = 8 π ρ r 4 dr 3 Integrate to obtain: I = 8 π ∫ ρ r 4 dr 3 R 0 R R ⎤ 8π (0.508)M ⎡ 1 1.22r 4 dr − ∫ r 5 dr ⎥ ⎢∫ 3 3R R0 ⎣0 ⎦ 4.26M ⎡1.22 5 1 5 ⎤ = R − R ⎥ 6 ⎦ R3 ⎢ 5 ⎣ = = 0.329MR 2 Rotation 651 55 ••• Picture the Problem Let the origin be at the apex of the cone, with the z axis along the cone’s symmetry axis. Then the radius of the elemental ring, at a distance z from the apex, can be obtained from the r R = . The mass dm of the z H elemental disk is ρdV = ρπr2dz. We’ll proportion integrate r2dm to find the moment of inertia of the disk in terms of R and H and then integrate dm to obtain a second equation in R and H that we can use to eliminate H in our expression for I. Express the moment of inertia of the cone in terms of the moment of inertia of the elemental disk: I= 1 2 = 1 2 ∫r 2 dm R2 2 ⎛ R2 ⎞ z ⎜ ρπ 2 z 2 ⎟dz ∫ H2 ⎜ H ⎟ ⎝ ⎠ 0 H = πρ R 4 2H 4 H 4 ∫ z dz = πρ R 4 H 10 0 H H Express the total mass of the cone in terms of the mass of the elemental disk: M = πρ ∫ r 2 dz = πρ ∫ Divide I by M, simplify, and solve for I to obtain: I= 56 ••• Picture the Problem Let the axis of rotation be the x axis. The radius r of the elemental area is R 2 − z 2 and its mass, dm, is σ dA = 2σ R 2 − z 2 dz . We’ll integrate z2 dm to determine I in terms of σ and then divide this result by M in order to eliminate σ and express I in terms of M and R. 0 0 = 1 πρ R H 3 2 3 10 MR 2 R2 2 z dz H2 652 Chapter 9 Express the moment of inertia about the x axis: I = ∫ z 2 dm = ∫ z 2σ dA R = ∫z 2 (2σ R 2 − z 2 dz ) −R = 1 σπR 4 4 The mass of the thin uniform disk is: Divide I by M, simplify, and solve for I to obtain: M = σπR 2 I= 1 4 MR 2 , a result in agreement with the expression given in Table 9-1 for a cylinder of length L = 0. 57 ••• Picture the Problem Let the origin be at the apex of the cone, with the z axis along the cone’s symmetry axis, and the axis of rotation be the x rotation. Then the radius of the elemental disk, at a distance z from the apex, can be obtained from the r R = . The mass dm of the z H elemental disk is ρdV = ρπr2dz. Each proportion elemental disk rotates about an axis that is parallel to its diameter but removed from it by a distance z. We can use the result from Problem 9-57 for the moment of inertia of the elemental disk with respect to a diameter and then use the parallel axis theorem to express the moment of inertia of the cone with respect to the x axis. Using the parallel axis theorem, express the moment of inertia of the elemental disk with respect to the x axis: In Problem 9-57 it was established that the moment of inertia of a thin uniform disk of mass M and radius R rotating about a diameter is 1 MR 2 . Express this result in 4 dI x = dI disk + dm z 2 (1) where dm = ρ dV = ρπ r 2 dz dI disk = 1 4 (ρπ r dz )r 2 2 2 ⎛ R2 ⎞ = ρπ ⎜ 2 z 2 ⎟ dz ⎜H ⎟ ⎝ ⎠ 1 4 Rotation 653 terms of our elemental disk: Substitute in equation (1) to obtain: ⎡ 1 ⎛ R 2 ⎞2 ⎤ dI x = πρ ⎢ ⎜ 2 z 2 ⎟ ⎥ dz ⎜ ⎟ ⎢4 ⎝ H ⎠ ⎥ ⎣ ⎦ ⎛ ⎛R + ⎜ πρ ⎜ ⎜ ⎝H ⎝ Integrate from 0 to H to obtain: 2 ⎞ ⎞ z ⎟ dz ⎟ z 2 ⎟ ⎠ ⎠ 2 ⎡1 ⎛ R2 R2 4 ⎤ 2⎞ I x = πρ ∫ ⎢ ⎜ 2 z ⎟ + 2 z ⎥ dz ⎟ 4⎜ H ⎥ ⎠ 0 ⎢ ⎝H ⎣ ⎦ 4 2 3 ⎛R H R H ⎞ = πρ ⎜ ⎟ ⎜ 20 + 5 ⎟ ⎠ ⎝ H H H R2 2 z dz H2 Express the total mass of the cone in terms of the mass of the elemental disk: M = πρ ∫ r 2 dz = πρ ∫ Divide Ix by M, simplify, and solve for Ix to obtain: ⎛ H 2 R2 ⎞ I x = 3M ⎜ ⎜ 5 + 20 ⎟ ⎟ ⎝ ⎠ 0 0 = πρ R H 1 3 2 Remarks: Because both H and R appear in the numerator, the larger the cones are, the greater their moment of inertia and the greater the energy consumption required to set them into motion. Rotational Kinetic Energy 58 • Picture the Problem The kinetic energy of this rotating system of particles can be calculated either by finding the tangential velocities of the particles and using these values to find the kinetic energy or by finding the moment of inertia of the system and using the expression for the rotational kinetic energy of a system. (a) Use the relationship between v and ω to find the speed of each particle: v3 = r3ω = (0.2 m )(2 rad/s ) = 0.4 m/s and v1 = r1ω = (0.4 m )(2 rad/s ) = 0.8 m/s 654 Chapter 9 Find the kinetic energy of the system: 2 K = 2 K 3 + 2 K1 = m3v3 + m1v12 = (3 kg )(0.4 m/s ) + (1kg )(0.8 m/s ) 2 2 = 1.12 J (b) Use the definition of the moment of inertia of a system of particles to obtain: Substitute numerical values and evaluate I: I = ∑ mi ri 2 i = m1r12 + m2 r22 + m3 r32 + m4 r42 I = (1 kg )(0.4 m ) + (3 kg )(0.2 m ) 2 2 + (1 kg )(0.4 m ) + (3 kg )(0.2 m ) 2 2 = 0.560 kg ⋅ m 2 Calculate the kinetic energy of the system of particles: K = 1 Iω 2 = 2 1 2 (0.560 kg ⋅ m )(2 rad/s) 2 2 = 1.12 J *59 • Picture the Problem We can find the kinetic energy of this rotating ball from its angular speed and its moment of inertia. We can use the same relationship to find the new angular speed of the ball when it is supplied with additional energy. (a) Express the kinetic energy of the ball: K = 1 Iω 2 2 Express the moment of inertia of ball with respect to its diameter: 2 I = 5 MR 2 Substitute for I: K = 1 MR 2ω 2 5 Substitute numerical values and evaluate K: K= 1 5 (1.4 kg )(0.075 m )2 ⎛ rev 2π rad 1 min ⎞ × ⎜ 70 ⎟ ⎜ min × rev × 60 s ⎟ ⎠ ⎝ = 84.6 mJ (b) Express the new kinetic energy with K′ = 2.0846 J: K ' = 1 Iω ' 2 2 Express the ratio of K to K′: K' 1 Iω' 2 ⎛ ω' ⎞ = 2 =⎜ ⎟ 2 1 K ⎝ω ⎠ 2 Iω ' 2 2 Rotation 655 Solve for ω′: ω' = ω Substitute numerical values and evaluate ω′: K' K ω' = (70 rev/min ) 2.0846 J 0.0846 J = 347 rev/min 60 • Picture the Problem The power delivered by an engine is the product of the torque it develops and the angular speed at which it delivers the torque. Express the power delivered by the engine as a function of the torque it develops and the angular speed at which it delivers this torque: P =τω Substitute numerical values and evaluate P: ⎛ rev 2π rad 1 min ⎞ ⎟ = 155 kW P = (400 N ⋅ m )⎜ 3700 × × ⎜ min rev 60 s ⎟ ⎝ ⎠ 61 •• Picture the Problem Let r1 and r2 be the distances of m1 and m2 from the center of mass. We can use the definition of rotational kinetic energy and the definition of the center of mass of the two point masses to show that K1/K2 = m2/m1. Iω12 m1r12ω 2 m1r12 = = 2 Iω 2 m2 r22ω 2 m2 r22 Use the definition of rotational kinetic energy to express the ratio of the rotational kinetic energies: K1 = K2 Use the definition of the center of mass to relate m1, m2, r1, and r2: r1 m1 = r2 m2 r Solve for 1 , substitute and r2 K1 m1 ⎛ m2 ⎞ m2 ⎜ = ⎜m ⎟ = m ⎟ K 2 m2 ⎝ 1 ⎠ 1 simplify to obtain: 1 2 1 2 2 62 •• Picture the Problem The earth’s rotational kinetic energy is given by K rot = 1 Iω 2 where I is its moment of inertia with respect to its axis of rotation. The 2 656 Chapter 9 center of mass of the earth-sun system is so close to the center of the sun and the earthsun distance so large that we can use the earth-sun distance as the separation of their centers of mass and assume each to be point mass. Express the rotational kinetic energy of the earth: K rot = 1 Iω 2 2 Find the angular speed of the earth’s rotation using the definition of ω: ω= From Table 9-1, for the moment of inertia of a homogeneous sphere, we find: I = 2 MR 2 5 Substitute numerical values in equation (1) to obtain: K rot = (1) 2π rad ∆θ = ∆t 24 h × 3600 s h −5 = 7.27 × 10 rad/s = 2 5 (6.0 ×10 24 )( kg 6.4 × 106 m ) 2 = 9.83 × 1037 kg ⋅ m 2 (9.83 ×10 kg ⋅ m ) × (7.27 × 10 rad/s ) 1 2 37 2 2 −5 = 2.60 × 10 29 J Express the earth’s orbital kinetic energy: 2 K orb = 1 Iω orb 2 Find the angular speed of the center of mass of the earth-sun system: ω= ∆θ ∆t (2) 2π rad = 365.25 days × 24 h 3600 s × day h = 1.99 × 10−7 rad/s Express and evaluate the orbital moment of inertia of the earth: 2 I = M E Rorb ( )( = 6.0 × 1024 kg 1.50 × 1011 m = 1.35 × 1047 kg ⋅ m 2 Substitute in equation (2) to obtain: K orb = (1.35 ×10 kg ⋅ m ) × (1.99 × 10 rad/s ) 1 2 47 −7 = 2.67 × 1033 J 2 2 ) 2 Rotation 657 Evaluate the ratio K orb : K rot K orb 2.67 × 1033 J = ≈ 10 4 29 K rot 2.60 × 10 J *63 •• Picture the Problem Because the load is not being accelerated, the tension in the cable equals the weight of the load. The role of the massless pulley is to change the direction the force (tension) in the cable acts. (a) Because the block is lifted at constant speed: (b) Apply the definition of torque at the winch drum: (c) Relate the angular speed of the winch drum to the rate at which the load is being lifted (the tangential speed of the cable on the drum): (d) Express the power developed by the motor in terms of the tension in the cable and the speed with which the load is being lifted: ( T = mg = (2000 kg ) 9.81m/s 2 ) = 19.6 kN τ = Tr = (19.6 kN )(0.30 m ) = 5.89 kN ⋅ m ω= v 0.08 m/s = = 0.267 rad/s r 0.30 m P = Tv = (19.6 kN )(0.08 m/s ) = 1.57 kW 64 •• Picture the Problem Let the zero of gravitational potential energy be at the lowest point of the small particle. We can use conservation of energy to find the angular velocity of the disk when the particle is at its lowest point and Newton’s 2nd law to find the force the disk will have to exert on the particle to keep it from falling off. (a) Use conservation of energy to relate the initial potential energy of the system to its rotational kinetic energy when the small particle is at its lowest point: Solve for ωf: ∆K + ∆U = 0 or, because Uf = Ki = 0, 1 2 (I disk ωf = + I particle )ωf2 − mg∆h = 0 2mg∆h I disk + I particle 658 Chapter 9 Substitute for Idisk, Iparticle, and ∆h and simplify to obtain: 2mg (2 R ) = 1 MR 2 + mR 2 2 ωf = (b) The mass is in uniform circular motion at the bottom of the disk, so the sum of the force F exerted by the disk and the gravitational force must be the centripetal force: F − mg = mRωf2 Solve for F and simplify to obtain: 8mg R(2m + M ) F = mg + mRω f2 ⎛ 8mg ⎞ = mg + mR⎜ ⎜ R(2mM ) ⎟ ⎟ ⎝ ⎠ 8m ⎞ ⎛ = mg ⎜1 + ⎟ ⎝ 2m + M ⎠ 65 •• Picture the Problem Let the zero of gravitational potential energy be at the center of mass of the ring when it is directly below the point of support. We’ll use conservation of energy to relate the maximum angular velocity and the initial angular velocity required for a complete revolution to the changes in the potential energy of the ring. (a) Use conservation of energy to relate the initial potential energy of the ring to its rotational kinetic energy when its center of mass is directly below the point of support: ∆K + ∆U = 0 or, because Uf = Ki = 0, 1 2 2 I Pω max − mg∆h = 0 Use the parallel axis theorem and Table 9-1 to express the moment of inertia of the ring with respect to its pivot point P: I P = I cm + mR 2 Substitute in equation (1) to obtain: 1 2 (1) Solve for ωmax: Substitute numerical values and evaluate ωmax: (mR 2 ) 2 + mR 2 ωmax − mgR = 0 ωmax = g R ωmax = 9.81 m/s 2 = 3.62 rad/s 0.75 m Rotation 659 (b) Use conservation of energy to relate the final potential energy of the ring to its initial rotational kinetic energy: Noting that the center of mass must rise a distance R if the ring is to make a complete revolution, substitute for IP and ∆h to obtain: Solve for ωi: Substitute numerical values and evaluate ωi: ∆K + ∆U = 0 or, because Ui = Kf = 0, − 1 I Pω i2 + mg∆h = 0 2 − 1 (mR 2 + mR 2 )ωi2 + mgR = 0 2 g R ωi = 9.81 m/s 2 ωi = = 3.62 rad/s 0.75 m 66 •• Picture the Problem We can find the energy that must be stored in the flywheel and relate this energy to the radius of the wheel and use the definition of rotational kinetic energy to find the wheel’s radius. Relate the kinetic energy of the flywheel to the energy it must deliver: Express the moment of inertia of the flywheel: Substitute for Icyl and solve for ω: Substitute numerical values and evaluate R: K rot = 1 I cylω 2 = (2 MJ/km )(300 km ) 2 = 600 MJ I cyl = 1 MR 2 2 R= 2 ω K rot M 2 R= 400 rev 2π rad × s rev 106 J 600 MJ × MJ 100 kg = 1.95 m 67 •• Picture the Problem We’ll solve this problem for the general case of a ladder of length L, mass M, and person of mass m. Let the zero of gravitational potential energy be at floor level and include you, the ladder, and the earth in the system. We’ll use 660 Chapter 9 conservation of energy to relate your impact speed falling freely to your impact speed riding the ladder to the ground. Use conservation of energy to relate the speed with which a person will strike the ground to the fall distance L: ∆K + ∆U = 0 or, because Ki = Uf = 0, 1 2 mvf2 − mgL = 0 Solve for vf2 : vf2 = 2 gL Letting ωr represent the angular velocity of the ladder+person system as it strikes the ground, use conservation of energy to relate the initial and final momenta of the system: ∆K + ∆U = 0 Substitute for the moments of inertia to obtain: Substitute vr for Lωf and solve for vr2 : Express the ratio vr2 : vf2 Solve for vr to obtain: or, because Ki = Uf = 0, 1 2 1 2 (I person L⎞ ⎛ + I ladder )ωr2 − ⎜ mgL + Mg ⎟ = 0 2⎠ ⎝ 1 ⎞ 2 2 ⎛ L⎞ ⎛ ⎜ m + M ⎟ L ω f − ⎜ mgL + Mg ⎟ = 0 3 ⎠ 2⎠ ⎝ ⎝ M⎞ ⎛ 2 gL⎜ m + ⎟ 2 ⎠ ⎝ v r2 = M m+ 3 v r2 = vf2 M 2 M m+ 3 m+ vr = vf 6m + 3M 6m + 2 M Unless M , the mass of the ladder, is zero, vr > vf . It is better to let go and fall to the ground. Rotation 661 Pulleys, Yo-Yos, and Hanging Things *68 •• Picture the Problem We’ll solve this problem for the general case in which the mass of the block on the ledge is M, the mass of the hanging block is m, and the mass of the pulley is Mp, and R is the radius of the pulley. Let the zero of gravitational potential energy be 2.5 m below the initial position of the 2-kg block and R represent the radius of the pulley. Let the system include both blocks, the shelf and pulley, and the earth. The initial potential energy of the 2-kg block will be transformed into the translational kinetic energy of both blocks plus rotational kinetic energy of the pulley. (a) Use energy conservation to relate the speed of the 2 kg block when it has fallen a distance ∆h to its initial potential energy and the kinetic energy of the system: Substitute for Ipulley and ω to obtain: Solve for v: Substitute numerical values and evaluate v: ∆K + ∆U = 0 or, because Ki = Uf = 0, 1 2 (m + M )v 2 + 1 I pulleyω 2 − mgh = 0 2 1 2 (m + M )v 2 + 1 (1 MR 2 ) v 2 − mgh = 0 2 2 2 R v= 2mgh M +m+ 1 Mp 2 v= 2(2 kg ) 9.81 m/s 2 (2.5 m ) 4kg + 2 kg + 1 (0.6 kg ) 2 ( ) = 3.95 m/s (b) Find the angular velocity of the pulley from its tangential speed: 69 •• Picture the Problem The diagrams show the forces acting on each of the masses and the pulley. We can apply Newton’s 2nd law to the two blocks and the pulley to obtain three equations in the unknowns T1, T2, and a. ω= v 3.95 m/s = = 49.3 rad/s R 0.08 m 662 Chapter 9 Apply Newton’s 2nd law to the two blocks and the pulley: ∑F =T = m a , ∑τ = (T − T ) r = I α , x p 1 (1) 4 2 1 (2) p and ∑F x = m2 g − T2 = m2 a Eliminate α in equation (2) to obtain: T2 − T1 = 1 M p a 2 Eliminate T1 and T2 between equations (1), (3) and (4) and solve for a: a= (3) (4) m2 g m2 + m4 + 1 M p 2 Substitute numerical values and evaluate a: (2 kg )(9.81m/s2 ) = a= 2 kg + 4 kg + 1 (0.6 kg ) 2 Using equation (1), evaluate T1: T1 = (4 kg ) 3.11 m/s 2 = 12.5 N Solve equation (3) for T2: T2 = m2 ( g − a ) Substitute numerical values and evaluate T2: ( 3.11 m/s 2 ) ( T2 = (2 kg ) 9.81 m/s 2 − 3.11 m/s 2 ) = 13.4 N 70 •• Picture the Problem We’ll solve this problem for the general case in which the mass of the block on the ledge is M, the mass of the hanging block is m, the mass of the pulley is Mp, and R is the radius of the pulley. Let the zero of gravitational potential energy be 2.5 m below the initial position of the 2-kg block. The initial potential energy of the 2-kg block will be transformed into the translational kinetic energy of both blocks plus rotational kinetic energy of the pulley plus work done against friction. (a) Use energy conservation to relate the speed of the 2 kg block when it has fallen a distance ∆h to its initial potential energy, the kinetic energy of the system and the work done against friction: Substitute for Ipulley and ω to obtain: ∆K + ∆U + Wf = 0 or, because Ki = Uf = 0, 1 2 (m + M )v 2 + 1 I pulleyω 2 2 − mgh + µ k Mgh = 0 (m + M )v 2 + 1 (1 M p ) v 2 2 2 2 1 2 R − mgh + µ k Mgh = 0 Rotation 663 Solve for v: 2 gh(m − µ k M ) M +m+ 1 Mp 2 v= Substitute numerical values and evaluate v: v= ( ) 2 9.81m/s 2 (2.5 m )[2 kg − (0.25)(4 kg )] = 2.79 m/s 4kg + 2 kg + 1 (0.6 kg ) 2 (b) Find the angular velocity of the pulley from its tangential speed: ω= v 2.79 m/s = = 34.9 rad/s R 0.08 m 71 •• Picture the Problem Let the zero of gravitational potential energy be at the water’s surface and let the system include the winch, the car, and the earth. We’ll apply energy conservation to relate the car’s speed as it hits the water to its initial potential energy. Note that some of the car’s initial potential energy will be transformed into rotational kinetic energy of the winch and pulley. Use energy conservation to relate the car’s speed as it hits the water to its initial potential energy: Express ωw and ωp in terms of the speed v of the rope, which is the same throughout the system: Substitute to obtain: Solve for v: Substitute numerical values and evaluate v: ∆K + ∆U = 0 or, because Ki = Uf = 0, 1 2 2 2 mv 2 + 1 I w ω w + 1 I pω p − mg∆h = 0 2 2 v2 v2 and ωp = 2 2 rw rp ωw = 1 2 v2 1 v2 mv + I w 2 + 2 I p 2 − mg∆h = 0 rw rp 2 v= v= 1 2 2mg∆h I I m+ w + p 2 rw rp2 ( ) 2(1200 kg ) 9.81 m/s 2 (5 m ) 320 kg ⋅ m 2 4 kg ⋅ m 2 1200 kg + + (0.8 m )2 (0.3 m )2 = 8.21 m/s 664 Chapter 9 *72 •• Picture the Problem Let the system include the blocks, the pulley and the earth. Choose the zero of gravitational potential energy to be at the ledge and apply energy conservation to relate the impact speed of the 30-kg block to the initial potential energy of the system. We can use a constant-acceleration equations and Newton’s 2nd law to find the tensions in the strings and the descent time. (a) Use conservation of energy to relate the impact speed of the 30-kg block to the initial potential energy of the system: Substitute for ωp and Ip to obtain: ∆K + ∆U = 0 or, because Ki = Uf = 0, 1 2 2 m30 v 2 + 1 m20 v 2 + 1 I pω p 2 2 + m20 g∆h − m30 g∆h = 0 1 2 m30 v + m20 v + 2 1 2 2 ( 1 1 2 2 M pr 2 ) ⎛ v2 ⎞ ⎜ 2⎟ ⎜r ⎟ ⎝ ⎠ + m20 g∆h − m30 g∆h = 0 Solve for v: Substitute numerical values and evaluate v: 2 g∆h(m30 − m20 ) m20 + m30 + 1 M p 2 v= ( ) 2 9.81 m/s 2 (2 m )(30 kg − 20 kg ) v= 20 kg + 30 kg + 1 (5 kg ) 2 = 2.73 m/s v 2.73 m/s = = 27.3 rad/s r 0.1 m (b) Find the angular speed at impact from the tangential speed at impact and the radius of the pulley: ω= (c) Apply Newton’s 2nd law to the blocks: ∑F ∑F Using a constant-acceleration equation, relate the speed at impact to the fall distance and the x = T1 − m20 g = m20 a (1) x = m30 g − T2 = m30 a (2) 2 v 2 = v0 + 2a∆h or, because v0 = 0, Rotation 665 acceleration and solve for and evaluate a: (2.73 m/s) = 1.87 m/s 2 v2 a= = 2∆h 2(2 m ) Substitute in equation (1) to find T1: T1 = m20 ( g + a ) 2 ( = (20 kg ) 9.81 m/s 2 + 1.87 m/s 2 ) = 234 N Substitute in equation (2) to find T2: T2 = m30 (g − a ) ( = (30 kg ) 9.81m/s 2 − 1.87 m/s 2 = 238 N (d) Noting that the initial speed of the 30-kg block is zero, express the time-of-fall in terms of the fall distance and the block’s average speed: ∆t = ∆h ∆h 2∆h = = vav 1 v v 2 Substitute numerical values and evaluate ∆t: ∆t = 2(2 m ) = 1.47 s 2.73 m/s ∑τ = TR = I sphereα (1) = mg − T = ma (2) 73 •• Picture the Problem The force diagram shows the forces acting on the sphere and the hanging object. The tension in the string is responsible for the angular acceleration of the sphere and the difference between the weight of the object and the tension is the net force acting on the hanging object. We can use Newton’s 2nd law to obtain two equations in a and T that we can solve simultaneously. (a)Apply Newton’s 2nd law to the sphere and the hanging object: 0 and ∑F x Substitute for Isphere and α in equation (1) to obtain: TR = ( 2 5 MR 2 a )R (3) ) 666 Chapter 9 Eliminate T between equations (2) and (3) and solve for a to obtain: (b) Substitute for a in equation (2) and solve for T to obtain: g 2M 1+ 5m a= T= 2mMg 5m + 2 M 74 •• Picture the Problem The diagram shows the forces acting on both objects and the pulley. By applying Newton’s 2nd law of motion, we can obtain a system of three equations in the unknowns T1, T2, and a that we can solve simultaneously. (a) Apply Newton’s 2nd law to the pulley and the two objects: ∑F =T −m g = m a, ∑ τ = (T − T )r = I α , x 0 1 2 1 1 (1) 1 0 (2) and ∑F x = m2 g − T2 = m2 a Substitute for I0 = Ipulley and α in equation (2) to obtain: (T2 − T1 ) r = (1 mr 2 ) a 2 Eliminate T1 and T2 between equations (1), (3) and (4) and solve for a to obtain: a= Substitute numerical values and evaluate a: a= r (3) (4) (m2 − m1 )g m1 + m2 + 1 m 2 (510 g − 500 g )(981cm/s2 ) 500 g + 510 g + 1 (50 g ) 2 = 9.478 cm/s 2 (b) Substitute for a in equation (1) and solve for T1 to obtain: T1 = m1 (g + a ) ( = (0.500 kg ) 9.81 m/s 2 + 0.09478 m/s 2 = 4.9524 N ) Rotation 667 Substitute for a in equation (3) and solve for T2 to obtain: T2 = m2 ( g − a ) ( = (0.510 kg ) 9.81m/s 2 − 0.09478 m/s 2 = 4.9548 N ∆T = T2 − T1 = 4.9548 N − 4.9524 N Find ∆T: = 0.0024 N (c) If we ignore the mass of the pulley, our acceleration equation is: a= (m2 − m1 )g Substitute numerical values and evaluate a: a= (510 g − 500 g )(981cm/s2 ) m1 + m2 500 g + 510 g = 9.713 cm/s 2 Substitute for a in equation (1) and solve for T1 to obtain: T1 = m1 ( g + a ) Substitute numerical values and evaluate T1: ( ) T1 = (0.500 kg ) 9.81 m/s 2 + 0.09713 m/s 2 = 4.9536 N From equation (4), if m = 0: T1 = T2 *75 •• Picture the Problem The diagram shows the forces acting on both objects and the pulley. By applying Newton’s 2nd law of motion, we can obtain a system of three equations in the unknowns T1, T2, and α that we can solve simultaneously. (a) Express the condition that the system does not accelerate: τ net = m1 gR1 − m2 gR2 = 0 ) 668 Chapter 9 R1 R2 Solve for m2: m2 = m1 Substitute numerical values and evaluate m2: m 2 = (24 kg ) (b) Apply Newton’s 2nd law to the objects and the pulley: ∑F ∑τ 1.2 m = 72.0 kg 0.4 m x = m1 g − T1 = m1 a , 0 = T1 R1 − T2 R2 = I 0α , (2) (1) and ∑F x Eliminate a in favor of α in equations (1) and (3) and solve for T1 and T2: Substitute for T1 and T2 in equation (2) and solve for α to obtain: = T2 − m2 g = m2 a T1 = m1 ( g − R1α ) (3) (4) and T2 = m2 ( g + R2α ) α= (5) (m1 R1 − m2 R2 )g 2 m1 R12 + m2 R2 + I 0 Substitute numerical values and evaluate α: α= [(36 kg )(1.2 m ) − (72 kg )(0.4 m )](9.81 m/s 2 ) (36 kg )(1.2 m )2 + (72 kg )(0.4 m )2 + 40 kg ⋅ m 2 = 1.37 rad/s 2 Substitute in equation (4) to find T1: [ ( )] ( )] T1 = (36 kg ) 9.81 m/s 2 − (1.2 m ) 1.37 rad/s 2 = 294 N Substitute in equation (5) to find T2: [ T2 = (72 kg ) 9.81 m/s 2 + (0.4 m ) 1.37 rad/s 2 = 746 N Rotation 669 76 •• Picture the Problem Choose the coordinate system shown in the diagram. By applying Newton’s 2nd law of motion, we can obtain a system of two equations in the unknowns T and a. In (b) we can use the torque equation from (a) and our value for T to findα. In (c) we use the condition that the acceleration of a point on the rim of the cylinder is the same as the acceleration of the hand, together with the angular acceleration of the cylinder, to find the acceleration of the hand. (a) Apply Newton’s 2nd law to the cylinder about an axis through its center of mass: ∑τ 0 = TR = I 0 a R (1) and ∑F x = Mg − T = 0 Solve for T to obtain: T = Mg (b) Rewrite equation (1) in terms of α: TR = I 0α Solve for α: α= TR I0 Substitute for T and I0 to obtain: α= MgR 2g = 2 1 R 2 MR (c) Relate the acceleration a of the hand to the angular acceleration of the cylinder: a = Rα Substitute for α to obtain: ⎛ 2g ⎞ a = R⎜ ⎟ = 2g ⎝ R ⎠ (2) 670 Chapter 9 77 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. By applying Newton’s 2nd law to the cylinder and the block we can obtain simultaneous equations in a, T, and α from which we can express a and T. By applying the conservation of energy, we can derive an expression for the speed of the block when it reaches the bottom of the incline. (a) Apply Newton’s 2nd law to the cylinder and the block: ∑τ 0 x (b) Substitute for a in equation (2) and solve for T: (1) and ∑F Substitute for α in equation (1), solve for T, and substitute in equation (2) and solve for a to obtain: = TR = I 0α a= T= = m2 g sin θ − T = m2 a g sin θ m 1+ 1 2m2 1 2 m1 g sin θ m 1+ 1 2m2 (c) Noting that the block is released from rest, express the total energy of the system when the block is at height h: E = U + K = m2 gh (d) Use the fact that this system is conservative to express the total energy at the bottom of the incline: Ebottom = m2 gh (e) Express the total energy of the system when the block is at the bottom of the incline in terms of its kinetic energies: Ebottom = K tran + K rot = 1 m2 v 2 + 1 I 0ω 2 2 2 (2) Rotation 671 Substitute for ω and I0 to obtain: Solve for v to obtain: (f) For θ = 0: For θ = 90°: 1 2 m2 v + 2 1 2 ( 1 2 2 ) 1 2 m1a , v2 m1r = m2 gh r2 2 gh m 1+ 1 2 m2 v= a =T =0 g , m 1+ 1 2 m2 a= T= m1 g = m 1+ 1 2 m2 1 2 and 2 gh m 1+ 1 2 m2 v= For m1 = 0: a = g sin θ , T = 0 , and v= 2 gh *78 •• Picture the Problem Let r be the radius of the concentric drum (10 cm) and let I0 be the moment of inertia of the drum plus platform. We can use Newton’s 2nd law in both translational and rotational forms to express I0 in terms of a and a constantacceleration equation to express a and then find I0. We can use the same equation to find the total moment of inertia when the object is placed on the platform and then subtract to find its moment of inertia. (a) Apply Newton’s 2nd law to the platform and the weight: ∑τ ∑F 0 = Tr = I 0α (1) x = Mg − T = Ma (2) 672 Chapter 9 Substitute a/r for α in equation (1) and solve for T: T= I0 a r2 Substitute for T in equation (2) and solve for a to obtain: I0 = Mr 2 ( g − a ) a Using a constant-acceleration equation, relate the distance of fall to the acceleration of the weight and the time of fall and solve for the acceleration: (3) ∆x = v0 ∆t + 1 a (∆t ) 2 2 or, because v0 = 0 and ∆x = D, a= 2D (∆t )2 Substitute for a in equation (3) to obtain: ⎛ g (∆t )2 ⎞ ⎛g ⎞ − 1⎟ I 0 = Mr 2 ⎜ − 1⎟ = Mr 2 ⎜ ⎜ 2D ⎟ ⎝a ⎠ ⎝ ⎠ Substitute numerical values and evaluate I0: I 0 = (2.5 kg )(0.1 m ) 2 ( ) ⎡ 9.81 m/s 2 (4.2 s )2 ⎤ ×⎢ − 1⎥ 2(1.8 m ) ⎣ ⎦ = 1.177 kg ⋅ m 2 (b) Relate the moments of inertia of the platform, drum, shaft, and pulley (I0) to the moment of inertia of the object and the total moment of inertia: ⎛g ⎞ I tot = I 0 + I = Mr 2 ⎜ − 1⎟ ⎝a ⎠ Substitute numerical values and evaluate Itot: I tot = (2.5 kg )(0.1 m ) ⎛ g (∆t )2 ⎞ = Mr ⎜ − 1⎟ ⎜ 2D ⎟ ⎝ ⎠ 2 2 ( ) ⎡ 9.81m/s 2 (6.8 s )2 ⎤ ×⎢ − 1⎥ 2(1.8 m ) ⎣ ⎦ = 3.125 kg ⋅ m 2 Solve for and evaluate I: I = I tot − I 0 = 3.125 kg ⋅ m 2 − 1.177 kg ⋅ m 2 = 1.948 kg ⋅ m 2 Rotation 673 Objects Rolling Without Slipping *79 •• Picture the Problem The forces acting on the yo-yo are shown in the figure. We can use a constant-acceleration equation to relate the velocity of descent at the end of the fall to the yo-yo’s acceleration and Newton’s 2nd law in both translational and rotational form to find the yo-yo’s acceleration. Using a constant-acceleration equation, relate the yo-yo’s final speed to its acceleration and fall distance: 2 v 2 = v0 + 2a∆h or, because v0 = 0, v = 2a∆h ∑F (1) x = mg − T = ma (2) 0 = Tr = I 0α (3) Use Newton’s 2nd law to relate the forces that act on the yo-yo to its acceleration: and Use a = rα to eliminate α in equation (3) Tr = I 0 Eliminate T between equations (2) and (4) to obtain: mg − Substitute 1 2 mR 2 for I0 in equation (5): ∑τ mg − Solve for a: a= Substitute numerical values and evaluate a: a= Substitute in equation (1) and evaluate v: a r (4) I0 a = ma r2 1 2 (5) mR 2 a = ma r2 g R2 1+ 2 2r 9.81 m/s 2 = 0.0864 m/s 2 2 (1.5 m ) 1+ 2 2(0.1 m ) ( ) v = 2 0.0864 m/s 2 (57 m ) = 3.14 m/s 674 Chapter 9 80 •• Picture the Problem The diagram shows the forces acting on the cylinder. By applying Newton’s 2nd law of motion, we can obtain a system of two equations in the unknowns T, a, and α that we can solve simultaneously. (a) Apply Newton’s 2nd law to the cylinder: ∑τ 0 = TR = I 0α (1) = Mg − T = Ma (2) and ∑F x ( ) ⎛a⎞ MR 2 ⎜ ⎟ ⎝R⎠ Substitute for α and I0 in equation (1) to obtain: TR = Solve for T: T = 1 Ma 2 Substitute for T in equation (2) and solve for a to obtain: a= (b) Substitute for a in equation (3) to obtain: T = 1 M (2 g ) = 2 3 2 3 1 2 (3) g 1 3 Mg 81 •• Picture the Problem The forces acting on the yo-yo are shown in the figure. Apply Newton’s 2nd law in both translational and rotational form to obtain simultaneous equations in T, a, and α from which we can eliminate α and solve for T and a. Apply Newton’s 2nd law to the yo-yo: ∑F x = mg − T = ma (1) 0 = Tr = I 0α (2) and ∑τ Use a = rα to eliminate α in equation (2) Tr = I 0 a r (3) Rotation 675 Eliminate T between equations (1) and (3) to obtain: Substitute 1 2 mR 2 for I0 in equation (4): mg − (4) mR 2 a = ma mg − r2 1 2 Solve for a: a= Substitute numerical values and evaluate a: a= Use equation (1) to solve for and evaluate T: I0 a = ma r2 g R2 1+ 2 2r 9.81 m/s 2 = 0.192 m/s 2 2 (0.1 m ) 1+ 2 2(0.01 m ) T = m( g − a ) ( = (0.1 kg ) 9.81 m/s 2 − 0.192 m/s 2 ) = 0.962 N *82 • Picture the Problem We can determine the kinetic energy of the cylinder that is due to its rotation about its center of mass by examining the ratio K rot K . ( )v r 2 Express the rotational kinetic energy of the homogeneous solid cylinder: K rot = 1 I cylω 2 = 2 Express the total kinetic energy of the homogeneous solid cylinder: K = K rot + K trans = 1 mv 2 + 1 mv 2 = 3 mv 2 4 2 4 Express the ratio K rot : K 1 2 1 2 mr 2 2 = 1 mv 2 4 K rot 1 mv 2 1 = 4 2 = 3 and (b) is correct. 3 K 4 mv 83 • Picture the Problem Any work done on the cylinder by a net force will change its kinetic energy. Therefore, the work needed to give the cylinder this motion is equal to its kinetic energy. Express the relationship between the work needed to stop the cylinder and its kinetic energy: W = ∆K = 1 mv 2 + 1 Iω 2 2 2 676 Chapter 9 Because the cylinder is rolling without slipping, its translational and angular speeds are related according to: v = rω Substitute for I (see Table 9-1) and ω and simplify to obtain: W = 1 mv 2 + 1 Iω 2 2 2 = 1 mv 2 + 1 2 2 ( 1 2 mr 2 )v r 2 2 = 3 mv 2 4 Substitute for m and v to obtain: W = 3 4 (60 kg )(5 m/s )2 = 1.13 kJ 84 • Picture the Problem The total kinetic energy of any object that is rolling without slipping is given by K = K trans + K rot . We can find the percentages associated with each motion by expressing the moment of inertia of the objects as kmr2 and deriving a general expression for the ratios of rotational kinetic energy to total kinetic energy and translational kinetic energy to total kinetic energy and substituting the appropriate values of k. Express the total kinetic energy associated with a rotating and translating object: K = K trans + K rot = 1 mv 2 + 1 Iω 2 2 2 ( = 1 mv 2 + 1 kmr 2 2 2 )v r 2 2 = 1 mv 2 + 1 kmv 2 = 1 mv 2 (1 + k ) 2 2 2 Express the ratio K rot : K Express the ratio K trans : K (a) Substitute k = 2/5 for a uniform sphere to obtain: 1 1 K rot kmv 2 k = 1 2 2 = = 1 1+ k 1+ 1 K 2 mv ( + k ) k 2 1 1 K trans mv = 1 22 = K 1 1+ k 2 mv ( + k ) 1 K rot = = 0.286 = 28.6% 1 K 1+ 0.4 and K trans 1 = = 0.714 = 71.4% K 1 + 0.4 Rotation 677 (b) Substitute k = 1/2 for a uniform cylinder to obtain: 1 K rot = = 33.3% 1 K 1+ 0.5 and K trans 1 = = 66.7% K 1 + 0.5 (c) Substitute k = 1 for a hoop to obtain: 1 K rot = = 50.0% 1 K 1+ 1 and K trans 1 = = 50.0% K 1+1 85 • Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. As the hoop rolls up the incline its translational and rotational kinetic energies are transformed into gravitational potential energy. We can use energy conservation to relate the distance the hoop rolls up the incline to its total kinetic energy at the bottom of the incline. Using energy conservation, relate the distance the hoop will roll up the incline to its kinetic energy at the bottom of the incline: Express Ki as the sum of the translational and rotational kinetic energies of the hoop: When a rolling object moves with speed v, its outer surface turns with a speed v also. Hence ω = v/r. Substitute for I and ω to obtain: ∆K + ∆U = 0 or, because Kf = Ui = 0, − Ki + Uf = 0 (1) K i = K trans + K rot = 1 mv 2 + 1 Iω 2 2 2 K i = 1 mv 2 + 2 1 2 (mr ) v r 2 2 2 = mv 2 Letting ∆h be the change in elevation of the hoop as it rolls up the incline and ∆L the distance it rolls along the incline, express Uf: U f = mg∆h = mg∆L sin θ Substitute in equation (1) to obtain: − mv 2 + mg∆L sin θ = 0 678 Chapter 9 v2 ∆L = g sin θ Solve for ∆L: Substitute numerical values and evaluate ∆L: ∆L = (15 m/s)2 (9.81m/s ) sin30° = 2 45.9 m *86 •• Picture the Problem From Newton’s 2nd law, the acceleration of the center of mass equals the net force divided by the mass. The forces acting on the sphere are its weight r r mg downward, the normal force Fn that balances the normal component of the weight, r and the force of friction f acting up the incline. As the sphere accelerates down the incline, the angular velocity of rotation must increase to maintain the nonslip condition. We can apply Newton’s 2nd law for rotation about a horizontal axis through the center of mass of the sphere to find α, which is related to the acceleration by the nonslip condition. r r r The only torque about the center of mass is due to f because both mg and Fn act through the center of mass. Choose the positive direction to be down the incline. r Apply ∑ F = ma to the sphere: r mg sin θ − f = macm Apply ∑τ = I α to the sphere: fr = I cmα cm Use the nonslip condition to eliminate α and solve for f: fr = I cm acm r and f = I cm acm r2 Substitute this result for f in equation (1) to obtain: mg sin θ − From Table 9-1 we have, for a solid sphere: I cm = 2 mr 2 5 I cm acm = macm r2 (1) Rotation 679 mg sin θ − 2 acm = macm 5 Substitute in equation (1) and simplify to obtain: Solve for and evaluate θ : ⎛ 7 acm ⎞ ⎟ ⎟ ⎝ 5g ⎠ θ = sin −1 ⎜ ⎜ ⎡ 7(0.2 g ) ⎤ = sin −1 ⎢ ⎥ = 16.3° ⎣ 5g ⎦ 87 •• Picture the Problem From Newton’s 2nd law, the acceleration of the center of mass equals the net force divided by the mass. The forces acting on the thin spherical shell are r r its weight mg downward, the normal force Fn that balances the normal component of the r weight, and the force of friction f acting up the incline. As the spherical shell accelerates down the incline, the angular velocity of rotation must increase to maintain the nonslip condition. We can apply Newton’s 2nd law for rotation about a horizontal axis through the center of mass of the sphere to find α, which is related to the acceleration by the nonslip r r condition. The only torque about the center of mass is due to f because both mg and r Fn act through the center of mass. Choose the positive direction to be down the incline. Apply r r mg sin θ − f = macm α to the thin fr = I cmα ∑ F = ma to the thin spherical shell: Apply ∑τ = I cm spherical shell: Use the nonslip condition to eliminate α and solve for f: fr = I cm Substitute this result for f in equation (1) to obtain: mg sin θ − From Table 9-1 we have, for a thin I cm = 2 mr 2 3 I acm and f = cm acm r2 r I cm acm = macm r2 (1) 680 Chapter 9 spherical shell: Substitute in equation (1) and simplify to obtain: Solve for and evaluate θ : mg sin θ − 2 acm = macm 3 5acm 3g 5(0.2 g ) = sin −1 = 19.5° 3g θ = sin −1 Remarks: This larger angle makes sense, as the moment of inertia for a given mass is larger for a hollow sphere than for a solid one. 88 •• Picture the Problem The three forces acting on the basketball are the weight of the ball, the normal force, and the force of friction. Because the weight can be assumed to be acting at the center of mass, and the normal force acts through the center of mass, the only force which exerts a torque about the center of mass is the frictional force. We can use Newton’s 2nd law to find a system of simultaneous equations that we can solve for the quantities called for in the problem statement. (a) Apply Newton’s 2nd law in both translational and rotational form to the ball: ∑F ∑F x = mg sin θ − f s = ma , (1) y = Fn − mg cosθ = 0 (2) and ∑τ = f s r = I 0α 0 (3) a r Because the basketball is rolling without slipping we know that: α= Substitute in equation (3) to obtain: fs r = I 0 From Table 9-1 we have: I 0 = 2 mr 2 3 Substitute for I0 and α in equation (4) and solve for fs: fs r = ( 2 3 a r mr 2 (4) )a ⇒ f r s = 2 ma 3 (5) Rotation 681 Substitute for fs in equation (1) and solve for a: a= 3 5 g sin θ (b) Find fs using equation (5): 3 f s = 2 m( 5 g sin θ ) = 3 (c) Solve equation (2) for Fn: Fn = mg cos θ Use the definition of fs,max to obtain: f s,max = µ s Fn = µ s mg cos θ max Use the result of part (b) to obtain: 2 5 Solve for θmax: 2 5 mg sin θ mg sin θ max = µs mg cos θ max θ max = tan −1 ( 5 µs ) 2 89 •• Picture the Problem The three forces acting on the cylinder are the weight of the cylinder, the normal force, and the force of friction. Because the weight can be assumed to be acting at the center of mass, and the normal force acts through the center of mass, the only force which exerts a torque about the center of mass is the frictional force. We can use Newton’s 2nd law to find a system of simultaneous equations that we can solve for the quantities called for in the problem statement. (a) Apply Newton’s 2nd law in both translational and rotational form to the cylinder: ∑F ∑F x = mg sin θ − f s = ma , (1) y = Fn − mg cosθ = 0 (2) and ∑τ 0 = f s r = I 0α (3) a r Because the cylinder is rolling without slipping we know that: α= Substitute in equation (3) to obtain: fs r = I 0 From Table 9-1 we have: I 0 = 1 mr 2 2 a r (4) 682 Chapter 9 Substitute for I0 and α in equation (4) and solve for fs: fs r = ( 1 2 2 3 g sin θ mr 2 )a ⇒ f r s = 1 ma 2 Substitute for fs in equation (1) and solve for a: a= (b) Find fs using equation (5): f s = 1 m( 2 g sin θ ) = 2 3 (c) Solve equation (2) for Fn: Fn = mg cos θ Use the definition of fs,max to obtain: f s,max = µ s Fn = µ s mg cos θ max Use the result of part (b) to obtain: 1 3 Solve for θmax: (5) θ max = tan −1 (3µs ) 1 3 mg sin θ mg sin θ max = µs mg cos θ max *90 •• Picture the Problem Let the zero of gravitational potential energy be at the elevation where the spheres leave the ramp. The distances the spheres will travel are directly proportional to their speeds when they leave the ramp. Express the ratio of the distances traveled by the two spheres in terms of their speeds when they leave the ramp: L' v'∆t v' = = L v∆t v Use conservation of mechanical energy to find the speed of the spheres when they leave the ramp: ∆K + ∆U = 0 or, because Ki = Uf = 0, Express Kf for the spheres: K f = K trans + K rot (1) Kf −Ui = 0 (2) = 1 mv 2 + 1 I cmω 2 2 2 ( = 1 mv 2 + 1 kmR 2 2 2 v )R 2 2 = 1 mv 2 + 1 kmv 2 2 2 = (1 + k ) 1 mv 2 2 where k is 2/3 for the spherical shell and 2/5 for the uniform sphere. Substitute in equation (2) to obtain: (1 + k ) 1 mv 2 = mgH 2 Rotation 683 Solve for v: 2 gH 1+ k v= 1+ 2 1+ k L' 3 = = = 1.09 2 1 + k' 1+ 5 L Substitute in equation (1) to obtain: or L' = 1.09 L 91 •• Picture the Problem Let the subscripts u and h refer to the uniform and thin-walled spheres, respectively. Because the cylinders climb to the same height, their kinetic energies at the bottom of the incline must be equal. Express the total kinetic energy of the thin-walled cylinder at the bottom of the inclined plane: K h = K trans + K rot = 1 mh v 2 + 1 I hω 2 2 2 Express the total kinetic energy of the solid cylinder at the bottom of the inclined plane: K u = K trans + K rot = 1 mu v'2 + 1 I uω 2 2 2 Because the cylinders climb to the same height: 3 4 = mh v + 2 1 2 1 2 = 1 mu v' 2 + 1 2 2 mu v' 2 = mu gh and mh v 2 = mh gh mu v' 2 mu gh = mh v 2 mh gh Divide the first of these equations by the second: 3 4 Simplify to obtain: 3v' 2 =1 4v 2 Solve for v′: v' = 4 v 3 ( ) v2 mh r = mh v 2 2 r ( 2 1 2 mu r 2 ) v' r 2 2 = 3 mu v' 2 4 684 Chapter 9 92 •• Picture the Problem Let the subscripts s and c refer to the solid sphere and thinwalled cylinder, respectively. Because the cylinder and sphere descend from the same height, their kinetic energies at the bottom of the incline must be equal. The force diagram shows the forces acting on the solid sphere. We’ll use Newton’s 2nd law to relate the accelerations to the angle of the incline and use a constant acceleration to relate the accelerations to the distances traveled down the incline. Apply Newton’s 2nd law to the sphere: ∑F ∑F x = mg sin θ − f s = ma s , (1) y = Fn − mg cosθ = 0 , (2) = f s r = I 0α (3) and ∑τ Substitute for I0 and α in equation (3) and solve for fs: Substitute for fs in equation (1) and solve for a: 0 fsr = ( 2 5 )a ⇒ f r mr 2 s 2 = 5 mas 5 as = 7 g sin θ Proceed as above for the thin-walled cylinder to obtain: a c = 1 g sin θ 2 Using a constant-acceleration equation, relate the distance traveled down the incline to its acceleration and the elapsed time: ∆s = v0 ∆t + 1 a(∆t ) 2 or, because v0 = 0, Because ∆s is the same for both objects: as t s2 = a c t c2 2 ∆s = 1 a(∆t ) 2 2 (4) where t c2 = (t s + 2.4 ) = t s2 + 4.8t s + 5.76 2 provided tc and ts are in seconds. Substitute for as and ac to obtain the quadratic equation: t s2 + 4.8t s + 5.76 = 10 t s2 7 Rotation 685 Solve for the positive root to obtain: t s = 12.3 s Substitute in equation (4), simplify, and solve for θ : θ = sin −1 ⎢ Substitute numerical values and evaluate θ : θ = sin −1 ⎢ ⎡14∆s ⎤ 2 ⎥ ⎣ 5 gts ⎦ ⎤ 14(3 m ) 2⎥ 2 ⎣ 5 9.81 m/s (12.3 s ) ⎦ ⎡ ( ) = 0.324° 93 ••• Picture the Problem The kinetic energy of the wheel is the sum of its translational and rotational kinetic energies. Because the wheel is a composite object, we can model its moment of inertia by treating the rim as a cylindrical shell and the spokes as rods. K = K trans + K rot Express the kinetic energy of the wheel: = 1 M tot v 2 + 1 I cmω 2 2 2 = 1 M tot v 2 + 1 I cm 2 2 v2 R2 where Mtot = Mrim + 4Mspoke Express the moment of inertia of the wheel: I cm = I rim + I spokes ( = M rim R 2 + 4 1 M spoke R 2 3 = (M rim + 4 M spoke )R 2 3 Substitute for Icm in the equation for K: [ K = 1 M tot v 2 + 1 (M rim + 4 M spoke )R 2 3 2 2 = Substitute numerical values and evaluate K: ) [ (M 1 2 tot + M rim ) + 2 M spoke v 2 3 vR 2 K = [ 1 (7.8 kg + 3 kg ) + 2 (1.2 kg )](6 m/s ) 2 3 2 = 223 J 2 686 Chapter 9 94 ••• Picture the Problem Let M represent the combined mass of the two disks and their connecting rod and I their moment of inertia. The object’s initial potential energy is transformed into translational and rotational kinetic energy as it rolls down the incline. The force diagram shows the forces acting on this composite object as it rolls down the incline. Application of Newton’s 2nd law will allow us to derive an expression for the acceleration of the object. (a) Apply Newton’s 2nd law to the disks and rod: ∑F ∑F x = Mg sin θ − f s = Ma , (1) y = Fn − Mg cos θ = 0 , (2) = f s r = Iα (3) and ∑τ Eliminate fs and α between equations (1) and (3) and solve for a to obtain: Express the moment of inertia of the two disks plus connecting rod: a= 0 Mg sin θ I M+ 2 r (4) I = 2 I disk + I rod ( ) = 2 1 mdisk R 2 + 1 mrod r 2 2 2 = mdisk R 2 + 1 mrod r 2 2 Substitute numerical values and evaluate I: I = (20 kg )(0.3 m ) + 1 (1 kg )(0.02 m ) 2 Substitute in equation (4) and evaluate a: a= 2 = 1.80 kg ⋅ m 2 (41kg )(9.81m/s 2 ) sin30° 41 kg + 1.80 kg ⋅ m 2 (0.02 m )2 = 0.0443 m/s 2 (b) Find α from a: α= a 0.0443 m/s 2 = = 2.21 rad/s 2 r 0.02 m 2 Rotation 687 (c) Express the kinetic energy of translation of the disks-plus-rod when it has rolled a distance ∆s down the incline: K trans = 1 Mv 2 2 Using a constant-acceleration equation, relate the speed of the disks-plus-rod to their acceleration and the distance moved: 2 v 2 = v0 + 2a∆s or, because v0 = 0, Substitute to obtain: K trans = Ma∆s v 2 = 2a∆s ( ) = (41 kg ) 0.0443 m/s 2 (2 m ) = 3.63 J (d) Express the rotational kinetic energy of the disks after rolling 2 m in terms of their initial potential energy and their translational kinetic energy: Substitute numerical values and evaluate Krot: K rot = U i − K trans = Mgh − K trans ( ) K rot = (41 kg ) 9.81m/s 2 (2 m )sin30° − 3.63 J = 399 J 95 ••• Picture the Problem We can express the coordinates of point P as the sum of the coordinates of the center of the wheel and the coordinates, relative to the center of the r wheel, of the tip of the vector r0 . Differentiation of these expressions with respect to time will give us the x and y components of the velocity of point P. (a) Express the coordinates of point P relative to the center of the wheel: Because the coordinates of the center of the circle are X and R: x = r0 cosθ and y = r0 sin θ (x P , y P ) = ( X + r0 cosθ , R + r0 sin θ ) 688 Chapter 9 (b) Differentiate xP to obtain: Note that d ( X + r0 cosθ ) dt dX dθ = − r0 sin θ ⋅ dt dt v Px = dX dθ V = V and = −ω = − so: dt dt R v Px = V + Differentiate yP to obtain: v Py = Because dθ V = −ω = − : dt R r r (c) Calculate v ⋅ r : r0V sin θ R d (R + r0 sin θ ) = r0 cos θ ⋅ dθ dt dt vPy = − r0V cosθ R r r v ⋅ r = v Px rx + v Py ry rV ⎛ ⎞ = ⎜V + 0 sin θ ⎟(r0 cos θ ) R ⎝ ⎠ ⎛rV ⎞ − ⎜ 0 cos θ ⎟(R + r0 sin θ ) ⎝ R ⎠ = 0 (d) Express v in terms of its components: 2 2 v = vx + v y 2 rV ⎞ ⎛ ⎞ ⎛ rV = ⎜V + 0 sin θ ⎟ + ⎜ − 0 cosθ ⎟ R ⎠ ⎝ ⎠ ⎝ R = V 1+ 2 Express r in terms of its components: r0 r2 sin θ + 02 R R r = rx2 + ry2 = (r0 cosθ )2 + (R + r0 sin θ )2 r0 r02 = R 1 + 2 sin θ + 2 R R Divide v by r to obtain: ω= v V = r R 2 Rotation 689 *96 ••• Picture the Problem Let the letter B identify the block and the letter C the cylinder. We can find the accelerations of the block and cylinder by applying Newton’s 2nd law and solving the resulting equations simultaneously. Apply ∑F = ma x to the block: F − f ' = maB (1) Apply ∑F = ma x to the cylinder: f = MaC , (2) Apply ∑τ fR = I CMα (3) Substitute for ICM in equation (3) and solve for f = f ′ to obtain: f = 1 MRα 2 (4) Relate the acceleration of the block to the acceleration of the cylinder: aC = aB + aCB x x CM = I CMα to the cylinder: or, because aCB = −Rα is the acceleration of the cylinder relative to the block, aC = aB − Rα and Equate equations (2) and (4) and substitute from (5) to obtain: Substitute equation (4) in equation (1) and substitute for aC to obtain: Solve for aB: 97 ••• Picture the Problem Let the letter B identify the block and the letter C the cylinder. In this problem, as in Problem 97, we can find the accelerations of the block and cylinder by applying Newton’s 2nd law and solving the resulting equations simultaneously. Rα = aB − aC aB = 3aC F − 1 Ma B = ma B 3 aB = 3F M + 3m (5) 690 Chapter 9 Apply ∑F = ma x to the block: F − f = maB (1) Apply ∑F = ma x to the cylinder: f = MaC , (2) Apply ∑τ fR = I CMα (3) Substitute for ICM in equation (3) and solve for f: f = 1 MRα 2 (4) Relate the acceleration of the block to the acceleration of the cylinder: aC = aB + aCB x x CM = I CMα to the cylinder: or, because aCB = −Rα, aC = aB − Rα and Rα = aB − aC (a) Solve for α and substitute for aB to obtain: (5) aB − aC 3aC − aC 2aC = = R R R 2F = R(M + 3m ) α= From the force diagram it is evident that the torque and, therefore, α is in the counterclockwise direction. (b) Equate equations (2) and (4) and substitute (5) to obtain: aB = 3aC From equations (1) and (4) we obtain: F − 1 Ma B = ma B 3 Solve for aB: aB = Substitute to obtain the linear acceleration of the cylinder relative to the table: 3F M + 3m aC = 1 a B = 3 F M + 3m Rotation 691 (c) Express the acceleration of the cylinder relative to the block: aCB = aC − aB = aC − 3aC = −2aC = − 2F M + 3m 98 ••• Picture the Problem Let the system include the earth, the cylinder, and the r block. Then F is an external force that changes the energy of the system by doing work on it. We can find the kinetic energy of the block from its speed when it has traveled a distance d. We can find the kinetic energy of the cylinder from the sum of its translational and rotational kinetic energies. In part (c) we can add the kinetic energies of the block and the cylinder to show that their sum is the work done by r F in displacing the system a distance d. (a) Express the kinetic energy of the block: 2 K B = Won block = 1 mvB 2 Using a constant-acceleration equation, relate the velocity of the block to its acceleration and the distance traveled: 2 2 v B = v0 + 2 a B d or, because the block starts from rest, Substitute to obtain: K B = 1 m(2aB d ) = maB d 2 2 vB = 2 a B d (1) Apply ∑F = ma x to the block: F − f = maB (2) Apply ∑F = ma x to the cylinder: f = MaC , (3) Apply ∑τ fR = I CMα (4) Substitute for ICM in equation (4) and solve for f: f = 1 MRα 2 (5) Relate the acceleration of the block to the acceleration of the cylinder: aC = aB + aCB x x CM = I CMα to the cylinder: or, because aCB = −Rα, 692 Chapter 9 aC = aB − Rα and Rα = aB − aC Equate equations (3) and (5) and substitute in (6) to obtain: aB = 3aC Substitute equation (5) in equation (2) and use aB = 3aC to obtain: (6) F − MaC = maB or F − 1 MaB = maB 3 Solve for aB: Substitute in equation (1) to obtain: (b) Express the total kinetic energy of the cylinder: F m+ 1M 3 aB = KB = mFd m+ 1M 3 2 K cyl = K trans + K rot = 1 MvC + 1 I CMω 2 2 2 2 = 1 MvC + 1 I CM 2 2 where vCB = vC − vB . In part (a) it was established that: aB = 3aC Integrate both sides of the equation with respect to time to obtain: (7) 2 vCB R2 vB = 3vC + constant Substitute the initial conditions to obtain: where the constant of integration is determined by the initial conditions that vC = 0 when vB = 0. constant = 0 and vB = 3vC Substitute in our expression for vCB to obtain: Substitute for ICM and vCB in equation (7) to obtain: vCB = vC − vB = vC − 3vC = −2vC 2 K cyl = 1 MvC + 1 2 2 = 3 Mv 2 2 C ( ) (− 2v ) R 2 1 2 MR 2 C 2 (8) Rotation 693 Because vC = 1 vB : 3 2 2 vC = 1 v B 9 It part (a) it was established that: 2 vB = 2 a B d and aB = Substitute to obtain: F m+ 1M 3 2 vC = 1 9 = ⎛ F 1 ⎝m+ 3M 2 (2aB d ) = 9 ⎜ ⎜ ⎞ ⎟d ⎟ ⎠ 2 Fd 9(m + 1 M ) 3 Substitute in equation (8) to obtain: ⎛ 2 Fd ⎞ K cyl = 3 M ⎜ 2 ⎜ 9(m + 1 M ) ⎟ ⎟ 3 ⎝ ⎠ MFd = 3(m + 1 M ) 3 (c) Express the total kinetic energy of the system and simplify to obtain: K tot = K B + K cyl = = mFd MFd + m + 1 M 3(m + 1 M ) 3 3 (3m + M ) Fd = 3(m + 1 M ) 3 Fd 99 •• Picture the Problem The forces responsible for the rotation of the gears are shown in the diagram to the right. The forces acting through the centers of mass of the two gears have been omitted because they produce no torque. We can apply Newton’s 2nd law in rotational form to obtain the equations of motion of the gears and the not slipping condition to relate their angular accelerations. (a) Apply ∑τ = Iα to the gears to obtain their equations of motion: Because the gears do not slip 2 N ⋅ m − FR1 = I1α1 (1) and FR2 = I 2α 2 (2) where F is the force keeping the gears from slipping with respect to each other. R1α1 = R2α 2 694 Chapter 9 relative to each other, the tangential accelerations of the points where they are in contact must be the same: or Divide equation (1) by R1 to obtain: 2 N⋅m I − F = 1 α1 R1 R1 Divide equation (2) by R2 to obtain: α2 = F= R1 α1 = 1 α1 2 R2 (3) I2 α2 R2 Add these equations to obtain: 2 N ⋅ m I1 I = α1 + 2 α 2 R1 R1 R2 Use equation (3) to eliminate α2: 2 N ⋅ m I1 I = α1 + 2 α1 R1 R1 2 R2 Solve for α1 to obtain: Substitute numerical values and evaluate α1: α1 = α1 = 2N⋅m R I1 + 1 I 2 2 R2 2N⋅m 0.5 m 1 kg ⋅ m 2 + 16 kg ⋅ m 2 2(1 m ) ( ) = 0.400 rad/s 2 (0.400 rad/s ) = Use equation (3) to evaluate α2: α2 = (b) To counterbalance the 2-N·m torque, a counter torque of 2 N·m must be applied to the first gear. Use equation (2) with α1 = 0 to find F: 2 N ⋅ m − FR1 = 0 and F= 1 2 2 0.200 rad/s 2 2N⋅m 2N⋅m = = 4.00 N R1 0.5 m Rotation 695 *100 •• Picture the Problem Let r be the radius of the marble, m its mass, R the radius of the large sphere, and v the speed of the marble when it breaks contact with the sphere. The numeral 1 denotes the initial configuration of the sphere-marble system and the numeral 2 is configuration as the marble separates from the sphere. We can use conservation of energy to relate the initial potential energy of the marble to the sum of its translational and rotational kinetic energies as it leaves the sphere. Our choice of the zero of potential energy is shown on the diagram. (a) Apply conservation of energy: ∆U + ∆K = 0 or U 2 − U 1 + K 2 − K1 = 0 Because U2 = K1 = 0: − mg [R + r − (R + r ) cos θ ] + 1 mv 2 + 1 Iω 2 = 0 2 2 or − mg [(R + r )(1 − cos θ )] + 1 mv 2 + 1 Iω 2 = 0 2 2 Use the rolling-without-slipping condition to eliminate ω: − mg [(R + r )(1 − cosθ )] + 1 mv 2 + 1 I 2 2 From Table 9-1 we have: Substitute to obtain: v2 =0 r2 2 I = 5 mr 2 − mg [(R + r )(1 − cosθ )] + 1 mv 2 + 1 2 2 ( 2 5 mr 2 )v r 2 2 =0 or − mg [(R + r )(1 − cos θ )] + 1 mv 2 + 1 mv 2 = 0 2 5 Solve for v2 to obtain: Apply ∑F r = mar to the marble as it separates from the sphere: v2 = 10 g (R + r )(1 − cosθ ) 7 mg cos θ = m or v2 R+r 696 Chapter 9 v2 cos θ = g (R + r ) Substitute for v2: cos θ = 1 ⎡10 ⎤ ⎢ 7 g (R + r )(1 − cos θ )⎥ g (R + r ) ⎣ ⎦ ⎡10 ⎤ = ⎢ (1 − cos θ )⎥ ⎣7 ⎦ Solve for and evaluate θ : ⎛ 10 ⎞ ⎟ = 54.0° ⎝ 17 ⎠ θ = cos −1 ⎜ The force of friction is always less than µs multiplied by the normal force on the marble. However, the normal force decreases to 0 at the (b) point where the ball leaves the sphere, meaning that the force of friction must be less than the force needed to keep the ball rolling without slipping before it leaves the sphere. Rolling With Slipping 101 • Picture the Problem Part (a) of this problem is identical to Example 9-16. In part (b) we can use the definitions of translational and rotational kinetic energy to find the ratio of the final and initial kinetic energies. (a) From Example 9-16: s1 = 2 12 v0 , 49 µ k g t1 = 2 v0 , and 7 µk g v1 = 5 µ k gt1 = 2 (b) When the ball rolls without slipping, v1 = rω. Express the final kinetic energy of the ball: 5 v0 7 K f = K trans + K rot = 1 Mv12 + 1 Iω 2 2 2 = 1 Mv12 + 1 2 2 ( 2 5 Mr 2 2 7 5 = 10 Mv12 = 14 Mv0 )v r 2 1 2 Rotation 697 2 Mv0 5 = 2 Mv0 7 Express the ratio of the final and initial kinetic energies: Kf = Ki (c) Substitute in the expressions in (a) to obtain: (8 m/s) 12 s1 = = 26.6 m 49 (0.06) 9.81 m/s 2 5 14 1 2 2 ( ) t1 = 2 8 m/s = 3.88 s 7 (0.06) 9.81 m/s 2 v1 = 5 (8 m/s) = 5.71 m/s 7 ( ) *102 •• Picture the Problem The cue stick’s blow delivers a rotational impulse as well as a translational impulse to the cue ball. The rotational impulse changes the angular momentum of the ball and the translational impulse changes its linear momentum. Express the rotational impulse Prot as the product of the average torque and the time during which the rotational impulse acts: Express the average torque it produces about an axis through the center of the ball: Substitute in the expression for Prot to obtain: The translational impulse is also given by: Substitute to obtain: Solve for ω0: Prot = τ av ∆t τ av = P0 (h − r ) sin θ = P0 (h − r ) where θ (= 90°) is the angle between F and the lever arm h − r. Prot = P0 (h − r )∆t = (P0 ∆t )(h − r ) = Ptrams (h − r ) = ∆L = Iω0 Ptrans = P0 ∆t = ∆p = mv0 2 mv0 (h − r ) = 5 mr 2ω0 ω0 = 5v0 (h − r ) 2r 2 698 Chapter 9 103 •• Picture the Problem The angular velocity of the rotating sphere will decrease until the condition for rolling without slipping is satisfied and then it will begin to roll. The force diagram shows the forces acting on the sphere. We can apply Newton’s 2nd law to the sphere and use the condition for rolling without slipping to find the speed of the center of mass when the sphere begins to roll without slipping. Relate the velocity of the sphere when it begins to roll to its acceleration and the elapsed time: v = a∆t (1) Apply Newton’s 2nd law to the sphere: ∑F ∑F x = f k = ma , (2) y = Fn − mg = 0 , (3) = f k r = I 0α (4) and ∑τ 0 Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a: a = µk g Substitute in equation (1) to obtain: v = a∆t = µ k g∆t Solve for α in equation (4): α= Express the angular speed of the sphere when it has been moving for a time ∆t: ω = ω0 − α ∆t = ω0 − Express the condition that the sphere rolls without slipping: v = rω Substitute from equations (5) and (6) and solve for the elapsed time until the sphere begins to roll: ∆t = (5) fkr mar 5 µk g = 2 2 = I0 2 r 5 mr 2 rω 0 7 µk g 5µ k g ∆t 2r (6) Rotation 699 Use equation (4) to find v when the sphere begins to roll: v = µ k g∆t = 2rω0 2 rω 0 µ k g = 7 µk g 7 104 •• Picture the Problem The sharp force delivers a rotational impulse as well as a translational impulse to the ball. The rotational impulse changes the angular momentum of the ball and the translational impulse changes its linear momentum. In parts (c) and (d) we can apply Newton’s 2nd law to the ball to obtain equations describing both the translational and rotational motion of the ball. We can then solve these equations to find the constant accelerations that allow us to apply constant-acceleration equations to find the velocity of the ball when it begins to roll and its sliding time. (a) Relate the translational impulse delivered to the ball to its change in its momentum: Ptrans = Fav ∆t = ∆p = mv0 Solve for v0: v0 = Substitute numerical values and evaluate v0: v0 = Fav ∆t m (20 kN )(2 ×10−4 s ) = 0.02 kg 200 m/s (b) Express the rotational impulse Prot as the product of the average torque and the time during which the rotational impulse acts: Prot = τ av ∆t Letting h be the height at which the impulsive force is delivered, express the average torque it produces about an axis through the center of the ball: τ av = Fl sin θ where θ is the angle between F and the lever arm l . Substitute h − r for l and 90° for θ τ av = F (h − r ) 700 Chapter 9 to obtain: Substitute in the expression for Prot to obtain: Because Ptrans = F∆t: Prot = F (h − r )∆t Prot = Ptrans (h − r ) = ∆L = Iω0 2 = 5 mr 2ω0 Express the translational impulse delivered to the cue ball: Ptrans = P0 ∆t = ∆p = mv0 Substitute for Ptrans to obtain: 2 5 mr 2ω0 = mv0 Solve for ω0: ω0 = 5v0 (h − r ) 2r 2 Substitute numerical values and evaluate ω0: ω0 = 5(200 m/s )(0.09 m − 0.05 m ) 2 2(.05 m ) = 8000 rad/s (c) and (d) Relate the velocity of the ball when it begins to roll to its acceleration and the elapsed time: v = a∆t Apply Newton’s 2nd law to the ball: ∑F ∑F (1) x = f k = ma , (2) y = Fn − mg = 0 , (3) = f k r = I 0α (4) and ∑τ 0 Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a: a = µk g Substitute in equation (1) to obtain: v = a∆t = µ k g∆t Solve for α in equation (4): α= fkr mar 5 µk g = 2 2 = I0 2 r 5 mr (5) Rotation 701 5µ k g ∆t 2r Express the angular speed of the ball when it has been moving for a time ∆t: ω = ω0 − α ∆t = ω0 − Express the speed of the ball when it has been moving for a time ∆t: v = v 0 + µ k g∆t Express the condition that the ball rolls without slipping: v = rω Substitute from equations (6) and (7) and solve for the elapsed time until the ball begins to roll: ∆t = 2 rω 0 − v0 7 µk g ∆t = 2 ⎡ (0.05 m )(8000 rad/s ) − 200 m/s ⎤ ⎥ (0.5) 9.81m/s 2 7⎢ ⎣ ⎦ Substitute numerical values and evaluate ∆t: (6) (7) ( ) = 11.6 s Use equation (4) to express v when the ball begins to roll: Substitute numerical values and evaluate v: v = v0 + µ k g∆t ( ) v = 200 m/s + (0.5) 9.81 m/s 2 (11.6 s ) = 257 m/s 105 •• Picture the Problem Because the impulse is applied through the center of mass, ω0 = 0. We can use the results of Example 9-16 to find the rolling time without slipping, the distance traveled to rolling without slipping, and the velocity of the ball once it begins to roll without slipping. (a) From Example 9-16 we have: t1 = 2 v0 7 µk g Substitute numerical values and evaluate t1: t1 = 2 4 m/s = 0.194 s 7 (0.6) 9.81 m/s 2 s1 = 2 12 v0 49 µ k g (b) From Example 9-16 we have: ( ) 702 Chapter 9 Substitute numerical values and evaluate s1: (4 m/s) 12 s1 = = 0.666 m 49 (0.6) 9.81 m/s 2 (c) From Example 9-16 we have: v1 = 5 v0 7 Substitute numerical values and evaluate v1: v1 = 5 (4 m/s) = 2.86 m/s 7 2 ( ) 106 •• Picture the Problem Because the impulsive force is applied below the center line, the spin is backward, i.e., the ball will slow down. We’ll use the impulsemomentum theorem and Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constantacceleration equations to relate these quantities to each other and to the elapsed time to rolling without slipping. (a) Express the rotational impulse delivered to the ball: Prot = mv0 r = mv0 = Solve for ω0: ω0 = ∑τ ∑F ( 2 5 ) 2R = I cmω0 3 mR 2 ω0 5 v0 3 R = f k R = I cmα , (1) y = Fn − mg = 0 , (2) x (b) Apply Newton’s 2nd law to the ball to obtain: = − f k = ma (3) 0 and ∑F µ k mgR µ k mgR Using the definition of fk and Fn from equation (2), solve for α: α= Using a constant-acceleration equation, relate the angular speed of the ball to its acceleration: ω = ω0 + α∆t = ω0 + I cm = 2 5 mR 2 = 5µ k g 2R 5µ k g ∆t 2R Rotation 703 Using the definition of fk and Fn from equation (2), solve equation (3) for a: a = −µ k g Using a constant-acceleration equation, relate the speed of the ball to its acceleration: v = v0 + a∆t = v0 − µ k g∆t Impose the condition for rolling without slipping to obtain: 5µ g ⎞ ⎛ R⎜ ω 0 + k ∆t ⎟ = v0 − µ k g∆t 2R ⎝ ⎠ Solve for ∆t: Substitute in equation (4) to obtain: ∆t = (4) 16 v0 21 µ k g ⎛ 16 v0 ⎞ 5 v = v0 − µ k g ⎜ ⎟ ⎜ 21 µ g ⎟ = 21 v0 k ⎠ ⎝ = 0.238v0 (c) Express the initial kinetic energy of the ball: 2 2 K i = K trans + K rot = 1 mv0 + 1 Iω0 2 2 = mv + 1 2 2 0 ( 2 ) ⎛ 5v ⎞ 19 2 mR ⎜ 0 ⎟ = mv0 ⎝ 3R ⎠ 18 2 1 2 2 5 2 = 1.056mv0 (d) Express the work done by friction in terms of the initial and final kinetic energies of the ball: Wfr = K i − K f Express the final kinetic energy of the ball: K f = 1 mv 2 + 1 I cmω 2 2 2 = 1 mv 2 + 2 1 2 ( 2 5 mR 2 v )R 2 2 7 = 10 mv 2 2 7 = 10 m(0.238v0 ) = 0.0397 mv0 2 Substitute to find Wfr: 2 2 Wfr = 1.056mv0 − 0.0397 mv 0 2 = 1.016mv0 704 Chapter 9 107 •• Picture the Problem The figure shows the forces acting on the bowling during the sliding phase of its motion. Because the ball has a forward spin, the friction force is in the direction of motion and will cause the ball’s translational speed to increase. We’ll apply Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constantacceleration equations to relate these quantities to each other and to the elapsed time to rolling without slipping. (a) and (b) Relate the velocity of the ball when it begins to roll to its acceleration and the elapsed time: v = v0 + a∆t (1) Apply Newton’s 2nd law to the ball: ∑F ∑F x = f k = ma , (2) y = Fn − mg = 0 , (3) = f k R = I 0α (4) and ∑τ 0 Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a: a = µk g Substitute in equation (1) to obtain: v = v 0 + a∆t = v 0 + µ k g∆t Solve for α in equation (4): α= Relate the angular speed of the ball to its acceleration: ω = ω0 − Apply the condition for rolling without slipping: 5 µk g ⎞ ⎛ v = Rω = R⎜ ω0 − ∆t ⎟ 2 R ⎝ ⎠ 5 µk g ⎞ ⎛ 3v = R⎜ 0 − ∆t ⎟ ⎝ R 2 R ⎠ fk R maR 5 µk g = 2 = 2 I0 2 R 5 mR 5 µk g ∆t 2 R (5) Rotation 705 ∴ v = 3v0 − 5 µ k g∆t 2 Equate equations (5) and (6) and solve ∆t: ∆t = Substitute for ∆t in equation (6) to obtain: v= (c) Relate ∆x to the average speed of the ball and the time it moves before beginning to roll without slipping: ∆x = vav ∆t = (6) 4 v0 7 µk g 11 v0 = 1.57v0 7 1 2 (v0 + v )∆t 11 ⎞⎛ 4v0 ⎞ ⎛ ⎟ = 1 ⎜ v0 + v0 ⎟⎜ 2 7 ⎠⎜ 7 µ k g ⎟ ⎝ ⎝ ⎠ = 2 v2 36 v0 = 0.735 0 49 µ k g µk g *108 •• Picture the Problem The figure shows the forces acting on the cylinder during the sliding phase of its motion. The friction force will cause the cylinder’s translational speed to decrease and eventually satisfy the condition for rolling without slipping. We’ll use Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constantacceleration equations to relate these quantities to each other and to the distance traveled and the elapsed time until the satisfaction of the condition for rolling without slipping. (a) Apply Newton’s 2nd law to the cylinder: ∑F ∑F x = − f k = Ma , (1) y = Fn − Mg = 0 , (2) = f k R = I 0α (3) and ∑τ Use fk = µkFn to eliminate Fn between equations (1) and (2) and solve for a: 0 a = −µ k g 706 Chapter 9 Using a constant-acceleration equation, relate the speed of the cylinder to its acceleration and the elapsed time: v = v0 + a∆t = v0 − µ k g∆t Similarly, eliminate fk between equations (2) and (3) and solve for α: α= Using a constant-acceleration equation, relate the angular speed of the cylinder to its acceleration and the elapsed time: ω = ω0 + α∆t = Apply the condition for rolling without slipping: ⎛ 2µ g ⎞ v = v0 − µ k g∆t = Rω = R⎜ k ∆t ⎟ ⎝ R ⎠ = 2 µ k g∆t Solve for ∆t: Substitute for ∆t in the expression for v: (b) Relate the distance the cylinder travels to its average speed and the elapsed time: 2µ k g R ∆t = 2µ k g ∆t R v0 3µ k g v = v0 − µ k g ∆x = vav ∆t = v0 3µ k g 1 2 = 2 v0 3 ⎛ v0 ⎞ ⎟ ⎟ ⎝ 3µ k g ⎠ (v0 + 2 v0 )⎜ 3 ⎜ 2 5 v0 = 18 µ k g (c) Express the ratio of the energy dissipated in friction to the cylinder’s initial mechanical energy: Wfr K i − K f = Ki Ki Express the kinetic energy of the cylinder as it begins to roll without slipping: K f = 1 Mv 2 + 1 I cmω 2 2 2 = 1 Mv 2 + 1 2 2 ( 1 2 MR 2 v )R 2 2 2 = 3 3 ⎛2 ⎞ 1 2 Mv 2 = M ⎜ v0 ⎟ = Mv0 4 4 ⎝3 ⎠ 3 Rotation 707 2 2 Wfr 1 Mv0 − 1 Mv0 1 3 2 = = 2 1 Ki 3 2 Mv0 Substitute for Ki and Kf and simplify to obtain: 109 •• Picture the Problem The forces acting on the ball as it slides across the floor are its r r weight mg, the normal force Fn exerted by v the floor, and the friction force f . Because the weight and normal force act through the center of mass of the ball and are equal in magnitude, the friction force is the net (decelerating) force. We can apply Newton’s 2nd law in both translational and rotational form to obtain a set of equations that we can solve for the acceleration of the ball. Once we have determined the ball’s acceleration, we can use constantacceleration equations to obtain its velocity when it begins to roll without slipping. (a) Apply r r ∑ F = ma to the ball: ∑F x = − f = ma (1) = Fn − mg = 0 (2) and ∑F y From the definition of the coefficient of kinetic friction we have: f = µ k Fn Solve equation (2) for Fn: Fn = mg Substitute in equation (3) to obtain: f = µ k mg Substitute in equation (1) to obtain: − µ k mg = ma (3) or a = −µk g Apply ∑τ = Iα to the ball: Solve for α to obtain: Assuming that the coefficient of kinetic friction is constant*, we can use constant-acceleration equations to describe how long it will take the ball to begin fr = Iα α= fr µ k mgr = I I vf − v = a∆t = − µ k g∆t and ωf = µ k gmr I ∆t (4) (5) 708 Chapter 9 rolling without slipping: Once rolling without slipping has been established, we also have: Equate equations (5) and (6): Solve for ∆t: Substitute in equation (4) to obtain: Solve for vf: (b) Express the total kinetic energy of the ball: ωf = vf r (6) vf µ k gmr = ∆t r I ∆t = vf I µ k gmr 2 ⎛ vf I ⎞ vf − v = − µ k g ⎜ ⎜ µ gmr 2 ⎟ ⎟ ⎝ k ⎠ I = − 2 vf mr vf = K= 1 I 1+ mr 2 v 1 2 1 2 mvf + Iωf 2 2 Because the ball is now rolling without slipping, v = rωf and: 2 2 2 2 1 ⎛ 1 1 1 2⎛ 1 ⎞ 2 1 ⎛ ⎞ v ⎞ ⎞ ⎜ 1 + I / mr 2 ⎛ ⎟ = mv K = m⎜ ⎟ v + I⎜ ⎟ ⎜ 2 ⎟ ⎜ 2 ⎝ 1 + I / mr 2 ⎠ 2 ⎝ 1 + I / mr 2 ⎠ r 2 2 ⎝ 1 + I / mr ⎠ ⎟ ⎝ ⎠ ( = ) 1 2⎛ 1 ⎞ mv ⎜ 2 ⎟ 2 ⎝ 1 + I / mr ⎠ * Remarks: This assumption is not necessary. One can use the impulse-momentum theorem and the related theorem for torque and change in angular momentum to prove that the result holds for an arbitrary frictional force acting on the ball, so long as the ball moves along a straight line and the force is directed opposite to the direction of motion of the ball. General Problems *110 • Picture the Problem The angular velocity of an object is the ratio of the number of revolutions it makes in a given period of time to the elapsed time. Rotation 709 The moon’s angular velocity is: 1rev 27.3 days 1rev 2π rad 1day 1h = × × × 27.3 days rev 24 h 3600 s ω= = 2.66 × 10−6 rad/s 111 • Picture the Problem The moment of inertia of the hoop, about an axis perpendicular to the plane of the hoop and through its edge, is related to its moment of inertia with respect to an axis through its center of mass by the parallel axis theorem. Apply the parallel axis theorem: I = I cm + Mh 2 = MR 2 + MR 2 = 2mR 2 112 •• Picture the Problem The force you exert on the rope results in a net torque that accelerates the merry-go-round. The moment of inertia of the merry-go-round, its angular acceleration, and the torque you apply are related through Newton’s 2nd law. (a) Using a constant-acceleration equation, relate the angular displacement of the merry-go-round to its angular acceleration and acceleration time: ∆θ = ω 0 ∆t + 1 α (∆t ) 2 2 or, because ω0 = 0, ∆θ = 1 α (∆t ) 2 2 2∆θ 2(2π rad ) = = 0.0873 rad/s 2 2 2 (∆t ) (12 s ) Solve for and evaluate α: α= (b) Use the definition of torque to obtain: τ = Fr = (260 N )(2.2 m ) = 572 N ⋅ m (c) Use Newton’s 2nd law to find the moment of inertia of the merry-goround: I= τ net 572 N ⋅ m = α 0.0873 rad/s 2 = 6.55 × 103 kg ⋅ m 2 710 Chapter 9 113 • Picture the Problem Because there are no horizontal forces acting on the stick, the center of mass of the stick will not move in the horizontal direction. Choose a coordinate system in which the origin is at the horizontal position of the center of mass. The diagram shows the stick in its initial raised position and when it has fallen to the ice. Express the displacement of the right end of the stick ∆x as the difference between the position coordinates x2 and x2: ∆x = x2 − x1 Using trigonometry, find the initial coordinate of the right end of the stick: x1 = l cos θ = (1 m ) cos30° = 0.866 m Because the center of mass has not moved horizontally: x2 = l = 1 m Substitute to find the displacement of the right end of the stick: ∆x = 1 m − 0.866 m = 0.134 m 114 •• Picture the Problem The force applied to the string results in a torque about the center of mass of the disk that accelerates it. We can relate these quantities to the moment of inertia of the disk through Newton’s 2nd law and then use constant-acceleration equations to find the disk’s angular velocity the angle through which it has rotated in a given period of time. The disk’s rotational kinetic energy can be found from its definition. (a) Use the definition of torque to obtain: τ ≡ FR = (20 N )(0.12 m ) = 2.40 N ⋅ m (b) Use Newton’s 2nd law to express the angular acceleration of the disk in terms of the net torque acting on it and its moment of inertia: α= Substitute numerical values and evaluate α: α= (c) Using a constant-acceleration equation, relate the angular velocity of the disk to its angular ω = ω 0 + α∆t τ net I = τ net 1 2 MR 2 2(2.40 N ⋅ m ) = 66.7 rad/s 2 2 (5 kg )(0.12 m ) or, because ω0 = 0, ω = α∆t Rotation 711 acceleration and the elapsed time: Substitute numerical values and evaluate ω: ω = (66.7 rad/s 2 )(5 s ) = 333 rad/s ( (d) Use the definition of rotational kinetic energy to obtain: K rot = 1 Iω 2 = 2 Substitute numerical values and evaluate Krot: K rot = (e) Using a constant-acceleration equation, relate the angle through which the disk turns to its angular acceleration and the elapsed time: ∆θ = ω 0 ∆t + 1 α (∆t ) 2 1 4 1 1 2 2 ) MR 2 ω 2 (5 kg )(0.12 m )2 (333 rad/s)2 = 2.00 kJ 2 or, because ω0 = 0, ∆θ = 1 α (∆t ) 2 2 (66.7 rad/s )(5 s) Substitute numerical values and evaluate ∆θ : ∆θ = (f) Express K in terms of τ and θ : ⎛τ ⎞ 2 2 K = 1 Iω 2 = 1 ⎜ ⎟(α∆t ) = 1 ατ (∆t ) 2 2 2 ⎝α ⎠ 1 2 2 = τ ∆θ 115 •• Picture the Problem The diagram shows the rod in its initial horizontal position and then, later, as it swings through its vertical position. The center of mass is denoted by the numerals 0 and 1. Let the length of the rod be represented by L and its mass by m. We can use Newton’s 2nd law in rotational form to find, first, the angular acceleration of the rod and then, from α, the acceleration of any point on the rod. We can use conservation of energy to find the angular velocity of the center of mass of the rod when it is vertical and then use this value to find its linear velocity. (a) Relate the acceleration of the center of the rod to the angular a = lα = L α 2 2 = 834 rad 712 Chapter 9 acceleration of the rod: Use Newton’s 2nd law to relate the torque about the suspension point of the rod (exerted by the weight of the rod) to the rod’s angular acceleration: L 3g τ α= =1 2 = 2 I 3 ML 2L Substitute numerical values and evaluate α: α= 3 9.81 m/s 2 = 18.4 rad/s 2 2(0.8 m ) Substitute numerical values and evaluate a: a= 1 2 (b) Relate the acceleration of the end of the rod to α: Mg ( ) (0.8 m )(18.4 rad/s 2 ) = ( 7.36 m/s 2 aend = Lα = (0.8 m ) 18.4 rad/s 2 ) = 14.7 m/s 2 (c) Relate the linear velocity of the center of mass of the rod to its angular velocity as it passes through the vertical: v = ω∆h = 1 ωL 2 Use conservation of energy to relate the changes in the kinetic and potential energies of the rod as it swings from its initial horizontal orientation through its vertical orientation: ∆K + ∆U = K1 − K 0 + U 1 − U 0 = 0 Substitute to obtain: Substitute for ∆h and solve for ω: Substitute to obtain: Substitute numerical values and evaluate v: or, because K0 = U1 = 0, K1 − U 0 = 0 1 2 I Pω 2 = mg∆h 3g L ω= v=1L 2 v= 1 2 3g = L ( 1 2 3 gL ) 3 9.81 m/s 2 (0.8 m ) = 2.43 m/s Rotation 713 116 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the track. The initial potential energy of the marble is transformed into translational and rotational kinetic energy as it rolls down the track to its lowest point and then, because the portion of the track to the right is frictionless, into translational kinetic energy and, eventually, into gravitational potential energy. ∆K + ∆U = 0 Using conservation of energy, relate h2 to the kinetic energy of the marble at the bottom of the track: or, because Kf = Ui = 0, Substitute for Ki and Uf to obtain: − 1 Mv 2 − Mgh2 = 0 2 Solve for h2: Using conservation of energy, relate h1 to the kinetic energy of the marble at the bottom of the track: − Ki + U f = 0 h2 = v2 2g (1) ∆K + ∆U = 0 or, because Ki = Uf = 0, Kf −Ui = 0 Mv 2 + 1 Iω 2 − Mgh1 = 0 2 Substitute for Kf and Ui to obtain: 1 2 Substitute for I and solve for v2 to obtain: v 2 = 10 gh1 7 Substitute in equation (1) to obtain: h2 = gh1 = 2g 10 7 5 7 h1 *117 •• Picture the Problem To stop the wheel, the tangential force will have to do an amount of work equal to the initial rotational kinetic energy of the wheel. We can find the stopping torque and the force from the average power delivered by the force during the slowing of the wheel. The number of revolutions made by the wheel as it stops can be found from a constant-acceleration equation. (a) Relate the work that must be done to stop the wheel to its kinetic energy: W = 1 Iω 2 = 2 ( 1 1 2 2 ) mr 2 ω 2 = 1 mr 2ω 2 4 714 Chapter 9 Substitute numerical values and evaluate W: W= 1 4 (120 kg )(1.4 m )2 ⎡ rev 2π rad 1min ⎤ × ⎢1100 × × min rev 60 s ⎥ ⎦ ⎣ 2 = 780 kJ (b) Express the stopping torque is terms of the average power required: Pav = τω av Solve for τ : τ= Substitute numerical values and evaluate τ : 780 kJ (2.5 min )(60 s/min ) τ= (1100 rev/min )(2π rad/rev)(1 min/60 s ) 2 Pav ωav = 90.3 N ⋅ m τ 90.3 N ⋅ m = 151 N 0.6 m Relate the stopping torque to the magnitude of the required force and solve for F: F= (c) Using a constant-acceleration equation, relate the angular displacement of the wheel to its average angular velocity and the stopping time: ∆θ = ωav ∆t Substitute numerical values and evaluate ∆θ: ⎛ 1100 rev/min ⎞ ∆θ = ⎜ ⎟ (2.5 min ) 2 ⎝ ⎠ R = = 1380 rev 118 •• Picture the Problem The work done by the four children on the merry-go-round will change its kinetic energy. We can use the work-energy theorem to relate the work done by the children to the distance they ran and Newton’s 2nd law to find the angular acceleration of the merry-go-round. Rotation 715 (a) Use the work-kinetic energy theorem to relate the work done by the children to the kinetic energy of the merry-go-round: Substitute for I and solve for ∆s to obtain: Substitute numerical values and evaluate ∆s: Wnet force = ∆K = Kf or 4 F∆s = 1 Iω 2 2 ∆s = Iω 2 1 mr 2ω 2 mr 2ω 2 = 2 = 8F 8F 16 F ⎡ ⎤ (240 kg )(2 m ) ⎢1rev × 2π rad ⎥ rev ⎦ ⎣ 2.8 s ∆s = 16(26 N ) 2 2 = 11.6 m τ net 4 Fr 8F = 2 mr mr (b) Apply Newton’s 2nd law to express the angular acceleration of the merry-go-round: α= Substitute numerical values and evaluate α: α= (c) Use the definition of work to relate the force exerted by each child to the distance over which that force is exerted: W = F∆s = (26 N )(11.6 m ) = 302 J (d) Relate the kinetic energy of the merry-go-round to the work that was done on it: Wnet force = ∆K = K f − 0 = 4 F∆s Substitute numerical values and evaluate Wnet force: Wnet force = 4(26 N )(11.6 m ) = 1.21 kJ I = 1 2 8(26 N ) = 0.433 rad/s 2 (240 kg )(2 m ) 119 •• Picture the Problem Because the center of mass of the hoop is at its center, we can use Newton’s second law to relate the acceleration of the hoop to the net force acting on it. The distance moved by the center of the hoop can be determined using a constantacceleration equation, as can the angular velocity of the hoop. (a) Using a constant-acceleration equation, relate the distance the ∆s = 1 a cm (∆t ) 2 2 716 Chapter 9 center of the travels in 3 s to the acceleration of its center of mass: Relate the acceleration of the center of mass of the hoop to the net force acting on it: Substitute to obtain: a cm = Fnet m F (∆t ) 2m 2 ∆s = Substitute numerical values and evaluate ∆s: (5 N )(3 s )2 ∆s = 2(1.5 kg ) (b) Relate the angular velocity of the hoop to its angular acceleration and the elapsed time: ω = α ∆t Use Newton’s 2nd law to relate the angular acceleration of the hoop to the net torque acting on it: α= Substitute to obtain: ω= F∆t mR Substitute numerical values and evaluate ω: ω= (5 N )(3 s ) = (1.5 kg )(0.65 m ) τ net I = = 15.0 m FR F = 2 mR mR 15.4 rad/s 120 •• Picture the Problem Let R represent the radius of the grinding wheel, M its mass, r the radius of the handle, and m the mass of the load attached to the handle. In the absence of information to the contrary, we’ll treat the 25-kg load as though it were concentrated at a point. Let the zero of gravitational potential energy be where the 25-kg load is at its lowest point. We’ll apply Newton’s 2nd law and the conservation of mechanical energy to determine the initial angular acceleration and the maximum angular velocity of the wheel. (a) Use Newton’s 2nd law to relate the acceleration of the wheel to the net torque acting on it: α= τ net I = 1 2 mgr MR 2 + mr 2 Rotation 717 Substitute numerical values and evaluate α: α= (25 kg )(9.81m/s2 )(0.65 m ) 2 2 1 2 (60 kg )(0.45 m ) + (25 kg )(0.65 m ) = 9.58 rad/s 2 (b) Use the conservation of mechanical energy to relate the initial potential energy of the load to its kinetic energy and the rotational kinetic energy of the wheel when the load is directly below the center of mass of the wheel: ∆K + ∆U = 0 or, because Ki = Uf = 0, K f,trans + K f,rot − U i = 0 . ( ) MR 2 ω 2 − mgr = 0 , 1 2 mv 2 + 1 2 1 2 Substitute and solve for ω: mr ω + MR 2ω 2 − mgr = 0 , 2 1 2 2 1 4 and ω= Substitute numerical values and evaluate ω: 4mgr 2mr 2 + MR 2 ω= 4(25 kg ) 9.81 m/s 2 (0.65 m ) 2 2 2(25 kg )(0.65 m ) + (60 kg )(0.45 m ) ( ) = 4.38 rad/s *121 •• Picture the Problem Let the smaller block have the dimensions shown in the diagram. Then the length, height, and width of the larger block are Sl, Sh, and Sw, respectively. Let the numeral 1 denote the smaller block and the numeral 2 the larger block and express the ratios of the surface areas, masses, and moments of inertia of the two blocks. (a) Express the ratio of the surface areas of the two blocks: A2 2(Sw)(Sl ) + 2(Sl )(Sh ) + 2(Sw)(Sh ) = A1 2wl + 2lh + 2wh = S2 (2wl + 2lh + 2wh ) 2 wl + 2lh + 2 wh = S2 718 Chapter 9 (b) Express the ratio of the masses of the two blocks: M 2 ρV2 V2 (Sw)(Sl )(Sh ) = = = M 1 ρV1 V1 w lh = (c) Express the ratio of the moments of inertia, about the axis shown in the diagram, of the two blocks: I2 = I1 = S3 (wlh ) = S3 wlh 1 12 [ M 2 (Sl ) + (Sh ) 2 2 1 12 M 1 l + h 2 [ 2 [ M 2 S2 l 2 + h 2 ⎛ M 2 ⎞ 2 =⎜ ⎟ ⎜M ⎟S M1 l2 + h2 ⎝ 1⎠ [ In part (b) we showed that: M2 = S3 M1 Substitute to obtain: ( ) I2 = (S3 )(S2 ) = S5 I1 122 •• Picture the Problem We can derive the perpendicular-axis theorem for planar objects by following the step-by-step procedure outlined in the problem. (a) and (b) ( ) I z = ∫ r 2 dm = ∫ x 2 + y 2 dm = ∫ x 2 dm + ∫ y 2 dm = Ix + Iy (c) Let the z axis be the axis of rotation of the disk. By symmetry: Ix = Iy Express Iz in terms of Ix: I z = 2I x Letting M represent the mass of the disk, solve for Ix: Ix = 1 Iz = 2 1 2 ( 1 2 ) MR 2 = 1 4 MR 2 123 •• Picture the Problem Let the zero of gravitational potential energy be at the center of the disk when it is directly below the pivot. The initial gravitational potential energy of the disk is transformed into rotational kinetic energy when its center of mass is directly below the pivot. We can use Newton’s 2nd law to relate the force exerted by the pivot to the weight of the disk and the centripetal force acting on it at its lowest point. Rotation 719 (a) Use the conservation of mechanical energy to relate the initial potential energy of the disk to its kinetic energy when its center of mass is directly below the pivot: ∆K + ∆U = 0 or, because Ki = Uf = 0, K f,rot − U i = 0 Iω 2 − Mgr = 0 Substitute for K f,rot and U i : 1 2 Use the parallel-axis theorem to relate the moment of inertia of the disk about the pivot to its moment of inertia with respect to an axis through its center of mass: I = I cm + Mh 2 Solve equation (1) for ω and substitute for I to obtain: (1) or I = 1 Mr 2 + Mr 2 = 3 Mr 2 2 2 ω= 4g 3r (b) Letting F represent the force exerted by the pivot, use Newton’s 2nd law to express the net force acting on the swinging disk as it passes through its lowest point: Fnet = F − Mg = Mrω 2 Solve for F and simplify to obtain: F = Mg + Mrω 2 = Mg + Mr = Mg + 4 Mg = 3 124 •• Picture the Problem The diagram shows a vertical cross-piece. Because we’ll need to take moments about the point of rotation (point P), we’ll need to use the parallelaxis theorem to find the moments of inertia of the two parts of this composite structure. Let the numeral 1 denote the vertical member and the numeral 2 the horizontal member. We can apply Newton’s 2nd law in rotational form to the structure to express its angular acceleration in terms of the net torque causing it to fall and its moment of inertia with respect to point P. 7 3 Mg 4g 3r 720 Chapter 9 (a) Taking clockwise rotation to be positive (this is the direction the structure is going to rotate), apply τ = I Pα : ⎛l ⎞ ⎛ w⎞ m2 g ⎜ 2 ⎟ − m1 g ⎜ ⎟ = I Pα ⎝ 2⎠ ⎝2⎠ ∑ Solve for α to obtain: α= or α= Convert l 1 , l 2 , and w to SI units: Using Table 9-1 and the parallelaxis theorem, express the moment of inertia of the vertical member about an axis through point P: m2 gl 2 − m1 gw 2I P g (m2 l 2 − m1w) 2(I1P + I 2 P ) (1) 1m = 3.66 m , 3.281ft 1m l 2 = 6 ft × = 1.83 m , and 3.281ft 1m w = 2 ft × = 0.610 m 3.281ft l 1 = 12 ft × ⎛ w⎞ = m l + m1 ⎜ ⎟ ⎝2⎠ 2 = m1 1 l 1 + 1 w 2 4 3 I1P 2 2 1 1 1 3 ( ) [ (3.66 m) + Substitute numerical values and evaluate I1P: I1P = (350 kg ) Using the parallel-axis theorem, express the moment of inertia of the horizontal member about an axis through point P: (0.610 m )2 ] I 2 P = I 2,cm + m2 d 2 Solve for d: 2 1 3 1 4 = 1.60 × 103 kg ⋅ m 2 (2) where d 2 = (l 1 + 1 w) + ( 1 l 2 − w) 2 2 2 2 (l 1 + 1 w)2 + ( 1 l 2 − w)2 2 2 d= Substitute numerical values and evaluate d: d= [3.66 m + 1 (0.610 m )] 2 + [1 (1.83 m ) − 0.610 m] 2 2 2 From Table 9-1 we have: 1 I 2,cm = 12 m2 l 2 2 Substitute in equation (2) to obtain: I2P = 1 12 m2l 2 + m2 d 2 2 = m2 ( 1 12 l2 + d 2 2 ) = 3.86 m Rotation 721 [ 1 I 2 P = (175 kg ) 12 (1.83 m ) + (3.86 m ) Evaluate I2P: 2 2 = 2.66 × 103 kg ⋅ m 2 Substitute in equation (1) and evaluate α: α= (9.81m/s )[(175 kg )(1.83 m) − (350 kg )(0.61m)] = 2 2(1.60 + 2.66)× 10 kg ⋅ m 3 (b) Express the magnitude of the acceleration of the sparrow: 2 0.123 rad/s 2 a = αR where R is the distance of the sparrow from the point of rotation and R 2 = (l 1 + w) + (l 2 − w) 2 Solve for R: 2 (l 1 + w)2 + (l 2 − w)2 R= Substitute numerical values and evaluate R: R= (3.66 m + 0.610 m )2 + (1.83 m − 0.610 m )2 Substitute numerical values and evaluate a: ( = 4.44 m ) a = 0.123 rad/s 2 (4.44 m ) = 0.546 m/s 2 (c) Refer to the diagram to express ax in terms of a: a x = a cos θ = a Substitute numerical values and evaluate ax: ax = 0.546 m/s 2 ( l1 + w R m+ ) 3.664.440.61m m = 0.525 m/s 2 722 Chapter 9 125 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. The initial potential energy of the spool is transformed into rotational and translational kinetic energy when the spool reaches the bottom of the incline. We can apply the conservation of mechanical energy to find an expression for its speed at that location. The force diagram shows the forces acting on the spool when there is enough friction to keep it from slipping. We’ll use Newton’s 2nd law in both translational and rotational form to derive an expression for the static friction force. The spool will move down the plane (a) In the absence of friction, the forces acting on the spool will be its weight, the normal force exerted by the incline, and the tension in the string. A component of its weight will cause the spool to accelerate down the incline and the tension in the string will exert a torque that will cause counterclockwise rotation of the spool. Use the conservation of mechanical energy to relate the speed of the center of mass of the spool at the bottom of the slope to its initial potential energy: Substitute for K f, trans , K f,rot and U i : Substitute for ω and solve for v to obtain: at constant acceleration, spinning in a counterclockwise direction as string unwinds. ∆K + ∆U = 0 or, because Ki = Uf = 0, K f,trans + K f,rot − U i = 0 . 1 2 1 2 Mv 2 + 1 Iω 2 − MgD sin θ = 0 2 v2 Mv + I 2 − MgD sin θ = 0 r 2 1 2 and v= 2MgD sin θ I M+ 2 r (1) Rotation 723 ∑F ∑τ Eliminate T between these equations to obtain: x fs = = Mg sin θ − T − f s = 0 0 (b) Apply Newton’s 2nd law to the spool: = Tr − f s R = 0 Mg sin θ , up the incline. R 1+ r 126 •• Picture the Problem While the angular acceleration of the rod is the same at each point along its length, the linear acceleration and, hence, the force exerted on each coin by the rod, varies along its length. We can relate this force the linear acceleration of the rod through Newton’s 2nd law and the angular acceleration of the rod. Fnet = mg − F ( x ) = ma( x ) Letting x be the distance from the pivot, use Newton’s 2nd law to express the force F acting on a coin: or Use Newton’s 2nd law to relate the angular acceleration of the system to the net torque acting on it: L τ 3g α = net = 1 2 = 2 I 2L 3 ML Relate a(x) and α: a( x ) = xα = x Substitute in equation (1) to obtain: F ( x ) = m( g − gx ) = mg (1 − x ) Evaluate F(0.25 m): F (0.25 m ) = mg (1 − 0.25 m ) = 0.75mg Evaluate F(0.5 m): F (0.5 m ) = mg (1 − 0.5 m ) = 0.5mg Evaluate F(0.75 m): F (0.75 m ) = mg (1 − 0.75 m ) = 0.25mg Evaluate F(1 m): F (1 m ) = F (1.25 m ) = F (1.5 m ) = 0 F ( x ) = m(g − a ( x )) (1) Mg 3g = gx 2(1.5 m ) *127 •• Picture the Problem The diagram shows the force the hand supporting the meterstick exerts at the pivot point and the force the earth exerts on the meterstick acting at the center of mass. We can relate the angular acceleration to the acceleration of the end of the meterstick using a = Lα and use Newton’s 2nd law in rotational form to relate α to the moment of inertia of the meterstick. 724 Chapter 9 (a) Relate the acceleration of the far end of the meterstick to the angular acceleration of the meterstick: Apply ∑τ P = I Pα to the meterstick: Solve for α: From Table 9-1, for a rod pivoted at one end, we have: Substitute to obtain: a = Lα (1) ⎛L⎞ Mg ⎜ ⎟ = I Pα ⎝2⎠ α= MgL 2I P 1 I P = ML2 3 α= 3MgL 3g = 2ML2 2 L a= 3g 2 Substitute numerical values and evaluate a: a= 3(9.81 m/s 2 ) = 14.7 m/s 2 2 (b) Express the acceleration of a point on the meterstick a distance x from the pivot point: a = αx = Substitute in equation (1) to obtain: Express the condition that the meterstick leaves the penny behind: a>g Substitute to obtain: 3g x 2L 3g x>g 2L Solve for and evaluate x: x> 2 L 2(1 m ) = = 66.7 cm 3 3 Rotation 725 128 •• Picture the Problem Let m represent the 0.2-kg mass, M the 0.8-kg mass of the cylinder, L the 1.8-m length, and x + ∆x the distance from the center of the objects whose mass is m. We can use Newton’s 2nd law to relate the radial forces on the masses to the spring’s stiffness constant and use the work-energy theorem to find the work done as the system accelerates to its final angular speed. (a) Express the net inward force acting on each of the 0.2-kg masses: Solve for k: Substitute numerical values and evaluate k: ∑F radial = k∆x = m(x + ∆x )ω 2 m( x + ∆x )ω k= ∆x k= 2 (0.2 kg )(0.8 m )(24 rad/s)2 0.4 m = 230 N/m (b) Using the work-energy theorem, relate the work done to the change in energy of the system: W = K rot + ∆U spring Express I as the sum of the moments of inertia of the cylinder and the masses: I = I M + I 2m From Table 9-1 we have, for a solid cylinder about a diameter through its center: 1 I = 1 mr 2 + 12 mL2 4 = 1 Iω 2 + 1 k (∆x ) 2 2 2 (1) 1 = 1 Mr 2 + 12 ML2 + 2 I m 2 where L is the length of the cylinder. For a disk (thin cylinder), L is small and: I = 1 mr 2 4 Apply the parallel-axis theorem to obtain: I m = 1 mr 2 + mx 2 4 Substitute to obtain: 1 I = 1 Mr 2 + 12 ML2 + 2(1 mr 2 + mx 2 ) 2 4 Substitute numerical values and evaluate I: 1 = 1 Mr 2 + 12 ML2 + 2m(1 r 2 + x 2 ) 2 4 726 Chapter 9 I= 1 2 1 (0.8 kg )(0.2 m )2 + 12 (0.8 kg )(1.8 m )2 + 2(0.2 kg )[1 (0.2 m )2 + (0.8 m )2 ] 4 = 0.492 N ⋅ m 2 Substitute in equation (1) to obtain: W = 1 2 (0.492 N ⋅ m )(24 rad/s) 2 2 + 1 (230 N/m )(0.4 m ) = 160 J 2 2 129 •• Picture the Problem Let m represent the 0.2-kg mass, M the 0.8-kg mass of the cylinder, L the 1.8-m length, and x + ∆x the distance from the center of the objects whose mass is m. We can use Newton’s 2nd law to relate the radial forces on the masses to the spring’s stiffness constant and use the work-energy theorem to find the work done as the system accelerates to its final angular speed. Using the work-energy theorem, relate the work done to the change in energy of the system: W = K rot + ∆U spring Express I as the sum of the moments of inertia of the cylinder and the masses: I = I M + I 2m From Table 9-1 we have, for a solid cylinder about a diameter through its center: 1 I = 1 mr 2 + 12 mL2 4 = 1 Iω 2 + 1 k (∆x ) 2 2 (1) 2 1 = 1 Mr 2 + 12 ML2 + 2 I m 2 where L is the length of the cylinder. For a disk (thin cylinder), L is small and: I = 1 mr 2 4 Apply the parallel-axis theorem to obtain: I m = 1 mr 2 + mx 2 4 Substitute to obtain: 1 I = 1 Mr 2 + 12 ML2 + 2 1 mr 2 + mx 2 2 4 ( ( 1 = 1 Mr 2 + 12 ML2 + 2m 1 r 2 + x 2 2 4 ) Substitute numerical values and evaluate I: I= 1 2 1 (0.8 kg )(0.2 m )2 + 12 (0.8 kg )(1.8 m )2 + 2(0.2 kg )[1 (0.2 m )2 + (0.8 m )2 ] 4 = 0.492 N ⋅ m 2 ) Rotation 727 Express the net inward force acting on each of the 0.2-kg masses: Solve for ω: Substitute numerical values and evaluate ω: Substitute numerical values in equation (1) to obtain: ∑F radial = k∆x = m(x + ∆x )ω 2 ω= k∆x m( x + ∆x ) ω= (60 N/m )(0.4 m ) = 12.2 rad/s (0.2 kg )(0.8 m ) W= 1 2 (0.492 N ⋅ m )(12.2 rad/s) 2 2 + 1 (60 N/m )(0.4 m ) 2 2 = 41.4 J 130 •• Picture the Problem The force diagram shows the forces acting on the cylinder. Because F causes the cylinder to rotate clockwise, f, which opposes this motion, is to the right. We can use Newton’s 2nd law in both translational and rotational forms to relate the linear and angular accelerations to the forces acting on the cylinder. τ net FR 2F = 2 1 MR 2 MR (a) Use Newton’s 2nd law to relate the angular acceleration of the center of mass of the cylinder to F: α= Use Newton’s 2nd law to relate the acceleration of the center of mass of the cylinder to F: acm = Express the rolling-without-slipping condition to the accelerations: α' = (b) Take the point of contact with the floor as the ″pivot″ point, express the net torque about that point, and solve for α: τ net = 2 FR = Iα I = Fnet F = M M acm F = = 2α R MR and α= 2 FR I 728 Chapter 9 Express the moment of inertia of the cylinder with respect to the pivot point: I = 1 MR 2 + MR 2 = 3 MR 2 2 2 Substitute to obtain: α= 3 2 2 FR 4F = 2 MR 3MR 4F 3M Express the linear acceleration of the cylinder: acm = Rα = Apply Newton’s 2nd law to the forces acting on the cylinder: ∑F Solve for f: f = Macm − F = x = = F + f = Ma cm 1 3 4F −F 3 F in the positive x direction. 131 •• Picture the Problem As the load falls, mechanical energy is conserved. As in Example 9-7, choose the initial potential energy to be zero. Apply conservation of mechanical energy to obtain an expression for the speed of the bucket as a function of its position and use the given expression for t to determine the time required for the bucket to travel a distance y. Apply conservation of mechanical energy: U f + Kf = U i + Ki = 0 + 0 = 0 Express the total potential energy when the bucket has fallen a distance y: U f = U bf + U cf + U wf (1) ⎛ y⎞ = −mgy − mc'g ⎜ ⎟ ⎝2⎠ where mc' is the mass of the hanging part of the cable. Assume the cable is uniform and express mc' in terms of mc, y, and L: mc' mc m = or mc' = c y y L L Substitute to obtain: mc gy 2 U f = −mgy − 2L Rotation 729 K f = K bf + K cf + K wf Noting that bucket, cable, and rim of the winch have the same speed v, express the total kinetic energy when the bucket is falling with speed v: = 1 mv 2 + 1 mc v 2 + 1 Iωf2 2 2 2 = 1 mv 2 + 1 mc v 2 + 1 (1 MR 2 ) 2 2 2 2 v2 R2 = 1 mv 2 + 1 mc v 2 + 1 Mv 2 2 2 4 Substitute in equation (1) to obtain: − mgy − Solve for v: v= mc gy 2 1 2 + 2 mv 2L + 1 mc v 2 + 1 Mv 2 = 0 2 4 2mc gy 2 L M + 2m + 2mc 4mgy + A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows: Cell Formula/Content D9 0 D10 D9+$B$8 E9 0 E10 ((4*$B$3*$B$7*D10+2*$B$7*D10^2/(2*$B$5))/ ($B$1+2*$B$3+2*$B$4))^0.5 F10 2mc gy 2 L M + 2m + 2mc 4mgy + ⎛v +v ⎞ t n−1 + ⎜ n−1 n ⎟∆y 2 ⎝ ⎠ F9+$B$8/((E10+E9)/2) J9 Algebraic Form y0 y + ∆y v0 0.5*$B$7*H9^2 1 2 3 4 5 6 7 8 9 10 11 12 13 15 A M= R= m= mc= L= B 10 0.5 5 3.5 10 g= 9.81 dy= 0.1 C kg m kg kg m m/s^2 m 1 2 D E F y 0.0 0.1 0.2 0.3 0.4 0.5 v(y) 0.00 0.85 1.21 1.48 1.71 1.91 t(y) 0.00 0.23 0.33 0.41 0.47 0.52 G gt 2 H I J t(y) 0.00 0.23 0.33 0.41 0.47 0.52 y 0.0 0.1 0.2 0.3 0.4 0.5 1/2gt^2 0.00 0.27 0.54 0.81 1.08 1.35 730 Chapter 9 105 106 107 108 109 9.6 9.7 9.8 9.9 10.0 9.03 9.08 9.13 9.19 9.24 2.24 2.25 2.26 2.27 2.28 2.24 2.25 2.26 2.27 2.28 9.6 9.7 9.8 9.9 10.0 24.61 24.85 25.09 25.34 25.58 The solid line on the graph shown below shows the position y of the bucket when it is in free fall and the dashed line shows y under the conditions modeled in this problem. 20 18 16 y' 14 free fall y (m) 12 10 8 6 4 2 0 0.0 0.4 0.8 1.2 1.6 t (s) 132 •• Picture the Problem The pictorial representation shows the forces acting on the cylinder when it is stationary. First, we note that if the tension is small, then there can be no slipping, and the system must roll. Now consider the point of contact of the cylinder with the surface as the “pivot” point. If τ about that point is zero, the system will not roll. This will occur if the line of action of the tension passes through the pivot point. From the diagram we see that: ⎛r⎞ ⎝R⎠ θ = cos −1 ⎜ ⎟ 2.0 Rotation 731 *133 •• Picture the Problem Free-body diagrams for the pulley and the two blocks are shown to the right. Choose a coordinate system in which the direction of motion of the block whose mass is M (downward) is the positive y direction. We can use the given µ ∆θ relationship T 'max = Te s to relate the tensions in the rope on either side of the pulley and apply Newton’s 2nd law in both rotational form (to the pulley) and translational form (to the blocks) to obtain a system of equations that we can solve simultaneously for a, T1, T2, and M. (a) Use T 'max = Te s to evaluate the maximum tension required to prevent the rope from slipping on the pulley: T 'max = (10 N ) e (0.3 )π = 25.7 N (c) Given that the angle of wrap is π radians, express T2 in terms of T1: T2 = T1e 0.3π = 2.57T1 µ ∆θ Because the rope doesn’t slip, we can relate the angular acceleration, α, of the pulley to the acceleration, a, of the hanging masses by: Apply ∑F y = ma y to the two blocks to obtain: Apply ∑τ = Iα to the pulley to obtain: α= (1) a r T1 − mg = ma (2) and Mg − T2 = Ma (3) (T2 − T1 ) r = I a (4) r Substitute for T2 from equation (1) in equation (4) to obtain: (2.57T1 − T1 ) r = I a Solve for T1 and substitute numerical values to obtain: I 0.35 kg ⋅ m 2 T1 = a= a 2 (5) 1.57 r 2 1.57(0.15 m ) r = (9.91 kg )a Substitute in equation (2) to obtain: (9.91 kg )a − mg = ma 732 Chapter 9 Solve for and evaluate a: (b) Solve equation (3) for M: mg g = 9.91 kg − m 9.91 kg − 1 m 2 9.81 m/s = = 1.10 m/s 2 9.91 kg −1 1 kg a= T2 g −a M = Substitute in equation (5) to find T1: T1 = (9.91 kg ) (1.10 m/s 2 ) = 10.9 N Substitute in equation (1) to find T2: T2 = (2.57 )(10.9 N ) = 28.0 N Evaluate M: 28.0 N = 3.21 kg 9.81 m/s 2 − 1.10 m/s 2 M = 134 ••• Picture the Problem When the tension is horizontal, the cylinder will roll forward and r the friction force will be in the direction of T . We can use Newton’s 2nd law to obtain equations that we can solve simultaneously for a and f. ∑F = T + f = ma (1) ∑τ = Tr − fR = Iα (a) Apply Newton’s 2nd law to the cylinder: (2) x and a 1 = 2 mRa (3) R Substitute for I and α in equation (2) to obtain: Tr − fR = 1 mR 2 2 Solve equation (3) for f: f = Tr 1 − ma R 2 (4) Substitute equation (4) in equation (1) and solve for a: a= 2T ⎛ r⎞ ⎜1 + ⎟ 3m ⎝ R ⎠ (5) Substitute equation (5) in equation (4) to obtain: f = T ⎛ 2r ⎞ ⎜ − 1⎟ 3⎝ R ⎠ (b) Equation (4) gives the acceleration of the center of mass: a= 2T ⎛ r⎞ ⎜1 + ⎟ 3m ⎝ R ⎠ Rotation 733 (c) Express the condition that a > T : m 2T ⎛ r⎞ T 2⎛ r⎞ ⎜1 + ⎟ > ⇒ ⎜ 1 + ⎟ > 1 3m ⎝ R ⎠ m 3⎝ R⎠ or r> 1 2 R r f > 0, i.e., in the direction of T . (d) If r > 1 R : 2 135 ••• Picture the Problem The system is shown in the drawing in two positions, with angles θ0 and θ with the vertical. The drawing also shows all the forces that act on the stick. These forces result in a rotation of the stick—and its center of mass—about the pivot, and a tangential acceleration of the center of mass. We’ll apply the conservation of mechanical energy and Newton’s 2nd law to relate the radial and tangential forces acting on the stick. Use the conservation of mechanical energy to relate the kinetic energy of the stick when it makes an angle θ with the vertical and its initial potential energy: Kf − Ki + U f − U i = 0 or, because Kf = 0, − 1 Iω 2 + Mg 2 L L cosθ − Mg cosθ 0 = 0 2 2 3g (cosθ − cosθ 0 ) L Substitute for I and solve for ω2: ω2 = Express the centripetal force acting on the center of mass: Fc = M L 2 ω 2 L 3g (cosθ − cosθ 0 ) =M 2 L 3Mg (cosθ − cosθ 0 ) = 2 r Express the radial component of Mg : (Mg )radial = Mg cosθ Express the total radial force at the hinge: F|| = Fc + (Mg)radial 734 Chapter 9 = = Relate the tangential acceleration of the center of mass to its angular acceleration: 3Mg (cos θ − cos θ 0 ) + Mg cos θ 2 1 2 Mg (5 cos θ − 3 cos θ 0 ) a⊥= 1 Lα 2 L sin θ 3g sin θ 2 = 2 1 2L 3 ML Use Newton’s 2nd law to relate the angular acceleration of the stick to the net torque acting on it: α= Express a⊥ in terms of α: a⊥= 1 Lα = 2 Solve for F⊥ to obtain: F⊥ = − 1 Mg sin θ where the minus sign 4 τ net I = Mg 3 4 gsinθ = gsinθ + F⊥/M indicates that the force is directed oppositely to the tangential component of r Mg. Chapter 10 Conservation of Angular Momentum Conceptual Problems *1 • r r r r ˆ (a) True. The cross product of the vectors A and B is defined to be A × B = AB sin φ n. r r If A and B are parallel, sinφ = 0. r (b) True. By definition, is along the axis. (c) True. The direction of a torque exerted by a force is determined by the definition of the cross product. 2 • r r Determine the Concept The cross product of the vectors A and B is defined to be r r ˆ A × B = AB sin φ n. Hence, the cross product is a maximum when sinφ = 1. This r r condition is satisfied provided A and B are perpendicular. (c) is correct. 3 • r r r r r Determine the Concept L and p are related according to L = r × p. From this r r definition of the cross product, L and p are perpendicular; i.e., the angle between them is 90°. 4 • r r r r r Determine the Concept L and p are related according to L = r × p. Because the motion is along a line that passes through point P, r = 0 and so is L. (b) is correct. *5 •• r r r r r Determine the Concept L and p are related according to L = r × p. r r r Doubling p doubles L. r r r Doubling r doubles L. (a) Because L is directly proportional r to p : (b) Because L is directly proportional r to r : 735 736 Chapter 10 6 •• Determine the Concept The figure shows a particle moving with constant speed in a straight line (i.e., with constant velocity and constant linear momentum). The magnitude of L is given by rpsinφ = mv(rsinφ). Referring to the diagram, note that the distance rsinφ from P to the line along which the particle is moving is constant. Hence, mv(rsinφ) is constant and so r L is constant. 7 • False. The net torque acting on a rotating system equals the change in the system’s angular momentum; i.e., τ net = dL dt , where L = Iω. Hence, if τ net is zero, all we can say for sure is that the angular momentum (the product of I and ω) is constant. If I changes, so mustω. *8 •• Determine the Concept Yes, you can. Imagine rotating the top half of your body with arms flat at sides through a (roughly) 90° angle. Because the net angular momentum of the system is 0, the bottom half of your body rotates in the opposite direction. Now extend your arms out and rotate the top half of your body back. Because the moment of inertia of the top half of your body is larger than it was previously, the angle which the bottom half of your body rotates through will be smaller, leading to a net rotation. You can repeat this process as necessary to rotate through any arbitrary angle. 9 • Determine the Concept If L is constant, we know that the net torque acting on the system is zero. There may be multiple constant or time-dependent torques acting on the system as long as the net torque is zero. (e) is correct. 10 •• Determine the Concept No. In order to do work, a force must act over some distance. In each ″inelastic collision″ the force of static friction does not act through any distance. 11 •• Determine the Concept It is easier to crawl radially outward. In fact, a radially inward force is required just to prevent you from sliding outward. *12 •• Determine the Concept The pull that the student exerts on the block is at right angles to r r r its motion and exerts no torque (recall that τ = r × F and τ = rF sin θ ). Therefore, we Conservation of Angular Momentum 737 can conclude that the angular momentum of the block is conserved. The student does, however, do work in displacing the block in the direction of the radial force and so the block’s energy increases. (b) is correct. *13 •• Determine the Concept The hardboiled egg is solid inside, so everything rotates with a uniform velocity. By contrast, it is difficult to get the viscous fluid inside a raw egg to start rotating; however, once it is rotating, stopping the shell will not stop the motion of the interior fluid, and the egg may start rotating again after momentarily stopping for this reason. 14 • r r False. The relationship τ = dL dt describes the motion of a gyroscope independently of whether it is spinning. 15 • Picture the Problem We can divide the expression for the kinetic energy of the object by the expression for its angular momentum to obtain an expression for K as a function of I and L. Express the rotational kinetic energy of the object: K = 1 Iω 2 2 Relate the angular momentum of the object to its moment of inertia and angular velocity: L = Iω Divide the first of these equations by the second and solve for K to obtain: K= L2 and so (b) is correct. 2I 16 • Determine the Concept The purpose of the second smaller rotor is to prevent the body of the helicopter from rotating. If the rear rotor fails, the body of the helicopter will tend to rotate on the main axis due to angular momentum being conserved. 17 •• Determine the Concept One can use a right-hand rule to determine the direction of the torque required to turn the angular momentum vector from east to south. Letting the fingers of your right hand point east, rotate your wrist until your fingers point south. Note that your thumb points downward. (b) is correct. 738 Chapter 10 18 •• Determine the Concept In turning east, the man redirects the angular momentum vector from north to east by exerting a clockwise torque (viewed from above) on the gyroscope. As a consequence of this torque, the front end of the suitcase will dip downward. (d ) is correct. 19 •• (a) The lifting of the nose of the plane rotates the angular momentum vector upward. It veers to the right in response to the torque associated with the lifting of the nose. (b) The angular momentum vector is rotated to the right when the plane turns to the right. In turning to the right, the torque points down. The nose will move downward. 20 •• r Determine the Concept If L points up and the car travels over a hill or through a valley, the force on the wheels on one side (or the other) will increase and car will tend to r tip. If L points forward and car turns left or right, the front (or rear) of the car will tend to lift. These problems can be averted by having two identical flywheels that rotate on the same shaft in opposite directions. 21 •• Determine the Concept The rotational kinetic energy of the woman-plus-stool system is given by K rot = 1 Iω 2 = L2 2 I . Because L is constant (angular momentum is conserved) 2 and her moment of inertia is greater with her arms extended, (b) is correct. *22 •• Determine the Concept Consider the overhead view of a tether pole and ball shown in the adjoining figure. The ball rotates counterclockwise. The torque about the center of the pole is clockwise and of magnitude RT, where R is the pole’s radius and T is the tension. So L must decrease and (e) is correct. 23 •• Determine the Concept The center of mass of the rod-and-putty system moves in a straight line, and the system rotates about its center of mass. Conservation of Angular Momentum 739 24 • (a) True. The net external torque acting a system equals the rate of change of the angular r r dL momentum of the system; i.e., ∑ τ i,ext = . dt i (b) False. If the net torque on a body is zero, its angular momentum is constant but not necessarily zero. Estimation and Approximation *25 •• Picture the Problem Because we have no information regarding the mass of the skater, we’ll assume that her body mass (not including her arms) is 50 kg and that each arm has a mass of 4 kg. Let’s also assume that her arms are 1 m long and that her body is cylindrical with a radius of 20 cm. Because the net external torque acting on her is zero, her angular momentum will remain constant during her pirouette. Express the conservation of her angular momentum during her pirouette: Li = Lf or I arms outω arms out = I arms inω arms in Express her total moment of inertia with her arms out: Treating her body as though it is cylindrical, calculate its moment of inertia of her body, minus her arms: (1) I arms out = I body + I arms I body = 1 mr 2 = 2 1 2 (50 kg )(0.2 m )2 = 1.00 kg ⋅ m 2 [ Modeling her arms as though they are rods, calculate their moment of inertia when she has them out: I arms = 2 1 (4 kg )(1 m ) 3 Substitute to determine her total moment of inertia with her arms out: I arms out = 1.00 kg ⋅ m 2 + 2.67 kg ⋅ m 2 Express her total moment of inertia with her arms in: I arms in = I body + I arms = 2.67 kg ⋅ m 2 2 = 3.67 kg ⋅ m 2 [ = 1.00 kg ⋅ m 2 + 2 (4 kg )(0.2 m ) = 1.32 kg ⋅ m 2 2 740 Chapter 10 Solve equation (1) for ω arms in and substitute to obtain: ω arms in = = I arms out I arms in ω arms out 3.67 kg ⋅ m 2 (1.5 rev/s) 1.32 kg ⋅ m 2 = 4.17 rev/s 26 •• Picture the Problem We can express the period of the earth’s rotation in terms of its angular velocity of rotation and relate its angular velocity to its angular momentum and moment of inertia with respect to an axis through its center. We can differentiate this expression with respect to I and then use differentials to approximate the changes in I and T. 2π Express the period of the earth’s rotation in terms of its angular velocity of rotation: T= Relate the earth’s angular velocity of rotation to its angular momentum and moment of inertia: ω= L I Substitute to obtain: T= 2π I L Find dT/dI: dT 2π T = = dI L I Solve for dT/T and approximate ∆T: dT dI ∆I = or ∆T ≈ T T I I Substitute for ∆I and I to obtain: Substitute numerical values and evaluate ∆T: ω ∆T ≈ mr 2 5m T= T 2 2 3M E 5 M E RE ∆T = 5 2.3 × 1019 kg (1d ) 3 6 × 10 24 kg 2 3 ( ( ) ) −6 = 6.39 × 10 d = 6.39 × 10 −6 d × = 0.552 s 24 h 3600 s × d h Conservation of Angular Momentum 741 27 • Picture the Problem We can use L = mvr to find the angular momentum of the particle. In (b) we can solve the equation L = l(l + 1)h for l(l + 1) and the approximate value of l. (a) Use the definition of angular momentum to obtain: L = mvr = (2 × 10−3 kg )(3 ×10 −3 m/s )(4 × 10−3 m ) = 2.40 ×10 −8 kg ⋅ m 2 /s L2 h2 (b) Solve the equation L = l(l + 1)h for l(l + 1) : l(l + 1) = Substitute numerical values and evaluate l(l + 1) : ⎛ 2.40 × 10 −8 kg ⋅ m 2 /s ⎞ l(l + 1) = ⎜ ⎜ 1.05 × 01−34 J ⋅ s ⎟ ⎟ ⎝ ⎠ 2 = 5.22 × 1052 Because l >>1, approximate its value with the square root of l(l + 1) : l ≈ 2.29 × 10 26 The quantization of angular momentum is not noticed in macroscopic (c) physics because no experiment can differentiate between l = 2 × 10 26 and l = 2 × 10 26 + 1. *28 •• Picture the Problem We can use conservation of angular momentum in part (a) to relate the before-and-after collapse rotation rates of the sun. In part (b), we can express the fractional change in the rotational kinetic energy of the sun as it collapses into a neutron star to decide whether its rotational kinetic energy is greater initially or after the collapse. (a) Use conservation of angular momentum to relate the angular momenta of the sun before and after its collapse: I bωb = I aωa Using the given formula, approximate the moment of inertia Ib of the sun before collapse: 2 I b = 0.059MRsun ( (1) )( = 0.059 1.99 × 1030 kg 6.96 × 105 km = 5.69 × 10 46 kg ⋅ m 2 ) 2 742 Chapter 10 Find the moment of inertia Ia of the sun when it has collapsed into a spherical neutron star of radius 10 km and uniform mass distribution: 2 I a = 5 MR 2 = 2 5 (1.99 ×10 30 ) kg (10 km ) 2 = 7.96 ×1037 kg ⋅ m 2 Substitute in equation (1) and solve for ωa to obtain: ωa = Ib 5.69 × 10 46 kg ⋅ m 2 ωb = ωb Ia 7.96 × 1037 kg ⋅ m 2 = 7.15 × 108 ωb Given that ωb = 1 rev/25 d, evaluate ωa: ⎛ 1 rev ⎞ ⎟ ⎟ ⎝ 25 d ⎠ ωa = 7.15 × 108 ⎜ ⎜ = 2.86 × 10 7 rev/d The additional rotational kinetic energy comes at the expense of gravitational potential energy, which decreases as the sun gets smaller. Note that the rotational period decreases by the same factor of Ib/Ia and becomes: Ta = 2π ωa = 2π = 3.02 × 10−3 s 2π rad 1d 1h 7 rev 2.86 × 10 × × × d rev 24 h 3600 s ∆K K a − K b K a = −1 = Kb Kb Kb (b) Express the fractional change in the sun’s rotational kinetic energy as a consequence of its collapse and simplify to obtain: I aωa2 = −1 2 I bωb 1 2 1 2 = I aωa2 −1 2 I bωb Substitute numerical values and evaluate ∆K/Kb: 2 7 1 ∆K ⎛ ⎞ ⎛ 2.86 × 10 rev/d ⎞ ⎟ − 1 = 7.15 × 108 (i.e., the rotational kinetic =⎜ ⎟⎜ K b ⎝ 7.15 × 108 ⎠ ⎜ 1 rev/25 d ⎟ ⎠ ⎝ energy increases by a factor of approximately 7×108.) 29 •• Picture the Problem We can solve I = CMR 2 for C and substitute numerical values in order to determine an experimental value of C for the earth. We can then compare this value to those for a spherical shell and a sphere in which the mass is uniformly distributed to decide whether the earth’s mass density is greatest near its core or near its crust. Conservation of Angular Momentum 743 (a) Express the moment of inertia of the earth in terms of the constant C: Solve for C to obtain: Substitute numerical values and evaluate C: I = CMR 2 C= I MR 2 C= 8.03 ×1037 kg ⋅ m 2 2 5.98 × 1024 kg (6370 km ) ( ) = 0.331 (b) If all of the mass were in the crust, the moment of inertia of the earth would be that of a thin spherical shell: I spherical shell = 2 MR 2 3 If the mass of the earth were uniformly distributed throughout its volume, its moment of inertia would be: I solid sphere = 2 MR 2 5 Because experimentally C < 2/5 = 0.4, the mass density must be greater near the center of the earth. *30 •• Picture the Problem Let’s estimate that the diver with arms extended over head is about 2.5 m long and has a mass M = 80 kg. We’ll also assume that it is reasonable to model the diver as a uniform stick rotating about its center of mass. From the photo, it appears that he sprang about 3 m in the air, and that the diving board was about 3 m high. We can use these assumptions and estimated quantities, together with their definitions, to estimate ω and L. ∆θ ∆t (1) L = Iω Express the diver’s angular velocity ω and angular momentum L: (2) ω= and Using a constant-acceleration equation, express his time in the air: ∆t = ∆t rise 3 m + ∆tfall 6 m = Substitute numerical values and evaluate ∆t: Estimate the angle through which he rotated in 1.89 s: ∆t = 2∆yup g + 2∆ydown g 2(3 m ) 2(6 m ) + = 1.89 s 2 9.81 m/s 9.81 m/s 2 ∆θ ≈ 0.5 rev = π rad 744 Chapter 10 Substitute in equation (1) and evaluate ω: ω= π rad 1.89 s = 1.66 rad/s Use the ″stick rotating about an axis through its center of mass″ model to approximate the moment of inertia of the diver: 1 I = 12 ML2 Substitute in equation (2) to obtain: 1 L = 12 ML2ω Substitute numerical values and evaluate L: 1 L = 12 (80 kg )(2.5 m ) (1.66 rad/s ) 2 = 69.2 kg ⋅ m 2 /s ≈ 70 kg ⋅ m 2 /s Remarks: We can check the reasonableness of this estimation in another way. Because he rose about 3 m in the air, the initial impulse acting on him must be about 600 kg⋅m/s (i.e., I = ∆p = Mvi). If we estimate that the lever arm of the force is roughly l = 1.5 m, and the angle between the force exerted by the board and a line running from his feet to the center of mass is about 5°, we obtain L = Ilsin5° ≈ 78 kg⋅m2/s, which is not too bad considering the approximations made here. 31 •• Picture the Problem First we assume a spherical diver whose mass M = 80 kg and whose diameter, when curled into a ball, is 1 m. We can estimate his angular velocity when he has curled himself into a ball from the ratio of his angular momentum to his moment of inertia. To estimate his angular momentum, we’ll guess that the lever arm l of the force that launches him from the diving board is about 1.5 m and that the angle between the force exerted by the board and a line running from his feet to the center of mass is about 5°. Express the diver’s angular velocity ω when he curls himself into a ball in mid-dive: Using a constant-acceleration equation, relate the speed with which he left the diving board v0 to his maximum height ∆y and our estimate of his angle with the vertical direction: Solve for v0: Substitute numerical values and evaluate v0: ω= L I (1) 2 0 = v0 y + 2a y ∆y where v0 y = v0 cos 5° v0 = 2 g∆y cos 2 5° v0 = 2 9.81m/s 2 (3 m ) = 7.70 m/s cos 5° ( ) Conservation of Angular Momentum 745 Approximate the impulse acting on the diver to launch him with the speed v0: I = ∆p = Mv0 Letting l represent the lever arm of the force acting on the diver as he leaves the diving board, express his angular momentum: L = Il sin 5° = Mv0l sin 5° Use the ″uniform sphere″ model to approximate the moment of inertia of the diver: 2 I = 5 MR 2 Substitute in equation (1) to obtain: Substitute numerical values and evaluate ω: ω= Mv0l sin 5° 5v0l sin 5° = 2 MR 2 2R2 5 ω= 5(7.70 m/s )(1.5 m )sin 5° 2 2(0.5 m ) = 10.1 rad/s *32 •• Picture the Problem We’ll assume that he launches himself at an angle of 45° with the horizontal with his arms spread wide, and then pulls them in to increase his rotational speed during the jump. We’ll also assume that we can model him as a 2-m long cylinder with an average radius of 0.15 m and a mass of 60 kg. We can then find his take-off speed and ″air time″ using constant-acceleration equations, and use the latter, together with the definition of rotational velocity, to find his initial rotational velocity. Finally, we can apply conservation of angular momentum to find his initial angular momentum. Using a constant-acceleration equation, relate his takeoff speed v0 to his maximum elevation ∆y: 2 v 2 = v0 y + 2a y ∆y or, because v0y = v0sin45°, v = 0, and ay = − g, 2 0 = v0 sin 2 45° − 2 g∆y Solve for v0 to obtain: v0 = 2 g∆y 2 g∆y = 2 sin 45° sin 45° Substitute numerical values and evaluate v0: v0 = 2 9.81m/s 2 (0.6 m ) = 4.85 m/s sin45° Use its definition to express Goebel’s angular velocity: ω= Use a constant-acceleration equation to express Goebel’s ″air time″ ∆t: ∆t = 2∆trise 0.6 m = 2 ( ) ∆θ ∆t 2∆y g 746 Chapter 10 Substitute numerical values and evaluate ∆t: ∆t = 2 Substitute numerical values and evaluate ω: ω= Use conservation of angular momentum to relate his take-off angular velocity ω0 to his average angular velocity ω as he performs a quadruple Lutz: Assuming that he can change his angular momentum by a factor of 2 by pulling his arms in, solve for and evaluate ω0: 2(0.6 m ) = 0.699 s 9.81 m/s 2 4 rev 2π rad × = 36.0 rad/s 0.699 s rev I 0ω0 = Iω ω0 = 1 I ω = (36 rad/s ) = 18.0 rad/s I0 2 Express his take-off angular momentum: L0 = I 0ω0 Assuming that we can model him as a solid cylinder of length l with an average radius r and mass m, express his moment of inertia with arms drawn in (his take-off configuration): I 0 = 2 1 mr 2 = mr 2 2 where the factor of 2 represents our assumption that he can double his moment of inertia by extending his arms. Substitute to obtain: L0 = mr 2ω0 Substitute numerical values and evaluate L0: L0 = (60 kg )(0.15 m ) (18 rad/s ) ( ) 2 = 24.3 kg ⋅ m 2 /s Vector Nature of Rotation 33 • r r ˆ j Picture the Problem We can express F and r in terms of the unit vectors i and ˆ and r then use the definition of the cross product to find τ . r r ˆ F = − Fi r r r = Rˆ j Express F in terms of F and the unit ˆ vector i : Express r in terms of R and the unit vector ˆ : j Conservation of Angular Momentum 747 r F: ( r r r τ = r × F = FR ˆ × −i j ˆ r Calculate the cross product of r and ( ) ) ˆ j ˆ = FR i × ˆ = FRk 34 • r r Picture the Problem We can find the torque is the cross product of r and F . r r Compute the cross product of r and F : ( )( ) ( ) r r r ˆ j τ = r × F = xi + yˆ − mgˆ j ˆ j = −mgx i × ˆ − mgy ˆ × ˆ j j ( ) ˆ = − mgxk 35 • r ˆ Picture the Problem The cross product of the vectors A = Ax i + Ay ˆ j r ˆ j and B = B x i + B y ˆ is given by ( ) ( ) ( ) () ( ) ( ) ( ) r r ˆ ˆ ˆ j A × B = Ax B x i × i + Ax B y i × ˆ + Ay B x ˆ × i + Ay B y ˆ × ˆ j ˆ j j ˆ ˆ = Ax B x (0) + Ax B y k + Ay B x − k + Ay B y (0) ˆ ˆ = A B k + A B −k x r y () r y x r ˆ (a) Find A × B for A = 4 i and r ˆ j B = 6i + 6ˆ : ( ) ( ) ( ) r r ˆ ˆ A × B = 4i × 6i + 6 ˆ j ˆ ˆ ˆ j = 24 i × i + 24 i × ˆ ˆ ˆ = 24(0) + 24k = 24k r r r r r r ˆ ˆ A × B = 2i + 3 ˆ × 3i + 2 ˆ j j ˆ ˆ ˆ j = 6 i ×i + 4 i × ˆ + 9 ˆ×i j ˆ j j + 6 ˆ× ˆ ˆ (c) Find A × B for A = 2 i + 3 ˆ j r ˆ and B =3 i + 2 ˆ : j ) ( ) ( ) j j = 24(0) + 24(− ˆ ) = − 24 ˆ r r ˆ ˆ ˆ A × B = 4i × 6i + 6k ˆ ˆ ˆ ˆ = 24 i × i + 24 i × k r ˆ ˆ B = 6i + 6k : r ( r ˆ (b) Find A × B for A = 4 i and ( )( ) ( ) ( ) ( ) ( ) ˆ ˆ = 6(0 ) + 4(k )+ 9(− k ) + 6(0 ) ˆ = − 5k 748 Chapter 10 *36 • r r Picture the Problem The magnitude of A × B is given by AB sin θ . r r AB sin θ = AB cos θ Equate the magnitudes of A × B r r and A ⋅ B : ∴ sin θ = cos θ or tan θ = ±1 Solve for θ to obtain: θ = tan −1 ± 1 = ± 45° or ± 135° 37 •• r Picture the Problem Let r be in the xy r plane. Then ω points in the positive z direction. We can establish the results called for in this problem by forming the appropriate cross products and by r differentiating v . r r ˆ ω =ωk (a) Express ω using unit vectors: r r ˆ r = ri Express r using unit vectors: r r Form the cross product of ω and r : ( ) r r ˆ ˆ ˆ ˆ ω × r = ω k × r i = r ω k × i = rω ˆ j = vˆ j r r r ∴ v = ω× r r (b) Differentiate v with respect to t to r express a : r r dv d r r a= = (ω × r ) dt dt r r dω r r dr = × r + ω× dt dt r dω r r r = × r + ω× v dt r r r r = a t + ω × (ω × r ) r r = a t + ac r r r r where a c = ω × (ω × r ) r r and a t and a c are the tangential and Conservation of Angular Momentum 749 centripetal accelerations, respectively. 38 •• r r ˆ j Picture the Problem Because Bz = 0, we can express B as B = B x i + B y ˆ and form its r cross product with A to determine Bx and By. r Express B in terms of its components: r r ˆ B = Bx i + B y ˆ j (1) ( ) Express A × B : r r ˆ ˆ ˆ ˆ A × B = 4i × Bx i + B y ˆ = 4 B y k = 12k j Solve for By: By = 3 Relate B to Bx and By: 2 B 2 = B x2 + B y Solve for and evaluate Bx: 2 Bx = B 2 − B y = 5 2 − 32 = 4 Substitute in equation (1): r ˆ B = 4i + 3 ˆ j r 39 • r r ˆ ˆ Picture the Problem We can write B in the form B = B x i + B y ˆ + B z k and use the dot j r r product of A and B to find By and their cross product to find Bx and Bz. r r ˆ ˆ B = Bx i + B y ˆ + Bz k j r r r r A ⋅ B = 3B y = 12 Express B in terms of its components: Evaluate A ⋅ B : (1) and By = 4 r r Evaluate A × B : ( r r ˆ ˆ A × B = 3 ˆ × Bx i + 4 ˆ + Bz k j j ˆ ˆ = −3B k + 3B i x r r ˆ Because A × B = 9 i : Bx = 0 and Bz = 3. Substitute in equation (1) to obtain: r ˆ B = 4 ˆ + 3k j z ) 750 Chapter 10 40 •• r r r Picture the Problem The dot product of A with the cross product of B and C is a scalar ax ay az quantity and can be expressed in determinant form as bx by bz . We can expand this cx c y cz r r r r r r determinant by minors to show that it is equivalent to A ⋅ ( B × C ) , C ⋅ ( A × B ) , and r r r B ⋅ (C × A) . r The dot product of A with the cross r r product of B and C is a scalar quantity and can be expressed in determinant form as: Expand the determinant by minors to obtain: ax r r r A ⋅ ( B × C ) = bx cx ay az by cy bz cz ax ay az bx cx by cy bz = a x b y c z − a x bz c y cz + a y bz c x − a y bx c z (1) + a z bx c y − a z b y c x r Evaluate the cross product of B and r C to obtain: r r ˆ B × C = (by cz − bz c y ) i ˆ + (bz cx − bx cz ) ˆ + (bx c y − by cx )k j r Form the dot product of A with r r B × C to obtain: ( ) r r r A ⋅ B × C = axby cz − axbz c y + a ybz cx − a ybx cz + az bx c y − az by cx Because (1) and (2) are the same, we can conclude that: ay az by bz cx Proceed as above to establish that: ax r r r A ⋅ ( B × C ) = bx cy cz ax r r r C ⋅ ( A × B ) = bx ay az by bz cx cy cz and (2) Conservation of Angular Momentum 751 ax r r r B ⋅ (C × A) = bx cx ay az by cy bz cz 41 •• r Picture the Problem Let, without loss of generality, the vector C lie along the x axis and r the vector B lie in the xy plane as shown below to the left. The diagram to the right shows the parallelepiped spanned by the three vectors. We can apply the definitions of r (r ) r the cross- and dot-products to show that A ⋅ B × C is the volume of the parallelepiped. r r Express the cross-product of B and C : ( ) r r ˆ B × C = (BC sin θ ) − k and r r B × C = (B sin θ )C = area of the parallelogram r r r cross-product of B and C to obtain: Form the dot-product of A with the ( ) r r r A ⋅ B × C = A(B sin θ )C cos φ = (BC sin θ )( A cos φ ) = (area of base )(height ) = Vparallelepiped *42 •• Picture the Problem Draw the triangle using the three vectors as shown below. r r r Note that A + B = C . We can find the r magnitude of the cross product of A and r r r B and of A and C and then use the cross r r r r r product of A and C , using A + B = C, to show that AC sin b = AB sin c or B/sin b = C/ sin c. Proceeding similarly, we can extend the law of sines to the third side of the triangle and the angle opposite it. 752 Chapter 10 Express the magnitude of the cross r r A × B = AB sin c Express the magnitude of the cross r r A × C = AC sin b r r product of A and B : r r product of A and C : r Form the cross product of A with r C to obtain: r r r r ( ) r r r r r A× C = A× A + B r r r r = A× A + A× B r r = A× B r r because A × A = 0 . r r r r A× C = A× B Because A × C = A × B : and AC sin b = AB sin c Simplify and rewrite this expression to obtain: B C = sin b sin c Proceed similarly to extend this result to the law of sines: A B C = = sin a sin b sin c Angular Momentum 43 • r r r r r r Picture the Problem L and p are related according to L = r × p. If L = 0, then r r examination of the magnitude of r × p will allow us to conclude that sin φ = 0 and that the particle is moving either directly toward the point, directly away from the point, or through the point. r r r r r r r × p = r × mv = mr × v = 0 r Because L = 0: or r r r ×v = 0 r r Express the magnitude of r × v : r r r × v = rv sin φ = 0 Because neither r nor v is zero: sin φ = 0 r r where φ is the angle between r and v . Solve for φ: φ = sin −1 0 = 0° or 180° Conservation of Angular Momentum 753 44 • Picture the Problem We can use their definitions to calculate the angular momentum and moment of inertia of the particle and the relationship between L, I, and ω to determine its angular speed. (a) Express and evaluate the r magnitude of L : L = mvr = (2 kg )(3.5 m/s )(4 m ) = 28.0 kg ⋅ m 2 /s (b) Express the moment of inertia of the particle with respect to an axis through the center of the circle in which it is moving: I = mr 2 = (2 kg )(4 m ) = 32 kg ⋅ m 2 (c) Relate the angular speed of the particle to its angular momentum and solve for and evaluate ω: ω= 2 L 28.0 kg ⋅ m 2 /s = = 0.875 rad/s 2 32 kg ⋅ m 2 I 45 • Picture the Problem We can use the definition of angular momentum to calculate the angular momentum of this particle and the relationship between its angular momentum and angular speed to describe the variation in its angular speed with time. (a) Express the angular momentum of the particle as a function of its mass, speed, and distance of its path from the reference point: L = rmv sin θ (b) Because L = mr2ω: ω∝ = (6 m )(2 kg )(4.5 m/s )sin90° = 54.0 kg ⋅ m 2 /s 1 and r2 ω increases as the particle approaches the point and decreases as it recedes. *46 •• Picture the Problem We can use the formula for the area of a triangle to find the area swept out at t = t1, add this area to the area swept out in time dt, and then differentiate this expression with respect to time to obtain the given expression for dA/dt. Express the area swept out at t = t1: A1 = 1 br1 cos θ1 = 1 bx1 2 2 r r where θl is the angle between r1 and v and 754 Chapter 10 r x1 is the component of r1 in the direction of r v. Express the area swept out at t = t1 + dt: Differentiate with respect to t: Because rsinθ = b: A = A1 + dA = 1 b( x1 + dx ) 2 = 1 b( x1 + vdt ) 2 dA 1 dx 1 = 2b = 2 bv = constant dt dt 1 2 bv = 1 2 (r sin θ )v = 1 (rp sin θ ) 2m L 2m = 47 •• Picture the Problem We can find the total angular momentum of the coin from the sum of its spin and orbital angular momenta. Lspin = I cmω spin (a) Express the spin angular momentum of the coin: From Problem 9-44: I = 1 MR 2 4 Substitute for I to obtain: Lspin = 1 MR 2ωspin 4 Substitute numerical values and evaluate Lspin: Lspin = 1 4 (0.015 kg )(0.0075 m )2 ⎛ rev 2π rad ⎞ × ⎜10 × ⎟ s rev ⎠ ⎝ = 1.33 × 10−5 kg ⋅ m 2 /s (b) Express and evaluate the total angular momentum of the coin: L = Lorbit + Lspin = 0 + Lspin (c) From Problem 10-14: Lorbit = 0 = 1.33 × 10 −5 kg ⋅ m 2 /s and L = 1.33 × 10 −5 kg ⋅ m 2 /s (d) Express the total angular momentum of the coin: L = Lorbit + Lspin Conservation of Angular Momentum 755 Find the orbital momentum of the coin: Lorbit = ± MvR = ±(0.015 kg )(0.05 m/s )(0.1 m ) = ±7.50 × 10−5 kg ⋅ m 2 /s where the ± is a consequence of the fact that the coin’s direction is not specified. L = ±7.50 × 10 −5 kg ⋅ m 2 /s Substitute to obtain: + 1.33 × 10 −5 kg ⋅ m 2 /s L = 8.83 × 10 −5 kg ⋅ m 2 /s The possible values for L are: or L = − 6.17 × 10 −5 kg ⋅ m 2 /s 48 •• Picture the Problem Both the forces acting on the particles exert torques with respect to an axis perpendicular to the page and through point O and the net torque about this axis is their vector sum. Express the net torque about an axis perpendicular to the page and through point O: r r r Because r1 − r2 points along − F1 : r r r r r r τ net = ∑ τ i = r1 × F1 + r2 × F2 i r r r = (r1 − r2 )× F1 r r because F2 = − F1 r r r (r1 − r2 ) × F1 = 0 Torque and Angular Momentum 49 • Picture the Problem The angular momentum of the particle changes because a net torque acts on it. Because we know how the angular momentum depends on time, we can find the net torque acting on the particle by differentiating its angular momentum. We can use a constant-acceleration equation and Newton’s 2nd law to relate the angular speed of the particle to its angular acceleration. (a) Relate the magnitude of the torque acting on the particle to the rate at which its angular momentum changes: τ net = dL d = [(4 N ⋅ m )t ] dt dt = 4.00 N ⋅ m 756 Chapter 10 (b) Using a constant-acceleration equation, relate the angular speed of the particle to its acceleration and time-in-motion: ω = ω0 + α t where ω0 = 0 Use Newton’s 2nd law to relate the angular acceleration of the particle to the net torque acting on it: α= τ net Substitute to obtain: ω= τ net Substitute numerical values and evaluate ω: I ω= mr 2 = τ net mr 2 t (4 N ⋅ m )t (1.8 kg )(3.4 m )2 ( ) = 0.192 rad/s 2 t provided t is in seconds. 50 •• Picture the Problem The angular momentum of the cylinder changes because a net torque acts on it. We can find the