hw1 - 01.9. The quantity 77 = 3.14159 . . . is a number...

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Unformatted text preview: 01.9. The quantity 77 = 3.14159 . . . is a number with no dimen— sions, since it is a ratio of two lengths. Describe two or three other geometrical or physical quantities that are dimensionless. 01.19. What are the units of volume? Suppose another student tells you that a cyiinder of radius r and height it has voiume given by firah. Explain why this cannot be right. {21.11. Three archers each fire four arrows at a target. Joe’s four arrows hit at points i0 cm above, 10 cm below, 10 cm to the left, and :0 cm to the right of the center of the target. All four of Moe’s arrows hit within 1 cm of a point 20 cm from the center, and Fio’s four arrows all hit within 1 cm of the center. The contest judge says that one of the archers is precise but not accurate, another archer is accu~ rate but not precise, and the third archer is both accurate and precise. Which description goes with which archer? Explain your reasoning. 01.12. A circular racetrack has a radius of 500 in. What is the dis- placement of a bicyclist when she travels around the track from the north side to the south side? When she makes one complete circle around the track? Expiain your reasoning. (21.13. Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain your reasoning. {31.14. One sometimes speaks of the “direction of time,” evolving from past to future. Does this mean that time is a vector quantity? Explain your reasoning. (11.15. Air traffic controllers give instructions to airline pilots teliing them in which direction they are to fly. These instructions are called “vectors.” If these are the only instructions given, is the name “vector” used correctly? Why or why not? 01.16. Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain. Can the magnitude of a vector be less than the magnitude of any of its com— ponents? Explain. 01.17. (a) Does it make sense to say that a vector is negative? Why? (b) Does it nitrite sense to say that one vector is the negative of another"? Why? Does your answer here contradict what you said in part (at? M w) A M _) {21.18. If C is the vector sum ofA and B, C = A + B, what must be true ifC 2 A + B? What must be true ifC 2 0? q1.19.w)1i‘ 2i and E are nonzero vectors, is it possible for 3 J5? and A x B both to be zero? Explain. 01.20. What doesmgi .174), the scalar product of a vector with itself, give? What about A X the vector product of a vectorwith itseif? 01.21. Let X represent any nonzero vector. Why is Iii/A a unit vec- tor and what is its direction? 1ft? is the angle that 3 makes with the +x—axis, explain why {Ail/i) ~i is called the direction cosine for that axis. 02.22. Whigh of the fpllowing are legitimate mathematicai opera- tions; (20ft - (ME, —~ C); (1)} (2i —— E) X E; (c) 3’ (f? x 6‘); (d) A x (B x (e) X x - In each case, give the reason for your answer. QI.23._)CODS}C§BI the two repeated vector products A: x (I? x 6') and (A x E) x E". Give an exampie that illustrates the generai rule that these two vector products do not have the same magni— tude or direction. Can you choose the vectors 3, B, and 6‘ such that these two vector products are equal? If so, give an example. 01.24. Show that, no matter what X and E are, E - {If x Ii) 2 0. (Hint: Do not look for an elaborate mathematical proof. Rather look at the deiimnition of the direction of the cross product.) 01.25. (a) If A 45; fl 0._ does it necessary foilow that A t—* 0 or B m 0? Expiain. (ii) .If A X ii; = 0, does it necessary foliow that A m 0 or B = 0? Explain. Exercises 29 01.26. If X m 0 for a vector in the xy piane, does it follow that A, = way? What can you say about A“. and 14).? Exercises Section 1.3 Standards and Units Section 1.4 Unit Consistency and Conversions 1.3. Starting with the definition 'i in. m 2.54 cm, find the number of (a) kilometers in i.00 mite and (b) feet in 1.00 ion. 1.2. According to the label on a bottle of salad dressing, the vol- ume of the contents is 0.473 liter (L). Using only the conversions 1 L = 1000 cm3 and lin. m 2.54 cm, express this volume in cubic inches. 1.3. How many nanoseconds does it taice tight to travel 1.00 ft in vacuum? (This result is a useful quantity to remember.) 1.13. The density of lead is 11.3 g/cm3. What is this value in idio— grarns per cubic meter? 1.5. The most powerqu engine availabie for the ciassic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a dispiacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions l L = 1000 cm3 and l in. = 2.54 cm. 1.6. A square field measuring 100.0 m by l00.0 in has an area of 1.00 hectare. An acre has an area or 43,600 n2. If a country lot has an area of 12.0 acres, what is the area in hectares? 1.7. How many years older wilt you be 1.00 biliion seconds from now? (Assume a 365—day year.) 1.3. While driving in an exotic foreign land you see a speed limit sign on a highway that reads i80,000 furlongs per fortnight. How many miles per hour is this? (One furlong is i miie, and a fortnight is 14 days. A furiong originally referred to the length of a plowed furrow.) 1.9. A certain fuel—efficient hybrid car gets gasoline mileage of 55.0 mpg (miles per gallon). (a) If you are driving this car in Europe and want to compare its miieage with that of other Euro pean cars, express this miieagc in km/L (L = liter). Use the con- version factors in Appendix E. (b) If this car’s gas tank holds 45 L, how many tanks of gas will you use to drive 1500 km? 130. The following conversions occur frequently in physics and are very useful. (a) Use 1 mi = 5280 ft and '1 h w 3600 s to con- vert 60 mph to units of ft/ s. (b) The acceleration of a freely falling object is 32 ft/sz. Lisa 1 ft x 30.48 cm to expreSs this acceieration in units of m/sz. (c) The density of water is 'i .0 g/crn3. Convert this density to units of icg/m3. 1.11. Neptunium. In the felt of 2002, a group of scientists at Los Alamos National Laboratory determined that the criticai mass of neptunium-ZB? is about 60 kg. The critical mass of a fissionable materiai is the minimum amount that must be brought together to start a chain reaction. This element has a density of 19.5 g/cma. What would be the radius of a sphere of this material that has a critical mass? Section 1.5 Uncertainty and Significant Figures 1.22. A useful and easy-to‘reniernber approximate vaiue for the number of seconds in a year is 17 X 107. Determine the percent error in this approximate value. {There are 365.24 days in one year.) 1.13. Figure 1.7 shows the result of unacceptable error in the stop ping position of a train. (a) If a train travels 890 ion from Beriin to Paris and then overshoots the end of the track by 10 in, what is the percent error in the total distance covered? (in) Is it correct to write the total distance covered by the train as 890,010 In? Explain. 30 CHAPTER 1 Units, Physical Quantities, and Vectors 1.14. With a wooden ruier you measure the length of a rectanguiar piece of sheet metal to be i2 mrn. You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 mm. Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (is) What is the ratio of the rectangle’s width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is the ratio of the length to the width? 1.15. Estimate the percent error in measuring (a) a distance of about 75 cm with a meter stick; (b) a mass of about 12 g with a chemical balance; (c) a time interval of about 6 min with a stopwatch. £26. A rectangular piece of aiuminum is 5.10 :1: 0.01 cm long and 1.90 t 0.01 cm wide. (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties in the iength and in the width. (This is a general result; see Challenge Problem 2.98.) 1.17. As you eat your way through a bag of chocolate chip cookies, you ohserve that each cookie is a circuiar riisk with a diameter of 8.50 i 0.02 cm and a thickness of 0.050 m 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (in) Find the ratio of the diameter to the thickness and the uncer— tainty in this ratio. Section 3.6 Estimates and Orders of Magnitude 138. How many gallons of gasoline are used in the United States in one day? Assume two cars for every three peopie, that each car is driven an average of i0,000 mi per year, and that the average car gets 20 miles per galion. 1.19. A rather ordinary middle—aged man is in the hospitai for a routine checkwup. The nurse writes the quantity 200 on his medical chart but forgets to include the units. Which of the following quanw tities could the 200 plausibly represent? (a) his mass in kilograms; (b) his height in meters; (c) his height in centimeters; (d) his height in millimeters; (e) his age in months. L20. How many kernels of corn does it take to fill a 241. soft drink bottle? 1.21. How many words are there in this hook? 1.22. Four astronauts are in a spherical space station. (a) if, as is typical, each of them breathes about 500 cm3 of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain ali this air? 1.23. How many times does a typical person blink her eyes in a iifettme? $.24. How many times does a human heart beat during a lifetime? How many gallons of blood does it pump? (Estimate that the heart pumps 50 cm3 of blood with each beat.) 1.25. in Wagner’s opera Dos Rheingoid, the goddess Freia is ran— somed for a pile of gold just tail enough and wide enough to hide her from sight. Estimate the monetary value of this pile. The den~ sity of gold is 19.3 g/cma, and its vaiue is about $10 per gram (although this varies). 1.26. You are using water to dilute small amounts of chemicals in the laboratory, drop try drop. How many drops of water are in a 1.0 L bottie? (Hint: Start by estimating the diameter of a drop of water.) 1.21. How many pizzas are consumed each academic year by stu« dents at your school? - 1.28. How many dollar biEls woutd you have to stack to reach the moon? Would that he cheaper than building and iaunching a space craft? (Him: Start by folding a dollar bill to see how many thick~ nesses make 'i.0 mm.) 1.29. How much would it cost to paper the entire United States {inciuding Aiaska and Hawaii) with dollar bills? What weald be the cost to each person in the United States? Section 1.7 Vectors and Vector Addition 1.30. Hearing rattles from a snake, you make two rapid displace— ments of magnitude 1.8 m and 2.4 m. In sketches (roughly to scale), show how your two displacements might add up to give a resuitant of magnitude (a) 4.2 rn; (b) 0.6 m; (c) 3.0 in. 1.31. A postal employee drives a delivery truck along the route shown in Fig. 1.33. Determine the magnitude and direction of the Figure 1.33 Exercises 1.3i and 1.38. resultant displacement by drawing a scale diagram. (See also Exerw cise 1.38 for a different approach to this same probiem.) 1.32. For the vectors 3 and 1.5 in Fig. 1.34, use a scale drawing to find the magnitude and direc- tion of (a) the vector sum K + Figure 1.34 Exercises 1.32, 1.35, 1.39, 1.47, 1.53, and 1.57, and Problem 1.72. and (min) the vector difference y A ~B. Use your answers to ,§(15.0 m) find the magnitude and direction "" of (c) —Ai - a and (chit —A. (See also Exercise i.39 for a dif— ferent approach to this problem.) 1.33. A spelunker is surveying a cave. She follows a passage 180 in straight west, then 2'10 m in a direction 45" east of south, and then 280 m at 30° east of 502.0111) north. After a fourth unmeasured dispiacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth dispiacement. (See aiso Probiem 1.73 for a different approach to this problem.) .1' (3.00 on) Section 1.8 Components of Vectors 1.34. Use a scale drawing to find the x- and y—components of the following vectors. For each vector the numbers given are the mag- nitude of the vector and the angle, measured in the sense from the +x~axis toward the +y-axis, that it makes with the "ex-axis: (a) magnitude 9.30 in, angle 60.0“; (b) magnitude 22.0 km, angie 135°; (c) magnitude 6.35 cm, angie 307°. we 1.35. Compute the x— and y-components of the vectors 3, E, C, and ES in Fig. 2.34. 1.36. Let the angle 6 be the angle that the vector 3 makes with the +x-axis, measured counterciockwise from that axis. Find the angle 6 for a vector that has the following components: (a) AA. m 2.00 m, Ay = ~t.00 m; (b) A,{ m 2.00 m, A). “m: 1.00 In; (c) A. 2 “200121, Ay m i.00 m; (d) Ax fl “0.00111, A). = ~l. .00 In. 1.37. A rocket fires two engines simultaneously. One produces a thrust of 725 N directly forward, While the other gives a 513-N thrust at 32.40 above the forward direction. Find the magnitude and direction (relative to the forward direction} of the resultant force that these engines exert on the rocket. 1.38. A postal emptoyee drives a delivery truck over the route shown in Fig. 1.33. Use the method of components to determine the magnitude and direction of her resuitant displacement. In a vectoraaddition diagram (roughiy to scale), show that the result- ant displacement found from your diagram is in qualitative agreement with the result you obtained using the method of components. 1.39. For the vectors 2i and ii in Fig. :34, use the method of com- ponents to find the magnitude and direction of (a) the vector sum .3 —i— E; (b) the vector sum fi +3); (0) the vector difference 3 -~ (d) the vector difference ii - 1.40. Find the magnitude and direction of the vector represented by the following pairs of components: (a) A,r 3 M860 cm, A). m 5.20 cm; {0) A, x M070 m, Ay = —2.45 in; (c) A, = 7.75 km, A). a “2.70 km. 1.41. A disoriented physics professor drives 3.25 km north, then 4.75 ion west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of com ponents. In a vector addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualita- h‘ve agreement with the i‘esuit you obtained using the method of components. my 1.42;} VectorA has components Ax fl 1.30 cm, A). m 2.25 cm; vec— tor B has components 13r = cm“; 3,. = ~3.75 cm. Find (a) the components gt" the vector sum A + B; (b) the magnitude anddireg tion of A + B; (c) the components of the vector difference B -— A; (d) the magnitude and direction of 3 ~ 3. 1.43. Vector Si is 2.80 cm long and is 600° above the xmaxis in the first quadrant. Vector B is 1.90 cm long and is 600° below the x-axis in the fourth quadrant (Fig. 1.35). Use com— ponents to find the magnitude and direction of {a} X + 3; (0)3 m a {0) it -~ 3. In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your stretch. 1.44. A river flows from south to north at 5.0 km/h. On this river, a boat is heading east to west perpendicular to the current at 7.0 km/h. As viewed by an eagle hovering at rest over the shore, how fast and in what direction is this boat traveling? MS. Use vector components to find the magnitude and direc— tion of the vector needed to balance the two vectors shown in Figure 1.35 Exercises 1.43 and 1.59. ' . 4' ‘4 A (2.80 cm) 000° (1.90 cm) Exercises 3i Figure 1.36 Exercise L45. Figure 1.36. Let the 625-N vector be along the ~y—axis and let the +x~axis be perpen~ dicuiar to it toward the right. 1.46. Two ropes in a vertical plane exert equal magnitude forces on a hanging weight but pull with an angle of 860° between them. What pull does each one exert if their resultant pull is 372 N directiy upward? Section 1.9 Unit Vectors 1.41. Write each vector in Fig. $.34 in terms of the unit vectors 2 and 1.48. In each case, find the x- and y-components of vector K: (202 e 5.02 ~ 6.3}; (b) A e 1L2} — 9.912; (c) it e —15.03 + 22.4}; (d) 21’ n 5.00, where r} z 42 m of. 1.49. (a) Write each vector in Fig. 'i.37 in terms of the unit vectors 2 and j. (0} Use unit vectors to express the vector “c. where E? fl 3.00:4 -— 4.003. {c) Find the magnitude and direc- tion of 1.50. Given two vectors Kw 4.003 + 3.00,? and E x 5.003 — 2.00}. (a) find the magnitude of each vector; (b) write an expres- sion for the vector difference 3 w ii using unit vectors; (c) find the magnitude and direc- tion of the vector difference K — (d) In a vector diagram show and Ii — and also show that your diagram agrees qualitatively with your answer in part (0}. £5}. (a) Is the vector (i +f + it) a unit vector? Justify your answer. (b) Can a unit vector have any components with magni- tude greater than unity? Can it have any negative components? In each case justify your answer. (c) If}; m 0(302 4: 413i), where a is a constant, determine the value of a that makes A) a unit vector. Figure 1.37 Exercise 1.49 and Problem 1.86. Section 1.10 Products of Vectors 1.52. (a) Use vector components to prove that two vectors com— mute for both addition and the scalar product. (b) Prove that two vectors anticommute for the vector product; that is, prove that X x is 2 ~13; x X. 1.53. For the vectors 2, E, and E in Fig. 1.30, find the scaiar prod— acts (a) Lid; an”)? .6; (mi of. 1.54. (a) Find the scalar product of the two vectors .3 and 1”}; given in Exercise 1.50. (b) Find the angle between these two vectors. 1.55. Find the angle between each of the foilowing pairs of vectors: (21} X m “2.003 + 6.00j and ii m 2.003 — 300;“ (b) it m 3.002 + 5.00,? ii a 10.002 + 6.00;" (o X = “4.002 + 2.00;“ ii = 7.002 a taooj 1.56. By making simple stretches of the appropriate vector prod- ucts, show that (a) X '3 can be interpreted as the product of the magnitude of X times the component of :0 along or the magni— tude of 13 times the component-of X along 3; (h) EX x fil can be interpreted as the product of the magnitude of X timesflthe compo- nent of ii perpendicuiar to X, or the magnitude of 5; times the component of perpendicular to E. and and VECTORS and SCALARS 13 (b) Here each vector is equal to but opposite in direction to the corresponding one in (a). The field there'- fore appears as in Fig.(b). In Fig.{a) the fieid has the appearance ot a £1de emerging from a point Source 0 and flowing in the directions indicated. For this reason the field is called a Source field and 0 is a source. in Fig.(b)_ the field seems to be flowing toward 0, and the field is therefore called a sink field and 0 is a sink. In three dimensions the corresponding interpretation is that a fluid is emerging radialiy from (or pro- ceeding radially'toward) a line source (or line sink). The vector field is called two dimensional since it is independent of z. (c) Since the magnitude of each vector is V942 -l- y2 + 22. all points on the sphere 262+ 3/2 + 22 = a2, a > 0 have vectors of magnitude a associated with them. The field therefore takes on the appearance of that of a fluid emerging from source 0 and proceeding in all directions in space. This is a three dimension- al course field. SUPPLEMENTARY PROBLEMS 31. Which of the following are scaiars and which are vectors? (:2) Kinetic energy, (6) electric field intensity. (c) entropy, (d) work, (a) centrifngal force, (f) temperature, (g) gravitational potential, (it) charge. (i) shear- ing stress, (1‘) frequency. AM“- (a) scalar. (5) vector, (a) scalar. (d) scalar, (9) vector, (f) scalar, {g3 scalar, (h) scalar, (i) vector (j) scalar 32. An airplane travels 200 miles due west and then 150 miles 60° north of west. Determine the resultant dis— placement (a) graphically, (b) analytically. Ans. magnitude 304.1 mi (501/3773, direction 2501'?" north of east (arc sin 31/111/74) 33. Find the resultant of the foliowing displacements: A, 20 miles 36°sonth of east; B, 50 miies due west; (3, 40 miles northeast; D, 30-miles 60° south of west. Ans. magnitude 20.9 mi, direction 21°39’ south of West 34;. Show graphically that -(A-—B) a —A + B. 35. An object P is acted upon by three coplanar forces as shown in Fig.(a) below. Determine the force needed to prevent P from moving. Arts. 323 lb directly opposite 150 1b force 36. Given vectors ALB, C and D (Fig.0) below). Construct (a) 3A— 213— (C MD) (b) é—C + gut—B + 21)). /\\ F1343) Fig.(b) ...
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hw1 - 01.9. The quantity 77 = 3.14159 . . . is a number...

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