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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