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344 Chapter 8 Rotational Motion
Substitute this into the Newton’s second law equation for the
sphere’s motion along the x axis:
Mg sin u - f = MaCM, x
2
MaCM, x = MaCM, x
5
2
7
Mg sin u = MaCM, x + MaCM, x = MaCM, x
5
5
5
aCM, x = g sin u
7
Mg sin u -
Since 5>7 (about 0.714) is greater than 2>3 (about 0.667), this is in
fact greater than the disk’s acceleration aCM, x = 12>32g sin u.
8-12 (c) The angular speed of the ball increases. Once the
ball has been hit and set in motion it has angular momentum
around the pole, and because we neglect resistive forces there
is no external torque on it. (Yes, there is a force on the ball due
to tension in the rope. This force is straight inward toward the
pole, however, so the angle f between sr and Fs in Equation
8-20, the definition of torque, is 180°. Hence the magnitude
of the torque is zero: t = rF sin f = rF sin 180° = 0.) Angular
momentum Lz is therefore conserved. As the rope loops around
the pole, the ball gets closer to the rotational axis and its
moment of inertia I decreases. Hence the angular speed (which
is the magnitude of the angular velocity vz) must increase in
order for the magnitude of Lz = Ivz to remain constant.
8-13 (d) The angular momentum L of a system is conserved
only if no net torque acts on the system. This isn’t the case for
either the pulley or the disk: A net torque due to the tension
force acts on the pulley as it rotates, and a net torque due to the
force of friction acts on the disk as it rolls downhill. For both
objects the moment of inertia I remains unchanged (the objects
don’t change shape) and the angular velocity vz increases, so
Lz = Ivz increases. Conservation laws are great when they
apply, but remember that they do not apply in all situations!
8-14 By bending in the middle, the cat divides its body into
two segments, each of which can rotate around a different
axis. The cat uses its muscles to rotate its front section into an
upright position while simultaneously rotating its hind quarters
slightly in the same direction. This small twist cancels the angular momentum generated by the rotation of the front part of his
body. Next, the opposite occurs: As the rear of the animal rotates
into alignment with the front, it slightly twists its forequarters.
Because the net angular momentum of the cat must remain zero,
the cat’s body must twist in order to cancel the angular momentum associated with the front and rear of its body separately.
Questions and Problems
In a few problems, you are given more data than you actually need;
in a few other problems, you are required to supply data from
your general knowledge, outside sources, or informed estimate.
Interpret as significant all digits in numerical values that have
trailing zeros and no decimal points.
For all problems, use g = 9.80 m>s2 for the free-fall acceleration due to gravity.
• Basic, single-concept problem
•• Intermediate-level problem, may require synthesis of concepts and multiple steps
••• Challenging problem
SSM Solution is in Student Solutions Manual
Conceptual Questions
1. •Describe any inconsistencies in the following statement: The
units of torque are N # m, but that’s not the same as the units of
energy.
s )? Why are factors
2. •What are the units of angular velocity (v
of 2p present in many equations describing rotational motion?
3. •Why is it critical to define the axis of rotation when you set
out to find the moment of inertia of an object? SSM
4. •Explain how an object moving in a straight line can have a
nonzero angular momentum.
5. •What are the units of the following quantities: (a) rotational
kinetic energy, (b) moment of inertia, and (c) angular momentum?
6. •Explain which physical quantities change when an ice
skater moves her arms in and out as she rotates in a pirouette.
What causes her angular velocity to change, if it changes at all?
7. •While watching two people on a seesaw, you notice that the
person at the top always leans backward, while the person at
the bottom always leans forward. (a) Why do the riders do this?
(b) Assuming they are sitting equidistant from the pivot point
Freed_c08_290-352_st_hr1.indd 344
of the seesaw, what, if anything, can you say about the relative
masses of the two riders? SSM
8. •The four solids shown in Figure 8-33 have equal heights,
widths, and masses. The axes of rotation are located at the center of each object and are perpendicular to the plane of the
paper. Rank the moments of inertia from greatest to least.
Hoop
Solid cylinder
Solid sphere
Hollow sphere
Figure 8-33 ​Problem 8
9. •Referring to the time-lapse photograph of a falling cat in
Figure 8-32, do you think that a cat will fall on her feet if she
does not have a tail? Explain your answer using the concepts
of this chapter.
10. •What is the ratio of rotational kinetic energy for two balls,
each tied to a light string and spinning in a circle with a radius
equal to the length of the string? The first ball has a mass m and a
string of length L, and rotates at a rate of v. The second ball has
a mass 2m and a string of length 2L, and rotates at a rate of 2v.
11. •A student cannot open a door at her school. She pushes
with ever-greater force, and still the door will not budge! Knowing that the door does push open, is not locked, and a minimum
torque is required to open the door, give a few reasons why this
might be occurring.
12. •Analyze the following statement and determine if there are
any physical inconsistencies: While rotating a ball on the end
of a string of length L, the rotational kinetic energy remains
constant as long as the length and angular speed are fixed.
When the ball is pulled inward and the length of the string is
4/9/13 12:45 PM
Questions and Problems 345
shortened, the rotational kinetic energy will remain constant
due to conservation of energy, but the angular momentum will
not because there is an external force acting on the ball to pull
it inward. The moment of inertia and angular speed will, of
course, remain the same throughout the process because the
ball is rotating in the same plane throughout the motion.
13. •A freely rotating turntable moves at a steady angular velocity. A glob of cookie dough falls straight down and
attaches to the very edge of the turntable. Describe which quantities (angular velocity, angular acceleration, torque, rotational
kinetic energy, moment of inertia, or angular momentum) are
conserved during the process and describe qualitatively what
happens to the motion of the turntable. SSM
14. •Describe what a “torque wrench” is and discuss any difficulties that a Canadian auto or bicycle mechanic might have
working with an American mechanic’s tools (and vice versa).
15. •In describing rotational motion, it is often useful to develop
an analogy with translational motion. First, write a set of equations describing translational motion. Then write the rotational
analogs (for example, u = u0 . . .) of the translational equations
(for example, x = x0 + v0 t + 12at 2) using the following legend:
x3u v3v a3a F3t m3I
p 3 L K translational 3 K rotational
16. •In Chapter 4, you learned that the mass of an object determines how that object responds to an applied force. Write a rotational analog to that idea based on the concepts of this chapter.
17. •Using the rotational concepts of this chapter, explain why
a uniform solid sphere beats a uniform solid cylinder which
beats a ring when the three objects “race” down an inclined
plane while rolling without slipping. SSM
18. •Which quantity is larger: the angular momentum of Earth
rotating on its axis each day or the angular momentum of Earth
revolving about the Sun each year? Try to determine the answer
without using a calculator.
19. •Define the SI unit radian. The unit appears in some physical quantities (for example, the angular velocity of a turntable is
3.5 rad>s) and it is omitted in others (for example, the translational velocity at the rim of a turntable is 0.35 m>s). Because the
formula relating rotational and translational quantities involves
multiplying by a radian (v = rv), discuss when it is appropriate
to include radians and when the unit should be dropped.
20. •A hollow cylinder rolls without slipping up an incline, stops,
and then rolls back down. Which of the following graphs in Figure
8-34 shows the (a) angular acceleration and (b) angular velocity
for the motion? Assume that up the ramp is the positive direction.
+
+
+
–
–
–
(a)
(b)
(c)
+
+
+
–
–
–
(d)
(e)
(f)
Figure 8-34 ​Problem 20
Freed_c08_290-352_st_hr1.indd 345
21. •Consider a situation in which a merry-go-round, starting
from rest, speeds up in the counterclockwise direction. It eventually reaches and maintains a maximum angular velocity. After
a short time the merry-go-round then starts to slow down and
eventually stops. Assume the accelerations experienced by the
merry-go-round have constant magnitudes. (a) Which graph
in Figure 8-35 describes the angular velocity as the merrygo-round speeds up? (b) Which graph describes the angular
position as the merry-go-round speeds up? (c) Which graph
describes the angular velocity as the merry-go-round travels at
its maximum velocity? (d) Which graph describes the angular
position as the merry-go-round travels at its maximum velocity?
(e) Which graph describes the angular velocity as the merry-goround slows down? (f) Which graph describes the angular position as the merry-go-round slows down? (g) Draw a graph of
the torque experienced by the merry-go-round as a function of
time during the scenario described in the problem. SSM
w
w
w
t
w
t
w
t
t
t
(a)
(b)
(c)
(d)
(e)
q
q
q
q
q
t
(i)
t
t
(ii)
(iii)
t
(iv)
t
(v)
Figure 8-35 ​Problem 21
22. •Rank the torques exerted on the bolts in A–D (Figure 8-36)
from least to greatest. Note that the forces in B and D make
an angle of 45° with the wrench. Assume the wrenches and the
magnitude of the force F are identical.
(a)
(c)
(b)
F
F
F
(d)
F
Figure 8-36 ​Problem 22
Multiple-Choice Questions
23. •A solid sphere of radius R, a solid cylinder of radius R,
and a rod of length R all have the same mass, and all three are
rotating with the same angular velocity. The sphere is rotating around an axis through its center. The cylinder is rotating
around its long axis, and the rod is rotating around an axis
through its center but perpendicular to the rod. Which one has
the greatest rotational kinetic energy?
A. the sphere
B. the cylinder
C. the rod
D. the rod and cylinder have the same rotational kinetic
energy
E. they all have the same kinetic energy
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346 Chapter 8 Rotational Motion
24. •How would a flywheel’s (spinning disk’s) kinetic energy
change if its moment of inertia were five times larger but its
angular speed were five times smaller?
A. 0.1 times as large as before
B. 0.2 times as large as before
C. same as before
D. 5 times as large as before
E. 10 times as large as before
25. •You have two steel spheres; sphere 2 has twice the radius
of sphere 1. What is the ratio of the moment of inertia I2:I1?
A. 2
B. 4
C. 8
D. 16
E. 32
26. •A solid ball, a solid disk, and a hoop, all with the same
mass and the same radius, are set rolling without slipping up an
incline, all with the same initial energy. Which goes farthest up
the incline?
A. the ball
B. the disk
C. the hoop
D. the hoop and the disk roll to the same height, farther
than the ball
E. they all roll to the same height
27. •A solid ball, a solid disk, and a hoop, all with the same
mass and the same radius, are set rolling without slipping up
an incline, all with the same initial linear speed. Which goes
farthest up the incline?
A. the ball
B. the disk
C. the hoop
D. the hoop and the disk roll to the same height, farther
than the ball
E. they all roll to the same height SSM
28. •Bob and Lily are riding on a merry-go-round. Bob rides
on a horse toward the outside of the circular platform, and
Lily rides on a horse toward the center of the circular platform.
When the merry-go-round is rotating at a constant angular
speed, Bob’s angular speed is
A. exactly half as much as Lily’s.
B. larger than Lily’s.
C. smaller than Lily’s.
D. the same as Lily’s.
E. exactly twice as much as Lily’s.
29. •Bob and Lily are riding on a merry-go-round. Bob rides
on a horse toward the outer edge of a circular platform. and
Lily rides on a horse toward the center of the circular platform.
When the merry-go-round is rotating at a constant angular
speed v, Bob’s speed v is
A. exactly half as much as Lily’s.
B. larger than Lily’s.
C. smaller than Lily’s.
D. the same as Lily’s.
E. exactly twice as much as Lily’s.
30. •While a gymnast is in the air during a leap, which of the
following quantities must remain constant for her?
A. position
B. velocity
C. momentum
Freed_c08_290-352_st_hr1.indd 346
D. angular velocity
E. angular momentum about her center of mass
31. •The moment of inertia of a thin ring about its symmetry
axis is ICM = MR2. What is the moment of inertia if you twirl a
large ring around your finger, so that in essence it rotates about
a point on the ring, about an axis parallel to the symmetry axis?
A. 5MR2
B. 2MR2
C. MR2
Pivot
D. 1.5MR2
E. 0.5MR2 SSM
32. •You give a quick push
to a ball at the end of a
massless, rigid rod, causing
the ball to rotate clockwise Top view
in a horizontal circle (Figure 8-37). The rod’s pivot is
frictionless. After the push
has ended, the ball’s angular
velocity
Figure 8-37 ​Problem 32
A. steadily increases.
B. increases for a while, then remains constant.
C. decreases for a while, then remains constant.
D. remains constant.
E. steadily decreases.
Push
Estimation/Numerical Analysis
33. •Estimate the angular speed of a car moving around a cloverleaf on-ramp of a typical freeway. Cloverleaf ramps extend
through approximately three-quarters of a circle to connect
two orthogonal freeways. SSM
34. •A fan is designed to last for a certain time before it will
have to be replaced (planned obsolescence). The fan only has
one speed (at a maximum of 750 rpm), and it reaches the speed
in 2 s (starting from rest). It takes the fan 10 s for the blade to
stop once it is turned off. If the manufacturer specifies that the
fan will operate up to 1 billion rotations, estimate how many
days you will be able to use the fan.
35. •Estimate the torque you apply when you open a door in
your house. Be sure to specify the axis to which your estimate
refers.
36. •Make a rough estimate of the moment of inertia of a pencil
that is spun about its center by a nervous student during an exam.
37. •Estimate the moment of inertia of a figure skater as she
rotates about the longitudinal axis that passes straight down
through the center of her body into the ice. Make this estimation for the extreme parts of a pirouette (arms fully extended
and arms drawn in tightly).
38. •Estimate the angular displacement (in radians and degrees)
of Earth in one day of its orbit around the Sun.
39. •Estimate the angular speed of the apparent passage of the
Sun across the sky of Earth (from dawn until dusk).
40. •Estimate the angular acceleration of a lone sock that is
inside a washing machine that starts from rest and reaches the
maximum speed of its spin cycle in typical fashion.
41. •Estimate the angular momentum about the center of rotation for a “skip-it ball” that is spun around on the ankle of a
small child (the child hops over the ball as it swings around and
around her feet). SSM
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Questions and Problems 347
42. •Estimate the moment of inertia of Earth about its central
axis as it rotates once in a day. Try to recall (or guess) the mass
and radius of Earth before you look up the data.
43. •Using a spreadsheet and the data below, calculate the
average angular speed of the rotating object over the first 10 s.
Calculate the average angular acceleration from 15 to 25 s. If
the object has a moment of inertia of 0.25 kg # m2 about the axis
of rotation, calculate the average torque during the following
time intervals: 0 < t < 10 s, 10 s < t < 15 s, and 15 s < t < 25 s.
t (s)
u (rad)
t (s)
u (rad)
t (s)
u (rad)
0
1
2
3
4
5
6
7
8
0
0.349
0.700
1.05
1.40
1.75
2.10
2.44
2.80
9
10
11
12
13
14
15
16
17
3.14
3.50
3.50
3.49
3.50
3.51
3.51
3.98
5.01
18
19
20
21
22
23
24
25
6.48
8.53
11.0
14.1
17.6
21.6
26.2
31.0
53. •What is the combined moment of inertia of three point
objects (m1 = 1.00 kg, m2 = 1.50 kg, m3 = 2.00 kg) tied together
with massless strings and rotating about the axis O as shown
in Figure 8-39?
O
m1
1m
m2
2m
m3
Figure 8-39
3m
Problem 53
54. ••A baton twirler in a marching band complains that her
baton is defective (Figure 8-40). The manufacturer specifies that
the baton should have an overall length of L = 60.0 cm and a
total mass between 940 and 950 g (there is one 350-g objects on
each end). Also according to the manufacturer, the moment of
inertia about the central axis passing through the baton should
fall between 0.0750 and 0.0800 kg # m2. The twirler (who has
completed a class in physics) claims this is impossible. Who’s
right? Explain your answer.
Problems
8-2 An object’s rotational kinetic energy is related to its
angular velocity and how its mass is distributed
44. •What is the angular speed of an object that completes
2.00 rev every 12.0 s? Give your answer in rad>s.
45. •A car rounds a curve with a translational speed of 12.0 m>s.
If the radius of the curve is 7.00 m, calculate the angular speed
in rad>s.
46. •Convert the following:
45.0 rev>min = ______rad>s
Figure 8-40
​Problem 54
L
55. ••What is the moment of inertia of a steering wheel about
the axis that passes through its center? Assume the rim of the
wheel has a radius R and a mass M. Assume that there are
five radial spokes that connect in the center as shown in Figure 8-41. The spokes are thin rods of length R and mass 12M,
and are evenly spaced around the wheel. SSM
3313 rpm = ______rad>s
2p rev>s = ______rad>s
47. •Calculate the angular speed of the Moon as it orbits
Earth (recall that the Moon completes one orbit about Earth in
27.4 days and the Earth–Moon distance is 3.84 * 108 m). SSM
48. •If a 0.250-kg point object rotates at 3.00 rev>s about an axis
that is 0.500 m away, what is the kinetic energy of the object?
49. •What is the rotational kinetic energy of an object that has
a moment of inertia of 0.280 kg # m2 about the axis of rotation
when its angular speed is 4.00 rad>s?
50. •What is the moment of inertia of an object that rotates at
13.0 rev>min about an axis and has a rotational kinetic energy
of 18.0 J?
51. •What is the angular speed of a rotating wheel that
has a moment of inertia of 0.330 kg # m2 and a rotational
kinetic energy of 2.75 J? Give your answer in both rad>s and
rev>min. SSM
Figure 8-41
Problem 55
56. •Using the parallel-axis theorem, calculate the moment of
inertia for a solid, uniform sphere about an axis that is tangent
to its surface (Figure 8-42).
I=
2
–
5
MR2
I=?
8-3 An object’s moment of inertia depends on its mass
distribution and the choice of rotation axis
52. •What is the combined moment of inertia for the three
point objects about the axis O in Figure 8-38?
O
2.4 kg
1.8 kg
28 cm
Freed_c08_290-352_st_hr1.indd 347
3.0 kg
42 cm
R
Figure 8-38
Problem 52
Figure 8-42 ​Problem 56
4/9/13 12:45 PM
348 Chapter 8 Rotational Motion
57. •Calculate the moment
of inertia for a uniform,
solid cylinder (mass M, I = 1–2 MR2
radius R) if the axis of rotation is tangent to the side of
the cylinder as shown in
Figure 8-43.
62. ••What is the moment
of inertia of the sphere–rod
system shown in Figure 8-48
where the sphere has a radius
R and a mass M and the rod
is thin and massless, and has a
length L? The sphere–rod system is spun about an axis A.
I=?
58. •Calculate the moment of inertia for a thin rod that is
1.25 m long and has mass of 2.25 kg. The axis of rotation
passes through the rod at a point one-third of the way from
the left end ­(Figure 8-44).
2
—
3L
59. •Calculate the moment
of inertia of a thin plate
that is 5.00 cm * 7.00 cm
in area and has a mass
density of 1.50 g>cm2. The
axis of rotation is located
at the left side, as shown in
Figure 8-45.
L
R
M
Figure 8-48 ​Problem 62
8-4 Conservation of mechanical energy also applies to
rotating objects
Figure 8-43 ​Problem 57
1
—
3L
A
Figure 8-44 Problem 58
63. •Calculate the final speed of a uniform, solid cylinder of
radius 5.00 cm and mass 3.00 kg that starts from rest at the top
of an inclined plane that is 2.00 m long and tilted at an angle
of 25.0° with the horizontal. Assume the cylinder rolls without
slipping down the ramp.
64. •Calculate the final speed of a uniform, solid sphere of
radius 5.00 cm and mass 3.00 kg that starts with a translational speed of 2.00 m>s at the top of an inclined plane that
is 2.00 m long and tilted at an angle of 25.0° with the horizontal. Assume the sphere rolls without slipping down the
ramp.
65. •••A spherical marble that has a mass of 50.0 g and a
radius of 0.500 cm rolls without slipping down a loop-the-loop
track that has a radius of 20.0 cm. The marble starts from rest
and just barely clears the loop to emerge on the other side of the
track. What is the minimum height that the marble must start
from to make it around the loop?
5.00 cm
7.00 cm
Figure 8-45 ​Problem 59
60. •Calculate the radius of a solid sphere of mass M that has
the same moment of inertia about an axis through its center of
mass as a second solid sphere of radius R and mass M which
has the axis of rotation passing tangent to the surface and parallel to the center of mass axis (Figure 8-46).
66. •••A billiard ball of mass 160 g and radius 2.50 cm starts
with a translational speed of 2.00 m>s at point A on the track
as shown in Figure 8-49. If point B is at the top of a hill that has
a radius of curvature of 60 cm, what is the normal force acting
on the ball at point B? Assume the billiard ball rolls without
slipping on the track.
A
B
10 cm
M
M
r=?
60 cm
R
Figure 8-46
Problem 60
61. ••Two uniform, solid
spheres (one has a mass M
M
and a radius R and the
M
other has a mass M and a
2R
radius 2R) are connected
M
by a thin, uniform rod of
3R
length 3R and mass M
(Figure 8-47). Find the
moment of inertia about Figure 8-47 ​Problem 61
the axis through the center
of the rod. SSM
Freed_c08_290-352_st_hr1.indd 348
R
Figure 8-49 ​Problem 66
67. •Sports A bowling ball that has a radius of 11.0 cm
and a mass of 5.00 kg rolls without slipping on a level
lane at 2.00 rad>s. Calculate the ratio of the translational
kinetic energy to the rotational kinetic energy of the bowling
ball. SSM
68. •Astronomy Earth is approximately a solid sphere, has
a mass of 5.98 * 1024 kg and a radius of 6.38 * 106 m, and
completes one rotation about its central axis each day. Calculate the rotational kinetic energy of Earth as it spins on its
axis.
69. •Astronomy Calculate the translational kinetic energy of
Earth as it orbits the Sun once each year (the Earth–Sun distance
4/9/13 12:45 PM
Questions and Problems 349
is 1.50 * 1011 m). Calculate the ratio of the translational kinetic
energy to the rotational kinetic energy calculated in the previous problem.
70. •A potter’s flywheel is made of a 5.00-cm-thick, round
slab of concrete that has a mass of 60.0 kg and a diameter of
35.0 cm. This disk rotates about an axis that passes through
its center, perpendicular to its round area. Calculate the angular speed of the slab about its center if the rotational kinetic
energy is 15.0 J. Express your answer in both rad>s and
rev>min.
8-6 Torque is to rotation as force is to translation
79. •What is the torque about your shoulder axis if you hold a
10-kg barbell in one hand straight out and at shoulder height?
Assume your hand is 75 cm from your shoulder. SSM
80. •A driver applies a horizontal force of 20.0 N (to the right)
to the top of a steering wheel, as shown in Figure 8-50. The
steering wheel has a radius of 18.0 cm and a moment of inertia of 0.0970 kg # m2. Calculate the angular acceleration of the
steering wheel about the central axis.
71. ••Sports A flying disk (160 g, 25.0 cm in diameter) spins at
a rate of 300 rpm with its center balanced on a fingertip. What
is the rotational kinetic energy of the Frisbee if the disc has 70%
of its mass on the outer edge (basically a thin ring 25.0-cm in
diameter) and the remaining 30% is a nearly flat disk 25.0-cm
in diameter? SSM
8-5 The equations for rotational kinematics are almost
identical to those for linear motion
72. •Suppose a roulette wheel is spinning at 1 rev>s. (a) How
long will it take for the wheel to come to rest if it experiences
an angular acceleration of - 0.02 rad>s 2? (b) How many rotations will it complete in that time?
73. •A spinning top completes 6000 rotations before it
starts to topple over. The average speed of the rotations is
800 rpm. Calculate how long the top spins before it begins
to topple.
74. •A child pushes a merry-go-round that has a diameter of
4.00 m and goes from rest to an angular speed of 18.0 rpm in
a time of 43.0 s. (a) Calculate the angular displacement and the
average angular acceleration of the merry-go-round. (b) What
is the maximum tangential speed of the child if she rides on the
edge of the platform?
75. •Jerry twirls an umbrella around its central axis so that it
completes 24.0 rotations in 30.0 s. (a) If the umbrella starts
from rest, calculate the angular acceleration of a point on the
outer edge. (b) What is the maximum tangential speed of a
point on the edge if the umbrella has a radius of 55.0 cm?
76. •Prior to the music CD, stereo systems had a phonographic turntable on which vinyl disk recordings were played.
A ­particular phonographic turntable starts from rest and
achieves a final constant angular speed of 3313 rpm in a time of
4.5 s. How many rotations did the turntable undergo during
that time? The classic Beatles album Abbey Road is 47 min
and 7 s in duration. If the turntable requires 8 s to come to
rest once the album is over, calculate the total number of rotations for the complete start-up, playing, and slowdown of the
album.
Figure 8-50
Problem 80
81. •Medical When
the palmaris longus
muscle in the forearm
is flexed, the wrist
moves back and forth
Palmaris longus muscle
(Figure 8-51). If the
­muscle generates a
force of 45.0 N and it
is acting with an effective lever arm of 22.0
cm, what is the torque
that the muscle produces on the wrist?
Curiously, over 15%
of all Caucasians lack
this muscle; a smaller Figure 8-51 ​Problem 81
percentage of Asians (around 5%) also lack it. Some
studies ­correlate the absence of the muscle with carpal tunnel
syndrome.
82. •A torque wrench is used to tighten a nut on a bolt. The
wrench is 25 cm long, and a force of 120 N is applied at the end
of the wrench as shown in Figure 8-52. Calculate the torque
about the axis that passes through the bolt.
F = 120 N
20°
77. •A CD player varies its speed as it moves to another circular track on the CD. A CD player is rotating at 300 rpm. To
read another track, the angular speed is increased to 450 rpm
in a time of 0.75 s. Calculate the average angular acceleration
in rad>s 2 for the change to occur. SSM
78. •Astronomy A communication satellite circles Earth in a
geosynchronous orbit such that the satellite remains directly
above the same point on the surface of Earth. (a) What angular
displacement (in radians) does the satellite undergo in 1 h of its
orbit? (b) Calculate the angular speed of the satellite in rev>min
and rad>s.
Freed_c08_290-352_st_hr1.indd 349
25 cm
Figure 8-52 ​Problem 82
83. •An 85.0-cm-wide door is pushed open with a force of F =
75.0 N. Calculate the torque about an axis that passes through
the hinges in each of the cases in Figure 8-53. SSM
4/9/13 12:45 PM
350 Chapter 8 Rotational Motion
moment of inertia of the wheel? The external torque is then
removed, and a brake is applied. If it takes the wheel 200 s to
come to rest after the brake is applied, what is the magnitude
of the torque exerted by the brake?
85 cm
(a)
(b)
Hinge
axis
F
F
(c)
25°
(d)
F
F
Figure 8-53
70°
Problem 83
84. •••A 50.0-g meter stick is balanced at its midpoint (50 cm).
Then a 200-g weight is hung with a light string from the 70.0cm point, and a 100-g weight is hung from the 10.0-cm point
(Figure 8-54). Calculate the clockwise and counterclockwise
torques acting on the board due to the four forces shown about
the following axes: (a) the 0-cm point, (b) the 50-cm point, and
(c) the 100-cm point.
88. •A flywheel of mass 35.0 kg and diameter 60.0 cm spins at
400 rpm when it experiences a sudden power loss. The flywheel
slows due to friction in its bearings during the 20.0 s the power
is off. If the flywheel makes 200 complete revolutions during
the power failure, (a) at what rate is the flywheel spinning when
the power comes back on? (b) How long would it have taken
for the flywheel to come to a complete stop?
89. ••A solid cylindrical pulley with
a mass of 1.00 kg and a radius of
0.25 m is free to rotate about its
axis. An object of mass 0.250 kg is
attached to the pulley with a light
string (Figure 8-56). Assuming the
string does not stretch or slip, calculate the tension in the string and the
angular acceleration of the pulley.
Fpivot
10 cm
100 g
50 cm
Figure 8-56 ​Problem 89
70 cm
mstickg
200 g
Figure 8-54 ​Problem 84
85. •A robotic arm lifts
a barrel of radioactive
waste (Figure 8-55). If
the maximum torque
delivered by the arm
about the axis O is
3000 N # m and the distance r in the diagram
is 3.00 m, what is the
maximum mass of the
barrel?
O
r
90. ••A block with mass m1 = 2.00 kg rests on a frictionless
table. It is connected with a light string over a pulley to a hanging
block of mass m2 = 4.00 kg. The pulley is a uniform disk with
a radius of 4.00 cm and a mass of 0.500 kg (Figure 8-57).
(a) Calculate the acceleration of each block
m1
and the tension in each
segment of the string.
(b) How long does it
take the blocks to move
a distance of 2.25 m?
(c) What is the angular
m2
speed of the pulley at
this time?
Figure 8-57 ​Problem 90
91. ••A yo-yo with a rolling radius of
r = 2.50 cm rolls down a string with a
­linear acceleration of 6.50 m>s 2 (Figure 8-58). (a) Calculate the tension
in the string and the angular acceleration of the yo-yo. (b) What is the
moment of inertia of this yo-yo?
w
r
Figure 8-55 ​Problem 85
86. •A typical adult can deliver about 10 N # m of torque when
attempting to open a twist-off cap on a bottle. What is the maximum force that the average person can exert with his fingers if
most bottle caps are about 2 cm in diameter?
Figure 8-58 ​Problem 91
8-8 Angular momentum is conserved when there is zero net
torque on a system
8-7 The techniques used for solving problems with
Newton’s second law also apply to rotation problems
92. •What is the angular momentum about the central axis of
a thin disk that is 18.0 cm in diameter, has a mass of 2.50 kg,
and rotates at a constant 1.25 rad>s?
87. •A potter’s wheel is initially at rest. A constant external
torque of 75.0 N # m is applied to the wheel for 15.0 s, giving
the wheel an angular speed of 500 rev>min. (a) What is the
93. •What is the angular momentum of a 300-g tetherball
when it whirls around the central pole at 60.0 rpm and at a
radius of 125 cm?
Freed_c08_290-352_st_hr1.indd 350
4/9/13 12:45 PM
Questions and Problems 351
94. •Astronomy Calculate the angular momentum of Earth as
it orbits the Sun. Recall that the mass of Earth is 5.98 * 1024 kg,
the distance between Earth and the Sun is 1.50 * 1011 m, and
the time for one orbit is 365.3 days.
95. •Astronomy Calculate the angular momentum of Earth
as it spins on its central axis once each day. Assume Earth is
approximately a uniform, solid sphere that has a mass of 5.98
* 1024 kg and a radius of 6.38 * 106 m.
96. •What is the speed of an electron in the lowest energy
orbital of hydrogen, of radius equal to 5.29 * 10211 m? The
mass of an electron is 9.11 * 10231 kg, and its angular momentum in this orbital is 1.055 * 10234 J # s.
97. •What is the angular momentum of a 70.0-kg person riding on a Ferris wheel that has a diameter of 35.0 m and rotates
once every 25.0 s? SSM
98. •A professor sits on a rotating stool that spins at 10.0 rpm
while she holds a 1.00-kg weight in each of her hands. Her outstretched arms are 0.750 m from the axis of rotation, which
passes through her head into the center of the stool. When
she draws the weights in toward her body, her angular speed
increases to 20.0 rpm. Neglecting the mass of her arms, how far
are the weights from the rotational axis at the increased speed?
extended horizontally. Suppose a 62.0-kg skater is 1.80 m tall,
has arms that are each 65.0 cm long (including the hands), and
a trunk that can be modeled as being 35.0 cm in diameter. If
the skater is initially spinning at 70.0 rpm with his arms outstretched, what will his angular velocity be (in rpm) when he
pulls in his arms until they are at his sides parallel to his trunk?
104. •Derive an expression for the moment of inertia of a
spherical shell (for example, the peel of an orange) that has a
mass M and a radius R, and rotates about an axis that is tangent to the surface.
105. •Because of your success in physics class you are selected
for an internship at a prestigious bicycle company in its research
and development division. Your first task involves designing a
wheel made of a hoop that has a mass of 1.00 kg and a radius
of 50.0 cm, and spokes with a mass of 10.0 g each. The wheel
should have a total moment of inertia 0.280 kg # m2. (a) How
many spokes are necessary to construct the wheel? (b) What is
the mass of the wheel? SSM
General Problems
106. ••Two beads that each have a mass M are attached to a
thin rod that has a length 2L and a mass M>8. Each bead is
initially a distance L>4 from the center of the rod. The whole
system is set into uniform rotation about the center of the rod,
with initial angular frequency v = 20p rad>s. If the beads are
then allowed to slide to the ends of the rod, what will the angular frequency become?
99. •A baton is constructed by attaching two small objects that
each have a mass M to the ends of a rod that has a length L and a
uniform mass M. Find an expression for the moment of inertia of
the baton when it is rotated around a point (3>8) L from one end.
107. ••A uniform disk that has a mass M of 0.300 kg and a
radius R of 0.270 m rolls up a ramp of angle u equal to 55.0°
with initial speed v of 4.8 m>s. If the disk rolls without slipping, how far up the ramp does it go?
100. •The outside diameter of the playing area of an optical
Blu-ray disc is 11.75 cm, and the inside diameter is 4.50 cm.
When viewing movies, the disc rotates so that a laser maintains
a constant linear speed relative to the disc of 7.50 m>s as it
tracks over the playing area. (a) What are the maximum and
minimum angular speeds (in rad>s and rpm) of the disc? (b) At
which location of the laser on the playing area do these speeds
occur? (c) What is the average angular acceleration of a Blu-ray
disc as it plays an 8.0-h set of movies?
108. •In a new model of a machine, a spinning solid spherical
part of radius R must be replaced by a ring of the same mass
which is to have the same kinetic energy. Both parts need to
spin at the same rate, the sphere about an axis through its center and the ring about an axis perpendicular to its plane at its
center. (a) What should the radius of the ring be in terms of R?
(b) Will both parts have the same angular momentum? If not,
which one will have more?
101. •A table saw has a 25.0-cm-diameter blade that rotates at
a rate of 7000 rpm. It is equipped with a safety mechanism that
can stop the blade within 5.00 ms if something like a finger is
accidentally placed in contact with the blade. (a) What average
angular acceleration occurs if the saw starts at 7000 rpm and
comes to rest in this time? (b) How many rotations does the
blade complete during the stopping period?
102. •In 1932 Albert Dremel of Racine, Wisconsin, created his
rotary tool that has come to be known as a dremel. (a) Suppose a dremel starts from rest and achieves an operating speed
of 35,000 rev>min. If it requires 1.20 s for the tool to reach
operating speed and it is held at that speed for 45.0 s, how many
rotations has the bit made? Suppose it requires another 8.50 s for
the tool to return to rest. (b) What are the average angular accelerations for the start-up and the slow-down periods? (c) How
many rotations does the tool complete from start to finish?
103. ••Medical, Sports On average, both arms and hands
together account for 13% of a person’s mass, while the head is
7.0% and the trunk and legs account for 80%. We can model a
spinning skater with his arms outstretched as a vertical cylinder
(head + trunk + legs) with two solid uniform rods (arms + hands)
Freed_c08_290-352_st_hr1.indd 351
109. •Many 2.5-in-diameter (6.35-cm) computer hard disks
spin at a constant 7200 rpm operating speed. The disks have
a mass of about 7.50 g and are essentially uniform throughout
with a very small hole at the center. If they reach their operating
speed 2.50 s after being turned on, what average torque does
the disk drive supply to the disk during the acceleration?
110. ••Sports At the 1984 Olympics, the great diver Greg
­Louganis won one of his 10 gold medals for the reverse
312 somersault tuck dive. In the dive, Louganis began his 312
turns with his body tucked in at a maximum height of approximately 2.0 m above the platform, which itself was 10.0 m
above the water. He spun uniformly 312 times and straightened out his body just as he reached the water. A reasonable
approximation is to model the diver as a thin uniform rod
2.0 m long when he is stretched out and as a uniform solid
cylinder of diameter 0.75 m when he is tucked in. (a) What
was Louganis’s average angular speed as he fell toward the
water with his body tucked in? Hint: How long did it take
him to reach the water from his highest point? (b) What was
his angular speed just as he stretched out? (c) How much did
Louganis’s rotational kinetic energy change while extending
his body if his mass was 75 kg?
4/9/13 12:45 PM
352 Chapter 8 Rotational Motion
111. ••Medical The bones of the forearm (radius and ulna) are
hinged to the humerus at the elbow (Figure 8-59). The biceps
muscle connects to the bones of the forearm about 2 cm beyond
the joint. Assume the forearm has a mass of 2 kg and a length of
0.4 m. When the humerus and the biceps are nearly vertical and
the forearm is horizontal, if a person wishes to hold an object
of mass M so that her forearm remains motionless, what is the
relationship between the force exerted by the biceps muscle and
the mass of the object?
Humerus
Radius
M
Elbow
Hand
Figure 8-59 ​Problem 111
112. ••Medical The femur of a human leg (mass 10 kg, length
0.9 m) is in traction (Figure 8-60). The center of gravity of the
leg is one-third of the distance from the pelvis to the bottom
of the foot. Two objects, with masses m1 and m2, are hung at
the ends of the leg using pulleys to provide upward support. A
third object of 8 kg is hung to provide tension along the leg.
The body provides tension as well. (a) What is the mathematical relationship between m1 and m2? Is this relationship unique
in the sense that there is only one combination of m1 and m2
that maintains the leg in static equilibrium? (b) How does the
relationship change if the tension force due to m1 is applied at
the leg’s center of mass?
m1
Fbody
m2
8 kg
Figure 8-60 ​Problem 112
113. •Astronomy It is estimated that 60,000 tons of meteors and other space debris accumulates on Earth each year.
Assume the debris is accumulated uniformly across the surface
of Earth. (a) How much does Earth’s rotation rate change per
year as a result of this accumulation? (That is, find the change
Freed_c08_290-352_st_hr1.indd 352
114. •Astronomy Suppose we decided to use the rotation of
Earth as a source of energy. (a) What is the maximum amount
of energy we could obtain from this source? (b) By the year 2020
the projected rate at which the world uses energy is expected
to be 6.4 * 1020 J>y. If energy use continues at that rate, for
how many years would the spin of Earth supply our energy
needs? Does this seem long enough to justify the effort and
expense involved? (c) How long would it take before our day
was extended to 48 h instead of 24 h? Assume that Earth is uniform throughout.
115. •Astronomy In a little over 5 billion years, our Sun will
collapse to a white dwarf approximately 16,000 km in diameter. (Ignore the fact that the Sun will lose mass as it ages.)
(a) What will our Sun’s angular momentum and rotation rate
be as a white dwarf? (Express your answers as multiples of its
present-day values.) (b) Compared to its present value, will the
Sun’s rotational kinetic energy increase, decrease, or stay the
same when it becomes a white dwarf? If it does change, by
what factor will it change? The radius of the Sun is presently
6.96 * 108 m.
Muscle
Ulna
in angular velocity.) (b) How long would it take the accumulation of debris to change the rotation period by 1 s? SSM
116. •Astronomy (a) If all the people in the world (~7 billion)
lined up along the equator, would Earth’s rotation rate increase
or decrease? Justify your answer. (b) How would the rotation
rate change if all people were no longer on Earth? Assume the
average mass of a human is 70.0 kg.
117. •A 1000-kg merry-go-round (a flat, solid cylinder) supports 10 children, each with a mass of 50.0 kg, located at the
axis of rotation (thus you may assume the children have no
angular momentum at that location). Describe a plan to move
the children such that the angular velocity of the merry-goround decreases to one-half its initial value.
118. •One way for pilots to train for the physical demands of
flying at high speeds is with a device called the “human centrifuge.” It involves having the pilots travel in circles at high
speeds so that they can experience forces greater than their own
weight. The diameter of the NASA device is 17.8 m. (a) Suppose
a pilot starts at rest and accelerates at a constant rate so that he
undergoes 30 rev in 2 min. What is his angular acceleration (in
rad>s 2)? (b) What is his angular velocity (in rad>s) at the end of
that time? (c) After the 2-min period, the centrifuge moves at a
constant speed. The g-force experienced is the centripetal force
keeping the pilot moving along a circular path. What is the
g-force experienced by the pilot? (1 g = mass * 9.80 m>s 2)
(d) The pilot can tolerate 12 g’s in the horizontal direction.
How long would it take the centrifuge to reach that state if it
starts at the angular speed found in part (c) and accelerates at
the rate found in part (a)?
119. •••The moment of inertia of a rolling marble is I = 25MR2,
where M is the mass of the marble and R is the radius. The
marble is placed in front of a spring that has a constant k and
has been compressed a distance xc. The spring is released, and
as the marble comes off the spring it begins to roll without
slipping. Note: The static friction that causes rolling without
slipping does not do work. (a) Derive an expression for the time
it takes for the marble to travel a distance D along the surface
after it has lost contact with the spring. (b) Show that your
answer for part (a) has the correct units. SSM
4/9/13 12:45 PM