6 raymond a serway, john w jewett physics for scientists and engineers with modern physics 14

294

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

Categorize We model the sphere and the Earth as an isolated system with no nonconservative forces acting. This
model is the one that led to Equation 10.30, so we can use that result.

Analyze Evaluate the speed of the center of mass of
the sphere from Equation 10.30:

(1)

v CM ϭ c

1ϩ

1

2gh

d

2

2
2
2
5 MR >MR

1>2

ϭ

1107gh2 1>2

This result is less than 12gh, which is the speed an object would have if it simply slid down the incline without
rotating. (Eliminate the rotation by setting ICM ϭ 0 in Equation 10.30.)
To calculate the translational acceleration of the center of mass, notice that the vertical displacement of the
sphere is related to the distance x it moves along the incline through the relationship h ϭ x sin u.
Use this relationship to rewrite Equation (1):

v CM2 ϭ 10
7 gx sin u

Write Equation 2.17 for an object starting from rest and
moving through a distance x:

v CM2 ϭ 2a CMx

Equate the preceding two expressions to find aCM:

a CM ϭ 57g sin u

Finalize Both the speed and the acceleration of the center of mass are independent of the mass and the radius of the
sphere. That is, all homogeneous solid spheres experience the same speed and acceleration on a given incline. Try to
verify this statement experimentally with balls of different sizes, such as a marble and a croquet ball.
If we were to repeat the acceleration calculation for a hollow sphere, a solid cylinder, or a hoop, we would obtain
similar results in which only the factor in front of g sin u would differ. The constant factors that appear in the expressions for vCM and aCM depend only on the moment of inertia about the center of mass for the specific object. In all
cases, the acceleration of the center of mass is less than g sin u, the value the acceleration would have if the incline

were frictionless and no rolling occurred.

E XA M P L E 1 0 . 1 4

Pulling on a Spool3

A cylindrically symmetric spool of mass m and radius R sits at rest on a horizontal
table with friction (Fig. 10.27). With your hand on a massless string wrapped
around the axle of radius r, you pull on the spool with a constant horizontal force
of magnitude T to the right. As a result, the spool rolls without slipping a distance
L along the table with no rolling friction.

L
R

T
r

(A) Find the final translational speed of the center of mass of the spool.
SOLUTION
Conceptualize Use Figure 10.27 to visualize the motion of the spool when you
pull the string. For the spool to roll through a distance L, notice that your hand on
the string must pull through a distance different from L.

Figure 10.27 (Example 10.14) A
spool rests on a horizontal table. A
string is wrapped around the axle
and is pulled to the right by a hand.

Categorize The spool is a rigid object under a net torque, but the net torque includes that due to the friction

force, about which we know nothing. Therefore, an approach based on the rigid object under a net torque model
will not be successful. Work is done by your hand on the spool and string, which form a nonisolated system. Let’s see
if an approach based on the nonisolated system model is fruitful.

3

Example 10.14 was inspired in part by C. E. Mungan, “A primer on work–energy relationships for introductory physics,” The Physics Teacher, 43:10,
2005.


Section 10.9

Rolling Motion of a Rigid Object

295

Analyze The only type of energy that changes in the system is the kinetic energy of the spool. There is no rolling
friction, so there is no change in internal energy. The only way that energy crosses the system’s boundary is by the
work done by your hand on the string. No work is done by the static force of friction on the bottom of the spool
because the point of application of the force moves through no displacement.
Write the appropriate reduction of the conservation
of energy equation, Equation 8.2:

(1)

W ϭ ¢K ϭ ¢Ktrans ϩ ¢Krot

where W is the work done on the string by your hand. To find this work, we need to find the displacement of your
hand during the process.
We first find the length of string that has unwound off the spool. If the spool rolls through a distance L, the total

angle through which it rotates is u ϭ L/R. The axle also rotates through this angle.
/ ϭ ru ϭ

Use Equation 10.1a to find the total arc length through
which the axle turns:

r
L
R

This result also gives the length of string pulled off the axle. Your hand will move through this distance plus the distance L through which the spool moves. Therefore, the magnitude of the displacement of the point of application of
the force applied by your hand is ᐉ ϩ L ϭ L(1 ϩ r/R).
Evaluate the work done by your hand on the string:

W ϭ TL a 1 ϩ

(2)
TL a 1 ϩ

Substitute Equation (2) into Equation (1):

r
b
R

r
b ϭ 12mv CM2 ϩ 12Iv 2
R

where I is the moment of inertia of the spool about its center of mass and vCM and v are the final values after the

wheel rolls through the distance L.
TL a 1 ϩ

Apply the nonslip rolling condition v ϭ vCM/R:

(3)

Solve for vCM:

v CM2
r
b ϭ 12mv CM2 ϩ 12 I 2
R
R

v CM ϭ

2TL 11 ϩ r>R2

B m 11 ϩ I>mR 2 2

(B) Find the value of the friction force f.
SOLUTION
Categorize Because the friction force does no work, we cannot evaluate it from an energy approach. We model the
spool as a nonisolated system, but this time in terms of momentum. The string applies a force across the boundary of
the system, resulting in an impulse on the system. Because the forces on the spool are constant, we can model the
spool’s center of mass as a particle under constant acceleration.
Analyze Write the impulse–momentum theorem (Eq.
9.40) for the spool:

(4)

1T Ϫ f 2 ¢t ϭ m 1v CM Ϫ 02 ϭ mv CM

For a particle under constant acceleration starting from rest, Equation 2.14 tells us that the average velocity of the
center of mass is half the final velocity.


296

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

Use Equation 2.2 to find the time interval for the center
of mass of the spool to move a distance L from rest to a
final speed vCM:

(5)

¢t ϭ

1T Ϫ f 2

Substitute Equation (5) into Equation (4):

L
2L
ϭ
vCM, avg

vCM

2L
ϭ mvCM
vCM

fϭTϪ

Solve for the friction force f:

fϭTϪ

Substitute vCM from Equation (3):

mvCM2
2L

m 2TL 11 ϩ r>R 2
c
d
2L m 11 ϩ I>mR 2 2

ϭTϪT

11 ϩ r>R2

11 ϩ I>mR 2 2

ϭ T c1 Ϫ

11 ϩ r>R 2

11 ϩ I>mR 2 2

d

Finalize Notice that we could use the impulse-momentum theorem for the translational motion of the spool while
ignoring that the spool is rotating! This fact demonstrates the power of our growing list of approaches to solving
problems.

Summary
Sign in at www.thomsonedu.com and go to ThomsonNOW to take a practice test for this chapter.
DEFINITIONS
The angular position of a rigid object is defined as the angle
u between a reference line attached to the object and a reference line fixed in space. The angular displacement of a particle moving in a circular path or a rigid object rotating about
a fixed axis is ⌬u ϵ uf Ϫ ui.
The instantaneous angular speed of a particle moving in a
circular path or of a rigid object rotating about a fixed axis is
vϵ

du
dt

(10.3)

The instantaneous angular acceleration of a particle moving in a circular path or of a rigid object rotating about a
fixed axis is
aϵ

dv

dt

(10.5)

When a rigid object rotates about a fixed axis, every part of
the object has the same angular speed and the same angular
acceleration.

The moment of inertia of a system of particles
is defined as
I ϵ a m ir i 2

(10.15)

i

where mi is the mass of the ith particle and ri is
its distance from the rotation axis.
The magnitude
of the torque associated with
S
a force F acting on an object at a distance r
from the rotation axis is
t ϵ r F sin f ϭ Fd

(10.19)

where f is the angle between the position vector of the point of application of the force and
the force vector, and d is the moment arm of
the force, which is the perpendicular distance

from the rotation axis to the line of action of
the force.


297

Summary

CO N C E P T S A N D P R I N C I P L E S
When a rigid object rotates about a
fixed axis, the angular position,
angular speed, and angular acceleration are related to the translational
position, translational speed, and
translational acceleration through
the relationships
s ϭ ru

(10.1a)

v ϭ rv

(10.10)

at ϭ r a

(10.11)

If a rigid object rotates about a fixed axis with angular speed v, its
rotational kinetic energy can be written
K R ϭ 12Iv 2

(10.16)

where I is the moment of inertia about the axis of rotation.

The moment of inertia of a rigid object is
Iϭ

Ύr
¬

2
¬dm

(10.17)

where r is the distance from the mass element dm to the axis of rotation.

The rate at which work is done by an external force in rotating a rigid object about a fixed axis, or the power
delivered, is
ᏼ ϭ tv

(10.23)

If work is done on a rigid object and the only result of the work is rotation about a fixed axis, the net work done
by external forces in rotating the object equals the change in the rotational kinetic energy of the object:
1
1
2
2

a W ϭ 2Iv f Ϫ 2Iv i

(10.24)

The total kinetic energy of a rigid object rolling on a rough surface without slipping equals the rotational kinetic
energy about its center of mass plus the translational kinetic energy of the center of mass:
K ϭ 12ICMv 2 ϩ 12Mv CM2

(10.28)

A N A LYS I S M O D E L S F O R P R O B L E M S O LV I N G

a ϭ constant

a

Rigid Object Under Constant Angular Acceleration. If a rigid
object rotates about a fixed axis under constant angular acceleration, one can apply equations of kinematics that are analogous
to those for translational motion of a particle under constant
acceleration:
vf ϭ vi ϩ at

(10.6)

u f ϭ u i ϩ vit ϩ 12at 2

(10.7)

v f 2 ϭ v i 2 ϩ 2a 1u f Ϫ u i 2

(10.8)

u f ϭ u i ϩ 12 1vi ϩ vf 2t

(10.9)

Rigid Object Under a Net Torque. If a rigid
object free to rotate about a fixed axis has a
net external torque acting on it, the object
undergoes an angular acceleration a, where
a t ϭ Ia

(10.21)

This equation is the rotational analog to
Newton’s second law in the particle under a
net force model.


298

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

Questions
Ⅺ denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question
1. What is the angular speed of the second hand of a clock?
S
What is the direction of V as you view a clock hanging on

a vertical wall? What is the magnitude of the angular
S
acceleration vector A of the second hand?
2. One blade of a pair of scissors rotates counterclockwise in
S
the xy plane. What is the direction of V? What is the
S
direction of A if the magnitude of the angular velocity is
decreasing in time?
3. O A wheel is moving with constant angular acceleration
3 rad/s2. At different moments its angular speed is
Ϫ2 rad/s, 0, and ϩ 2 rad/s. At these moments, analyze
the magnitude of the tangential component of acceleration and the magnitude of the radial component of acceleration for a point on the rim of the wheel. Rank the following six items from largest to smallest: (a) |at| when v ϭ
Ϫ2 rad/s (b)|ar| when v ϭ Ϫ2 rad/s (c) |at| when v ϭ 0
(d) |ar| when v ϭ 0 (e)|at| when v ϭ 2 rad/s (f) |ar| when
v ϭ 2 rad/s If two items are equal, show them as equal
in your ranking. If a quantity is equal to zero, show that in
your ranking.
4. O (i) Suppose a car’s standard tires are replaced with tires
1.30 times larger in diameter. Then what will the
speedometer reading be? (a) 1.69 times too high (b) 1.30
times too high (c) accurate (d) 1.30 times too low
(e) 1.69 times too low (e) inaccurate by an unpredictable
factor (ii) What will be the car’s fuel economy in miles per
gallon or km/L? (a) 1.69 times better (b) 1.30 times better (c) essentially the same (d) 1.30 times worse (e) 1.69
times worse
5. O Figure 10.8 shows a system of four particles joined by
light, rigid rods. Assume a ϭ b and M is somewhat larger
than m. (i) About which of the coordinate axes does the
system have the smallest moment of inertia? (a) the x axis

(b) the y axis (c) the z axis (d) The moment of inertia has
the same small value for two axes. (e) The moment of
inertia is the same for all axes. (ii) About which axis does
the system have the largest moment of inertia? (a) the x
axis (b) the y axis (c) the z axis (d) The moment of inertia has the same large value for two axes. (e) The moment
of inertia is the same for all axes.
6. Suppose just two external forces act on a stationary rigid
object and the two forces are equal in magnitude and
opposite in direction. Under what condition does the
object start to rotate?
7. O As shown in Figure 10.19, a cord is wrapped onto a
cylindrical reel mounted on a fixed, frictionless, horizontal axle. Two experiments are conducted. (a) The cord is
pulled down with a constant force of 50 N. (b) An object
of weight 50 N is hung from the cord and released. Are
the angular accelerations equal in the two experiments? If
not, in which experiment is the angular acceleration
greater in magnitude?
8. Explain how you might use the apparatus described in
Example 10.10 to determine the moment of inertia of the
wheel. (If the wheel does not have a uniform mass density, the moment of inertia is not necessarily equal to
1
2
2 MR .)

9. O A constant nonzero net torque is exerted on an object.
Which of the following can not be constant? Choose all that
apply. (a) angular position (b) angular velocity (c) angular
acceleration (d) moment of inertia (e) kinetic energy
(f) location of center of mass
10. Using the results from Example 10.10, how would you calculate the angular speed of the wheel and the linear

speed of the suspended counterweight at t ϭ 2 s, assuming the system is released from rest at t ϭ 0? Is the expression v ϭ R v valid in this situation?
11. If a small sphere of mass M were placed at the end of the
rod in Figure 10.21, would the result for v be greater
than, less than, or equal to the value obtained in Example
10.11?
12. O A solid aluminum sphere of radius R has moment of
inertia I about an axis through its center. What is the
moment of inertia about a central axis of a solid aluminum sphere of radius 2R ? (a) I (b) 2I (c) 4I (d) 8I
(e) 16I (f) 32I
13. Explain why changing the axis of rotation of an object
changes its moment of inertia.
14. Suppose you remove two eggs from the refrigerator, one
hard-boiled and the other uncooked. You wish to determine which is the hard-boiled egg without breaking the
eggs. This determination can be made by spinning the
two eggs on the floor and comparing the rotational
motions. Which egg spins faster? Which egg rotates more
uniformly? Explain.
15. Which of the entries in Table 10.2 applies to finding the
moment of inertia of a long, straight sewer pipe rotating
about its axis of symmetry? Of an embroidery hoop rotating about an axis through its center and perpendicular to
its plane? Of a uniform door turning on its hinges? Of a
coin turning about an axis through its center and perpendicular to its faces?
16. Is it possible to change the translational kinetic energy of
an object without changing its rotational energy?
17. Must an object be rotating to have a nonzero moment of
inertia?
18. If you see an object rotating, is there necessarily a net
torque acting on it?
19. O A decoration hangs from the ceiling of your room at
the bottom end of a string. Your bored roommate turns

the decoration clockwise several times to wind up the
string. When your roommate releases it, the decoration
starts to spin counterclockwise, slowly at first and then
faster and faster. Take counterclockwise as the positive
sense and assume friction is negligible. When the string is
entirely unwound, the ornament has its maximum rate of
rotation. (i) At this moment, is its angular acceleration
(a) positive, (b) negative, or (c) zero? (ii) The decoration
continues to spin, winding the string counterclockwise as
it slows down. At the moment it finally stops, is its angular
acceleration (a) positive, (b) negative, or (c) zero?
20. The polar diameter of the Earth is slightly less than the
equatorial diameter. How would the moment of inertia of


Problems

the Earth about its axis of rotation change if some material from near the equator were removed and transferred
to the polar regions to make the Earth a perfect sphere?
21. O A basketball rolls across a floor without slipping, with its
center of mass moving at a certain velocity. A block of ice
of the same mass is set sliding across the floor with the
same speed along a parallel line. (i) How do their energies
compare? (a) The basketball has more kinetic energy.
(b) The ice has more kinetic energy. (c) They have equal
kinetic energies. (ii) How do their momenta compare?
(a) The basketball has more momentum. (b) The ice has
more momentum. (c) They have equal momenta. (d) Their
momenta have equal magnitudes but are different vectors.
(iii) The two objects encounter a ramp sloping upward.

(a) The basketball will travel farther up the ramp. (b) The
ice will travel farther up the ramp. (c) They will travel
equally far up the ramp.
22. Suppose you set your textbook sliding across a gymnasium
floor with a certain initial speed. It quickly stops moving
because of a friction force exerted on it by the floor.
Next, you start a basketball rolling with the same initial
speed. It keeps rolling from one end of the gym to the
other. Why does the basketball roll so far? Does friction
significantly affect its motion?
23. Three objects of uniform density—a solid sphere, a solid
cylinder, and a hollow cylinder—are placed at the top of
an incline (Fig. Q10.23). They are all released from rest

299

at the same elevation and roll without slipping. Which
object reaches the bottom first? Which reaches it last? Try
this experiment at home and notice that the result is
independent of the masses and the radii of the objects.
24. Figure Q10.24 shows a side view of a child’s tricycle with
rubber tires on a horizontal concrete sidewalk. If a string
is attached to the upper pedal on the far side and pulled
forward horizontally, the tricycle rolls forward. Instead,
assume a string is attached to the lower pedal on the near
side and pulled forward horizontally as shown by A. Does
the tricycle start to roll? If so, which way? Answer the
same questions if (b) the string is pulled forward and
upward as shown by B, (c) the string is pulled straight
down as shown by C, and (d) the string is pulled forward

and downward as shown by D. (e) What if the string is
instead attached to the rim of the front wheel and pulled
upward and backward as shown by E? (f) Explain a pattern of reasoning, based on the diagram, that makes it
easy to answer questions such as all of these. What physical quantity must you evaluate?

E
B

A

D
C
Figure Q10.24

Figure Q10.23

Problems
The Problems from this chapter may be assigned online in WebAssign.
Sign in at www.thomsonedu.com and go to ThomsonNOW to assess your understanding of this chapter’s topics
with additional quizzing and conceptual questions.
1, 2, 3 denotes straightforward, intermediate, challenging; Ⅺ denotes full solution available in Student Solutions Manual/Study
Guide ; ᮡ denotes coached solution with hints available at www.thomsonedu.com; Ⅵ denotes developing symbolic reasoning;
ⅷ denotes asking for qualitative reasoning;
denotes computer useful in solving problem
Section 10.1 Angular Position, Velocity, and Acceleration
1. During a certain period of time, the angular position of a
swinging door is described by u ϭ 5.00 ϩ 10.0t ϩ 2.00t 2,
where u is in radians and t is in seconds. Determine the
angular position, angular speed, and angular acceleration
of the door (a) at t ϭ 0 and (b) at t ϭ 3.00 s.

2. A bar on a hinge starts from rest and rotates with an
angular acceleration a ϭ (10 ϩ 6t) rad/s2, where t is in
seconds. Determine the angle in radians through which
the bar turns in the first 4.00 s.
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

Section 10.2 Rotational Kinematics: The Rigid Object
Under Constant Angular Acceleration
3. A wheel starts from rest and rotates with constant angular
acceleration to reach an angular speed of 12.0 rad/s in
3.00 s. Find (a) the magnitude of the angular acceleration
of the wheel and (b) the angle in radians through which
it rotates in this time interval.
4. A centrifuge in a medical laboratory rotates at an angular
speed of 3 600 rev/min. When switched off, it rotates

= ThomsonNOW;

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning


5.

6.

7.

8.

9.

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

through 50.0 revolutions before coming to rest. Find the
constant angular acceleration of the centrifuge.
ᮡ An electric motor rotating a grinding wheel at
100 rev/min is switched off. The wheel then moves with
constant negative angular acceleration of magnitude
2.00 rad/s2. (a) During what time interval does the wheel
come to rest? (b) Through how many radians does it turn
while it is slowing down?
A rotating wheel requires 3.00 s to rotate through 37.0
revolutions. Its angular speed at the end of the 3.00-s
interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?
(a) Find the angular speed of the Earth’s rotation on its
axis. As the Earth turns toward the east, we see the sky
turning toward the west at this same rate.
(b) The rainy Pleiads wester
And seek beyond the sea
The head that I shall dream of

That shall not dream of me.
—A. E. Housman (© Robert E. Symons)
Cambridge, England is at longitude 0°, and Saskatoon,
Saskatchewan, Canada is at longitude 107° west. How
much time elapses after the Pleiades set in Cambridge
until these stars fall below the western horizon in Saskatoon?
A merry-go-round is stationary. A dog is running on the
ground just outside the merry-go-round’s circumference,
moving with a constant angular speed of 0.750 rad/s. The
dog does not change his pace when he sees what he has
been looking for: a bone resting on the edge of the
merry-go-round one third of a revolution in front of him.
At the instant the dog sees the bone (t ϭ 0), the merrygo-round begins to move in the direction the dog is
running, with a constant angular acceleration equal to
0.015 0 rad/s2. (a) At what time will the dog reach the
bone? (b) The confused dog keeps running and passes
the bone. How long after the merry-go-round starts to
turn do the dog and the bone draw even with each other
for the second time?
The tub of a washing machine goes into its spin cycle,
starting from rest and gaining angular speed steadily for
8.00 s, at which time it is turning at 5.00 rev/s. At this
point, the person doing the laundry opens the lid and a
safety switch turns off the machine. The tub smoothly
slows to rest in 12.0 s. Through how many revolutions
does the tub turn while it is in motion?

Section 10.3 Angular and Translational Quantities
10. A racing car travels on a circular track of radius 250 m.
Assuming the car moves with a constant speed of 45.0 m/s,

find (a) its angular speed and (b) the magnitude and
direction of its acceleration.
11. Make an order-of-magnitude estimate of the number of
revolutions through which a typical automobile tire turns
in 1 yr. State the quantities you measure or estimate and
their values.
12. ⅷ Figure P10.12 shows the drive train of a bicycle that has
wheels 67.3 cm in diameter and pedal cranks 17.5 cm long.
The cyclist pedals at a steady cadence of 76.0 rev/min. The
chain engages with a front sprocket 15.2 cm in diameter
and a rear sprocket 7.00 cm in diameter. (a) Calculate the
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

speed of a link of the chain relative to the bicycle frame.
(b) Calculate the angular speed of the bicycle wheels.
(c) Calculate the speed of the bicycle relative to the road.
(d) What pieces of data, if any, are not necessary for the
calculations?

Crank
Sprocket
Chain
Figure P10.12

13. A wheel 2.00 m in diameter lies in a vertical plane and
rotates with a constant angular acceleration of
4.00 rad/s2. The wheel starts at rest at t ϭ 0, and the
radius vector of a certain point P on the rim makes an
angle of 57.3° with the horizontal at this time. At t ϭ
2.00 s, find (a) the angular speed of the wheel, (b) the
tangential speed and the total acceleration of the point P,
and (c) the angular position of the point P.
14. A discus thrower (Fig. P10.14) accelerates a discus from
rest to a speed of 25.0 m/s by whirling it through 1.25 rev.
Assume the discus moves on the arc of a circle 1.00 m in
radius. (a) Calculate the final angular speed of the discus.
(b) Determine the magnitude of the angular acceleration
of the discus, assuming it to be constant. (c) Calculate the
time interval required for the discus to accelerate from
rest to 25.0 m/s.

Bruce Ayers/Stone/Getty

300

Figure P10.14

15. A small object with mass 4.00 kg moves counterclockwise
with constant speed 4.50 m/s in a circle of radius 3.00 m
centered at the origin. It starts at the point with position
vector 13.00ˆi ϩ 0ˆj 2 m. Then it undergoes an angular displacement of 9.00 rad. (a) What it its position vector? Use
unit–vector notation for all vector answers. (b) In what
quadrant is the particle located, and what angle does its
position vector make with the positive x axis? (c) What is

its velocity? (d) In what direction is it moving? Make a
sketch of its position, velocity, and acceleration vectors.
(e) What is its acceleration? (f) What total force is
exerted on the object?

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ⅷ = qualitative reasoning


Problems

16. A car accelerates uniformly from rest and reaches a speed
of 22.0 m/s in 9.00 s. The tires have diameter 58.0 cm and
do not slip on the pavement. (a) Find the number of revolutions each tire makes during this motion. (b) What is
the final angular speed of a tire in revolutions per second?
17. ᮡ A disk 8.00 cm in radius rotates at a constant rate of
1 200 rev/min about its central axis. Determine (a) its
angular speed, (b) the tangential speed at a point 3.00 cm
from its center, (c) the radial acceleration of a point on
the rim, and (d) the total distance a point on the rim
moves in 2.00 s.
18. ⅷ A straight ladder is leaning against the wall of a house.
The ladder has rails 4.90 m long, joined by rungs 0.410 m
long. Its bottom end is on solid but sloping ground so
that the top of the ladder is 0.690 m to the left of where it
should be, and the ladder is unsafe to climb. You want to
put a rock under one foot of the ladder to compensate

for the slope of the ground. (a) What should be the thickness of the flat rock? (b) Does using ideas from this chapter make it easier to explain the solution to part (a)?
Explain your answer.
19. A car traveling on a flat (unbanked) circular track accelerates uniformly from rest with a tangential acceleration
of 1.70 m/s2. The car makes it one-quarter of the way
around the circle before it skids off the track. Determine
the coefficient of static friction between the car and track
from these data.
20. In part (B) of Example 10.2, the compact disc was modeled as a rigid object under constant angular acceleration
to find the total angular displacement during the playing
time of the disc. In reality, the angular acceleration of a
disc is not constant. In this problem, let us explore the
actual time dependence of the angular acceleration.
(a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h.
Show that the radius r of a given portion of the track is
given by

r ϭ ri ϩ

h
u
2p

y
3.00 kg

2.00 kg

4.00 kg

Figure P10.21

22. ⅷ Rigid rods of negligible mass lying along the y axis connect three particles (Fig. P10.22). The system rotates
about the x axis with an angular speed of 2.00 rad/s. Find
(a) the moment of inertia about the x axis and the total
rotational kinetic energy evaluated from 12Iv 2 and (b) the
tangential speed of each particle and the total kinetic
energy evaluated from © 12m iv i 2. (c) Compare the answers
for kinetic energy in parts (a) and (b).
y
y ϭ 3.00 m

4.00 kg

x

O
2.00 kg

y ϭ Ϫ2.00 m

3.00 kg

y ϭ Ϫ4.00 m

Figure P10.22

23. Two balls with masses M and m are connected by a rigid
rod of length L and negligible mass as shown in Figure
P10.23. For an axis perpendicular to the rod, show that
the system has the minimum moment of inertia when the

axis passes through the center of mass. Show that this
moment of inertia is I ϭ mL2, where m ϭ mM/(m ϩ M).

L
m

M
x

LϪx
Figure P10.23

where v is the constant speed with which the disc surface
passes the laser. (c) From the result in part (b), use integration to find an expression for the angle u as a function
of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
Section 10.4 Rotational Kinetic Energy
21. ᮡ The four particles in Figure P10.21 are connected by
rigid rods of negligible mass. The origin is at the center
of the rectangle. The system rotates in the xy plane about
the z axis with an angular speed of 6.00 rad/s. Calculate
(a) the moment of inertia of the system about the z axis
and (b) the rotational kinetic energy of the system.
Ⅺ = SSM/SG;

x

O

4.00 m

v
du
ϭ
dt
ri ϩ 1h>2p2u

3 = challenging;

2.00 kg
6.00 m

where ri is the radius of the innermost portion of the
track and u is the angle through which the disc turns to
arrive at the location of the track of radius r. (b) Show
that the rate of change of the angle u is given by

2 = intermediate;

301

ᮡ

24. As a gasoline engine operates, a flywheel turning with the
crankshaft stores energy after each fuel explosion to provide the energy required to compress the next charge of
fuel and air. In the engine of a certain lawn tractor, suppose a flywheel must be no more than 18.0 cm in diameter. Its thickness, measured along its axis of rotation, must
be no larger than 8.00 cm. The flywheel must release
60.0 J of energy when its angular speed drops from
800 rev/min to 600 rev/min. Design a sturdy steel flywheel to meet these requirements with the smallest mass
you can reasonably attain. Assume the material has the
density listed for iron in Table 14.1. Specify the shape and

mass of the flywheel.

= ThomsonNOW;

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning


302

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

25. ⅷ A war-wolf or trebuchet is a device used during the Middle Ages to throw rocks at castles and now sometimes
used to fling large vegetables and pianos as a sport. A simple trebuchet is shown in Figure P10.25. Model it as a stiff
rod of negligible mass, 3.00 m long, joining particles of
mass 60.0 kg and 0.120 kg at its ends. It can turn on a frictionless, horizontal axle perpendicular to the rod and
14.0 cm from the large-mass particle. The rod is released
from rest in a horizontal orientation. (a) Find the maximum speed that the 0.120-kg object attains. (b) While the
0.120-kg object is gaining speed, does it move with constant acceleration? Does it move with constant tangential
acceleration? Does the trebuchet move with constant
angular acceleration? Does it have constant momentum?
Does the trebuchet-Earth system have constant mechanical energy?

Figure P10.25

tread wall of uniform thickness 2.50 cm and width
20.0 cm. Assume the rubber has uniform density equal to

1.10 ϫ 103 kg/m3. Find its moment of inertia about an
axis through its center.
28. ⅷ A uniform, thin solid door has height 2.20 m, width
0.870 m, and mass 23.0 kg. Find its moment of inertia for
rotation on its hinges. Is any piece of data unnecessary?
29. Attention! About face! Compute an order-of-magnitude estimate for the moment of inertia of your body as you stand
tall and turn about a vertical axis through the top of your
head and the point halfway between your ankles. In your
solution, state the quantities you measure or estimate and
their values.
30. Many machines employ cams for various purposes such as
opening and closing valves. In Figure P10.30, the cam is a
circular disk rotating on a shaft that does not pass
through the center of the disk. In the manufacture of the
cam, a uniform solid cylinder of radius R is first
machined. Then an off-center hole of radius R/2 is
drilled, parallel to the axis of the cylinder, and centered
at a point a distance R/2 from the cylinder’s center. The
cam, of mass M, is then slipped onto the circular shaft
and welded into place. What is the kinetic energy of the
cam when it is rotating with angular speed v about the
axis of the shaft?

Section 10.5 Calculation of Moments of Inertia
26. Three identical thin rods, each of length L and mass m, are
welded perpendicular to one another as shown in Figure
P10.26. The assembly is rotated about an axis that passes
through the end of one rod and is parallel to another.
Determine the moment of inertia of this structure.

R
2R

z
Figure P10.30

31. Following the procedure used in Example 10.4, prove
that the moment of inertia about the yЈ axis of the rigid
rod in Figure 10.9 is 13ML2.

y

Section 10.6 Torque
32. The fishing pole in Figure P10.32 makes an angle of 20.0°
with the horizontal. What is the torque exerted by the fish
about an axis perpendicular to the page and passing
through the angler’s hand?

x

Axis of
rotation
Figure P10.26

27. Figure P10.27 shows a side view of a car tire. Model it as
having two sidewalls of uniform thickness 0.635 cm and a
2.00 m
Sidewall

20.0Њ

20.0Њ
37.0Њ

33.0 cm
100 N
16.5 cm

Figure P10.32
30.5 cm
Tread

33.

Figure P10.27

2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

ᮡ Find the net torque on the wheel in Figure P10.33
about the axle through O, taking a ϭ 10.0 cm and b ϭ
25.0 cm.

= ThomsonNOW;

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning


Problems

39. An electric motor turns a flywheel through a drive belt
that joins a pulley on the motor and a pulley that is
rigidly attached to the flywheel as shown in Figure P10.39.
The flywheel is a solid disk with a mass of 80.0 kg and a
diameter of 1.25 m. It turns on a frictionless axle. Its pulley has much smaller mass and a radius of 0.230 m. The
tension in the upper (taut) segment of the belt is 135 N,
and the flywheel has a clockwise angular acceleration of
1.67 rad/s2. Find the tension in the lower (slack) segment
of the belt.

10.0 N

30.0Њ

303

a
O

12.0 N
b
9.00 N
Figure P10.33

Section 10.7 The Rigid Object Under a Net Torque
34. A grinding wheel is in the form of a uniform solid disk of
radius 7.00 cm and mass 2.00 kg. It starts from rest and
accelerates uniformly under the action of the constant
torque of 0.600 Nиm that the motor exerts on the wheel.
(a) How long does the wheel take to reach its final operating speed of 1 200 rev/min? (b) Through how many
revolutions does it turn while accelerating?
35. ᮡ A model airplane with mass 0.750 kg is tethered by a
wire so that it flies in a circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net
thrust produces about the center of the circle. (b) Find
the angular acceleration of the airplane when it is in level
flight. (c) Find the translational acceleration of the airplane tangent to its flight path.
36. The combination of an applied force and a friction force
produces a constant total torque of 36.0 Nиm on a wheel
rotating about a fixed axis. The applied force acts for
6.00 s. During this time, the angular speed of the wheel
increases from 0 to 10.0 rad/s. The applied force is then
removed, and the wheel comes to rest in 60.0 s. Find
(a) the moment of inertia of the wheel, (b) the magnitude of the frictional torque, and (c) the total number of
revolutions of the wheel.
37. A block of mass m1 ϭ 2.00 kg and a block of mass m2 ϭ
6.00 kg are connected by a massless string over a pulley in
the shape of a solid disk having radius R ϭ 0.250 m and
mass M ϭ 10.0 kg. These blocks are allowed to move on a
fixed wedge of angle u ϭ 30.0° as shown in Figure P10.37.
The coefficient of kinetic friction is 0.360 for both blocks.
Draw free-body diagrams of both blocks and of the pulley.
Determine (a) the acceleration of the two blocks and
(b) the tensions in the string on both sides of the pulley.

m1

I, R

Figure P10.39

40. ⅷ A disk having moment of inertia 100 kgиm2 is free to
rotate without friction, starting from rest, about a fixed
axis through its center as shown at the top of Figure
10.19. A tangential force whose magnitude can range
from T ϭ 0 to T ϭ 50.0 N can be applied at any distance
ranging from R ϭ 0 to R ϭ 3.00 m from the axis of rotation. Find a pair of values of T and R that cause the disk
to complete 2.00 revolutions in 10.0 s. Does one answer
exist, or no answer, or two answers, or more than two, or
many, or an infinite number?
Section 10.8 Energy Considerations in Rotational Motion
41. In a city with an air-pollution problem, a bus has no combustion engine. It runs on energy drawn from a large,
rapidly rotating flywheel under the floor of the bus. At
the bus terminal, the flywheel is spun up to its maximum
rotation rate of 4 000 rev/min by an electric motor. Every
time the bus speeds up, the flywheel slows down slightly.
The bus is equipped with regenerative braking so that the
flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1 600 kg and
radius 0.650 m. The bus body does work against air resistance and rolling resistance at the average rate of 18.0 hp
as it travels with an average speed of 40.0 km/h. How far
can the bus travel before the flywheel has to be spun up
to speed again?
42. Big Ben, the Parliament tower clock in London, has an
hour hand 2.70 m long with a mass of 60.0 kg and a
minute hand 4.50 m long with a mass of 100 kg (Fig.

m2
u

38. A potter’s wheel—a thick stone disk of radius 0.500 m and
mass 100 kg—is freely rotating at 50.0 rev/min. The potter can stop the wheel in 6.00 s by pressing a wet rag
against the rim and exerting a radially inward force of
70.0 N. Find the effective coefficient of kinetic friction
between wheel and rag.
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

= ThomsonNOW;

John Lawrence/Getty

Figure P10.37

Figure P10.42

Problem 42 and 76.

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning

304

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

P10.42). Calculate the total rotational kinetic energy of
the two hands about the axis of rotation. (You may model
the hands as long, thin rods.)
43. The top in Figure P10.43 has a moment of inertia equal to
4.00 ϫ 10Ϫ4 kg · m2 and is initially at rest. It is free to
rotate about the stationary axis AAЈ. A string, wrapped
around a peg along the axis of the top, is pulled in such a
manner as to maintain a constant tension of 5.57 N. If the
string does not slip while it is unwound from the peg,
what is the angular speed of the top after 80.0 cm of
string has been pulled off the peg?
AЈ

Figure P10.45

F

A

Figure P10.43

44. ⅷ Consider the system shown in Figure P10.44 with m1 ϭ

20.0 kg, m2 ϭ 12.5 kg, R ϭ 0.200 m, and the mass of the
uniform pulley M ϭ 5.00 kg. Object m2 is resting on the
floor, and object m1 is 4.00 m above the floor when it is
released from rest. The pulley axis is frictionless. The
cord is light, does not stretch, and does not slip on the
pulley. Calculate the time interval required for m1 to hit
the floor. How would your answer change if the pulley
were massless?

M

R

m1

m2
Figure P10.44

46. A cylindrical rod 24.0 cm long with mass 1.20 kg and
radius 1.50 cm has a ball of diameter 8.00 cm and mass
2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top. The
system is free to pivot about the bottom end of the rod
after being given a slight nudge. (a) After the rod rotates
through 90°, what is its rotational kinetic energy?
(b) What is the angular speed of the rod and ball?
(c) What is the linear speed of the ball? (d) How does
this speed compare with the speed if the ball had fallen
freely through the same distance of 28 cm?
47. An object with a weight of 50.0 N is attached to the free
end of a light string wrapped around a reel of radius

0.250 m and mass 3.00 kg. The reel is a solid disk, free to
rotate in a vertical plane about the horizontal axis passing
through its center. The suspended object is released
6.00 m above the floor. (a) Determine the tension in the
string, the acceleration of the object, and the speed with
which the object hits the floor. (b) Verify your last answer
by using the principle of conservation of energy to find
the speed with which the object hits the floor.
48. A horizontal 800-N merry-go-round is a solid disk of
radius 1.50 m, started from rest by a constant horizontal
force of 50.0 N applied tangentially to the edge of the
disk. Find the kinetic energy of the disk after 3.00 s.
49. This problem describes one experimental method for
determining the moment of inertia of an irregularly
shaped object such as the payload for a satellite. Figure
P10.49 shows a counterweight of mass m suspended by a
cord wound around a spool of radius r, forming part of a
turntable supporting the object. The turntable can rotate
without friction. When the counterweight is released from
rest, it descends through a distance h, acquiring a speed
v. Show that the moment of inertia I of the rotating apparatus (including the turntable) is mr 2(2gh/v 2 Ϫ 1).

45. In Figure P10.45, the sliding block has a mass of 0.850 kg,
the counterweight has a mass of 0.420 kg, and the pulley
is a hollow cylinder with a mass of 0.350 kg, an inner
radius of 0.020 0 m, and an outer radius of 0.030 0 m.
The coefficient of kinetic friction between the block and
the horizontal surface is 0.250. The pulley turns without
friction on its axle. The light cord does not stretch and
does not slip on the pulley. The block has a velocity of

0.820 m/s toward the pulley when it passes through a
photogate. (a) Use energy methods to predict its speed
after it has moved to a second photogate, 0.700 m away.
(b) Find the angular speed of the pulley at the same
moment.
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

= ThomsonNOW;

m
Figure P10.49

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning


Problems

50. The head of a grass string trimmer has 100 g of cord
wound in a light cylindrical spool with inside diameter
3.00 cm and outside diameter 18.0 cm, as shown in Figure
P10.50. The cord has a linear density of 10.0 g/m. A single strand of the cord extends 16.0 cm from the outer
edge of the spool. (a) When switched on, the trimmer

speeds up from 0 to 2 500 rev/min in 0.215 s. (a) What
average power is delivered to the head by the trimmer
motor while it is accelerating? (b) When the trimmer is
cutting grass, it spins at 2 000 rev/min and the grass
exerts an average tangential force of 7.65 N on the outer
end of the cord, which is still at a radial distance of
16.0 cm from the outer edge of the spool. What is the
power delivered to the head under load?

16.0 cm

55.

56.

3.0 cm

57.

18.0 cm
Figure P10.50

51. (a) A uniform solid disk of radius R and mass M is free to
rotate on a frictionless pivot through a point on its rim
(Fig. P10.51). If the disk is released from rest in the position shown by the blue circle, what is the speed of its center of mass when the disk reaches the position indicated
by the dashed circle? (b) What is the speed of the lowest
point on the disk in the dashed position? (c) What If?
Repeat part (a) using a uniform hoop.

Pivot

R

g

58.

305

The cube then moves up a smooth incline that makes an
angle u with the horizontal. A cylinder of mass m and
radius r rolls without slipping with its center of mass moving with speed v and encounters an incline of the same
angle of inclination but with sufficient friction that the
cylinder continues to roll without slipping. (a) Which
object will go the greater distance up the incline?
(b) Find the difference between the maximum distances
the objects travel up the incline. (c) Explain what
accounts for this difference in distances traveled.
(a) Determine the acceleration of the center of mass of a
uniform solid disk rolling down an incline making angle u
with the horizontal. Compare this acceleration with that
of a uniform hoop. (b) What is the minimum coefficient
of friction required to maintain pure rolling motion for
the disk?
A uniform solid disk and a uniform hoop are placed side
by side at the top of an incline of height h. If they are
released from rest at the same time and roll without slipping, which object reaches the bottom first? Verify your
answer by calculating their speeds when they reach the
bottom in terms of h.
ⅷ A metal can containing condensed mushroom soup

has mass 215 g, height 10.8 cm, and diameter 6.38 cm. It
is placed at rest on its side at the top of a 3.00-m-long
incline that is at 25.0° to the horizontal and is then
released to roll straight down. It reaches the bottom of
the incline after 1.50 s. Assuming mechanical energy conservation, calculate the moment of inertia of the can.
Which pieces of data, if any, are unnecessary for calculating the solution?
ⅷ A tennis ball is a hollow sphere with a thin wall. It is set
rolling without slipping at 4.03 m/s on a horizontal section of a track as shown in Figure P10.58. It rolls around
the inside of a vertical circular loop 90.0 cm in diameter
and finally leaves the track at a point 20.0 cm below the
horizontal section. (a) Find the speed of the ball at the top
of the loop. Demonstrate that it will not fall from the
track. (b) Find its speed as it leaves the track. What If?
(c) Suppose static friction between ball and track were
negligible so that the ball slid instead of rolling. Would its
speed then be higher, lower, or the same at the top of the
loop? Explain.

Figure P10.51

Section 10.9 Rolling Motion of a Rigid Object
52. ⅷ A solid sphere is released from height h from the top
of an incline making an angle u with the horizontal. Calculate the speed of the sphere when it reaches the bottom
of the incline (a) in the case that it rolls without slipping
and (b) in the case that it slides frictionlessly without
rolling. (c) Compare the time intervals required to reach
the bottom in cases (a) and (b).
53. ᮡ A cylinder of mass 10.0 kg rolls without slipping on a
horizontal surface. At a certain instant its center of mass
has a speed of 10.0 m/s. Determine (a) the translational

kinetic energy of its center of mass, (b) the rotational
kinetic energy about its center of mass, and (c) its total
energy.
54. ⅷ A smooth cube of mass m and edge length r slides with
speed v on a horizontal surface with negligible friction.
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

Figure P10.58

Additional Problems
59. As shown in Figure P10.59, toppling chimneys often break
apart in midfall because the mortar between the bricks
cannot withstand much shear stress. As the chimney
begins to fall, shear forces must act on the topmost sections to accelerate them tangentially so that they can keep
up with the rotation of the lower part of the stack. For
simplicity, let us model the chimney as a uniform rod of

= ThomsonNOW;

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning

306

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

Jerry Wachter/Photo Researchers, Inc.

length ᐉ pivoted at the lower end. The rod starts at rest in
a vertical position (with the frictionless pivot at the bottom) and falls over under the influence of gravity. What
fraction of the length of the rod has a tangential acceleration greater than g sin u, where u is the angle the chimney
makes with the vertical axis?

Figure P10.59 A building demolition site in Baltimore, Maryland. At
the left is a chimney, mostly concealed by the building, that has broken
apart on its way down. Compare with Figure 10.18.

60. Review problem. A mixing beater consists of three thin
rods, each 10.0 cm long. The rods diverge from a central
hub, separated from one another by 120°, and all turn in
the same plane. A ball is attached to the end of each rod.
Each ball has cross-sectional area 4.00 cm2 and is so
shaped that it has a drag coefficient of 0.600. Calculate the
power input required to spin the beater at 1 000 rev/min
(a) in air and (b) in water.
61. A 4.00-m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg.
The spool is mounted on a frictionless axle and is initially
at rest. The cord is pulled from the spool with a constant
acceleration of magnitude 2.50 m/s2. (a) How much work
has been done on the spool when it reaches an angular

speed of 8.00 rad/s? (b) Assuming there is enough cord
on the spool, how long does it take the spool to reach this
angular speed? (c) Is there enough cord on the spool?
62. ⅷ An elevator system in a tall building consists of an 800kg car and a 950-kg counterweight, joined by a cable that
passes over a pulley of mass 280 kg. The pulley, called a
sheave, is a solid cylinder of radius 0.700 m turning on a
horizontal axle. The cable has comparatively small mass
and constant length. It does not slip on the sheave. The
car and the counterweight move vertically, next to each
other inside the same shaft. A number n of people, each
of mass 80.0 kg, are riding in the elevator car, moving
upward at 3.00 m/s and approaching the floor where the
car should stop. As an energy-conservation measure, a
computer disconnects the electric elevator motor at just
the right moment so that the sheave-car-counterweight
system then coasts freely without friction and comes to
rest at the floor desired. There it is caught by a simple
latch rather than by a massive brake. (a) Determine the
distance d the car coasts upward as a function of n. Evaluate the distance for (b) n ϭ 2, (c) n ϭ 12, and (d) n ϭ 0.
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

(e) Does the expression in part (a) apply for all integer
values of n or only for what values? Explain. (f) Describe
the shape of a graph of d versus n. (g) Is any piece of data

unnecessary for the solution? Explain. (h) Contrast the
meaning of energy conservation as it is used in the statement of this problem and as it is used in Chapter 8.
(i) Find the magnitude of the acceleration of the coasting
elevator car, as it depends on n.
63. ⅷ Figure P10.63 is a photograph of a lawn sprinkler. Its
rotor consists of three metal tubes that fill with water
when a hose is connected to the base. As water sprays out
of the holes at the ends of the arms and the hole near the
center of each arm, the assembly with the three arms
rotates. To analyze this situation, let us make the following assumptions: (1) The arms can be modeled as thin,
straight rods, each of length L. (2) The water coming
from the hole at distance ᐉ from the center sprays out
horizontally, parallel to the ground and perpendicular to
the arm. (3) The water emitted from the holes at the
ends of the arms sprays out radially away from the center
of the rotor. When filled with water, each arm has mass m.
The center of the assembly is massless. The water ejected
from a hole at distance ᐉ from the center causes a thrust
force F on the arm containing the hole. The mounting
for the three-arm rotor assembly exerts a frictional torque
that is described by t ϭ Ϫb v, where v is the angular
speed of the assembly. (a) Imagine that the sprinkler is in
operation. Find an expression for the constant angular
speed with which the assembly rotates after it completes
an initial period of angular acceleration. Your expression
should be in terms of F, ᐉ, and b. (b) Imagine that the
sprinkler has been at rest and is just turned on. Find an
expression for the initial angular acceleration of the rotor,
that is, the angular acceleration when the arms are filled
with water and the assembly just begins to move from rest.

Your expression should be in terms of F, ᐉ, m, and L.
(c) Now, take a step toward reality from the simplified
model. The arms are actually bent as shown in the photograph. Therefore, the water from the ends of the arms is
not actually sprayed radially. How will this fact affect the
constant angular speed with which the assembly rotates in
part (a)? In reality, will it be larger, smaller, or
unchanged? Provide a convincing argument for your
response. (d) How will the bend in the arms, described in
part (c), affect the angular acceleration in part (b)? In
reality, will it be larger, smaller, or unchanged? Provide a
convincing argument for your response.

= ThomsonNOW;

L

ᐉ

Figure P10.63

Ⅵ = symbolic reasoning;

ⅷ = qualitative reasoning


Problems

64. A shaft is turning at 65.0 rad/s at time t ϭ 0. Thereafter,
its angular acceleration is given by
a ϭ Ϫ10.0 rad>s2 Ϫ 5.00t rad>s3

where t is the elapsed time. (a) Find its angular speed at t
ϭ 3.00 s. (b) How far does it turn in these 3 s?
65. A long, uniform rod of length L and mass M is pivoted
about a horizontal, frictionless pin through one end. The
rod is released, almost from rest in a vertical position as
shown in Figure P10.65. At the instant the rod is horizontal, find (a) its angular speed, (b) the magnitude of its
angular acceleration, (c) the x and y components of the
acceleration of its center of mass, and (d) the components of the reaction force at the pivot.
y

observes that drops of water fly off tangentially. She measures the height reached by drops moving vertically (Fig.
P10.67). A drop that breaks loose from the tire on one
turn rises a distance h1 above the tangent point. A drop
that breaks loose on the next turn rises a distance h2 Ͻ h1
above the tangent point. The height to which the drops
rise decreases because the angular speed of the wheel
decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.
69. A uniform, hollow, cylindrical spool has inside radius R/2,
outside radius R, and mass M (Fig. P10.69). It is mounted
so that it rotates on a fixed, horizontal axle. A counterweight of mass m is connected to the end of a string
wound around the spool. The counterweight falls from
rest at t ϭ 0 to a position y at time t. Show that the torque
due to the friction forces between spool and axle is
tf ϭ R c m a g Ϫ

L
Pivot

307

x

2y
t

2

b ϪM

5y
4t 2

d

M

Figure P10.65

66. A cord is wrapped around a pulley of mass m and radius r.
The free end of the cord is connected to a block of mass
M. The block starts from rest and then slides down an
incline that makes an angle u with the horizontal. The
coefficient of kinetic friction between block and incline is
m. (a) Use energy methods to show that the block’s speed
as a function of position d down the incline is
4gdM 1sin u Ϫ m cos u 2
vϭ
B
m ϩ 2M

m
R/2
R/2

y
Figure P10.69

(b) Find the magnitude of the acceleration of the block
in terms of m, m, M, g, and u.
67. A bicycle is turned upside down while its owner repairs a
flat tire. A friend spins the other wheel, of radius 0.381 m,
and observes that drops of water fly off tangentially. She
measures the height reached by drops moving vertically
(Fig. P10.67). A drop that breaks loose from the tire on
one turn rises h ϭ 54.0 cm above the tangent point. A
drop that breaks loose on the next turn rises 51.0 cm
above the tangent point. The height to which the drops
rise decreases because the angular speed of the wheel
decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

70. (a) What is the rotational kinetic energy of the Earth
about its spin axis? Model the Earth as a uniform sphere
and use data from the endpapers. (b) The rotational
kinetic energy of the Earth is decreasing steadily because
of tidal friction. Find the change in one day, assuming the
rotational period increases by 10.0 ms each year.
71. Two blocks as shown in Figure P10.71 are connected by a
string of negligible mass passing over a pulley of radius
0.250 m and moment of inertia I. The block on the frictionless incline is moving up with a constant acceleration
of 2.00 m/s2. (a) Determine T1 and T2, the tensions in

the two parts of the string. (b) Find the moment of inertia of the pulley.
2.00 m/s2
T1

h

15.0 kg
m1

T2
m 2 20.0 kg

37.0Њ
Figure P10.71

Figure P10.67

Problems 67 and 68.

68. A bicycle is turned upside down while its owner repairs a
flat tire. A friend spins the other wheel, of radius R, and
2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

72. The reel shown in Figure P10.72 has radius R and

moment of inertia I. One end of the block of mass m is
connected to a spring of force constant k, and the other
end is fastened to a cord wrapped around the reel. The
reel axle and the incline are frictionless. The reel is

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308

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

wound counterclockwise so that the spring stretches a distance d from its unstretched position and the reel is then
released from rest. (a) Find the angular speed of the reel
when the spring is again unstretched. (b) Evaluate the
angular speed numerically at this point, taking I ϭ
1.00 kg · m2, R ϭ 0.300 m, k ϭ 50.0 N/m, m ϭ 0.500 kg,
d ϭ 0.200 m, and u ϭ 37.0°.
R
m

k
u

Figure P10.72

73. As a result of friction, the angular speed of a wheel
changes with time according to

du
ϭ v0eϪst
dt
where v0 and s are constants. The angular speed changes
from 3.50 rad/s at t ϭ 0 to 2.00 rad/s at t ϭ 9.30 s. Use
this information to determine s and v0. Then determine
(a) the magnitude of the angular acceleration at t ϭ 3.00 s,
(b) the number of revolutions the wheel makes in the
first 2.50 s, and (c) the number of revolutions it makes
before coming to rest.
74. A common demonstration, illustrated in Figure P10.74,
consists of a ball resting at one end of a uniform board of
length ᐉ, hinged at the other end, and elevated at an
angle u. A light cup is attached to the board at rc so that it
will catch the ball when the support stick is suddenly
removed. (a) Show that the ball will lag behind the falling
board when u is less than 35.3°. (b) Assuming the board is
1.00 m long and is supported at this limiting angle, show
that the cup must be 18.4 cm from the moving end.

rc

Support
stick

u

h
R

Figure P10.77

Hinged end
r

Figure P10.74

75. ⅷ A tall building is located on the Earth’s equator. As the
Earth rotates, a person on the top floor of the building
moves faster than someone on the ground with respect to
an inertial reference frame because the latter person is
closer to the Earth’s axis. Consequently, if an object is
dropped from the top floor to the ground a distance h
below, it lands east of the point vertically below where it
was dropped. (a) How far to the east will the object land?

2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

M

78. A uniform solid sphere of radius r is placed on the inside

surface of a hemispherical bowl with much larger radius
R. The sphere is released from rest at an angle u to the
vertical and rolls without slipping (Fig. P10.78). Determine the angular speed of the sphere when it reaches the
bottom of the bowl.

Cup

ᐉ

Express your answer in terms of h, g, and the angular
speed v of the Earth. Ignore air resistance, and assume
the free-fall acceleration is constant over this range of
heights. (b) Evaluate the eastward displacement for h ϭ
50.0 m. (c) In your judgment, were we justified in ignoring this aspect of the Coriolis effect in our previous study of
free fall?
76.
The hour hand and the minute hand of Big Ben, the
Parliament tower clock in London, are 2.70 m and 4.50 m
long and have masses of 60.0 kg and 100 kg, respectively
(see Figure P10.42). (i) Determine the total torque due to
the weight of these hands about the axis of rotation when
the time reads (a) 3:00, (b) 5:15, (c) 6:00, (d) 8:20, and
(e) 9:45. (You may model the hands as long, thin uniform
rods.) (ii) Determine all times when the total torque
about the axis of rotation is zero. Determine the times to
the nearest second, solving a transcendental equation
numerically.
77. A string is wound around a uniform disk of radius R and
mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.77).
Show that (a) the tension in the string is one-third of the

weight of the disk, (b) the magnitude of the acceleration
of the center of mass is 2g/3, and (c) the speed of the
center of mass is (4gh/3)1/2 after the disk has descended
through distance h. Verify your answer to part (c) using
the energy approach.

ᮡ

u

R

Figure P10.78

79. A solid sphere of mass m and radius r rolls without slipping along the track shown in Figure P10.79. It starts
from rest with the lowest point of the sphere at height h

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309

Problems

above the bottom of the loop of radius R, much larger
than r. (a) What is the minimum value of h (in terms of

R) such that the sphere completes the loop? (b) What are
the components of the net force on the sphere at the
point P if h ϭ 3R ?

m

h

R
P

force of friction is to the right and equal in magnitude to
F/3. (c) If the cylinder starts from rest and rolls without
slipping, what is the speed of its center of mass after it has
rolled through a distance d ?
84. A plank with a mass M ϭ 6.00 kg rides on top of two identical solid cylindrical rollers that have R ϭ 5.00 cm and m
ϭ 2.00 kg (Fig. P10.84).
The plank is pulled by a constant
S
horizontal force F of magnitude 6.00 N applied to the
end of the plank and perpendicular to the axes of the
cylinders (which are parallel). The cylinders roll without
slipping on a flat surface. There is also no slipping
between the cylinders and the plank. (a) Find the acceleration of the plank and of the rollers. (b) What friction
forces are acting?
M

Figure P10.79

80. A thin rod of mass 0.630 kg and length 1.24 m is at rest,

hanging vertically from a strong fixed hinge at its top
end. Suddenly a horizontal impulsive force 114.7ˆi 2 N is
applied to it. (a) Suppose the force acts at the bottom
end of the rod. Find the acceleration of its center of mass
and the horizontal force the hinge exerts. (b) Suppose
the force acts at the midpoint of the rod. Find the acceleration of this point and the horizontal hinge reaction.
(c) Where can the impulse be applied so that the hinge
will exert no horizontal force? This point is called the center of percussion.
81. (a) A thin rod of length h and mass M is held vertically
with its lower end resting on a frictionless horizontal surface. The rod is then released to fall freely. Determine the
speed of its center of mass just before it hits the horizontal surface. (b) What If? Now suppose the rod has a fixed
pivot at its lower end. Determine the speed of the rod’s
center of mass just before it hits the surface.
82. Following Thanksgiving dinner your uncle falls into a
deep sleep, sitting straight up facing the television set. A
naughty grandchild balances a small spherical grape at
the top of his bald head, which itself has the shape of a
sphere. After all the children have had time to giggle, the
grape starts from rest and rolls down without slipping.
The grape loses contact with your uncle’s scalp when the
radial line joining it to the center of curvature makes
what angle with the vertical?
83. A spool of wire ofSmass M and radius R is unwound under
a constant force F (Fig. P10.83). Assuming the spool is a
uniform solid cylinder that doesn’t slip,S show that (a) the
acceleration of the center of mass is 4F/3M and (b) the

F
M
R

Figure P10.83

2 = intermediate;

3 = challenging;

Ⅺ = SSM/SG;

ᮡ

m

R

F
m

R

Figure P10.84

85. A spool of thread consists of a cylinder of radius R1 with
end caps of radius R2 as shown in the end view illustrated
in Figure P10.85. The mass of the spool, including the
thread, is m, and its moment of inertia about an axis
through its center is I. The spool is placed on a rough
horizontal
surface so that it rolls without slipping when a
S

force T acting to the right is applied to the free end of
the thread. Show that the magnitude of the friction force
exerted by the surface on the spool is given by

fϭ a

I ϩ mR1R2
bT
I ϩ mR22

Determine the direction of the force of friction.

R2
R1
T

Figure P10.85

86. ⅷ A large, cylindrical roll of tissue paper of initial radius
R lies on a long, horizontal surface with the outside end
of the paper nailed to the surface. The roll is given a
slight shove (vi ഠ 0) and commences to unroll. Assume
the roll has a uniform density and that mechanical energy
is conserved in the process. (a) Determine the speed of
the center of mass of the roll when its radius has diminished to r. (b) Calculate a numerical value for this speed
at r ϭ 1.00 mm, assuming R ϭ 6.00 m. (c) What If? What
happens to the energy of the system when the paper is
completely unrolled?

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310

Chapter 10

Rotation of a Rigid Object About a Fixed Axis

Answers to Quick Quizzes
10.1 (i), (c). For a rotation of more than 180°, the angular
displacement must be larger than p ϭ 3.14 rad. The
angular displacements in the three choices are (a) 6 rad
Ϫ 3 rad ϭ 3 rad, (b) 1 rad Ϫ (Ϫ1) rad ϭ 2 rad, and
(c) 5 rad Ϫ 1 rad ϭ 4 rad. (ii), (b). Because all angular
displacements occur in the same time interval, the displacement with the lowest value will be associated with
the lowest average angular speed.
10.2 (b). In Equation 10.8, both the initial and final angular
speeds are the same in all three cases. As a result, the
angular acceleration is inversely proportional to the
angular displacement. Therefore, the highest angular
acceleration is associated with the lowest angular displacement.
10.3 (i), (b). The system of the platform, Alex, and Brian is a
rigid object, so all points on the rigid object have the
same angular speed. (ii), (a). The tangential speed is
proportional to the radial distance from the rotation
axis.

10.4 (a). Almost all the mass of the pipe is at the same distance from the rotation axis, so it has a larger moment
of inertia than the solid cylinder.

10.5 (i), (b). The fatter handle of the screwdriver gives you a
larger moment arm and increases the torque you can
apply with a given force from your hand. (ii), (a). The
longer handle of the wrench gives you a larger moment
arm and increases the torque you can apply with a given
force from your hand.
10.6 (b). With twice the moment of inertia and the same frictional torque, there is half the angular acceleration.
With half the angular acceleration, it will require twice
as long to change the speed to zero.
10.7 (b). All the gravitational potential energy of the box–
Earth system is transformed to kinetic energy of translation. For the ball, some of the gravitational potential
energy of the ball–Earth system is transformed to rotational kinetic energy, leaving less for translational kinetic
energy, so the ball moves downhill more slowly than the
box does.


11.1 The Vector Product and Torque
11.2 Angular Momentum: The Nonisolated System
11.3 Angular Momentum of a Rotating Rigid Object
11.4 The Isolated System: Conservation of Angular Momentum
11.5 The Motion of Gyroscopes and Tops
A competitive diver undergoes a rotation during a dive. She spins at a
higher rate when she folds her body into a smaller package due to the
principle of conservation of angular momentum, as discussed in this
chapter. (The Image Bank/Getty Images)

11

Angular Momentum

The central topic of this chapter is angular momentum, a quantity that plays a key
role in rotational dynamics. In analogy to the principle of conservation of linear
momentum for an isolated system, the angular momentum of a system is conserved if no external torques act on the system. Like the law of conservation of linear momentum, the law of conservation of angular momentum is a fundamental
law of physics, equally valid for relativistic and quantum systems.

11.1

z

tϭ r ؋F

r

An important consideration in defining angular momentum is the process of multiplying two vectors by means of the operation called the vector product. We will
introduce the vectorSproduct by considering the vector nature of torque.
S
Consider a force F acting on a rigid object at the vector position r (Active Fig.
11.1). As we saw in Section 10.6, the magnitude of the torque due to this force
S
S
about an axis through the
origin is rF sin f, where f is the angle between r and F.
S
The axis about which
F tends to produce rotation is perpendicular to the plane
S
S

formed by r and F.
S
S
S
The torque vector T is related to the two vectors
r and F. We can establish a
S
S
S
mathematical relationship between T, r , and F using a mathematical operation
called the vector product, or cross product:
S

Tϵr ؋F
S

S

y

O

The Vector Product and Torque

(11.1)

P
f

x

F

ACTIVE FIGURE 11.1
S

The torque vector T lies in a direction
perpendicular to the plane formed by
S
the position Svector r and the applied
force vector F.
Sign in at www.thomsonedu.com and
go to ThomsonNOW to moveS point P
and change the force vector F to see
the effect on the torque vector.

311


312

Chapter 11

Angular Momentum
S

PITFALL PREVENTION 11.1
The Cross Product Is a Vector
Remember that the result of taking
a cross product between two vectors is a third vector. Equation 11.3

gives only the magnitude of this
vector.

We
now give a formal definition
of the vector product. Given
any two vectors A
S
S
S
S
and B, the vector product A ؋ B is defined as aSthird Svector C, which
has a magniS
tude of AB sin u, where u is the angle between A and B. That is, if C is given by
S

S

S

CϭA؋B

(11.2)

its magnitude is
C ϭ AB sin u

(11.3)
S

S

The quantity AB sin u is equal to the areaS of the parallelogram formed by A and B
as
shown
in Figure 11.2. The direction of C is perpendicular to the plane formed by
S
S
A and B, and the best way to determine this direction is to use the right-hand rule
S
illustrated in Figure 11.2. The
four fingers of the right hand are pointed along A
S
and then “wrapped” into BS through
the angle u. The direction Sof the
upright
S
S
S
thumbS is the Sdirection of A ؋ B ϭ C. Because of the notation, A ؋ B is often
read “A cross B,” hence the term cross product.
Some properties of the vector product that follow from its definition are as follows:
Properties of the vector
product

ᮣ

1. Unlike the scalar product, the vector product is not commutative. Instead, the
order in which the two vectors are multiplied in a cross product is important:
S

S

S

S

A ؋ B ϭ ϪB ؋ A

(11.4)

Therefore, if you change the order of the vectors in a cross product, you
must change the sign. You can easily verify this relationship with the righthand
rule.
S
S
S
S
A
2. If
is
parallel to B 1u ϭ 0 or 180°2, then A ؋ B ϭ 0; therefore, it follows that
S
S
A؋
A ϭ 0.
S
S
S
S

3. If A is perpendicular to B, then 0 A ؋ B 0 ϭ AB.
4. The vector product obeys the distributive law:
A ؋ 1B ϩ C 2 ϭ A ؋ B ϩ A ؋ C
S

S

S

S

S

S

S

(11.5)

5. The derivative of the cross product with respect to some variable such as t is
S

S

S
S
S
d S
dA
dB

1A ؋ B 2 ϭ
؋BϩA؋
dt
dt
dt

(11.6)

where it is important to preserve the multiplicative order of the terms on the
right side in view of Equation 11.4.
It is left as an exercise (Problem 10) to show from Equations 11.3 and 11.4 and
from the definition of unit vectors that the cross products of the unit vectors ˆi , ˆj ,
and ˆ
k obey the following rules:
Cross products of unit
vectors

ᮣ

ˆi ؋ ˆi ϭ ˆj ؋ ˆj ϭ ˆ
k؋ˆ
kϭ0

(11.7a)

ˆi ؋ ˆj ϭ Ϫ ˆj ؋ ˆi ϭ ˆ
k

(11.7b)

Right-hand rule
C‫؍‬A؋B

A
u
B

؊C ‫ ؍‬B ؋ A
S

S

S

Figure 11.2 The vector product A ؋ B is a third vector
C having a magnitude AB sin u equalSto theS
S
area of the parallelogram shown. The direction of C is perpendicular to the plane formed by A and B,
and this direction is determined by the right-hand rule.


Section 11.1

ˆj ؋ ˆ
ˆ ؋ ˆj ϭ ˆi
k ϭ Ϫk

The Vector Product and Torque

313

(11.7c)

ˆ
k ؋ ˆi ϭ Ϫ ˆi ؋ ˆ
k ϭ ˆj

(11.7d)

Signs are interchangeable in cross products. For example, A ؋ 1ϪB 2 ϭ ϪA ؋ B
and ˆi ؋ 1Ϫ ˆj 2 ϭ Ϫ ˆi ؋ ˆj .
S
S
The cross product of any two vectors A and B can be expressed in the following
determinant form:
S

S

S

S

ˆi ˆj ˆ
k
A A
A A
A A
A ؋ B ϭ † A x A y A z † ϭ ` y z ` ˆi ϩ ` z x ` ˆj ϩ ` x y ` ˆ
k

By Bz
Bz Bx
Bx By
Bx By Bz
S

S

Expanding these determinants gives the result
A ؋ B ϭ 1A yBz Ϫ A zBy 2 ˆi ϩ 1A zBx Ϫ A xBz 2 ˆj ϩ 1A xBy Ϫ A yBx 2 ˆ
k
S

S

(11.8)

Given the definition of the cross product, we can now assign a direction to the
torque vector. If the force lies in the xy plane, as in Active Figure 11.1, the torque
S
T is represented by a vector parallel to the z axis. The force in Active Figure 11.1
creates a torque that tends to rotate the object counterclockwise about the z axis;
S
S
the direction of T is toward increasing
z, and T is therefore in the positive z direcS
S
tion. If we reversed the direction of F in Active Figure 11.1, T would be in the negative z direction.

Quick Quiz 11.1 Which of the following statements about the relationship

between the magnitude of the cross product
of
two vectors and the product
ofS the
S
S
S
magnitudes of the vectors
is
true?
(a)
is
larger
than
AB.
(b)
0
A
؋
B
0
A
؋
B 0 is
0
S
S
smaller than AB. (c) 0 A ؋ B 0 couldSbe larger
or
smaller

than
AB,
depending
on
S
the angle between the vectors. (d) 0 A ؋ B 0 could be equal to AB.

E XA M P L E 1 1 . 1

The Vector Product

Two vectors lying in the xy plane are given by the equations A ϭ 2iˆ ϩ 3jˆ and B ϭ Ϫˆi ϩ 2jˆ. Find A ؋ B and verify
S
S
S
S
that A ؋ B ϭ ϪB ؋ A.
S

S

S

S

SOLUTION
Conceptualize Given the unit–vector notations of the vectors, think about the directions the vectors point in space.
Imagine the parallelogram shown in Figure 11.2 for these vectors.
Categorize Because we use the definition of the cross product discussed in this section, we categorize this example
as a substitution problem.

A ؋ B ϭ 12ˆi ϩ 3ˆj 2 ؋ 1Ϫ ˆi ϩ 2ˆj 2
S

Write the cross product of the two vectors:

A ؋ B ϭ 2ˆi ؋ 1Ϫ ˆi 2 ϩ 2ˆi ؋ 2ˆj ϩ 3ˆj ؋ 1Ϫ ˆi 2 ϩ 3ˆj ؋ 2ˆj
S

Perform the multiplication:

S

ˆ ϩ 3k
ˆ ϩ 0 ϭ 7k
ˆ
A ؋ B ϭ 0 ϩ 4k
S

Use Equations 11.7a through 11.7d to evaluate the
various terms:
S

S

S

S

S

S

B ؋ A ϭ 1Ϫ ˆi ϩ 2ˆj 2 ؋ 12ˆi ϩ 3ˆj 2

S

S

To verify that A ؋ B ϭ ϪB ؋ A, evaluate B ؋ A:
Perform the multiplication:

S

S

B ؋ A ϭ 1Ϫ ˆi 2 ؋ 2ˆi ϩ 1Ϫ ˆi 2 ؋ 3ˆj ϩ 2ˆj ؋ 2ˆi ϩ 2ˆj ؋ 3ˆj
S

S