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Chapter: Chapter 9

Multiple Choice

1. Complete the following statement: When a net torque is applied to a rigid object, it always
produces a
A) constant acceleration.
B) rotational equilibrium.
C) constant angular velocity.
D) constant angular momentum.
E) change in angular velocity.

Ans: E
Difficulty: Easy
SectionDef: Section 9-1

2. A wrench is used to tighten a nut as shown in the figure. A 12-N force is applied 7.0 cm
from the axis of rotation. What is the magnitude of the torque due to the applied force?
A) 0.58 N  m
B) 0.84 N  m
C) 1.71 N  m
D) 14 N  m
E) 58 N  m

Ans: B
Difficulty: Medium
SectionDef: Section 9-1

3. A string is tied to a doorknob 0.72 m from the hinge as illustrated in the figure. At the
instant shown, the force applied to the string is 5.0 N. What is the magnitude of the torque on
the door?

A) 2.1 N  m
B) 3.0 N  m
C) 1.0 N  m
D) 0.78 N  m
E) 0.60 N  m

Ans: B
Difficulty: Hard
SectionDef: Section 9-1

4. A uniform 13-kg trap door is oriented horizontally and hinged as shown. What is the
magnitude of the torque on the door at the instant that the release is activated and the door can
freely rotate?

A) 4.9 N  m
B) 9.8 N  m
C) 48 N  m
D) 72 N  m
E) 96 N  m

Ans: C
Difficulty: Easy
SectionDef: Section 9-1

5. Complete the following statement: A body is in translational equilibrium

A) only if it is at rest.
B) only if it is moving with constant velocity.
C) only if it is moving with constant acceleration.
D) if it is either at rest or moving with constant velocity.
E) if it is moving with either constant velocity or constant acceleration.

Ans: D
Difficulty: Easy
SectionDef: Section 9-2 and 9-3
6. A horizontal, 10-m plank weighs 100 N. It rests on two supports that are placed 1.0 m from
each end as shown in the figure. How close to one end can an 800-N person stand without
causing the plank to tip?

A) 0m
B) 0.3 m
C) 0.5 m
D) 0.7 m
E) 0.9 m

Ans: C
Difficulty: Hard
SectionDef: Section 9-2 and 9-3

7. A 3.0-kg ball and a 1.0-kg ball are placed at opposite ends of a massless beam so that the
system is in equilibrium as shown. Note: The drawing is not drawn to scale. What is the value
of the ratio of the lengths, b/a?

A) 2.0
B) 2.5
C) 3.0
D) 4.0
E) 5.0
Ans: D
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

8. One end of a rope is tied to the handle of a horizontally-oriented and uniform door. A force
is applied to the other end of the rope as shown in the drawing. The door has a weight of 145
N and is hinged on the right. What is the maximum magnitude of for which the door will
remain at rest?

A) 145 N
B) 265 N
C) 381 N
D) 424 N
E) 530 N

Ans: B
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

9. A meter stick is pivoted at the 0.50-m line. A 3.0-kg object is hung from the 0.15-m line.
Where should a 5.0-kg object be hung to achieve equilibrium (the meter stick oriented horizontal
and motionless)?
A) 0.06-m line
B) 0.24-m line
C) 0.56-m line
D) 0.71-m line
E) A 5.0-kg object cannot be placed anywhere on the meter stick to result in equilibrium in this
system.
Ans: D
Difficulty: Hard
SectionDef: Section 9-2 and 9-3

10. In the drawing shown, the large wheel has a radius of 7.1 m. A rope is wrapped around the
edge of the wheel and a 7.6 kg-box hangs from the rope. A smaller disk of radius 1.38 m is
attached to the wheel. A rope is wrapped around the edge of the disk as shown. An axis of
rotation passes through the center of the wheel-disk system. What is the value of the mass M
that will prevent the wheel from rotating?

A) 34 kg
B) 12 kg
C) 17 kg
D) 39 kg
E) 46 kg

Ans: D
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

Reference: Ref 9-1

An 80-kg man balances the boy on a teeter-totter as shown. Note: Ignore the weight of the
board.

11. What is the approximate mass of the boy?

A) 10 kg
B) 20 kg
C) 40 kg
D) 45 kg
E) 50 kg

Ans: B
Refer To: Ref 9-1
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

12. What, approximately, is the magnitude of the downward force exerted on the fulcrum?
A) zero newtons
B) 100 N
C) 600 N
D) 800 N
E) 1000 N

Ans: E
Refer To: Ref 9-1
Difficulty: Easy
SectionDef: Section 9-2 and 9-3
13. Which one of the following statements most accurately describes the center of gravity of an
object?
A) It is the point where gravity acts on the object.
B) It is the point where all the mass is concentrated.
C) It must be experimentally determined for all objects.
D) It is the point on the object where all the weight is concentrated.
E) It is the point from which the torque produced by the weight of the object can be calculated.

Ans: E
Difficulty: Easy
SectionDef: Section 9-2 and 9-3

14. Consider four point masses located as shown in the sketch. The acceleration due to gravity
is the same everywhere. What is the x coordinate of the center of gravity for this system?

A) 2.0 m
B) 2.7 m
C) 3.0 m
D) 3.3 m
E) 3.8 m

Ans: C
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

15. Three objects are positioned along the x axis as follows: 4.4 kg at x = + 1.1 m, 3.7 kg at x =
–0.80 m, and 2.9 kg at x = –1.6 m. The acceleration due to gravity is the same everywhere.
What is the distance from the location of the center of gravity to the location of the center of
mass for this system?
A) zero meters
B) –0.52 m
C) –0.26 m
D) +0.26 m
E) +0.52 m

Ans: A
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

16. A 14-kg beam is hinged at one end. A 6.0-kg triangular object and a 7.5-kg I-shaped object
are positioned as shown. Dots indicate the individual centers of gravity of the beam and the two
objects. What is the distance from the axis of rotation to the center of gravity for this system?

A) 1.3 m
B) 1.1 m
C) 0.96 m
D) 0.89 m
E) 0.71 m

Ans: A
Difficulty: Medium
SectionDef: Section 9-2 and 9-3

17. Consider the following four objects: a hoop, a flat disk, a solid sphere, and a hollow sphere.
Each of the objects has mass M and radius R. The axis of rotation passes through the center of
each object, and is perpendicular to the plane of the hoop and the plane of the flat disk. Which
of these objects requires the largest torque to give it the same angular acceleration?
A) the solid sphere
B) the hollow sphere
C) the hoop
D) the flat disk
E) both the solid and the hollow spheres

Ans: C
Difficulty: Hard
SectionDef: Section 9-4

18. A 66-N  m torque acts on a wheel with a moment of inertia 175 kg  m2. If the wheel starts
from rest, how long will it take the wheel to make one revolution?
A) 0.37 s
B) 0.71 s
C) 2.4 s
D) 5.8 s
E) 6.9 s

Ans: D
Difficulty: Hard
SectionDef: Section 9-4

19. A string is wrapped around a pulley of radius 0.05 m and moment of inertia 0.2 kg  m2.
If the string is pulled with a force , the resulting angular acceleration of the pulley is 2 rad/s2.
Determine the magnitude of the force .

A) 0.4 N
B) 2N
C) 8N
D) 16 N
E) 40 N
Ans: C
Difficulty: Medium
SectionDef: Section 9-4

20. A massless frame in the shape of a square with 2-m sides has a 1-kg ball at each corner.
What is the moment of inertia of the four balls about an axis through the corner marked O and
perpendicular to the plane of the paper?

A) 4 kg  m2
B) 8 kg  m2
C) 10 kg  m2
D) 12 kg  m2
E) 16 kg  m2

Ans: E
Difficulty: Medium
SectionDef: Section 9-4

21. A certain merry-go-round is accelerated uniformly from rest and attains an angular speed of
1.2 rad/s in the first 18 seconds. If the net applied torque is 1200 N  m, what is the moment
of inertia of the merry-go-round?
A) 1400 kg  m2
B) 18 000 kg  m2
C) 9000 kg  m2
D) 500 kg  m2
E) This cannot be determined since the radius is not specified.

Ans: B
Difficulty: Hard
SectionDef: Section 9-4

22. The drawing shows the top view of a door that is 1.68 m wide. Two forces are applied to
the door as indicated. What is the magnitude of the net torque on the door with respect to the
hinge?

A) 0Nm
B) 4.4 N  m
C) 8.3 N  m
D) 9.1 N  m
E) 11 N  m

Ans: C
Difficulty: Medium
SectionDef: Section 9-4

23. Which one of the following statements concerning the moment of inertia I is false?
A) I may be expressed in units of kg  m2.
B) I depends on the angular acceleration of the object as it rotates.
C) I depends on the location of the rotation axis relative to the particles that make up the object.
D) I depends on the orientation of the rotation axis relative to the particles that make up the
object.
E) Of the particles that make up an object, the particle with the smallest mass may contribute
the greatest amount to I.

Ans: B
Difficulty: Medium
SectionDef: Section 9-4

24. Two uniform solid spheres, A and B have the same mass. The radius of sphere B is twice
that of sphere A. The axis of rotation passes through each sphere. Which one of the following
statements concerning the moments of inertia of these spheres is true?
A) The moment of inertia of A is one-fourth that of B.
B) The moment of inertia of A is one-half that of B.
C) The moment of inertia of A is 5/4 that of B.
D) The moment of inertia of A is 5/2 that of B.
E) The two spheres have equal moments of inertia.

Ans: A
Difficulty: Medium
SectionDef: Section 9-4

25. Three objects are attached to a massless rigid rod that has an axis of rotation as shown.
Assuming all of the mass of each object is located at the point shown for each, calculate the
moment of inertia of this system.

A) 1.3 kg  m2
B) 3.1 kg  m2
C) 5.3 kg  m2
D) 7.2 kg  m2
E) 9.1 kg  m2

Ans: E
Difficulty: Medium
SectionDef: Section 9-4

26. Three children are pulling on a rotatable platform on a playground. The platform has a
radius of 3.80 m. In the picture, two children are pulling with equal forces of 40.0 N in an
attempt to make the platform rotate clockwise. The third child applies a force of 60 N as shown.
What is the net torque on the platform? Note: “ccw” is counterclockwise and “cw” is
clockwise.

A) 76 N  m, ccw
B) 76 N  m, cw
C) 290 N  m, ccw
D) 290 N  m, cw
E) 0Nm

Ans: B
Difficulty: Medium
SectionDef: Section 9-4

27. A string is wrapped around a pulley of radius 0.20 m and moment of inertia 0.40 kg  m2.
The string is pulled with a force of 28 N. What is the magnitude of the resulting angular
acceleration of the pulley?
A) 22 rad/s2
B) 14 rad/s2
C) 68 rad/s2
D) 0.55 rad/s2
E) 8.0 rad/s2

Ans: B
Difficulty: Medium
SectionDef: Section 9-4

28. A 45-N brick is suspended by a light string from a 2.0-kg pulley. The brick is released
from rest and falls to the floor below as the pulley rotates through 5.0 rad. The pulley may be
considered a solid disk of radius 1.5 m. What is the angular speed of the pulley?

A) 17 rad/s
B) 15 rad/s
C) 9.4 rad/s
D) 8.1 rad/s
E) 7.3 rad/s

Ans: A
Difficulty: Hard
SectionDef: Section 9-4

29. A hollow cylinder of mass M and radius R rolls down an inclined plane. A block of mass
M easily slides down an identical inclined plane. Complete the following statement: If both
objects are released at the same time from the top of their inclined planes,
A) the cylinder will reach the bottom first.
B) the block will reach the bottom first.
C) the block will reach the bottom with the greater kinetic energy.
D) the cylinder will reach the bottom with the greater kinetic energy.
E) both the block and the cylinder will reach the bottom at the same time.

Ans: B
Difficulty: Easy
SectionDef: Section 9-5

30. A solid sphere and a hollow sphere each of mass M and radius R are released at the same
time from the top of an inclined plane. Which one of the following statements is necessarily
true?
A) The solid sphere will reach the bottom first.
B) The hollow sphere will reach the bottom first.
C) Both spheres will reach the bottom at the same time.
D) The solid sphere will reach the bottom with the greater kinetic energy.
E) The hollow sphere will reach the bottom with the greater kinetic energy.

Ans: A
Difficulty: Easy
SectionDef: Section 9-5

31. Consider the following three objects, each of the same mass and radius:
(1) a solid sphere (2) a solid disk (3) a hoop
All three are released from rest at the top of an inclined plane. The three objects proceed down
the incline undergoing rolling motion without slipping. In which order do the objects reach the
bottom of the incline?
A) 3, 1, 2
B) 2, 3, 1
C) 1, 2, 3
D) 3, 2, 1
E) All three reach the bottom at the same time.

Ans: C
Difficulty: Medium
SectionDef: Section 9-5
32. A rider on a motorcycle that has a mass of 195 kg is traveling with a speed of 24 m/s.
Each of the two wheels of the motorcycle has a radius of 0.29 m and a moment of inertia of 0.46
kg  m2. What is the approximate total rotational kinetic energy of the wheels?
A) 800 J
B) 1100 J
C) 1500 J
D) 3200 J
E) 4100 J

Ans: D
Difficulty: Medium
SectionDef: Section 9-5

33. A 1.0-kg wheel in the form of a solid disk rolls along a horizontal surface with a speed of
6.0 m/s. What is the total kinetic energy of the wheel?
A) 9.0 J
B) 18 J
C) 27 J
D) 36 J
E) 54 J

Ans: C
Difficulty: Medium
SectionDef: Section 9-5

34. A 2.0-kg solid cylinder of radius 0.5 m rotates at a rate of 40 rad/s about its cylindrical axis.
What power is required to bring the cylinder to rest in 10 s?
A) 20 W
B) 40 W
C) 160 W
D) 200 W
E) 400 W

Ans: A
Difficulty: Medium
SectionDef: Section 9-5

35. A solid cylinder of radius 0.35 m is released from rest from a height of 1.8 m and rolls down
the incline as shown. What is the angular speed of the cylinder when it reaches the horizontal
surface?

A) 8.2 rad/s
B) 14 rad/s
C) 34 rad/s
D) 67 rad/s
E) This cannot be determined because the mass is unknown.

Ans: B
Difficulty: Hard
SectionDef: Section 9-5

36. A solid sphere rolls without slipping along a horizontal surface. What percentage of its
total kinetic energy is rotational kinetic energy?
A) 33 %
B) 50 %
C) 12 %
D) 75 %
E) 29 %

Ans: E
Difficulty: Medium
SectionDef: Section 9-5

37. A hollow sphere of radius 0.25 m is rotating at 13 rad/s about an axis that passes through its
center. The mass of the sphere is 3.8 kg. Assuming a constant net torque is applied to the
sphere, how much work is required to bring the sphere to a stop?
A) 1.0 J
B) 3.8 J
C) 13 J
D) 25 J
E) 38 J

Ans: C
Difficulty: Medium
SectionDef: Section 9-5

38. A ceiling fan has five blades, each with a mass of 0.34 kg and a length of 0.66 m. The fan
is operating in its “low” setting at which the angular speed is 9.4 rad/s. If the blades can be
approximated as uniform thin rods that rotate about one end, what is the total rotational kinetic
energy of the five blades?
A) 35 J
B) 29 J
C) 23 J
D) 17 J
E) 11 J

Ans: E
Difficulty: Medium
SectionDef: Section 9-5

39. A solid cylinder with a mass m and radius r is mounted so that it can be rotated about an
axis that passes through the center of both ends. At what angular speed  must the cylinder
rotate to have the same total kinetic energy that it would have if it were moving horizontally with
a speed v without rotation?

A)

B)

C)

D)

E)

Ans: D
Difficulty: Hard
SectionDef: Section 9-5

40. A child standing on the edge of a freely spinning merry-go-round moves quickly to the
center. Which one of the following statements is necessarily true concerning this event and
why?
A) The angular speed of the system decreases because the moment of inertia of the system has
increased.
B) The angular speed of the system increases because the moment of inertia of the system has
increased.
C) The angular speed of the system decreases because the moment of inertia of the system has
decreased.
D) The angular speed of the system increases because the moment of inertia of the system has
decreased.
E) The angular speed of the system remains the same because the net torque on the merry-go-
round is zero N  m.

Ans: D
Difficulty: Easy
SectionDef: Section 9-6
41. What happens when a spinning ice skater draws in her outstretched arms?
A) Her angular momentum decreases.
B) Her angular momentum increases.
C) Her moment of inertia decreases causing her to speed up.
D) Her moment of inertia decreases causing her to slow down.
E) The torque that she exerts increases her moment of inertia.

Ans: C
Difficulty: Easy
SectionDef: Section 9-6

42. A spinning star begins to collapse under its own gravitational pull. Which one of the
following occurs as the star becomes smaller?
A) The star's angular velocity decreases.
B) The star's angular momentum remains constant.
C) The star's angular momentum increases.
D) The star's angular velocity remains constant.
E) Both the star's angular momentum and its angular velocity remain constant.

Ans: B
Difficulty: Easy
SectionDef: Section 9-6

43. A spinning skater draws in her outstretched arms thereby reducing her moment of inertia by
a factor of 2. Determine the ratio of her final kinetic energy to her initial kinetic energy.
A) 0.5
B) 1
C) 2
D) 4
E) 16

Ans: C
Difficulty: Medium
SectionDef: Section 9-6

44. A 1500-kg satellite orbits a planet in a circular orbit of radius 6.2 × 106 m. What is the
angular momentum of the satellite in its orbit around the planet if the satellite completes one
orbit every 1.5 × 104 s?
A) 3.9 × 106 kg  m2/s
B) 1.4 × 1014 kg  m2/s
C) 6.2 × 108 kg  m2/s
D) 8.1 × 1011 kg  m2/s
E) 2.4 × 1013 kg  m2/s

Ans: E
Difficulty: Hard
SectionDef: Section 9-6

45. A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the
position of her arms, the skater decreases her moment of inertia to one-half its initial value.
What is the skater's final angular speed?
A) 3.0 rad/s
B) 4.5 rad/s
C) 9.0 rad/s
D) 12 rad/s
E) 18 rad/s

Ans: D
Difficulty: Medium
SectionDef: Section 9-6

46. Two equal spheres, labeled A and B in the figure, are attached to a massless rod with a
frictionless pivot at the point P. The system is made to rotate clockwise with angular speed  on
a horizontal, frictionless tabletop. Sphere A collides with and sticks to another equal sphere that
is at rest on the tabletop. Note: the masses of all three spheres are equal.
What is the angular speed of the system immediately after the collision?

A) 
B) 0.82
C) 0.60
D) 0.56
E) 0.29

Ans: D
Difficulty: Medium
SectionDef: Section 9-6

47. Planets A and B are uniform solid spheres that rotate at a constant speed about axes through
their centers. Although B has twice the mass and three times the radius of A, each planet has
the same rotational kinetic energy. What is the ratio B/A of their angular speeds?
A) 0.055
B) 0.093
C) 0.165
D) 0.191
E) 0.236

Ans: E
Difficulty: Hard
SectionDef: Section 9-6

48. A solid sphere of radius R rotates about a diameter with an angular speed . The sphere
then collapses under the action of internal forces to a final radius R/2. What is the final angular
speed of the sphere?
A) /4
B) /2
C) 
D) 2
E) 4

Ans: E
Difficulty: Medium
SectionDef: Section 9-6

49. A ball of mass M moves in a circular path on a horizontal, frictionless surface. It is

attached to a light string that passes through a hole in the center of the table. If the string is
pulled down, thereby reducing the radius of the path of the ball, the speed of the ball is observed
to increase. Complete the following sentence: This occurs because

A) the linear momentum of the ball is conserved.

B) it is required by Newton's first law of motion.
C) the angular momentum of the ball is conserved.
D) the angular momentum of the ball must increase.
E) the total mechanical energy of the ball must remain constant.

Ans: C
Difficulty: Easy
SectionDef: Section 9-6

50. A 4.0-kg ball moves in a straight line at 8.0 m/s as shown in the figure. At the instant
shown, what is its angular momentum about the point P?
A) 33 kg  m2/s
B) 85 kg  m2/s
C) 110 kg  m2/s
D) 160 kg  m2/s
E) zero kg  m2/s

Ans: B
Difficulty: Hard
SectionDef: Section 9-6

Reference: Ref 9-2

Two skaters, each of mass 40 kg, approach each other along parallel paths that are separated by a
distance of 2 m. Both skaters have a speed of 10 m/s. The first skater carries a 2-m pole that
may be considered massless. As he passes the pole, the second skater catches hold of the end.
The two skaters then go around in a circle about the center of the pole.

51. What is the angular speed of the skaters after they have linked together?
A) 4 rad/s
B) 5 rad/s
C) 10 rad/s
D) 20 rad/s
E) 40 rad/s

Ans: C
Refer To: Ref 9-2
Difficulty: Hard
SectionDef: Section 9-6

52. What is their combined angular momentum about the center of the pole?
A) 2 kg  m2/s
B) 40 kg  m2/s
C) 80 kg  m2/s
D) 400 kg  m2/s
E) 800 kg  m2/s

Ans: E
Refer To: Ref 9-2
Difficulty: Medium
SectionDef: Section 9-6

Reference: Ref 9-3

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the
right with a constant linear speed of 6.0 m/s.

6 m/s
2.0 kg

53. Which one of the following statements is true concerning the angular momentum of this
hoop?
A) It points into the paper.
B) It points out of the paper.
C) It points to the left.
D) It points to the right.
E) It varies from point to point on the hoop.

Ans: A
Refer To: Ref 9-3
Difficulty: Easy
SectionDef: Section 9-6

54. What is the total kinetic energy of the hoop?

A) 36 J
B) 54 J
C) 72 J
D) 96 J
E) 140 J

Ans: C
Refer To: Ref 9-3
Difficulty: Medium
SectionDef: Section 9-6

55. A compact disc rotates about its center at constant angular speed. Which one of the
following quantities is constant and non-zero for a dust particle near the edge of the disc?
A) linear velocity
B) torque about the center of the disc
C) centripetal acceleration
D) angular momentum
E) angular acceleration

Ans: D
Difficulty: Medium
SectionDef: Additional Problems
56. A steady horizontal force of magnitude 21 N is applied at the axle of a solid disk as
shown. The disk has mass 2.0 kg and diameter 0.10 m. What is the linear speed of the center
of the disk after it has moved 12 m?

A) 9.0 m/s
B) 13 m/s
C) 16 m/s
D) 22 m/s
E) 44 m/s

Ans: B
Difficulty: Hard
SectionDef: Additional Problems

57. A uniform disk of radius 1.2 m and mass 0.60 kg is rotating at 25 rad/s around an axis that
passes through its center and is perpendicular to the disk. A rod makes contact with the rotating
disk with a force of 4.5 N at a point 0.75 m from the axis of rotation as shown. The disk is
brought to a stop in 5.0 s. What is the coefficient of kinetic friction for the two materials in
contact?

A) 0.22
B) 0.15
C) 0.64
D) 0.37
E) 0.55

Ans: C
Difficulty: Hard
SectionDef: Additional Problems