Ch 9 Rotational Dynamics

These are our class notes from Chapter 9 of Cutnell and Johnson's Physics

1. Rotational Dynamics
Chapters 9

2. Learning Objectives
Objectives for this chapter fall under applications of
previous objectives

3. Table Of Contents
1.The Action of Forces and Torques on Rigid
Objects
2.Rigid Objects in Equilibrium
3.Center of Gravity
4.Newton’s Second Law for rotational Motion
About a Fixed Axis
5.Rotational Work and Energy
6.Angular Momentum

4. Chapter 9:
Rotational Dynamics
Section 1:
The Action of Forces and Torques
on Rigid Objects

5. In pure translational motion, all points on an
object travel on parallel paths.
The most general motion is a combination of
translation and rotation.
Translation vs. Rotation

6. What causes an object to have an
angular acceleration?
According to Newton’s
second law, a net force
causes an object to have
an acceleration.
TORQUE
The amount of torque
depends on where and in
what direction the force is
applied, as well as the
location of the axis of
rotation.

7. Magnitude of Torque = (Magnitude of the force) x (Lever arm)
´=Ft
Direction: The torque is positive when the force tends to produce a
counterclockwise rotation about the axis.
SI Unit of Torque: newton x meter (N·m)
DEFINITION OF TORQUE

8. Example 2 The Achilles Tendon
The tendon exerts a force of magnitude
790 N. Determine the torque (magnitude
and direction) of this force about the
ankle joint.
´=Ft
m106.3
55cos
2-
´
=


( )( )
mN 15
55cosm106.3N 720
2
×=
´=
- 
t

9. 9.1.1. You are using a wrench in an attempt to
loosen a nut by applying a force as shown. But
this fails to loosen the nut. Which one of the
following choices is most likely to loosen
this tough nut?
•Tie a rope of length 2L to the wrench at the same location and apply the same
force as shown.
•Place a pipe of length 2L over the handle of the wrench and apply the same force
to the opposite end (farthest from the nut).
c) Double the force.
o)Doubling the length or doubling the force will have the same result, but doubling
the length is easier.
e) Continue applying the same force as in the drawing and eventually the nut will
loosen.

10. 9.1.2. A 1.5-kg ball is tied to the end of a string. The ball is then
swung at a constant angular velocity of 4p rad/s in a horizontal
circle of radius 2.0 m. What is the torque on the stone?
a) 18 Nm
b) 29 Nm
c) 36 Nm
d) 59 Nm
e) zero Nm

11. 9.1.3. A 1.0-m long steel bar is suspended from a rope from the ceiling as shown.
The rope is attached to the bar at its mid-point. A force directed at an
angle q is applied at one end. At the other end, a force is applied
perpendicular to the bar. If the magnitudes of the two forces are equal, for
which one of the following values of the angle q will the net torque on the
bar have the smallest magnitude? The net torque is the sum of the torques on
the bar.
a) 0°
b) 90°
c) 135°
d) 180°
e) 270°
1F
uur
2F
uur
1F
uur
2F
uur

12. 9.1.4. An interesting method for exercising a dog is to have it walk on
the rough surface of a circular platform that freely rotates about its
center as shown. When the dog begins walking near the outer
edge of the platform as shown, how will the platform move, if at
all? Assume the bearing on which the platform can rotate is
frictionless.
a) When the dog walks, the platform will rotate counterclockwise
when viewed from above.
b) When the dog walks, the platform will rotate clockwise when
viewed from above.
c) When the dog walks, the platform will not rotate.

13. 9.1.5. When using pruning shears, such as the pair shown,
to cut a branch from a tree, it is better to insert the branch
closer to the hinge than near the end of the shears. Which
one of the following statements best explains the reason
this observation is true?
a) The torque acting on the branch is smallest near the hinge.
b) The torque acting on the branch is largest near the hinge.
c) The torque exerted on the shears yields the greatest force on the branch near the
hinge.
d) The long handles determine the force exerted on the branch, which is the same
no matter where on the shears the branch is placed.
e) The same torque is exerted on the shears and the branch, regardless of the force
applied to the handles.

14. 9.1.6. An object with a triangular cross-section is free to rotate about the axis
represented by the black dot shown. Four forces with identical
magnitudes are exerted on the object. Which one of the forces, if any,
exerts the largest torque on the object?
a) 1
b) 2
c) 3
d) 4
e) The same torque is exerted by each force.

15. Chapter 9:
Rotational Dynamics
Section 2:
Rigid Objects in Equilibrium

16. If a rigid body is in equilibrium, neither its linear motion nor its
rotational motion changes.
0=åxF å=0
yF
å=0t
0==
yx
aa 0=a
Rigid Objects in Equilibrium

17. A rigid body is in equilibrium if it has zero translational
acceleration and zero angular acceleration. In equilibrium,
the sum of the externally applied forces is zero, and the
sum of the externally applied torques is zero.
å=0tå=0
yF0=åxF
EQUILIBRIUM OF A RIGID BODY

18. Reasoning Strategy
Select the object to which the equations for equilibrium are to
be applied.
Draw a free-body diagram that shows all of the external
forces acting on the object.
Choose a convenient set of x, y axes and resolve all forces
into components that lie along these axes.
Apply the equations that specify the balance of forces at
equilibrium.
(Set the net force in the x and y directions equal to zero.)
6.Select a convenient axis of rotation. Set the sum of the
torques about this axis equal to zero.
7.Solve the equations for the desired unknown quantities.

19. Example 3 A Diving Board
A woman whose weight is 530 N is poised at
the right end of a diving board with length 3.90
m. The board has negligible weight and is
supported by a fulcrum 1.40 m away from the
left end. Find the forces that the bolt and the
fulcrum exert on the board.
0
22 =-=å WWF t
( )( )
m 1.40
m 90.3N 530
2
=F
2
2


W
W
F= 0
21 =-+-=å WFFF
y
0N 530N 1480
1 =-+-F
N 950
1=F
N 1480=

20. The arm is horizontal
and weighs 31.0 N. The
deltoid muscle can
supply 1840 N of force.
What is the weight of the
heaviest dumbbell he
can hold?
Example 5 Bodybuilding
0=+--=å Mddaa MWW t
( )

 0.13sinm 150.0=
M
d
Maa
d
MW
W

+-
=
( )( )( )( )
N 1.86
m 620.0
0.13sinm 150.0N 1840m 280.0N 0.31
=
+-
=

dW

21. 9.2.1. At the circus, a clown balances a step ladder on his forehead.
A few people in the audience notice that he is continually moving to
keep the ladder from falling off his forehead. Why is this movement
necessary?
f)The clown is trying to apply a torque to the ladder in the
direction opposite to other torques on the ladder.
b) The clown is trying to keep the center of mass of the ladder directly above his
head so that the torque due to the gravitational force is zero Nm.
c) By rocking the ladder on his forehead, the ladder will be more stable than if it
were stationary. This is similar to riding a bicycle. You can easily balance a
bicycle when it’s rolling, but not when it’s stationary.
d) This movement is not necessary. The clown is trying to make this look harder
than it really is for entertainment value. The ladder will easily balance on the
clown’s forehead.

22. 9.2.2. In the seventeenth century, French mathematician Gilles de Roberval
developed a balance, shown in part A in the figure, for commercial weighing;
and it is still in use today. A variation of this device, shown part B of the
figure, is used for physics demonstrations. In this case, the two triangular
objects have equal mass and rest on the two horizontal arms at an equal distance
from the vertical bars. When the system is released, there is no movement
because the system is in equilibrium. One of the objects is then slid to the right
as shown in part C, what will happen when the system is released?
a) The arm on the right
will go up.
b) The arm on the left
will go up.
c) Neither arm will move.

23. 9.2.3. Consider the three situations shown in the figure. Three forces act
on the triangular object in different ways. Two of the forces have
magnitude F and one of the forces has a magnitude 2F. In which
case(s), if any, will the object be in equilibrium? In each case, the
forces may act at the center of gravity or at the center of a corner.
a) A only
b) B only
c) C only
d) A and C
e) A and B

24. 9.2.4. A 4.0-m board is resting directly on top of a 4.0-m long table.
The weight of the board is 340 N. An object with a weight of
170 N is placed at the right end of the board. What is the
maximum horizontal distance that the board can be moved
toward the right such that the board remains in equilibrium?
a) 0.75 m
b) 1.0 m
c) 1.3 m
d) 1.5 m
e) 2.0 m

25. 9.2.5. Jack is moving to a new apartment, so he has loaded a hand truck with four
boxes: box A is full of books and weighs 133 N, box B has more books and
weighs 111 N, box C contains his music collection on CDs and weighs 65 N,
and box D contains clothes and weighs 47 N. The height of each box is 0.30 m.
The center of gravity of each of the boxes is located at its center. In preparing
to pull the hand truck up the ramp of the moving truck he rotates it to the
position shown. What is the magnitude of the force that Jack is applying to the
hand truck at a distance of 1.4 m from the axel of the wheel?
a) 360 N
b) 200 N
c) 150 N
d) 96 N
e) 69 N
F
ur

26. 9.2.6. A pair of fuzzy dice is hanging from the rearview mirror of a sports
car. As the car accelerates smoothly, the strings of the dice are tilted
slightly toward the rear of the car. From the perspective of the driver,
which one of the following statements is true, if the dice are stationary?
a) The dice are in static equilibrium.
b) The dice are not in equilibrium because the torque on the dice is not zero.
c) The dice are in equilibrium, but not static equilibrium.
d) The dice are not in equilibrium because the linear momentum of the dice
is not zero.
e) None of the above statements are true.

27. 9.2.7. Which one of the following pictures best represents the forces
that prevent the ladder from slipping while someone is standing on
it?

28. Chapter 9:
Rotational Dynamics
Section 3:
Center of Gravity

29. The center of gravity of a rigid
body is the point at which
its weight can be considered
to act when the torque due
to the weight is being calculated.
DEFINITION OF CENTER OF GRAVITY

30. When an object has a symmetrical shape and its weight is distributed
uniformly, the center of gravity lies at its geometrical center.
CENTER OF GRAVITY

31. CENTER OF GRAVITY


++
++
=
21
2211
WW
xWxW
x
cg

32. Example 6 The Center of Gravity of an Arm
The horizontal arm is composed
of three parts: the upper arm (17 N),
the lower arm (11 N), and the hand
(4.2 N).
Find the center of gravity of the
arm relative to the shoulder joint.


++
++
=
21
2211
WW
xWxW
x
cg
( )( )( )( )( )( )
N 2.4N 11N 17
m 61.0N 2.4m 38.0N 11m 13.0N 17
++
++
=
cg
x m 28.0=

33. Conceptual Example 7 Overloading a Cargo Plane
This accident occurred because the plane was overloaded toward
the rear. How did a shift in the center of gravity of the plane cause
the accident?

34. Finding the center of gravity of an
irregular shape

35. 9.3.1. Six identical bricks are stacked on top of one another.
Note that the vertical dashed line indicates that the left edge
of the top brick is located to the right of the right side of the
bottom brick. Is the equilibrium configuration shown
possible, why or why not?
a) Yes, this is possible as long as the combined center of gravity of the blocks
above a given brick does not extend beyond the right side of the brick below.
b) Yes, this is possible as long as the left side of each block is directly above the
center of gravity of the brick directly below it.
c) Yes, this is possible as long as the center of gravity of the blocks above a given
brick remains directly above the center of gravity of the blocks below that brick.
d) No, this is not possible because the center of gravity of the top two blocks
extends beyond the right edge of the bottom two blocks.
e) No, because the center of gravity of the top block is to the right of the third block
from the top.

36. 9.3.2. Consider the diamond-shaped object shown that is designed to balance
on a thin thread like a tight rope walker at a circus. At the bottom of the
diamond, there is a narrow notch that is as wide as the thickness of the
thread. The mass of each of the metal spheres at the ends of the wires
connected to the diamond is equal to the mass of the diamond. Which
one of the points indicated is the most likely location of the center of
gravity for this object?
a) A
b) B
c) C
d) D
e) E

37. 9.3.3. Consider the object shown. A bottle is inserted into a board that
has a hole in it. The bottle and board are then set up on the table
and are in equilibrium. Which of the points indicated is the most
likely location for the center of mass for the bottle and board
system?
a) A
b) B
c) C
d) D
e) E

38. Chapter 9:
Rotational Dynamics
Section 4:
Newton’s Second Law for Rotational Motion About
a Fixed Axis

39. displacement
velocity
elapsed time
acceleration
Dx
v
t
a
Dq
w
t
a
inertia m I
Cause “a/a” F t
Translation vs. Rotation

40. TTmaF=
ara
T
=
rF
T´=t
atI=
Moment of Inertia, I=kmr
2
k depends on shape and axis
Newton’s Second Law for Rotational Motion About
a Fixed Axis
a
t
mr
r
=

41. ROTATIONAL ANALOG OF NEWTON’S SECOND LAW
FOR
A RIGID BODY ROTATING ABOUT A FIXED AXIS
÷
÷
ø
ö
ç
ç
è
æ
´
÷
÷
ø
ö
ç
ç
è
æ
=
onaccelerati
Angular
inertia
ofMoment
torqueexternalNet
atI=å
( )å=
2
mrIRequirement: Angular acceleration
must be expressed in radians/s
2
.

42. Example 9 The Moment of Inertial Depends on Where
the Axis Is.
Two particles each have mass and are fixed at the ends of a
thin rigid rod. The length of the rod is L. Find the moment of
inertia when this object rotates relative to an axis that is
perpendicular to the rod at (a) one end and (b) the center.

43. (b)
( )
2
22
2
11
2
rmrmmrI +==å
22
21
LrLr ==mmm ==
21
2
2
1
mLI=
(a)
( )
2
22
2
11
2
rmrmmrI +==å
Lrr ==
21
0mmm ==
21
2
mLI=
() ()
22
0 LmmI +=
( ) ( )
22
22 LmLmI +=

44. Common values of I
Slender
Rod
Rectangular
Plane
Sphere
Cylinder
axis
through
center
axis
through
center
thin-walled
hollow
hollow
thin-walled
hollow
axis
through
end
axis
along
edge
solid
solid

45. Example 12 Hoisting a Crate
The combined moment of inertia of the dual pulley is 50.0 kg·m
2
. The
crate weighs 4420 N. A tension of 2150 N is maintained in the cable
attached to the motor. Find the angular acceleration of the dual
pulley.

46. at ITT =-=å 2211 
yy mamgTF =-
¢
=å 2
massless rope, assume equal
y
mamgT +=
2
a
2=
ya
( )aImamgT
y
=+-
211

( )aa ImmgT =+-
2211

2
srad3.6=
2
2
211


mI
mgT
+
-
=a
( )( )( )( )( )
( )( )
22
2
m 200.0kg 451mkg 46.0
m 200.0sm80.9kg 451m 600.0N 2150
+×
-
=a

47. 9.4.1. Two solid disks, which are free to rotate independently about the same axis
that passes through their centers and perpendicular to their faces, are initially at
rest. The two disks have the same mass, but one of has a radius R and the other
has a radius 2R. A force of magnitude F is applied to the edge of the larger
radius disk and it begins rotating. What force must be applied to the edge of the
smaller disk so that the angular acceleration is the same as that for the larger
disk? Express your answer in terms of the force F applied to the larger disk.
a) 0.25F
b) 0.50F
c) F
d) 1.5F
e) 2F

48. 9.4.2. The corner of a rectangular piece of wood is attached to a rod that
is free to rotate as shown. The length of the longer side of the
rectangle is 4.0 m, which is twice the length of the shorter side.
Two equal forces with magnitudes of 22 N are applied to two of the
corners. What is the magnitude of the net torque on the block and
direction of rotation, if any?
a) 44 Nm, clockwise
b) 44 Nm, counterclockwise
c) 88 Nm, clockwise
d) 88 Nm, counterclockwise
e) zero Nm, no rotation
F
ur
F
ur

49. 9.4.3. 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) 1, 2, 3
b) 2, 3, 1
c) 3, 1, 2
d) 3, 2, 1
e) All three reach the bottom at the same time.

50. 9.4.4. A long board is free to rotate about the pivot shown in each of
the four configurations shown. Weights are hung from the board
as indicated. In which of the configurations, if any, is the net
torque about the pivot axis the largest?
a) 1
b) 2
c) 3
d) 4
e) The net torque is the same for
all four situations.

51. 9.4.5. The drawing shows a yo-yo in contact with a tabletop. A string
is wrapped around the central axle. How will the yo-yo behave if
you pull on the string with the force shown?
a) The yo-yo will roll to the left.
b) The yo-yo will roll to the right.
c) The yo-yo will spin in place, but not roll.
d) The yo-yo will not roll, but it will move to the left.
e) The yo-yo will not roll, but it will move to the right.

52. Chapter 9:
Rotational Dynamics
Section 5:
Rotational Work and Energy

53. sFW ×=
qrs=
rF´=t
tq=W
Rotational Work
qFr=

54. The rotational work done by a constant torque in
turning an object through an angle is
tq=
RW
Requirement: The angle must
be expressed in radians.
SI Unit of Rotational Work: joule (J)
DEFINITION OF ROTATIONAL WORK

55. ( )å=
22
2
1
wmrKE
2
2
1
T
mvKE=
wrv
T
=
Rotational Energy
22
2
1
wmr=
( )
22
2
1
wå= mr
2
2
1
wI=

56. 2
2
1
wIKE
R=
The rotational kinetic energy of a rigid rotating object is
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)
DEFINITION OF ROTATIONAL KINETIC
ENERGY

57. Example 13 Rolling Cylinders
A thin-walled hollow cylinder (mass = m
h
, radius = r
h
) and
a solid cylinder (mass = m
s
, radius = r
s
) start from rest at
the top of an incline.
Determine which cylinder
has the greatest translational
speed upon reaching the
bottom.

58. mghImvE
total ++=
2
2
12
2
1
w
iiifff mghImvmghImv ++=++
2
2
12
2
12
2
12
2
1
ww
iff mghImv =+
2
2
12
2
1
w
ENERGY CONSERVATION
rv
ff=w
iff mghrvImv =+
22
2
12
2
1
2
2
rIm
mgh
v
o
f
+
=
The cylinder with the smaller moment of inertia will
have a greater final translational speed.

59. 9.5.1. Four objects start from rest and roll without slipping down a ramp.
The objects are a solid sphere, a hollow cylinder, a solid cylinder,
and a hollow sphere. Each of the objects has the same radius and the
same mass, but they are made from different materials. Which
object will have the greatest speed at the bottom of the ramp?
a) Since they are all starting from rest, all of the objects will have the
same speed at the bottom as a result of the conservation of
mechanical energy.
b) solid sphere
c) hollow cylinder
d) solid cylinder
e) hollow sphere

60. 9.5.2. A bowling ball is rolling without slipping at constant speed
toward the pins on a lane. What percentage of the ball’s total
kinetic energy is translational kinetic energy?
a) 50 %
b) 71 %
c) 46 %
d) 29 %
e) 33 %

61. 9.5.3. A hollow cylinder is rotating about an axis that passes through the center of
both ends. The radius of the cylinder is r. At what angular speed w must the
this 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)
2
2
v
r
w=
2
v
r
w=
v
r
w=
2
v
r
w=
2
2
v
r
w=

62. 9.5.4. Two solid cylinders are rotating about an axis that passes
through the center of both ends of each cylinder. Cylinder A has
three times the mass and twice the radius of cylinder B, but they
have the same rotational kinetic energy. What is the ratio of the
angular velocities, w
A
/w
B
, for these two cylinders?
a) 0.25
b) 0.50
c) 1.0
d) 2.0
e) 4.0

63. 9.5.5. Consider the drawing. A rope is wrapped around one-third of the
circumference of a solid disk of radius R = 2.2 m that is free to rotate about an
axis that passes through its center. The force applied to the rope has a
magnitude of 35 N; and the disk has a mass M of 7.5 kg. Assuming the force is
applied horizontally as shown and the disk is initially at rest, determine the
amount of rotational work done until the time when the end of the rope reaches
the top of the disk?
a) 140 N
b) 160 N
c) 180 N
d) 210 N
e) 250 N

64. Chapter 9:
Rotational Dynamics
Section 6:
Angular Momentum

65. 9.6 Angular Momentum
DEFINITION OF ANGULAR MOMENTUM
The angular momentum L of a body rotating about a
fixed axis is the product of the body’s moment of
inertia and its angular velocity with respect to that
axis:
wIL=
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Angular Momentum: kg·m
2
/s

66. Translation vs Rotation
displacement
velocity
elapsed time
acceleration
Dx
v
t
a
Dq
w
t
a
inertia m I
Cause “a/a” F t
momentum p L

67. PRINCIPLE OF CONSERVATION OFANGULAR MOMENTUM
The angular momentum of a system remains constant (is
conserved) if the net external torque acting on the system
is zero.

68. Conceptual Example 14 A Spinning Skater
An ice skater is spinning with both
arms and a leg outstretched. She
pulls her arms and leg inward and
her spinning motion changes
dramatically.
Use the principle of conservation
of angular momentum to explain
how and why her spinning motion
changes.

69. Example 15 A Satellite in an Elliptical Orbit
An artificial satellite is placed in an
elliptical orbit about the earth. Its point
of closest approach is 8.37x10
6
m
from the center of the earth, and
its point of greatest distance is
25.1x10
6
m from the center of
the earth.
The speed of the satellite at the
perigee is 8450 m/s. Find the speed
at the apogee.

70. wIL=
angular momentum conservation
PPAA II ww=
rv=w
P
P
P
A
A
A
r
v
mr
r
v
mr
22
=
PPAA vrvr=
A
PP
A
r
vr
v=
( )( )
m 1025.1
sm8450m 1037.8
6
6
´
´
= sm2820=
2
mrI=

71. 9.6.1. A star is rotating about an axis that passes through its center.
When the star “dies,” the balance between the inward pressure due to
the force of gravity and the outward pressure from nuclear processes
is no longer present and the star collapses inward; and its radius
decreases with time. Which one of the following choices best
describes what happens as the star collapses?
a) The angular velocity of the star remains constant.
b) The angular momentum of the star remains constant.
c) The angular velocity of the star decreases.
d) The angular momentum of the star decreases.
e) Both angular momentum and angular velocity increase.

72. 9.6.2. A solid sphere of radius R rotates about an axis that is
tangent to the sphere with an angular speed w. Under the
action of internal forces, the radius of the sphere increases to
2R. What is the final angular speed of the sphere?
a) w/4
b) w/2
c) w
d) 2w
e) 4w

73. 9.6.3. While excavating the tomb of Tutankhamen (d. 1325 BC),
archeologists found a sling made of linen. The sling could hold a stone
in a pouch, which could then be whirled in a horizontal circle. The stone
could then be thrown for hunting or used in battle. Imagine the sling
held a 0.050-kg stone; and it was whirled at a radius of 1.2 m with an
angular speed of 2.0 rev/s. What was the angular momentum of the
stone under these circumstances?
a) 0.14 kg × m
2
/s
b) 0.90 kg × m
2
/s
c) 1.2 kg × m
2
/s
d) 2.4 kg × m
2
/s
e) 3.6 kg × m
2
/s

74. 9.6.4. Joe has volunteered to help out in his physics class by sitting on a stool that easily
rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a
sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close
to his body and he rotates faster. Why does his speed increase?
a) By bringing the dumbbells inward, Joe exerts a torque on the stool.
b) By bringing the dumbbells inward, Joe decreases the moment of inertia.
c) By bringing the dumbbells inward, Joe increases the angular momentum.
d) By bringing the dumbbells inward, Joe increases the moment of inertia.
e) By bringing the dumbbells inward, Joe decreases the angular momentum.

75. 9.6.5. Joe has volunteered to help out in his physics class by sitting on
a stool that easily rotates. Joe holds the dumbbells out as shown as
the stool rotates. Then, Joe drops both dumbbells. How does the
rotational speed of stool change, if at all?
a) The rotational speed increases.
b) The rotational speed decreases,
but Joe continues to rotate.
c) The rotational speed remains
the same.
d) The rotational speed quickly
decreases to zero rad/s.

76. 9.6.6. Joe has volunteered to help out in his physics class by sitting on a stool that easily
rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both
dumbbells. Then, the angular momentum of Joe and the stool changes, but the angular
velocity does not change. Which of the following choice offers the best explanation?
a) The force exerted by the dumbbells acts in
opposite direction to the torque.
b) Angular momentum is conserved, when no
external forces are acting.
c) Even though the angular momentum decreases,
the moment of inertia also decreases.
d) The decrease in the angular momentum is balanced
by an increase in the moment of inertia.
e) The angular velocity must increase when the
dumbbells are dropped.

77. 9.6.7. Sarah has volunteered to help out in her physics class by sitting on a stool
that easily rotates. The drawing below shows the view from above her head.
She holds the dumbbells out as shown as the stool rotates. Then, she drops both
dumbbells. Which one of the four trajectories illustrated best represents the
motion of the dumbbells after they are dropped?

78. 9.6.8. Two ice skaters are holding hands and spinning around their combined center
of mass, represented by the small black dot in Frame 1, with an angular
momentum L. When the skaters are at the position shown in Frame 2, they
release hands and move
in opposite directions as
shown in Frame 3.
What is the angular
momentum of the
skaters in Frame 3?
a) zero kg × m
2
/s
•a value that is greater than zero kg × m
2
/s, but less than L
•a value less than L and decreasing as they move further apart
d) a value that is greater than L
e) L