PSE9e ISM chapter11 final tủ tài liệu training

11
Angular Momentum
CHAPTER OUTLINE
11.1

The Vector Product and Torque

11.2

Analysis Model: Nonisolated System (Angular Momentum)

11.3

Angular Momentum of a Rotating Rigid Object

11.4

Analysis Model: Isolated System (Angular Momentum)

11.5

The Motion of Gyroscopes and Tops

* An asterisk indicates a question or problem new to this edition.

ANSWERS TO OBJECTIVE QUESTIONS
OQ11.1

Answer (b). Her angular momentum stays constant as I is cut in half
1
and ω doubles. Then Iω 2 doubles.

2

OQ11.2

The angular momentum of the mouse-turntable system is initially
zero, with both at rest. The frictionless axle isolates the mouseturntable system from outside torques, so its angular momentum
must stay constant with the value of zero.
(i)

Answer (a). The mouse makes some progress north, or
counterclockwise.

(ii)

Answer (b). The turntable will rotate clockwise. The turntable
rotates in the direction opposite to the motion of the mouse, for
the angular momentum of the system to remain zero.

(iii) No. Mechanical energy changes as the mouse converts some
chemical into mechanical energy, positive for the motions of
both the mouse and the turntable.
(iv) No. Linear momentum is not conserved. The turntable has zero
momentum while the mouse has a bit of northward momentum.
Initially, momentum is zero; later, when the mouse moves
584
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Chapter 11

585

north, the fixed axle prevents the turntable from moving south.
(v)

Yes. Angular momentum is constant, with the value of zero.

OQ11.3

(i) Answer (a), (ii) Answer (e), (3 m, down) × (2 N, toward you) =
6 N · m, left

OQ11.4

Answer c = e > b = d > a = 0. The unit vectors have magnitude 1, so
the magnitude of each cross product is |1 · 1 · sin θ| where θ is the
angle between the factors. Thus for (a) the magnitude of the cross
product is sin 0° = 0. For (b), |sin 135°| = 0.707, (c) sin 90° = 1, (d) sin
45° = 0.707, (e) sin 90° = 1.

OQ11.5

(a) No. (b) No. An axis of rotation must be defined to calculate the
torque acting on an object. The moment arm of each force is
measured from the axis, so the value of the torque depends on the
location of the axis.

OQ11.6

(i)

( )

Answer (e). Down–cross–left is away from you: − ˆj × − ˆi = − kˆ ,
as in the first picture.

(ii)

( )

Answer (d). Left–cross–down is toward you: − ˆi × − ˆj = kˆ , as in
the second picture.

ANS FIG. OQ11.6
OQ11.7

(i)

Answer (a). The angular momentum is constant. The moment of
inertia decreases, so the angular speed must increase.

(ii)

No. Mechanical energy increases. The ponies must do work to
push themselves inward.

(iii) Yes. Momentum stays constant, with the value of zero.
(iv) Yes. Angular momentum is constant with a nonzero value. No
outside torque can influence rotation about the vertical axle.
OQ11.8

Answer (d). As long as no net external force, or torque, acts on the
system, the linear and angular momentum of the system are
constant.

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586

Angular Momentum

ANSWERS TO CONCEPTUAL QUESTIONS
CQ11.1

The star is isolated from any outside torques, so its angular
momentum is conserved as it changes size. As the radius of the star
decreases, its moment of inertia decreases, resulting in its angular
speed increasing.

CQ11.2

The suitcase might contain a spinning gyroscope. If the gyroscope is
spinning about an axis that is oriented horizontally passing through
the bellhop, the force he applies to turn the corner results in a torque
that could make the suitcase swing away. If the bellhop turns quickly
enough, anything at all could be in the suitcase and need not be
rotating. Since the suitcase is massive, it will tend to follow an
inertial path. This could be perceived as the suitcase swinging away
by the bellhop.

CQ11.3

The long pole has a large moment of inertia about an axis along the
rope. An unbalanced torque will then produce only a small angular
acceleration of the performer-pole system, to extend the time
available for getting back in balance. To keep the center of mass
above the rope, the performer can shift the pole left or right, instead
of having to bend his body around. The pole sags down at the ends
to lower the system’s center of gravity.

CQ11.4

(a)

Frictional torque arises from kinetic friction between the inside
of the roll and the child’s fingers. As with all friction, the
magnitude of the friction depends on the normal force between
the surfaces in contact. As the roll unravels, the weight of the
roll decreases, leading to a decrease in the frictional force, and,
therefore, a decrease in the torque.

(b)

As the radius R of the paper roll shrinks, the roll’s angular
v
speed ω =
must increase because the speed v is constant.
R

(c)

If we think of the roll as a uniform disk, then its moment of
1
inertia is I = MR 2 . But the roll’s mass is proportional to its
2
base area π R 2 ; therefore, the moment of inertia is proportional
to R4. The moment of inertia decreases as the roll shrinks. When
the roll is given a sudden jerk, its angular acceleration may not
be great enough to set the roll moving in step with the paper, so
the paper breaks. The roll is most likely to break when its radius
is large, when its moment of inertia is large, than when its
radius is small, when its moment of inertia is small.

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Chapter 11
CQ11.5

587

Work done by a torque results in a change in rotational kinetic
energy about an axis. Work done by a force results in a change in
translational kinetic energy. Work by either has the same units:

W = FΔx = [ N ][ m ] = N ⋅ m = J
W = τ Δθ = [ N ⋅ m ][ rad ] = N ⋅ m = J
CQ11.6

Suppose we look at the motorcycle moving to the right. Its drive
wheel is turning clockwise. The wheel speeds up when it leaves the
ground. No outside torque about its center of mass acts on the
airborne cycle, so its angular momentum is conserved. As the drive
wheel’s clockwise angular momentum increases, the frame of the
cycle acquires counterclockwise angular momentum. The cycle’s
front end moves up and its back end moves down.

CQ11.7

Its angular momentum about that axis is constant in time. You
cannot conclude anything about the magnitude of the angular
momentum.

CQ11.8

No. The angular momentum about any axis that does not lie along
the instantaneous line of motion of the ball is nonzero.

CQ11.9

The Earth is an isolated system, so its angular momentum is
conserved when the distribution of its mass changes. When its mass
moves away from the axis of rotation, its moment of inertia increases,
its angular speed decreases, so its period increases. Most of the mass
of Earth would not move, so the effect would be small: we would not
have more hours in a day, but more nanoseconds.

CQ11.10

As the cat falls, angular momentum must be conserved. Thus, if the
upper half of the body twists in one direction, something must get an
equal angular momentum in the opposite direction. Rotating the
lower half of the body in the opposite direction satisfies the law of
conservation of angular momentum.

CQ11.11

Energy bar charts are useful representations for keeping track of the
various types of energy storage in a system: translational and
rotational kinetic energy, various types of potential energy, and
internal energy. However, there is only one type of angular
momentum. Therefore, there is no need for bar charts when
analyzing a physical situation in terms of angular momentum.

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588

Angular Momentum

SOLUTIONS TO END-OF-CHAPTER PROBLEMS
Section 11.1

The Vector Product and Torque

P11.1

ˆi ˆj kˆ

 
ˆ
M × N = 2 −3 1 = ˆi(6 − 5) − ˆj(−4 − 4) + k(10
+ 12) = ˆi + 8.00ˆj + 22.0kˆ
4 5 −2

P11.2

(a)

 
area = A × B = ABsin θ = ( 42.0 cm ) ( 23.0 cm ) sin ( 65.0° − 15.0° )
= 740 cm 2

(b)

The longer diagonal is equal to the sum of the two vectors.
 
A + B = [(42.0 cm)cos15.0° + (23.0 cm)cos65.0°]ˆi

+[(42.0 cm)]sin 15.0° + (23.0 cm)sin 65.0°]ˆj

P11.3

 
A + B = ( 50.3 cm ) ˆi + ( 31.7 cm ) ˆj
 
2
2
length = A + B = ( 50.3 cm ) + ( 31.7 cm ) = 59.5 cm



We take the cross product of each term of A with each term of B,
using the cross-product multiplication table for unit vectors. Then we
use the identification of the magnitude of the cross product as
AB sin θ to find θ . We assume the data are known to three significant
digits.
(a)

We use the definition of the cross product and note that
ˆi × ˆi = ˆj × ˆj = 0:
 
A × B = 1ˆi + 2 ˆj × 2 ˆi + 3ˆj
 
A × B = 2 ˆi × ˆi + 3ˆi × ˆj − 4ˆj × ˆi + 6ˆj × ˆj

(

(b)

) (

)

= 0 + 3kˆ − 4 ( −kˆ ) + 0 = 7.00kˆ
 
Since A × B = ABsin θ , we have
 
⎛ A×B ⎞
⎞

7
−1 ⎛
θ = sin ⎜
= 60.3°
⎟ = sin ⎜ 2
⎝ 1 + 2 2 2 2 + 32 ⎟⎠
⎝ AB ⎠
−1

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Chapter 11
P11.4

589

ˆi × ˆi = 1 ⋅ 1 ⋅ sin 0° = 0
ˆj × ˆj and kˆ × kˆ are zero
similarly since the vectors
being multiplied are parallel.

ˆi × ˆj = 1 ⋅ 1 ⋅ sin 90° = 1
ANS. FIG. P11.4
P11.5

We first resolve all of
the forces shown in
Figure P11.5 into
components parallel to

and perpendicular to
the beam as shown in
ANS. FIG. P11.5.
(a)

ANS. FIG. P11.5

The torque about an axis
through point O is given by

τ O = + [( 25 N ) cos 30°⎤⎦ ( 2.0 m )

− [( 10 N ) sin 20°]( 4.0 m ) = +30 N ⋅m

or
(b)

τ 0 = 30 N ⋅ m counterclockwise

The torque about an axis through point C is given by

τ C = + ⎡⎣( 30 N ) sin 45° ⎤⎦ (2.0 m)

− ⎡⎣( 10 N ) sin 20° ⎤⎦ (2.0 m) = + 36 N ⋅ m

or
P11.6

τ C = 36 N ⋅ m counterclockwise

 
A ⋅ B = −3.00 ( 6.00 ) + 7.00 ( −10.0 ) + ( −4.00 ) ( 9.00 )
= −124
AB =

(a)

( −3.00)2 + (7.00)2 + ( −4.00)2 ⋅ (6.00)2 + ( −10.0)2 + ( 9.00)2

= 127
 
⎛
⎞
−1 A ⋅ B
cos ⎜
= cos −1 ( −0.979 ) = 168°
⎟
⎝ AB ⎠

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590

Angular Momentum

(b)

ˆi
 

A × B = −3.00

ˆj
kˆ
7.00 −4.00 = 23.0ˆi + 3.00ˆj − 12.0kˆ

6.00 −10.0

9.00

 
2
2
2
A × B = ( 23.0 ) + ( 3.00 ) + ( −12.0 ) = 26.1
 
⎛ A×B ⎞
−1
sin −1 ⎜
⎟ = sin ( 0.206 ) = 11.9° or 168°
⎜⎝ AB ⎟⎠
(c)

P11.7

Only the first method gives the angle between the vectors

unambiguously because sin(180° – θ ) = sin θ but cos (180° – θ )
= – cos θ ; in other words, the vectors can only be at most 180°
apart and using the second method cannot distinguish θ from

180° – θ.
 
 
We are given the condition A × B = A ⋅ B.
This says that ABsin θ = ABcosθ
so

tan θ = 1

θ = 45.0° satisfies this condition.
P11.8

(a)

The torque acting on the particle about the origin is

ˆi ˆj kˆ

 
τ = r × F = 4 6 0 = ˆi ( 0 − 0 ) − ˆj( 0 − 0 ) + kˆ ( 8 − 18 )
3 2 0
= ( −10.0 N ⋅ m ) kˆ
(b)

Yes. The point or axis must be on the other side of the line of
action of the force, and half as far from this line along which
the force acts. Then the lever arm of the force about this new
axis will be half as large and the force will produce counterclockwise instead of clockwise torque.

(c)

Yes. There are infinitely many such points, along a line that
passes through the point described in (b) and parallel the
line of action of the force.

(d)

Yes, at the intersection of the line described in (c) and the y axis.

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Chapter 11
(e)

(f)

591

No, because there is only one point of intersection of the line
described in (d) with the y axis.
Let (0, y) represent the coordinates of the special axis of rotation
located on the y axis of Cartesian coordinates. Then the
displacement from this point to the particle feeling the force is

rnew = 4ˆi + (6 − y)ˆj in meters. The torque of the force about this
new axis is

ˆi
ˆj

kˆ



τ new = rnew × F = 4 6 − y 0
3

2

0

= ˆi ( 0 − 0 ) − ˆj( 0 − 0 ) + kˆ ( 8 − 18 + 3y )
= ( +5 N ⋅ m ) kˆ
Then,

8 − 18 + 3y = 5

→

3y = 15

→

y=5

The position vector of the new axis is 5.00ˆj m .
P11.9

(a)

The lever arms of the forces about O are all the same, equal to
length OD, L.




If F3 has a magnitude F3 = F1 + F2 , the net torque is zero:

∑ τ = F1L + F2 L − F3 L = F1L + F2 L − ( F1 + F2 ) L = 0

(b)


The torque produced by F3 depends on the perpendicular


distance OD, therefore translating the point of application of F3 to

any other point along BC will not change the net torque .

ANS. FIG. P11.9
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592
P11.10

Angular Momentum
(a)
(b)

No.

The cross-product vector must be perpendicular to both of the
factors, so its dot product with either factor must be zero. To
check:

(2ˆi − 3ˆj + 4kˆ ) ⋅ ( 4ˆi + 3ˆj − kˆ ) = (ˆi ⋅ ˆi ) 8 + −9(ˆj ⋅ ˆj) − 4(kˆ ⋅ kˆ )
= 8 − 9 − 4 = −5
The answer is not zero.

No. The cross product could not work out that way.

Section 11.2
P11.11

Analysis Model: Nonisolated System
(Angular Momentum)

Taking the geometric center of the compound object to be the pivot,
the angular speed and the moment of inertia are

ω = v/r = (5.00 m/s)/0.500 m = 10.0 rad/s
and

I = ∑ mr 2 = ( 4.00 kg )( 0.500 m ) + ( 3.00 kg )( 0.500 m )
2

2

= 1.75 kg · m 2
By the right-hand rule, we find that the angular velocity is directed out
of the plane. So the object’s angular momentum, with magnitude

)

L = I ω = ( 1.75 kg ⋅ m 2 (10.0 rad/s)
is the vector

L = ( 17.5 kg ⋅ m 2 /s ) kˆ

ANS. FIG. P11.11

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Chapter 11
P11.12

  
We use L = r × p:

L = 1.50ˆi + 2.20ˆj m × ( 1.50 kg ) 4.20ˆi − 3.60ˆj m/s

L = −8.10kˆ − 13.9kˆ kg ⋅ m 2 /s = ( −22.0 kg ⋅ m 2 /s ) kˆ

(
(

P11.13

593

)

  
We use L = r × p:

L=

ˆi
x
mvx

(

)

)

kˆ
0 = ˆi ( 0 − 0 ) − ˆj ( 0 − 0 ) + kˆ mxvy − myvx
0

ˆj
y
mvy

(

)


L = m xvy − yvx kˆ

(

P11.14

)

Whether we think of the Earth’s surface as curved or flat, we interpret
the problem to mean that the plane’s line of flight extended is precisely
tangent to the mountain at its peak, and nearly parallel to the wheat
field. Let the positive x direction be eastward, positive y be northward,
and positive z be vertically upward.

(a) r = ( 4.30 km ) kˆ = ( 4.30 × 103 m ) kˆ

(

)



p = mv = ( 12 000 kg ) −175ˆi m/s = −2.10 × 106 ˆi kg ⋅ m/s
  
L = r × p = 4.30 × 103 kˆ m × −2.10 × 106 ˆi kg ⋅ m/s

(

=
(b)

No.

( −9.03 × 10

9

) (

kg ⋅ m 2 /s ) ˆj

)

 
L = r p sin θ = mv ( r sin θ ) , and r sin θ is the altitude of the

plane. Therefore, L = constant as the plane moves in level flight
with constant velocity.

P11.15

(c)

Zero. The position vector from Pike’s Peak to the plane is anti-

(a)

parallel to the velocity of the plane. That is, it is directed along the
same line and opposite in direction. Thus, L = mvr sin 180° = 0.
  

Zero because L = r × p and r = 0.

(b)

At the highest point of the trajectory,

1
vi2 sin 2θ
and
x= R=
2
2g

y = hmax

2
vi sin θ )
(
=

2g

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594

Angular Momentum
The angular momentum is then



L1 = r1 × mv 1
2
⎡ v 2 sin 2θ
vi sin θ ) ˆ ⎤
(
i
ˆ
=⎢
i+
j ⎥ × mvxi ˆi
2g
2g
⎢⎣
⎥⎦

−mvi3 sin θ 2 cos θ ˆ
=
k
2g

ANS. FIG. P11.15
(c)



v 2 sin 2θ vi2 ( 2 sin θ cos θ )

L 2 = Rˆi × mv 2 , where R = i
=
g
g

(

= mRˆi × vi cos θ ˆi − vi sin θ ˆj

)

−2mvi3 sin θ sin θ ˆ
= −mRvi sin θ kˆ =
k
g

(d)

The downward force of gravity exerts a torque
in the − z direction.

P11.16

We start with the particle under a net force model
in the x and y directions:

mv 2
r

∑ Fx = max :

T sin θ =

∑ Fy = may :

T cosθ = mg

So

sin θ v 2
=
cosθ rg

then

L = rmv sin 90.0° = rm rg

and v = rg

sin θ
cosθ

ANS. FIG. P11.16

sin θ
sin θ
= m2 gr 3
cosθ

cosθ

and since r = sin θ ,
L=

m2 g3

sin 4 θ
cosθ

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Chapter 11
P11.17

595

The angular displacement of the particle around the circle is
vt
θ = ωt = .
R
The vector from the center of the circle to the mass is then
R cosθ ˆi + R sin θ ˆj, where R is measured from the +x axis.

The vector from point P to the mass is

r = Rˆi + R cos θ ˆi + R sin θ ˆj

⎡⎛


⎛ vt ⎞ ⎞
⎛ vt ⎞ ⎤
r = R ⎢⎜ 1 + cos ⎜ ⎟ ⎟ ˆi + sin ⎜ ⎟ ˆj ⎥
⎝ R ⎠⎠
⎝ R⎠ ⎦
⎣⎝
The velocity is


 dr
⎛ vt ⎞
⎛ vt ⎞
v=
= −v sin ⎜ ⎟ ˆi + v cos ⎜ ⎟ ˆj
⎝
⎠
⎝ R⎠
dt
R

So

 

L = r × mv

L = mvR ⎡⎣( 1 + cos ω t ) ˆi + sin ω tˆj ⎤⎦ × ⎡⎣ − sin ω tˆi + cos ω tˆj ⎤⎦

⎡

⎤
⎛ vt ⎞
L = mvR ⎢ cos ⎜ ⎟ + 1⎥ kˆ
⎝
⎠
R
⎣
⎦
P11.18

(a)

The net torque on the counterweight-cord-spool system is
 
τ = r × F = Rmg sin θ

τ = 8.00 × 10−2 m ( 4.00 kg ) ( 9.80 m/s 2 ) sin 90.0° = 3.14 N ⋅ m
(b)



L = ∑ r × mi v i = Rmv + RMv = R ( m + M ) v
i

L = ( 0.080 0 m )( 4.00 kg + 2.00 kg ) v = (0.480 kg ⋅ m)v

(c)
P11.19

τ=

dL
= ( 0.480 kg ⋅ m ) a
dt

→

(

a=

3.14 N ⋅ m
= 6.53 m/s 2
0.480 kg ⋅ m

)


Differentiating r = 6.00ˆi + 5.00tˆj m with respect to time gives

 dr
v=
= 5.00ˆj m/s
dt


so
p = mv = ( 2.00 kg ) 5.00ˆj m/s = 10.0ˆj kg ⋅ m/s

(

)

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596

Angular Momentum
ˆi
  
L = r × p = 6.00

and

0

P11.20

(a)

∫


r

0


dr =

∫

t

0



vdt = r − 0 =

ˆj
5.00t
10.0
t

kˆ
0 =
0

(60.0 kg ⋅ m /s ) kˆ

∫ (6t ˆi + 2tˆj)dt =
2

0

2



r = ( 6t 3 /3 ) ˆi + ( 2t 2 /2 ) ˆj

= 2t 3 ˆi + t 2 ˆj in meters, where t is in seconds.

(b)

The particle starts from rest at the origin, starts moving into
the first quadrant, and gains speed faster and faster while
turning to move more and more nearly parallel to the x axis.

(c)



a = (dv/dt) = (d/dt)(6t 2 ˆi + 2t ˆj) = (12t ˆi + 2 ˆj) m/s 2

(d)



F = ma = (5 kg)(12t ˆi + 2 ˆj) m/s 2 = (60t ˆi + 10 ˆj) N

(e)

  
τ = r × F = (2t 3 ˆi + t 2 ˆj) × (60tˆi + 10ˆj) = 20t 3 kˆ − 60t 3 kˆ
= −40t 3 kˆ N ⋅ m

(f)

 

ˆ
L = r × mv = (5 kg)(2t 3 ˆi + t 2 ˆj) × (6t 2 ˆi + 2tˆj) = 5(4t 4 kˆ − 6t 4 k)
= −10t 4 kˆ kg ⋅ m 2 /s

(g)

K=

1   1
mv ⋅ v = (5 kg)(6t 2 ˆi + 2tˆj)⋅(6t 2 ˆi + 2tˆj) = (2.5)(36t 4 + 4t 2 )
2
2

= (90t 4 + 10t 2 ) J

(h)

P = (d/dt)(90t 4 + 10t 2 ) J = (360t 3 + 20t) W , all where t is in
seconds.

P11.21

(a)

The vector from P to the falling ball is

1
  

r = ri + v it + at 2
2

⎛1
⎞
r =  cos θ ˆi +  sin θ ˆj + 0 − ⎜ gt 2 ⎟ ˆj
⎝2
⎠

(

The velocity of the ball is
  
v = v i + at = 0 − gtˆj
 

So
L = r × mv

)

ANS. FIG. P11.21

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Chapter 11


⎡

⎛1
⎞ ⎤
L = m ⎢  cos θ ˆi +  sin θ ˆj + 0 − ⎜ gt 2 ⎟ ˆj ⎥ × − gtˆj
⎝2
⎠ ⎦
⎣

L = −mg t cos θ kˆ

(

(

)

The Earth exerts a gravitational torque on the projectile in the
negative z direction.

(b)

(c)

)

597

Differentiating with respect to time, we have −mg cos θ kˆ for
the rate of change of angular momentum, which is also the torque
due to the gravitational force on the ball.

Section 11.3
P11.22

Angular Momentum of a Rotating Rigid Object

The moment of inertia of the sphere about an axis through its center is
I=

2
2
2
MR 2 = ( 15.0 kg ) ( 0.500 m ) = 1.50 kg ⋅ m 2
5
5

Therefore, the magnitude of the angular momentum is

L = Iω = ( 1.50 kg ⋅ m 2 ) ( 3.00 rad/s ) = 4.50 kg ⋅ m 2 /s
Since the sphere rotates counterclockwise about the vertical axis, the
angular momentum vector is directed upward in the +z direction.
Thus,


L = ( 4.50 kg ⋅ m 2 /s ) kˆ
P11.23

The total angular momentum about the center point is given by

L = I hω h + I mω m
For the hour hand:

mh L2h 60.0 kg ( 2.70 m )
Ih =
=
= 146 kg ⋅ m 2
3
3

For the minute hand:

mm L2m 100 kg ( 4.50 m )
Im =
=
= 675 kg ⋅ m 2
3
3

In addition,

ωh =

2

2

while

ωm =

2π rad ⎛ 1 h ⎞

−4
⎜⎝
⎟⎠ = 1.45 × 10 rad/s
12 h 3 600 s

2π rad ⎛ 1 h ⎞
−3
⎜⎝
⎟⎠ = 1.75 × 10 rad/s
1h
3600 s

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598

Angular Momentum
Thus,

L = ( 146 kg ⋅ m 2 ) ( 1.45 × 10−4 rad s )

+ ( 675 kg ⋅ m 2 ) ( 1.75 × 10−3 rad/s )

or L = 1.20 kg ⋅ m 2 /s . The hands turn clockwise, so their vector
angular momentum is perpendicularly into the clock face.
P11.24

We begin with
K=

1 2
Iω
2

And multiply the right-hand side by

I
:
I

1 2 1 I 2ω 2
K = Iω =
2
2 I
Substituting L = Iω then gives
K=

P11.25

(a)

1 2 1 I 2ω 2
L2
Iω =
=
2
2 I
2I

For an axis of rotation passing through the center of mass, the
magnitude of the angular momentum is given by

1
2
⎛1
⎞
L = Iω = ⎜ MR 2 ⎟ ω = ( 3.00 kg )( 0.200 m ) ( 6.00 rad/s )
⎝2
⎠
2
= 0.360 kg ⋅ m 2 /s
(b)

For a point midway between the center and the rim, we use the
parallel-axis theorem to find the moment of inertia about this
point. Then,
2
⎡1
⎛ R⎞ ⎤
2
L = Iω = ⎢ MR + M ⎜ ⎟ ⎥ ω
⎝ 2⎠ ⎦
⎣2
3
2
= ( 3.00 kg ) ( 0.200 m ) ( 6.00 rad/s ) = 0.540 kg ⋅ m 2 /s
4

P11.26

(a)

Modeling the Earth as a sphere, we first calculate its moment of
inertia about its rotation axis.
2
2
2
MR 2 = ( 5.98 × 1024 kg ) ( 6.37 × 106 m )
5
5
37
= 9.71 × 10 kg ⋅ m 2

I=

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Chapter 11

599

Completing one rotation in one day, Earth’s rotational angular
speed is

ω=

1 rev 2π rad
=

= 7.27 × 105 s −1
24 h 86 400 s

the rotational angular momentum of the Earth is then

L = Iω = ( 9.71 × 1037 kg ⋅ m 2 ) ( 7.27 × 105 s −1 )
= 7.06 × 1033 kg ⋅ m 2 /s
The Earth turns toward the east, counterclockwise as seen from
above north, so the vector angular momentum points north along
the Earth’s axis, towards the north celestial pole or nearly toward
the star Polaris.
(b)

In this case, we model the Earth as a particle, with moment of
inertia
I = MR 2 = ( 5.98 × 1024 kg ) ( 1.496 × 1011 m )

2

= 1.34 × 10 47 kg ⋅ m 2

Completing one orbit in one year, Earth’s orbital angular speed is

ω=

1 rev
2π rad
=
= 1.99 × 10−7 s −1
365.25 d ( 365.25 d ) ( 86 400 s/d )

the angular momentum of the Earth is then

L = Iω = ( 1.34 × 10 47 kg ⋅ m 2 ) ( 1.99 × 10−7 s −1 )
= 2.66 × 10 40 kg ⋅ m 2 /s
The Earth plods around the Sun, counterclockwise as seen from
above north, so the vector angular momentum points north
perpendicular to the plane of the ecliptic,
toward the north ecliptic pole or 23.5° away from Polaris, toward
the center of the circle that the north celestial pole moves in as the
equinoxes precess. The north ecliptic pole is in the constellation
Draco.
(c)

The periods differ only by a factor of 365 (365 days for orbital
motion to 1 day for rotation). Because of the huge distance
from the Earth to the Sun, however, the moment of inertia of
the Earth around the Sun is six orders of magnitude larger
than that of the Earth about its axis.

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600
P11.27

Angular Momentum
Defining the distance from the pivot to the particle as d, we first find
the rotational inertia of the system for each case, from the information
M = 0.100 kg, m = 0.400 kg, and D = 1.00 m.

(a)

For the meterstick rotated about its center, I m =

1
MD2 .
12

1
For the additional particle, I w = md 2 = m ⎛⎜ D2 ⎞⎟ .
⎝2 ⎠
Together, I = I m + I w =

1
1
MD2 + mD2 , or
12
4

(0.100 kg)(1.00 m)2 (0.400 kg)(1.00 m)2
I=
+
= 0.108 kg ⋅ m 2
12
4
And the angular momentum is
L = Iω = ( 0.108 kg ⋅ m 2 ) ( 4.00 rad/s ) = 0.433 kg ⋅ m 2 /s

(b)

For a stick rotated about a point at one end,
1
1
2
I m = mD2 = ( 0.100 kg )( 1.00 m ) = 0.033 3 kg ⋅ m 2
3
3

For a point mass, Iw = mD2 = (0.400 kg)(1.00 m)2 = 0.400 kg · m2
so together they have rotational inertia
I = Im + Iw = 0.433 kg · m2
and angular momentum
L = Iω = ( 0.433 kg ⋅ m 2 ) (4.00 rad/s) = 1.73 kg ⋅ m 2 /s

P11.28

We assume that the normal force n = 0 on the
front wheel. On the bicycle,

∑ Fx = max :
∑ Fy = may :

+ f s = max
+ n − Fg = 0 → n = mg

We must use the center of mass as the axis in

∑ τ = Iα :
Fg (0) − n (77.5 cm) + fs (88 cm) = 0

ANS. FIG. P11.28

We combine the equations by substitution:

−mg ( 77.5 cm ) + max ( 88 cm ) = 0

( 9.80 m/s ) 77.5 cm =
=
2

ax

88 cm

8.63 m/s 2

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Chapter 11

P11.29

We require ac = g =

ω=

g
=
r

601

v2
= ω 2 r:
r

( 9.80 m/s ) = 0.313 rad/s
2

100 m

I = Mr = ( 5 × 10 4 kg ) ( 100 m ) = 5 × 108 kg ⋅ m 2
2

2

(a)

L = Iω = ( 5 × 108 kg ⋅ m 2 ) ( 0.313 rad/s ) = 1.57 × 108 kg ⋅ m 2 /s

(b)

Δt =

Section 11.4
P11.30

(a)

Lf − 0

∑τ

=

1.57 × 108 kg ⋅ m 2 /s
= 6.26 × 103 s = 1.74 h
2 ( 125 N )( 100 m )

Analysis Model: Isolated System
(Angular Momentum)
From conservation of angular momentum for the isolated system
of two disks:

( I1 + I 2 )ω f

= I1ω i

or

ωf =

I1
ωi
I1 + I 2

This is an example of a totally inelastic collision.
(b)

Kf =

1
( I1 + I2 )ω 2f
2

Ki =

and

1
I1ω i2
2

1
( I1 + I 2 ) ⎛ I1 ⎞ 2
I1
2
so
=
ωi⎟ =
⎜
1
Ki
I1 + I 2
I1ω i2 ⎝ I1 + I 2 ⎠
2
From conservation of angular momentum,
Kf

P11.31

I iω i = I f ω f : ( 250 kg ⋅ m 2 ) ( 10.0 rev/min ) =

⎡ 250 kg ⋅ m 2 + ( 25.0 kg ) ( 2.00 m )2 ⎤ ω 2
⎣
⎦

ω 2 = 7.14 rev/min
P11.32

(a)

Angular momentum is conserved in the puck-rod-putty system
because there is no net external torque acting on the system.

Iω initial = Iω final :
⎛ vf ⎞
⎛v ⎞
mR 2 ⎜ i ⎟ + mp R 2 (0) = mR 2 + mp R 2 ⎜ ⎟
⎝ R⎠
⎝ R⎠

(

)

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

602

Angular Momentum

(

)

mRvi = m + mp Rv f
Solving for the final velocity gives

⎛ m ⎞
⎛
⎞
2.40 kg
vf = ⎜
vi = ⎜
( 5.00 m/s ) = 3.24 m/s
⎟
⎝ 2.40 kg + 1.30 kg ⎟⎠
⎝ m + mp ⎠
Then,

T=

P11.33

2π R 2π ( 1.50 m )
=
= 2.91 s

vf
3.24 m/s

(b)

Yes, because there is no net external torque acting on the puckrod-putty system.

(c)

No, because the pivot pin is always pulling on the rod to change
the direction of the momentum.

(d)

No. Some mechanical energy is converted into internal energy.
The collision is perfectly inelastic.

(a)

Mechanical energy is not constant; some chemical potential
energy in the woman’s body is transformed into mechanical
energy.

(b)

Momentum is not constant. The turntable bearing exerts an
external northward force on the axle to prevent the axle from
moving southward because of the northward motion of the
woman.

(c)

Angular momentum is constant because the system is isolated
from torque about the axle.

(d) From conservation of angular momentum for the system of the
woman and the turntable, we have Lf = Li = 0,
so,

L f = I womanω woman + I tableω table = 0

⎛ I
⎞
⎛ m
r2 ⎞ ⎛ v
⎞
and ω table = ⎜ − woman ⎟ ω woman = ⎜ − woman ⎟ ⎜ woman ⎟
I table ⎠ ⎝ r ⎠
⎝ I table ⎠
⎝
=−

mwoman rvwoman
I table

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Chapter 11

ω table = −
or
(e)

603

60.0 kg ( 2.00 m ) ( 1.50 m/s )
= −0.360 rad/s
500 kg ⋅ m 2

ω table = 0.360 rad/s ( counterclockwise )

Chemical energy converted into mechanical energy is equal to
ΔK = K f − 0 =

ΔK =

1
1 2
2
mwoman vwoman
+ Iω table
2
2

1
1
2
2
60 kg ) ( 1.50 m/s ) + ( 500 kg ⋅ m 2 ) ( 0.360 rad/s )

(
2
2

= 99.9 J

P11.34

(a)

The total angular momentum of the system of the student, the
stool, and the weights about the axis of rotation is given by

I total = I weights + I student = 2 ( mr 2 ) + 3.00 kg ⋅ m 2
Before:

r = 1.00 m

Thus,

Ii = 2(3.00 kg)(1.00 m)2 + 3.00 kg · m2 = 9.00 kg · m2

After:

r = 0.300 m

Thus,

If = 2(3.00 kg)(0.300 m)2 + 3.00 kg · m2 = 3.54 kg · m2

We now use conservation of angular momentum.

I f ω f = I iω i

(b)

P11.35

(a)

or

⎛I ⎞
⎛ 9.00 ⎞
ω f = ⎜ i ⎟ ωi = ⎜
( 0.750 rad/s ) = 1.91 rad/s
⎝ 3.54 ⎟⎠
⎝ If ⎠

Ki =

1
1
2
I iω i2 = ( 9.00 kg ⋅ m 2 ) ( 0.750 rad/s ) = 2.53 J
2
2

Kf =

1
1
2
I f ω 2f = ( 3.54 kg ⋅ m 2 ) ( 1.91 rad/s ) = 6.44 J
2
2

We solve by using conservation of angular momentum for the
turntable-clay system, which is isolated from outside torques:

Iω initial = Iω final :
1
⎛1
⎞
mR 2ω i = ⎜ mR 2 + mc r 2 ⎟ ω f
⎝2
⎠
2

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


604

Angular Momentum
Solving for the final angular velocity gives

ωf =

1

mR 2ω i
2

1
2
30.0 kg )( 1.90 m ) ( 4π rad/s )
(
2

=
1
1
mR 2 + mc r 2
( 30.0 kg )(1.90 m )2 + ( 2.25 kg )(1.80 m )2
2
2
= 11.1 rad/s counterclockwise

(b)

No. The initial energy is

1 2 1⎛ 1
⎞
Iω i = ⎜ mR 2 ⎟ ω i2
⎝
⎠
2
2 2
1 1

2
2
= ⎡⎢ ( 30.0 kg )( 1.90 m ) ⎤⎥ ( 4π rad/s )
2 ⎣2
⎦
= 4 276 J

Ki =

The final mechanical energy is

1 2 1⎛ 1
⎞
Iω f = ⎜ mR 2 + mc r 2 ⎟ ω 2f
⎠
2
2⎝ 2
1 1
2
2
= ⎡⎢ ( 30.0 kg )( 1.90 m ) + ( 2.25 kg )( 1.80 m ) ⎤⎥
2 ⎣2
⎦

Kf =

× ( 11.1 rad/s )

2

= 3 768 J
Thus 507 J of mechanical energy is transformed into internal
energy. The “angular collision” is completely inelastic.
(c)

P11.36

No. The original horizontal momentum is zero. As soon as the
clay has stopped skidding on the turntable, the final momentum
is (2.25 kg)(1.80 m)(11.1 rad/s) = 44.9 kg · m/s north. This is the
amount of impulse injected by the bearing. The bearing thereafter
keeps changing the system momentum to change the direction of
the motion of the clay. The turntable bearing promptly imparts
an impulse of 44.9 kg · m/s north into the turntable-clay system,
and thereafter keeps changing the system momentum.

When they touch, the center of mass is distant from the center of the
larger puck by
y CM =

(a)

0 + ( 80.0 g ) ( 4.00 cm + 6.00 cm )
120 g + 80.0 g

= 4.00 cm

L = r1m1 v1 + r2 m2 v2 = 0 + ( 6.00 × 10−2 m ) ( 80.0 × 10−3 kg ) ( 1.50 m/s )
= 7.20 × 10−3 kg ⋅ m 2 /s

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Chapter 11
(b)

605

The moment of inertia about the CM is

⎛1
⎞ ⎛1
⎞
I = ⎜ m1r12 + m1d12 ⎟ + ⎜ m2 r22 + m2 d22 ⎟
⎝2
⎠ ⎝2
⎠
2
2
1
0.120 kg ) ( 6.00 × 10−2 m ) + ( 0.120 kg ) ( 4.00 × 10−2 )
(
2
2
1
+ ( 80.0 × 10−3 kg ) ( 4.00 × 10−2 m )
2

I=

+ ( 80.0 × 10−3 kg ) ( 6.00 × 10−2 m )

2

I = 7.60 × 10−4 kg ⋅ m 2
Angular momentum of the two-puck system is conserved:
L = Iω

L 7.20 × 10−3 kg ⋅ m 2 /s
=
= 9.47 rad/s
I
7.60 × 10−4 kg ⋅ m 2

Taking the origin at the pivot point, note that r is perpendicular

to v, so sin θ = 1 and L f = Li = mr sin θ = mv vertically down.

ω=

P11.37

(a)
(b)

Taking vf to be the speed of the bullet and the block together, we
first apply conservation of angular momentum: Li = Lf becomes

⎛ m ⎞
mv =  ( m + M ) v f or v f = ⎜

v
⎝ m + M ⎟⎠
The total kinetic energies before and after the collision are,
respectively,
Ki =

1 2
mv
2
2

2
1
1
⎛ m ⎞ 2 1⎛ m ⎞ 2
2
v = ⎜
v
and K f = ( m + M ) v f = ( m + M ) ⎜
⎝ m + M ⎟⎠
2
2
2 ⎝ m + M ⎟⎠

So the fraction of the kinetic energy that is converted into internal
energy will be

1 2 1 ⎛ m2 ⎞ 2
mv − ⎜
v

2 ⎝ m + M ⎟⎠
−ΔK K i − K f 2
M
Fraction =
=
=
=
1 2
Ki
Ki
m+ M
mv
2

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


606

Angular Momentum

ANS. FIG. P11.37
P11.38

(a)

Let ω be the angular speed of the signboard
when it is vertical.
1 2
Iω = Mgh

2
1⎛ 1
1
2⎞
2
⎜⎝ ML ⎟⎠ ω = Mg L ( 1 − cosθ )
2 3
2

ω=
=

3g ( 1 − cosθ )
L
3 ( 9.80 m/s 2 )( 1 − cos 25.0° )

ANS. FIG. P11.38

0.500 m

= 2.35 rad/s
(b)

I iω i − mvL = I f ω f represents angular momentum conservation for
the sign-snowball system. Substituting into the above equation,

1
⎛1
2
2⎞

2
⎜⎝ ML + mL ⎟⎠ ω f = ML ω i − mvL
3
3
Solving,
1
MLω i − mv
3
ωf =
⎛ 1 M + m⎞ L
⎝3
⎠
1
( 2.40 kg )( 0.500 m )( 2.347 rad/s ) − ( 0.400 kg )(1.60 m/s )
=3
⎡ 1 2.40 kg + 0.400 kg ⎤ ( 0.500 m )
)
⎢⎣ 3 (
⎥⎦
= 0.498 rad/s

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Chapter 11
(c)

607

Let hCM = distance of center of mass from the axis of rotation.

hCM =

( 2.40 kg )( 0.250 m ) + ( 0.400 kg )( 0.500 m ) = 0.285 7 m
2.40 kg + 0.400 kg

Applying conservation of mechanical energy,

1 1
( M + m) ghCM ( 1 − cosθ ) = ⎛⎜⎝ ML2 + mL2 ⎞⎟⎠ ω 2
2 3

Solving for θ then gives
⎡ ⎛1
⎤
M + m⎞ L2ω 2 ⎥
⎢
⎝
⎠
3
θ = cos −1 ⎢1 −
⎥
2 ( M + m) ghCM ⎥
⎢
⎢⎣
⎥⎦
⎧ ⎡1
2 ⎫
2
2.40 kg ) + 0.400 kg ⎤⎥ ( 0.500 m ) ( 0.498 rad/s ) ⎪
(

⎪
⎢
⎪
⎪
3
⎦
= cos −1 ⎨1 − ⎣
⎬
2
2 ( 2.40 kg + 0.400 kg ) ( 9.80 m/s )( 0.285 7 m ) ⎪
⎪
⎪⎩
⎪⎭
= 5.58°

P11.39

(a)

Consider the system to consist of
the wad of clay and the cylinder.
No external forces acting on this
system have a torque about the
center of the cylinder. Thus,
angular momentum of the system
is conserved about the axis of the
cylinder.

Iω = mvi d

Lf = Li:
or

ANS. FIG. P11.39

⎡1
2
2⎤
⎢⎣ 2 MR + mR ⎥⎦ ω = mvi d

Thus,

ω=

2mvi d
( M + 2m) R 2

(b)

No; some mechanical energy of the system (the kinetic energy
of the clay) changes into internal energy.

(c)

The linear momentum of the system is not constant. The axle
exerts a backward force on the cylinder when the clay strikes.

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

608
P11.40

Angular Momentum
The rotation rate of the station is such that at its rim the centripetal
acceleration, ac, is equal to the acceleration of gravity on the Earth’s
surface, g. Thus, the normal force from the rim’s floor provides
centripetal force on any person equal to that person’s weight:

∑ Fr = mar :

n=

mv 2
g
→ mg = mω i2 r → ω i2 =
r
r

The space station is isolated, so its angular momentum is conserved.
When the people move to the center, the station’s moment of inertia
decreases, its angular speed increases, and the effective value of
gravity increases.
From angular momentum conservation: I iω i = I f ω f →

ωf
ωi

=

Ii
, where
If

I i = I station + I people, i
2
= ⎡⎣ 5.00 × 108 kg ⋅ m 2 + 150 ( 65.0 kg ) ( 100 m ) ⎤⎦
= 5.98 × 108 kg ⋅ m 2

I f = I station + I people, f
2
= ⎡⎣ 5.00 × 108 kg ⋅ m 2 + 50 ( 65.0 kg ) ( 100 m ) ⎤⎦
= 5.32 × 108 kg ⋅ m 2

The centripetal acceleration is the effective value of gravity: ac ∝ g.
Comparing values of acceleration before and during the union
meeting, we have
gf
gi

=

ac , f
ac , i

2

2
⎛ Ii ⎞
⎛ 5.98 × 108 ⎞

= 2 =⎜ ⎟ =⎜
= 1.26 → g f = 1.26g i
ωi ⎝ I f ⎠
⎝ 5.32 × 108 ⎟⎠

ω 2f

When the people move to the center, the angular speed of the
station increases. This increases the effective gravity by 26%.
Therefore, the ball will not take the same amount of time to drop.
P11.41

(a)

Yes , the bullet has angular momentum about an axis through
the hinges of the door before the collision.

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.