Chapter 8pages 869–870

1. The rotational velocity of a merry-go-round is increased at a constant rate from 1.5 rad/s to 3.5 rad/s in a time of 9.5 s.What is the rotational acceleration of the merry-go-round?

� � �

� 0.21 rad/s2

2. A record player’s needle is 6.5 cm from thecenter of a 45-rpm record. What is thevelocity of the needle?

First convert rpm to rad/s.

� �� �� � � 4.71 rad/s

v � r� � (6.5 cm)� �(4.71 rad/s)

� 0.31 m/s

3. Suppose a baseball rolls 3.2 m across thefloor. If the ball’s angular displacement is 82 rad, what is the circumference of the ball?

d � r�

r � � � 0.039 m

c � 2�r � 2�(0.039 m) � 0.25 m

4. A painter uses a 25.8-cm long screwdriver topry the lid off of a can of paint. If a force of85 N is applied to move the screwdriver 60.0°from the perpendicular, calculate the torque.

� � Fr sin � � (85 N)(0.258 m)(sin 60.0°)

� 19 N�m

5. A force of 25 N is applied vertically at theend of a wrench handle that is 45 cm longto tighten a bolt in the clockwise direction.What torque is needed by the bolt to keepthe wrench from turning?

In order to keep the wrench from turn-ing, the torque on the bolt must be equalin magnitude but opposite in direction ofthe torque applied by the wrench.

� � Fr sin � � (25 N)(0.45 m)(sin 90.0°)

� 11 N�m counterclockwise

6. A 92-kg man uses a 3.05-m board to attemptto move a boulder, as shown in the diagrambelow. He pulls the end of the board with aforce equal to his weight and is able to moveit to 45° from the perpendicular. Calculatethe torque applied.

� � Fr sin �

� mgr sin �

� (92 kg)(9.80 m/s2)(3.05 m)(sin 45°)

� 1.9�103 N�m

7. If a 25-kg child tries to apply the same torqueas in the previous question using only his orher weight for the applied force, what wouldthe length of the lever arm need to be?

L � r sin �

� � �

� 7.8 m

8. Logan, whose mass is 18 kg, sits 1.20 mfrom the center of a seesaw. If Shiro mustsit 0.80 m from the center to balanceLogan, what is Shiro’s mass?

FLrL � FSrS

mLgrL � mSgrS

mS �

�

� 27 kg

9. Two forces—55-N clockwise and 35-Ncounterclockwise—are applied to a merry-go-round with a diameter of 4.5 m. What isthe net torque?

(18 kg)(1.20 m)��

0.80 m

mLrL�rS

1.9�103 N�m���(25 kg)(9.80 m/s2)

��mg

��F

45°

3.2 m�82 rad

d��

1 m�100 cm

1 min�60 s

2� rad�1 rev

45 rev�1 min

(3.5 rad/s) � (1.5 rad/s)���

9.5 s����t

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The net torque is the sum of the individual torques.

�net � �ccw � �cw

� Fccwr sin � � Fcwr sin �

� (Fccw � Fcw)r sin �

� (35 N � 55 N)� �(sin 90.0°)

� �45 N�m

45 N�m clockwise

10. A student sits on a stool holding a 5.0-kgdumbbell in each hand. He extends his armssuch that each dumbbell is 0.60 m from theaxis of rotation. The student’s moment ofinertia is 5.0 kg�m2. What is the moment ofinertia of the student and the dumbbells?

Isingle dumbbell � mr2 � (5.0 kg)(0.60 m)2

� 1.8 kg�m2

Itotal � 2Isingle dumbbell � I student

� (2)(1.8 kg�m2) � 5.0 kg�m2

� 8.6 kg�m2

11. A basketball player spins a basketball with aradius of 15 cm on his finger. The mass ofthe ball is 0.75 kg. What is the moment ofinertia about the basketball?

I � mr2 � � �(0.75 kg)(0.15 m)2

� 6.8�10�3 kg�m2

12. A merry-go-round in the park has a radiusof 2.6 m and a moment of inertia of 1773 kg�m2. What is the mass of the merry-go-round?

I � mr2

m � �

� 5.2�102 kg

13. The merry-go-round described in the previ-ous problem is pushed with a constant forceof 53 N. What is the angular acceleration?

� �

�

�

� 7.8�10�2 rad/s2

14. What is the angular velocity of the merry-go-round described in problems 12 and 13after 85 s, if it started from rest?

� � �

�i � 0 since it started from rest

�f � ��t

� (7.8�10�2 rad/s2)(85 s)

� 6.6 rad/s

15. An ice-skater with a moment of inertia of1.1 kg�m2 begins to spin with her armsextended. After 25 s, she has an angularvelocity of 15 rev/s. What is the net torqueacting on the skater?

�net � I�

�

� � �

� �� 4.1 N�m

2� rad�

rev

(1.1 kg�m2)(15 rev/s � 0 rev/s)����

255

I(�f � �i)���t

�f � �i��t

����t

(53 N)(2.6 m)(sin 90.0°)���

1773 kg�m2

Fr sin ��

1

�net�I

(2)(1773 kg�m2)��

(2.6 m)22I�r 2

1�2

2�5

2�5

4.5 m�

2

35 N55 N

4.5 m

Chapter 8 continued

16. A board that is 1.5 m long is supported in two places. If the force exerted by thefirst support is 25 N and the forced exerted by the second is 62 N, what is themass of the board?

Fnet � F1 � F2 � (�Fg)

Since the system is in equilibrium,

Fnet � 0.

0 � F1 � F2 � Fg

Fg � F1 � F2

mg � F1 � F2

m � �

� 8.9 kg

17. A child begins to build a house of cards by laying an 8.5-cm-long playing cardwith a mass of 0.75 g across two other playing cards: support card A and supportcard B. If support card A is 2.0 cm from the end and exerts a force of 1.5�10�3 N,how far from the end is support card B located? Let the axis of rotation be at thepoint support card A comes in contact with the top card.

Fnet � FA � FB � (�Fg)

Since the system is in equilibrium,

Fnet � 0.

0 � FA � FB � Fg

FB � Fg � FA

� mg � FA

Since the axis of rotation is about support card A, �A � 0.so �net � �B � �g

The system is in equilibrium, so �net � 0.

0 � �B � �g

�B � � �g

�B � rBFB and �g � �rgFg

rBFB � rgFg

rB � �

�

� 2.8�10�2 m

� 28 cm

���12

��(0.085 m) � 0.020 m�(7.5�10�4 kg)(9.80 m/s2)������

(7.5�10�4 kg)(9.80 m/s2) � 1.5�10�3 N

rgmg��mg � FA

rgFg�FB

25 N � 62 N��

9.80 m/s2F1 � F2��

g

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Chapter 8 continued

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18. If support card A in the previous problem was moved so that it now is 2.5 cmfrom the end, how far from the other end does support card B need to be toreestablish equilibrium?

rB �

�

� 2.2�10�2 m

� 22 cm

���12

��(0.085 m) � 0.025 m�(7.5�10�4 kg)(9.80 m/s2)������

(7.5�10�4 kg)(9.80 m/s2) � 1.5�10�3 N

rgmg��mg � FA

Chapter 8 continued

Chapter 9pages 870–871

1. A ball with an initial momentum of 6.00 kg�m/s bounces off a wall and travels inthe opposite direction with a momentum of 4.00 kg�m/s. What is the magnitudeof the impulse acting on the ball?

Choose the direction away from the wall to be positive.

pf � �4.00 kg�m/s

pi � �6.00 kg�m/s

Impulse � pf � pi � (4.00 kg�m/s) � (�6.00 kg�m/s)

� 10.0 kg�m/s

2. If the ball in the previous problem interacts with the wall for a time interval of0.22 s, what is the average force exerted on the wall?

Impulse � F�t

F � � � 45 N

3. A 42.0-kg skateboarder traveling at 1.50 m/s hits a wall and bounces off of it. If the magnitude of the impulse is 150.0 kg·m/s, calculate the final velocity of theskateboarder.

Choose the direction away from the wall to be positive.

Impulse � pf � pi � mvf � (�mvi)

� m(vf � vi)

vf � � vi

� � 1.50 m/s

� 2.07 m/s

4. A 50.0-g toy car traveling with a velocity of 3.00 m/s due north collides head-onwith an 180.0-g fire truck traveling with a velocity of 0.50 m/s due south. Thetoys stick together after the collision. What are the magnitude and direction oftheir velocity after the collision?

Choose north to be positive.

pi � pf

pci � pti � pf

mcvci � mtvti � (mc � mt)vf

vf �

�

� 0.26 m/s, due north

(50.0 g)(3.00 m/s) � (180.0 g)(�0.50 m/s)�����

50.0 g � 180.0 g

mcvci � mtvti��mc � mt

150.0 kg�m/s��

42.0 kg

Impulse��

m

10.0 kg�m/s��

0.22 sImpulse��

�t

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5. A 0.040-kg bullet is fired into a 3.50-kgblock of wood, which was initially at rest. The bullet remains embedded within theblock of wood after the collision. The bullet and the block of wood move at avelocity of 7.40 m/s. What was the original velocity of the bullet?

pi � pf

pbi � pwi � pf

mbvbi � mwvwi � (mb � mw)vf

where vwi � 0. Thus,

vbi �

� 6.5�102 m/s

6. Ball A, with a mass of 0.20 kg, strikes ball B,with a mass of 0.30 kg. The initial velocityof ball A is 0.95 m/s. Ball B is initially atrest. What are the final speed and directionof ball A and B after the collision if theystick together?

pi � pf

mAvAi � (mA � mB)vf

vf �

�

� 0.38 m/s in the same direction as ball A’s initial velocity

7. An ice-skater with a mass of 75.0 kg pushesoff against a second skater with a mass of42.0 kg. Both skaters are initially at rest.After the push, the larger skater moves offwith a speed of 0.75 m/s eastward. What is the velocity (magnitude and direction) ofthe smaller skater after the push?

pi � pf

� m1vf1 � m2vf2

vf2 �

�

� �1.3 m/s

The second skater moves west with avelocity of 1.3 m/s.

8. Suppose a 55.0-kg ice-skater, who was initially at rest, fires a 2.50-kg gun. The0.045-kg bullet leaves the gun at a velocityof 565.0 m/s. What is the velocity of theice-skater after she fires the gun?

pi � pf

0 � pfs � pfb

� (ms � mg)vfs � mbvfb

because the final mass of the skaterincludes the mass of the gun held bythe skater. Then,

vfs �

�

� �0.44 m/s

9. A 1200-kg cannon is placed at rest on an ice rink. A 95.0-kg cannonball is shotfrom the cannon. If the cannon recoils at a speed of 6.80 m/s, what is the speed ofthe cannonball?

pi � pf

Since the cannon and cannonball are atrest before the blast, pi � 0.00 kg�m/s.

So pfc � �pfb

mcvfc � �mbvfb

vfb �

�

� �86 m/s

10. An 82-kg receiver, moving 0.75 m/s north,is tackled by a 110.0-kg defensive linemanmoving 0.15 m/s east. The football playershit the ground together. Calculate their finalvelocity (magnitude and direction).

pri � mrvri, y � (82 kg)(0.75 m/s)

� 62 kg�m/s north

pdi � mdvdi, x � (110.0 kg)(0.15 m/s)

� 16 kg�m/s, east

�(1200 kg)(6.80 m/s)���

95.0 kg

�mcvfc�mb

�(0.045 kg)(565.0 m/s)���

55.0 kg � 2.50 kg

�mbvfb��ms � mg

�(75.0 kg)(0.75 m/s)���

42.0 kg

�m1vf1�m2

(0.20 kg)(0.95 m/s)���0.20 kg � 0.30 kg

mAvAi��mA � mB

(0.040 kg � 3.50 kg)(7.40 m/s)����

0.040 kg

Chapter 9 continued

Law of conservation of momentum states

pi � pf

pf, x � pi, x � 16 kg�m/s

pf, y � pi, y � 62 kg�m/s

pf � (pf, x2 � pf, y

2)�12�

� ((16 kg�m/s)2 � (62 kg�m/s)2)�12�

� 64 kg�m/s

vf � �

� 0.33 m/s

� � tan�1� �

� tan�1� �

� tan�1� �� 75°

11. A 985-kg car traveling south at 29.0 m/s hits a truck traveling 18.0 m/s west, as shown in the figurebelow. After the collision, the vehiclesstick together and travel with a finalmomentum of 4.0�104 kg�m/s at an angle of 45°. What is the mass of the truck?

pf2 � pfx

2 � pfy2

pf2 � pix

2 � piy2

pf2 � mc

2vci2 � mt

2vti2

mt � � ��12�

� � ��12�

� 1.6�103 kg

(4.0�104 kg�m/s) � (985 kg)2(29.0 m/s)2�����

(18.0 m/s)2

pf2 � mc

2vci2

��vti

2

29.0 m/s

18.0 m/s

p � 4.0�104 kg�m/s

45°

(82 kg)(0.75 m/s)���(110.0 kg)(0.15 m/s)

mrvri, y�mdvdi, x

pf,y�pf,x

64 kg�m/s���82 kg � 110.0 kg

pf��mr � md

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12. A 77.0-kg woman is walking 0.10 m/s east in the gym. A man throws a 15.0-kgball south and accidentally hits the woman. The woman and the ball movetogether with a velocity of 0.085 m/s. Calculate the direction the woman and the ball move.

� � cos�1� �

� cos�1� �

� cos�1� �

� cos�1� �� 1.0�101 degrees south of east

(77.0 kg)(0.10 m/s)����(77.0 kg � 15.0 kg)(0.085 m/s)

mwviw��(mw � mb)vf

pix�pf

pfx�pf

Chapter 9 continued