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Potential Energy
1. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What potential energy does the woman record after the ball
has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
ANS: b
2. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What potential energy does the man record after the ball
has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
ANS: d
3. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What kinetic energy does the woman record after the ball
has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
127
128
Potential Energy
ANS: c
Chapter 7
129
4. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What change in potential energy does the woman record
after the ball has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
ANS: b
5. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What change in potential energy does the man record after
the ball has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
ANS: b
6. The motion of a ball of mass m dropped off a building is observed by a man at ground level and a woman
at the top of the building. The man’s origin for measuring gravitational potential energy is at ground level
and the woman’s origin for measuring gravitational potential energy is at the top of the building. The
building’s height is h and air friction is neglected. What is the sum of the potential energy and the kinetic
energy of the ball from the woman’s perspective after the ball has fallen a distance d?
a. mgh
b. –mgd
c. mgd
d. mg h  d
e. zero
a f
ANS: e
7. An object of mass m is lifted a height y from the floor to a table by a woman. What is its change in
gravitational potential energy?
a. mgy
b. –mgy
1
c. mv 2
2
1
d.  mv 2
2
e. zero
130
Potential Energy
ANS: a
Chapter 7
131
8. An object of mass m is lifted a height y from the floor to a table by a woman. How much work did she do?
a. mgy
b. –mgy
1
c. mv 2
2
1
d.  mv 2
2
e. zero
ANS: a
9. An object of mass m is lifted a distance y from the floor to a table by a man. Ignore air resistance. The work
done on the object by all forces acting on it is
a. mgy.
b. –mgy.
1
c. mv 2 .
2
1
d.  mv 2 .
2
e. zero.
ANS: e
10. An object of mass m is lifted a distance y from the floor to a table by a man. The system on which he
performs work consists of the object and the earth. How much work has been performed on the system by
external forces?
a. mgy
b. –mgy
1
c. mv 2
2
1
d.  mv 2
2
e. zero
ANS: a
11. A woman lifts a shoebox of mass m a distance y from the floor to a closet shelf. How much work was done
by gravity?
a. mgy
b. –mgy
1
c. mv 2
2
1
d.  mv 2
2
e. zero
ANS: b
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Potential Energy
12. A car of weight mg is driven up an incline of angle  until it reaches a height h above the earth’s surface.
What is its gravitational potential energy with respect to the Earth’s surface?
1
a. mv 2
2
b. mgh
c. –mgh
d. mghsin 
e. 0
ANS: b
13. When a truck of gravitational potential energy mgy is loaded to a total weight of 2mg, its new gravitational
potential energy is
a. 2mgy
b. mgy
c. –mgy
d. –2mgy
e. 0
ANS: a
14. Which of the following is in stable equilibrium?
a. A ball on the floor
b. A pyramid resting on its base
c. A spinning top
d. A coin standing on edge
e. A nail balanced on its point
ANS: b
15. Which of the following is in neutral equilibrium?
a. A ball on the floor
b. A pyramid resting on its base
c. A spinning top
d. A coin standing on edge
e. A nail balanced on its point
ANS: a
16. Which of the following is in unstable equilibrium?
a. A marble on the floor
b. A brick lying on the table
c. A spinning top
d. A pyramid resting on its base
e. A coin lying on its face
ANS: c
Chapter 7
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17. What kinetic energy in J must a 0.25 kg mass have to ascend 3.2 m vertically in the Earth’s gravitational
field?
a. 9.8
b. 8.2
c. 7.8
d. 7.5
e. 6.9
ANS: c
18. What kinetic energy in J does an 8.24  10 4 kg airliner have when moving at a speed of 630 km/hr?
a. 1.93  10 6
b. 1.78  10 7
c. 1.83  10 8
d. 1.26  10 9
e. 1.87  1010
ANS: d
19. A 0.25 kg mass is located 24 m above the surface of the Earth. What is its potential energy in J with respect
to the Earth’s surface?
a. 29
b. 34
c. 42
d. 49
e. 59
ANS: e
20. An 80 kg man walks down a ramp that makes a 40° angle with the horizontal. By how much does his
gravitational potential energy change (in kJ) when he walks 15 m down the ramp?
a. 7.6
b. 7.9
c. 8.3
d. 8.5
e. 8.7
ANS: a
21. A 50 kg boy is on a massless swing that has an effective length of 3.0 m. His potential energy is zero when
the angle between the swing and the vertical is zero. What is his potential energy (in kJ) when the angle
between the swing and the vertical is 35 ?
a. 125
b. 136
c. 245
d. 266
e. 263
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Potential Energy
ANS: d
Chapter 7
135
22. A 50 kg boy is on a massless swing that has an effective length of 3.0 m. His potential energy is zero when
the angle between the swing and the vertical is zero. The maximum angle between the swing and the vertical
is 35°? What is his speed in m/s at the bottom of the swing?
a. 6.9
b. 6.2
c. 5.1
d. 4.2
e. 3.3
ANS: e
23. A 5.0 kg particle has 160 J of net work done on it. What is the change in kinetic energy in J?
a. 160
b. 80
c. 320
d. 40
e. 32
ANS: a
24. A 5.0 kg particle has 160 J of work done on it. If its initial speed is zero, what is its final speed in m/s?
a. 5.0
b. 6.0
c. 7.0
d. 8.0
e. 9.0
ANS: d
25. A 5.0 kg particle has 160 J of work done on it. If its initial speed is 6.0 m/s, what is its final speed in m/s?
a. 8.0
b. 9.0
c. 10
d. 11
e. 12
ANS: c
26. The conservative force F   4.0x  3.0 N does work on a particle moving along the x-axis. How much
work in J is done on the particle by this force when the particle moves from x  2.0 m to x  3.0 m ?
a. 12
b. 13
c. 14
d. 15
e. 16
ANS: b
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Potential Energy
Chapter 7
137
27. The conservative force F   3.00x  4.00 N does work on a particle moving along the x axis. What is the
magnitude of the change in potential energy (in J) of the particle when the particle moves from x  2.0 m to
x  3.0 m ?
a. 10.5
b. 11.5
c. 12.5
d. 13.5
e. 14.5
ANS: b
28. When released, a bead slides without friction down a wire and makes a loop–the–loop as shown in the
diagram. What is its speed in m/s as it leaves the bottom of the circular loop?
bead
6.0 m
2.0 m
a. 6.3
b. 7.4
c. 8.9
d. 11
e. 13
ANS: d
29. When released, a bead slides without friction down a wire and makes a loop–the–loop as shown in the
diagram. What is its speed in m/s at the top of the circular loop?
bead
6.0 m
2.0 m
a. 6.3
b. 7.4
c. 8.9
d. 11
e. 13
ANS: a
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Potential Energy
Chapter 7
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30. A 1.50 kg rock is thrown horizontally from the top of a 38.1 m high building with a speed of 6.10 m/s.
How much work in J is done by gravity on the falling rock in its descent?
a. 540
b. 545
c. 558
d. 560
e. 562
ANS: d
31. A 1.5 kg rock is thrown horizontally from the top of a 25 m high building with a speed of 16 m/s. What is
the rock’s speed in m/s just before it strikes the ground?
a. 16
b. 21
c. 25
d. 27
e. 29
ANS: d
32. A 0.44 kg ball is thrown upward and reaches a maximum height of 23 m. What is its initial speed in m/s
when air resistance is neglected?
a. 16
b. 21
c. 25
d. 27
e. 29
ANS: b
33. A 2.6 kg block starts sliding down an inclined plane from a height of 3.0 m. It slides 5.0 m down the plane.
The coefficient of friction between the block and the plane is 0.18. How much work (in J) is done on the block
by gravity during its 5.0 m displacement?
5m
4m
a. 18
b. 36
c. 58
d. 76
e. 94
3m
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Potential Energy
ANS: d
Chapter 7
141
34. A gun that shoots corks of mass 0.0046 kg uses a spring of spring constant 9.8 N/m. The 0.22 m long spring
is compressed to 0.11 m. Its 0.22 m long barrel exerts a constant 0.25 N force of friction on the cork when the
gun is fired. What is the cork’s speed in m/s at the muzzle of the gun?
0.22 m
0.11 m
a. 3.5
b. 3.72
c. 6.5
d. 7.2
e. 7.5
ANS: b
35. What is the potential energy in J of two 12.0  10 6 C charges located 1.90 m apart?
k e  8.99  10 9 Nm 2 C 2
e
j
a. 0.281
b. 0.381
c. 0.481
d. 0.581
e. 0.681
ANS: e
36. Three charges of 3.01  10 6 C are assembled in an equilateral triangle with sides of 0.0121 m. How much
work in J is needed to do this? k e  8.99  10 9 Nm 2 C 2
e
j
a. 3.61
b. 6.73
c. 13.5
d. 20.2
e. 21.0
ANS: d
37. A charge of 8.20  10-6 C is placed at the origin and a second charge of 4.80  10 6 C is placed on the xaxis at x  0.0811 m . What is the electric potential energy of these charges in J? k e  8.99  10 9 Nm 2 C 2
e
a. 2.18
b. 3.09
c. 4.36
d. 5.02
e. 6.28
j
142
Potential Energy
ANS: c
Chapter 7
143
38. A 2.0 kg mass swings at the end of a light string (length  3.0 m ). Its speed at the lowest point on its circular
path is 6.0 m/s. What is its kinetic energy in J at an instant when the string makes an angle of 50° with the vertical?
a. 21
b. 15
c. 28
d. 36
e. 23
ANS: b
39. A 2.5 kg object suspended from the ceiling by a string that has a length of 2.5 m is released from rest with
the string 40° below the horizontal position. What is the tension in N in the string at the instant the object
passes through its lowest position?
a. 11
b. 25
c. 42
d. 18
e. 32
ANS: c
40. A certain pendulum consists of a 1.5 kg mass swinging at the end of a string (length  2.0 m ). At the
lowest point in the swing the tension in the string is equal to 20 N. To what maximum height in cm above this
lowest point will the mass rise during its oscillation?
a. 77
b. 50
c. 63
d. 36
e. 95
ANS: d
41. A 0.80 kg object tied to the end of a 2.0 m string swings as a pendulum. At the lowest point of its swing,
the object has a kinetic energy of 10 J. Determine the speed of the object in m/s at the instant when the string
makes an angle of 50° with the vertical.
a. 5.6
b. 4.4
c. 3.3
d. 5.0
e. 6.1
ANS: c
a
f
42. A single conservative force Fx  6x  12 N (x is in m) acts on a particle moving along the x axis. The
potential energy associated with this force is assigned a value of +20 J at x  0 . What is the potential energy in
J at x  3.0 m ?
a. +11
b. +29
144
Potential Energy
c. +9.0
d. –9.0
e. +20
ANS: b
Chapter 7
145
43. As a particle moves along the x axis it is acted upon by a single conservative force given by
Fx  20  4x N where x is in m. The potential energy associated with this force has the value +30 J at the
origin x  0 . What is the value of the potential energy in J at x  4.0 m ?
a. –48
b. +78
c. –18
d. +48
e. +80
a f
a f
ANS: c
44. A 0.40 kg particle moves under the influence of a single conservative force. At point A where the particle
has a speed of 10 m/s, the potential energy associated with the conservative force is +40 J. As the particle
moves from A to B, the force does +25 J of work on the particle. What is the value of the potential energy in J
at point B?
a. +65
b. +15
c. +35
d. +45
e. –40
ANS: b
45. As a 1.0 kg object moves from point A to point B, it is acted upon by a single conservative force that does
–40 J of work during this motion. At point A the speed of the particle is 6.0 m/s and the potential energy
associated with the force is +50 J. What is the potential energy in J at point B?
a. +72
b. +10
c. +90
d. +28
e. +68
ANS: c
46. As a 1.5 kg mass moves along the x axis, it is acted upon by a single conservative force given by
Fx  6x 2 N , where x is in m. At x  0 (where its speed is 4.0 m/s), the potential energy associated with the
force is +30 J. What is the potential energy in J at x  2.0 m ?
a. –16
b. +46
c. +36
d. +14
e. –28
ANS: d
146
Potential Energy
e
j
47. A single conservative force Fx  6 x 2  12 N (x is in m) acts on a particle moving along the x axis. The
potential energy associated with this force is assigned a value of +30 J at x  0 . What is the potential energy in
J at x  3.0 m ?
a. 18
b. –18
c. 48
d. 12
e. –48
ANS: d
48. A 7.0 kg block on a horizontal frictionless surface is attached to a light spring (force constant  1.2 kN m ).
The block is initially at rest at its equilibrium position when a force (magnitude P) acting parallel to the
surface is applied to the block, as shown. When the block is 8.0 cm from the equilibrium position, it has a
speed of 0.80 m/s. How much work in J is done on the block by the force P as the block moves the 8.0 cm?
P
a. 7.4
b. 5.4
c. 6.1
d. 6.7
e. 4.9
ANS: c
49. A 0.60 kg object is suspended from the ceiling at the end of a 2.0 m string. When swinging, it has a speed
of 4.0 m/s at the lowest point of its path. What is the maximum angle that the string can make with the
vertical axis through the point of suspension of the string?
a. 61°
b. 54°
c. 69°
d. 77°
e. 47°
ANS: b
50. A 2.0 kg object with K  2.0 J is falling straight down when 10 m above the surface of the Earth. What is its
kinetic energy in J when it is 5.0 m above the surface of the Earth?
a. 198
b. 98
c. 46
Chapter 7
d. 100
e. 50
ANS: d
147
148
Potential Energy
51. A rock is thrown with speed v 0 at an angle of 45° and gets to a maximum height of y. What is its speed
y
when it is at height ?
2
a. v  v 0  2 ay
b. v  v 02  gy
c. v  v 0  ay
d. v  Fd  v 0
1
e. v  mv 02
2
ANS: b
af
52. When U x has a relative minimum value, we say an object at that position is in
a. stable equilibrium.
b. unstable equilibrium.
c. neutral equilibrium.
d. Newtonian equilibrium.
e. Einsteinian equilibrium.
ANS: a
53. The equation that states mathematically the principle of conservation of mechanical energy for an isolated
system is
a. K f  Ki .
b. U f  U i .
c. K f  U f  Ki  U i .
d. K f  Ki  U f  U i .
e. K f  U f  W  Ki  U i .
ANS: c
54. The zero level of the gravitational potential energy of a body near the surface of the Earth is located at
a. the surface of the Earth.
b. the initial position of the body.
c. the final position of the body.
d. the center of the Earth.
e. a position chosen for convenience.
ANS: e
55. What is the minimum speed (in m/s) with which a spaceship must leave the surface of the Earth in order
to be able to escape the gravitational attraction of the Earth? G  6.67  10 11 Nm 2 kg 2 ,
M Earth  5.98  10 24 kg , R Earth  6.37  10 6 m , m ship  2000 kg
a. 4.21  10 4
b. 1.12  10 4
c. 5.00  10 4
d. 4.43
Chapter 7
e. 2.05  10 7
ANS: b
149
150
Potential Energy
56. For a body A of mass 50 g with a 1.00 C charge to escape from a body B with a 2.00 C charge when body
e
j
A is 40 m distant from body B, body A’s minimum speed must be k e  8.99  10 9 Nm 2 C 2 .
a. 21.2 m/s
b. 134 m/s
c. 134 km/s
d. 21.2 km/s
e. 141 m/s
ANS: b
57. A space-mapping mission encounters a simple binary star system in which the two stars revolve around
their common center of mass. The distance between them at any instant is R AB . The spaceship, of mass m sh ip ,
is distant R A from star A of mass M A and R B from star B of mass M B . The gravitational potential energy of
the spaceship relative to the star system when U gravitational  0 at infinity is
Gm ship M A Gm ship M B
a. 
.

RA
RB
Gm ship M A Gm ship M B GM A M B
b. 
.


RA
RB
R AB
GM A M B
c. 
.
R AB
Gm ship M A Gm ship M B
d. 
.

R AB
R AB
GM A M B GM A M B
e. 
.

RA
RB
ANS: a
58. An example of a non-isolated system in a steady state or a quasi-steady state is
a. a house maintained at constant temperature.
b. the Earth.
c. your body.
d. all of the above.
e. only (a) and (b) above.
ANS: d
59. Two masses, M A and M B , with M B  2 M A , are released at the same time and allowed to fall straight
down. Neglect air resistance. When we compare their kinetic energies after they have fallen equal distances,
we find that
a. K B  K A .
b. K B  2K A .
c. K B  4K A
d. K A  2K B .
e. K A  4K B .
ANS: b
Chapter 7
151
152
Potential Energy
60. Two masses, M A and M B , with M B  2 M A , are released at the same time and allowed to fall straight
down. Neglect air resistance. When we compare their kinetic energies after they have fallen for equal times,
we find that
a. K B  K A .
b. K B  2K A .
c. K B  4K A
d. K A  2K B .
e. K A  4K B .
ANS: b
61. A system consists of the Earth and of a ball of mass m at rest at height h above the Earth. When the ball is
h
released and falls a distance , the change in total mechanical energy of the system is
2
a. 0.
mgh
b.
.
2
c. mgh.
mgh 1
 mv 2 .
d.
2
2
1
e. mgh  mv 2 .
2
ANS: a
62. A person of mass m stands in an elevator that is at rest at the position where gravitational potential energy
is taken to be zero. The elevator then accelerates upwards with acceleration a. When the elevator has risen a
distance h, the person’s total mechanical energy is
a. 0.
b. mgh.
c. mah.
d. a  g mh .
b g
e. b
a  gg
mh .
ANS: d
63. A person of mass m stands in an elevator that is at rest at the position where gravitational potential energy
is taken to be zero. The elevator then descends with acceleration a. When the elevator has descended a
distance h, the person’s total mechanical energy is
a. 0.
b. –mgh.
c. mah.
d. a  g mh .
b g
e. b
a  gg
mh .
ANS: e
Chapter 7
153
64. A mechanism catches a train car initially moving at speed v 0 at the instant when it has compressed a
heavy horizontal spring of spring constant k to maximum compression x m ax . At that instant the total
mechanical energy of the system of train plus spring is
1 2
a. kx max
.
2
1
1 2
b. mv 02  kx max
.
2
2
1 2
1
c. kx max
 mv 02 .
2
2
1
1 2
2
d. mv 0  kx max .
2
2
e. a quantity that cannot be determined from the information given.
ANS: a
65. A toy car of total mass m but with wheels of negligible mass, is started up an inclined plane of angle 
with initial velocity v 0 . It travels a distance x up the plane before starting to roll down again. Assuming its
initial gravitational potential energy is 0, we can find its gravitational potential energy at the highest point
where the vertical displacement is h, by using
a. h  x .
b. h  x cos  .
c. h  x sin  .
d. h  x tan  .
e. h  x csc .
ANS: c
66. A car starts at height h above ground and rolls around the loop-the-loop of radius R. We want to calculate
h in terms of R for the car just to make it around without losing contact with the track. Assume the track is
frictionless. Which equation below must be one of the equations we use to solve this problem? (Use v top as
the speed at the top of the circular loop and v bottom as the speed at the bottom of the circular loop.)
bead
h
2.0 m
2
2
a. v bottom
 v top
 gR
2
2
b. v bottom
 v top
 2 gR
2
2
c. v bottom
 v top
 gR
2
2
d. v bottom
 v top
 2 gR
2
2
e. v bottom
 v top
 5 gR
ANS: b
154
Potential Energy
Chapter 7
155
67. A pendulum of mass m and length L is raised to a point that is L 3 higher than its initial position. It is then
allowed to fall and strike a spring of spring constant k that is located at the pendulum’s lowest position. If the
spring were to absorb all the pendulum’s kinetic energy, it would have compression x m ax such that
1 mgL
.
3 k
2 mgL
b. x 
.
3 k
3 mgL
c. x 
.
2 k
3mgL
d. x 
.
k
e. x  L .
a. x 
ANS: b
68. When the gravitational potential energy of a body of mass m initially moving at speed v 0 changes by
U   J, the final velocity of the body, v f , can be found from
1
1
a. mv 2f  J  mv 02 .
2
2
1
1
2
b. mv f  mv 02   J .
2
2
1
1
2
c. mv f  mv 02   J .
2
2
1
1
2
d. mv f  mv 02   J .
2
2
1
1
2
e. mv f  mv 02  J  0 .
2
2
ANS: c
69. When the gravitational potential energy of a body of mass m initially moving at speed v 0 changes by
U   J, the final velocity of the body, v f , can be found from
1
1
a. mv 2f  J  mv 02 .
2
2
1
1
2
b. mv f  mv 02   J .
2
2
1
1
2
c. mv f  mv 02   J .
2
2
1
1
2
d. mv f  mv 02   J .
2
2
1
1
2
e. mv f  mv 02  J  0 .
2
2
ANS: e
156
Potential Energy
70. A light string suspended over a frictionless pulley of negligible mass connects a 20 kg and a 12 kg mass as
shown below. When they are released from rest, and the 20 kg mass has fallen 1.0 m, the speed of the 12 kg
mass, in m/s, is
12 kg
20 kg
a. 1.1.
b. 1.6.
c. 2.2.
d. 3.1.
e. 4.9.
ANS: c
71. A 70 kg person who runs up a 10-story flight of stairs to a point 25 m above his starting point in 5.0
minutes has a power expenditure of
a. 29 W.
b. 57 W.
c. 570 W.
d. 3400 W.
e. 17, 100 W.
ANS: b
73. For a 70 kg person to match the 100 W power expenditure of champion athletes when running up a 10story flight of stairs to a point 25 m above his starting point, he has to reach the top in
a. 60 s.
b. 86 s.
c. 100 s.
d. 172 s.
e. 300 s.
ANS: d