CP7e: Ch. 1 Problems

Chapter 1 Problems
4. Each of the following
equations was given by a student
during an examination:
1 2 1 2
mv = mv0
+
2 2
v = v + at
0
2
ma = v
2
mgh
Do a dimensional analysis of each
equation and explain why the
equation can’t be correct.
5. Newton’s law of universal
gravitation is represented by
Mm
F = G 2
r
where F is the gravitational force, M
and m are masses, and r is a length.
Force has the SI units kg ∙ m/s 2 . What
are the SI units of the proportionality
constant G?
Section 1.4 Uncertainty in
Measurement and Significant
Figures
7. How many significant figures
are there in (a) 78.9 ± 0.2, (b) 3.788 ×
10 9 , (c) 2.46 × 10 –6 , (d) 0.0032?
11. A farmer measures the
perimeter of a rectangular field. The
length of each long side of the
rectangle is found to be 38.44 m, and
the length of each short side is found
to be 19.5 m. What is the perimeter
of the field?
Section 1.5 Conversion of Units
15. A fathom is a unit of length,
usually reserved for measuring the
depth of water. A fathom is
approximately 6 ft in length. Take
the distance from Earth to the Moon
to be 250 000 miles, and use the
given approximation to find the
distance in fathoms.
17. A rectangular building lot
measures 100 ft by 150 ft. Determine
the area of this lot in square meters
(m 2 ).
21. The speed of light is about
3.00 × 10 8 m/s. Convert this figure to
miles per hour.
23. The amount of water in
reservoirs is often measured in acre‐
ft. One acre‐ft is a volume that covers
an area of one acre to a depth of one
foot. An acre is 43 560 ft 2 . Find the
volume in SI units of a reservoir
containing 25.0 acre‐ft of water.
26. (a) Find a conversion factor to
convert from miles per hour to
kilometers per hour. (b) For a while,

federal law mandated that the
maximum highway speed would be
55 mi/h. Use the conversion factor
from part (a) to find the speed in
kilometers per hour. (c) The
maximum highway speed has been
raised to 65 mi/h in some places. In
kilometers per hour, how much of an
increase is this over the 55‐mi/h
limit?
Section 1.7 Coordinate Systems
35. A point is located in a polar
coordinate system by the coordinates
r = 2.5 m and θ = 35°. Find the x‐ and
y‐coordinates of this point, assuming
that the two coordinate systems have
the same origin.
38. Two points in a rectangular
coordinate system have the
coordinates (5.0, 3.0) and (–3.0, 4.0),
where the units are centimeters.
Determine the distance between
these points.
Section 1.8 Trigonometry
39. For the triangle shown in Figure
P1.39, what are (a) the length of the
unknown side, (b) the tangent of θ,
and (c) the sine of φ?
Figure P1.39
40. A ladder 9.00 m long leans
against the side of a building. If the
ladder is inclined at an angle of 75.0°
to the horizontal, what is the
horizontal distance from the bottom
of the ladder to the building?
42. A right triangle has a
hypotenuse of length 3.00 m, and
one of its angles is 30.0°. What are
the lengths of (a) the side opposite
the 30.0° angle and (b) the side
adjacent to the 30.0° angle?
43. In Figure P1.43, find (a) the
side opposite θ, (b) the side adjacent
to φ, (c) cos θ, (d) sin φ, and (e) tan
φ.
Figure P1.43

46. A surveyor measures the
distance across a straight river by the
following method: Starting directly
across from a tree on the opposite
bank, he walks 100 m along the
riverbank to establish a baseline.
Then he sights across to the tree. The
angle from his baseline to the tree is
35.0°. How wide is the river?

Chapter 2 Problems
Section 2.1 Displacement Section
Section 2.2 Velocity
1. A person travels by car from one city
to another with different constant speeds
between pairs of cities. She drives for 30.0
min at 80.0 km/h, 12.0 min at 100 km/h, and
45.0 min at 40.0 km/h and spends 15.0 min
eating lunch and buying gas. (a) Determine
the average speed for the trip. (b)
Determine the distance between the initial
and final cities along the route.
2. (a) Sand dunes on a desert island
move as sand is swept up the windward
side to settle in the leeward side. Such
“walking” dunes have been known to
travel 20 feet in a year and can travel as
much as 100 feet per year in particularly
windy times. Calculate the average speed in
each case in m/s. (b) Fingernails grow at the
rate of drifting continents, about 10 mm/yr.
Approximately how long did it take for
North America to separate from Europe, a
distance of about 3 000 mi?
3. Two boats start together and race
across a 60‐km‐wide lake and back. Boat A
goes across at 60 km/h and returns at 60
km/h. Boat B goes across at 30 km/h, and its
crew, realizing how far behind it is getting,
returns at 90 km/h. Turnaround times are
negligible, and the boat that completes the
round trip first wins. (a) Which boat wins
and by how much? (Or is it a tie?) (b) What
is the average velocity of the winning boat?
5. A motorist drives north for 35.0
minutes at 85.0 km/h and then stops for
15.0 minutes. He then continues north,
traveling 130 km in 2.00 h. (a) What is his
total displacement? (b) What is his average
velocity?
6. A graph of position versus time for a
certain particle moving along the x‐axis is
shown in Figure P2.6. Find the average
velocity in the time intervals from (a) 0 to
2.00 s, (b) 0 to 4.00 s, (c) 2.00 s to 4.00 s, (d)
4.00 s to 7.00 s, and (e) 0 to 8.00 s.
Figure P2.6 (Problems 6 and 15)
11. A person takes a trip, driving with a
constant speed of 89.5 km/h, except for a
22.0‐min rest stop. If the person’s average
speed is 77.8 km/h, how much time is spent
on the trip and how far does the person
travel?
12. A tortoise can run with a speed of
0.10 m/s, and a hare can run 20 times as
fast. In a race, they both start at the same
time, but the hare stops to rest for 2.0
minutes. The tortoise wins by a shell (20

cm). (a) How long does the race take? (b)
What is the length of the race?
15. A graph of position versus time for a
certain particle moving along the x‐axis is
shown in Figure P2.6. Find the
instantaneous velocity at the instants (a) t =
1.00 s, (b) t = 3.00 s, (c) t = 4.50 s, and (d) t =
7.50 s.
Section 2.3 Acceleration
22. The velocity vs. time graph for an
object moving along a straight path is
shown in Figure P2.22. (a) Find the average
acceleration of the object during the time
intervals 0 to 5.0 s, 5.0 s to 15 s, and 0 to 20
s. (b) Find the instantaneous acceleration at
2.0 s, 10 s, and 18 s.
Figure P2.22
Section 2.5 One‐Dimensional Motion
with Constant Acceleration
26. A truck covers 40.0 m in 8.50 s while
smoothly slowing down to a final speed of
2.80 m/s. (a) Find the truck’s original speed.
(b) Find its acceleration.
30. A truck on a straight road starts from
rest and accelerates at 2.0 m/s 2 until it
reaches a speed of 20 m/s. Then the truck
travels for 20 s at constant speed until the
brakes are applied, stopping the truck in a
uniform manner in an additional 5.0 s. (a)
How long is the truck in motion? (b) What
is the average velocity of the truck during
the motion described?
33. A driver in a car traveling at a speed
of 60 mi/h sees a deer 100 m away on the
road. Calculate the minimum constant
acceleration that is necessary for the car to
stop without hitting the deer (assuming
that the deer does not move in the
meantime).
35. A train is traveling down a straight
track at 20 m/s when the engineer applies
the brakes, resulting in an acceleration of
–1.0 m/s 2 as long as the train is in motion.
How far does the train move during a 40‐s
time interval starting at the instant the
brakes are applied?
37. A car starts from rest and travels for
5.0 s with a uniform acceleration of +1.5
m/s 2 . The driver then applies the brakes,
causing a uniform acceleration of –2.0 m/s 2 .
If the brakes are applied for 3.0 s, (a) how
fast is the car going at the end of the
braking period, and (b) how far has the car
gone?

Section 2.6 Freely Falling Objects
43. A ball is thrown vertically upward
with a speed of 25.0 m/s. (a) How high does
it rise? (b) How long does it take to reach its
highest point? (c) How long does the ball
take to hit the ground after it reaches its
highest point? (d) What is its velocity when
it returns to the level from which it started?
47. A small mailbag is released from a
helicopter that is descending steadily at 1.50
m/s. After 2.00 s, (a) what is the speed of the
mailbag, and (b) how far is it below the
helicopter? (c) What are your answers to
parts (a) and (b) if the helicopter is rising
steadily at 1.50 m/s?
49. A model rocket is launched straight
upward with an initial speed of 50.0 m/s. It
accelerates with a constant upward
acceleration of 2.00 m/s 2 until its engines
stop at an altitude of 150 m. (a) What is the
maximum height reached by the rocket? (b)
How long after lift‐off does the rocket reach
its maximum height? (c) How long is the
rocket in the air?

Chapter 3 Problems
Section 3.1 Vectors and Their Properties
2. An airplane flies 200 km due west
from city A to city B and then 300 km in the
direction of 30.0° north of west from city B
to city C. (a) In straight‐line distance, how
far is city C from city A? (b) Relative to city
A, in what direction is city C?
4. A jogger runs 100 m due west, then
changes direction for the second leg of the
run. At the end of the run, she is 175 m
away from the starting point at an angle of
15.0° north of west. What were the direction
and length of her second displacement? Use
graphical techniques.
5. A plane flies from base camp to lake
A, a distance of 280 km at a direction of
20.0° north of east. After dropping off
supplies, the plane flies to lake B, which is
190 km and 30.0° west of north from lake A.
Graphically determine the distance and
direction from lake B to the base camp.
8. Each of the displacement vectors A r
and B r shown in Figure P3.8 has a
magnitude of 3.00 m. Graphically find (a)
+ , (b) A r – B r , (c) B r – A r A , and (d) A
r
B r r –
2 B r .
Figure P3.8
Section 3.2 Components of a Vector
13. A vector has an x‐component of
–25.0 units and a y‐component of 40.0 units.
Find the magnitude and direction of the
vector.
17. A commuter airplane starts from an
airport and takes the route shown in Figure
P3.17. The plane first flies to city A, located
175 km away in a direction 30.0° north of
east. Next, it flies for 150 km 20.0° west of
north, to city B. Finally, the plane flies 190
km due west, to city C. Find the location of
city C relative to the location of the starting
point.

Figure P3.17
18. The helicopter view in Figure P3.18
shows two people pulling on a stubborn
mule. Find (a) the single force that is
equivalent to the two forces shown and (b)
the force that a third person would have to
exert on the mule to make the net force
equal to zero. The forces are measured in
units of newtons (N).
Figure P3.18
24. A student stands at the edge of a cliff
and throws a stone horizontally over the
edge with a speed of 18.0 m/s. The cliff is
50.0 m above a flat, horizontal beach, as
shown in Figure P3.24. How long after
being released does the stone strike the
beach below the cliff? With what speed and
angle of impact does the stone land?
Figure P3.24
28. An artillery shell is fired with an
initial velocity of 300 m/s at 55.0° above the
horizontal. To clear an avalanche, it
explodes on a mountainside 42.0 s after
firing. What are the x‐ and y‐coordinates of
the shell where it explodes, relative to its
firing point?
29. A brick is thrown upward from the
top of a building at an angle of 25° to the
horizontal and with an initial speed of 15
m/s. If the brick is in flight for 3.0 s, how tall
is the building?

Chapter 4 Problems
Section 4.1 Forces
Section 4.2 Newton’s First Law
Section 4.3 Newton’s Second Law
Section 4.4 Newton’s Third Law
1. A 6.0‐kg object undergoes an
acceleration of 2.0 m/s 2 . (a) What is the
magnitude of the resultant force acting on
it? (b) If this same force is applied to a 4.0‐
kg object, what acceleration is produced?
6. A freight train has a mass of 1.5 × 10 7
kg. If the locomotive can exert a constant
pull of 7.5 × 10 5 N, how long does it take to
increase the speed of the train from rest to
80 km/h?
8. A 5.0‐g bullet leaves the muzzle of a
rifle with a speed of 320 m/s. What force
(assumed constant) is exerted on the bullet
while it is traveling down the 0.82‐m‐long
barrel of the rifle?
12. Two forces are applied to a car in an
effort to move it, as shown in Figure P4.12.
(a) What is the resultant of these two
forces? (b) If the car has a mass of 3 000 kg,
what acceleration does it have? Ignore
friction.
Figure P4.12
13. After falling from rest from a height
of 30 m, a 0.50‐kg ball rebounds upward,
reaching a height of 20 m. If the contact
between ball and ground lasted 2.0 ms,
what average force was exerted on the ball?
Section 4.5 Applications of Newton’s
Laws
15. Find the tension in each cable
supporting the 600‐N cat burglar in Figure
P4.15.

Figure P4.15
16. Find the tension in the two wires
that support the 100‐N light fixture in
Figure P4.16.
Figure P4.16
19. Two blocks are fastened to the
ceiling of an elevator as in Figure P4.19. The
elevator accelerates upward at 2.00 m/s 2 .
Find the tension in each rope.
Figure P4.19
tension does the bird produce in the wire?
Ignore the weight of the wire.
25. A 2 000‐kg car is slowed down
uniformly from 20.0 m/s to 5.00 m/s in 4.00
s. (a) What average force acted on the car
during that time, and (b) how far did the
car travel during that time?
26. Two packing crates of masses 10.0 kg
and 5.00 kg are connected by a light string
that passes over a frictionless pulley as in
Figure P4.26. The 5.00‐kg crate lies on a
smooth incline of angle 40.0°. Find the
acceleration of the 5.00‐kg crate and the
tension in the string.

Figure P4.26
Section 4.6 Forces of Friction
37. A 1 000‐N crate is being pushed
across a level floor at a constant speed by a
force F of 300 N at an angle of 20.0° below
the horizontal, as shown in Figure P4.37a.
(a) What is the coefficient of kinetic friction
between the crate and the floor? (b) If the
300‐N force is instead pulling the block at
an angle of 20.0° above the horizontal, as
shown in Figure P4.37b, what will be the
acceleration of the crate? Assume that the
coefficient of friction is the same as that
found in (a).
r
Figure P4.37
38. A hockey puck is hit on a frozen lake
and starts moving with a speed of 12.0 m/s.
Five seconds later, its speed is 6.00 m/s. (a)
What is its average acceleration? (b) What is
the average value of the coefficient of
kinetic friction between puck and ice? (c)
How far does the puck travel during the
5.00‐s interval?
41. The coefficient of static friction
between the 3.00‐kg crate and the 35.0°
incline of Figure P4.41 is 0.300. What
minimum force F r must be applied to the
crate perpendicular to the incline to prevent
the crate from sliding down the incline?
Figure P4.41
47. A 3.00‐kg block starts from rest at
the top of a 30.0° incline and slides 2.00 m
down the incline in 1.50 s. Find (a) the
acceleration of the block, (b) the coefficient
of kinetic friction between the block and the
incline, (c) the frictional force acting on the
block, and (d) the speed of the block after it
has slid 2.00 m.
58. (a) What is the minimum force of
friction required to hold the system of
Figure P4.58 in equilibrium? (b) What
coefficient of static friction between the 100‐
N block and the table ensures equilibrium?
(c) If the coefficient of kinetic friction

etween the 100‐N block and the table is
0.250, what hanging weight should replace
the 50.0‐N weight to allow the system to
move at a constant speed once it is set in
motion?
Figure P4.58

Chapter 5 Problems
Section 5.1 Work
1. A weight lifter lifts a 350‐N set of
weights from ground level to a position
over his head, a vertical distance of 2.00 m.
How much work does the weight lifter do,
assuming he moves the weights at constant
speed?
2. If a man lifts a 20.0‐kg bucket from a
well and does 6.00 kJ of work, how deep is
the well? Assume that the speed of the
bucket remains constant as it is lifted.
6. A horizontal force of 150 N is used to
push a 40.0‐kg packing crate a distance of
6.00 m on a rough horizontal surface. If the
crate moves at constant speed, find (a) the
work done by the 150‐N force and (b) the
coefficient of kinetic friction between the
crate and surface.
7. A sledge loaded with bricks has a
total mass of 18.0 kg and is pulled at
constant speed by a rope inclined at 20.0°
above the horizontal. The sledge moves a
distance of 20.0 m on a horizontal surface.
The coefficient of kinetic friction between
the sledge and surface is 0.500. (a) What is
the tension in the rope? (b) How much
work is done by the rope on the sledge? (c)
What is the mechanical energy lost due to
friction?
Section 5.2 Kinetic Energy and the
Work–Energy Theorem
9. A mechanic pushes a 2.50 × 10 3 ‐kg
car from rest to a speed of v, doing 5 000 J
of work in the process. During this time, the
car moves 25.0 m. Neglecting friction
between car and road, find (a) v and (b) the
horizontal force exerted on the car.
10. A 7.00‐kg bowling ball moves at 3.00
m/s. How fast must a 2.45‐g Ping‐Pong ball
move so that the two balls have the same
kinetic energy?
12. A crate of mass 10.0 kg is pulled up a
rough incline with an initial speed of 1.50
m/s. The pulling force is 100 N parallel to
the incline, which makes an angle of 20.0°
with the horizontal. The coefficient of
kinetic friction is 0.400, and the crate is
pulled 5.00 m. (a) How much work is done
by gravity? (b) How much mechanical
energy is lost due to friction? (c) How much
work is done by the 100‐N force? (d) What
is the change in kinetic energy of the crate?
(e) What is the speed of the crate after being
pulled 5.00 m?
15. A 2.0‐g bullet leaves the barrel of a
gun at a speed of 300 m/s. (a) Find its
kinetic energy. (b) Find the average force
exerted by the expanding gases on the
bullet as it moves the length of the 50‐cm‐
long barrel.
30. A bead of mass m = 5.00 kg is
released from point and slides on the
frictionless track shown in Figure P5.30.
Determine (a) the bead’s speed at points
and and (b) the net work done by the
force of gravity in moving the bead from
to .

Figure P5.30
Figure P5.32 (Problems 32 and 89)
33. The launching mechanism of a toy gun
consists of a spring of unknown spring
constant, as shown in Figure P5.33a. If the
spring is compressed a distance of 0.120 m
and the gun fired vertically as shown, the
gun can launch a 20.0‐g projectile from rest
to a maximum height of 20.0 m above the
starting point of the projectile. Neglecting
all resistive forces, determine (a) the spring
constant and (b) the speed of the projectile
as it moves through the equilibrium
position of the spring (where x = 0), as
shown in Figure P5.33b.
Figure P5.33
41. A 2.1 × 10 3 ‐kg car starts from rest at
the top of a 5.0‐m‐long driveway that is
inclined at 20° with the horizontal. If an
average friction force of 4.0 × 10 3 N impedes
the motion, find the speed of the car at the
bottom of the driveway.
43. Starting from rest, a 10.0‐kg block
slides 3.00 m down to the bottom of a
frictionless ramp inclined 30.0° from the
floor. The block then slides an additional
5.00 m along the floor before coming to a
stop. Determine (a) the speed of the block at
the bottom of the ramp, (b) the coefficient of
kinetic friction between block and floor,
and (c) the mechanical energy lost due to
friction.
Section 5.6 Power

51. The electric motor of a model train
accelerates the train from rest to 0.620 m/s
in 21.0 ms. The total mass of the train is 875
g. Find the average power delivered to the
train during its acceleration.
52. An electric scooter has a battery
capable of supplying 120 Wh of energy.
[Note that an energy of 1 Wh = (1 J/s)(3600
s) = 3600 J] If frictional forces and other
losses account for 60.0% of the energy
usage, what change in altitude can a rider
achieve when driving in hilly terrain if the
rider and scooter have a combined weight
of 890 N?
54. A 650‐kg elevator starts from rest
and moves upwards for 3.00 s with
constant acceleration until it reaches its
cruising speed, 1.75 m/s. (a) What is the
average power of the elevator motor during
this period? (b) How does this amount of
power compare with its power during an
upward trip with constant speed?
Section 5.7 Work Done by a Varying
Force
55. The force acting on
a particle varies as in Figure P5.55. Find the
work done by the force as the particle
moves (a) from x = 0 to x = 8.00 m, (b) from
x = 8.00 m to x = 10.0 m, and (c) from x = 0 to
x = 10.0 m.
Figure P5.55
56. An object is subject to a force Fx that
varies with position as in Figure P5.56. Find
the work done by the force on the object as
it moves (a) from x = 0 to x = 5.00 m, (b)
from x = 5.00 m to x = 10.0 m, and (c) from x
= 10.0 m to x = 15.0 m. (d) What is the total
work done by the force over the distance x
= 0 to x = 15.0 m?
Figure P5.56

Chapter 6 Problems
Section 6.1 Momentum and Impulse
1. A ball of mass 0.150 kg is dropped
from rest from a height of 1.25 m. It
rebounds from the floor to reach a height of
0.960 m. What impulse was given to the ball
by the floor?
2. A tennis player receives a shot with
the ball (0.060 0 kg) traveling horizontally
at 50.0 m/s and returns the shot with the
ball traveling horizontally at 40.0 m/s in the
opposite direction. (a) What is the impulse
delivered to the ball by the racquet? (b)
What work does the racquet do on the ball?
4. A 0.10‐kg ball is thrown straight up
into the air with an initial speed of 15 m/s.
Find the momentum of the ball (a) at its
maximum height and (b) halfway to its
maximum height.
8. A 75.0‐kg stuntman jumps from a
balcony and falls 25.0 m before colliding
with a pile of mattresses. If the mattresses
are compressed 1.00 m before he is brought
to rest, what is the average force exerted by
the mattresses on the stuntman?
11. The force shown in the force vs. time
diagram in Figure P6.11 acts on a 1.5‐kg
object. Find (a) the impulse of the force, (b)
the final velocity of the object if it is initially
at rest, and (c) the final velocity of the object
if it is initially moving along the x‐axis with
a velocity of –2.0 m/s.
Figure P6.11
12. A force of magnitude Fx acting in the
x‐direction on a 2.00‐kg particle varies in
time as shown in Figure P6.12. Find (a) the
impulse of the force, (b) the final velocity of
the particle if it is initially at rest, and (c) the
final velocity of the particle if it is initially
moving along the x‐axis with a velocity of
–2.00 m/s.
Figure P6.12

15. The front 1.20 m of a 1 400‐kg car is
designed as a “crumple zone” that collapses
to absorb the shock of a collision. If a car
traveling 25.0 m/s stops uniformly in 1.20
m, (a) how long does the collision last, (b)
what is the magnitude of the average force
on the car, and (c) what is the acceleration
of the car? Express the acceleration as a
multiple of the acceleration of gravity.
Section 6.2 Conservation of Momentum
18. A 730‐N man stands in the middle of
a frozen pond of radius 5.0 m. He is unable
to get to the other side because of a lack of
friction between his shoes and the ice. To
overcome this difficulty, he throws his 1.2‐
kg physics textbook horizontally toward
the north shore at a speed of 5.0 m/s. How
long does it take him to reach the south
shore?
20. A rifle with a weight of 30 N fires a
5.0‐g bullet with a speed of 300 m/s. (a)
Find the recoil speed of the rifle. (b) If a 700‐
N man holds the rifle firmly against his
shoulder, find the recoil speed of the man
and rifle.
22. A 65.0‐kg person throws a 0.045 0‐kg
snowball forward with a ground speed of
30.0 m/s. A second person, with a mass of
60.0 kg, catches the snowball. Both people
are on skates. The first person is initially
moving forward with a speed of 2.50 m/s,
and the second person is initially at rest.
What are the velocities of the two people
after the snowball is exchanged? Disregard
friction between the skates and the ice.
Section 6.3 Collisions
Section 6.4 Glancing Collisions
27. A railroad car of mass 2.00 × 10 4 kg
moving at 3.00 m/s collides and couples
with two coupled railroad cars, each of the
same mass as the single car and moving in
the same direction at 1.20 m/s. (a) What is
the speed of the three coupled cars after the
collision? (b) How much kinetic energy is
lost in the collision?
30. An 8.00‐g bullet is fired into a 250‐g
block that is initially at rest at the edge of a
table of height 1.00 m (Fig. P6.30). The
bullet remains in the block, and after the
impact the block lands 2.00 m from the
bottom of the table. Determine the initial
speed of the bullet.
Figure P6.30
32. A 1 200‐kg car traveling initially with
a speed of 25.0 m/s in an easterly direction
crashes into the rear end of a 9 000‐kg truck
moving in the same direction at 20.0 m/s
(Fig. P6.32). The velocity of the car right
after the collision is 18.0 m/s to the east. (a)
What is the velocity of the truck right after
the collision? (b) How much mechanical
energy is lost in the collision? Account for
this loss in energy.

Figure P6.32
34. (a) Three carts of masses 4.0 kg, 10
kg, and 3.0 kg move on a frictionless
horizontal track with speeds of 5.0 m/s, 3.0
m/s, and 4.0 m/s, as shown in Figure P6.34.
The carts stick together after colliding. Find
the final velocity of the three carts. (b) Does
your answer require that all carts collide
and stick together at the same time?
Figure P6.34
40. A billiard ball rolling across a table
at 1.50 m/s makes a head‐on elastic collision
with an identical ball. Find the speed of
each ball after the collision (a) when the
second ball is initially at rest, (b) when the
second ball is moving toward the first at a
speed of 1.00 m/s, and (c) when the second
ball is moving away from the first at a
speed of 1.00 m/s.
43. A 2 000‐kg car moving east at 10.0
m/s collides with a 3 000‐kg car moving
north. The cars stick together and move as a
unit after the collision, at an angle of 40.0°
north of east and a speed of 5.22 m/s. Find
the speed of the 3 000‐kg car before the
collision.

Chapter 7 Problems
Section 7.1 Angular Speed and Angular
Acceleration
1. The tires on a new compact car have
a diameter of 2.0 ft and are warranted for 60
000 miles. (a) Determine the angle (in
radians) through which one of these tires
will rotate during the warranty period. (b)
How many revolutions of the tire are
equivalent to your answer in (a)?
2. A wheel has a radius of 4.1 m. How
far (path length) does a point on the
circumference travel if the wheel is rotated
through angles of 30°, 30 rad, and 30 rev,
respectively?
4. A potter’s wheel moves from rest to
an angular speed of 0.20 rev/s in 30 s. Find
its angular acceleration in radians per
second per second.
Section 7.2 Rotational Motion under
Constant Angular Acceleration
Section 7.3 Relations between Angular
and Linear Quantities
5. A dentist’s drill starts from rest.
After 3.20 s of constant angular
acceleration, it turns at a rate of 2.51 × 10 4
rev/min. (a) Find the drill’s angular
acceleration. (b) Determine the angle (in
radians) through which the drill rotates
during this period.
7. A machine part rotates at an angular
speed of 0.60 rad/s; its speed is then
increased to 2.2 rad/s at an angular
acceleration of 0.70 rad/s 2 . Find the angle
through which the part rotates before
reaching this final speed.
9. The diameters of the main rotor and tail
rotor of a single‐engine helicopter are 7.60
m and 1.02 m, respectively. The respective
rotational speeds are 450 rev/min and 4 138
rev/min. Calculate the speeds of the tips of
both rotors. Compare these speeds with the
speed of sound, 343 m/s.
12. A coin with a diameter of 2.40 cm is
dropped on edge onto a horizontal surface.
The coin starts out with an initial angular
speed of 18.0 rad/s and rolls in a straight
line without slipping. If the rotation slows
with an angular acceleration of magnitude
1.90 rad/s 2 , how far does the coin roll before
coming to rest?
13. A rotating wheel requires 3.00 s to
rotate 37.0 revolutions. Its angular velocity
at the end of the 3.00‐s interval is 98.0 rad/s.
What is the constant angular acceleration of
the wheel?
Section 7.4 Centripetal Acceleration
14. It has been suggested that rotating
cylinders about 10 mi long and 5.0 mi in
diameter be placed in space and used as
colonies. What angular speed must such a
cylinder have so that the centripetal
acceleration at its surface equals the free‐fall
acceleration on Earth?

17. (a) What is the tangential
acceleration of a bug on the rim of a 10‐in.‐
diameter disk if the disk moves from rest to
an angular speed of 78 rev/min in 3.0 s? (b)
When the disk is at its final speed, what is
the tangential velocity of the bug? (c) One
second after the bug starts from rest, what
are its tangential acceleration, centripetal
acceleration, and total acceleration?
18. A race car starts from rest on a
circular track of radius 400 m. The car’s
speed increases at the constant rate of 0.500
m/s 2 . At the point where the magnitudes of
the centripetal and tangential accelerations
are equal, determine (a) the speed of the
race car, (b) the distance traveled, and (c)
the elapsed time.
19. A 55.0‐kg ice‐skater is moving at 4.00
m/s when she grabs the loose end of a rope,
the opposite end of which is tied to a pole.
She then moves in a circle of radius 0.800 m
around the pole. (a) Determine the force
exerted by the horizontal rope on her arms.
(b) Compare this force with her weight.
21. A certain light truck can go around a
flat curve having a radius of 150 m with a
maximum speed of 32.0 m/s. With what
maximum speed can it go around a curve
having a radius of 75.0 m?
22. The cornering performance of an
automobile is evaluated on a skid pad,
where the maximum speed that a car can
maintain around a circular path on a dry,
flat surface is measured. Then the
centripetal acceleration, also called the
lateral acceleration, is calculated as a
multiple of the free‐fall acceleration g. The
main factors affecting the performance of
the car are its tire characteristics and
suspension system. A Dodge Viper GTS can
negotiate a skid pad of radius 61.0 m at 86.5
km/h. Calculate its maximum lateral
acceleration.
23. A 50.0‐kg child stands at the rim of a
merry‐go‐round of radius 2.00 m, rotating
with an angular speed of 3.00 rad/s. (a)
What is the child’s centripetal acceleration?
(b) What is the minimum force between her
feet and the floor of the carousel that is
required to keep her in the circular path? (c)
What minimum coefficient of static friction
is required? Is the answer you found
reasonable? In other words, is she likely to
stay on the merry‐go‐round?

Chapter 8 Problems
Section 8.1 Torque
1. If the torque required to loosen a nut
that is holding a flat tire in place on a car
has a magnitude of 40.0 N ∙ m, what
minimum force must be exerted by the
mechanic at the end of a 30.0‐cm lug
wrench to accomplish the task?
2. A steel band exerts a horizontal force
of 80.0 N on a tooth at point B in Figure
P8.2. What is the torque on the root of the
tooth about point A?
Figure P8.2
3. Calculate the net torque (magnitude
and direction) on the beam in Figure P8.3
about (a) an axis through O perpendicular
to the page and (b) an axis through C
perpendicular to the page.
Figure P8.3
4. Write the necessary equations of
equilibrium of the object shown in Figure
P8.4. Take the origin of the torque equation
about an axis perpendicular to the page
through the point O.
Figure P8.4

Section 8.2 Torque and the Two
Conditions for Equilibrium
Section 8.3 The Center of Gravity
Section 8.4 Examples of Objects in
Equilibrium
7. The arm in Figure P8.7 weighs 41.5
N. The force of gravity acting on the arm
acts through point A. Determine the
magnitudes of the tension force t Fr in the
deltoid muscle and the force s Fr exerted by
the shoulder on the humerus (upper‐arm
bone) to hold the arm in the position
shown.
Figure P8.7
8. A water molecule consists of an
oxygen atom with two hydrogen atoms
bound to it as shown in Figure P8.8. The
bonds are 0.100 nm in length, and the angle
between the two bonds is 106°. Use the
coordinate axes shown, and determine the
location of the center of gravity of the
molecule. Take the mass of an oxygen atom
to be 16 times the mass of a hydrogen atom.
Figure P8.8
9. A cook holds a
2.00‐kg carton of milk at arm’s length (Fig.
P8.9). What force FB r must be exerted by the
biceps muscle? (Ignore the weight of the
forearm.)
Figure P8.9

17. A 500‐N uniform rectangular sign
4.00 m wide and 3.00 m high is suspended
from a horizontal, 6.00‐m‐long, uniform,
100‐N rod as indicated in Figure P8.17. The
left end of the rod is supported by a hinge,
and the right end is supported by a thin
cable making a 30.0° angle with the vertical.
(a) Find the tension T in the cable. (b) Find
the horizontal and vertical components of
force exerted on the left end of the rod by
the hinge.
Figure P8.17
19. The chewing muscle, the masseter, is
one of the strongest in the human body. It is
attached to the mandible (lower jawbone)
as shown in Figure P8.19a. The jawbone is
pivoted about a socket just in front of the
auditory canal. The forces acting on the
jawbone are equivalent to those acting on
the curved bar in Figure P8.19b: c Fr is the
force exerted by the food being chewed
against the jawbone, T is the force of
r
tension in the masseter, and R r is the force
exerted by the socket on the mandible. Find
T r and R r for a person who bites down on a
piece of steak with a force of 50.0 N.
Figure P8.19
20. A hungry 700‐N bear walks out on a
beam in an attempt to retrieve some
“goodies” hanging at the end (Fig. P8.20).
The beam is uniform, weighs 200 N, and is
6.00 m long; the goodies weigh 80.0 N. (a)
Draw a free‐body diagram of the beam. (b)
When the bear is at x = 1.00 m, find the
tension in the wire and the components of
the reaction force at the hinge. (c) If the
wire can withstand a maximum tension of

900 N, what is the maximum distance the
bear can walk before the wire breaks?
Figure P8.20
21. A uniform semicircular sign 1.00 m
in diameter and of weight w is supported
by two wires as shown in Figure P8.21.
What is the tension in each of the wires
supporting the sign?
Figure P8.21
22. A 20.0‐kg floodlight in a park is
supported at the end of a horizontal beam
of negligible mass that is hinged to a pole,
as shown in Figure P8.22. A cable at an
angle of 30.0° with the beam helps to
support the light. Find (a) the tension in the
cable and (b) the horizontal and vertical
forces exerted on the beam by the pole.
Figure P8.22
23. A uniform plank of length 2.00 m
and mass 30.0 kg is supported by three
ropes, as indicated by the blue vectors in
Figure P8.23. Find the tension in each rope
when a 700‐N person is 0.500 m from the
left end.

Figure P8.23
24. A 15.0‐m, 500‐N uniform ladder rests
against a frictionless wall, making an angle
of 60.0° with the horizontal. (a) Find the
horizontal and vertical forces exerted on the
base of the ladder by the Earth when an
800‐N firefighter is 4.00 m from the bottom.
(b) If the ladder is just on the verge of
slipping when the firefighter is 9.00 m up,
what is the coefficient of static friction
between ladder and ground?
25. An 8.00‐m, 200‐N uniform ladder
rests against a smooth wall. The coefficient
of static friction between the ladder and the
ground is 0.600, and the ladder makes a
50.0° angle with the ground. How far up
the ladder can an 800‐N person climb
before the ladder begins to slip?
26. A 1 200‐N uniform boom is
supported by a cable perpendicular to the
boom as in Figure P8.26. The boom is
hinged at the bottom, and a 2 000‐N weight
hangs from its top. Find the tension in the
supporting cable and the components of the
reaction force exerted on the boom by the
hinge.
Figure P8.26
28. One end of a uniform 4.0‐m‐long rod
of weight w is supported by a cable. The
other end rests against a wall, where it is
held by friction. (See Fig. P8.28.) The
coefficient of static friction between the wall
and the rod is μs = 0.50. Determine the
minimum distance x from point A at which
an additional weight w (the same as the
weight of the rod) can be hung without
causing the rod to slip at point A.
Figure P8.2

Chapter 10 Problems
Section 10.1 Temperature and the Zeroth
Law of Thermodynamics
Section 10.2 Thermometers and
Temperature Scales
1. For each of the following
temperatures, find the equivalent
temperature on the indicated scale: (a)
–273.15°C on the Fahrenheit scale, (b) 98.6°F
on the Celsius scale, and (c) 100 K on the
Fahrenheit scale.
2. The pressure in a constant‐volume
gas thermometer is 0.700 atm at 100°C and
0.512 atm at 0°C. (a) What is the
temperature when the pressure is 0.0400
atm? (b) What is the pressure at 450°C?
Section 10.3 Thermal Expansion of Solids
and Liquids
10. A cylindrical brass sleeve is to be
shrink‐fitted over a brass shaft whose
diameter is 3.212 cm at 0°C. The diameter of
the sleeve is 3.196 cm at 0°C. (a) To what
temperature must the sleeve be heated
before it will slip over the shaft? (b)
Alternatively, to what temperature must
the shaft be cooled before it will slip into
the sleeve?
11. The New River Gorge bridge in West
Virginia is a 518‐m‐long steel arch. How
much will its length change between
temperature extremes of –20°C and 35°C?
12. A grandfather clock is controlled by
a swinging brass pendulum that is 1.3 m
long at a temperature of 20°C. (a) What is
the length of the pendulum rod when the
temperature drops to 0.0°C? (b) If a
pendulum’s period is given by
T = 2π L / g , where L is its length, does the
change in length of the rod cause the clock
to run fast or slow?
14. A cube of solid aluminum has a
volume of 1.00 m 3 at 20°C. What
temperature change is required to produce
a 100‐cm 3 increase in the volume of the
cube?
15. A brass ring of diameter 10.00 cm at
20.0°C is heated and slipped over an
aluminum rod of diameter 10.01 cm at
20.0°C. Assuming the average coefficients
of linear expansion are constant, (a) to what
temperature must the combination be
cooled to separate the two metals? Is that
temperature attainable? (b) What if the
aluminum rod were 10.02 cm in diameter?
17. A gold ring has an inner diameter of
2.168 cm at a temperature of 15.0°C.
Determine its inner diameter at 100°C (αgold
= 1.42 × 10 –5 °C –1 ).

21. An automobile fuel tank is filled to
the brim with 45 L (12 gal) of gasoline at
10°C. Immediately afterward, the vehicle is
parked in the sunlight, where the
temperature is 35°C. How much gasoline
overflows from the tank as a result of the
expansion? (Neglect the expansion of the
tank.)
23. The average coefficient of volume
expansion for carbon tetrachloride is 5.81 ×
10 –4 (°C) –1 . If a 50.0‐gal steel container is
filled completely with carbon tetrachloride
when the temperature is 10.0°C, how much
will spill over when the temperature rises
to 30.0°C?
Section 10.4 Macroscopic Description of
an Ideal Gas
27. One mole of oxygen
gas is at a pressure of 6.00 atm and a
temperature of 27.0°C. (a) If the gas is
heated at constant volume until the
pressure triples, what is the final
temperature? (b) If the gas is heated so that
both the pressure and volume are doubled,
what is the final temperature?
28. Gas is contained in an 8.0‐L vessel at
a temperature of 20°C and a pressure of 9.0
atm. (a) Determine the number of moles of
gas in the vessel. (b) How many molecules
are in the vessel?
29. (a) An ideal gas occupies a volume
of 1.0 cm 3 at 20°C and atmospheric
pressure. Determine the number of
molecules of gas in the container. (b) If the
pressure of the 1.0‐cm 3 volume is reduced
to 1.0 × 10 –11 Pa (an extremely good
vacuum) while the temperature remains
constant, how many moles of gas remain in
the container?
30. A tank having a volume of 0.100 m 3
contains helium gas at 150 atm. How many
balloons can the tank blow up if each filled
balloon is a sphere 0.300 m in diameter at
an absolute pressure of 1.20 atm?
31. A cylinder with a movable piston
contains gas at a temperature of 27.0°C, a
volume of 1.50 m 3 , and an absolute pressure
of 0.200 × 10 5 Pa. What will be its final
temperature if the gas is compressed to
0.700 m 3 and the absolute pressure
increases to 0.800 × 10 5 Pa?

Chapter 11 Problems
Section 11.1 Heat and Internal Energy
Section 11.2 Specific Heat
1. Water at the top of Niagara Falls has
a temperature of 10.0°C. If it falls a distance
of 50.0 m and all of its potential energy goes
into heating the water, calculate the
temperature of the water at the bottom of
the falls.
3. Lake Erie contains roughly 4.00 × 10 11
m 3 of water. (a) How much energy is
required to raise the temperature of that
volume of water from 11.0°C to 12.0°C? (b)
How many years would it take to supply
this amount of energy by using the 1 000‐
MW exhaust energy of an electric power
plant?
4. An aluminum rod is 20.0 cm long at
20°C and has a mass of 350 g. If 10 000 J of
energy is added to the rod by heat, what is
the change in length of the rod?
5. How many joules of energy are
required to raise the temperature of 100 g of
gold from 20.0°C to 100°C?
6. As part of an exercise routine, a 50.0‐
kg person climbs 10.0 meters up a vertical
rope. How many (food) Calories are
expended in a single climb up the rope? (1
food Calorie = 10 3 calories)
8. The apparatus shown in Figure P11.8
was used by Joule to measure the
mechanical equivalent of heat. Work is
done on the water by a rotating paddle
wheel, which is driven by two blocks
falling at a constant speed. The temperature
of the stirred water increases due to the
friction between the water and the paddles.
If the energy lost in the bearings and
through the walls is neglected, then the loss
in potential energy associated with the
blocks equals the work done by the paddle
wheel on the water. If each block has a
mass of 1.50 kg and the insulated tank is
filled with 200 g of water, what is the
increase in temperature of the water after
the blocks fall through a distance of 3.00 m?
Figure P11.8 The falling weights rotate the
paddles, causing the temperature of the
water to increase.

9. A 5.00‐g lead bullet traveling at 300
m/s is stopped by a large tree. If half the
kinetic energy of the bullet is transformed
into internal energy and remains with the
bullet while the other half is transmitted to
the tree, what is the increase in temperature
of the bullet?
10. A 1.5‐kg copper block is given an
initial speed of 3.0 m/s on a rough
horizontal surface. Because of friction, the
block finally comes to rest. (a) If the block
absorbs 85% of its initial kinetic energy as
internal energy, calculate its increase in
temperature. (b) What happens to the
remaining energy?
11. A 200‐g aluminum cup contains 800
g of water in thermal equilibrium with the
cup at 80°C. The combination of cup and
water is cooled uniformly so that the
temperature decreases by 1.5°C per minute.
At what rate is energy being removed?
Express your answer in watts.
Section 11.3 Calorimetry
12. Lead pellets, each of mass 1.00 g, are
heated to 200°C. How many pellets must be
added to 500 g of water that is initially at
20.0°C to make the equilibrium temperature
25.0°C? Neglect any energy transfer to or
from the container.
15. An aluminum cup contains 225 g of
water and a 40‐g copper stirrer, all at 27°C.
A 400‐g sample of silver at an initial
temperature of 87°C is placed in the water.
The stirrer is used to stir the mixture until it
reaches its final equilibrium temperature of
32°C. Calculate the mass of the aluminum
cup.
16. It is desired to cool iron parts from
500°F to 100°F by dropping them into water
that is initially at 75°F. Assuming that all
the heat from the iron is transferred to the
water and that none of the water
evaporates, how many kilograms of water
are needed per kilogram of iron?
17. A 100‐g aluminum calorimeter
contains 250 g of water. The two substances
are in thermal equilibrium at 10°C. Two
metallic blocks are placed in the water. One
is a 50‐g piece of copper at 80°C. The other
sample has a mass of 70 g and is originally
at a temperature of 100°C. The entire
system stabilizes at a final temperature of
20°C. Determine the specific heat of the
unknown second sample.
Section 11.4 Latent Heat and Phase
Change
20. A 50‐g ice cube at 0°C is heated until
45 g has become water at 100°C and 5.0 g
has become steam at 100°C. How much
energy was added to accomplish the
transformation?
21. A 100‐g cube of ice at 0°C is dropped
into 1.0 kg of water that was originally at
80°C. What is the final temperature of the
water after the ice has melted?
22. How much energy is required to
change a 40‐g ice cube from ice at –10°C to
steam at 110°C?

31. Steam at 100°C is added to ice at 0°C.
(a) Find the amount of ice melted and the
final temperature when the mass of steam
is 10 g and the mass of ice is 50 g. (b) Repeat
with steam of mass 1.0 g and ice of mass 50
g.
Section 11.5 Energy Transfer
32. The average thermal conductivity of
the walls (including windows) and roof of a
house in Figure P11.32 is 4.8 × 10 –4 kW/m ∙
°C, and their average thickness is 21.0 cm.
The house is heated with natural gas, with a
heat of combustion (energy released per
cubic meter of gas burned) of 9300 kcal/m 3 .
How many cubic meters of gas must be
burned each day to maintain an inside
temperature of 25.0°C if the outside
temperature is 0.0°C? Disregard radiation
and energy loss by heat through the
ground.
Figure P11.32
33. (a) Find the rate of energy flow
through a copper block of cross‐sectional
area 15 cm 2 and length 8.0 cm when a
temperature difference of 30°C is
established across the block. Repeat the
calculation, assuming that the material is
(b) a block of stagnant air with the given
dimensions; (c) a block of wood with the
given dimensions.
35. A steam pipe is covered with 1.50‐cm‐
thick insulating material of thermal
conductivity 0.200 cal/cm ∙ °C ∙ s. How
much energy is lost every second when the
steam is at 200°C and the surrounding air is
at 20.0°C? The pipe has a circumference of
800 cm and a length of 50.0 m. Neglect
losses through the ends of the pipe.

Chapter 12 Problems
Section 12.1 Work in Thermodynamic
Processes
2. Sketch a PV diagram and find the
work done by the gas during the following
stages: (a) A gas is expanded from a volume
of 1.0 L to 3.0 L at a constant pressure of 3.0
atm. (b) The gas is then cooled at constant
volume until the pressure falls to 2.0 atm.
(c) The gas is then compressed at a constant
pressure of 2.0 atm from a volume of 3.0 L
to 1.0 L. (Note: Be careful of signs.) (d) The
gas is heated until its pressure increases
from 2.0 atm to 3.0 atm at a constant
volume. (e) Find the net work done during
the complete cycle.
5. A gas expands from I to F along the
three paths indicated in Figure P12.5.
Calculate the work done on the gas along
paths (a) IAF, (b) IF, and (c) IBF.
Figure P12.5 (Problems 5 and 15)
9. One mole of an ideal gas initially at a
temperature of Ti = 0°C undergoes an
expansion at a constant pressure of 1.00 atm
to four times its original volume. (a)
Calculate the new temperature Tf of the gas.
(b) Calculate the work done on the gas
during the expansion.
10. (a) Determine the work done on a
fluid that expands from i to f as indicated in
Figure P12.10. (b) How much work is done
on the fluid if it is compressed from f to i
along the same path?
Figure P12.10
Section 12.2 The First Law of
Thermodynamics
13. A gas is compressed at a constant
pressure of 0.800 atm from 9.00 L to 2.00 L.
In the process, 400 J of energy leaves the gas
by heat. (a) What is the work done on the
gas? (b) What is the change in its internal
energy?
16. A gas is taken through the cyclic
process described by Figure P12.16. (a) Find
the net energy transferred to the system by
heat during one complete cycle. (b) If the
cycle is reversed—that is, the process

follows the path ACBA—what is the net
energy transferred by heat per cycle?
Figure P12.16 (Problems 16 and 18)
22. One mole of gas initially at a
pressure of 2.00 atm and a volume of 0.300
L has an internal energy equal to 91.0 J. In
its final state, the gas is at a pressure of 1.50
atm and a volume of 0.800 L, and its
internal energy equals 180 J. For the paths
IAF, IBF, and IF in Figure P12.22, calculate
(a) the work done on the gas and (b) the net
energy transferred to the gas by heat in the
process.
Figure P12.22
Section 12.3 Heat Engines and the Second
Law of Thermodynamics
23. A heat engine operates between two
reservoirs at temperatures of 20°C and
300°C. What is the maximum efficiency
possible for this engine?
24. A steam engine has a boiler that
operates at 300°F, and the temperature of
the exhaust is 150°F. Find the maximum
efficiency of this engine.
25. The energy absorbed by an engine is
three times greater than the work it
performs. (a) What is its thermal efficiency?
(b) What fraction of the energy absorbed is
expelled to the cold reservoir?
27. One of the most efficient engines
ever built is a coal‐fired steam turbine
engine in the Ohio River valley, driving an
electric generator as it operates between 1
870°C and 430°C. (a) What is its maximum
theoretical efficiency? (b) Its actual

efficiency is 42.0%. How much mechanical
power does the engine deliver if it absorbs
1.40 × 10 5 J of energy each second from the
hot reservoir?
29. An engine absorbs 1 700 J from a hot
reservoir and expels 1 200 J to a cold
reservoir in each cycle. (a) What is the
engine’s efficiency? (b) How much work is
done in each cycle? (c) What is the power
output of the engine if each cycle lasts for
0.300 s?
31. In one cycle, a heat engine absorbs
500 J from a high‐temperature reservoir and
expels 300 J to a low‐temperature reservoir.
If the efficiency of this engine is 60% of the
efficiency of a Carnot engine, what is the
ratio of the low temperature to the high
temperature in the Carnot engine?
32. A heat engine operates in a Carnot
cycle between 80.0°C and 350°C. It absorbs
21 000 J of energy per cycle from the hot
reservoir. The duration of each cycle is 1.00
s. (a) What is the mechanical power output
of this engine? (b) How much energy does
it expel in each cycle by heat?
33. A nuclear power plant has an
electrical power output of 1 000 MW and
operates with an efficiency of 33%. If excess
energy is carried away from the plant by a
river with a flow rate of 1.0 × 10 6 kg/s, what
is the rise in temperature of the flowing
water?

Chapter 13 Problems
Section 13.1 Hooke’s Law
1. A 0.40‐kg object is attached to a
spring with force constant 160 N/m so that
the object is allowed to move on a
horizontal frictionless surface. The object is
released from rest when the spring is
compressed 0.15 m. Find (a) the force on the
object and (b) its acceleration at that instant.
2. A load of 50 N attached to a spring
hanging vertically stretches the spring 5.0
cm. The spring is now placed horizontally
on a table and stretched 11 cm. (a) What
force is required to stretch the spring by
that amount? (b) Plot a graph of force (on
the y‐axis) versus spring displacement from
the equilibrium position along the x‐axis.
Section 13.2 Elastic Potential Energy
7. A slingshot consists of a light leather
cup containing a stone. The cup is pulled
back against two parallel rubber bands. It
takes a force of 15 N to stretch either one of
these bands 1.0 cm. (a) What is the potential
energy stored in the two bands together
when a 50‐g stone is placed in the cup and
pulled back 0.20 m from the equilibrium
position? (b) With what speed does the
stone leave the slingshot?
10. An automobile having a mass of 1
000 kg is driven into a brick wall in a safety
test. The bumper behaves like a spring with
constant 5.00 × 10 6 N/m and is compressed
3.16 cm as the car is brought to rest. What
was the speed of the car before impact,
assuming that no energy is lost in the
collision with the wall?
12. A 1.50‐kg block at rest on a tabletop
is attached to a horizontal spring having
constant 19.6 N/m, as in Figure P13.12. The
spring is initially unstretched. A constant
20.0‐N horizontal force is applied to the
object, causing the spring to stretch. (a)
Determine the speed of the block after it has
moved 0.300 m from equilibrium if the
surface between the block and tabletop is
frictionless. (b) Answer part (a) if the
coefficient of kinetic friction between block
and tabletop is 0.200.
Figure P13.12
13. A 10.0‐g bullet is fired into, and
embeds itself in, a 2.00‐kg block attached to
a spring with a force constant of 19.6 N/m
and whose mass is negligible. How far is
the spring compressed if the bullet has a
speed of 300 m/s just before it strikes the
block and the block slides on a frictionless
surface? [Note: You must use conservation
of momentum in this problem. Why?]
Section 13.3 Comparing Simple Harmonic
Motion with Uniform Circular Motion

Section 13.4 Position, Velocity, and
Acceleration as a Function of Time
15. A 0.40‐kg object connected to a light
spring with a force constant of 19.6 N/m
oscillates on a frictionless horizontal
surface. If the spring is compressed 4.0 cm
and released from rest, determine (a) the
maximum speed of the object, (b) the speed
of the object when the spring is compressed
1.5 cm, and (c) the speed of the object when
the spring is stretched 1.5 cm. (d) For what
value of x does the speed equal one‐half the
maximum speed?
16. An object–spring system oscillates
with an amplitude of 3.5 cm. If the spring
constant is 250 N/m and the object has a
mass of 0.50 kg, determine (a) the
mechanical energy of the system, (b) the
maximum speed of the object, and (c) the
maximum acceleration of the object.
18. A 50.0‐g object is attached to a
horizontal spring with a force constant of
10.0 N/m and released from rest with an
amplitude of 25.0 cm. What is the velocity
of the object when it is halfway to the
equilibrium position if the surface is
frictionless?
23. A spring stretches 3.9 cm when a 10‐
g object is hung from it. The object is
replaced with a block of mass 25 g that
oscillates in simple harmonic motion.
Calculate the period of motion.
26. The motion of an object is described
by the equation
x =
⎛π
t ⎞
( 0.
30 m)
cos⎜
⎟
⎝ 3 ⎠
Find (a) the position of the object at t = 0
and t = 0.60 s, (b) the amplitude of the
motion, (c) the frequency of the motion, and
(d) the period of the motion.
27. A 2.00‐kg object on a frictionless
horizontal track is attached to the end of a
horizontal spring whose force constant is
5.00 N/m. The object is displaced 3.00 m to
the right from its equilibrium position and
then released, initiating simple harmonic
motion. (a) What is the force (magnitude
and direction) acting on the object 3.50 s
after it is released? (b) How many times
does the object oscillate in 3.50 s?
Section 13.8 Frequency, Amplitude, and
Wavelength
36. A cork on the surface of a pond bobs
up and down two times per second on
ripples having a wavelength of 8.50 cm. If
the cork is 10.0 m from shore, how long
does it take a ripple passing the cork to
reach the shore?
37. A wave traveling in the positive x‐
direction has a frequency of 25.0 Hz, as in
Figure P13.37. Find the (a) amplitude, (b)
wavelength, (c) period, and (d) speed of the
wave.

Figure P13.37
39. If the frequency of oscillation of the
wave emitted by an FM radio station is 88.0
MHz, determine (a) the wave’s period of
vibration and (b) its wavelength. (Radio
waves travel at the speed of light, 3.00 × 10 8
m/s.)