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Section 5.1
1.
Work
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.
3.
A tugboat exerts a constant force of 5.00 × 103 N on a ship moving at constant speed through a harbor.
How much work does the tugboat do on the ship if each moves a distance of 3.00 km?
4.
A shopper in a supermarket pushes a cart with a force of 35 N directed at an angle of 25° downward
from the horizontal. Find the work done by the shopper as she moves down a 50-m length of aisle.
5.
Starting from rest, a 5.00-kg block slides 2.50 m down a rough 30.0° incline. The coefficient of kinetic
friction between the block and the incline is μk = 0.436. Determine (a) the work done by the force of gravity, (b)
the work done by the friction force between block and incline, and (c) the work done by the normal force.
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?
8.
A block of mass 2.50 kg is pushed 2.20 m along a frictionless horizontal table by a constant 16.0-N force
directed 25.0° below the horizontal. Determine the work done by (a) the applied force, (b) the normal force
exerted by the table, (c) the force of gravity, and (d) the net force on the block.
Section 5.2
9.
Kinetic Energy and the Work–Energy Theorem
A mechanic pushes a 2.50 × 103-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?
11.
A person doing a chin-up weighs 700 N, exclusive of the arms. During the first 25.0 cm of the lift, each
arm exerts an upward force of 355 N on the torso. If the upward movement starts from rest, what is the
person’s velocity at that point?
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?
13.
A 70-kg base runner begins his slide into second base when he is moving at a speed of 4.0 m/s. The
coefficient of friction between his clothes and Earth is 0.70. He slides so that his speed is zero just as he reaches
the base. (a) How much mechanical energy is lost due to friction acting on the runner? (b) How far does he
slide?
14.
An outfielder throws a 0.150-kg baseball at a speed of 40.0 m/s and an initial angle of 30.0°. What is the
kinetic energy of the ball at the highest point of its motion?
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.
16.
A 0.60-kg particle has a speed of 2.0 m/s at point A and a kinetic energy of 7.5 J at point B. What is (a)
its kinetic energy at A? (b) its speed at point B? (c) the total work done on the particle as it moves from A to B?
17.
A 2 000-kg car moves down a level highway under the actions of two forces: a 1 000-
N forward force exerted on the drive wheels by the road and a 950-N resistive force. Use the work–energy
theorem to find the speed of the car after it has moved a distance of 20 m, assuming that it starts from rest.
18.
On a frozen pond, a 10-kg sled is given a kick that imparts to it an initial speed of v0 = 2.0 m/s. The
coefficient of kinetic friction between sled and ice is μk = 0.10. Use the work–energy theorem to find the
distance the sled moves before coming to rest.
Section 5.3
Gravitational Potential Energy
Section 5.4
Spring Potential Energy
19.
Find the height from which you would have to drop a ball so that it would have a speed of 9.0 m/s just
before it hits the ground.
20.
A flea is able to jump about 0.5 m. It has been said that if a flea were as big as a human, it would be able
to jump over a 100-story building! When an animal jumps, it converts work done in contracting muscles into
gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is
proportional to its cross-sectional area, and the work done by the muscle is this force times the length of
contraction. If we magnified a flea by a factor of 1 000, the cross section of its muscle would increase by 1 0002
and the length of contraction would increase by 1 000. How high would this “superflea” be able to jump?
(Don’t forget that the mass of the “superflea” increases as well.)
21.
An athlete on a trampoline leaps straight up into the air with an initial speed of 9.0 m/s. Find (a) the
maximum height reached by the athlete relative to the trampoline and (b) the speed of the athlete when she is
halfway up to her maximum height.
22.
Truck suspensions often have “helper springs” that engage at high loads. One such arrangement is a
leaf spring with a helper coil spring mounted on the axle, as shown in Figure P5.22. When the main leaf spring
is compressed by distance y0, the helper spring engages and then helps to support any additional load.
Suppose the leaf spring constant is 5.25 × 105 N/m, the helper spring constant is 3.60 × 105 N/m, and y0 = 0.500
m. (a) What is the compression of the leaf spring for a load of 5.00 = 105 N? (b) How much work is done in
compressing the springs?
Figure P5.22
23.
A daredevil on a motorcycle leaves the end of a ramp with a speed of 35.0 m/s as in Figure P5.23. If his
speed is 33.0 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore
friction and air resistance.
Figure P5.23
24.
A softball pitcher rotates a 0.250-kg ball around a vertical circular path of radius 0.600 m before
releasing it. The pitcher exerts a 30.0-N force directed parallel to the motion of the ball around the complete
circular path. The speed of the ball at the top of the circle is 15.0 m/s. If the ball is released at the bottom of the
circle, what is its speed upon release?
25.
The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a
force of approximately 800 N as it contracts a distance of 7.5 cm in a 75-kg male. How much work can the
biceps muscles (one in each arm) perform in a single contraction? Compare this amount of work with the
energy required to lift a 75-kg person 40 cm in performing a chin-up. Do you think the biceps muscle is the
only muscle involved in performing a chin-up?
Section 5.5
26.
Systems and Energy Conservation
A 50-kg pole vaulter running at 10 m/s vaults over the bar. Her speed when she is above the bar is 1.0
m/s. Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she
crosses the bar.
27.
A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts
from rest and has a speed of 3.00 m/s at the bottom, what is the height of the hill?
28.
A 0.400-kg bead slides on a curved wire, starting from rest at point
frictionless, find the speed of the bead (a) at
and (b) at
.
Figure P5.28 (Problems 28 and 36)
in Figure P5.28. If the wire is
29.
A 5.00-kg steel ball is dropped onto a copper plate from a height of 10.0 m. If the ball leaves a dent 3.20
mm deep in the plate, what is the average force exerted by the plate on the ball during the impact?
30.
A bead of mass m = 5.00 kg is released from point
P5.30. Determine (a) the bead’s speed at points and
moving the bead from to
and slides on the frictionless track shown in Figure
and (b) the net work done by the force of gravity in
.
Figure P5.30
31.
Tarzan swings on a 30.0-m-long vine initially inclined at an angle of 37.0° with the vertical. What is his
speed at the bottom of the swing (a) if he starts from rest? (b) if he pushes off with a speed of 4.00 m/s?
32.
Three objects with masses m1 = 5.0 kg, m2 = 10 kg, and m3 = 15 kg, respectively, are attached by strings
over frictionless pulleys, as indicated in Figure P5.32. The horizontal surface is frictionless and the system is
released from rest. Using energy concepts, find the speed of m3 after it moves down a distance of 4.0 m.
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
34.
A projectile is launched with a speed of 40 m/s at an angle of 60° above the horizontal. Use
conservation of energy to find the maximum height reached by the projectile during its flight.
35.
A 0.250-kg block is placed on a light vertical spring (k = 5.00 × 103 N/m) and pushed downwards,
compressing the spring 0.100 m. After the block is released, it leaves the spring and continues to travel
upwards. What height above the point of release will the block reach if air resistance is negligible?
36.
The wire in Problem 28 (Fig. P5.28) is frictionless between points
. The 0.400-kg bead starts from rest at
. (a) Find its speed at
in mechanical energy as it goes from
to
37.
and
and rough between
. (b) If the bead comes to rest at
and
, find the loss
.
(a) A child slides down a water slide at an amusement park from an initial height h. The slide can be
considered frictionless because of the water flowing down it. Can the equation for conservation of mechanical
energy be used on the child? (b) Is the mass of the child a factor in determining his speed at the bottom of the
slide? (c) The child drops straight down rather than following the curved ramp of the slide. In which case will
he be traveling faster at ground level? (d) If friction is present, how would the conservation-of-energy equation
be modified? (e) Find the maximum speed of the child when the slide is frictionless if the initial height of the
slide is 12.0 m.
38.

(a) A block with a mass m is pulled along a horizontal surface for a distance x by a constant force F at
an angle θ with respect to the horizontal. The coefficient of kinetic friction between block and table is μk. Is the
force exerted by friction equal to μkmg? If not, what is the force exerted by friction? (b) How much work is

done by the friction force and by F ? (Don’t forget the signs.) (c) Identify all the forces that do no work on the
block. (d) Let m = 2.00 kg, x = 4.00 m, θ = 37.0°, F = 15.0 N, and μk = 0.400, and find the answers to parts (a) and
(b).
39.
A 70-kg diver steps off a 10-m tower and drops from rest straight down into the water. If he comes to
rest 5.0 m beneath the surface, determine the average resistive force exerted on him by the water.
40.
An airplane of mass 1.5 × 104 kg is moving at 60 m/s. The pilot then revs up the engine so that the
forward thrust by the air around the propeller becomes 7.5 × 104 N. If the force exerted by air resistance on the
body of the airplane has a magnitude of 4.0 × 104 N, find the speed of the airplane after it has traveled 500 m.
Assume that the airplane is in level flight throughout this motion.
41.
A 2.1 × 103-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 × 103 N impedes the motion, find the speed of the car at the bottom
of the driveway.
42.
A 25.0-kg child on a 2.00-m-long swing is released from rest when the ropes of the swing make an
angle of 30.0° with the vertical. (a) Neglecting friction, find the child’s speed at the lowest position. (b) If the
actual speed of the child at the lowest position is 2.00 m/s, what is the mechanical energy lost due to friction?
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.
44.
A child slides without friction from a height h along a curved water slide (Fig. P5.44). She is launched
from a height h/5 into the pool. Determine her maximum airborne height y in terms of h and the launch angle
θ.
Figure P5.44
45.
A skier starts from rest at the top of a hill that is inclined 10.5° with respect to the
horizontal. The hillside is 200 m long, and the coefficient of friction between snow and skis is 0.075 0. At the
bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier glide
along the horizontal portion of the snow before coming to rest?
46.
In a circus performance, a monkey is strapped to a sled and both are given an initial speed of 4.0 m/s
up a 20° inclined track. The combined mass of monkey and sled is 20 kg, and the coefficient of kinetic friction
between sled and incline is 0.20. How far up the incline do the monkey and sled move?
47.
An 80.0-kg skydiver jumps out of a balloon at an altitude of 1 000 m and opens the parachute at an
altitude of 200.0 m. (a) Assuming that the total retarding force on the diver is constant at 50.0 N with the
parachute closed and constant at 3 600 N with the parachute open, what is the speed of the diver when he
lands on the ground? (b) Do you think the skydiver will get hurt? Explain. (c) At what height should the
parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How
realistic is the assumption that the total retarding force is constant? Explain.
Section 5.6
48.
Power
A skier of mass 70 kg is pulled up a slope by a motor-driven cable. (a) How much work is required to
pull him 60 m up a 30° slope (assumed frictionless) at a constant speed of 2.0 m/s? (b) What power must a
motor have to perform this task?
49.
Columnist Dave Barry poked fun at the name “The Grand Cities,” adopted by Grand Forks, North
Dakota, and East Grand Forks, Minnesota. Residents of the prairie towns then named a sewage pumping
station for him. At the Dave Barry Lift Station No. 16, untreated sewage is raised vertically by 5.49 m in the
amount of 1 890 000 liters each day. With a density of 1 050 kg/m3, the waste enters and leaves the pump at
atmospheric pressure through pipes of equal diameter. (a) Find the output power of the lift station. (b) Assume
that a continuously operating electric motor with average power 5.90 kW runs the pump. Find its efficiency. In
January 2002, Barry attended the outdoor dedication of the lift station and a festive potluck supper to which
the residents of the different Grand Forks sewer districts brought casseroles, Jell-O® salads, and “bars”
(desserts).
50.
While running, a person dissipates about 0.60 J of mechanical energy per step per kilogram of body
mass. If a 60-kg person develops a power of 70 W during a race, how fast is the person running? (Assume a
running step is 1.5 m long.)
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?
53.
A 1.50 × 103-kg car starts from rest and accelerates uniformly to 18.0 m/s in 12.0 s. Assume that air
resistance remains constant at 400 N during this time. Find (a) the average power developed by the engine and
(b) the instantaneous power output of the engine at t = 12.0 s, just before the car stops accelerating.
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
55.
Work Done by a Varying Force
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
57.
The force acting on an object is given by Fx = (8x – 16) N, where x is in meters. (a) Make a plot of this
force versus x from x = 0 to x = 3.00 m. (b) From your graph, find the net work done by the force as the object
moves from x = 0 to x = 3.00 m.
Additional Problems
58.
A 2.0-m-long pendulum is released from rest when the support string is at an angle of 25° with the
vertical. What is the speed of the bob at the bottom of the swing?
59.
An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to
230 N. (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do in
pulling the bow?
60.
A block of mass 12.0 kg slides from rest down a frictionless 35.0° incline and is stopped by a strong
spring with k = 3.00 × 104 N/m. The block slides 3.00 m from the point of release to the point where it comes to
rest against the spring. When the block comes to rest, how far has the spring been compressed?
61.
(a) A 75-kg man steps out a window and falls (from rest) 1.0 m to a sidewalk. What is his speed just
before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for
his fall is an approximately 0.50-cm give in the pads of his feet. Calculate the average force exerted on him by
the ground in this situation. This average force is sufficient to cause damage to cartilage in the joints or to
break bones.
62.
A toy gun uses a spring to project a 5.3-g soft rubber sphere horizontally. The spring constant is 8.0
N/m, the barrel of the gun is 15 cm long, and a constant frictional force of 0.032 N exists between barrel and
projectile. With what speed does the projectile leave the barrel if the spring was compressed 5.0 cm for this
launch?
63.
Two objects are connected by a light string passing over a light, frictionless pulley as in Figure P5.63.
The 5.00-kg object is released from rest at a point 4.00 m above the floor. (a) Determine the speed of each object
when the two pass each other. (b) Determine the speed of each object at the moment the 5.00-kg object hits the
floor. (c) How much higher does the 3.00-kg object travel after the 5.00-kg object hits the floor?
Figure P5.63
64.
Two blocks, A and B (with mass 50 kg and 100 kg, respectively), are connected by a string, as shown in
Figure P5.64. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block
A and the incline is μk = 0.25. Determine the change in the kinetic energy of block A as it moves from
distance of 20 m up the incline if the system starts from rest.
to
,a
Figure P5.64
65.
A 200-g particle is released from rest at point A on the inside of a smooth hemispherical bowl of radius
R = 30.0 cm (Fig. P5.65). Calculate (a) its gravitational potential energy at A relative to B, (b) its kinetic energy
at B, (c) its speed at B, (d) its potential energy at C relative to B, and (e) its kinetic energy at C.
Figure P5.65
66.
Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is 1
kilocalorie, which we define in Chapter 11 as 1 kcal = 4 186 J. Metabolizing 1 gram of fat can release 9.00 kcal. A
student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football
stadium as fast as she can and as many times as necessary. Is this in itself a practical way to lose weight? To
evaluate the program, suppose she runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For simplicity,
ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human
muscles is 20.0%. This means that when your body converts 100 J from metabolizing fat, 20 J goes into doing
mechanical work (here, climbing stairs). The remainder goes into internal energy. Assume the student’s mass
is 50.0 kg. (a) How many times must she run the flight of stairs to lose 1 pound of fat? (b) What is her average
power output, in watts and in horsepower, as she is running up the stairs?
67.
In terms of saving energy, bicycling and walking are far more efficient means of transportation than is
travel by automobile. For example, when riding at 10.0 mi/h, a cyclist uses food energy at a rate of about 400
kcal/h above what he would use if he were merely sitting still. (In exercise physiology, power is often
measured in kcal/h rather than in watts. Here, 1 kcal = 1 nutritionist’s Calorie = 4 186 J.) Walking at 3.00 mi/h
requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for
travel by car. Gasoline yields about 1.30 × 108 J/gal. Find the fuel economy in equivalent miles per gallon for a
person (a) walking and (b) bicycling.
68.
An 80.0-N box is pulled 20.0 m up a 30° incline by an applied force of 100 N that points upwards,
parallel to the incline. If the coefficient of kinetic friction between box and incline is 0.220, calculate the change
in the kinetic energy of the box.
69.
A ski jumper starts from rest 50.0 m above the ground on a frictionless track and flies off the track at an
angle of 45.0° above the horizontal and at a height of 10.0 m above the level ground. Neglect air resistance. (a)
What is her speed when she leaves the track? (b) What is the maximum altitude she attains after leaving the
track? (c) Where does she land relative to the end of the track?
70.
A 5.0-kg block is pushed 3.0 m up a vertical wall with constant speed by a constant force of magnitude
F applied at an angle of θ = 30° with the horizontal, as shown in Figure P5.70. If the coefficient of kinetic

friction between block and wall is 0.30, determine the work done by (a) F , (b) the force of gravity, and (c) the
normal force between block and wall. (d) By how much does the gravitational potential energy increase during
the block’s motion?
Figure P5.70
71.
The ball launcher in a pinball machine has a spring with a force constant of 1.20 N/cm (Fig. P5.71). The
surface on which the ball moves is inclined 10.0° with respect to the horizontal. If the spring is initially
compressed 5.00 cm, find the launching speed of a 0.100-kg ball when the plunger is released. Friction and the
mass of the plunger are negligible.
Figure P5.71
72.
The masses of the javelin, discus, and shot are 0.80 kg, 2.0 kg, and 7.2 kg, respectively, and record
throws in the corresponding track events are about 98 m, 74 m, and 23 m, respectively. Neglecting air
resistance, (a) calculate the minimum initial kinetic energies that would produce these throws, and (b) estimate
the average force exerted on each object during the throw, assuming the force acts over a distance of 2.0 m. (c)
Do your results suggest that air resistance is an important factor?
73.
Jane, whose mass is 50.0 kg, needs to swing across a river filled with crocodiles in order to rescue

Tarzan, whose mass is 80.0 kg. However, she must swing into a constant horizontal wind force F on a vine
that is initially at an angle of θ with the vertical. (See Fig. P5.73.) In the figure, D = 50.0 m, F = 110 N, L = 40.0 m,
and θ = 50.0°. (a) With what minimum speed must Jane begin her swing in order to just make it to the other
side? (Hint: First determine the potential energy that can be associated with the wind force. Because the wind
force is constant, use an analogy with the constant gravitational force.) (b) Once the rescue is complete, Tarzan
and Jane must swing back across the river. With what minimum speed must they begin their swing?
Figure P5.73
74.
A hummingbird is able to hover because, as the wings move downwards, they exert a downward force
on the air. Newton’s third law tells us that the air exerts an equal and opposite force (upwards) on the wings.
The average of this force must be equal to the weight of the bird when it hovers. If the wings move through a
distance of 3.5 cm with each stroke, and the wings beat 80 times per second, determine the work performed by
the wings on the air in 1 minute if the mass of the hummingbird is 3.0 grams.
75.
A child’s pogo stick (Fig. P5.75) stores energy in a spring (k = 2.50 × 104 N/m). At position
m), the spring compression is a maximum and the child is momentarily at rest. At position
is relaxed and the child is moving upwards. At position
(x1 = –0.100
(x = 0), the spring
, the child is again momentarily at rest at the top of
the jump. Assuming that the combined mass of child and pogo stick is 25.0 kg, (a) calculate the total energy of
the system if both potential energies are zero at x = 0, (b) determine x2, (c) calculate the speed of the child at x =
0, (d) determine the value of x for which the kinetic energy of the system is a maximum, and (e) obtain the
child’s maximum upward speed.
Figure P5.75
76.
A 2.00-kg block situated on a rough incline is connected to a spring of negligible mass having a spring
constant of 100 N/m (Fig. P5.76). The block is released from rest when the spring is unstretched, and the pulley
is frictionless. The block moves 20.0 cm down the incline before coming to rest. Find the coefficient of kinetic
friction between block and incline.
Figure P5.76
77.
In the dangerous “sport” of bungee jumping, a daring student jumps from a hot-air balloon with a
specially designed elastic cord attached to his waist, as shown in Figure P5.77. The unstretched length of the
cord is 25.0 m, the student weighs 700 N, and the balloon is 36.0 m above the surface of a river below.
Calculate the required force constant of the cord if the student is to stop safely 4.00 m above the river.
© Jamie Budge/Corbis
Figure P5.77 Bungee jumping. (Problems 77 and 82)
78.
An object of mass m is suspended from the top of a cart by a string of length L as in Figure P5.78a. The
cart and object are initially moving to the right at a constant speed v0. The cart comes to rest after colliding and
sticking to a bumper, as in Figure P5.78b, and the suspended object swings through an angle θ. (a) Show that
the initial speed is v0  2 gL1  cos  . (b) If L = 1.20 m and θ = 35.0°, find the initial speed of the cart. (Hint:
The force exerted by the string on the object does no work on the object.)
Figure P5.78
79.
A truck travels uphill with constant velocity on a highway with a 7.0° slope. A 50-kg package sits on
the floor of the back of the truck and does not slide, due to a static frictional force. During an interval in which
the truck travels 340 m, what is the net work done on the package? What is the work done on the package by
the force of gravity, the normal force, and the friction force?
80.
As part of a curriculum unit on earthquakes, suppose that 375 000 British schoolchildren stand on their
chairs and simultaneously jump down to the floor. Seismographers around the country see whether they can
detect the resulting ground tremor. (This experiment was actually based on a suggestion by the children
themselves.) (a) Find the energy released in the experiment. Model the children as having average mass 36.0
kg and as stepping from chair seats 38.0 cm above the floor. (b) Most of the energy is converted very rapidly
into internal energy within the bodies of the children and the floors of the school buildings. Assume that 1% of
the energy is carried away by a seismic wave. The magnitude of an earthquake on the Richter scale is given by
M 
log E  4.8
1.5
where E is the seismic wave energy in joules. According to this model, what is the magnitude of the
demonstration quake?
81.
A loaded ore car has a mass of 950 kg and rolls on rails with negligible friction. It starts from rest and is
pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 30.0° above the horizontal. The
car accelerates uniformly to a speed of 2.20 m/s in 12.0 s and then continues at constant speed. (a) What power
must the winch motor provide when the car is moving at constant speed? (b) What maximum power must the
motor provide? (c) What total energy transfers out of the motor by work by the time the car moves off the end
of the track, which is of length 1 250 m?
82.
A daredevil wishes to bungee-jump from a hot-air balloon 65.0 m above a carnival midway (Fig. P5.77).
He will use a piece of uniform elastic cord tied to a harness around his body to stop his fall at a point 10.0 m
above the ground. Model his body as a particle and the cord as having negligible mass and a tension force
described by Hooke’s force law. In a preliminary test, hanging at rest from a 5.00-m length of the cord, the
jumper finds that his body weight stretches it by 1.50 m. He will drop from rest at the point where the top end
of a longer section of the cord is attached to the stationary balloon. (a) What length of cord should he use? (b)
What maximum acceleration will he experience?
83.
The system shown in Figure P5.83 consists of a light, inextensible cord, light frictionless pulleys, and
blocks of equal mass. Initially, the blocks are at rest the same height above the ground. The blocks are then
released. Find the speed of block A at the moment when the vertical separation of the blocks is h.
Figure P5.83
84.
A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral
springs under tension, one near each corner of the shelf. Each tray has a mass of 580 g and is rectangular, 45.3
cm by 35.6 cm, and 0.450 cm thick. (a) Show that the top tray in the stack can always be at the same height
above the floor, however many trays are in the dispenser. (b) Find the spring constant each spring should have
in order for the dispenser to function in this convenient way. Is any piece of data unnecessary for this
determination?
85.
In bicycling for aerobic exercise, a woman wants her heart rate to be between 136 and 166 beats per
minute. Assume that her heart rate is directly proportional to her mechanical power output. Ignore all forces
on the woman-plus-bicycle system, except for static friction forward on the drive wheel of the bicycle and an
air resistance force proportional to the square of the bicycler’s speed. When her speed is 22.0 km/h, her heart
rate is 90.0 beats per minute. In what range should her speed be so that her heart rate will be in the range she
wants?
86.
In a needle biopsy, a narrow strip of tissue is extracted from a patient with a hollow needle. Rather than
being pushed by hand, to ensure a clean cut the needle can be fired into the patient’s body by a spring.
Assume the needle has mass 5.60 g, the light spring has force constant 375 N/m, and the spring is originally
compressed 8.10 cm to project the needle horizontally without friction. The tip of the needle then moves
through 2.40 cm of skin and soft tissue, which exerts a resistive force of 7.60 N on it. Next, the needle cuts 3.50
cm into an organ, which exerts a backward force of 9.20 N on it. Find (a) the maximum speed of the needle and
(b) the speed at which a flange on the back end of the needle runs into a stop, set to limit the penetration to
5.90 cm.
87.
The power of sunlight reaching each square meter of the Earth’s surface on a clear day in the tropics is
close to 1 000 W. On a winter day in Manitoba, the power concentration of sunlight can be 100 W/m2. Many
human activities are described by a power-per-footprint-area on the order of 102 W/m2 or less. (a) Consider, for
example, a family of four paying $80 to the electric company every 30 days for 600 kWh of energy carried by
electric transmission to their house, with floor area 13.0 m by 9.50 m. Compute the power-per-area measure of
this energy use. (b) Consider a car 2.10 m wide and 4.90 m long traveling at 55.0 mi/h using gasoline having a
“heat of combustion” of 44.0 MJ/kg with fuel economy 25.0 mi/gallon. One gallon of gasoline has a mass of
2.54 kg. Find the power-per-area measure of the car’s energy use. It can be similar to that of a steel mill where
rocks are melted in blast furnaces. (c) Explain why the direct use of solar energy is not practical for a
conventional automobile.
88.
In 1887 in Bridgeport, Connecticut, C. J. Belknap built the water slide shown in Figure P5.88. A rider on
a small sled, of total mass 80.0 kg, pushed off to start at the top of the slide (point
) with a speed of 2.50 m/s.
The chute was 9.76 m high at the top, 54.3 m long, and 0.51 m wide. Along its length, 725 wheels made friction
negligible. Upon leaving the chute horizontally at its bottom end (point
), the rider skimmed across the
water of Long Island Sound for as much as 50 m, “skipping along like a flat pebble,” before at last coming to
rest and swimming ashore, pulling his sled after him. (a) Find the speed of the sled and rider at point
. (b)
Model the force of water friction as a constant retarding force acting on a particle. Find the work done by water
friction in stopping the sled and rider. (c) Find the magnitude of the force the water exerts on the sled. (d) Find
the magnitude of the force the chute exerts on the sled at point
.
Engraving from Scientific American, July 1888.
Figure P5.88
89.
Three objects with masses m1 = 5.0 kg, m2 = 10 kg, and m3 = 15 kg, respectively, are attached by strings
over frictionless pulleys as indicated in Figure P5.32. The horizontal surface exerts a force of friction of 30 N on
m2. If the system is released from rest, use energy concepts to find the speed of m3 after it moves down 4.0 m.