Download CP7e: Ch. 5 Problems

Chapter 5 Problems
1, 2, 3 = straightforward, intermediate,
challenging
= full solution available in Student
Solutions Manual/Study Guide
= coached solution with
hints available at www.cp7e.com
= biomedical application
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.
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 Kinetic Energy and the
Work–Energy Theorem
9.
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-cmlong 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
Energy
Gravitational Potential
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.)
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.
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.
Figure P5.23
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
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 Systems and Energy
Conservation
26.
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
in Figure
P5.28. If the wire is frictionless, find the
speed of the bead (a) at
and (b) at .
Figure P5.28 (Problems 28 and 36)
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
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
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)
Figure P5.33
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.
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
and and
rough between
and . The 0.400-kg
bead starts from rest at . (a) Find its speed
at . (b) If the bead comes to rest at , find
the loss in mechanical energy as it goes
from to .
37.
(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.
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
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
Power
48.
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.
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
Force
Work Done by a Varying
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.
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?
Figure P5.55
53.
A 1.50 × 10 -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.
3
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
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?
comes to rest, how far has the spring been
compressed?
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
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?
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 P5.63
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 to , a
distance of 20 m up the incline if the system
starts from rest.
Figure P5.64
65.
A 200-g particle is released from rest
at point A on the inside of a smooth
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
(x1 = –0.100 m), the
spring compression is a maximum and the
child is momentarily at rest. At position
(x = 0), the spring is relaxed and the child is
moving upwards. At position , 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 P5.76
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.
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.
© 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.)
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?
Figure P5.78
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 .
Figure P5.88
Engraving from Scientific American, July 1888.
© Copyright 2004 Thomson. All rights reserved.
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.