Download Chapter 5 Problems

Chapter 5 Problems
exerted by the table, (c) the force of gravity, and (d) the
net force on the block.
1, 2, 3 = straightforward, intermediate, challenging
Section 5.1 Work
Section 5.2 Kinetic Energy and the WorkKinetic Energy Theorem
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?
9. A mechanic pushes a 2.50 x 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.
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.
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?
3. A tugboat exerts a constant force of 5.00 x 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?
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 this point?
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.
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?
6. A scraper is drawn over a tooth 20 times, each time
moving a distance of 0.75 cm. The scraper is held
against the tooth with a normal force of 5.0 N.
Assuming a coefficient of kinetic friction of 0.90
between the scraper and the tooth, determine the work
done to clean the tooth.
13. A 70-kg base runner begins his slide into second
base when 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?
7. A sledge loaded with bricks has a total mass of 18.0
kg and is pulled at constant speed by a rope. The rope is
inclined at 20.0° above the horizontal, and 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 of the rope? (b)
How much work is done by the rope on the sledge? (c)
What is the mechanical energy lost due to friction?
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?
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
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 the
bullet 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 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. One is a 1 000-N forward
force exerted on the drive wheels by the road; the other
is a 950-N resistive force. Use the work-kinetic energy
theorem to find the speed of the car after it has moved a
distance of 20 m, assuming it starts from rest.
22. The spring attached to the cart in Figure P5.22 has a
spring constant k = 250 N/m. How much work is done
to straighten the legs? Disregard gravity and assume the
spring is initially unstretched.
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-kinetic energy theorem to find the
distance the sled moves before coming to rest.
Section 5.3 Potential Energy
Section 5.4 Conservative and Nonconservative
Forces
19. A person’s heart and head are 1.3 m and 1.8 m
above the feet, respectively. Determine the potential
energy associated with 0.50 kg of blood in the heart
relative to (a) the feet, (b) the head.
20. A 2.00-kg ball is attached to a ceiling by a 1.00-mlong string. The height of the room is 3.00 m. What is
the gravitational potential energy associated with the
ball relative to (a) the ceiling? (b) the floor? (c) a point
at the same elevation as the ball?
21. The arm, including the hand, in Figure P5.21 has a
mass of 7.0 kg. Treating the arm as if it were a single
point mass m attached to a rigid massless rod at point A
as shown, determine the work that must be done by the
deltoid muscle to raise the arm from position 1 to
position 2.
FIGURE P5.22
23. The triceps muscle can be modeled as a spring of
such a strength that a force of 105 N is required to
stretch it 2.00 cm. Determine the work done to stretch
this muscle 2.00 cm.
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. A 40-N toy is placed in a light swing that is attached
to ropes 2.0 m long. Find the gravitational potential
energy associated with the toy relative to its lowest
position (a) when the ropes are horizontal, (b) when the
ropes make a 30° angle with the vertical, and (c) at the
bottom of the circular arc.
Section 5.5 Conservation of Mechanical Energy
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 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?
FIGURE P5.21
28. A 0.400-kg bead slides on a curved wire, starting
from rest at point A in Figure P5.28. If the wire is
frictionless, find the speed of the bead (a) at B and (b) at
C.
FIGURE P5.30
FIGURE P5.28
29. A gymnast swings on the high bar as shown in
Figure P5.29. Starting from rest directly over the bar, he
swings around the bar while keeping his arms and legs
outstretched. Treating the gymnast as though his entire
mass were concentrated at a point 1.20 m from the bar,
determine his speed as he passes under the bar at
position A.
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, 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 4.0 m.
FIGURE P5.32
FIGURE P5.29
30. A bead of mass m = 5.00 kg is released from point A
and slides on the frictionless track shown in Figure
P5.30. Determine (a) the bead’s speed at points B and C
and (b) the net work done by the force of gravity in
moving the bead from A to C.
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.
travels 0.48 m before coming to rest. Assuming the
attendant exerts a horizontal force on the gurney,
determine the total weight of the gurney plus patient.
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 x 104 kg is moving at 60
m/s. The pilot then revs up the engine so that the
forward thrust exerted by the air around the propeller
becomes 7.5 x 104 N. If the force exerted by air
resistance on the body of the airplane has a magnitude
of 4.0 x 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.
FIGURE P5.33
34. A projectile is launched with a speed of 40 m/s at an
angle of 60° above the horizontal. Find the maximum
height reached by the projectile during its flight by
using conservation of energy.
35. A 0.250-kg block is placed on a light vertical spring
(k = 5.00 x 103 N/m) and pushed downward,
compressing the spring 0.100 m. After the block is
released, it leaves the spring and continues to travel
upward. What height above the point of release will the
block reach if air resistance is negligible?
Section 5.6 Nonconservative Forces,
Nonisolated Systems, and Conservation of
Energy
36. If the wire in Problem 28 (Fig. P5.28) is frictionless
between points A and B and rough between B and C, and
if the 0.400-kg bead starts from rest at A, (a) find its
speed at B. (b) If the bead comes to rest at C, find the
loss in mechanical energy as it goes from B to C.
37. A morsel of food with a mass of 4.2 g is injected
into the esophagus with an initial speed of 2.5 cm/s. On
the way down to the stomach, the walls of the
esophagus exert an upward resistive force of 2.7 x 10 –3
N on the morsel. If the esophagus is 20.0 cm long, with
what speed does the morsel of food enter the stomach?
38. An attendant pushes a patient on a gurney 20.7 m
down a hall with a constant speed of 0.88 m/s, doing 2
000 J of work on the gurney. If the gurney is released, it
41. A 2.1 x 103-kg car starts from rest at the top of a 5.0m-long driveway that is sloped at 20° with the
horizontal. If an average friction force of 4.0 x 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 a frictionless ramp (inclined at 30.0° from the
floor) to the bottom. 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 θ.
FIGURE P5.44
45. A skier starts from rest at the top of a hill that is
inclined at 10.5° with 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 sky diver 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 sky diver will
get hurt? Explain. (c) At what height should the
parachute be opened so that the final speed of the sky
diver 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 5.7 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. A 50.0-kg student climbs a 5.00-m-long rope and
stops at the top. (a) What must her average speed be in
order to match the power output of a 200-W light bulb?
(b) How much work does she do?
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. Water flows over a section of Niagara Falls at the
rate of 1.2 x 106 kg/s and falls 50 m (Fig. P5.51). How
much power is generated by the falling water?
52. A 1.50 x 103–kg car accelerates uniformly from rest
to 10.0 m/s in 3.00 s. Find (a) the work done on the car
in this time interval, (b) the average power delivered by
the engine in this time interval, and (c) the instantaneous
power delivered by the engine at t = 2.00 s.
53. A 1.50 x 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. It moves upward
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 interval? (b) How does
this compare with its power during an upward cruise
with constant speed?
Section 5.8 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.
spring with k = 3.00 x 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?
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?
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 cartilage damage 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 P5.56
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 this force as the object moves
from x = 0 to x = 3.00 m.
Additional Problems
58. A 98.0-N grocery cart is pushed 12.0 m by a shopper
who exerts a constant horizontal force of 40.0 N. If all
frictional forces are neglected and the cart starts from
rest, what is its final speed?
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
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 C to
D, a distance of 20 m up the incline if the system starts
from rest.
68. A catcher “gives” with the ball when he catches a
0.15-kg baseball moving at 25 m/s. (a) If he moves his
glove a distance of 2.0 cm, what is the average force
acting on his hand? (b) Repeat for the case in which his
glove and hand move 10 cm.
FIGURE P5.64
65. A 700-N Marine in basic training climbs a 10.0-m
vertical rope at a constant speed in 8.00 s. What is his
power output?
66. Energy is conventionally measured in Calories as
well as in joules. One Calorie in nutrition is one
kilocalorie, which we define in Chapter 11 as 1 kcal = 4
186 J. Metabolizing one 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 one pound of fat? (b) What
is her average power output, in watts and in horsepower,
as she is running up the stairs?
67. For 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 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 x 108 J/gal. Find the fuel
economy in equivalent miles per gallon for a person (a)
walking, and (b) bicycling.
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 his speed when he leaves the track? (b)
What is the maximum altitude he attains after leaving
the track? (c) Where does he 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
this motion?
FIGURE P5.70
71. The ball launcher in a pinball machine has a spring
that has 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 89 m, 69 m,
and 21 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 man-eating crocodiles in order to rescue
Tarzan. 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). 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? Assume that Tarzan has a mass
of 80.0 kg.
FIGURE P5.73
74. As it plows a parking lot, a snowplow pushes an
ever-growing pile of snow in front of it. Suppose a car
moving through air is similarly modeled as a cylinder
pushing a growing plug of air in front of it. The
originally stationary air is set into motion at the constant
speed v of the cylinder, as in Figure P5.74. In a time ∆t,
a new disk of air of area A and mass ∆m must be moved
a distance v∆t and hence must be given kinetic energy
½(∆m)v2. Using this model, show that the power loss
due to air resistance is A 3/2 and the resistive force is
ρAv2 /2, where ρ is the density of air.
75. A child’s pogo stick (Fig. P5.75) stores energy in a
spring (k = 2.50 x 104 N/m). At position A (x1 = –0.100
m), the spring compression is a maximum and the child
is momentarily at rest. At position B (x = 0), the spring
is relaxed and the child is moving upward. At position
C, 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 the block and incline.
FIGURE P5.76
77. In the dangerous “sport” of bungee jumping, a
daring student jumps from a 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.
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 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 =
2gL( 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