7
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R E V I E W & S U M MA RY
Kinetic Energy The kinetic energy K associated with the motion of a particle of mass m and speed v, where v is well below the speed of light, is
(7-1)
Work Work W is energy transferred to or from an object via a force acting on the object. Energy transferred to the object is positive work, and from the object,
negative work.
Work Done by a Constant Force The work done on a particle by a constant force
during displacement
is
(7-7,7-8)
in which is the constant angle between the directions of
and
. Only the component of
that is along the displacement
can do work on the object. When
two or more forces act on an object, their net work is the sum of the individual works done by the forces, which is also equal to the work that would be done on the
object by the net force
of those forces.
Work and Kinetic Energy For a particle, a change K in the kinetic energy equals the net work W done on the particle:
(7-10)
in which Ki is the initial kinetic energy of the particle and Kf is the kinetic energy after the work is done. Equation 7-10 rearranged gives us
(7-11)
Work Done by the Gravitational Force The work Wg done by the gravitational force
a displacement
on a particle-like object of mass m as the object moves through
is given by
(7-12)
in which is the angle between
and
.
Work Done in Lifting and Lowering an Object The work Wa done by an applied force as a particle-like object is either lifted or lowered is related to the
work Wg done by the gravitational force and the change K in the object's kinetic energy by
(7-15)
If Kf = Ki, then Eq. 7-15 reduces to
(7-16)
which tells us that the applied force transfers as much energy to the object as the gravitational force transfers from it.
Spring Force The force
from a spring is
(7-20)
where
is the displacement of the spring's free end from its position when the spring is in its relaxed state (neither compressed nor extended), and k is the spring
constant (a measure of the spring's stiffness). If an x axis lies along the spring, with the origin at the location of the spring's free end when the spring is in its relaxed
state, Eq. 7-20 can be written as
(7-21)
A spring force is thus a variable force: It varies with the displacement of the spring's free end.
Work Done by a Spring Force If an object is attached to the spring's free end, the work Ws done on the object by the spring force when the object is moved
from an initial position xi to a final position xf is
(7-25)
If xi = 0 and xf = x, then Eq. 7-25 becomes
(7-26)
Work Done by a Variable Force When the force
on a particle-like object depends on the position of the object, the work done by
on the object while
the object moves from an initial position ri with coordinates (xi, yi, zi) to a final position rf with coordinates (xf, yf, zf) must be found by integrating the force. If we
assume that component Fx may depend on x but not on y or z, component Fy may depend on y but not on x or z, and component Fz may depend on z but not on x or y,
then the work is
(7-36)
If
has only an x component, then Eq. 7-36 reduces to
(7-32)
Power The power due to a force is the rate at which that force does work on an object. If the force does work W during a time interval t, the average power due to
the force over that time interval is
(7-42)
Instantaneous power is the instantaneous rate of doing work:
(7-43)
For a force
at an angle to the direction of travel of the instantaneous velocity
, the instantaneous power is
(7-47,7-48)
CHAPETER 7 PROBLEMS
PROBLEMS
sec. 7-3 Kinetic Energy
1
A proton (mass m = 1.67 10-27 kg) is being accelerated along a straight line at 3.6 1015 m/s2 in a
machine. If the proton has an initial speed of 2.4 107 m/s and travels 3.5 cm, what then is (a) its speed and
(b) the increase in its kinetic energy?
Answer:
(a) 2.9 107 m/s; (b) 2.1 10-13 J
2 If a Saturn V rocket with an Apollo spacecraft attached had a combined mass of 2.9 105 kg and reached a
speed of 11.2 km/s, how much kinetic energy would it then have?
3
On August 10, 1972, a large meteorite skipped across the atmosphere above the western United
States and western Canada, much like a stone skipped across water. The accompanying fireball was so
bright that it could be seen in the daytime sky and was brighter than the usual meteorite trail. The
meteorite's mass was about 4 106 kg; its speed was about 15 km/s. Had it entered the atmosphere
vertically, it would have hit Earth's surface with about the same speed. (a) Calculate the meteorite's loss of
kinetic energy (in joules) that would have been associated with the vertical impact. (b) Express the energy
as a multiple of the explosive energy of 1 megaton of TNT, which is 4.2 10 15 J. (c) The energy associated
with the atomic bomb explosion over Hiroshima was equivalent to 13 kilotons of TNT. To how many
Hiroshima bombs would the meteorite impact have been equivalent?
Answer:
(a) 5 1014 J; (b) 0.1 megaton TNT; (c) 8 bombs
4 A bead with mass 1.8 10-2 kg is moving along a wire in the positive direction of an x axis. Beginning at
time t = 0, when the bead passes through x = 0 with speed 12 m/s, a constant force acts on the bead. Figure
7-22 indicates the bead's position at these four times: t0 = 0, t1 = 1.0 s, t2 = 2.0 s, and t3 = 3.0 s. The bead
momentarily stops at t = 3.0 s. What is the kinetic energy of the bead at t = 10 s?
Figure 7-22 Problem 4.
5 A father racing his son has half the kinetic energy of the son, who has half the mass of the father. The
father speeds up by 1.0 m/s and then has the same kinetic energy as the son. What are the original speeds
of (a) the father and (b) the son?
Answer:
(a) 2.4 m/s; (b) 4.8 m/s
6
A force
is applied to a bead as the bead is moved along a straight wire through displacement +5.0 cm.
The magnitude of
is set at a certain value, but the angle between
be chosen. Figure 7-23 gives the work W done by
much work is done by
and the bead's displacement can
on the bead for a range of values; W0 = 25 J. How
if is (a) 64 and (b) 147?
Figure 7-23 Problem 6.
sec. 7-5 Work and Kinetic Energy
7
A 3.0 kg body is at rest on a frictionless horizontal air track when a constant horizontal force acting in
the positive direction of an x axis along the track is applied to the body. A stroboscopic graph of the
position of the body as it slides to the right is shown in Fig. 7-24. The force is applied to the body at t =
0, and the graph records the position of the body at 0.50 s intervals. How much work is done on the body
by the applied force
between t = 0 and t = 2.0 s?
Figure 7-24 Problem 7.
Answer:
0.96 J
8
A ice block floating in a river is pushed through a displacement
embankment by rushing water, which exerts a force
work does the force do on the block during the displacement?
along a straight
on the block. How much
9 The only force acting on a 2.0 kg canister that is moving in an xy plane has a magnitude of 5.0 N. The
canister initially has a velocity of 4.0 m/s in the positive x direction and some time later has a velocity of
6.0 m/s in the positive y direction. How much work is done on the canister by the 5.0 N force during this
time?
Answer:
20 J
10 A coin slides over a frictionless plane and across an xy coordinate system from the origin to a point with
xy coordinates (3.0 m, 4.0 m) while a constant force acts on it. The force has magnitude 2.0 N and is
directed at a counterclockwise angle of 100 from the positive direction of the x axis. How much work is
done by the force on the coin during the displacement?
11 A 12.0 N force with a fixed orientation does work on a particle as the particle moves through the threedimensional displacement
m. What is the angle between the force and
the displacement if the change in the particle's kinetic energy is (a) +30.0 J and (b) -30.0 J?
Answer:
(a) 62.3; (b) 118
12 A can of bolts and nuts is pushed 2.00 m along an x axis by a broom along the greasy (frictionless) floor
of a car repair shop in a version of shuffleboard. Figure 7-25 gives the work W done on the can by the
constant horizontal force from the broom, versus the can's position x. The scale of the figure's vertical
axis is set by Ws = 6.0 J. (a) What is the magnitude of that force? (b) If the can had an initial kinetic
energy of 3.00 J, moving in the positive direction of the x axis, what is its kinetic energy at the end of the
2.00 m?
Figure 7-25 Problem 12.
13 A luge and its rider, with a total mass of 85 kg, emerge from a downhill track onto a horizontal straight
track with an initial speed of 37 m/s. If a force slows them to a stop at a constant rate of 2.0 m/s2, (a) what
magnitude F is required for the force, (b) what distance d do they travel while slowing, and (c) what work
W is done on them by the force? What are (d) F, (e) d, and (f) W if they, instead, slow at 4.0 m/s2?
Answer:
(a) 1.7 102 N; (b) 3.4 102 m; (c) - 5.8 104 J; (d) 3.4 102 N; (e) 1.7 102 m; (f) - 5.8 104 J
14
Figure 7-26 shows an overhead view of three horizontal forces acting on a cargo canister that was
initially stationary but now moves across a frictionless floor. The force magnitudes are F1 = 3.00 N, F2 =
4.00 N, and F3 = 10.0 N, and the indicated angles are 2 = 50.0 and 3 = 35.0. What is the net work
done on the canister by the three forces during the first 4.00 m of displacement?
Figure 7-26 Problem 14.
15
Figure 7-27 shows three forces applied to a trunk that moves leftward by 3.00 m over a frictionless
floor. The force magnitudes are F1 = 5.00 N, F2 = 9.00 N, and F3 = 3.00 N, and the indicated angle is =
60.0. During the displacement, (a) what is the net work done on the trunk by the three forces and (b)
does the kinetic energy of the trunk increase or decrease?
Figure 7-27 Problem 15.
Answer:
(a) 1.50 J; (b) increases
16
An 8.0 kg object is moving in the positive direction of an x axis. When it passes through x = 0, a
constant force directed along the axis begins to act on it. Figure 7-28 gives its kinetic energy K versus
position x as it moves from x = 0 to x = 5.0 m; K0 = 30.0 J. The force continues to act. What is v when the
object moves back through x = -3.0 m?
Figure 7-28 Problem 16.
sec. 7-6 Work Done by the Gravitational
Force
17
A helicopter lifts a 72 kg astronaut 15 m vertically from the ocean by means of a cable.
The acceleration of the astronaut is g/10. How much work is done on the astronaut by (a) the force from
the helicopter and (b) the gravitational force on her? Just before she reaches the helicopter, what are her
(c) kinetic energy and (d) speed?
Answer:
(a) 12 kJ; (b) - 11 kJ; (c) 1.1 kJ; (d) 5.4 m/s
18
(a) In 1975 the roof of Montreal's Velodrome, with a weight of 360 kN, was lifted by 10 cm so
that it could be centered. How much work was done on the roof by the forces making the lift? (b) In 1960
a Tampa, Florida, mother reportedly raised one end of a car that had fallen onto her son when a jack failed.
If her panic lift effectively raised 4000 N (about
force do on the car?
19
of the car's weight) by 5.0 cm, how much work did her
In Fig. 7-29, a block of ice slides down a frictionless ramp at angle = 50 while an ice worker pulls
on the block (via a rope) with a force
that has a magnitude of 50 N and is directed up the ramp. As the
block slides through distance d = 0.50 m along the ramp, its kinetic energy increases by 80 J. How much
greater would its kinetic energy have been if the rope had not been attached to the block?
Figure 7-29 Problem 19.
Answer:
25 J
20 A block is sent up a frictionless ramp along which an x axis extends upward. Figure 7-30 gives the kinetic
energy of the block as a function of position x; the scale of the figure's vertical axis is set by Ks = 40.0 J.
If the block's initial speed is 4.00 m/s, what is the normal force on the block?
Figure 7-30 Problem
20.
21
A cord is used to vertically lower an initially stationary block of mass M at a constant downward
acceleration of g/4. When the block has fallen a distance d, find (a) the work done by the cord's force on
the block, (b) the work done by the gravitational force on the block, (c) the kinetic energy of the block,
and (d) the speed of the block.
Answer:
(a) - 3Mgd/4; (b) Mgd; (c) Mgd/4; (d) (gd/2)0.5
22 A cave rescue team lifts an injured spelunker directly upward and out of a sinkhole by means of a motordriven cable. The lift is performed in three stages, each requiring a vertical distance of 10.0 m: (a) the
initially stationary spelunker is accelerated to a speed of 5.00 m/s; (b) he is then lifted at the constant
speed of 5.00 m/s; (c) finally he is decelerated to zero speed. How much work is done on the 80.0 kg
rescuee by the force lifting him during each stage?
23
In Fig. 7-31, a constant force
of magnitude 82.0 N is applied to a 3.00 kg shoe box at angle = 53.0,
causing the box to move up a frictionless ramp at constant speed. How much work is done on the box by
when the box has moved through vertical distance h = 0.150 m?
Figure 7-31 Problem 23.
Answer:
4.41 J
24
In Fig. 7-32, a horizontal force
of magnitude 20.0 N is applied to a 3.00 kg psychology book as
the book slides a distance d = 0.500 m up a frictionless ramp at angle = 30.0. (a) During the
displacement, what is the net work done on the book by
, the gravitational force on the book, and the
normal force on the book? (b) If the book has zero kinetic energy at the start of the displacement, what is
its speed at the end of the displacement?
Figure 7-32 Problem 24.
25
In Fig. 7-33, a 0.250 kg block of cheese lies on the floor of a 900 kg elevator cab that is being pulled
upward by a cable through distance d1 = 2.40 m and then through distance d2 = 10.5 m. (a) Through d1, if
the normal force on the block from the floor has constant magnitude FN = 3.00 N, how much work is
done on the cab by the force from the cable? (b) Through d2, if the work done on the cab by the
(constant) force from the cable is 92.61 kJ, what is the magnitude of FN?
Figure 7-33 Problem 25.
Answer:
(a) 25.9 kJ; (b) 2.45 N
sec. 7-7 Work Done by a Spring Force
26 In Fig. 7-9, we must apply a force of magnitude 80 N to hold the block stationary at x = -2.0 cm. From
that position, we then slowly move the block so that our force does +4.0 J of work on the springblock
system; the block is then again stationary. What is the block's position? ( Hint: There are two answers.)
27 A spring and block are in the arrangement of Fig. 7-9. When the block is pulled out to x = +4.0 cm, we
must apply a force of magnitude 360 N to hold it there. We pull the block to x = 11 cm and then release it.
How much work does the spring do on the block as the block moves from xi = +5.0 cm to (a) x = +3.0 cm,
(b) x = -3.0 cm, (c) x = -5.0 cm, and (d) x = -9.0 cm?
Answer:
(a) 7.2 J; (b) 7.2 J; (c) 0; (d) - 25 J
28 During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one
another with large catapults that are made with surgical hose mounted on a window frame. A balloon filled
with dyed water is placed in a pouch attached to the hose, which is then stretched through the width of the
room. Assume that the stretching of the hose obeys Hooke's law with a spring constant of 100 N/m. If the
hose is stretched by 5.00 m and then released, how much work does the force from the hose do on the
balloon in the pouch by the time the hose reaches its relaxed length?
29 In the arrangement of Fig. 7-9, we gradually pull the block from x = 0 to x = +3.0 cm, where it is
stationary. Figure 7-34 gives the work that our force does on the block. The scale of the figure's vertical
axis is set by Ws = 1.0 J. We then pull the block out to x = +5.0 cm and release it from rest. How much
work does the spring do on the block when the block moves from xi = +5.0 cm to (a) x = +4.0 cm, (b) x =
-2.0 cm, and (c) x = -5.0 cm?
Figure 7-34 Problem 29.
Answer:
(a) 0.90 J; (b) 2.1 J; (c) 0
30 In Fig. 7-9a, a block of mass m lies on a horizontal frictionless surface and is attached to one end of a
horizontal spring (spring constant k) whose other end is fixed. The block is initially at rest at the position
where the spring is unstretched (x = 0) when a constant horizontal force in the positive direction of the
x axis is applied to it. A plot of the resulting kinetic energy of the block versus its position x is shown in
Fig. 7-35. The scale of the figure's vertical axis is set by Ks = 4.0 J. (a) What is the magnitude of
What is the value of k?
? (b)
Figure 7-35 Problem
30.
31
The only force acting on a 2.0 kg body as it moves along a positive x axis has an x
component Fx = -6x N, with x in meters. The velocity at x = 3.0 m is 8.0 m/s. (a) What is the velocity of
the body at x = 4.0 m? (b) At what positive value of x will the body have a velocity of 5.0 m/s?
Answer:
(a) 6.6 m/s; (b) 4.7 m
32 Figure 7-36 gives spring force Fx versus position x for the springblock arrangement of Fig. 7-9. The
scale is set by Fs = 160.0 N. We release the block at x = 12 cm. How much work does the spring do on the
block when the block moves from xi = +8.0 cm to (a) x = +5.0 cm, (b) x = -5.0 cm, (c) x = -8.0 cm, and
(d) x = -10.0 cm?
Figure 7-36 Problem 32.
33 The block in Fig. 7-9a lies on a horizontal frictionless surface, and the spring constant is 50 N/m.
Initially, the spring is at its relaxed length and the block is stationary at position x = 0. Then an applied
force with a constant magnitude of 3.0 N pulls the block in the positive direction of the x axis, stretching
the spring until the block stops. When that stopping point is reached, what are (a) the position of the
block, (b) the work that has been done on the block by the applied force, and (c) the work that has been
done on the block by the spring force? During the block's displacement, what are (d) the block's position
when its kinetic energy is maximum and (e) the value of that maximum kinetic energy?
Answer:
(a) 0.12 m; (b) 0.36 J; (c) - 0.36 J; (d) 0.060 m; (e) 0.090 J
sec. 7-8 Work Done by a General
Variable Force
34
A 10 kg brick moves along an x axis. Its acceleration as a function of its position is shown in Fig. 737. The scale of the figure's vertical axis is set by as = 20.0 m/s2. What is the net work performed on the
brick by the force causing the acceleration as the brick moves x from = 0 to x = 8.0 m?
Figure 7-37 Problem 34.
35
The force on a particle is directed along an x axis and given by F = F0(x/x0 - 1). Find the
work done by the force in moving the particle from x = 0 to x = 2x0 by (a) plotting F(x) and measuring the
work from the graph and (b) integrating F(x).
Answer:
(a) 0; (b) 0
36 A 5.0 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force
that varies with position as shown in Fig. 7-38.
Figure 7-38 Problem
36.
The scale of the figure's vertical axis is set by Fs = 10.0 N. How much work is done by the force as the
block moves from the origin to x = 8.0 m?
37
Figure 7-39 gives the acceleration of a 2.00 kg particle as an applied force
moves it from rest along
an x axis from x = 0 to x = 9.0 m. The scale of the figure's vertical axis is set by as = 6.0 m/s2. How much
work has the force done on the particle when the particle reaches (a) x = 4.0 m, (b) x = 7.0 m, and (c) x =
9.0 m? What is the particle's speed and direction of travel when it reaches (d) x = 4.0 m, (e) x = 7.0 m,
and (f) x = 9.0 m?
Figure 7-39 Problem 37.
Answer:
(a) 42 J; (b) 30 J; (c) 12 J; (d) 6.5 m/s, + x axis; (e) 5.5 m/s, + x axis; (f) 3.5 m/s, + x axis
38 A 1.5 kg block is initially at rest on a horizontal frictionless surface when a horizontal force along an x
axis is applied to the block. The force is given by
N, where x is in meters and the
initial position of the block is x = 0. (a) What is the kinetic energy of the block as it passes through x =
2.0 m? (b) What is the maximum kinetic energy of the block between x = 0 and x = 2.0 m?
39
A force
acts on a particle as the particle moves along an x axis, with in
newtons, x in meters, and c a constant. At x = 0, the particle's kinetic energy is 20.0 J; at x = 3.00 m, it is
11.0 J. Find c.
Answer:
4.00 N/m
40 A can of sardines is made to move along an x axis from x = 0.25 m to x = 1.25 m by a force with a
magnitude given by F = exp(-4x2), with x in meters and F in newtons. (Here exp is the exponential
function.) How much work is done on the can by the force?
41 A single force acts on a 3.0 kg particle-like object whose position is given by x = 3.0t - 4.0t2 + 1.0t3, with
x in meters and t in seconds. Find the work done on the object by the force from t = 0 to t = 4.0 s.
Answer:
5.3 102 J
42 Figure 7-40 shows a cord attached to a cart that can slide along a frictionless horizontal rail aligned
along an x axis. The left end of the cord is pulled over a pulley, of negligible mass and friction and at
cord height h = 1.20 m, so the cart slides from x1 = 3.00 m to x2 = 1.00 m. During the move, the tension
in the cord is a constant 25.0 N. What is the change in the kinetic energy of the cart during the move?
Figure 7-40 Problem 42.
sec. 7-9 Power
43
A force of 5.0 N acts on a 15 kg body initially at rest. Compute the work done by the force in (a)
the first, (b) the second, and (c) the third seconds and (d) the instantaneous power due to the force at the
end of the third second.
Answer:
(a) 0.83 J; (b) 2.5 J; (c) 4.2 J; (d) 5.0 W
44 A skier is pulled by a towrope up a frictionless ski slope that makes an angle of 12 with the horizontal.
The rope moves parallel to the slope with a constant speed of 1.0 m/s. The force of the rope does 900 J of
work on the skier as the skier moves a distance of 8.0 m up the incline. (a) If the rope moved with a
constant speed of 2.0 m/s, how much work would the force of the rope do on the skier as the skier moved
a distance of 8.0 m up the incline? At what rate is the force of the rope doing work on the skier when the
rope moves with a speed of (b) 1.0 m/s and (c) 2.0 m/s?
45
A 100 kg block is pulled at a constant speed of 5.0 m/s across a horizontal floor by an
applied force of 122 N directed 37 above the horizontal. What is the rate at which the force does work on
the block?
Answer:
4.9 102 W
46 The loaded cab of an elevator has a mass of 3.0 103 kg and moves 210 m up the shaft in 23 s at constant
speed. At what average rate does the force from the cable do work on the cab?
47 A machine carries a 4.0 kg package from an initial position of
at t = 0 to a final position of
at t = 12 s. The constant force applied by the machine on
the package is
For that displacement, find (a) the work
done on the package by the machine's force and (b) the average power of the machine's force on the
package.
Answer:
(a) 1.0 102 J; (b) 8.4 W
48 A 0.30 kg ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring ( k
= N/m) whose other end is fixed. The ladle has a kinetic energy of 10 J as it passes through its
equilibrium position (the point at which the spring force is zero). (a) At what rate is the spring doing
work on the ladle as the ladle passes through its equilibrium position? (b) At what rate is the spring doing
work on the ladle when the spring is compressed 0.10 m and the ladle is moving away from the
equilibrium position?
49
A fully loaded, slow-moving freight elevator has a cab with a total mass of 1200 kg, which is
required to travel upward 54 m in 3.0 min, starting and ending at rest. The elevator's counter-weight has a
mass of only 950 kg, and so the elevator motor must help. What average power is required of the force
the motor exerts on the cab via the cable?
Answer:
7.4 102 W
50 (a) At a certain instant, a particle-like object is acted on by a force
while the object's velocity is
What is the instantaneous rate at which the force does work on the
object? (b) At some other time, the velocity consists of only a y component. If the force is unchanged and
the instantaneous power is -12 W, what is the velocity of the object?
51
A force
initial position of
acts on a 2.00 kg mobile object that moves from an
to a final position of
in 4.00 s. Find (a) the work done on the object by the
force in the 4.00 s interval, (b) the average power due to the force during that interval, and (c) the angle
between vectors
and
.
Answer:
(a) 32.0 J; (b) 8.00 W; (c) 78.2
52 A funny car accelerates from rest through a measured track distance in time T with the engine operating
at a constant power P. If the track crew can increase the engine power by a differential amount dP, what
is the change in the time required for the run?
Additional Problems
53 Figure 7-41 shows a cold package of hot dogs sliding right-ward across a frictionless floor through a
distance d = 20.0 cm while three forces act on the package. Two of them are horizontal and have the
magnitudes F1 = 5.00 N and F2 = 1.00 N; the third is angled down by = 60.0 and has the magnitude F3 =
4.00 N. (a) For the 20.0 cm displacement, what is the net work done on the package by the three applied
forces, the gravitational force on the package, and the normal force on the package? (b) If the package has
a mass of 2.0 kg and an initial kinetic energy of 0, what is its speed at the end of the displacement?
Figure 7-41 Problem 53.
Answer:
(a) 1.20 J; (b) 1.10 m/s
54 The only force acting on a 2.0 kg body as the body moves along an x axis varies as shown in Fig. 7-42. The
scale of the figure's vertical axis is set by Fs = 4.0 N. The velocity of the body at x = 0 is 4.0 m/s. (a) What
is the kinetic energy of the body at x = 3.0 m? (b) At what value of x will the body have a kinetic energy of
8.0 J? (c) What is the maximum kinetic energy of the body between x = 0 and x = 5.0 m?
Figure 7-42 Problem 54.
55
A horse pulls a cart with a force of 40 lb at an angle of 30 above the horizontal and moves along at
a speed of 6.0 mi/h. (a) How much work does the force do in 10 min? (b) What is the average power (in
horsepower) of the force?
Answer:
(a) 1.8 105 ft lb; (b) 0.55 hp
56 An initially stationary 2.0 kg object accelerates horizontally and uniformly to a speed of 10 m/s in 3.0 s. (a)
In that 3.0 s interval, how much work is done on the object by the force accelerating it? What is the
instantaneous power due to that force (b) at the end of the interval and (c) at the end of the first half of the
interval?
57 A 230 kg crate hangs from the end of a rope of length L = 12.0 m. You push horizontally on the crate with a
varying force to move it distance d = 4.00 m to the side (Fig. 7-43). (a) What is the magnitude of
when the crate is in this final position? During the crate's displacement, what are (b) the total work done on
it, (c) the work done by the gravitational force on the crate, and (d) the work done by the pull on the crate
from the rope? (e) Knowing that the crate is motionless before and after its displacement, use the answers
to (b), (c), and (d) to find the work your force does on the crate. (f) Why is the work of your force not
equal to the product of the horizontal displacement and the answer to (a)?
Figure 7-43 Problem 57.
Answer:
(a) 797 N; (b) 0; (c) - 1.55 kJ; (d) 0; (e) 1.55 kJ; (f) F varies during displacement
58 To pull a 50 kg crate across a horizontal frictionless floor, a worker applies a force of 210 N, directed 20
above the horizontal. As the crate moves 3.0 m, what work is done on the crate by (a) the worker's force,
(b) the gravitational force on the crate, and (c) the normal force on the crate from the floor? (d) What is the
total work done on the crate?
59
An explosion at ground level leaves a crater with a diameter that is proportional to the energy of
the explosion raised to the power; an explosion of 1 megaton of TNT leaves a crater with a 1 km
diameter. Below Lake Huron in Michigan there appears to be an ancient impact crater with a 50 km
diameter. What was the kinetic energy associated with that impact, in terms of (a) megatons of TNT (1
megaton yields 4.2 1015 J) and (b) Hiroshima bomb equivalents (13 kilotons of TNT each)? (Ancient
meteorite or comet impacts may have significantly altered Earth's climate and contributed to the extinction
of the dinosaurs and other life-forms.)
Answer:
(a) 1 105 megatons TNT; (b) 1 107 bombs
60 A frightened child is restrained by her mother as the child slides down a frictionless playground slide. If the
force on the child from the mother is 100 N up the slide, the child's kinetic energy increases by 30 J as she
moves down the slide a distance of 1.8 m. (a) How much work is done on the child by the gravitational
force during the 1.8 m descent? (b) If the child is not restrained by her mother, how much will the child's
kinetic energy increase as she comes down the slide that same distance of 1.8 m?
61
How much work is done by a force
a position to a position
Answer:
-6J
with x in meters, that moves a particle from
to a position
?
62 A 250 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 2.5 N/cm (Fig. 744). The block becomes attached to the spring and compresses the spring 12 cm before momentarily
stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational
force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring?
(Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum
compression of the spring?
Figure 7-44 Problem 62.
63
To push a 25.0 kg crate up a frictionless incline, angled at 25.0 to the horizontal, a worker exerts a
force of 209 N parallel to the incline. As the crate slides 1.50 m, how much work is done on the crate by (a)
the worker's applied force, (b) the gravitational force on the crate, and (c) the normal force exerted by the
incline on the crate? (d) What is the total work done on the crate?
Answer:
(a) 314 J; (b) - 155 J; (c) 0; (d) 158 J
64 Boxes are transported from one location to another in a ware-house by means of a conveyor belt that
moves with a constant speed of 0.50 m/s. At a certain location the conveyor belt moves for 2.0 m up an
incline that makes an angle of 10 with the horizontal, then for 2.0 m horizontally, and finally for 2.0 m
down an incline that makes an angle of 10 with the horizontal. Assume that a 2.0 kg box rides on the belt
without slipping. At what rate is the force of the conveyor belt doing work on the box as the box moves (a)
up the 10 incline, (b) horizontally, and (c) down the 10 incline?
65 In Fig. 7-45, a cord runs around two massless, frictionless pulleys. A canister with mass m = 20 kg hangs
from one pulley, and you exert a force on the free end of the cord. (a) What must be the magnitude of
if you are to lift the canister at a constant speed? (b) To lift the canister by 2.0 cm, how far must you pull
the free end of the cord? During that lift, what is the work done on the canister by (c) your force (via the
cord) and (d) the gravitational force? (Hint: When a cord loops around a pulley as shown, it pulls on the
pulley with a net force that is twice the tension in the cord.)
Figure 7-45 Problem 65.
Answer:
(a) 98 N; (b) 4.0 cm; (c) 3.9 J; (d) - 3.9 J
66 If a car of mass 1200 kg is moving along a highway at 120 km/h, what is the car's kinetic energy as
determined by someone standing alongside the highway?
67
A spring with a pointer attached is hanging next to a scale marked in millimeters. Three different
packages are hung from the spring, in turn, as shown in Fig. 7-46. (a) Which mark on the scale will the
pointer indicate when no package is hung from the spring? (b) What is the weight W of the third package?
Figure 7-46 Problem 67.
Answer:
(a) 23 mm; (b) 45 N
68 An iceboat is at rest on a frictionless frozen lake when a sudden wind exerts a constant force of 200 N,
toward the east, on the boat. Due to the angle of the sail, the wind causes the boat to slide in a straight line
for a distance of 8.0 m in a direction 20 north of east. What is the kinetic energy of the iceboat at the end
of that 8.0 m?
69 If a ski lift raises 100 passengers averaging 660 N in weight to a height of 150 m in 60.0 s, at constant
speed, what average power is required of the force making the lift?
Answer:
165 kW
70
A force
acts on a particle as the particle goes through displacement
(Other forces also act on the particle.) What is c if the work done on the
particle by force is
is (a) 0, (b) 17 J, and (c) -18 J?
71 A constant force of magnitude 10 N makes an angle of 150 (measured counterclockwise) with the positive
x direction as it acts on a 2.0 kg object moving in an xy plane. How much work is done on the object by the
force as the object moves from the origin to the point having position vector
Answer:
- 37 J
?
72 In Fig. 7-47a, a 2.0 N force is applied to a 4.0 kg block at a downward angle as the block moves
rightward through 1.0 m across a frictionless floor. Find an expression for the speed vf of the block at the
end of that distance if the block's initial velocity is (a) 0 and (b) 1.0 m/s to the right. (c) The situation in
Fig. 7-47b is similar in that the block is initially moving at 1.0 m/s to the right, but now the 2.0 N force is
directed downward to the left. Find an expression for the speed vf of the block at the end of the 1.0 m
distance. (d) Graph all three expressions for vf versus downward angle for = 0 to = 90. Interpret the
graphs.
Figure 7-47 Problem 72.
73
A force
in the positive direction of an x axis acts on an object moving along the axis. If the magnitude of
the force is F = 10e-x/2.0 N, with x in meters, find the work done by as the object moves from x = 0 to x =
2.0 m by (a) plotting F(x) and estimating the area under the curve and (b) integrating to find the work
analytically.
Answer:
(a) 13 J; (b) 13 J
74
A particle moves along a straight path through displacement
while force
acts on it. (Other forces also act on the particle.) What is the value of c if the work
done by
75
on the particle is (a) zero, (b) positive, and (c) negative?
An elevator cab has a mass of 4500 kg and can carry a maximum load of 1800 kg. If the cab is
moving upward at full load at 3.80 m/s, what power is required of the force moving the cab to maintain that
speed?
Answer:
235 kW
76 A 45 kg block of ice slides down a frictionless incline 1.5 m long and 0.91 m high. A worker pushes up
against the ice, parallel to the incline, so that the block slides down at constant speed. (a) Find the
magnitude of the worker's force. How much work is done on the block by (b) the worker's force, (c) the
gravitational force on the block, (d) the normal force on the block from the surface of the incline, and (e)
the net force on the block?
77 As a particle moves along an x axis, a force in the positive direction of the axis acts on it. Figure 7-48
shows the magnitude F of the force versus position x of the particle. The curve is given by F = a/x2, with a
= 9.0 N m2. Find the work done on the particle by the force as the particle moves from x = 1.0 m to x =
3.0 m by (a) estimating the work from the graph and (b) integrating the force function.
Figure 7-48 Problem 77.
Answer:
(a) 6 J; (b) 6.0 J
78
A CD case slides along a floor in the positive direction of an x axis while an applied force
acts on the
case. The force is directed along the x axis and has the x component Fax = 9x-3x2 with x in meters and Fax in
newtons. The case starts at rest at the position x = 0, and it moves until it is again at rest. (a) Plot the work
does on the case as a function of x. (b) At what position is the work maximum, and (c) what is that
maximum value? (d) At what position has the work decreased to zero? (e) At what position is the case
again at rest?
79
A 2.0 kg lunchbox is sent sliding over a frictionless surface, in the positive direction of an x axis
along the surface. Beginning at time t = 0, a steady wind pushes on the lunchbox in the negative direction
of the x axis. Figure 7-49 shows the position x of the lunchbox as a function of time t as the wind pushes on
the lunch-box. From the graph, estimate the kinetic energy of the lunchbox at (a) t = 1.0 s and (b) t = 5.0 s.
(c) How much work does the force from the wind do on the lunchbox from t = 1.0 s to t = 5.0 s?
Figure 7-49 Problem 79.
Answer:
(a) 0.6 J; (b) 0; (c) - 0.6 J
80 Numerical integration. A breadbox is made to move along an x axis from x = 0.15 m to x = 1.20 m by a
force with a magnitude given by F = exp(-2x2), with x in meters and F in newtons. (Here exp is the
exponential function.) How much work is done on the breadbox by the force?
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