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314 Chapter 7 Conservation of Energy and an Introduction to Energy and Work

The work that you do to compress the same spring by 20cm(so x 2 =−20 cm =
−0.20 m) is
1 1 1
2
2
2
−
W = kx − k (0) = (1000 N/m)(0.20m) = 20J
2 2 2 2
So you have to do positive work to compress a relaxed spring as well as to stretch one.
THE TAKEAWAY for Section 7-5

✔ The work-energy theorem for an object applies even when ✔ If the point of contact where a force is exerted on an
the object follows a curved path or the forces that are exerted object or a system does not move, there is no work done by
on the object are not constant. that force on that object or system.

✔ A force that is always exerted perpendicular to an object’s ✔ An ideal spring exerts a force proportional to the distance
path does zero work on the object. that it is stretched or compressed (Hooke’s law). The work

✔ The work-energy theorem for a system that cannot be required to stretch or compress a spring by a given distance
modeled as an object gives the total energy change of the from its equilibrium length is proportional to the square of
system on which the force is exerted, including internal and that distance.
potential energies. Objects can only have kinetic energy.
Prep for the AP ® Exam
Building Blocks (b) Does the speed of the sled increase or decrease as it

x

travels from = 30.0mx to = 50.0m? Your predic-

1. A gravitational force exerted on an object of mass m tion must be based on scientific theories.
attached to a vertical spring will cause the spring to stretch. (c) Calculate the velocity of the sled at = 100.0m.x

When the object is in equilibrium, the gravitational force EX 3. A boy swings a ball with a 6-N weight on a string at
exerted on the object is equal in magnitude to the spring 7-6 constant speed in a horizontal circle that has a diameter

force exerted on the object: mg =∆ . This table provides equal to 1 m. What is the work done on the ball by the
kx
data for a spring for eight different values of mass. 10-N tension force in the string as the ball travels through
one-half of the circular path?
Mass, m (g) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
(A) 31.4 J (B) 10 J (C) 6 J (D) 0 J

∆
Stretch, x (cm) 0.5 1.1 1.7 2.3 2.6 3.1 3.6 4.2 4. An ideal spring with a constant =k 1N/m is attached to
∆

∆

(a) Make a graph of mg versus x . a stationary beam so that it hangs vertically. At the lower
(b) Use your graph to calculate the value of the spring end of the spring a 100-g object is attached. The object is

constant, k using all of the data values. supported by a hand so that the spring force exerted on
,

(c) Shade the area in the graph that shows the work done the object is initially zero (the spring is at its equilibrium
to stretch the spring by 2.5 cm and calculate the work length). The object is then released and as it falls it passes
done by the object on the spring resulting in that stretch. through a position where the extension of the spring is
Justify your use of the graph to calculate the work. 0.5 m from its initial equilibrium length. Take the value

2. A sled of mass = 22.0kgm is accelerated from rest on of the gravitational field strength to be 10N/kg.

a horizontal surface to a velocity of = 12.5m/s,v as it (a) What is the work done on the object by the gravitational
travels from =x 0 to = 30.0m The net force in the force as it travels through this 0.5-m displacement?

x

.

,
direction of the displacement, in terms of max is graphed (A) 1 J (B) 0.5 J (C) −0.5J (D) −1J
F

below as a function of displacement for the sled. (b) What is the work done on the object by the spring
force as it travels through this 0.5-m displacement?
F(x) (N) (A) 1/8 J (B) 1/10J (C) −1/8J (D) −1/10 J
) (N)
x

F F F ma EX 5. An object resting on a smooth table is attached to the free
max
7-7 end of a horizontal spring that has a spring constant equal to
450 N/m. The object is initially held at rest 12 cm beyond the
equilibrium length of the spring, and is then pulled to a posi-
x (m) tion 18 cm beyond equilibrium, where it is again held at rest.
(m)
70
60
80
100
90
50
10
10 20 30 40 50 60 70 80 900100
20
40
30
0
100
Calculate the work the spring does on the object between
these two positions, both algebraically and graphically.

(a) Use the graph and the values given to find the value of Skill Builders
the work done on the sled as it moves from =x 0 to

x
= 30.0m and use the work to calculate the value of 6. Two forces are simultaneously exerted on an object over

,

F . the range of positions, ,0x ≤ x ≤ 2m The forces are
.
the maximum net force exerted on the sled, max
Uncorrected proofs have been used in this sample. Copyright © Bedford, Freeman & Worth Publishers.
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