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

energy, so we can use the work-energy theorem to write the change in potential energy
AP ® Exam Tip
of an ideal spring in terms of the change in its length. We are assuming that it is an
The equation sheet on the ideal spring, which means no energy is going into heating or deforming it and that its
AP ® Physics 1 exam lists the mass is negligible, so the only type of energy it has due to it being a system is poten-

U
K

spring potential energy as an tial energy, so we can rewrite Equation 7-12 as W =∆ +∆ . ∆K is zero, because it
extension x from its relaxed is at rest at both the initial and final lengths. W is the amount of work you do on the
2
1
equilibrium, but does not list spring, kx 2 f − kx Substituting, kx 2 f − kx i 2 = 0 +∆U .
.

1
1
1

s
i
the change in potential energy. 2 2 2 2 2 2
1
Understand why the change in ∆U s = 1 2 kx f − kx
i
2
potential energy when you are We can rewrite this:
stretching a spring is as given
1
2
in the equation to the right and ∆U s = U s,f − U s,i = 1 2 kx 2 f − kx
2
i
cannot be written in terms of
U is the value of s at x if we set = 0 we can just call ,xx In this case x
i

∆ ,x unless i x is the equilibrium Because s,f U f , x f .
x

length of the spring, = 0x . represents the extension of the spring from its relaxed equilibrium length ( i = 0) and
the quantity s is the spring potential energy :

U

Spring potential energy of a stretched or compressed spring Spring constant of the spring
EQUATION IN WORDS 1
Spring potential energy ( 7-16 ) U = 2 kx 2
s
Extension of the spring, when the equilibrium position of the end of the spring
when it is relaxed is de ned as x = 0 (x > 0 if spring is stretched, x < 0 if spring
is compressed)

The spring potential energy is zero if the spring is relaxed ( =x 0) and positive if the

U s

spring is stretched ( >x 0) or compressed ( < 0) ( Figure 7-20 ). This says that we have to

x

do work to either stretch the spring or compress it, and the work that we do goes into
the spring potential energy.
> 0:
x < 0: x > 0: While a human tendon is not an ideal spring, we can think of it as storing spring
spring is
spring is spring is potential energy when it is stretched. When you are running as you move over your
spring is
compressed stretched foot in contact with the ground, the Achilles tendon at the back of that leg stretches.
stretched
compressed
x The spring potential energy stored in that tendon is part of “springing” you back in the
0 0
air, helping you to sustain your running pace.
Figure 7-20 Spring potential Yes, you still get tired. Remember, the human tendon is not an ideal spring, so you
energy The potential energy in a had to put more energy into stretching it out from equilibrium than you get back when
spring is proportional to the square of it relaxes back to equilibrium. Even though the approximation is not perfect, it still
its extension d . (See Equation 7-16 .) gives us insight into how such a complicated system, the human body, functions!
THE TAKEAWAY for Section 7-6

✔ The work done by a conservative force depends only ✔ Potential energy is used to describe conservative
on the initial and final positions of the object, not on the interactions inside a system; a conservative force is a way to
path the object followed from one position to the other. The describe conservative interactions with objects or systems
gravitational force and the force exerted by an ideal spring are outside the system of interest.
examples of conservative forces.
Prep for the AP ® Exam

Building Blocks 3. A spring that is compressed by 12.5 cm stores 3.33 J of
potential energy. Calculate the value of the spring constant.
1. What does it mean when a force is referred to as
conservative? Skill Builders
EX 2. Over 630 m in height, the Burj Khalifa is the world’s
7-8 tallest skyscraper. Calculate the change in gravitational 4. A 350-kg box initially at rest is pulled 7.00 m up a 30.0 °

3
F

potential energy of the coin–Earth system when a inclined plane by an external force, ext = 5.00 × 10 N ,
1- dirham coin (6.1 g) is carried from ground level to the that is exerted parallel to the plane. Assume friction is
top of the Burj Khalifa. Calculate the speed of the coin negligible.

just before it hits the ground if it is dropped from rest at (a) Construct a representation of the scenario that
the top of the skyscraper and air resistance is neglected. includes the ramp, the inclination of the ramp, the
Based on the speed you calculated, do you feel it was a external force exerted on the box, the displacement,
good idea to neglect air resistance? the box, and the coordinate system to be used for the
Uncorrected proofs have been used in this sample. Copyright © Bedford, Freeman & Worth Publishers.
Distributed by Bedford, Freeman & Worth Publishers. For review purposes only. Not for redistribution.
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