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

In Equation 7-4 x f − x i is the displacement of the object along the x axis. Using
Equation 7-2, we multiply this by the x component of the net force exerted on the
Σ
object, F x —that is, by the net force component in the direction of the displacement
and get a slightly different expression for W , the work done by the net force on the
net
object. (This assumes that the net force on the object is constant, which is consistent
with our assumption that the object’s acceleration is constant.) That is,
(7-5) W net = ∑ ( F x ) x( f − x ) i
Σ
Newton’s second law tells us that F x = ma x , where m is the mass of the object. So
we can rewrite Equation 7-5 as

Work done on an object by the
net force exerted on that object Mass of the object
EQUATION IN WORDS
Calculating the work done on
an object by a constant net (7-6) W = ma (x − x )
i
net
f
x
force, linear motion
Constant acceleration of the object Displacement of the object
We can get another expression for the quantity ma x( f − x ) i by multiplying both
x
sides of Equation 7-4 by /2m and rearranging:
1 1
mv 2 = mv 2 + ma x ( − x ) i
x
2 f 2 i f
1 1
(
ma x f − x ) i = mv 2 − mv 2
x
2 f 2 i
The left-hand side of this equation is the work done on the object by the net force (see
Equation 7-6). So
1 1
2
(7-7) W net = mv 2 − mv
2 f 2 i
1
2
The right-hand side of Equation 7-7 is the change in the quantity mv over the course
2
v ,
of the displacement (the value at the end of the displacement, where the speed is f
1
2
minus the value at the beginning where the speed is i v ). We call this quantity, mv , the
2
kinetic energy K of the object:
Kinetic energy of an object Mass of the object
EQUATION IN WORDS 1
Kinetic energy of an object (7-8) K = mv 2
2
Speed of the object
2
2
2
The units of kinetic energy are ⋅ m/s . Because 1 J = 1 Nm and 1 N = 1 kg ⋅ m/s ,
kg
you can see that kinetic energy is measured in joules, the same as work. Using the defi-
nition of kinetic energy given in Equation 7-8, we can rewrite Equation 7-7 as
Work done on an object by the net force exerted on that object
EQUATION IN WORDS
The work-energy theorem for (7-9) W = K − K
an object net f i
Kinetic energy of the object Kinetic energy of the object
after the work is done on it before the work is done on it

When an object undergoes a displacement, the work done on it by the net force
equals the change in the object’s kinetic energy (its kinetic energy at the end of the
displacement minus its kinetic energy at the beginning of the displacement). This state-
ment is called the work-energy theorem for an object. It is valid as long as the object
model is appropriate, such as when a system is rigid—that is, it doesn’t deform like

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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