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240 CHAPTER 8 Conservation of Energy

As we will see in later chapters, the inverse-square force plays
EXAMPLE 2
a large role in physics—gravitational forces are inverse square,
and electric forces are inverse square. If we consider a particle that can move in
only one dimension along the positive x axis, this force has the form

A
F (x) (8.18)
x 2
x
where A is a constant. The point x 0 is called the center of force. If A is pos-
itive, the force is repulsive (F is positive, and the force therefore pushes a par-
x
ticle on the positive x axis away from the center of force); if A is negative, the
force is attractive (F is negative, and the force pulls the particle toward the
x
center of force). The magnitude of the force is very large near x 0, and it
decreases as the distance from this point increases (Figs. 8.4a and b). What is
the potential energy for this force?

(a) (b)
F x F x
0 1 2 3 4 m
1N
1
4
3 Force is positive – N
4 (away from origin;
repulsive). 1
1 – 2
2
Force is negative
(toward origin;
3
1 – 4 attractive).
4
–1
x
0 1 2 3 4 m
2
FIGURE 8.4 The inverse-square force A/x as a function of x, (a) for a
2
positive value of A (repulsive force; A 1 N
m ) and (b) for a negative value
.
2
of A (attractive force; A 1 N m ).
SOLUTION: According to Eq. (8.14),
x A
U (x) 2 dx
x 0 x
2
In the compact notation of tables of integrals, (1/x ) dx 1/x . Hence
x
A A A A A
U (x) B R 2 B a b R
x x x 0 x x 0
x 0
It is usually convenient to take x as the reference point, with U 0 at
0 0
x . With this choice,
A
U (x) (8.19)
x

COMMENT: Note that for a repulsive force (A 0), the potential energy decreases
with x (see Fig. 8.5a), and for an attractive force (A 0), the potential energy
increases with x (the potential energy is large and negative near x 0, and it increases
toward zero as x increases; see Fig. 8.5b).