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8.1 Potential Energy of a Conservative Force 239

force always opposes the motion, and the work done by the friction force is always neg-
ative. Thus, the work done by the friction force cannot be expressed as a difference
between two potential energies, and we cannot formulate a law of conservation of
mechanical energy if friction forces are acting. However, as we will see in Section 8.3,
we can formulate a more general law of conservation of energy, involving kinds of energy
other than mechanical, which remains valid even when there is friction.
In the case of one-dimensional motion, a force is conservative whenever it can be
expressed as an explicit function of position, F F (x). (Note that the friction force
x x
does not fit this criterion; the sign of the friction force depends on the direction of
motion, and therefore the friction force is not uniquely determined by the position x.)
For any such force F (x), we can construct the potential energy function by integra-
x
tion. We take a point x as reference point at which the potential energy is zero. The
0
potential energy at any other point x is constructed by evaluating an integral (in the
following equations, the integration variables are indicated by primes to distinguish
them from the upper limits of integration):

x (8.14) potential energy as integral of force
U (x) F (x ) dx
x
x 0

To check that this construction agrees with Eq. (8.12), we examine U U :
1
2
U U U (x ) U (x ) x 1 F (x ) dx x 2 F (x ) dx
1 2 1 2 x x
x 0 x 0
By one of the basic rules for integrals (see Appendix 4), the integral changes sign when
we reverse the limits of integration. Hence
2
U U x 0 F (x ) dx x 2 F (x ) dx
1
x
x
x 1
x 0
And by another basic rule, the sum of an integral from x to x and an integral from
1 0
x to x is equal to a single integral from x to x . Thus
0 2 1 2
2
U U x 2 F (x ) dx (8.15)
x
1
x 1
Here the right side is exactly the work done by the force as the particle moves from x
1
to x , in agreement with Eq. (8.12).This confirms that our construction of the poten-
2
tial energy is correct.
In the special case of the spring force F (x) kx, our general construction (8.14)
x
of the potential energy immediately yields the result (8.6), provided we take x 0.
0
For a particle moving under the influence of any conservative force, the total
mechanical energy is the sum of the kinetic energy and the potential energy; as before,
this total mechanical energy is conserved:
E K U [constant] (8.16)

or

2
1
E mv U [constant] (8.17) conservation of mechanical energy
2