Our important example of a spatially varying force is a spring.
It turns out quite generally that in a limited range of deformation
any body in compression or expansion produces a counteracting restoring force
which is simply proportional to the deformation away from equilibrium.
Usually the deformations can be hard to perceive. This is not the case
however for a spring. We see that an elastic solid produces a force which
depends on the distance and so represents a good opportunity
for us to test our expressions for the work produced by such a force.
Combining Eq. 3 and 4 and using the result from
mathematical analysis:
we get
