Explain why the work required to stretch a spring or ...?

Question

Explain why the work required to stretch a spring or other elastic object with a linear restoring force, of form F = - kx, from its equilibrium position to displacement x is `dW = .5 k x^2, and why we hence say that this is the elastic potential energy of the object in this position.

Answer 1

The work done is F(x) * dx.
The infinitesimal work done in stretching a spring by length dx is
-kx * dx
The work done in stretching the spring from the mean position to an end y is found by integrating dW with limits 0 to y,
This integration yields 0.5 k y^2 where y is the displacement of the spring.