Recall that Hooke's law for springs states that there is a restoring force given by [math]\vec{F}=-kx\hat{x}[/math] where x is measured from equilibrium position of the spring. Let's use this force to find an expression for the associated potential energy of a spring.[br][br]The place to start is always to calculate the work done by the conservative force - in this case using Hooke's law for springs. Once we have this result, we will flip its sign and call it the change in potential energy [math](W_c=-\Delta U)[/math] just as we saw in the second section of this chapter when potential energy was derived.

The calculation goes like this:[br][br][center][math]W=\int \vec{F}\cdot\vec{ds} \\[br]W=\int -kx\hat{i}\cdot(dx\hat{i}+dy\hat{j}+dz\hat{k}) \\[br]W=\int_{x_i}^{x_f} -kx \; dx \\[br]W=-\frac{k}{2}(x_f^2-x_i^2) \\[br]\Delta U=\frac{k}{2}(x_f^2-x_i^2)\[/math][/center]