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General Physics I:

Subsection 8.1.3 Elastic Potential Energy

When a force is applied to deform the shape of a material work has been done against the restoring force of a material hence this work is stored in a material in the form of elastic potential energy. Suppose a wire is stretched to a distance \(\,dx\) by applying a deforming force, then work done by the force is given by
\begin{equation*} \,dW=F\,dx. \end{equation*}
But for a restoring force,
\begin{equation*} F=-kx. \end{equation*}
Hence work done by a restoring force is given by
\begin{equation*} W_{r} = \int\,dW = -\int\limits_{0}^{x}kx\,dx = -\frac{1}{2}kx^{2} \end{equation*}
and the work done by a deforming force is given by
\begin{equation*} W_{d}= \frac{1}{2}kx^{2}. \end{equation*}
This work is stored in the spring as potential energy. That is,
\begin{equation*} U=\frac{1}{2}kx^{2} \end{equation*}