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*}