Calculating the Potential Energy of a Spring

In this wiki, we shall derive an expression to find the potential energy stored in a spring and look through a few examples where this can be used as well. The fundamental idea which we are going to use here is a bit of Hooke's law.

The elongation produced in an ideal spring is directly proportional to the spring force:

\[F=-kx.\]

Here \(k\) is called the spring constant.

The potential energy stored in the spring is given by

\[U=-\dfrac{1}{2}kx^2.\]

Let's start with the proof.

Let's start with the derivation of the above equation. Let the spring be stretched through a small distance \(dx\).

Then work done in stretching the spring through a distance \(dx\) is \(dW=Fdx,\) where \(F\) is the force applied to stretch the spring.

Total work done in stretching the spring from the interval \(x=0\) to \(x=x\) is obtained by integrating the expression:

\[\int dW=\int_0^xF\,dx.\]

Substituting \(F=-kx\), we get

\[W=\int_0^x-kx\,dx=-k\int_0^xx\,dx=-k\left[\dfrac{x^2}{2}\right]_0^x=-\dfrac{1}{2}kx^2.\]

This work done is nothing but the elastic potential energy of the spring.

\[U=-\dfrac{1}{2}kx^2.\ _\square\]

Let's go though a few simple examples.