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.

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