# Calculating the Potential Energy of a Spring

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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.

## Hooke's Law

The

elongationproduced in an ideal spring is directly proportional to the spring force:$F=-kx.$

Here $k$ is called the spring constant.

The

potential energystored 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.

**Cite as:**Calculating the Potential Energy of a Spring.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/calculating-the-potential-energy-of-a-spring-2/