# Derive an expression for the potential energy of elastic stretched spring

Last updated on April 14th, 2021 at 04:56 pm

Here we will obtain or **derive an expression for the potential energy of elastic stretched spring**. This energy is also known as *elastic potential energy*. To get this equation, we’ll calculate the work done to stretch or compress a spring that obeys Hooke’s law.

Hooke’s law states that the magnitude of force F on the spring and the resulting deformation ΔL are proportional, **F = kΔL, where k is a constant.**

So let’s go forward to obtain an expression for the potential energy of a spring. For our spring, we will replace ΔL (the amount of deformation produced by a force F) by the distance x that the spring is stretched or compressed along its length.

So the force needed to stretch the spring has magnitude F = kx, where k is the spring’s force constant.

The force increases linearly from 0 at the start to kx in the fully stretched position.

The average force is *kx/2*.

**Thus the work done in stretching or compressing the spring is Ws = Fd = force . displacement = (kx/2).x = (1/2)kx ^{2} **

Thus we have completed the derivation of the expression for the Potential Energy stored in the spring.

## Derive an expression for potential energy of a spring – ** Using Area under a graph of F versus x**

Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of F vs. x is the work done by the force. In Figure (c) we see that this area is also (1/2)kx^{2} .

**We, therefore, derive the equation of the potential energy of a spring, also known as the elastic potential energy as follows: PE=(1/2)kx ^{2} **where k is the spring’s force constant and x is the displacement from its undeformed position.

## Summary | Take Away

We have **derive**d **an expression for the potential energy stored in the spring**, also known as the elastic potential energy. We have shown 2 different ways to derive the equation. Here are a few points to remember.

– **The elastic potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it by a distance of x. **

– **The potential energy of the spring does not depend on the path taken; it depends only on the stretch or compression x in the final configuration.****Related Study:** Equation of time period for the mass-spring system with horizontal oscillation with derivation