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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.
The elongation produced in an ideal spring is directly proportional to the spring force:
Here \(k\) is called the spring constant.
The potential energy stored in the spring is given by
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:
Substituting \(F=-kx\), we get
This work done is nothing but the elastic potential energy of the spring.
Let's go though a few simple examples.