Question

Derive the expression for the potential energy stored in a spring?

CLASS - XI PHYSICS (Work, Energy and Power)

Answer 1

The characteristic of a spring is that the force needed to stretch or compress a spring by an amount x is equal to k x and is in the direction opposite to the extension or compression. Here k is called the spring constant.
 
     Restoration Force of the spring (compression or tensile) Fs = - k x  = - external force F
 
     Work done by the external force in compressing the spring from x = 0 to x = L can be obtained by:

$W = \int\limits^L_0 {F .} \, ds = \int\limits^L_0 {( k x)} \, dx,\ \ \ \ as\ displacement\ s= x\\ \\ W = k\int\limits^L_0 {x} \, dx = \frac{1}{2}k [x^2 ]_0^L = \frac{1}{2}k\ L^2 =\frac{1}{2}k x^2$  is the energy given by the external force to the spring with an extension x. 

This energy is stored as potential energy in the spring.