**Elastic Potential Energy**

Within elastic limit, if a force is applied on a body it becomes strained and when the force is withdrawn the body gets back its original shape. In order to strain or deform a body, we need to do work on the body. This work remains stored as potential energy. This energy is called elastic potential energy. Elastic potential energy is energy stored as a result of applying a force to deform an elastic object. Let find the equation of elastic potential energy.

**The potential energy of a stretched spring**

Let one end of an ideal horizontal spring be fixed at a wall and a block of mass m is attached to the other end. The block can move on a frictionless horizontal surface (Figure). We neglect the mass of the spring compared to the mass of the block.

Fig: Elastic Potential Energy

Now, the spring is stretched along its length by a distance x. Due to elastic property, the spring will exert a force called restoring force which will try to oppose the deformation. This restoring force is proportional to the displacement and acts opposite to the displacement i.e.,

**F ∞ – k**

or, **F = kx.**

where k is a constant of proportionality and is called force constant of the spring. Now, to stretch the spring we must do work by applying a force F’ equal but opposite to the force F exerted by the spring. So,

Now, to stretch the spring we must do work by applying a force F’ equal but opposite to the force F exerted by the spring. So,

F’ = – F = – (- kx) = kx

This work remains stored as potential energy in the spring. So, potential energy;

U = W = ^{x}∫_{0} Fdx = ^{x}∫_{0} kx dx

= k ^{x}∫_{0} x dx = k [x^{2}/2]^{x}_{0}

so, **U = ½ kx ^{2}**

While compressing the spring by x, the same amount of energy will remain stored as potential energy in the spring.