Q.Derive the formula for the potential energy of a spring.

Elastic potential energy stored in a spring stretched/compressed by x is U = (1/2) k x², obtained by integrating the work done against the spring's restoring force F = −kx.

By Hooke's law, when a spring is stretched or compressed by a displacement x from its natural length, it exerts a restoring force proportional to x and directed opposite to the displacement:

F(x) = −k x

where k is the spring (force) constant. To stretch the spring quasi-statically by a further small displacement dx, an external agent must apply a force equal and opposite to the spring force, i.e. +kx, and the small work done is:

dW = k x dx

This work is stored as elastic potential energy in the spring. The total potential energy when the spring has been displaced from x=0 to x is obtained by integrating:

U(x) = ∫[0 to x] k x' dx' = k [x'²/2] from 0 to x = (1/2) k x²

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