Work done by a Spring

Let mass m be at position x = 0. The spring is compressed (or elongated) by an external force F. An internal force F_s is called into play in the spring due to its elastic property. This force F_s keeps increasing with increasing x and becomes equal to F when the compression (or elongation) is maximum at x = x_m.

According to Hooke’s law (true for small x only), |F_s| = kx, where k is known as spring constant. Since the direction of F_s is always opposite to compression (or extension),

F = F_s = – kx

In the event of compression of the spring, the external force F is directed towards left and the displacement x is also towards left. Hence, the work done by the external force is positive. However, for the same direction of displacement, the restoring force generated in the spring is towards right, i.e. F and x are oppositely directed. The work done by the spring force is negative.

The work done by the external force is positive but the work done by the spring force is negative and its magnitude is (½)kx_m².