Physics Complete: Potential Energy of Spring

The spring force is an example of a variable force which is conservative. Fig. one shows a block attached to a spring and resting on a smooth horizontal surface. The other end of the spring is attached to a rigid wall. The spring is light and may be treated as mass less. In an ideal spring, the spring force Fs is proportional to x where x is the displacement of the block from the equilibrium position. The displacement could be either positive [Fig. 1(b)] or negative [Fig. 1(c)]. This force law for the spring is called Hooke’s law and is mathematically stated as

F_s = − kx The constant k is called the spring constant. Its unit is N m-1. The spring is said to be stiff if k is large and soft if k is small.

Suppose that we pull the block outwards as in Fig. 1(b). If the extension is xm, the work done by the spring force is

This expression may also be obtained by considering the area of the triangle as in Fig. 1(d). The work done by the external pulling force F is positive since it overcomes the spring force.