# The Potential Energy of a Spring

Consider a very light (practically of zero mass) and perfectly elastic spring fixed firmly at one end to a rigid support at point O and the other end attached to a block of mass M. When the block is pulled to the point B from its equilibrium position A through distance AB = r (say), then, due to elasticity, restoring force a set up in the spring. It, therefore, returns to the equilibrium position A and due to inertia, it moves upto point C and so on.

In case, the horizontal surface OX, over which the block moves is smooth and frictionless, the distance CA will be equal to AB i.e. equal to r. The work done in stretching the spring from A to B is stored in the system in the form of potential energy of the block of mass M.

Let us calculate the potential energy of the mass M, when spring is pulled from mean position A upto any point, say P such that AP = x (figure 6.9). As the spring gets stretched; due to elasticity, restoring force F is set up in the spring, such that
F x
F = - k x

where the constant of proportionality k is called force constant of the spring or simply spring constant. The negative sigh shows that the restoring force acts in a direction opposite to the one in which the displacement increases. Suppose that the block is further displaced through a very small distance PQ = dx [figure 6.9]. The restoring force can be supposed to remain practically unchanged, as the increase in length is infinitesimally small. The small work done in increasing the length of the spring by dx is given by
dW = F dx = k x dx

The work done in increasing the length of the spring by an amount x can be calculated by integrating the above between the limits x = 0 to x = x i.e.
W = = k = k = k
W =

This work done is stored in the system as its potential energy at point P i.e., when displacement of block from equilibrium position is x. Potential energy of the system, when the block is pulled upto the point B can be obtained by setting x = r in equation (1.9). Thus,
P.E. of the system at point B = =
At point B, block is at rest. Hence,
K.E of the system at point B = 0
At point A, x = 0. Therefore,
P.E. of the system at point A = = 0

The loss in P.E. when system returns to the position A appears as equal increase in kinetic energy, so that
K.E of the system at the point A = - 0 =The P.E. of the system of the point C can be found by setting x = -r in equation.
Therefore
P.E. of the system at point C = =
Obviously, the K.E of the system at the point C = 0
As said above, P.E. of the system at the point P =

The decrease in P.E. when system returns from the position A to P appears as the equal increase in kinetic energy. Therefore,
K.E. of the system at the point P = -

It we plot the P.E. and K.E. against the displacement x, the graph will be as depicted by the two dotted curves as shown in figure. The sum of K.E. and P.E. is shown by thick horizontal straight line and is always equal to .

Energy can manifest itself in many different forms. So far we have studied mechanical energy which exists in two forms, namely, kinetic energy and potential energy. Mechanical energy is not the only form in which energy can exist. The other forms of energy are described below.