21.1 Nature of equilibrium

We encounter different kinds of equilibrium. Take the case of a tennis ball and a paper weight, which are placed on a table. A slight force on tennis ball makes it to roll and finally drop off the table, whereas paper weight hardly moves from its position on the application of same force. Two bodies, therefore, are equilibrium of different nature. In this module, we shall not go for details of every aspects of equilibrium; but will limit ourselves to broader classification, which is based on energy concept.

We shall further limit the analysis of equilibrium to the context of equilibrium in conservative force field like gravity. A body in conservative force field possesses mechanical energy in the form of kinetic and potential energy. Potential energy, as we know, is exclusively defined for conservative force field and is a function of position. In this module, we shall attempt to correlate potential energy with the nature of equilibrium in following categories :

• Stable
• Unstable
• Neutral

Disturbance and equilibrium

In order to appreciate the role of potential energy about equilibrium, we first need to visualize equilibrium in the light of external disturbance. We should keep this in mind that disturbance that we talk about is a relatively small force.

A typical set of example to illustrate the nature of equilibrium consist of three settings of a small ball (i) inside a spherical shell (ii) over the top of a sphere/ spherical shell and (iii) over a horizontal surface. These three settings are shown in the figure.

What do we expect when the ball inside the shell is slightly disturbed to its left. A component of gravity acts to decelerate the motion; brings the ball to a stop; and then accelerates the ball back to its original position and beyond. Restoration by gravity continues till the ball is static at the original position, depending upon the friction. The equilibrium of the ball inside the shell is “stable” equilibrium as it is unable to move out of its setting. The identifying nature of this equilibrium is that a restoring force comes into picture to restore the position of the object.

Let us now consider the second case in which the ball is placed over the shell. We can easily visualize that it is difficult to achieve this equilibrium in the first place. Secondly, when the ball is disturbed with a smallest touch, it starts falling down. The gravity here plays a different role altogether. It aids in destabilizing the equilibrium by pulling the ball down. The equilibrium of the ball over the top of the sphere is called “unstable” equilibrium. The identifying nature of this equilibrium is that once equilibrium ends, there is no returning back to original position as there is no restoring mechanism available.