21.1 Nature of equilibrium  (Page 2/4)

In the last case, when ball is disturbed, it moves on the horizontal surface. If the surface is smooth it maintains the small velocity so imparted. The distinguishing aspect is that component of gravity in horizontal direction (direction of motion) is zero. As such gravity neither plays the role of restoring force nor that of an aid to the disturbance. The equilibrium of the ball on the horizontal surface is called “neutral” equilibrium. The identifying nature of this equilibrium is that once equilibrium ends, there is neither the tendency of returning back nor the tendency of moving away from the original position.

Potential energy and equilibrium

We are interested here to establish the characterizing features of equilibrium. As such, we shall keep our discussion limited to one dimension and attempt to find the required correlation.

For the conservative force system, force is related to potential energy as :

$$F=-\frac{dU}{dx}$$

For translational equilibrium,

$$F=-\frac{dU}{dx}=0$$

This relationship can be used to interpret equilibrium, if we have values of potential energy function, U(x), with respect to displacement “x”. A plot of U(x) .vs. x will indicate position of equilibrium, where tangent to the plot is parallel to x-axis so that slope of the curve is zero at that point.

Stable equilibrium

In order to correlate, potential energy with stable equilibrium, we draw an indicative potential energy plot of a pendulum bob, which is displaced through a maximum angle of 15. Let pendulum of length, L = 1 m and mass of pendulum bob (of high density material) = 10 kg. The maximum potential energy corresponds to maximum angular displacement.

$$U_{\max }=mgh=mgL\left(1-\cos \theta \right)$$

$$\Rightarrow U_{\max }=10X10X1\left(1-\cos {15}^0\right)=100X\left(1-0.966\right)=3.4J$$

The maximum potential energy is equal to maximum mechanical energy that the bob can have in the setup. The important thing to note about the plot is that we measure “x” from point “O” – from the extreme point in the left. The indicative plot is shown here :

Further, as force is negative of the slope of the potential energy curve, it is first positive when slope of potential energy curve is negative; negative when slope is positive. An indicative force - displacement plot corresponding to potential energy curve is shown here.

These two pairs of plot let us analyze the equilibrium of pendulum bob. For this, we consider motion of the bob towards left from its mean position. The bob gains potential energy at the expense of kinetic energy. Further, the bob has negative velocity as it is moving in opposite to the reference direction. From "F-x" plot, we see that force is acting in positive x-direction. This means that velocity and force are in opposite direction. As such, pendulum bob is decelerated. Ultimately bob comes to a stop and then reverses direction towards point “B”. In this reverse journey, it acquires kinetic energy at the expense of loosing potential energy.