21.1 Nature of equilibrium  (Page 4/4)

The position of equilibrium at point “B”, where slope of the curve is zero corresponds to maximum potential energy, which in turn is equal to maximum mechanical energy allowable for the body. We can refer this plot to the case of a ball placed over a spherical shell as shown below.

It is easy to realize that the ball has maximum potential energy at the top as is shown in potential energy plot. Further, as force is negative of the slope of the potential energy curve, it is first negative, 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.

The two pairs of plots let us analyze the equilibrium of ball, placed on the shell. For this we consider motion of the ball towards left. The ball gains kinetic energy at the expense of potential energy. The projection of velocity in x-direction is negative. From the figure below, we see that component of gravitational force is constrained to be tangential to sphere. Its component in x-direction is also acting in the negative direction.

Thus, both components of velocity and acceleration are in the opposite direction to the reference x-axis direction. As such, the spherical ball is accelerated and keeps gaining kinetic energy without any possibility of restoration to original energy state. Ultimately, the ball lands on the horizontal surface, which we have considered to be at zero reference potential. There is no component of gravitational force in horizontal direction on the surface. As such, it is stopped after some distance due to friction or keeps moving with uniform velocity if surface is smooth.

Similar is the case, when ball moves to the right from its equilibrium position.

From the discussion so far, we conclude that :

1: First derivative of potential energy function with respect to displacement is zero.

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

2: Potential energy of the body is maximum for stable equilibrium for a given potential energy function and maximum allowable mechanical energy.

$$U\left(x\right)=U_{\max }$$

For this, the second derivative of potential energy function is negative at equilibrium point,

$$\frac{d^{2}U}{d{x}^2}<0$$

3: There is no bounding pairs of turning points like in the case of stable equilibrium.

4: There is no restoring force the body. External force (gravity) aids in acquiring kinetic energy by the body.

Neutral equilibrium

The nature of the potential energy plot for neutral equilibrium is easy to visualize. Let us consider equilibrium of the ball, which is lying on horizontal surface. If we consider the horizontal surface to be the zero reference potential level, then potential energy plot is simply the x-axis itself; if not, it is a straight line parallel to x-axis.

On the other hand, Force – displacement plot is essentially x-axis as component of gravity in horizontal direction is zero. When ball is disturbed from its position, it merely moves till friction stops it. If the surface is smooth, ball keeps moving with the velocity imparted during disturbance.

From the discussion so far, we conclude that :

1: First derivative of potential energy function with respect to displacement is zero.

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

2: The second derivative of potential energy function is equal to zero.

$$\frac{d^{2}U}{d{x}^2}=0$$

3: There is no bounding pairs of turning points like in the case of stable equilibrium.

4: There is no restoring force on the body . External force (gravity) neither acts to restore the body or aid in acquiring kinetic energy by the body.