Equilibrium and Stability

Equilibrium and Stability:

Consider one dimensional motion of a particle of mass m under the influence of a force F which depends only on the position x of the particle. The potential energy U is related to the force F by the relation-

F = -du/dx

and the total energy E of the particle is given by-

(1/2) mν² + U = E

Suppose the variation of U with x in a typical case is as shown in the figure above. Since for a real situation E ≥ U (otherwise K.E. would become negative which is not possible) so the lowest total energy E₀ of the particle occurs where kinetic energy, (1/2) mν² is zero and potential energy, E₀ is minimum. This happens at x = x_A. At this point ν = 0, so the particle is at rest. It is a case of equilibrium. Here, since U(x) is minimum, so the slope of F-x curve is also zero at this point, i.e.,

If the particle is displaced slightly from this position in either direction, a force F appears on the particle.

For x > x_A, since the slope du/dx is positive, the force F that appears on the particle is negative which tries to bring the particle back to the equilibrium position x = x_A. Similarly for x < x_A, the slope is negative and the force is positive, which tries to bring the particle back to its equilibrium position. The position x = x_A, therefore, corresponds to the case of stable equilibrium. Here the potential energy of the particle is minimum.

Consider now a point B where the potential energy of the particle is maximum. Since at this point x = x_B, the slope of the F-x curve is zero and so the force is zero i.e.-

This is also a position of equilibrium. From this position, if the particle is displaced even the slightest, a force will act on it which will push the particle farther away from the position of equilibrium x = x_B. This can be seen from the fact that when x > x_B, the slope is negative and the force is positive. The position B, therefore, corresponds to the case of unstable equilibrium. When a particle is in unstable equilibrium it has a maximum potential energy.

In the horizontal region, U(x) is constant. For example near C, x = x_C, the slope of the F-x curve is zero. So the force-