Question

A particle oscillates about the equilibrium position $x_0$ subject to a force that has an associated potential energy U(x). Which of the following statements about U(x) is/are true:

• A. u(x) must be symmetric about x${}_0$
• B. u(x) must have a minimum at x${}_0$
• C. u(x) may have a minimum at x${}_0$
• D. u(x) must be positive in the vicinity of x${}_0$

Answer 1

The correct option is B u(x) must have a minimum at x${}_0$

Consider a general example of a system. Which has a stable equilibrium at x = x0 and a restoring force F(x) which maintains an oscillation about x = x${}_0$, as shown-

When a system oscillates about a point of equilibrium, it must be a stable equilibrium, which implies u(x) must have a minimum at that point, x${}_0$ in this case.

In the case drawn, a mass m drops down a curved wedge, reaches the bottom, and continues upwards due to inertia, till it loses all kinetic energy. It shows from a height y${}_0$on the right side as well as it is clear from the conservation of total mechanical energy. There will be oscillations about x${}_0$ with the tangential, component of the weight acting as the restoring force since the potential of m at a height of h is mgh, the u-x graph will look like-

We can practically set the level for zero potential energy at any height from the ground. The u(x) function is clearly not symmetric about x${}_0$, has negative values in the vicinity of x${}_0$.

Hence, correct