I encountered a physics problem which inquired about a ball rolling inside a parabolic bowl (i.e. a bowl where any cross section through the vertex would make a parabolic shape given by $y = kx^2$). The ball was to be released from some initial height $z_0$ and allowed to roll back and forth inside the bowl.
At first, I thought that the motion would be simple harmonic for any displacement, since $U(x) = mgh = mgkx^2$, and simple harmonic motion is characterized by a potential energy which is proportional to the square of the distance from some equilibrium point. However, the answer key said that the motion was not truly simple harmonic (and the rolling ball only appeared to be so for small amplitudes of motion).
Why is motion with the a potential energy proportional to the square of displacement from equilibrium not simple harmonic in this case?
EDIT: To clarify, the "ball" should have really been a point mass.