Condition for Simple Harmonic Motion

1. The problem statement, all variables and given/known data
a small object is mounted to the perimeter of a hoop of radius r. The mass of the object and the hoop is the same. The hoop is placed into a fixed semi-cylinder shaped rough trough of radius R, such that the small mass is at the top. Find the least R/r ratio such that the object performs simple harmonic motion.

3. The attempt at a solution

For the object to perform SHM it must be in a stable equilibrium position. So, its potential energy must be minimum at the mean position. But how do I express it mathematically?

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