Condition for Simple Harmonic Motion

[utkarshakash]

Homework Statement

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.

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?

 

[Saitama]

You have the right idea. I discussed the very same problem in past here.

Look: https://www.physicsforums.com/showthread.php?t=663246

 

[utkarshakash]

You have the right idea. I discussed the very same problem in past here.

Look: https://www.physicsforums.com/showthread.php?t=663246

Hmm that's a really long post and I think I should better leave this question as it would be too time consuming. btw Thanks for help!