Finding the natural frequency of a cylinder oscillating on a circular surface

[theBEAST]

Homework Statement

Here is the problem with my 3 attempts:

1. Alright, so the first attempt, I did not take into account the rotation of the cylinder ω_c so I think it is wrong. But it seems to make sense because the whole cylinder rotates about O, so if I find that energy using parallel axis theorem it should get me the correct answer?

2. Here I said the energy from the cylinder is due to the rotational energy of the cylinder itself and also the velocity of the cylinder. I think this one is correct, what do you guys think?

3. I am not sure how to use moment to solve for the natural frequency. For some reason it gave me the same answer as 1. which I think is wrong. Is this one correct?

I know this is a long question, but if anyone could give me some insight on whether or not I did it right, it would be greatly appreciated!

 

[haruspex]

Your first equation in (1) is missing a factor 1/2 on the θ², but it appears in the following line. As you say, the energy expression is incomplete, missing the rotational energy. There are several ways to express the total KE. E.g. you can consider the cylinder as rotating about its point of contact with the dish and use the parallel axis theorem. In this view, there is no other KE term.
Writing down the energy equation then differentiating doesn't seem to me to be using the 'energy method'. That transforms it into moment method. You could get the equation into the form θ-dot = etc. then do a trig substitution to integrate.
(2) is the better effort, but it seems to me you have again lost the factor 1/2 on the θ², and this time it does not reappear.