\(E = mgh + \frac{1}2} m v^{2} + \frac{1}{2} I \omega^{2} = mgh + \frac{1}2} m r ^{2} \omega ^{2} + \frac{1}{2} I \omega^{2}\)
for a solid cylinder: \(I = \frac{1}{2} m r^{2}\)
for a hollow cylinder: \(I = mr^{2}\)
I will look at the case of a hollow cylinder:
\(E = mgh + I \omega ^{2} = constant \\ \\ I = \frac{mgh}{ \omega^{2} }\)
That is as far as i get.