A disk of radius r and mass M is oscillating inside a cylinder with a bigger radius R, without slipping. The goal is to find the dependency on $\omega$, the angular velocity of the disk, and $\frac{d\theta}{dt}$.
I've seen this problem solved using the relation of the length traveled by the center of mass $s=(R-r) \theta$ and it's velocity $s= v_{cm}t$. So:
$v_{cm} t= (R-r) \theta\\ \Rightarrow \frac{\omega r}{(R-r)} = \frac{d\theta}{dt}$
But this solution doesn't consider aceleration of the center of mass. As the normal force isn't even constant, either should be angular acceleration. So my solution would have started from:
$v_{cm} t + \frac{1}{2} a_{cm} t^2 = (R-r) \theta$
and gets impossible complicated. I wonder if there is anything that I'm not aware about that makes the first (and simpler) solution ok rather than mine, or if it's just a simpler aproximation.