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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.


Solve by Conservation of Energy:


The kinetic energy has two components, a translational one and a rotational one:

$$K=\frac12 Mv^2+\frac12 I\dot{\theta}^2$$

For slipping without sliding, $v=r\dot{\theta}$

The potential energy (gravity):


Insert everything into $(1)$ and take the time derivative:


This the Newtonian equation of motion of the system. When solved it yields $\theta(t)$.


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