25.4: Cylinder Rolling Inside another Cylinder

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Now consider a solid cylinder radius a rolling inside a hollow cylinder radius R, angular distance from the lowest point \(\theta\), the solid cylinder axis moving at \(V=(R-a) \dot{\theta}\) and therefore having angular velocity (compute about the point of contact) \(\Omega=V / a\).

The kinetic energy is

\begin{equation}
\frac{1}{2} M V^{2}+\frac{1}{2} I(V / a)^{2}=\frac{1}{2}\left(M+\frac{I}{a^{2}}\right)(R-a)^{2} \dot{\theta}^{2}
\end{equation}

The potential energy is \(-M g(R-a) \cos \theta\).

The Lagrangian \(L=T-V\), the equation of motion is

\begin{equation}
\left(M+\frac{I}{a^{2}}\right)(R-a)^{2} \ddot{\theta}=-M g(R-a) \sin \theta \cong-M g(R-a) \theta
\end{equation}

so small oscillations are at frequency \(\omega=\sqrt{\frac{g}{\left(1+\frac{I}{M a^{2}}\right)(R-a)}}\).