= Small oscillations inside a circular cylinder
{title2=$\omega_0^2=g/[(1+\gamma)(R-r)]$}
For <rolling without slipping> inside a stationary circular cylinder, take the inner body's axial <moment of inertia> to be $I=\gamma mr^2$. Its <center of mass> follows a circle of radius $d=R-r$, and the reduced <Lagrangian> is $L=m(1+\gamma)d^2\dot\theta^2/2-mgd(1-\cos\theta)$. The <Euler-Lagrange equation> gives $(1+\gamma)d\ddot\theta+g\sin\theta=0$. Linearizing near the lowest position yields the displayed <small oscillation> frequency. A uniform solid cylinder has $\gamma=1/2$. This reduction requires sustained contact and enough <static friction> to prevent slipping.
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