= Rolling inside a fixed circular cylinder
{title2=$\Omega=-\frac{R_1-R_0}{R_0}\dot\theta$}
A round body of radius $R_0$ in a fixed cylinder of radius $R_1$ has <center of mass> path radius $a=R_1-R_0$. With orbital angle $\theta$ and spin <angular velocity> $\Omega$ both positive about the same axis, <rolling without slipping> gives $a\dot\theta+R_0\Omega=0$. For <moment of inertia> $I=\gamma mR_0^2$, its <kinetic energy> is $\frac12ma^2(1+\gamma)\dot\theta^2$. Release from rest at angle $\alpha$ gives, by <conservation of energy>, a descent time $\sqrt{1+\gamma}$ times that of frictionless sliding along the same <center of mass> path. The spin sign reverses if its positive direction is chosen opposite to the orbital angle.
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