Rolling ball between concentric races ($v_{\mathrm{cm}}=R\omega /2$)

If an inner race of radius $R$ rotates at angular velocity $\omega$ and an outer race is stationary, a ball of radius $r$ rolling at both contacts has centre speed $R\omega /2$ and spin $-R\omega /(2r)$. Its centre follows a circle of radius $R+r$. The angular momentum about the common race axis includes both orbital angular momentum and spin.