= Rolling disk with a rim point mass
{title2=$L=MR^2(5-2\cos\theta)\dot\theta^2/2-MgR(1-\cos\theta)$}
A uniform disk of <mass> $2M$ and radius $R$ carries an extra <point mass> $M$ on its rim. For <rolling without slipping>, let $\theta=0$ place that <mass> at the contact. Its position is $[R(\theta-\sin\theta),R(1-\cos\theta)]$. Combining disk translation, disk rotation and point motion gives the displayed <Lagrangian>. Its small-amplitude <angular frequency> is $\sqrt{g/(3R)}$, whereas the same object pivoted at its centre has <angular frequency> $\sqrt{g/(2R)}$.
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