= Tabletop mass coupled to a hanging mass
{title2=$U_J(r)=J^2/(2mr^2)+Mgr-Mgl$}
A tabletop <mass> $m$ joined through a hole by a <massless inextensible string> of length $l$ to a hanging <mass> $M$ has radial inertia $M+m$ and conserved <angular momentum> $J=mr^2\dot\theta$. The radial <energy> is $\tfrac12(M+m)\dot r^2+U_J(r)$, with $0<r<l$. Its <effective potential> diverges at the hole for $J\ne0$ and has a unique minimum at $r_0^3=J^2/(mMg)$. Since $U_J''(r_0)=3Mg/r_0$, small radial oscillations at fixed $J$ have squared angular frequency $3Mg/[(M+m)r_0]$.
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