Tabletop mass coupled to a hanging mass ($U_J(r)=J^2/(2m{r}^2)+Mgr-Mgl$)

A mass $m$ moves on a smooth horizontal table and is joined through a hole to a hanging mass $M$ by a massless inextensible string of length $l$. If the table radius is $r$, the hanging length is $l-r$, so the physical coordinate range is $0<r<l$. Their kinetic energy and potential energy areThe cyclic coordinate $\theta$ gives conserved angular momentum $J=m{r}^2\dot{\theta }$. At fixed $J$, the radial energy isFor $J\ne 0$, the effective potential prevents approach to the hole, and $U_J^{\prime }=0$ gives $r_0^3=J^2/(mMg)$. This is a physical interior equilibrium only if $r_0<l$, equivalently $J^2<mMg{l}^3$. If the minimum lies beyond the coordinate range, the formal effective potential minimum does not describe an accessible circular motion.

Since $U_J^{\prime \prime }(r_0)=3Mg/r_0$, small radial oscillations about an admissible $r_0$ haveThe hanging mass contributes radial inertia as well as weight. Reaching $r=l$ brings it to the guide, where the original smooth-coordinate model needs a new contact condition.