= Tabletop mass coupled to a hanging mass
{title2=$U_J(r)=J^2/(2mr^2)+Mgr-Mgl$}
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> are
$$
T=\tfrac12(M+m)\dot r^2+\tfrac12mr^2\dot\theta^2,\qquad V=Mgr-Mgl.
$$
The <cyclic coordinate> $\theta$ gives conserved <angular momentum> $J=mr^2\dot\theta$. At fixed $J$, the radial <energy> is
$$
E=\tfrac12(M+m)\dot r^2+U_J(r),\qquad U_J=\frac{J^2}{2mr^2}+Mgr-Mgl.
$$
For $J\ne0$, the <effective potential> prevents approach to the hole, and $U_J'=0$ gives $r_0^3=J^2/(mMg)$. This is a physical interior <equilibrium> only if $r_0<l$, equivalently $J^2<mMgl^3$. If the minimum lies beyond the coordinate range, the formal <effective potential> minimum does not describe an accessible circular motion.
Since $U_J''(r_0)=3Mg/r_0$, small radial <oscillations> about an admissible $r_0$ have
$$
\omega_r^2=\frac{3Mg}{(M+m)r_0}.
$$
The 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.
Back to article page
