Particle on a cone coupled to a hanging mass ($(1+\sqrt{2}){\dot{r}}^2+gr+h^2/(2r^2)=E$)

A particle of mass $m$ constrained to the cone $z=-r$ is joined through the apex by a light inextensible ideal string to a hanging particle of mass $\sqrt{2}m$. The cylindrical radius is $r>0$ and gravitational acceleration is $g$. If the total ideal string length is $s_0$, the second particle has vertical coordinate $z_2=-s_0+\sqrt{2}r$. The ideal string and normal reaction exert no azimuthal force, so conservation of angular momentum gives $r^2\dot{\theta }=h$.

The first particle has kinetic energy $m{\dot{r}}^2+m{h}^2/(2r^2)$, while the second has kinetic energy $\sqrt{2}m{\dot{r}}^2$. Their combined gravitational potential energy is $mgr$ plus a constant. Hence conservation of energy gives the displayed equation, with effective potential $U(r)=gr+h^2/(2r^2)$. For $h\ne 0$, $U$ is strictly convex and has its unique minimum at $r=(h^2/g{)}^{1/3}$. The energy level has two radial turning points unless the motion is a circular orbit at that minimum.

The model maintains a mechanical constraint on the surface. For unilateral contact with one side of the cone, a normal reaction computed with a fixed outward sign must also have the physically allowed sign. A finite length of ideal string additionally restricts $r<s_0/\sqrt{2}$ before the hanging particle reaches the hole.