= Particle on a cone coupled to a hanging mass
{title2=$(1+\sqrt2)\dot r^2+gr+h^2/(2r^2)=E$}
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> $\sqrt2m$. 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 $z_2=-s_0+\sqrt2r$. 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> $m\dot r^2+mh^2/(2r^2)$, while the second has <kinetic energy> $\sqrt2m\dot r^2$. 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 $h\ne0$, $U$ 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 $r<s_0/\sqrt2$ before the hanging particle reaches the hole.
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