Sliding on the inside of a Conical Surface

We analyze the frictionless motion of a point-like particle that slides under gravity on an inverted conical surface. This motion is studied for arbitrary initial conditions and a general relation, valid within 13%, between the periods of radial and angular oscillations, that holds for both small and high energy trajectories, is obtained. This relation allows us to identify the closed orbits of the system. The virtues of this model to illustrate pedagogically, how in a physical system, the energy is transferred between different modes, are also emphasized. Two easy identification criteria for this type of motion with potential interest in industrial design are obtained.