Equilibrium of an Inextensible Heavy Thread (Chain) on Cone or Sphere

Abstract

The problem of finding the equilibrium configurations of a heavy inextensible closed thread (chain) on a smooth surface is considered. The surface is a unilateral constraint for the thread and represents either a circular cone, the axis of which is in parallel to the vertical (gravity) or a sphere. It is shown that the equilibrium nonhorizontal configurations of the thread-chain are possible only when the cone half-angle is in the range from 30° to 45°. In the case of a sphere, it is shown that equilibrium nonhorizontal configurations of the thread-chain are impossible.