An explicit solution to the spinning ring problem

Abstract

A ring may be regarded as a torus with r << R, where R is the major radius and r is the minor radius. When such a ring is placed on a rough rod and released with some angular velocity, it may continue to vertically spin around the rod for some time instead of falling down immediately. It was observed that two different kinds of motion for the rod exist, which are referred to as single point and double point contact motion, based on the number of contact points of the ring that are in contact with the rod. Single point contact motion was observed for rings and double point contact motion was observed in the case of a washer. We investigated the characteristics of the single point contact motion. An explanation is provided for the single point contact motion of the ring and an analysis of the forces on the ring is made. We observe a hyperbolic decay in the angular velocity of the ring. An explicit solution is determined for the single point contact motion of the ring and the obtained solutions match the observed motion of the ring.