Qualitative analysis of a rolling hoop with mass unbalance

Abstract

The dynamical behavior of a rolling hoop with an unbalanced point mass under the influence of gravity is discussed. The whole process from rolling to hopping of the hoop is analyzed qualitatively. The conditions of slipping, hopping and touching down of the hoop are obtained. It is shown that the hoop cannot maintain a pure rolling before hopping up, and the slippage is unavoidable. The hoop has neither vertical velocity nor vertical acceleration at the moment when the normal constraint force vanishes. The hopping motion of the hoop can occur only when the derivative of the vertical acceleration with respect to time is positive. It requires that the angular velocity of the hoop should be larger than a critical value, and the mass point should be located in the fourth quadrant of the hoop circle at the moment of hopping. The whole process of the pure rolling, rolling with slipping, hopping and falling motions of the hoop is shown in the phase plane, and the physical explanation of the hopping motion is given.