A sphere rolling on the inside surface of a cone

Abstract

We analyse the motion of a sphere that rolls without slipping on the inside of a conical surface having its axis in the direction of the constant gravitational field of the Earth. This non-holonomic system admits a solution in terms of quadratures. We exhibit that for each set of conditions defining the motion there is only one circular orbit which is stable. We show that its solutions can be found using an analogy with central force problems. We also discuss the case of motion with no gravitational field, that is, of motion on a freely falling cone.