Introduction to Classical Mechanics · Problem 9.56
Angular momentum, Part II (General L) — Morin
Physics — Mechanics · ball-in-a-cone
Problem
A ball (with I=(2/5)MR2) rolls without slipping on the inside surface of a fixed cone, whose tip points downward. The half-angle at the vertex of the cone is θ. Initial conditions have been set up so that the contact point on the cone traces out a horizontal circle of radius ℓ≫R, at frequency Ω, while the contact point on the ball traces out a circle of radius r (not necessarily equal to R, as would be the case for a great circle). Assume that the coefficient of friction between the ball and the cone is sufficiently large to prevent slipping. What is the frequency of precession, Ω? It turns out that Ω can be made infinite if r/R takes on a particular value; what is this value? Work in the approximation where R≪ℓ.