### 18th Putnam 1958

**Problem A2**

A rough sphere radius R rests on top of a fixed rough sphere radius R. It is displaced slightly and starts to roll off. At what point does it lose contact?

**Solution**

Let the top of the fixed sphere be T and its centre O. Let X be the point of contact. Take angle XOT = θ. Let the mass be M and the normal force between the two spheres N. Resolving radially, MR(d<θ/dt)^{2} = Mg cos θ - N (1). Taking moments about an axis through X, MgR sin θ = I d^{2}θ/dt^{2} (2), where I is the moment of inertia of the sphere about X = the moment about a diameter + MR^{2} = 7/5 MR^{2}.

Integrating equation (2) gives g(1 - cos θ) = 7/10 R (dθ/dt)^{2}. Substituting in (1) gives N = g cos θ - 10/7 (1 - cos θ). Contact is lost when N becomes 0. In other words at the angle θ = cos^{-1} 10/17.

18th Putnam 1958

© John Scholes

jscholes@kalva.demon.co.uk

25 Feb 2002