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?
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