6th Putnam 1946 Problem B3 The density of a solid sphere depends solely on the radial distance. The gravitational force at any point inside the sphere, a distance r from the center, is kr 2 (where k is a constant). Find the density (in terms of G, k and r), and the gravitational force at a point outside the sphere. [You may assume the usual results about the gravitational attraction of a spherical shell.]
A shell exerts no net attraction on a point inside and acts on a point outside as if all its mass was concentrated at its center. So let the density at a radial distance r be ρ(r). Then G/r 2 ∫ 0 r 4πt 2 ρ(t) dt = kr 2 . Hence ∫ 0 r t 2 ρ(t) dt = k/(4Gπ) r 4 . Differentiating: r 2 ρ(r) = k/(Gπ) r 3 . So ρ(r) = k/(Gπ) r. The force at the surface of the sphere (R) is kR 2 . Hence the force at a distance r > R from the center of the sphere is kR 4 /r 2 . [We know the force is a simple inverse square law outside the sphere and it must be continuous at the surface.] 6th Putnam 1946 © John Scholes jscholes@kalva.demon.co.uk 14 Dec 1999