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 kr2 (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/r2 ∫0r 4πt2ρ(t) dt = kr2. Hence ∫0r t2ρ(t) dt = k/(4Gπ) r4. Differentiating: r2ρ(r) = k/(Gπ) r3. So ρ(r) = k/(Gπ) r.
The force at the surface of the sphere (R) is kR2. Hence the force at a distance r > R from the center of the sphere is kR4/r2. [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
14 Dec 1999