Paper 3, Section II, C

Vector Calculus | Part IA, 2010

Given a spherically symmetric mass distribution with density ρ\rho, explain how to obtain the gravitational field g=ϕ\mathbf{g}=-\nabla \phi, where the potential ϕ\phi satisfies Poisson's equation

2ϕ=4πGρ\nabla^{2} \phi=4 \pi G \rho

The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells SnS_{n} labelled by integers n1n \geqslant 1. The shell SnS_{n} has uniform density 2n1ρ02^{n-1} \rho_{0}, where ρ0\rho_{0} is a constant, and occupies the volume between radius 2n+12^{-n+1} and 2n2^{-n}.

Obtain a closed form expression for the mass of Geometria.

Obtain a closed form expression for the gravitational field g\mathbf{g} due to Geometria at a distance r=2Nr=2^{-N} from its centre of mass, for each positive integer N1N \geqslant 1. What is the potential ϕ(r)\phi(r) due to Geometria for r>1r>1 ?

Typos? Please submit corrections to this page on GitHub.