The hollow Moon

Problem

The hollow Moon. Newton proved that a sphere of uniform density attracts an external body as if all its mass were concentrated at the centre; he then extended the result to spherical shells of uniform density. Using these two results (and without resorting to integrals), show that a hollow sphere of outer radius $R$, inner radius $r$ and total mass $M$ distributed uniformly in the shell attracts a body on its outer surface with exactly the same force as a solid sphere of equal mass $M$ and radius $R$. Surprisingly the result does not depend on the inner radius $r$: even a very thin “shell” of mass $M$ has the same effect.

Solution

Idea: shell as the difference of two solid spheres. As suggested, we think of the shell as the difference between a large solid sphere (radius $R$) and a small solid sphere (radius $r$), both with the same uniform density $\rho$ as the shell material. Subtracting the small one from the large one leaves exactly the hollow shell (Povey 2015, §10.1).

The masses of the two fictitious spheres. With common density $\rho$: $M_R = \rho\,\tfrac{4}{3}\pi R^3, \qquad M_r = \rho\,\tfrac{4}{3}\pi r^3, \qquad M = M_R - M_r$

Force on a body outside the surface ($d = R$). A body of mass $m$ on the outer surface is at distance $R$ from the centre. It is external to both solid spheres, so by Newton’s theorem each acts as a point mass concentrated at the centre. The two forces are opposite (one attracts, the other is subtracted): $F = \frac{G M_R\, m}{R^2} - \frac{G M_r\, m}{R^2} = \frac{G(M_R - M_r)\,m}{R^2} = \frac{G M m}{R^2}$

Conclusion. The force depends only on the total mass $M$ and the outer radius $R$: identical to that of a solid sphere of equal mass. The surface acceleration is $\ev{g = \frac{G M}{R^2}, \quad\text{independent of the inner radius } r}$

Limiting case (thought experiment). In the limit $r \to R$ the shell becomes an extremely thin cap: as long as the total mass $M$ is conserved, the density grows, but the external force remains $GMm/R^2$. A “hollow Moon” would appear indistinguishable from a solid one, as long as we stay outside it.

Links

Topics: Gravitazione Concepts: Legge di gravitazione universale · Forza peso Skills: Impostazione simbolica · Analisi di casi limite e fantafisica