5.4.9: Solid Sphere
A solid sphere is just lots of hollow spheres nested together. Therefore, the field at an external point is just the same as if all the mass were concentrated at the centre, and the field at an internal point \(\text{P}\) is the same is if all the mass interior to \(\text{P}\), namely \(M_r\), were concentrated at the centre, the mass exterior to \(\text{P}\) not contributing at all to the field at \(\text{P}\). This is true not only for a sphere of uniform density, but of any sphere in which the density depends only of the distance from the centre – i.e., any spherically symmetric distribution of matter.
If the sphere is uniform, we have \(\frac{M_r}{M} = \frac{r^3}{a^3}\), so the field inside is
\[g = \frac{G M_r}{r^2} = \frac{GMr}{a^3}. \label{5.4.24} \tag{5.4.24}\]
Thus, inside a uniform solid sphere, the field increases linearly from zero at the centre to \(GM / a^2\) at the surface, and thereafter it falls off as \(GM/r^2\).
If a uniform hollow sphere has a narrow hole bored through it, and a small particle of mass \(m\) is allowed to drop through the hole, the particle will experience a force towards the centre of \(GMmr/a^3\), and will consequently oscillate with period \(P\) given by
\[P^2 = \frac{4 \pi^2}{GM}a^3. \label{5.4.25} \tag{5.4.25}\]