Question

Inside a uniform sphere of density $\rho$ there is a spherical cavity whose centre is at a distance $l$ from the centre of the sphere. Find the strength $G$ of the gravitational field inside the cavity.

Answer 1

Treating the cavity as negative mass of density $-\rho$ in a uniform sphere density $+\rho$ and using the superposition principle, the sought field strength is,
$\overrightarrow{G}=\overrightarrow{G_1}+\overrightarrow{G_2}$
or $\overrightarrow{G}=-\frac{4}{3}\pi \gamma \rho \overrightarrow{r_{+}}+-\frac{4}{3}\gamma \pi (-\rho )\overrightarrow{r_{-}}$
(where $\overrightarrow{r_{+}}$ and $\overrightarrow{r_{-}}$ are the position vectors of an arbitrary point $P$ inside the cavity with respect to centre of sphere and cavity respectively.)
Thus $\overrightarrow{G}=-\frac{4}{3}\pi \gamma \rho (\overrightarrow{r_{+}}-\overrightarrow{r_{-}})=-\frac{4}{3}\pi \gamma \rho \overrightarrow{l}$