Question

A uniform sphere has a mass $M$ and radius $R$. Find the gravitational pressure $P$ inside the sphere, as a function of the distance from its centre.

Answer 1

Consider a layer of thickness or at a distance $r$ from the centre of the sphere.
Now force due to the layer $dF=4\pi r^{2}dP$
$\Rightarrow dP4\pi r^2=\frac{G\left(\frac{4}{3}\pi r^2\rho \right)\left(4\pi r^{2}dr\rho \right)}{r^2}$
(Where $\rho$ is the mean density of sphere)
or, $dP=G\frac{4}{3}\pi {\rho }^2\pi {\rho }^{2}rdr\therefore p=G{\int }_r^R\frac{4}{3}\pi {\rho }^{2}rdr$
or $P=G\frac{2\pi }{3}{\rho }^2(R^2-r^2)$
$=\frac{3}{8}G\frac{\{1-r^2/R^2\}{M}^2}{\pi R^4}[\because \rho =M/(4/3\left)\pi R^3\right]$.