Question

A hypothetical planet in the shape of a sphere is completely made of an incompressible fluid and has a mass $M$ and radius $R$ . If the pressure at the surface of the planet is zero, then the pressure at the centre of the planet is [ $G$ = universal constant of gravitation]

• A. $P=\frac{3GM^{2}}{8\pi R^{4}}$
• B. $P=\frac{3GM^{2}}{4\pi R^{4}}$
• C. $P=\frac{3GM^{2}}{8\pi ^{2}R^{4}}$
• D. $P=\frac{3GM^{2}}{4\pi ^{2}R^{4}}$

Answer 1

Answer: (A)

The pressure gradient at a distance $r$ from the centre of the spherical planet is
$\frac{d P}{d r}=-\rho g$ , where $g$ is the acceleration due to gravity at that point
$\frac{d P}{d r}=-\left(\frac{3M}{4\pi R^{3}}\right)\left(\frac{GM}{R^{3}} r\right)$
$dP=-\left(\frac{3GM^{2}}{4\pi R^{6}} r\right)dr$
$\displaystyle \int _{P}^{0}dP=-\frac{3GM^{2}}{4\pi R^{6}}\displaystyle \int _{0}^{R}rdr$
$P=\frac{3GM^{2}}{8\pi R^{4}}$