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

Answer 1

Answer: (A)

The pressure gradient at a distance $r$ from the centre of the spherical planet is
$\frac{dP}{dr}=-\rho g$ , where $g$ is the acceleration due to gravity at that point
$\frac{dP}{dr}=-\left(\frac{3M}{4\pi R^3}\right)\left(\frac{GM}{R^3}r\right)$
$dP=-\left(\frac{3G{M}^2}{4\pi R^6}r\right)dr$
${\int }_P^{0}dP=-\frac{3G{M}^2}{4\pi R^6}{\int }_0^{R}rdr$
$P=\frac{3G{M}^2}{8\pi R^4}$