Question

A star of mass $M$ and radius $R$ is made up of gases. The average gravitational pressure compressing the star due to gravitational pull of the gases making up the star depends on $R$ as

• A. $1/R^4$
• B. $1/R$
• C. $1/R^2$
• D. $1/R^6$

Answer 1

Answer: (A)

For a spherical shell of radius $r$ and thickness $dr$, weight of layer is balanced by pressure on layer.

$\therefore $ Weight $=\left\{p-\left(p+\Delta p\right)\right\}4\pi r^{2}$
$\Rightarrow mg=-\Delta p.4\pi r^{2}$
$\Rightarrow 4\pi r^{2}.dr.p.g.=-\Delta p.4\pi r^{2}$
$\Rightarrow -\Delta p=pgdr$
$\Rightarrow -\Delta p=p^{2}.\frac{4}{3}\pi Rdr$
So, $P_{av}=\frac{1}{R}\int\limits^{R}_{0}\Delta p=\frac{4}{3}\pi p^{2}R\int\limits^{R}_{0}dr$
$=\frac{4}{3}\frac{\pi MR.R}{\left(V\right)^{2}}$
$=\frac{4}{3}\pi.\frac{MR^{2}}{\left(\frac{4}{3}\pi R^{3}\right)^{2}}$
$\Rightarrow P_{av}=\frac{3}{4\pi}.\frac{M}{R^{4}}$
$\Rightarrow P_{av}\alpha\frac{1}{R^{4}}$