Question

A star is modeled as a uniform spherical distribution of matter.How gravitational pressure on surface depends on volume of the star?

• A. $P\propto V$
• B. $P\propto V^{-1/3}$
• C. $P\propto V^{-2/3}$
• D. $P\propto V^{-4/3}$

Answer 1

Answer: (D)

$P\propto V^{-4/3}$

Gravitational potential energy of star

$U=-\frac{3G{M}^2}{5R}$

$VolumeV=\frac{4}{3}\pi R^3\Rightarrow R={\left(\frac{3V}{4\pi }\right)}^{1/3}$

$\Rightarrow U=-\frac{3}{5}{\left(\frac{4\pi }{3}\right)}^{1/3}G{M}^2{V}^{-1/3}$

As dW=PdV=-dU

$\Rightarrow P=-\frac{dU}{dV}=\frac{1}{5}{\left(\frac{4\pi }{3}\right)}^{1/3}{V}^{-4/3}$