Question

The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance ${}^{\prime }{r}^{\prime }$ from its centre. In that region, a small star is in a circular orbit of radius $R$. Then the period of revolution, $T$ depends on $R$ as

• A. $T\propto R$
• B. $T^2\propto \frac{1}{R^3}$
• C. $T^2\propto R$
• D. $T^2\propto R^3$

Answer 1

Answer: (C)

$dm=\rho dv$
$dm=\left(\frac{k}{r}\right)\left(4\pi r^{2}dr\right)$
$dm=4\pi krdr$
$M=\underset{0}{\overset{R}{\int }}dm=\underset{0}{\overset{R}{\int }}4\pi krdr$
$M={4\pi k\frac{r^2}{2}|}_0^R$
$M=2\pi k\left(R^2-0\right)$
$M=2\pi k{R}^2$
for circular motion gravitational force will provide required centripital force or
$\frac{GMm}{R^2}=\frac{m{v}^2}{R}$
$\frac{G\left(2\pi k{R}^2\right)m}{R^2}=\frac{m{v}^2}{R}$
$\Rightarrow v=\sqrt{2\pi GkR}$
Time period $T=\frac{2\pi R}{v}$
$T=\frac{2\pi R}{\sqrt{2\pi GkR}}\propto \sqrt{R}$
or $T^2\propto R$