Question

The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance $r$ 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. $T^2\propto R$
• B. $T^2\propto R^3$
• C. $T^2\propto \frac{1}{R^3}$
• D. $T\propto R$

Answer 1

The correct option is A $T^2\propto R$

According to question, mass density of a spherical galaxy varies as $\frac{k}{r}$.

$\text{Mass,}M=\int \rho dV$

$\text{As,}V=\frac{4}{3}\pi r^3\Rightarrow dV=4\pi r^{2}dr$

$\Rightarrow M={\int }_0^{r=R}\frac{k}{r}4\pi r^{2}dr$

$\Rightarrow M=4\pi k{\int }_0^{R}rdr$

$\Rightarrow M=\frac{4\pi k{R}^2}{2}=2\pi k{R}^2$


$F_G=\frac{GMm}{R^2}=m{\omega }_0^{2}R(=F_C)$

$\Rightarrow \frac{G\left(2\pi k{R}^2\right)}{R^2}={\omega }_0^2$

$\Rightarrow {\omega }_0=\sqrt{\frac{2\pi kG}{R}}$

$\because {\omega }_0=\frac{2\pi }{T}$

$\therefore T=\frac{2\pi }{{\omega }_0}=\frac{2\pi \sqrt{R}}{\sqrt{2\pi KG}}=\sqrt{\frac{2\pi R}{kG}}$

$\Rightarrow T^2=\frac{2\pi R}{kG}$

$\because 2\pi ,K\text{and}G\text{are constant}$

$\therefore T^2\propto R$.

Hence, option $\left(A\right)$ is correct.