Question 22 of 223

The mass density of a spherical galaxy varies as $\dfrac{K}{r}$ over a large distance $r$ from its center. 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${T}^{2}\propto \dfrac{1}{{R}^{3}}$D$T\propto R$

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Answer: (A) ${T}^{2}\propto R$

$M=\int \rho dV$
$M={\int }_{0}^{R}\dfrac{k}{r}4\pi {r}^{2}dr$
$M=4\pi k{\int }_{0}^{R}rdr$
$M=\dfrac{4\pi k{R}^{2}}{2}=2\pi k{R}^{2}$

${F}_{G}=\dfrac{GMm}{{R}^{2}}=m{\omega }_{0}^{2}R$
$\Rightarrow \dfrac{G\dfrac{4\pi k{R}^{2}}{2}}{{R}^{2}}={\omega }_{0}^{2}R\Rightarrow {\omega }_{0}=\sqrt{\dfrac{2\pi kG}{R}}$
$∵T=\dfrac{2\pi }{{\omega }_{0}}=\dfrac{2\pi \sqrt{R}}{\sqrt{2\pi kG}}=\sqrt{\dfrac{2\pi R}{kG}}$
$\Rightarrow {T}^{2}\propto R$