Question 222 of 223

A very long (length $L$) cylindrical galaxy is made of uniformly distributed mass and has radius $R(R\lt \lt L)$. A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of the star is $T$ and its distance from the galaxy's axis is $r$, then

A$T\propto \sqrt{r}$B$T\propto r$C$T\propto {r}^{2}$D${T}^{2}\propto {r}^{3}$

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Answer: (B) $T\propto r$

Due to a long solid cylinder gravitational field strength can be given as,

$g=\dfrac{2G\lambda }{r}$

Where,

$\lambda =$ linear mass density of galaxy.

For the orbital motion around the galaxy,



${f}_{g}={f}_{centripetal}$

$\Rightarrow mg=m{\omega }^{2}r$

$\Rightarrow \dfrac{2G\lambda }{r}={\omega }^{2}r$

$\Rightarrow {\omega }^{2}\propto \dfrac{1}{{r}^{2}}$

$\Rightarrow \omega \propto \dfrac{1}{r}$

$\Rightarrow \dfrac{2\pi }{T}\propto \dfrac{1}{r}\Rightarrow T\propto r$

So option $B$ is correct.