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A very long (length) cylindrical galaxy is made of uniformly distributed mass and has radius $R (R < < L) .$ 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 star is T and its distance from the galaxy's axis is r, then

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$T \propto \sqrt{r}$

$T^{2} \propto r^{3}$

$T \propto r^{2}$

$T \propto r$

answer is C.

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Detailed Solution

Centripetal force is provided by gravitational force $F = \frac{2 G M}{L r} m; F = (\frac{k}{r}) m k$ is some constant.

$\begin{array}{l} S o, \frac{m v^{2}}{r} = \frac{k m}{r} \\ \Rightarrow v = c o n s \tan t \\ T = \frac{2 π r}{v} ∴ T \propto r \end{array}$

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