A very long (length L ) cylindrical galaxy is made of uniformly distributed mass and has radius R ( R

Question

A very long (length L ) 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 the star is T and its distance from the galaxy's axis is r , then

TestHub · Accepted Answer

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

$g = \frac{2 G λ}{r}$

Where,

$λ =$ linear mass density of galaxy.

For the orbital motion around the galaxy,

$f_{g} = f_{c e n t r i p e t a l}$

$\Rightarrow m g = m ω^{2} r$

$\Rightarrow \frac{2 G λ}{r} = ω^{2} r$

$\Rightarrow ω^{2} \propto \frac{1}{r^{2}}$

$\Rightarrow ω \propto \frac{1}{r}$

$\Rightarrow \frac{2 π}{T} \propto \frac{1}{r} \Rightarrow T \propto r$

So option $B$ is correct.

Physics - Gravitation Question with Solution | TestHub

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