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

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

  1. T ∝ r
  2. T ∝r
  3. T ∝r2
  4. T2∝r3

Solution

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

g= 2 G λr

Where,

λ= linear mass density of galaxy.  

For the orbital motion around the galaxy,



fg=fcentripetal

⇒       mg=m ω2r 

⇒       2Gλr=ω2r

⇒      ω2 ∝1r2

⇒      ω ∝1r

⇒      2πT ∝1r     ⇒     T∝r

So option B is correct.

Asked in: JEE Main 2015 (10 Apr Online)

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