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

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

Solution

Gravitational force supplies the necessary centripetal force to hold a star in orbit.

Gravitational force = Centripetal force

F = 2GMm/Lr

mr(2π/T)2 = 2GMm/Lr

T ∝ r

The correct option is C.