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

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, T ∝ r, T ∝ r, T ∝ r 2, T 2 ∝ r 3

MARKS App · Accepted Answer

Due to a long solid cylinder gravitational field strength can be given as, g = 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 ⇒ m g = m ω 2 r ⇒ 2 G λ r = ω 2 r ⇒ ω 2 ∝ 1 r 2 ⇒ ω ∝ 1 r ⇒ 2 π T ∝ 1 r ⇒ T ∝ r So option B is correct.

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