A spherically symmetric gravitational system of particles has a mass density ρ = { ρ 0 for r ≤ R 0 for r R where ρ 0 is a constant. A test mass can undergo circular motion under the influence

Question

A spherically symmetric gravitational system of particles has a mass density ρ = { ρ 0 ​ for r ≤ R 0 for r > R ​ where ρ 0 ​ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed v as a function of distance r ( 0 < r < ∞ ) from the centre of the system is represented by

TestHub · Accepted Answer

For r ≤ R : r m v 2 ​ = r 2 G ⋅ m m ′ ​ Here, m ′ = ( 3 4 ​ π r 3 ) ρ 0 ​ Substituting in Eq. (i) we get, v ∝ r i.e., v − r graph is a straight line passing through origin. For r > R : r m v 2 ​ = r 2 G ⋅ m ( 3 4 ​ π R 3 ) ρ 0 ​ ​ or V ∝ r ​ 1 ​ The corresponding v − r graph will be as shown in option (c). ∴ correct option is (c).

Physics - Gravitation Question with Solution | TestHub

Options:

Doubts & Discussion

JEE Main + Advanced (1 year 12th class)