A spherically symmetric gravitational systom of particles has 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

Step-by-step solution

When r ≤ R , then force on the test mass n at the surface of the sphere = m g Force on the test mass at distance r from the centre of sphere is If r < R , then F = G M m R 2 · r R = G M m R 3 · r ∴ m v 2 r = G M m R 3 · r ∴ v   α   r If r > R then F = G M m r 2 ∴ M v 2 r = G M m r 2 ∴   v   α   1 r