A spherically symmetric gravitational system of particles has a mass density` rho={(rho_0,for, r,lt,R),(0,for,r,gt,R):}` where`rho_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 distahce `r(0ltrltOO)` form the centre of the system is represented by

Options

A.
B.
C.
D.

Answer

C

Explanation

When `r le R`, then force on the test mass m at the surface of the sphere =mg. Force on the test mass at distance r from the centre of sphere is

If `r lt R`, then `F= (GMm)/(R^(2)).(r )/(R )= (GMm)/(R^(3)).r`
`therefore (mv^(2))/(r )= (GMm)/(R^(3)).r`
`therefore v xx r`
If `r gt R`, then `F= (GMm)/(r^(2))`
`therefore (Mv^(2))/(r )= (GMm)/(r^(2))`
`therefore vxx (1)/(sqrtr)`
Hence, option (c ) is correct