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

The system is a solid sphere.
For r > R , gravitational field `=(GM)/r^2`
`therefore (mGM)/r^2 =(mv^2)/r rArr v prop 1/sqrtr`
For `r le R` , gravitational field =`(GM)/R^3 r`
`therefore m(GM)/R^3 r = (MV^2)/r rArr v prop r`
The corresponding v – r graph will be as shown in option (C)