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

For `r le R, (mv^(2))/r=(GmM)/(r^(2))to`(1)
here `M=(r/3pi r^(3))rho_(0)`
Sunstituting in Eq (1) we get `voor`
i.e. `v-r` graphs is a straight line passeing through origin. For `r gt R`
`(mv^(2))/r=(Gm(3/4piR^(3))rho_(0))/(r^(2)0` or `voo1/(sqrt(r))`
The corresponding v-r graph will be as shown in option 3.