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

B

Explanation

If Mis the total mass of the system of particles, then similar to a satellite revolving around the earth, the orbital velocity of the test mass is given by
`v = sqrt((GM)/(r ))`
For `r lt= R , v = sqrt((G.(4)/(3) pi r^(3)rho_(0))/(r ))`
i.e v ` prop sqrt(r^(2)) ` or v `prop ` r
i.e v Increases linearly with r up to r = R
Thus (1) and (4) are wrong but (2) and (3) satisfy this
`:.` For ` r gt R v = sqrt((GM)/(r ))` or `v prop (1)/(sqrt(r ))`
Thus Is correctly shown in figure (2). Figure (3) Is wrong because for r `gt` R, vis shown as constant.