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

Inside sphere `(r lt R), E=(GMr)/(R^(3))`
`F=mE=(GMmr)/(R^(3))=(Grho_(0).4/3piR^(3)mr)/(R^(3))`
`(mv^(2))/r=4/3pi rho_(0) Gmr implies v propr` (straight line)
Outside sphere `(r gt R)`
`(mv^(2))/r=mE=(mGM)/(r^(2))=(m.G.rho_(0).4/3piR^(3))/(r^(2))`
`v prop1/(sqrt(r))` (rectangular hyperbola type)