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 ge R`
Force on the test mass m is `F = mxx|E_g|`
Where `E_g` is the gravitational field intensity at the point
of observation
`:.(mv^2)/(r ) = mxx[(GM)/(r^2)]` where M is the total mass of the
spherical system.
`:. V prop (1)/(sqrtr)`
`Foorrlt R Again F' = m|E_g|`
`:. (mv^2)/(r ) = m [(GM)/(R^3)xxr]`
`rArr vprop r`