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

Case I : `r le R`
force on the test mass can be given by `F = "mg" (r )/(R )`
Where mg is the force on the mass at the surface of the sphere.
`"mg" (r )/(R )=(mv^(2))/(r ) rArr v = sqrt((g)/(R )) r rArr v prop r" for " r le R`
Case II : `r ge R`
When the test mass is outside the sphere
`therefore (GMm)/(r^(2)) = (mv^(2))/(r ), v = sqrt((GM)/(r )) = sqrt((g)/(r ))R`
`rArr v prop (1)/(sqrt(r ))`
therefore, choice C is correct the graph is asshiwn in alternative