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

When `r le R`, the force on the test mass `m` at the surface of the sphere `= mg`. Force on the test mas at distance `r` from the centre of sphere is
`F = mg.(r )/(R )`.
Therefore, `mg .(r )/(R ) = (m upsilon^(2))/(r )` or `upsilon sqrt((g)/(R ))`
or `upsilon prop r` for `r le R`
When `r gt R`, the test mass is outside the sphere
Then, `(GM m)/(r^(2)) = (m upsilon^(2))/(r )`
or `upsilon = sqrt((GM)/(r )) = sqrt((g)/(r )) R` or `upsilon prop (1)/(sqrt(r))`
Therefore, the graph shown in option (c ) is correct.