Question

A spherically symmetric gravitational system of particles has a mass density $\rho =\{\begin{array}{c}{\rho }_0\text{for}r\le R\\ 0\text{for}r>R\end{array}$
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 distance $r(0<r<\infty )$ from the centre of the system is represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is C



Using the equation for centripetal force on test mass for a point inside the sphere
For $r<R$
$\frac{GMr}{R^3}=\frac{m{V}^2}{r}$
$V\propto r$
Using the equation for centripetal force on test mass for a point outside the sphere
for $r\ge R\Rightarrow \frac{GM}{r^2}=\frac{m{V}^2}{r}$
$\Rightarrow V^2\propto \frac{1}{r}$