Question

A spherically symmetric gravitational system of particles has a mass density,
$\rho =\{\begin{array}{c}{\rho }_0\text{for}r\le R_0\\ 0\text{for}r>R\end{array}$
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

For $r\le R,E=\frac{GMr}{R^3}$

Centripetal force = Gravitational force

$\frac{m{v}^2}{r}=\frac{GmMr}{R^2}......\left(1\right)$

Here $M=\left(\frac{4}{3}\pi R^3\right){\rho }_0$

Substituting in equation (1) we get

$v\propto r$

i.e., $v-r$ graph is a straight line passing through the origin.

For $r>R,E=\frac{GM}{r^2}$

$\frac{m{v}^2}{r}=\frac{Gm\left(\frac{4}{3}\pi R^3\right){\rho }_0}{r^2}$

$\Rightarrow v\propto \frac{1}{\sqrt{r}}$

i.e., $v-r$ graph is a rectangular hyperbola.

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(C\right)$ is the correct answer.