Question

A spherically symmetric gravitational system of particles has a mass density $\rho ={\rho }_0$ for $r\le R$ and $\rho =0$ for $r>R$ , where ${\rho }_0$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of the particles. Which figure represents its speed $v$ as a function of distance $r\left(0<r<\infty \right)$ from the center of the system?

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

If $M$ is the total mass of the system of particles, then similar to a satellite revolving around the earth, the orbital velocity of the test mass is given by
$v=\sqrt{\frac{GM}{r}}$
For $r\le R,v=\sqrt{\frac{G\cdot \frac{4}{3}\pi r^3{\rho }_0}{r}}$
i.e. $v\propto \sqrt{r^2}$ or $v\propto r$
i.e. $v$ increases linearly with $r$ up to $r=R$
Thus $(1)$ and $(4)$ are wrong but $(2)$ and $(3)$ satisfy this.
$\therefore \text{For }r>R,v=\sqrt{\frac{GM}{r}}\text{or }v\propto \frac{1}{\sqrt{r}}$
Thus is correctly shown in figure $(2)$ . Figure $(3)$ is wrong because for $r>R,v$ is shown as constant.