Question

A spherically symmetric gravitational system of particles has a mass density \[ \rho = \begin{cases} \rho_0 & \text{for } r \le R \\ 0 & \text{for } r > R \end{cases} \] 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. Graph (a)
• B. Graph (b)
• C. Graph (c)
• D. Graph (d)

Hint:

Inside a uniform sphere, gravitational field is proportional to distance from the centre.

Answer 1

The Correct Option is C

Step 1: For \(r \le R\), enclosed mass \(M(r) \propto r^3\).

Step 2: Gravitational force provides centripetal force:
\[ \frac{v^2}{r} = \frac{GM(r)}{r^2} \Rightarrow v \propto r \]
Step 3: For \(r > R\), total mass is constant:
\[ v = \sqrt{\frac{GM}{r}} \Rightarrow v \propto \frac{1}{\sqrt{r}} \]
Thus, velocity increases linearly inside and decreases outside.