Question

"A spherically symmetric gravitational system of particles has a mass density $\rho =\Biggl \{ \begin{array} \ \rho_0 \ \text{for} \ r \le R\\ 0 \ \text{for} r\ > \ R \\ \end{array}$ , 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 from the centre of the system is represented by"

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The Correct Option is C

"For $ r \le R$ $ \, \, \, \, \, \, \, \frac{mv^2}{r}= \frac{GmM}{r^2} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(i)$ $Here, \, \, \, \, \, \, \, \, \, M= \bigg( \frac{4}{3} \pi r ^3 \bigg) \rho_0$ Substituting in E (i), we get v $\propto$ r For r> R, $\, \, \, \, \, \, \, \, \, \, \, \, \frac{mv^2}{r}= \frac{Gm \big( \frac{4}{3} \pi R^3 \big) \rho_0 } {r^2}$ or $\, \, \, \, \, \, \, \, \, v \propto \frac{1}{ \sqrt{r} }$ The corresponding v-r graph will be as shown in option (c). $\therefore$ Correct option is (c)."