Question

There exists a spherically symmetric gravitational system of particles of radius $R$. 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 the first region: $0<r<R$
Total force= $\frac{4}{3}\frac{\pi r^{3}Gm\rho }{r^2}=\frac{4}{3}\pi rGm\rho$

Here, gravitation force is equal to the centripetal force,
$\frac{4}{3}\pi rGm\rho =\frac{m{v}^2}{r}{v}^2\propto r^{2}orv\propto r$


In the second condition: $R<r<\infty$
$F=\frac{4}{3}\frac{\pi R^{3}Gm\rho }{r^2}$
again,
Gravitational force $=$ centripetal force
$\frac{4}{3}\frac{\pi R^{3}Gm\rho }{r^2}=\frac{m{v}^2}{r}$
so, $v\propto R^{3/2}v\propto \frac{1}{r^{1/2}}$
hence,
So, option C is correct.