Question

A sperically symmetric granitational syatem of particles has a mass density $s$ { $\rho$ for r $\le$ R & O for r > R } where $\rho$ is a const.
A test mass can undergo circular motion under the influence of granitational field of particles. Its speed is v as a function of hit r from centre of system is represented by

Answer 1

R.E.F image

If $r\le R,$

$f=\frac{Gm}{r^2}\left(P\frac{4}{3}\pi r^3\right)=\frac{4Gm\int nr}{3}$

So, $f=\frac{m{v}^2}{r}$ [centripetal force]

$\Rightarrow \frac{m{v}^2}{r}=\frac{4Gm\int nr}{3}\Rightarrow v=r\sqrt{\frac{4Gm}{3}\int }$

If $r>R,$

$f=\frac{Gm}{r^2}(\frac{4\pi R^3}{3}\int )=\frac{4Gm}{{30}^2}\int n{R}^3$

$f=\frac{m{v}^2}{r}\Rightarrow \frac{m{v}^2}{r}=\frac{4gm}{3r^2}fn{R}^3\Rightarrow v=\frac{1}{r}\sqrt{\frac{4G\delta n{R}^3}{3}}$