Question 50: Gravitation

Question

A spherically symmetric gravitational system of particles has a mass density
$$\rho  = \left\{ {\matrix{
   {{\rho _0}} & {for} & {r \le R}  \cr 
   0 & {for} & {r > R}  \cr

} } \right.$$
Where $$\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

Accepted Answer

For $$r\ge R$$,
Force on test mass m is $$F=m	imes |E_g|$$
where, $$E_g$$ = Gravitational field intensity at the point of observation
$$	herefore$$ $${{m{v^2}} \over r} = m 	imes \left[ {{{GM} \over {{r^2}}}} ight]$$,
Where, M = Total mass of spherical system
$$	herefore$$ $$v \propto {1 \over {\sqrt r }}$$
For $$r < R$$,
$$F' = m|E{'_g}|$$
$$	herefore$$ $${{m{V^2}} \over r} = m\left[ {{{GM} \over {{R^3}}} 	imes r} ight]$$
Hence, $$V \propto r$$

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