The mass density of a spherical galaxy varies as K/r over a large distance 'r' from its center.

Question

The mass density of a spherical galaxy varies as $K\over r$ over a large distance 'r' from its center. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as :

(1) $T^2\propto{1\over R^3}$

(2) $T^2\propto R$

(3) $T\propto R$

(4) $T^2\propto R^3$

Answer 1

Ans. (2) $T^2\propto R$

Sol. $M=\int\rho\mathrm{d}V$

$M=\int\limits_0^{r=R_0}{k\over r}4\pi r^2\mathrm{d}r$

$M=4\pi k\int\limits_0^{R_0}r\mathrm{d}r$

$M={4\pi kR_0^2\over2}=2\pi kR^2$

$F_G={GMm\over R_0^2}=m\omega_0^2R$

$\implies{G2\pi kR^2\over R^2}=\omega_0^2R$

$\implies\omega_0=\sqrt{2\pi KG\over R}$

$\because T={2\pi\over\omega_0}={2\pi\sqrt{R}\over\sqrt{2\pi KG}}=\sqrt{2\pi R\over KG}$

$\implies T^2\propto R$