The mass density of a spherical galaxy varies as (K)/(r) over a...

$M = ∫ ρdV$

$M = {∫}_{0}^{R}\dfrac{k}{r}4\pi{r}^{2}dr$

$M = 4\pik{∫}_{0}^{R}rdr$

$M = \dfrac{4\pik{R}^{2}}{2} = 2\pik{R}^{2}$

${F}_{G} = \dfrac{GMm}{{R}^{2}} = m{\omega}_{0}^{2}R$

$⇒ \dfrac{G\dfrac{4\pik{R}^{2}}{2}}{{R}^{2}} = {\omega}_{0}^{2}R ⇒ {\omega}_{0} = \sqrt{\dfrac{2\pikG}{R}}$

$∵ T = \dfrac{2\pi}{{\omega}_{0}} = \dfrac{2\pi\sqrt{R}}{\sqrt{2\pikG}} = \sqrt{\dfrac{2\piR}{kG}}$

$⇒ {T}^{2} ∝ R$