The time period (T) of the artificial satellite of earth depends on the density (d) of the earth as

Orbital velocity of a satellite

${\text{M}}_{\text{E}}$ mass of earth

${\text{R}}_{\text{E}}$  Radius of earth

${\text{v}}_{\text{0}}=\sqrt{\frac{{\text{GM}}_{\text{E}}}{\text{R}}}\to \left(1\right)$

Here we are asked to sought out dependence of time period of satellite on densityof earth

We know${\text{v}}_0=\frac{2\pi \text{R}}{\text{T}},{\text{M}}_{\text{E}}=\frac{4}{3}\pi {\text{R}}^{\text{3}}\text{d}$

d is density of earth

On substituting in eq – (1)

$\frac{2\pi \text{R}}{\text{T}}=\sqrt{\frac{\text{G}\frac{4}{3}\pi {\text{R}}^{\text{3}}\text{d}}{\text{R}}}$

$\frac{4{\pi }^2{\text{R}}^2}{{\text{T}}^2}=\text{G}\times \frac{4}{3}\pi {\text{R}}^{\text{2}}\text{d}$

${\text{T}}^{\text{2}}=\frac{12\pi }{\text{Gd}}$

$\therefore \text{T}\propto \frac{1}{\sqrt{\text{d}}}$