Question

Assuming density $d$ of a planet to be uniform, we can say that the time period of its artificial satellite is proportional to

• A. $ d $
• B. $ \sqrt{d} $
• C. $ 1/\sqrt{d} $
• D. $ 1/d $

Answer 1

The Correct Option is C

Density of the Planet can be written as
$\rho=\frac{m}{v}=\frac{m}{\frac{4}{3} \pi a^{3}}$
$\Rightarrow \rho \propto \frac{1}{a^{3}}$
According to Kepler's law, $T^{2} \propto a^{3}$
$\Rightarrow T^{2} \propto \frac{1}{\rho}$
or $T \propto \frac{1}{\sqrt{\rho}}=\frac{1}{\sqrt{d}}$
$[\because \rho=d]$