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. $\frac{1}{\sqrt{d}}$
• D. $\frac{1}{d}$

Answer 1

Answer: (C)

The time period of an artificial satellite revolving very close to the planet's surface is
$T=2\pi \sqrt{\frac{R^3}{GM}}$
where $M$ is the mass of the planet and $R$ its radius.
Assuming the planet to be of uniform density $d$, so its mass is
$M=\frac{4}{3}\pi R^{3}d$
$($ as density $=\frac{\text{ mass }}{\text{ volume }})$
$\therefore dT=2\pi \sqrt{\frac{R^3}{G\left(\frac{4}{3}\pi R^{3}d\right)}}$
$=\sqrt{\frac{3\pi }{Gd}}$ or $T\propto \frac{1}{\sqrt{d}}$