Question

The orbital period of a satellite in a circular orbit of radius r about a spherical planet of mass M and density $\rho$ for a low altitude orbit r will be....

• A. $\sqrt{\frac{3\pi }{G\rho }}$
• B. $\sqrt{3\pi G\rho }$
• C. $\sqrt{\frac{\pi }{G\rho }}$
• D. $\sqrt{2G\rho }$

Answer 1

The correct option is A $\sqrt{\frac{3\pi }{G\rho }}$

Applying newton's second law to the circular orbit

We have,

$\frac{m{v}^2}{r}=\frac{GMm}{r^2}$

But $V=\frac{2\pi r}{r}$

$m\frac{4{\pi }^2{r}^2}{r{T}^2}=\frac{GMm}{r^2};T^2=\frac{4{\pi }^2{r}^3}{GM};T=\frac{4\pi r^{\frac{3}{2}}}{\sqrt{GM}}$

but $M=\frac{4}{3}\pi r^3\rho$ and $r=r_p$

therefore,

$\begin{array}{c}T=\frac{2\pi {r^{\frac{3}{2}}}_p}{\sqrt{G\frac{4}{3}\pi {r^3}_p\rho }}=2\pi \sqrt{\frac{3}{4\pi Gp}}\\ =>\sqrt{\frac{3\pi }{Gp}}\end{array}$