Question

A satellite is orbiting just above the surface of a planet of average density $\rho$ with a time period T, then universal constant of gravitation is given by

• A. $\frac{1}{T^2\rho }$
• B. $\frac{3\pi }{T^2\rho }$
• C. $\frac{4{\pi }^2}{T^2\rho }$
• D. $\frac{\rho T^2}{4{\pi }^2}$
• E. None of the above

Answer 1

Answer: (B)

$\frac{3\pi }{T^2\rho }$

density is $\rho$ and let radius be $r$
So, $mass=\frac{4\pi \rho r^3}{3}$ and let mass of body be $m$.
Time period $T$ is given by $T=2\pi \sqrt{\frac{r^3}{GM}}=2\pi \sqrt{\frac{r^3}{\frac{4G\pi \rho r^3}{3}}}$
Solving the above equation we get $G=\frac{3\pi }{T^2\rho }$