Question

What is the radius of the circular orbit of a stationary satellite which remains motionless with respect to earth's surface ?

• A. ${\left(\frac{g{R}^2{T}^2}{4{\pi }^2}\right)}^{\frac{1}{3}}$
• B. ${\left(\frac{gR}{4{\pi }^2{T}^2}\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{gRT}{2\pi }\right)}^{\frac{1}{3}}$
• D. ${\left(\frac{gR}{2\pi T}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is A ${\left(\frac{g{R}^2{T}^2}{4{\pi }^2}\right)}^{\frac{1}{3}}$

let orbital radius be $r_o$

centipetal force for radius $r_o$: $m{\omega }^2{r}_o=\frac{(2\pi {)}^2}{T^2}{r}_o$

Gravitational force by earth = $\frac{G{m}_1{m}_2}{r^2}=\frac{GMm}{r_o^2}$

there is no motion with respect to surface

centripetal force = Gravitational force for a satellite.

$\frac{(2\pi {)}^2}{T^2}{r}_o=\frac{GMm}{r_o^2}$

using $g=\frac{GM}{R^2}⟹GM=g{R}^2$

$\frac{(2\pi {)}^2}{T^2}{r}_o=\frac{g{R}^{2}m}{r_o^2}$

$r_o^3=\frac{g{R}^2{T}^2}{4{\pi }^2}$

$r_o=\sqrt[3]{3\frac{g{R}^2{T}^2}{4{\pi }^2}}$

Hence option A is correct.