Question

The answer is 1 day. I need to know how to solve this step by step. thank you

The moon has a period of 27.3 d and is an average distance from Earth of 3.84 × 105km. A communications satellite is placed in an Earth orbit at 4.23 × 104 km from the center of Earth. What is the period of this satellite?

Answer 1

We know for satelite motion,

$m\omega^2 R = \frac{GMm}{R^2} \\ \Rightarrow \omega = \sqrt{\frac{GM}{R^3}} \\ \therefore T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R^3}{GM}}$

Therefore ,

$\frac{T_{sat}}{T_{moon}}=\sqrt{\frac{R^3_{sat}}{R^3_{moon}}} = \sqrt{\frac{(3.84\times10^5)^3}{(4.23\times 10^4)^3}}= 27.35\\ \Rightarrow T_{sat}=27.35\times 27.3 day = 746.70 day.$

So, time period if the satellite = 746.70 day.