Answer to Question #132258 in Physics for rose auburn

Question

A 200 kg communications satellite is placed into a circular orbit around the Earth with a radius of 4.23 x 10^7 m (26,300 miles). Find the gravitational force on the satellite. Use the equation for centripetal force to compute the speed of the satellite. Show that the period of the satellite – the time it takes to complete one orbit – is 1 day.

Answer 1

The gravitational force of interaction with earth of mass M will be

$$F_g=\frac{GMm}{R^2}=45\text{ N}.$$

Since at equilibrium of forces

$$m\frac{v^2}{R}=mg,$$

where

$$mg=G\frac{mM}{R^2},\\\space\\ v=\sqrt{gR}=\sqrt{\frac{GM}{R}}=3077\text{ m/s}.$$

The period is

$$T=\frac{d}{v}=\frac{2\pi R}{v}=24\text{ h}.$$