Question

A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite. If a geosynchronous satellite is in an equatorial orbit, its position appears stationary with respect to a ground station, and it is known as a geostationary satellite. Find the radius R of the orbit of a geosynchronous satellite that circles the earth.(Note that R is measured from the center of the earth, not the surface.) You mayuse the following constants:

The universal gravitational constant G is 6.67 x 10^-11 N m2 / kg2.
The mass of the earth is 5.98x 10^24kg
The radius of the earth is 6.38 x 10^6 m

Answer 1

Step-by-step explanation:

Given that,

A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite, T = 24 hours = 86400 s

We need to find the radius R of the orbit of a geosynchronous satellite that circles the earth. It can be calculated using Kepler's third law of motion as :

`T^2=(\pi^2)/(GM)R^3`

R is the distance from the center of the earth.

`T^2=(\pi^2)/(GM)R^3\\\\R^3=(GMT^2)/(\pi^2)`

G is universal gravitational constant

M is mass of earth

`R^3=(6.67* 10^(-11)* 5.98* 10^(24)* (86400)^2)/(\pi^2)\\\\R=\left((6.67*10^(-11)*5.98*10^(24)*(86400)^(2))/(\pi^(2))\right)^{(1)/(3)}\\\\R=67.06* 10^6\ m`

So, the satellite distance from the earth's surface is :

`h=67.01* 10^6-6.38* 10^6\\\\h=6.06* 10^7\ m`