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

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 :

$T^2=\dfrac{\pi^2}{GM}R^3$

R is the distance from the center of the earth.

$T^2=\dfrac{\pi^2}{GM}R^3\\\\R^3=\dfrac{GMT^2}{\pi^2}$

G is universal gravitational constant

M is mass of earth

$R^3=\dfrac{6.67\times 10^{-11}\times 5.98\times 10^{24}\times (86400)^2}{\pi^2}\\\\R=\left(\frac{6.67\times10^{-11}\times5.98\times10^{24}\times(86400)^{2}}{\pi^{2}}\right)^{\frac{1}{3}}\\\\R=67.06\times 10^6\ m$

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

$h=67.01\times 10^6-6.38\times 10^6\\\\h=6.06\times 10^7\ m$