Final answer:
The radius of a geosynchronous satellite's orbit can be calculated using Kepler's Third Law, which involves the known orbital period of 1 day and a gravitational constant adjusted for Earth. Subtract the Earth's radius from the orbit radius to obtain the satellite's height above the Earth.
Step-by-step explanation:
Calculating the Orbit Radius of a Geosynchronous Satellite
A geosynchronous Earth satellite is one with an orbital period that matches the Earth's rotation period, which is 1 day or 24 hours. To calculate the orbit radius of such a satellite, we apply Kepler's Third Law, stating that the radius cubed divided by the period squared is constant for all satellites orbiting the same parent body. Given that we consider the ratio for the moon to be 1.01 × 10¹⁸ km³/s² as a reference, we need to adjust this constant for Earth's smaller mass. Since the satellite must orbit in 24 hours, the radius can be found by rearranging Kepler's Law and solving for the radius of the geosynchronous orbit.
The radius of Earth is given as 6370 km, which is needed to calculate the actual altitude of the satellite above the Earth's surface. To find the satellite's height, one would subtract the radius of Earth from the orbital radius. The orbital speed and atmospheric conditions are not factors in this specific calculation, as the satellite is high enough to be free of atmospheric drag.
Communications satellites, which are typically geosynchronous, are crucial for activities such as weather observation and global positioning systems because they maintain their position relative to a fixed point on Earth's surface.