Question

A satellite with an orbital period of exactly 24.0 h is always positioned over the same spot on Earth.This is known as a geosynchronous orbit. Television, communication, and weather satellites use geosynchronous orbits. At what distance would a satellite have to orbit Earth in order to have a geosynchronous orbit?

Answer 1

35870474.30504 m

Step-by-step explanation:

r = Distance from the surface

T = Time period = 24 h

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

m = Mass of the Earth = 5.98 × 10²⁴ kg

Radius of Earth =
`6.38* 10^6\ m`

From Kepler's law we have relation

`T^2=(4\pi^2r^3)/(GM)\\\Rightarrow r^3=(T^2GM)/(4\pi^2)\\\Rightarrow r=\left(((24* 3600)^2* 6.67* 10^(-11)* 5.98* 10^(24))/(4\pi^2)\right)^{(1)/(3)}\\\Rightarrow r=42250474.30504\ m`

Distance from the center of the Earth would be

`42250474.30504-6.38* 10^6=\mathbf{35870474.30504\ m}`

35870474.30504 m