Answer:
h = 35857 km
Explanation:
A geosynchronous orbit can be defined as circular orbit which lies on the Earth's equatorial plane and follows the direction of the Earth's rotation in a period that's equal to the Earth's rotational period and thereby appearing motionless, at a fixed position in the sky relative to the ground observers.
We are given;
Radius of earth(R) = 6.37 x 10^(6) m
Mass of earth (Me) = 5.97 x 10^(24) kg
Gravitational constant = 6.67 × 10^(-11) m³/kg.s²
The earth has a rotational period of 24 hours per day. This gives in seconds
T = 24 × 60 × 60
T = 86400 s
Let's make the height of the orbit from Earth's surface to be h
Also, let ω be the uniform angular velocity in rad/s with which the satellite rotates in the geosynchronous orbit
Now, equating the centripetal force with the gravitational force gives us;
mω²(R + h) = G•Me•m/(R + h)²
m will cancel out. Also ω can be written as 2π/T
Thus,we now have;
(R + h) = ∛(G•Me•T²/(4π²))
Plugging in the relevant values, we have;
(R + h) = ∛(6.67 × 10^(-11) × 5.97 x 10^(24) × 86400²/(4π²))
(R + h) = 42227 Km
Since R = 6.37 x 10^(6)m = 6370 km
Thus;. h = 42227 - 6370 = 35857 km
