A geosynchronous Earth satellite is one that has an orbital period equal to the rotational period of the Earth. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth, provided it orbits in the equatorial plane in the same direction as Earth’s rotation. Calculate (a) the radius of such an orbit and (b) the altitude of the satellite above the surface of the Earth. Note that the rotational period of the Earth is

23 h 56 m 4 s, or 23.93 hours, not 24 hours!

To calculate the radius of a geosynchronous Earth satellite's orbit, we need to use the formula for the period of an object in circular orbit:

T = 2π√(r³/GM)

Where:
T is the period of the orbit
r is the radius of the orbit
G is the gravitational constant (6.67430 × 10^-11 m³ kg^-1 s^-2)
M is the mass of the Earth (5.972 × 10^24 kg)

Given that the period of the orbit is equal to the rotational period of the Earth (23.93 hours), we can convert this to seconds by multiplying by 60 and then 60 again:

T = 23.93 hours = 23.93 × 60 × 60 seconds = 86148 seconds

Substituting the known values into the formula, we can solve for r:

86148 = 2π√(r³/ (6.67430 × 10^-11 × 5.972 × 10^24))

Let's simplify this equation:

86148 = 2π√(r³/3.9859 × 10^14)

To solve for r, we need to isolate it using algebraic manipulation. First, square both sides of the equation to eliminate the square root:

(86148)² = (2π)²(r³/3.9859 × 10^14)

7438352704 = 4π²(r³/3.9859 × 10^14)

Now, divide both sides of the equation by 4π² and multiply by 3.9859 × 10^14 to isolate r³:

r³ = (7438352704 × 3.9859 × 10^14) / 4π²

Now, take the cube root of both sides to find r:

r = ∛[(7438352704 × 3.9859 × 10^14) / 4π²]

Using a calculator, evaluate the right-hand side of the equation to find the value of r.

To calculate the altitude of the satellite above the surface of the Earth, subtract the radius of the Earth (approximately 6,371 kilometers) from the radius of the orbit obtained from the previous calculation. This will give you the altitude of the satellite above the surface of the Earth.