A 5,000 kg satellite is orbiting the Earth in a geostationary orbit. The height of the satellite above the surface of the Earth is (M_E = 5.98 × 10²⁴ kg, R_E = 6.37 × 10⁶ m, G = 6.67 × 10⁻¹¹ N·m²/kg²) A) 8.29 × 10⁷ m. B) 6.35 × 10⁷ m. C) 3.59 × 10⁷ m. D) 2.95 × 10⁷ m. E) 1.40 × 10⁷ m.

Question

Quizplus · Accepted Answer

To maintain geostationary orbit, the satellite must have the same angular velocity as the Earth, which is given by:

ω = sqrt(GM/R^3)

where M is the mass of the Earth, R is the distance between the center of the Earth and the satellite, G is the gravitational constant, and ω is the angular velocity.

We can rearrange this equation to solve for R:

R = (GM/ω^2)^(1/3)

We are given the mass of the satellite, but we need to find the mass of the Earth. The mass of the Earth is given in the problem as 5.98 × 10^24 kg.

We can find the angular velocity of the Earth by using the formula:

ω = 2π/T

where T is the period of the Earth's rotation, which is equal to 24 hours = 86,400 seconds.

ω = 2π/86,400 = 7.27 × 10^-5 rad/s

Plugging in the values for M, ω, and G, we get:

R = (6.67 × 10^-11 × 5.98 × 10^24 / (7.27 × 10^-5)^2)^(1/3)
R = 4.23 × 10^7 m

Finally, we need to subtract the radius of the Earth from this distance to get the height of the satellite above the surface:

h = R - 6.37 × 10^6 m
h = 3.59 × 10^7 m

Therefore, the correct answer is choice C.

A 5,000 Kg Satellite Is Orbiting the Earth in a Geostationary