Question

g The international space station has an orbital period of 93 minutes at an altitude (above Earth's surface) of 410 km. A geosynchronous satellite has an orbital period of 24 hours. The radius of Earth is 6400 km. Use Kepler's 3rd Law to compare these two and find the orbital altitude of the geosynchronous satellite.

Answer 1

Answer:

r = 4.21 10⁷ m

Explanation:

Kepler's third law It is an application of Newton's second law where the forces of the gravitational force, obtaining

            T² = ($\frac{4\pi }{G M_s}$ ) r³             (1)

           

in this case the period of the season is

            T₁ = 93 min (60 s / 1 min) = 5580 s

            r₁ = 410 + 6370 = 6780 km

            r₁ = 6.780 10⁶ m

for the satellite

           T₂ = 24 h (3600 s / 1h) = 86 400 s

if we substitute in equation 1

            T² = K r³

            K = T₁²/r₁³

            K = $\frac{ 5580^2}{ (6.780 10^6)^2}$

            K = 9.99 10⁻¹⁴ s² / m³

we can replace the satellite values

            r³ = T² / K

            r³ = 86400² / 9.99 10⁻¹⁴

            r = ∛(7.4724 10²²)

            r = 4.21 10⁷ m

this distance is from the center of the earth