Question: Some communicationssatellites are put into a circular geosynchronous o

Question

Question:
        Some communicationssatellites are put into a circular geosynchronous orbit inwhich the satellite remains above a certain position on the Earth'sequator as it orbits. Such a satellite would always be in the sameposition in the sky as seen from the transmitting or receivingequipment of a communications broadcasting network.

A) Find the angular speed (in radian / s) of the satellite's orbitnecessary for it to remain above the same point of the equator.
B) What must be the orbital radius of such a satellite?
C) What minimum speed must be given to such a satellite to launchit from the surface of the Earth and place it in such an orbit?Neglect friction and any rotational effects of the Earth.
D) How much energy must be given to a 500 kg satellite to launch itfrom the Space Shuttle in a typical orbit of radius 6.8 X106 m to a geosynchronous orbit?

Explanation / Answer

the period of a satellite which remains above a certain position onthe Earth's equator as it orbits must be equal to the period of theearth, that is 24 hours = T a) angular speed = 2/T = 7.27*10-5rad/s b) mass of the satellite = m, mass of earth = M GmM/r2 = m2r r = (GM/2)1/3 =4.23*107 m c) speed on the orbit v = r = 3.07*103 m/s minimum speed = u energy conservation: mu2/2 - GmM/R = mv2/2 - GmM/r (where R =radius of earth) u = [v2 + 2GM*(1/R - 1/r)] =1.08*104 m/s d) m = 500 kg, r' = 6.8*106 m energy E + (-GmM/r') = mv2/2 - GmM/r E = m[v2/2 + GM(1/r' - 1/r)] =2.79*1010 J

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