(Figure 1) A satellite that goes around the earth once every 24 hours is called

Question

(Figure 1) A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite. If a geosynchronous satellite is in an equatorial orbit, its position appears stationary with respect to a ground station, and it is known as a geostationary satellite. Find the radius R of the orbit of a geosynchronous satellite that circles the earth. (Note that R is measured from the center of the earth, not the surface.) You may use the following constants: The universal gravitational constant G is 6.67 x 10^-11 N m^2/kg^2. The mass of the earth is 5.98 x 10^24 kg. The radius of the earth is 6.38 x 10^6 m. Give the orbital radius in meters to three significant digits. R = m

Explanation / Answer

mv^2/r=GmM/r^2, where M is the mass of the earth, and m is the mass of satellite v^2=GM/r r=GM/v^2 To find v, we have w=v/r So, v=w r Here, Angular velocity, w=2*pi/ T where, T is 24 hours=3600*24seconds T= 86400s w=7.27e-5 v=7.27e-5 r r = GM/(7.27e-5*r)^2 r^3 = GM/(7.27e-5)^2 r = (GM)^1/3*1/(7.27e-5)^2/3 By substituting, G = 6.67 x 10^-11 N * m^2/kg^2, and M = 5.98 x 10^24 kg, we get r=4.22e7m=42211.85km

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