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A 200-kg communications satellite is placed into a circular orbit around Earth with a radius of 4.23×107 m4.23 \times 10^7 \mathrm{~m} (26,300 miles) (a) Find the gravitational force on the satellite. (There is some useful information in Section 2.7.) (b) Use the equation for centripetal force to compute the speed of the satellite. (c) Show that the period of the satellite-the time it takes to complete one orbit-is 1 day. (The distance it travels during one orbit is 2π2 \pi, or 6.286.28, times the radius.) This is a geasynchronous orbit: the satellite stays above a fixed point on Earth's equator.

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In this problem we are given a communications satellite with mass m=200 kgm = 200 ~\mathrm{kg} placed into a circular orbit around the Earth, with radius r=4.23107 mr = 4.23 \cdot 10^7 ~\mathrm{m}. We must find:

(a)  (a)~~ Gravitational force FgF_g acting on the satellite

(b)  (b)~~ Speed υ\upsilon of the satellite

(c)  (c)~~ Period TT of the satellite (prove that T=1 dayT = 1 ~\mathrm{day})

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