Consider a satellite of mass m in circular orbit of radius r

Consider a satellite of mass m in circular orbit of radius r about the earth of mass M.

1. What force provides the centripetal force required to keep the satellite in its circular orbit?

2. Derive Kepler’s third law (an equation relating the period T of the orbit to the radius r of the orbit) for the satellite.

3. A synchronous satellite is one which always remains above the same point on the earth’s equator. Such synchronous satellites are useful for telecommunications applications. Use your result from question 2 to find the radius r for synchronous satellite orbiting earth.

Solution

1) Grvitational force of earth on satellite

2) period T = 2*pi*r/v

v is the speed of the satellite in an orbit of radius r is v = sqrt(G*M/r)

M is the mass of the earth and G is Universal gravitational constant

then T = 2*pi*r/sqrt(G*M/r) = 2*pi*r^(3/2)/sqrt(G*M)

squaring on both sides

T^2 = 4*pi^2*r^3/(G*M)

So T^2 is proportional to r^3

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3) for synchronous satellite ,the period of the satellite is T = 24 hours = 24*60*60 = 86400 sec

then r^3 = T^2*G*M/(4*pi^2) = 86400^2*6.67*10^-11*5.98*10^24/(4*3.142^2)

radius of the orbit is r = 42.2*10^6 m