Question

Consider a satellite of mass m in a circular orbit of radius r around the Earth of mass ME and radius RE.

1.

What is the gravitational force (magnitude and direction) on the satellite from Earth?

2.

If we define g(r) to be the force of gravity on a mass m at a radial distance r from the center of the Earth, divided by the mass m, then evaluate the ratio g(r)/g(RE)to see how g varies with radial distance. If we call h the height above the surface of the Earth, then what is the numerical value of g at a height h = RE compared to the value of 9.80 m/s2 at the Earth’s surface?

3.

Assume the satellite is high enough above the atmosphere that there is no friction, so that the gravitational force is the only one acting on the satellite. Thus, the gravitational force results in a centripetal acceleration. Use this to derive a relation between the expression you found in part 1 and the speed of the satellite. Write this in such a way to highlight the dependence of the speed on the radial distance r.

4.

Now suppose two satellites of the same mass orbit the Earth, one with r = RE and another with r = 4RE. Which has the slowest speed? By what factor?

5.

Return to part 3 and rewrite the speed in terms of the period T, the time it takes for a satellite to complete one entire orbit. Now derive an expression which relates how the distance ? depends on the time T. This is Kepler’s third law!

6.

Now consider again the two satellites from the question at the top of this page. What is the ratio of the periods of the two satellites?

Answer 1