Question

To find some of the parameters characterizing an object moving in a circular orbit. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit: a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M. For all parts of this problem, where appropriate, use G for the universal gravitational constant.

The potential energy U of an object of mass m that is separated by a distance R from an object of mass M is given by
U=−GMmR.

a. What is the kinetic energy K of the satellite?
b. Find an expression for the square of the orbital period.

Answer 1

G M m / R^2 = m v^2 / R centripetal force equals gravitational force

1/2 m v^2 = G M m / R = KE rearranging the above equation

T = 2 pi R / v time for one revolution (period)

T^2 = 4 pi^2 R^2 / v^2

From the very first equation v^2 = G M / R

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