How Is the Orbital Period of a Satellite Determined?

[mogley76]

Homework Statement

by using Newtons law of universal gravitaion and the relationship for the centripetal force, show that the orbital period T for a satellite circling a planet of mass M in orbit of radius r is given by :
T= 2pi sqrt r^3/GM

Homework Equations

none given

The Attempt at a Solution

consider a satellite of mass m orbiting the Earth of mass M with a constant speed v along a circular path of radius r.

the centripetal force : mv^2/r is provided by the gravitational attraction of the earth

so F= mv^2/r = GmM/r^2

v of robit = sgrt GM/r

the orbital period T = 2pi/w
since v=wr T now eqauls : 2pi r /v

therefore T= 2 pi sqrt r^3/GM
as both G and M are constants it can be seen that both the orbital speed and period depend only on the orbital radius.


have i derived this correctly?

 

[Delphi51]

Perfect!

 

[mogley76]

yay!

 

[Delphi51]

When you start out with Fc = Fg as you did, you nearly always get the formula you want pretty quickly. It is useful to have the alternate form of Fc= 4π²mR/T² handy. Use it if you are interested in T, and mv²/R if you are interested in v.