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.

 

[rwisz]

Yes, your derivation is correct. Using Newton's law of universal gravitation and the relationship for centripetal force, you have derived the equation for the orbital period of a satellite. This equation shows that the orbital period is directly proportional to the square root of the orbital radius and inversely proportional to the square root of the mass of the planet. This is a fundamental relationship in orbital mechanics and is used to calculate the orbital period of satellites and other objects in orbit around planets. Good job!