Orbiting Planet Homework: Deriving Formula for Moon's Orbit Period

[ahrog]

Homework Statement

When a moon orbits a planet, it can be shown that the period of the moon's orbit depends only on the mass of the planet and the radius of the moon's orbit.
a) Draw and label a diagram to illustrate the variables.
b) Derive a formula for the period of a moon's orbit. The only variables in the final answer should be the mass of the planet and the radius of the orbit. All other values should be constants.

Homework Equations

Fg= Gm1m2/r^2
g=Gm1/r^2

The Attempt at a Solution

Okay, I don't really understand how to make formula's to get an answer like this. I figure that the variables in the formula would be the period of orbit, mass and radius. The constants would probably be pi and G.

Could someone give me a hint at how to do this?

 

[LowlyPion]

If it's orbiting then you know that centrifugal is balanced by gravitational attraction.

You also know that v = ωr = 2πf*r = 2πr/T

 

[ahrog]

I don't get it still. How do I derive a formula out of that?

 

[Saladsamurai]

F=ma right? What else does F equal? What is the formula for Force of gravity?

Set them equal to each other and let's see what happens.

 

[ahrog]

Fg= Gm1m2/r^2...

 

[Saladsamurai]

Fg= Gm1m2/r^2...

Yep... and? Like I said set it equal to ma and let's see what happens.

$$m_{moon}a=\frac{GM_{planet}m_{moon}}{r^2}$$

now what?

 

[ahrog]

So I cancel out the two moons and get a=GM/r^2.

 

[Saladsamurai]

Yes. And when a "particle" is experiencing radial acceleration, how else can a be written?

 

[ahrog]

v^2/r=GM/r^2 And then if I cancel out the 2 r's I would get v^2=Gm/r...

Would I use v=d/t here? If so, then for (d/t)^2=GM/r^2 what would cancel out the d?

 

[Saladsamurai]

You are very close, but not exactly. Instead of distance we should use something else... what is the distance called when we are talking about a circle?

 

[ahrog]

Circumference, which is 2pir?

Then (2pir/t)^2=GM/r^2
Which would end up as

2pir^3/Gm=T^2

Would that be right?

 

[Saladsamurai]

$$(\frac{2\pi r}{T})^2=\frac{2^2*\pi^2*r^2}{T^2}=\frac{GM}{r}$$

Don't forget to square everything in the parentheses. And then you've got it.

 

[ahrog]

Thank you sooo much for going through this with me C: I really appreciate it

 

[Saladsamurai]

No probs