Kepler's Laws and orbiting satellite

[G-reg]

Homework Statement

A orbiting satellite stays over a certain spot on the equator of (rotating) Mercury. What is the altitude of the orbit (called a "synchronous orbit")?

Mercury mass = 3.3022 e23 kg
Mercury Radius = 2439.7 km

Homework Equations

$$T^2$$ = (4$$\Pi^2$$/GM) * r^3

and that's all I can think of which is why I'm stuck

The Attempt at a Solution

I'm sorry that I have no attempt, I'm really lost on this one and would greatly appreciate any help!

 

[mgb_phys]

Do you know what that equation means?

What T are you looking for?

 

[G-reg]

Yes, the equation is telling you the period of the motion squared.

But we're not looking for the period or the mass or the radius..we're looking for the altitude but i can't find an equation in my textbook that would help me out with that.

 

[mgb_phys]

What's 'r' in that equation?

 

[G-reg]

The radius..
Are you hinting that the period is the same thing as the altitude?

 

[ideasrule]

T is the time it takes for the satellite to make one orbit. r isn't the radius; it's the distance from the center of Mercury to the satellite. So to use that equation, you'll need to find Mercury's rotation period.

 

[G-reg]

Oh ok, I think I understand now. So we are looking for 'r'.
And we're using the "law of periods" to solve for 'r'.
Correct?

 

[G-reg]

Well I guess this guess is wrong because I got a wrong answer..
So where should I go from here?

 

[buffordboy23]

What is the definition of altitude? In other words, where is the reference point for which we determine the altitude?

 

[G-reg]

So it should be..

GM * (T/2pi)^2 = x

x - radius of mercury = answer?

 

[buffordboy23]

Hi G-reg,

You are really lost. You don't have enough data. A geosynchronous orbit simply means that the satellite remains fixed over a certain point on the surface of the planet. In other words, it's period is the same as the planet's period (length of Mercury's day). You need to obtain or look up this data for Mercury.

The second point to understand is in regards to your last post. You want to use "x", which represents Mercury's radius. What your saying here is that the satellite's is position is on the surface of Mercury, and this doesn't correspond to the satellite being in orbit. The satellite has some altitude; i.e., it is some distance above Mercury's surface. How can write an expression for the altitude h in terms of Mercury's radius R and the satellite's radial distance r from Mercury's center of mass?