Distance of geosynchronous satellite from earth

[hangontoyourego]

Hello, I had a bit of trouble figuring out this problem:

1. Homework Statement

Given the following, determine the distance in miles above the Earth's surface of a geosynchronous satellite.

M_Earth=5.98E24 kg
R_Earth=4,000 miles
1 mile=1604 m

Homework Equations

F_g=(Gm₁m₂)/r²

F_C=(mv²)/r

The Attempt at a Solution

((6.67E-11m^3/kg x s^2)(m_Satellite)(5.98E24kg))/(6416000m+x)^2 = (m_Satellitev_Satellite^2)/(6416000m+x)

For v of the satellite, I said the velocity is equal to the distance of the orbit divided by 86,400 seconds, so I have:

(3.98866E14)/(4.1165056E13+12832000x+x^2) = (((6.2832x)/(86400))^2)/(6416000+x)

and then

http://www4b.wolframalpha.com/Calculate/MSP/MSP94291ci3c60ee01fbifd00000i565236dhh0d08b?MSPStoreType=image/gif&s=2&w=476.&h=66.

Then, I found that x equaled 40,216,400 meters or 11,658 miles. Is this correct?

 

[TSny]

Hello, and welcome to PF!

Your expression for the distance traveled during one orbit is not quite correct. See if you can spot the problem.

It would be much nicer if you first solved the equations symbolically and then plug in numbers. I would also recommend that you first solve (in symbols) for the radius r and then find x.

 

[hangontoyourego]

Hi TSny,

I'm not so sure about this, but should I have calculated the orbital distance as x + radius of earth, or x+6416000?

So,

((G)(M_Earth))/(R_Earth+x)² = ((x+R_Earth)^2)/(R_Earth+x)

((G)(M_Earth))/(R_Earth+x)² = (x+R_Earth)

((G)(M_Earth)) = (x+R_Earth)³

((G)(M_Earth))^(1/3) = (x+R_Earth)³)^(1/3)

which, substituting the numbers in, would be:

((6.67E-11 m³/kg⋅s²)(5.98E24 kg))^(1/3) = x + 6416000

73610.93588 = x + 6416000

This can't be right though, since it yields a negative answer, right? Or do you take the absolute value of x and the answer is 6342389.064/1604 or 3,954 miles?

 

[SteamKing]

Hi TSny,

I'm not so sure about this, but should I have calculated the orbital distance as x + radius of earth, or x+6416000?

So,

((G)(M_Earth))/(R_Earth+x)² = ((x+R_Earth)^2)/(R_Earth+x)

What's the deal with this equation?

You basically have written GM/r² = r. That doesn't make sense. What happened to the orbital period of the satellite?

This can't be right though, since it yields a negative answer, right? Or do you take the absolute value of x and the answer is 6342389.064/1604 or 3,954 miles?

Only a desperate man takes the absolute value of something which is wrong, hoping to make it right.

Draw a sketch of the problem, and then apply your formulas. What's the formula for the velocity of an object traveling a circular path of radius R and period T?

What's the R for a satellite in a geosynchronous orbit? Remember, the satellite must orbit the entire earth, you know, center and all.

 

[TSny]

Hi TSny,

I'm not so sure about this, but should I have calculated the orbital distance as x + radius of earth, or x+6416000?

Yes, the radius of the circular orbit is r = R +x where R is the radius of the earth. So, the distance traveled in one orbit is the circumference of a circle with radius r. Note that the relation r = R + x means that you can easily solve for x if you first solve for r.

Your attempt at a solution in the first post shows that you have the right idea.

If you let T be the time for one orbit, how would you write v in terms of the symbols r and T?

You were correct to set the gravitational force equal to mv²/r where m is the mass of the satellite. So, in symbols, you have

GMm/r² = mv²/r where M is the mass of the earth.

Combine this with your expression for v in terms of r and T and see if you can solve symbolically for r.