# 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.

MEarth=5.98E24 kg
REarth=4,000 miles
1 mile=1604 m

Fg=(Gm1m2)/r2

FC=(mv2)/r

## The Attempt at a Solution

((6.67E-11m^3/kg x s^2)(mSatellite)(5.98E24kg))/(6416000m+x)^2 = (mSatellitevSatellite^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. [Broken]

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

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Homework Helper
Gold Member
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,

So,

((G)(MEarth))/(REarth+x)2 = ((x+REarth)^2)/(REarth+x)

((G)(MEarth))/(REarth+x)2 = (x+REarth)

((G)(MEarth)) = (x+REarth)3

((G)(MEarth))(1/3) = (x+REarth)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?

Staff Emeritus
Homework Helper
Hi TSny,

So,

((G)(MEarth))/(REarth+x)2 = ((x+REarth)^2)/(REarth+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.

Homework Helper
Gold Member
Hi TSny,