What is the radius of a geosynchronous satellite?

1. The problem statement, all variables and given/known data
Find the radius R of the orbit of a geosynchronous satellite that circles the earth. (Note that R is measured from the center of the earth, not the surface.) You may use the following constants:

* The universal gravitational constant G is [tex]6.67 \times 10^{-11}\;{\rm N \; m^2 / kg^2}[/tex].
* The mass of the earth is [tex]5.98 \times 10^{24}\;{\rm kg}[/tex].
* The radius of the earth is [tex]6.38 \times 10^{6}\;{\rm m}[/tex].


The correct answer is [tex]4.23\times10^7\;{\rm m}[/tex], but I get a different answer.

2. Relevant equations

[tex]T=2\pi\sqrt{\frac{R^3}{GM}}[/tex]

3. The attempt at a solution

Since T is measured in seconds, and there are 86,400 seconds in a day, some simple algebra gives me the answer of 1,994,400,816 m. What am I doing wrong?
 

ehild

Homework Helper
15,279
1,707
The formula is correct, do your calculation again. Use the normal form of the numbers.

ehild
 
what is the "normal form" for these numbers?

EDIT: I did the calculation a different way, and I got the right answer, but the only thing I did differently was the order that I did the algebra in. I don't know where my error lies.

I normally did:

[tex]\frac{86400\times\sqrt{GM}}{2\pi}=\sqrt{R^3}[/tex]

and solve for R, but instead, this time I did:

[tex]86400^2=4{\pi^2}\frac{R^3}{GM}[/tex]
[tex]\frac{86400^2\times{GM}}{4{\pi^2}}=R^3[/tex]

and it worked..... what did i do wrong the first time?

if you square the first equation, you get the same thing......


2nd edit, nevermind, i know what I did wrong.

this has been a nice lesson for me in making sure I properly place parenthesis in a calculator to ensure teh correct order of operations. because [tex]x/2\pi[/tex] is not interpreted as [tex]x/(2\pi)[/tex], but instead as [tex]\frac{x}{2}\pi[/tex]
 
Last edited:

ehild

Homework Helper
15,279
1,707
this has been a nice lesson for me in making sure I properly place parenthesis in a calculator to ensure teh correct order of operations. because [tex]x/2\pi[/tex] is not interpreted as [tex]x/(2\pi)[/tex], but instead as [tex]\frac{x}{2}\pi[/tex]
Never forget the parentheses in the denominator!!!!:rofl:

ehild
 

Borek

Mentor
27,852
2,424
No idea why radius of a satellite should depend on the Earth mass.
 

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