Homework Help: Satellite Motion - This doesnt seem right

As far as I can tell, I think the problem is that you used the wrong mass in the formula
[tex] \mbox{g}=\frac{GM}{r^2}[/tex]
You used the mass of the weight, but it should be the mass of the earth, which is approximately [tex]5.977\times10^{24} \mbox{ kg}[/tex]. The equation should be as follows:
[tex] 5 = \frac{(6.67\times10^{-11})(5.977\times10^{24})}{r^2}[/tex]

[tex] r = \sqrt{\frac{(6.67\times10^{-11})(5.977\times10^{24})}{5}} [/tex]

[tex] r = 8.93\times10^6 m [/tex]

Does that agree with the answer you have been given?

 

You also need to be careful about what r represents. In your formula, r is the distance from the center of the earth. The problem asks for the distance from the earth's surface.

You don't actually need to do all that calculation. You know that the object weighs 10 N at the earths surface: GM/R^2= 10. You are looking for r such that GM/r^2= 5. Dividing the first equation by the second, r^2/R^2= 2 so r= [itex]\sqrt{2}[/itex] R.

Again, that r is distance from the center of earth. Since R is the radius of the earth, the distance from the surface of the earth is [itex]\sqrt{2}[/itex]R- R= ([itex]\sqrt{2}[/itex]- 1)R.

 

My apologies, HallsofIvy is absolutely right. The answer I gave you was incomplete.

The formula calculated the distance from the centre of the earth to the object, however the question asks for the distance from the surface of the earth to the object. This means we must subtract the earth's radius from the answer we calculated, as this will give the distance from the centre of the earth.

[tex] r_{surface} = r - r_{earth} [/tex]
[tex] r_{surface} = (8.93\times10^6)-(6.38\times10^6)[/tex]

Object is [tex] 2.55\times10^6[/tex] metres away from the earth's surface. Does that make sense?