Gravitational Potential Energy Problem

[sheepcountme]

Homework Statement

The magnitude of the attractive force of gravity between two massive bodies is F=GMm/r^2, where G is a constant, M and m are the masses, and r is the distance between the centers of the two bodies. The radius of the Earth is 6.38×10^6 m and its mass is 5.97×10^24 kg. A satellite of mass 1.13e+3 kg is propelled from the surface of the Earth to a height of 35,786 km above the surface of the Earth. What is its change in gravitational potential energy?

Homework Equations

W=deltaPE
PE=mgdeltah

The Attempt at a Solution

I plugged all the numbers into the given equation to get F=1.6268x10^15
I thought this represented mg in PE=mgh, but after multiplying it with h (and changing h to m rather than km), I didn't come up with the correct answer (which should be 5.96x10^10). Can you tell me where I went wrong?

 

[aim1732]

Well g is not constant over such large distances.
Instead you can use

delta(P.E)= - integral(F.dx) where F is the conservative force.(gravity here).

 

[magwas]

the problem is that g in high altitudes is significantly different.
Work is F*h when F is constant. But F varies with height, and in this case the work is the area below F in a graph where F is drawn against h. Formally:
$$\int_{r1}^{r2} F dr = \int_{r1}^{r2} \frac{G m_{1} m_{2}}{r^{2}} dr$$
The area under $$\frac{1}{r^2}$$ from r=1 to r=r1 is $$1 - \frac{1}{r1}$$
From now on you can compute it even if you don't know what an integral is.

 

[sheepcountme]

Okay, so my integral will be GMm x integral 1/rinitial - 1/rfinal

which gives me GMm (1/6.38e6)-(1/4.138e7), so GMm (1.326e-7), but if I times this by the masses I get G(8.945e20)...but I'm stuck, and this doesn't seem anywhere near the correct answer :/ alas

 

[aim1732]

Hang on- that IS the correct answer.

 

[sheepcountme]

Unfortunately my book says the right answer is 5.96x10^10.

How would I possibly get rid of the G in the answer I got (G(8.945e20)) if I don't know what it is? Could I set this equal to something appropriate and make it cancel out?

 

[magwas]

G is the gravitational constant
$$G = 6.67428\left(67\right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$
http://en.wikipedia.org/wiki/Gravitational_constant

 

[sheepcountme]

BLAST! Textbook semantics have tricked me again! I assumed we weren't supposed to be able to know this since it's not mentioned in the book. However, it is obviously necessary. Thank you!