Gravitional Potential Energy and Mass of Earth

In summary, there are two equations for calculating the change in gravitational potential energy between a system of mass m and the Earth: U=mgy and U=GMm/r. The first equation is only accurate if the gravitational force is constant over the change in height, while the second equation is always correct. To find the value of h for which the first equation is off by 1%, we can use an error formula and solve for h. After correcting a mistake in the formula, the final answer is 0.01Re, or 64 km.
  • #1
Tirokai
2
0
This question is 12.59 from University Physics 11e.

"There are two equations from which a change in the gravitational potential energy U of the system of mass m and the Earth can be calculated. One is U=mgy. The other is U=GMm/r (M=mass of earth). The first equation is correct only if the gravitational force is a constant over the change in height delta-y. THe second is always correct. Actually, the gravitational force is never exactly cosntant over any change in height, but if the variation is small, we can ignore it. Consider the difference in U between a mass at the Earth's surface and a distance h above it using both equations, and find the value of h for which mgy is in error by 1%. Express this value of h as a fraction of the Earth's radius, and also obtain a numerical value for it."

The correct answer, from the rear of the book, is 0.01Re and 64 km.

My strategy, thus far, was to use the two gravitational potential energy formulae in an error formula. So I tried several different variations on (mgh-(-GMm/(Re+h)^2+Gmm/(Re)^2))/mgh=.01

Re= radius of Earth M=mass of earth

Having set up the formula, I used the solve function of a TI-89 to punch them out, all of them eventually coming out to be some random number in the millions or tens of millions.

Any help is greatly appreciated :D
 
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  • #2
In your formula you are squaring the R, why? For potential energy you don't square the r... :(
 
  • #3
You know what, I accidentally copied that from a formula someone else gave me (a physics major, too! shame on them) and so let me update the formula I used.

(mgh-((-GMm/Re+h)+(GMm/Re)))/(mgh)=.01

I simplified that down to

((gRe^2+GMe)h+(gRe)h^2)/((gRe^2)h+(gRe)h^2)=.01

Which gave me some bogus answer like -1.3x10^7

Thanks for the response, at any rate.
 
  • #4
.01 = (mgh - accepted value)/accepted value

thats it... you have an idea how to find the accepted value i think so that shouldn't eba problem... but i still see those squares there... there shoudnt be squares anywhere... if I am wrong please prove me wrong because I've got a test on this and waves tomorrow (curses at waves) :P
 

1. What is gravitational potential energy?

Gravitational potential energy is the energy that an object possesses due to its position in a gravitational field. It is the energy an object would have if it were at a certain height above a reference point, typically the surface of the Earth.

2. How is gravitational potential energy related to the mass of the Earth?

The gravitational potential energy of an object is directly proportional to the mass of the Earth. This means that as the mass of the Earth increases, so does the gravitational potential energy of an object at a certain height above its surface.

3. How is the mass of the Earth calculated using gravitational potential energy?

The mass of the Earth can be calculated using the formula for gravitational potential energy, which is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above the Earth's surface. By rearranging this formula, we can solve for the mass of the Earth (m = PE/gh).

4. Does the mass of the Earth affect the strength of its gravitational field?

Yes, the mass of the Earth directly affects the strength of its gravitational field. The greater the mass of the Earth, the stronger its gravitational field and the more gravitational potential energy objects have at a certain height above its surface.

5. How does the mass of the Earth impact the motion of objects on its surface?

The mass of the Earth determines the strength of its gravitational pull, which affects the motion of objects on its surface. Objects with greater mass will experience a stronger gravitational force and will accelerate towards the Earth at a faster rate than objects with less mass.

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