# Derive Potential Energy from Newtons graviational law

#### Gogsey

3. Beginning from Newton’s rule for the gravitational force = -GMm/r^2
derive an expression for the potential energy of a particle of mass m above the surface of the earth, and show that it can be written as:

U(r) = -g0 RE/r

where r is the distance from the centre of the earth, RE is the radius of the earth, and g0 is the value of the
gravitational field at the surface of the earth.

To cut to the chase I am at this point.

U(r) - U(RE) = -GMm(1/r - 1/RE) + C

and rewrote it as U(r) - U(RE) = -GMm(RE-R/REr) + C

But I can't seem to figure out how to get it in the form g0 RE/r.

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#### Doc Al

Mentor
To cut to the chase I am at this point.

U(r) - U(RE) = -GMm(1/r - 1/RE) + C
(a) You should be integrating from ∞ to r, not RE to r.
(b) Since it's a definite integral, there's no integration constant.

Once you fix these, then you can try to get the answer in terms of g0. (Start by figuring out what g0 equals!)

#### Gogsey

Ok, just need to clarify something. Why is it from infinity to r?

Also, the U(infinity) term goes to zero, since U(infinity) = -GMm/infinity as if this was the expression on its own.

#### Doc Al

Mentor
Ok, just need to clarify something. Why is it from infinity to r?
By convention, the potential is taken to be zero at infinity.

Also, the U(infinity) term goes to zero
That's true.

#### Gogsey

Ok, but we care about the potential above the surface. Then if we subtract the potential energy at the surface from the potential energy at distance r from the centre of the earth, don't we get the expression I original had?

#### gabbagabbahey

Homework Helper
Gold Member
Ok, but we care about the potential above the surface. Then if we subtract the potential energy at the surface from the potential energy at distance r from the centre of the earth, don't we get the expression I original had?
You seem to be misinterpreting "potential energy of a particle of mass m above the surface of the earth" as the difference in potential energy between the point r and the surface.

I'm pretty sure that the question wants you to find the potential (at a point above the surface) relative to some reference point and then show that if the reference point is at infinity, you get the desired expression.

The part about the point you are finding the potential at being above the surface is important because the force is only GmM/r^2 above the surface.

#### Gogsey

Ah ok, I thought the reference point we were supposed to use was at the centre of the earth.

So we should have that U(r) = -GMm/r, but I'm still clueless about getting it into that expression U(r) = g0 RE/r.

#### gabbagabbahey

Homework Helper
Gold Member
Ah ok, I thought the reference point we were supposed to use was at the centre of the earth.

So we should have that U(r) = -GMm/r, but I'm still clueless about getting it into that expression U(r) = g0 RE/r.
Well, if U(r) = -GMm/r , what is U(RE)? ....what does that make g0? (Remember, g0 is by definition the value of the gravitational potential at the surface)

#### Gogsey

I thought U(RE) should be U(infinity), since were doing from infinity to r, which is zero.

#### gabbagabbahey

Homework Helper
Gold Member
I thought U(RE) should be U(infinity), since were doing from infinity to r, which is zero.
That makes no sense. U(infinity)=0 since it is the reference point. U(RE) is what you get by plugging r=RE into your equation; U(r) = -GMm/r. Certainly, you don't get U(RE)=0 right?

#### Gogsey

Lol, no I don't get that.

In an earlier post Doc Al said we integrating from infinity to r, which I understand now. So I thought that if we were going from infinity to r for the right side, we should be doing the same for the left side a get:

U(r) - U(infinity) = -GMm/r + GMm/infinity

since infinity is the reference point, and then both these terms would go to zero, and we'd be left with:

U(r) = -GMm/r

Obviously its not since your telling me its wrong. Could you explain this for me?

#### Doc Al

Mentor
That's not wrong. Just be careful how you express things. When you wrote:
I thought U(RE) should be U(infinity), since were doing from infinity to r, which is zero.
That sure sounds like you are saying U(RE) = U(infinity) = 0, which is not what you meant at all.

#### Gogsey

So are we going from infinity to r for the left side or from RE to r?

#### Doc Al

Mentor
So are we going from infinity to r for the left side or from RE to r?
From ∞ to r.

"Derive Potential Energy from Newtons graviational law"

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