Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum Gravitational Potential Energy What is it exactly?

Gravitational Potential Energy. What is it exactly?

In the thread

http://www.scienceagogo.com/forum/ubbthreads.php?ubb=showflat&Number=34462&page=all

it has become clear that ImagingGeek and preearth need the definition of gravitational potential energy, nailed down tight.

Lets see a definition,... and lets discuss what the concept is useful for.

Last I heard you have to choose an arbitrary zero point. In school you say "on the ground" is zero potential energy. But if you dig a hole in the ground that something can fall into, you can redefine the zero to be the bottom of the hole, or go to negative potential energies.

It doesn't actually matter, because like most potentials you only calculate measurable quantities using the difference between two potentials, or the gradient, or something that is unaffected by where your arbitrary zero point is.

For atomic electrons I think they choose zero to be infinitely far away, that way bound electrons always have negative potential energies. But it doesn't matter because you only ever subtract the potential energies at different places.

It's meaningless to say "I have xJ of gravitational potential energy".

I would agree with all of that;

In particular, I am glad you pointed out that;

"The choice of zero point doesn't actually matter, because like most potentials you only calculate measurable quantities using the difference between two potentials." and;

"It's meaningless to say "I have x Joules of gravitational potential energy".

Below I've tried to put together a reasonable definition of gravitational potential energy.

If you disagree with anything, just say so.

Gravitational Potential.

Let A and B be two points in space.

Suppose, work is required to move a mass, M, from point A to point B.

Then, we say that there is a gravitational potential difference between A and B and that B has the higher potential.

We define the gravitational potential difference between (the mass M at) A and (the same mass at) B to be the work required to move the mass M from A to B.

Note that, we have defined only the potential difference between two points.

To define some sort of potential at a point A, we need to define it with respect to some fixed point Z.

It is convenient to define the potential to be zero at the chosen point Z.

We define the gravitational potential (of the mass M at) at A (with respect to Z) to be the work required to move the mass M from A to Z.

In celestial mechanics;

1) The point Z is chosen to be an infinite distance from any bodies of interest.

2) Z is generally called infinity.

3) The value of the gravitational potential at infinity is defined to be zero.

So, the gravitational potential (of the mass M) at A (with respect to infinity) is the work required to move the mass M from A to infinity.

Now, lets introduce a second mass, m.

We define the mutual gravitational potential energy of the two masses, at a distance r, to be the work required to bring them to a distance r apart, from an infinite initial separation.

What exactly, is the amount of this work?

The gravitational force between the two, at a distance x apart, is;

F = [G * M * m]/x^2

The work done in bringing them from an infinite initial separation, to a distance r, is;

W = Integral F.dx (the integral is from r to infinity)

= [G * M * m] Integral 1/x^2 dx

= [G * M * m] [-1/x] (evaluated from r to infinity)

= - [G * M * m]/r

So the mutual gravitational potential energy, U, of the two masses, at a distance r apart, is given by the formula;

U = - [G * M * m]/r

It is the energy required to bring the masses to a distance r apart, from an infinite initial separation.

No, you haven't. You derived the formula for gravitational potential from the formula for gravitational force, without understanding what it is your final formula means, or how it is used.

And, as kellog already pointed out, it is dependent on your reference point.

The scientifically accepted definition of Gravitational potential energy (GPE from now on) is the potential energy an object possesses because of its position in a gravitational field. The calculation of GPE for a static point can be interpreted as the potential energy lost as an object moves from infinity to its current position (this is why GPE is always negative). But when moving masses in a gravitational field, changes in GPE represent energy released or consumed in that movement.

So where you are making your mistake is in how gravitational potential energy can be used to determine the energy released/consumed when moving and object from one position in a gravitational field to another - and since we're talking about moving objects this value is critical.

In the case of a moving object - which is exactly what your two merging planets are - the difference in GPE from the start/end points represents the amount of energy (not potential energy) released (if moving closer to the reference point), or consumed (if moving away from the reference point).

Gravitational potential energy = U = -G*m1*m2/r, where r is the distance separating the object from the reference point.

And now where you've gone wrong - the reference point and how potential energy changes based on changes in its position relative to a reference point.

------------------------------------------

We have your two masses - lets call them 'A' (with a mass of mA) and 'B' (with a mass of mB), whose centers of mass are separated by distance 'r'. The gravitational potential energy of 'B', relative to 'A' is equal to: U = -[G*mA*mB]/r

[G*mA*mB] will be a constant in our system - G and the mass of our planets do not change. For the sake of simplicity, lets pretend that [G*mA*mB] = 1, and r is 1m. This means we have a potential energy of: U = -1/1m = -1J.

Keep in mind that negative sign - its important, and where you are going wrong.

Lets pretend we now move B closer to A such that they are half as far apart as they were before:
U = -1/0.5m
U = -2J

We've had a change in potential energy of:
deltaU = U(initial) - U(final)
deltaU = [-1J] - [-2J] = 1J

So in moving B 0.5m closer to A we release 1J worth of potential energy, which we can use to preform work. In the case of gravity, this work will be in the form of gravitational acceleration, meaning that B will now have a kinetic energy of 1J, relative to A. In other words, when you move two objects towards each other in a gravitational field the gravitational potential energy gets converted into work - as in you can then use that work to do stuff - say destroy a planet.

-----------------------------------

Now, what if we do the opposite - move B away from A, to a distance of 2m?
U = -1/2m
U = -0.5J

deltaU = [-1J] - [-0.5J] = -0.5J

Note the negative sign. This means we've had to add potential energy to move B farther away from A, which means we've had to use work to move B.

Now, what about your case of separating things to infinity:
U = -1/infinity
U = 0J

deltaU = -1J-0J = -1J

So to separate our two planets to infinity, we would have to do 1J worth of work. In other words, when you move two objects away from each other in a gravitational field, you must preform work to separate two masses.

--------------------------------------

The above shows where you went wrong in this, and the other, thread. In looking back I realize it was partly my fault - I assumed as a self-professed mathematician you would understand the impact of inverse relationships and how GPE translates into energy, and thus skipped the (obvious to me) deltaU calculation.

For that I apologize - but it doesn't change the fact you are wrong, and the numbers I calculated were correct.

In fact, I was overly generous and gave you a huge amount of wiggle room, in terms of the amount of movement - and thus the change in GPE (and thus the amount of energy released).

Going back to the original thread, where we had two massive planets immediately juxtaposed to each other. The gravitational potential energy of that initial state was -5.95 X 10^31 J. Since gravity is acting on the system, those two planets are drawn together. As you can see from the above examples, that means that after the merger the GPE will be even more negative, and the difference in GPE will represent the work done on the planet. Since the relationship is an inverse one...well, we'll get back to that.

The exact amount of work that gets conducted depends on just how far the two centers of mass can move before equilibrium is achieved (i.e. they merge to form a new center of mass). As I stated (and you never disagreed) they would move in ~50% of the way, simply because they are 2 ~equal spheres moving, and thus their merged center of mass should be at a point roughly where they first meet.

Keep in mind, GPE is essentially -a/r, where a is a constant determined by the (unchanging) mass in the system and the (unchanging) gravitational constant. So a movement of 50% the radius would mean the change in GPE is deltaU = -a - [-a/0.5] = -a - -2a = a.

Which is the value I used in my explanation. I assumed a self-professed mathematician would be aware of the impact of inverse relationships - apparently I was wrong.

So to go back to the point I made in the last thread - in your system, starting with the two planets touching and NO OTHER ENERGY, and assuming the center of mass of the final planet is located roughly where the two planets first meet, you will release 5.95 X 10^31 J of energy due to the change in gravitational potential energy.

Whining that if you were to apply 5.95 X 10^31 J of energy you would separate the masses to infinity is pointless - we're not pushing these things apart; gravity is moving them together.

Bryan

You truly are weird.

I think you are a propagandist. You are definitely not a scientist in the true sense of the word.

Don't get me wrong,... I sorta like you.

Anyway, your statement;

"Gravitational potential energy = U = -G*m1*m2/r, where r is the distance separating the object from the reference point."

is obviously nonsense.

In case you don't see the obvious nonsense; It doesn't make sense to talk about a distance r from the reference point to two different points.

Why don't you correct that first and we will go from there.

I can't wait to see your ramblings on the other thread.

The opening of the Indian Ocean.



Yet another cool animation, eh?

From: http://preearth.net/

I can no more edit your posts than you can edit mine.

I think I speak for us all when I say "don't let the door hit you on the way out"

Bryan

I think you speak for yourself , bryan.
this man has a idea , if it is or is not fully understood by you is your problem not his.

further , your inability to understand the least of his concepts does not translate into your inteligence , nor does it show that he is incorrect.



well should you ever cross that threshold bryan , I for one
would make sure the door lock was changed and you were not given a key.

Well paul, in post 34511 and 34514 you agreed with me - his model is a physical impossibility.

Now I think you're disagreeing for the sake of disagreeing.

Bryan

I wasnt talking about his hypothesis , bryan.

I was talking about your claim below.



you dont speak for me or anyone else in this forum.

you may imagine or think that you do.

but you dont.

Hi Paul;

I have started up my own bulletin board at:

http://www.preearth.net/phpBB3/search.php?st=365&search_id=active_topics

See if there are any topics that interest you.

http://www.preearth.net/phpBB3/search.php?st=365&search_id=active_topics

Virtually no one has made use of this board and am wondering if it is fully functional.

Would you be kind enough to leave a few comments so I can see that all is O.K.

It certainly seems functional from where I am.

Thanks a lot.

Paul, you have not bothered to leave any comment at

http://www.preearth.net/phpBB3/search.php?st=365&search_id=active_topics

Is there a reason for this?

Paul, how about a comment, or two, from you for my little forum

http://www.preearth.net/phpBB3/search.php?st=365&search_id=active_topics

It longs for your presence.

You certainly spoke for me, Bryan. FYI, you might want to use this link to ignore the offending user:
http://www.scienceagogo.com/forum/ubbthreads.php?ubb=toggleignore&User=3062

.
Charming. You must be really scared of something.

But why would I want to ignore him? Its fun, beating the trolls. Its really fun when they're like preearth - all the denial, conspiracy-ism, etc.

Bryan

Well yes, Bryan, do enjoy. There's a dearth of dead horses these days so by all means flog away. Though I were Murray Gell-Mann, there'd be nothing I could add to convince pre that he's barking up the wrong tree, so I'll go play with Jaws - he's my dog (I'm sure he'd love to eat meet pre).

preearth. It is a long time since the last post on this thread, but if you are still looking at this site, there is one thing I would like to say. I am neither a scientist nor a mathematician, so I struggle with a lot of your exchanges with Bryan and others. However, I like to consider opinions that are outside the norm.

Elsewhere, some time ago, I made a comment about Louis Savain (www, rebelscience.org)

"My feeling is that Savain has something interesting to say, for this reason I think it is sad that his sniping, ad hominem style of attack is distinctly off-putting, and seriously detracts from any real insights he might have to offer".

Its sad, but I feel that this comment also applies to you.

I think that behaviour shows an inability to understand other people's points of view or be critical of their own. If somebody has that way of thinking, then it's doubtful they'll actually produce any good ideas because they won't be able to critically examine their own ideas. I suppose you might get one flash of insight that happens to be right without making any adjustments. But hardly ever!

I wonder if the Geek has figured out exactly what "gravitational potential energy" is yet.

I doubt it.

It seems that bandying mathematical formulae about gets nowhere significant.
Here is a non-mathematical suggestion:

At the time of the Big Bang we are told that all the matter and energy of the Universe occupied a very small area. The energy of the BB forced this matter and energy to expand. GPE is simply the equal and opposite force which, provided there were no "interference", would tend to bring everything back together.

Of course, this implies acceptance of the BB, the expanding Universe and equal and opposite forces, but, heck, where would we be without a little "faith"?

BTW, preearth, it's good to see you haven't deserted us. It's been very quite around here, and you do tend to liven things up.

Hi Bill; as you know I am pushing a new theory which is encountering some significant resistance.

I get a lot of crap from a lot of people and tend to get a bit cranky at times.

The Geek, in particular, makes me cranky,... particularly, his making up "facts" he feels no one can check.

For example, he has made up "facts" which he claims come from papers that are not available to the public.

In one case, when the paper was made available, his "facts" were not to be found in it.

Of course, if his "facts" were actually in the paper, he could have included the paper as an attachment to the thread, so that all could see it,... but he didn't.

Also, in the "Einstein was NOT first to publish E=mc^2" thread, he gave as "proof" of his claims, a passage in Italian.

If he understood the passage he was quoting, then he had to be able to read Italian. If he could read Italian, then he could have translated the passage for us and more clearly verified his claim,... but he didn't.

These instances show he is a dishonest debater and that annoys me.

I have to say I am not surprised that your "new theory" is raising a few objections. Perhaps it is best to see this resistance as the refiner's fire in which your theory can be assayed.

Regarding your comments about Bryan; I prefer to eschew "ad hominem" attacks, and stick to discussion of the relevant issues.

Bill S. said; "I prefer to eschew "ad hominem" attacks, and stick to discussion of the relevant issues."

If you don't want to call a liar, a liar, and end up believing his lies, that's your concern.

I am not that stupid.

Thanks.

How did we get from name calling to "believing his lies"?

Seems a natural progression to me.

If you are trying to draw me into some sort of slanging match, you really are wasting your time.

If you are unwilling to discuss things in a reasonable manner, I am wasting my time trying.

The ball is in your court.

Whatever happened to the original point of all this, which I seem to remember (could be wrong!) was something like:

Preearth says the gravitational potential energy of two planets moving from very close to combining may not be converted into enough heat to melt the crust.

Somebody else saying the gravitational potential energy of the two planets moving from very close to combining would have to be converted to enough heat to melt the crust.

How about a simple calculation like this:

1) Find the gravitational potential energy of the two planets just touching (using several methods if need be)
2) Find the gravitational potential energy of the combined planet (using the same method(s) as in 1).
3) Find the difference, and make sure it's the same for all methods. If not, got back and identify the wrong ones.
4) Assume all the energy is converted to heat and so find the amount of heat energy created
5) Estimate an upper bound on the time taken to combine.
6) Apply a radiative cooling model (or a suitable approximation), and calculate the temperature gradient through the surface of the combined planet.
7) See if the surface of the combined planet has a layer of similar thickness to continental crust which never exceeds the melting point of that material.
8) Conclude that the energy calculation either disproves Preearth's hypothesis, or is compatible with it.

Easy peasy!! Who wants to work through this with me?

Bonus part) If it disproves it, then increase the time taken to combine. For a slow enough process it won't disprove the hypothesis. This can give a quantified lower limit on the time taken to combine.

Sounds like a fascinating project. I don't have the maths to be much help, but I look forward to watching things happen.

I reckon. I hope PreEarth will be keen too. I'm happy to have a good go at the maths. It seems a shame to see arguments degenerate into irrelevant details that aren't going to make a difference either way.

This potential energy -> melting crust seems like something so easy to quantify. I'm sure it won't require any supercomputer mantle convection modelling, just a best/worst case scenario.



"No sense of harmony, no sense of time ..."

Go for it, Kallog. Preearth seems like the sort of person who would have the maths to make it a lively exchange. I may have to limit myself to the occasional "What?".

Here is something on this from way back.

Here is a cute calculation that comes up in showing the impact between the two planets (i.e., Heaven and PreEarth) would not necessarily melt the entire surface of the larger planet that coalesced from them (i.e., Earth).

It is simple enough for high schools students to understand and it illustrates the power of (even simple) mathematics.

The problem is to estimate the rise in the temperature caused by placing the planet PreEarth next to the planet Heaven and letting gravity transform them into the Earth.

(Note, that if two large enough planets are placed side by side and left alone, then gravity will pull them into one planet. So here we are placing PreEarth next to Heaven and gravity is pulling these two into a new planet which just happens to be the Earth.)

We will make the simplifying assumption that both the planets Heaven and PreEarth were spherical with uniform density.

This makes the math simple and the argument easy to follow.

The gravitational binding energy of a planet, U, is the energy released by the assembly of the planet from atoms which were originally an infinite distance away. Or, alternatively, it is the energy needed to disassemble the planet into atoms by moving each an infinite distance away.

The gravitational binding energy of a spherical planet with uniform density, is given by the formula;

U = 0.6GM^2/R, where

G = 6.67428 x 10^-20 km^3/(kg s^2) is the gravitational constant,
M is the mass of the planet in kg,
R is its radius in km.

U is here measured in megajoules, MJ.

Earth Radius R_E = 6371 km.
Earth Mass M_E = 5.97369 x 10^24 kg.
Approximate Earth Binding Energy = 0.6*G*M_P^2/R_P = 22.430 x 10^25.

PreEarth Radius R_P = 5200 km.
PreEarth Mass M_P = 3.48280 x 10^24 kg.
Approximate PreEarth Binding Energy = 0.6*G*M_P^2/R_P = 9.341 x 10^25 MJ.

Heaven Radius R_H = 4680 km.
Heaven Mass M_H = 2.48456 x 10^24 kg.
Approximate Heaven Binding Energy = 0.6*G*M_H^2/R_H = 5.282 x 10^25 MJ.

The energy necessary to separate PreEarth and Heaven to infinity, is:

G*M_P*M_H/(5200+4680) = G*M_P*M_H/9880 = 5.846 x 10^25 MJ.

The idea is to take PreEarth and Heaven at the point of first contact, that is, when they are just 9,880 kilometers apart, dissemble them to infinity, then bring everything back from infinity and assemble Earth.

So, the energy released from the point of contact through the formation of the Earth, is:

Energy Released = (22.430 - 9.341 - 5.282 - 5.846) x 10^25 = 1.961 x 10^25 MJ.

This is (1.961 x 10^25)/(5.97369 x 10^24) = 3.2827 megajoules per kilogram.

Suppose an average specific heat of 1330 J/kg°K.

Then we have that the Earth experiences a 3282700/1330 = 2,468 degree (average) rise in the temperature.

I emphasize that this temperature rise is due solely to the energy released by just placing PreEarth next to Heaven and letting gravity transform them into the Earth.

It is worth mentioning that the temperature rise will not be uniform.

Any pre-impact kinetic energy of Heaven (relative to PreEarth) will cause an additional rise in temperature.

To be continued,...

Note that in the above calculation, the latent heat of melting of the rock has not been taken into account. This will reduce the temperature rise somewhat.

Sorry some of this doesn't make any sense to me because of my level of english or there is extreme confusion?

What I can't understand is why all the Newtonian equations trying to understand Gravitational Potential Energy we all do realize they are wrong and won't make sense at this level ... right?

If I am understanding you correctly what you really need to look at is Bill S's favourite the somewhat scary Friedmann equations
(http://en.wikipedia.org/wiki/Friedmann_equations)

Go down to the bottom section they do a rescale for you if you can't do the maths and there is effective potential energy for something in space.

I think that's what you are trying to work.

Trying any of this with newtonian law is ridiculous it's a given it's wrong.

For "favourite", read "nemesis"!

BTW. Towards the close of last year I offered a non-mathematical suggestion as a possible answer to the original question.

"At the time of the Big Bang we are told that all the matter and energy of the Universe occupied a very small area. The energy of the BB forced this matter and energy to expand. GPE is simply the equal and opposite force which, provided there were no "interference", would tend to bring everything back together.

Of course, this implies acceptance of the BB, the expanding Universe and equal and opposite forces, but, heck, where would we be without a little "faith"?"

No one responded; is that because it is too naive to deserve comment?

The problem is what is the force in effect you are asking the age old question question of the direction of entropy which leads directly into the "arrow of time".

I accept the argument but it becomes a bit like discussing religion origins I have no basis to discuss.

I suspect if you actually do the analysis it ends a bit like Doug Sweetsers analysis of Julian Barbour's theory that there is no such thing as time it is an illussion (http://www.science20.com/standup_physicist/julian_barbour_versus_minkowski_and_me-80765).