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Talk:Centers of gravity in non-uniform fields

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Previous Talk Archived GenKnowitall (talk) 16:45, 5 June 2011 (UTC)

Comment on Article as of 22 Sep 2011
The article demonstrates by result that Wikipedia is not a competent source of scientific theory and information. The process demonstrates why. This is unfortunate and disappointing. Despite renewed discussion and some acknowledgments of fundamental deficiencies the discussion below was abandoned August 15, 2011 without material change in the article. The article therefore remains incompetent in content and presentation and should not be used as a physics reference. Students should consult appropriate general physics references for introductory and advanced discussion of the subject in physics. Historical development of concept is also important, and scholarly articles on Archimedes will be instructive for historical development, although please be aware these will not likely be the earliest historical sources. The subject is interesting and also as useful to disciplining a physics mind as learning "balance" is to a dancer or martial arts student. "Balance", "Imbalance", go and study it, just not here, regretfully. GenKnowitall (talk) 20:26, 22 September 2011 (UTC)

 * GenKnowitall: I quite agree. This article makes some statements that are just plain incorrect.
 * I am not familiar with the reference
 * • Symon, Keith R. (1971), Mechanics, Addison-Wesley, ISBN 978-0-201-07392-8
 * but every time it is cited, the statement seems to be fallacious. Either the Symon book is garbage, or the person citing the book misconstrued what was there.
 * There is no mention in the article of a gravity gradient (strong close in, and weak farther out) which seems to me to be the most pertinent distinction between mass-center and center-of-gravity. The distinction comes into play when satellites are in orbit and there are no larger torques applied.  The gravity gradient will result in a torque that will tend to align a satellite with its long axis vertical.  Also, the mass-center is a fixed point within a rigid body, whereas in a non-uniform gravitational field (i.e. one where the gravity gradient effect is not negligible), the center of gravity will depend upon the orientation of the rigid body within the non-uniform gravity field.  Even so, the 'center-of-gravity' can be as well defined within that body as a center of buoyancy is for a ship.  It depends upon having some reference orientation and then looking at perturbations about that reference orientation (which is not necessarily at equilibrium).  Some of the citations of Symon seem to contradict that basic characterization of the distinction between mass-center (fixed) and center-of-gravity (may depend on orientation within a non-uniform gravitational field).  As for spherical vs. 'uniform' gravity, having an oblate spheroidal representation of the Earth does not cause center-of-gravity for an object near the Earth to become undefined.  ???  The treatment of cross product vs. orthogonal is garbage.  ???  This article is worse than not having an article, especially if someone thinks that they know more about the topic after reading it than they did before.  So I'm not even talking about Wiki conventions and standards.  Some of this material is both wrong and misleading.
 * And I have no idea what history there is to this article in terms of edit wars. Did 'either side' really have a clue?  Were there multiple sides?  The principles are really not that complicated.

 * PoqVaUSA (talk) 19:47, 14 April 2015 (UTC)

Comment on this Article as of 5 June 2011

 * There have now been two edit-wars over this topic (there should have been none). I anticipated this possibility when I entered with an article so look for no sympathy, but will conclude. While there are problems with the article I will proceed no further with revision and correction of it. The problem is not merely content, but article history demonstrates that the article cannot be effectively corrected in the Wikipedia editing environment.

 * In particular, prior discussion demonstrates something that should not happen in the field of Physics: refusal to discuss and collaborate, and abusive edit-war tactics intended to suppress collaboration and impose a viewpoint. What is particularly concerning that two admins actually participated in the last edit-war but failed to intervene to control obvious and even notorious sniping and disruptor conduct (one disruptor has been since banned by a third admin), and additionally took actions which were apparently arbitrary or biased (article lock). This admin conduct is not a solution to anything, it is itself a problem.  This failure brings into question the editorial process of the Wikipedia, its supervision, and whether any serious person should attempt contributions in such a "foodfight" environment.

 * Concerning the article content, NO STUDENT SHOULD CITE OR RELY UPON THIS ARTICLE AS WRITTEN, for while there is some merit here and the content is worthy of an article, it reflects the poor process used to create it and, in my opinion, the article as of June 5 2011 is replete with problems, including original research, misrepresentation of authority, and a non-balanced view of subject. Students should refer to your appropriate text. In any event, Physics is about THINKING as clearly as possible, as deeply as possible, about understanding the natural world. That is the tradition of Newton, the tradition of Einstein, the tradition of Physics. You do not need to be a "Physicist" to do it, you merely need to THINK like one, and it is that training and refinement that creates physics and  "physicists". Do that and you will be fine.

 * Unfortunately, this article's problems do not stand singularly, for I find many fundamental problems within Wikipedia articles, including Physics articles, problems of concept which are often hallmarks of STUDENT level understanding, scholarship, and analysis. There is nothing wrong with this either, for revealing misunderstanding or alternate views is what learning is all about. What is of grave concern is the abusive editing environment created by the Wikipedia itself in the administration of its supposed policies, practices which obstruct article editing. I say "supposed policies" because it is obvious that the practice does not match the professed policy, the "walk" does not match the "talk".  This is not a failure of policies alone, but a clear a failure of people charged with administering them. Wikipedia is what it is, but an authoritative encyclopedia it proves it is not. Let us instead call it what it is, a COMPENDIUM of scribbling.

 * I will, for a brief time, respond to civil comment.

GenKnowitall (talk) 21:42, 24 June 2011 (UTC)

 * For a start, no student should cite any Wikipedia article, full stop. Students should be taught where to find reliable sources, and Wikipedia cannot be one, by its very nature. It can however be a useful resource for finding reliable sources. As for the remainder of your comments, I have to ask whether you are proposing any substantive changes to the way Wikipedia works, or merely venting your frustrations? AndyTheGrump (talk) 23:41, 24 June 2011 (UTC)
 * I dropped a note at WikiProject Physics, maybe someone from there can help sort all this out. Herostratus (talk) 00:19, 25 June 2011 (UTC)
 * Thank you both for comments.
 * TO ANDYTHEGRUMP - I agree with your first three sentences, and reply to the last. I propose nothing, I am a mere messenger. There appears to be a gross disconnect between the policy and the practice. I am not the only one to have noticed it, but demonstrate it with a relatively clear example. I state only what is plain. Yet if there is a problem to fix I am not the one to propose or implement its solution, someone high will need to acknowledge it and examine how best to address it. GenKnowitall (talk) 03:13, 25 June 2011 (UTC)
 * TO HEROSTRATUS- I made no complaint about physics, in any event its already been sorted, the article is the result. That is the process, that is the result.  Anyway, Wikipedia Physics project is no authority for any proposition in physics or any other subject. For example, someone purporting to represent the group is assigning "importance" to subjects, don't you think that odd?  Instead my comments went beyond physics to the heart of the Wikipedia, and that is where any response lies. So fear not my good Herostratus, physics survived the inquisition, it will survive the Wikipedia. The open question is whether the Wikipedia will survive the Wikipedia. GenKnowitall (talk) 03:13, 25 June 2011 (UTC)

Would it be the proper time to propose a new lead for the article? (Feel free to improve my English, and please check the last sentence with somebody in the field.)

"In physics, center of gravity (CG) of a material body is a point that may be used for a summary description of gravitational interaction. That point may be defined either in the context of an external gravitational field acting on the body, or in the context of the gravitational field produced by the body and acting on other objects.

In the external field context, CG is the application point of the resultant gravitational force on the body, if it can be determined. In the homogenous gravitational field, CG coincides with the center of mass (CM) of the body. This is a very good approximation for smaller bodies near the surface of Earth, so there is no practical need to distinguish CG from CM in technical applications.

In the context of the gravitational field produced by the body, CG is the point which is the apparent source of the field as seen by the outside observers (i.e. felt by the other bodies), if it can be determined. In general, it can only be determined for each point of observation separately. However, a body with spherically symmetric distribution of mass has its CG in its center (i.e. in its CM) for all outside observers."--Ilevanat (talk) 02:06, 5 August 2011 (UTC)

The article is not all that bad
considering other Wikipedia articles on elementary (highschool) concepts in Physics. The recognition of 2 general meanings is fine, though each of them could be presented with somewhat better understanding (simpler presentation, correction of minor errors etc.). For the case of a body in the field, the elementary story on ballancing (stable, indifferent ...) would be helpful especially for discussion of the inhomogenous field. For the case of the field source, it should be clarified that the unique CoG refers only to the selected observation point. And even in this form, I believe it deserves to be back in Wiki, and should have a link in the CoM article (which happens to be of a comparable quality).

As for the students (from the above discussion), some of them do and will use Wiki regardless of the teachers warnings. It would be nice if Wikipedia would provide some more pronounced disclaimers for these less academically inclined souls when they attempt to "fast google" some simple physics definition...--Ilevanat (talk) 00:24, 12 July 2011 (UTC)

 * The article IS APPALLING from the standpoint of balance, scholarship, clarity, and general presentation. The CM article is not much better, alas, but this isn't about that article. Instead THIS article violates nearly every fundamental policy of the wikipedia, (although those polices are obviously more for show than do).  On the technical merits, for those who think center of gravity is really just the same concept as center of mass, that a google URL reference dump is scholarship, that misrepresenting and misunderstanding the teaching of a very fine scientist (ie Feynman), presenting a biochemist (Asimov) as a physics authority, treating popular science articles as representing a weight of authority by repetitively quoting them, ignoring and down-playng and deliberately bungling an 'opposing view' supported by authorities in the field, and other failings of the article rather too numerous to list much less correct... if all that is a "not all that bad"... well, I can only be thankful that science has always been blessed and advanced by a few capable thinkers even as they struggle for foothold in the great Mire of Stupid. Still, I'd like to at least say that this wasn't all that bad work "for a fifth grader", but even that mild approval refuses to pass my keyboard. The article doesn't even mention Newton! This isn't physics, it isn't teaching, it isn't knowledge, it isn't scholarship, it is... entirely something else. It is instead, apparently, the Brave New World. It is, apparently, the Wikipedia. GenKnowitall (talk) 21:00, 23 July 2011 (UTC)

Sorry if my disscusion lead to any missunderstanding regarding my general views on Wikipedia. I have just posted a contribution to the Village pump policy section, claiming that professionals should be motivated to edit Wiki articles if Wikipedia is ever to become an encyclopedia.

However, regarding the CG article, it appears (after reading the archived disscusion) that you are for some reason unwilling to recognize the fact that the term CG is generally used in two different meanings: a) as a point related to a body in an external field, and b) as a point related to the body that caauses the field. The current article covers both meanings, and that is why I think "it is not all that bad", although it should be significantly improved in the presentation of these meanings.--Ilevanat (talk) 01:49, 31 July 2011 (UTC)

 * If you read my comments, then perhaps you did not understand them. They plainly and specifically were objecting to process, not content. I opposed no alternernative view, instead requested alternatives be presented and discussed. Look again, you will see. To that I can add that it may ALSO be, and let us assume arguendo there IS, a serious problem with the main article definition.   [We might also assume - from your comments-, at least arguendo, that you don't see the flaw. ] Yet, because of a failure in process I cannot, as a practical matter, demonstrate it, and contribute to improve Wikipedia content directly.  The flaw remains. No student may rely on Wikipedia, no serious scholar may cite it. So what is it?
 * As for professionals... we have journals, we have textbooks, we have teachers, we have a peer review system. Wikipedia is a populist attempt to end-run that system, but without the safeguards. It could still work,  perhaps, including anonymously, but would have to have better trained admins to enforce disciplined editing process. They aren't. They don't. It fails. This article plainly demonstrates that . And there you are. GenKnowitall (talk) 06:57, 2 August 2011 (UTC)
 * As a basic rule, nobody should ever rely solely on Wikipedia. Wikipedia is a good place to start one's research, and find other trustworthy sources, but Wikipedia, a reliable source? It is not, and it never will be. That will be inherent in Wikipedia as long as IPs are allowed to edit this. Citizendium flopped, though they had a serious chance of being a trustworthy user-contributed encyclopedia. The problem with Citizendium is that nobody wants to join, although if we could somehow move every active, enthusiastic, smart, and registered user to Citizendium, I believe it would become a rival to Wikipedia. -- Σ  talk contribs   07:07, 2 August 2011 (UTC)
 * Sigma, you hold the wrong end of the stick. Its not that no student should rely solely on Wikipedia, but that they should not be allowed to access Wikipedia at all! Certainly I am serious. Only a master can use Wikipedia safely, and even then must keep their wits about them. All that twirling swordplay looks and is dangerous, especially for students, especially when the teacher is a fool. Students (and bystanders) are too precious to expend so. Rarely we are blessed with quick and original thinkers who need little, but mostly we have capable people who need to learn, to be formally trained, not just "Wikipedified". We are shown how in a structured way by an "authority" in the field, we learn the moves, carefully, in order, and practice until we become the master. An encyclopedia may "summarize" this "knowledge", but must in no circumstance subvert the learning process.  Here, and for multiple reasons, Wikipedia fails disastrously. Here college students and young PhD's practice their twirling swordplay to an audience of peers before middle and high-school bystanders.  No supervisor or mentor or responsible adult stands by to moderate and avert disaster.  It isn't that students shouldn't rely on it, it is that they shouldn't even witness it. Its not that Wikipedia has failed its objective, it is that perhaps it should be shut down and those responsible executed as a continuing danger to their fellow man.  Well, perhaps that is too far to go. Perhaps. Whether Wikipedia can evolve, or another structure supplant it, is to be seen, and even hoped. Despite my negative comments I still favor the idea of building a "Great Library" filled with treasure and accessible to all. But if the building is to be a trash heap containing all the refuse of the world's mind, then a razor and hot bath could seem a more worthy endpoint for man... GenKnowitall (talk) 20:06, 2 August 2011 (UTC)

Lead section
Would it be the proper time to propose a new lead for the article? (Feel free to improve my English, and please check the last sentence with somebody in the field.)

"In physics, center of gravity (CG) of a material body is a point that may be used for a summary description of gravitational interaction. That point may be defined either in the context of an external gravitational field acting on the body, or in the context of the gravitational field produced by the body and acting on other objects.

In the external field context, CG is the application point of the resultant gravitational force on the body, if it can be determined. In the homogenous gravitational field, CG coincides with the center of mass (CM) of the body. This is a very good approximation for smaller bodies near the surface of Earth, so there is no practical need to distinguish CG from CM in technical applications.

In the context of the gravitational field produced by the body, CG is the point which is the apparent source of the field as seen by the outside observers (i.e. felt by the other bodies), if it can be determined. In general, it can only be determined for each point of observation separately. However, a body with spherically symmetric distribution of mass has its CG in its center (i.e. in its CM) for all outside observers."--Ilevanat (talk) 02:11, 5 August 2011 (UTC)

 * (adding section header)
 * Sounds good, and the last sentence is correct; it shouldn't be hard to provide a citation. I just have a couple of issues:
 * Saying "the application point of the resultant" makes it sounds as if it's unique, but any point on the line of action is acceptable as a center of gravity. I think I've read a couple sources that try to nail down the "vertical" coordinate, but there doesn't seem to be a well-established convention.
 * In the external field context, there's the resultant/torque definition and then there's the average-by-weight definition, and the two definitions aren't compatible. If we mention one in the lead, I think we need to mention both.
 * Thoughts? Melchoir (talk) 03:27, 5 August 2011 (UTC)
 * Thoughts??? Ah, man... are just no more tears. Yes, several thoughts do occur, Melchoir. Let us consider a group of Neanderthals with big clubs all standing around a huge steaming pile of crap. The outcome is surely apparent? Foresight compels a wise man to stand aside. Yet I am not entirely unsympathetic, so offer this: Dump the present debacle entirely and start anew, carefully. This is a classical concept with ancient roots. Start with classical physics, then reconcile with ancient authority. You will need someone who actually understands classical physics, which will prove the necessary advantage. I have reason to doubt your ability, but will be sincerely glad to have misjudged. Show me. GenKnowitall (talk) 19:14, 5 August 2011 (UTC)

 * GenKnowitall, with apologies to Hardy Cross, not insulting other editors is "essential, but otherwise unimportant". It is likely going to prove impossible for anything to get done with insults (like the ones I've removed above) flying around.  Once they've stopped, then you can all ignore the unimportant civility issues, and concentrate on the important content issues. --Floquenbeam (talk) 21:53, 5 August 2011 (UTC)

 * Floquenbeam, I have reverted your redaction of my comments and have taken note of your comments which themselves may violate wikipedia policy. You are responsible for your comments to the same extent anyone else is. Troubling you wouldn't see that, though. An advisory given as a courtesy:  You should not edit an author's comments. If you disagree with something said,  believe it sanctionable,  and think something should be done about it,  then proceed through Wikipedia procedures. You aren't the courtesy police, but if you disagree in this case take it up as a complaint, which I may answer, but don't edit my comments.  Oh, as a formal matter Cross was applying Newton, not Archimedes, just as we might hope to God any competent structural engineer would modernly do. yer welcome, no charge. GenKnowitall (talk) 21:21, 8 August 2011 (UTC)
 * Will comment on your my talk page in a bit. However, a quick note now, to the other editors here: please let me know if GenKnowitall makes any more personal attacks on anyone on this page, and I will block his account. --Floquenbeam (talk) 21:53, 8 August 2011 (UTC)

 * Your threats are improper, Floquenbeam. perhaps also defamatory. I advise I have previously replied on your page. You may not care but perhaps you should consider whether to continue as an admin? Anyway, there really ought to be a streamlined way to remove admins who have gone over the edge, don't you think? I'll look into it. GenKnowitall (talk) 20:47, 12 August 2011 (UTC)

Signs of intelligence in the universe, God is Great! However, I have revised this section to include your earlier comments which were left outside this section. Your approach of excluding your comments while including mine was unnecessarily provocative, but I will accept with this revision. Hope I don't regret this courtesy. GenKnowitall (talk) 01:38, 13 August 2011 (UTC)

The average-by-weight definition first source in the article, Beatty (2006 p.48), can be viewed on Google books. It is easily seen that the definition is not a general one: Beatty approximates the Earth field as consisting of parallel vectors with variable strength (the only way the so called "scalar weight" can be introduced). Besides, on p.47 he says this definition is "equipollent" with the torque definition for the same field. Such definitions can be mentioned in the Wiki article, but I do not think they should be included in the lead. P.S. The application point of the resultant is determined by the torque rule, and can be anywhere along the line of action.--Ilevanat (talk) 02:42, 6 August 2011 (UTC)
 * I can see that Beatty might not intend to state a general definition, but there are other sources, most importantly Jong & Rogers, who write their equation (6.25) without preconditions and without justifying it in terms of torque. There are also a few other sources that speak vaguely of an "average" without writing down equations.
 * You and I probably agree that a "scalar weight" shouldn't be introduced except in a parallel field, but that doesn't mean that it can't be introduced. One can simply take $$g = |\mathbf g|$$. Of course, this definition makes us wince because we know it isn't likely to result in a well-behaved quantity, but that doesn't stop it from being a popular idea. Melchoir (talk) 05:34, 6 August 2011 (UTC)
 * ...Actually, I thought of a better say to state the situation with the weighted averages. A real gravitational field can't be both parallel and non-constant. And neither taking the absolute value of the field strength (Jong & Rogers), nor projecting it onto a single axis (Beatty), is going to give you an average that precisely lies on the line of action for the real field. So if torque is what we care about, I would guess that both methods are probably correct to first order in the size of the body, but not to second order.
 * This is kind of side-tracking from the original question, I guess. Tell you what, why don't you just write what you think best, and I'll probably edit it, and we can see if we actually agree. Melchoir (talk) 06:58, 6 August 2011 (UTC)

Let me hesitate a bit. First, I was only proposing the lead. Although it should determine much of the rest of the article, there would still remain a lot of detailed work. And second, I do really care about "an orderly process" of a potential edit discussion (perhaps even more than our colleague GenKnowitall), and I do not feel it is quite over yet.

Let us assume an arbitrary field gives the body a non-zero total weight, and let us set the Cartesian z-axix in that direction. The total weight torque will only have x and y non-zero components, from which the y and x coordinates of CG can be calculated (the line of total weight action). There remains the issue of CG z-coordinate.

In the "Beatty field" (the unit normal is in z-direction), the x and y torque equation components are identical to the y and x scalar weight CG equations. The z torque equation is a null identity, whereas the z CG equation gives an intuitively acceptable CG z coordinate, such that weight above is equal to the weight below.

However, this z coordinate may be perhaps attainable in a more general procedure (giving the Beatty result for the Beatty field). This would involve the static equilibrim consideration (stable, indifferent, unstable point of support), typical for the elementary CG applications. Imagine an infinitesimal rotation around, say, the x axis. We are now looking for a CG z coordinate such that that the sum of particle weight torques arround CG is zero (indiffernt equilibrium, but only in the limit of the zero angle of rotation). It would obviously work fine for a vertical thin rod (which is actually the Beatty field - and give the Beatty result). But is it really a more general approach? I never had to really calculate such things, is there anybody with some ready hands-on experience?--Ilevanat (talk) 02:41, 7 August 2011 (UTC)

 * (Actually I think it's easier to find consensus on the body before writing the lead, since there's more room to explain and provide citations in the body, while the lead is just a summary of the body.)
 * Anyway, if we're starting at the beginning, I have a problem with this statement: "The total weight torque will only have x and y non-zero components". In other words, you're saying that the torque is perpendicular to the weight. That's not generally true. I expect that it's almost never true, unless you happen to have some convenient symmetry in the system. Melchoir (talk) 18:52, 7 August 2011 (UTC)

Do not quite understand the "problem": torque is always perpendicular to the force by which it is produced: mathematically, because of the cross product r×F; physically, because it is parallel to the axis around which the force produces angular acceleration (parallel to the change of angular momentum). The coordinate system I introduced was not really necessary, but I thought it might facilitate understanding that the torque equation and the Beatty equation have two identical components for the Beatty field.

My actual concern is this: which general definitions of CG (no special field or mass configuration arrangements) are meaningful? These should be mentioned in the lead. Special cases and various literature references can at will be added in the body.

The apparent source of the field definition (GenKnowitall - Symon) is clearly meaningful and well-defined, although its practical usefulness may be questioned. (Once you calculate the force by the body on some material particle, why bother about the apparent point-source position: it cannot be used for calculating the force on another particle at different position, or can it?)

CG of the body in external field may be useful for the static equilibrium or even the equations - of - motion calculations; and that is how it all started (and the term used in some other languages suggests only this meaning for CG). But can it be meaningfully and clearly defined for a general field? E.g. by torque equation + something (weighted average or an infinitesimal rotation)? If not, it should be emphasized that it is used only for special fields (where Beatty field is a significantly better approximation for the surface of the Earth case than the homogenous field). And this should be noted ni the lead.--Ilevanat (talk) 02:01, 8 August 2011 (UTC)

 * Well hang on, r×F only applies to a point particle. One you sum this expression over an extended body where both r and F are changing, the result doesn't have to behave nicely.
 * Consider, for example, a cylindrical field along the z axis and a barbell consisting of two equal masses at r1 = (1, 0, 0) and r2 = (0, 1, 1). The forces are F1 = (-1, 0, 0) and F2 = (0, -1, 0). The torques are T1 = (0, 0, 0) and T2 = (1, 0, 0). The total force on the barbell is F = (-1, -1, 0) and the total torque is T = (1, 0, 0). The vectors T and F are not perpendicular; their dot product is -1. Agreed? Melchoir (talk) 04:59, 8 August 2011 (UTC)

You gave a nice example of a body in external field configuration for which there is no CG. There is no point of application (i.e. no line of action) for the total weight F, such that its torque would be equal to the sum of the actual particle wights torques. F is the net force (vectorial sum), but cannot be the resultant force (the net force with such a point of application that it gives the total torque), which is a distinction apparently unknown to English Wiki (despite of its widely praised article on force).

Even in serious English textbooks on physics, "point of application" of a force is only sparsely mentioned (though some other languages use a dedicated name, something like "grip"); in engineering, fortunately, this is not quite so, beacuse these people have to do some actual calculations with forces and their products must perform. Such "policy in physics" may lead to various misconceptions, which is obvious in e.g. English Wiki articles on torque and work, because these quantities can only be properly defined for "point forces" (forces that have a single point of application). Forces acting on extended parts of a body (macroscopic volumes, surfaces etc) must be docomposed into point forces acting on particles, unless they can be substituted by some resultant force (not just the net force) in the rigid body approximation. But these particles themselves have nothing to do with the following clear-cut definitions:

Torque of the force F with respect to some origin is the cross product r×F of the position vector r of the application point of F by F itself. There is no particle or body involved (as in Wiki). And the torque is always perpendicular to both vectors r and F. No complications. Except for the obvious property of the cross product: it will not change if r is position vector of any point along the line of force (therefore the application point of the resultant force, e.g. CG, can be moved anywhere along the line of force, if it is determined only by a torque equation).

Work of a force is the line integral of its scalar tangential component along the path of its application point. No complications with material points, bodies, displacements etc. The only assumption (conventional) is that the tangent is oriented in the direction of motion (of the application point). (I did make some very cautios interventions in the work article, approved in discussion, towards such definitions; but the result remains far from clear-cut.)

Faced with such problems in Wiki articles (and in general uderstanding) regarding very basic physical concepts, one might wonder whether CG deserves much further effort. I am going on vacations (which also caused the delay in my answer) and am not likely to engage in the further article editing, beyond the lead already proposed here. Use it at will.--Ilevanat (talk) 02:49, 11 August 2011 (UTC)

 * That's some unfortunate timing; it looks like we finally found common ground here. We agree that the torque-definition center of gravity need not exist. The problem is finding a source that says so explicitly. The nearest contender is Symon, who says on page 260, "For two extended bodies... The system of gravitational forces on either body due to the other may or may not have a resultant...". Unfortunately he doesn't come out and write "Other authors' definitions of the center of gravity are not well-defined", and in fact he doesn't mention other definitions at all. But maybe the citation can be pressed into service in a way that doesn't violate the No original research policy... Melchoir (talk) 05:28, 11 August 2011 (UTC)

Well, I did find few moments for a closing note. The problem is that CG is not really "my thing", and the authors cited here did not really care to fully explore its possible meanings and applications. So, I will only add here a few remarks of which I am certain (related only to the body in external field case, the other case is clear); but would not dare to rewrite the article.

CG is the concept historically derived from the equilibrium considerations related to a body near the surface of Earth, i.e. for a special case of a body in an external field. Before the concepts of (resultant) force and torque were fully developed, CG could only be described as a point in which as if the total weight of the body were contained. In present terms, this can be reformulated as the definition that CG is "the application point of the resultant force by the external gravitational field, if it can be determined".

The torque equation is no definition; it is a necessary condition that determines the position of the line of action of the resultant force for a given position/orientation of the body in the field. In the homogenous field these lines of action (as drawn on the body) intersect at a single point which is both CG and CM, so the torque equation alone is sufficient and can be taken as a defining equation in this case.

All other cases of a body in an external field may be problematic: CG may depend on the orientation of the body, or may not exist for some or all orientations. But if it does exist, it must satisfy the torque equation. For the parallel Beatty field, his formula is OK (and CG depends on the orientation of the body). For a central field, the Symon formula may be used (by the law of action and reaction); but it is impractical for the Earth field, as it gives the vertical CG coordinate as the distance from the center of Earth (so you cannot actually calculate the small distance between CG and CM).--Ilevanat (talk) 02:15, 15 August 2011 (UTC)

 * Well, I've updated the article with much of the above material. Hopefully it's not necessary to step through how it all relates. Melchoir (talk) 09:47, 16 October 2011 (UTC)

I believe this is a considerable improvement
There are still some weak points that may be improved (I might try though I may not be the most qualified person and may not have enough time to study all details). E.g. CG and CM have nothing in common conceptually (they coincide only in the homogenous outer field, and not in the spherically symmetric one); and the concept of CG as the apparent point source of the field deserves a separate subtitle. And the resultant force link is inappropriate before the target article makes distinction between the net force and the resultant force. Also, the net gravitational torque may deserve some explanation, so that it becomes clearer that the net weight may or may not be positioned in such a way as to produce an equal torque (and why should weight sometimes be denoted by W and sometimes by F in the same article?). (By the way, spherical harmonics are not likely to be helpful in any circumstance other than a spherically symmdetric one.)--Ilevanat (talk) 00:11, 26 October 2011 (UTC)
 * Glad to hear it! Some thoughts on individual points:
 * CG and CM have nothing in common conceptually
 * I think that's overstating the point.
 * (they coincide only in the homogenous outer field and not in the spherically symmetric one)
 * True, although in the spherically symmetric case, they do coincide for a spherically symmetric body, and the CG limits to the CM as the center of symmetry goes to infinity. I'll work this into the article.
 * the concept of CG as the apparent point source of the field deserves a separate subtitle
 * Maybe. The apparent-source treatment should be expanded upon if it's going to have its own subtitle. Since it's mathematically equivalent to the central-field treatment, I'm skeptical that there would be very much to say about it separately.
 * the resultant force link is inappropriate before the target article makes distinction between the net force and the resultant force
 * Yeah, currently Resultant force just redirects to Net force, which doesn't actually say anything about resultants. This is a shame, since it's a frequently viewed article. Would you like to fix it up? Meanwhile, I'll remove the link for now.
 * the net gravitational torque may deserve some explanation
 * I think this probably should be a separate article, maybe titled Gravitational torque or Gravity gradient torque or even Tidal torque. There are already articles on Tidal locking and Gravity-gradient stabilization, but I don't think these articles cover the topic with enough generality.
 * why should weight sometimes be denoted by W and sometimes by F in the same article?
 * It's a little clumsy, but here's a couple reasons. The cited sources use $W$. Currently $F$ is used for the total vector force, while $W$ is used for the sum of the magnitudes of the forces on all the particles. In the given context of a parallel field, $W$ is the magnitude of $F$, but in general this wouldn't be true.
 * spherical harmonics are not likely to be helpful in any circumstance other than a spherically symmetric one
 * Well, the second harmonic can be useful for tides and oblateness, but I don't see it in engineering sources. I'll remove that phrase.
 * Melchoir (talk) 01:18, 26 October 2011 (UTC)

Thank you for the welcome back note! But as you know, I am stuborn, slow, meticulous, lacking time etc... So, let us start with your first "thought":

I have noticed you had been editinig the CM article - one which is very relevant to this article. And this might be a good starting point for me to articulate few most general considerations regarding wiki articles. As in any encyclopedia, these articles should present all most relevant facts, and also various relevant attitudes/opinions if there are differing ones. However, wiki articles differ from other encyclopedias at least in the following aspects:
 * They are not fully polished and harmonized products of paid professionals.
 * This can be compensated by a more lenghty exposition, the material does not have to be selected for the most concise form. Less relevant things can be included (respecting some order of importance); alternative (and even repetitive) explanations can be included (as long as they are correct).
 * Anybody can change and re-write whatever we agree should be the most appropriate content.
 * It is my experience (however limited) that people will not change (or somebody will revert the changes) the statements that are clear, easy to understand, and supported by comprehensive arguments/proofs. (Of course, the references are important, but they are frequently inconsistent if not contradictory.) The problem here is how to avoid turning an encyclopedia item into a textbook article. But I believe that can be resolved by reasonable limitation of the proof length and complexity.

As an illustration, let me here discuss the center of mass concept. It is my understanding that it has been developed in order extend direct application of Newton laws of motion from point particles to macroscopic bodies (which can be supported by references such as e.g. the Feynman lectures):

The Newton second law (as well as the first) originally refers to a point particle; say, for a particle 1 it can be written as:
 * $$\scriptstyle \vec {F}_{total-on-1} = m_1 \vec {a}_{1}$$

Consider a system of e.g. three particles. Then one adds three such equations to obtain (internal forces cancel, and the total force is the net external force):
 * $$\scriptstyle \vec {F}_{net-ext} = m_1 \vec {a}_{1} + m_2 \vec {a}_{2} + m_3 \vec {a}_{3}$$

It is then obvious how a fictitious point C (the center of mass) should be defined in order to obtain the right-hand side of the above equation after two consecutive time derivatives:
 * $$ \scriptstyle m\, \vec {r}_{c} = m_1 \vec {r}_{1} + m_2 \vec {r}_{2} + m_3 \vec {r}_{3}$$

 * $$ \scriptstyle m\, \vec {v}_{c} = m_1 \vec {v}_{1} + m_2 \vec {v}_{2} + m_3 \vec {v}_{3}$$

 * $$ \scriptstyle m\, \vec {a}_{c} = m_1 \vec {a}_{1} + m_2 \vec {a}_{2} + m_3 \vec {a}_{3}$$

And then we get
 * $$ \vec {F}_{net-ext} = m\, \vec {a}_{C} $$.

The result is that the Newton second law for a particle can be stated in the same form for a system of particles (e.g. for a body), using CM. In addition, the equation with velocities shows that total momentum is (if you prefer to use the other form of the law of motion):
 * $$ \vec p = m\, \vec {v}_{C} $$.

And this is when and why CM appears in the history of physics: so that Newton laws of motion can have a simple formulation for extended bodies (to loosely quote Feynman, the nature was benevolent to physicists). Archimedes did not have the slightest idea about all that. He was analyzing the center of gravity. Which does very well coincide with the CM, but I fail to see any conceptual link.

The above text on CM is intended to explain the origin and meaning of the concept through a brief derivation (and I believe something like that should be presented in the main article). A phrase like "weighted average" should be chipped in as an afterthought, but it does not explain anything (in particular, neither purpose nor properties of CM). And the concept of "material point" should be mentioned as a "fancy mathematical abstraction" that is more likely to confuse a novice than to help.--Ilevanat (talk) 23:50, 28 October 2011 (UTC)

 * I think I agree with your general considerations. I'm not so sure about the historical understanding, though. We have Commandino and Galileo talking about the point through which any plane cuts the body into parts of equal moment. Is that not the center of mass? Feynman doesn't claim that his development follows the historical development; maybe you have a more specific reference? Melchoir (talk) 20:48, 29 October 2011 (UTC)

Oh, I did not mean to implicate Feynman in historical trivia (it is not his field, e.g. I believe he even referred to Magnesia as a Greek island). And I have no idea as to what Commandino and Galileo were talking about, nor do I have any other specific references regarding historical developments. However, I do not see how the concept of CM could have attained its present meaning before the Newton laws of motion became a generally understood and operational knowledge. What could CM possibly designate before that? People could have only been describing the equilibrium properties of material bodies CG (or possibly the centroid of geometrical objects). The present concepts of mass and force were just emerging through the Aristotle haze of misconceptions.

Anyway, I only wanted to underline what the primary meaning of CM is in "serious" physics. Surely, any less specific descriptions (including the "average mass location") can be used in the lead. But later in the body text, a straightforward derivation could be helpful to support the indisputable meaning of CM.

By the way - coming back to CG - if Moon were spherically symetric (which it is not), its CG (in the field of Earth) should be in its center (Symon argument is OK). What about scalar weight arguments that it should be closer to Earth (or did I miss something)?--Ilevanat (talk) 01:52, 30 October 2011 (UTC)

 * Here's all I'm saying:
 * The CM can be defined without reference to Newton's laws.
 * The fact that the CM is useful for extending Newton's laws from particles to extended bodies is then a theorem, not a definition.
 * Writers before Newton did, in fact, make use of such definitions.
 * I wouldn't argue that CM attained its present meaning before Newton. On the other hand, "momentum" didn't attain its present meaning until Lagrange and Hamilton, "energy" until Einstein, "conservation" until Noether, etc. We learn more about these concepts, but we don't throw away their historical motivations. The CM is useful for many purposes, including gravity.
 * So, I'm reluctant to allow any application to be claimed as "the indisputable meaning". At a practical level, of course, we have to choose a layout for any given article, so we have to decide what to discuss first and what to discuss second. I have no objections to discussing Newton's laws before any other application. We should just be careful not to project that editorial decision onto the history.
 * As for the CG of a spherically symmetric Moon, it depends on the definition you're working with. If you approximate the external field as a central field and use Symon's definition, you get the center of the Moon. If you approximate the external field as a variable, parallel field and use Beatty's definition, you get a point closer to the Earth. Of course, both points are on the line of action, so they're equally good for representing torque. Melchoir (talk) 05:19, 31 October 2011 (UTC)

I see you do not like the phrase "indisputable meaning"; and I agree it was not a good choice. I wrote it under wrong assumption that we agree the separation of CG and CM articles is justified by the conceptual difference: And, except for the uniform external field (and possibly for the parallel field, but I would leave that open), and for the spherically symmetric body as a source, the position of CG is not unique relative to the body (or at all) even if it can be determined.
 * 1) CM refers to the center of inertia, e.g. the point in which as if all mass is contained for purposes of the (classical) equations of motion;
 * 2) CG refers to the point where the resultant weight (produced by external fields) may have its application point (if it can be determined), or to the point which is the apparent source of the gravitational field produced by the body (if it can be determined).

It is, of course, possible to take another approach (but then a separate article on CG seems to me to be less justifiable):
 * 1) CG and CM are the same thing in a uniform external field, and for all practical purposes in the field near the surface of Earth as well. Some authors use a concept of a "generalized" CG to account for gravitational torques on a free body in a non-uniform field etc.

And of course the CM can be defined by the "average location of mass formula", apparently urelated to anything else. Any number of irrelevant or meaningless quantities can also be defined unrelated to anything else (and I am sure that some anonimous people have been doing that throughout the history).

But I believe that any people relevant to the history of physics - if they used the "average location of mass formula" or some equivalent before Newton times - had in mind some relevant usage of the concept. Also, I believe it probably had to do something with the application point of the resultant weight and the equilibrium conditions, i.e. with the CG. As I have already mentioned. Though, I cannot be quite sure... so, if you know otherwise, could you please describe an example in appropriate detail?

Anyway, as I am sure you know, I am not going to undertake any editing of the CG or CM article by myself. But I am willing (in response to your suggestion) to write a new article on net force, which I believe should change the title into "Net force and resultant force". However, I shall first propose the new version at the discussion page (and notify you).

But before that, let me revisit the issue of the Moon CG. By Symon definition, if the Moon is spherically symmetric, it acts on any other body (e.g. the Earth, and regardless of the shape of the Earth field) by a gravitational force which "originates" in the center of the Moon. As if the Moon were a point mass located at the center of the Moon. And this is the "source" CG of the Moon. Then using the action-reaction law, it becomes the "weight" CG. There are no assumptions about the field of the other body (i.e. of the Earth).--Ilevanat (talk) 00:59, 1 November 2011 (UTC)

 * I think that's a great description of the difference between our approaches! From what I've read while referencing the History section of the CM article, I think my approach is closer to the historical development. But I have to admit that I don't have a proven case either.
 * The other practical consideration, which I haven't mentioned yet, is incoming links from other Wikipedia articles. Whatever we think is the best conceptual foundation, most authors use "center of mass" and "center of gravity" interchangeably. Right now, a reader who clicks on a link to center of mass will see a different treatment than a reader who clicks on a link to center of gravity, even though the author generally didn't intend for their choice of words to mean anything. We could go through Special:WhatLinksHere/Center_of_mass and Special:WhatLinksHere/Center_of_gravity and try to impose some kind of consistency, but I think that's a losing effort.
 * So yes, I would like to move this article to Center of gravity relative to a field and make Center of gravity a redirect to Center of mass once again. However, I'm not in a rush, and I don't want to claim "ownership" of these articles. As long as no inaccuracies creep in, I'm happy to sit back, wait, and see if any other editors will come through and improve the articles in ways that would make the move unnecessary. If that doesn't work, I'll probably come back someday to try and impose my own preference.
 * Back to the Moon... the Symon definition absolutely depends on the existence of an external reference point - either the location of a test particle or the center of symmetry of a gravitating body like the Earth. In order for the Symon definition to pick out a unique CG on the line of action, we need to be able to measure distances from that reference point. Any discussion of a Symon CG in the absence of a reference point doesn't make sense. Melchoir (talk) 06:06, 1 November 2011 (UTC)

It certainly is helpful to clearly specify editorial preferences; and yours are clearly different from mine. Popular usage, and even perhaps historical development (in some sense), might be on your side. But my "last line of defense" in favor of separate articles (with appropriate cross-referencing already in the lead) and clear-cut conceptual difference between CM and CG is the following:
 * Many of the highly respected university physics textbooks uderline the conceptual difference between CM and CG. Admittedly, some do not. I believe it should be the prevalence of usage within such sources that wikipedia should follow.

As for the Moon... After the derivation of the spherical shell potential, Symon says: "...a spherically symmetric distribution of mass attracts (and therefore is attracted by) any other mass outside it as if all its mass were at its center". Perhaps he was a bit careless there, although I do not see any flaw. If correct, it would mean that the Moon (assumed symmetrical) is attracted by anything (including the Earth) "as if all mass of the Moon were at its center".

But let me make you more comfortable by choosing the center of the Earth as the reference point. Clearly, the field of the Earth (in the vicinity of the Moon) is best appoximated as the inverse square central field originating at that reference point (certainly by many orders of magnitude better than by the "scalar weight" parallel field). So, we are looking for the CG of the Moon while standing in the center of the Earth (where all mass of the Earth can be assumed to be concentrated for the purpose of describing the Earth field). That CG must be in the center of the Moon.

To better see how this conclusion is plausible, consider a thin rod of the legth equal to the radius of the Earth - or, even better, only two equal point masses at its ends (a bar-bell). If you put it vertically on the surface of the Earth, and calculate its CG by the Symon method, the CG is well below its center. But if you rotate it (say arround its center) into the horizontal position, the CG will be considerably above the center. Then one can intuitively guess that in a spherical shell, drawn through the rod endpoints, such CG upward and downward displacements (relative to the center) will cancel out. (The only purpose of this illustration is to avoid the actual spherical shell calculation.)

In conclusion, the statement (ascribed to Assimov) that CG of the Moon in the field of the Earth is below the Moon CM (because the lower portion is more strongly attracted) is clearly false. It might have been obtained in the Beaty "scalar" field, but there is no justification for such inadequate model, when much better field approximation can be used.--Ilevanat (talk) 22:18, 2 November 2011 (UTC)

 * Well, many textbooks mention a difference between CM and CG, but only for a few sentences and not in any detail. Very few textbooks explore CG in its own right; I'm aware of only Symon 1964, Goodman & Warner 2001, and Hamill 2009. The beauty of Wikipedia is that we have lots of space, so we can have an article that explores the intricacies of the instantaneous CG relative to a non-uniform field. But we don't have to make the phrase "center of gravity" link directly to that article when it usually means something else.
 * On the Moon: Don't worry, I agree with you about the facts, and I agree that Symon is correct, and I agree that the parallel-field approximation is a poor approximation. But IF we make that approximation, then the Beatty CG may be calculated, and it is closer to the Earth than the CM.
 * I want to stress two distinctions here. First, the Symon CG and the Beatty CG are incomparable; there is never a situation in which both points are defined. Second, the Symon CG and the Beatty CG make different claims. The Symon CG is a point from which both torque and net force may be calculated as if the body were a point mass. The Beatty CG is a point from which only torque may be calculated, and only if we already know the net force from a separate calculation. It is never claimed that a point mass at the Beatty CG would experience a force or a torque equal to that on the body. (If that claim were made, it would be false for exactly the reason you just described.) All that is claimed is that the Beatty CG lies on the line of force. It does not necessarily enjoy any particular advantage over any other point on the line of force. Melchoir (talk) 01:25, 3 November 2011 (UTC)

We obviously agree on most points, and it is only the matter of how we rank their relevance. And obviously, my ranking is:
 * 1) Coceptual difference between CM and CG is noted in the majority of the serious textbooks (though not in all). When the difference is noted, the concepts are described as different in principle.
 * 2) In practical applications there is little use (if any) for CG, except in cases where it coincides with CM. Therefore, most textbooks do not elaborate CG concept.
 * 3) In popular usage, CM and CG are usually treated as one and the same thing.

I wrote separate CM and CG articles for the Croatian wiki. The CM article ends with a note: "In different wikipedias, however, most frequently only one article appears, entitled either as CM or CG. In some of these articles, the difference between CM and CG is correctly described, in some it is at least mentioned, and in some it is not mentioned at all." So, whatever happens in English wikipedia, I have got it covered.

But I would prefer separate articles, and clearly separated concepts. Let us wait and see if there are any other opinions.

Note 1: In those rare cases when the position of CG (as different from CM) is actually calculated, the best available approximation should be used and explained. However, that is not always the case. For example, one of the widely used textbooks in USA, Young H. D., Freedman R. A., Sears and Zemansky University Physics, Addison-Wesley, claims that CG in Petronas towers in Malesia (about 450m high) is about 2 cm below their CM. But that is just a stand-alone note below a photograph of the towers.

Note 2: I fail to follow your arguments regarding Symon/Beatty. Beaty formula gives all coordinates of the Moon CG in a parallel field, and CG is the unique application point of the resultant gravitational force on the Moon, it is not just any point on the line of force. (Only it is a false field, and it is a false force.) With the assumption much closer to reality, namely that the Earth and the Moon are spherically symmetric, Symon approach uses the law of gravitation, to locate the Moon CG in its center. Again, this CG is the unique application point of the resultant gravitational force on the Moon, due to the action-reaction ("and therefore is attracted by"). There are no requirements that the Moon should be a point mass.--Ilevanat (talk) 01:48, 4 November 2011 (UTC)

 * I think I see the issue. What is the difference between "the unique application point of the resultant gravitational force" and "any point on the line of force"? Melchoir (talk) 02:33, 4 November 2011 (UTC)

Good question! I did try to address it earlier (under the setion "New lead", I believe). If we are talking about the application point of a resultant force in general, there is no difference (there is no "unique" application point, there is only the line of action). But if we are talking about the weight CG in the uniform external field - the usual approximation of the Earth field, in which the CG concept had been developing since the ancient times (before any clear concept of mass, let alone its center) - it is associated with the well-known considerations of static equilibrium. And in that context CG is the unique application point of the resultant weight (if the body is supported above CG, it is in the stable equilibrium, if ...etc.)

It is for me an unresolved question whether external field CG in general should be associated with such equilibrium considerations, but I am inclined to assume so. If not, then the Beatty formula ambition to give the vertical CG coordinate is useless/meaningless, as are any applications of the Symon approach to the external field CG (Symon formula should then be strictly applied to the apparent field source location point, which can also be called CG, but has nothing to do with the extarnal field CG). The torque requirement is all that is needed to determine the line of action, when there is one.

And it is because of that dilemma - whether external field CG should be associated with any equilibrium considerations - that I declined to engage in the editing of the article.

And that leaves us with the open question: is it correct to claim that the Moon CG is closer to the Earth than its CM (and if it is, on what grounds)? And I do not know the answer.

At the end of the above section, I supported the Symon careless claim about action-reaction, which would lead to the conclusion that the Moon CG in the Earth field is in the center of the Moon. But after additional considerations in the meantime, I am inclined to think that this may be not correct. Which does not mean that Beatty approximation is good enough.

My "additional considerations" consisted of a simple calculation: two equal point masses, one at the surface of the Earth, and the other one Earth radius (R) above (a vertical barbell). The Beatty formula gives the vertical coordinate of the barbell CG at 1.2R from the center of the Earth. And this CG satisfies equilibrium conditions (for any support above this CG, the barbell is in stable equilibrium). (And the Beatty field is entirely correct for this example.) Symon formula gives the barbell CG at 1.265R from the center of the Earth. This CG is the apparent source of the barbell field as seen from the center of the Earth. But now I do not think that the Newton 3 law is sufficient support to claim that the same point is the barbell CG in the Earth field. Any thoughts?--Ilevanat (talk) 03:02, 5 November 2011 (UTC)

 * Okay, let me just focus on the "stability CG" you're talking about. I like this concept. Let's calculate its location in the case of a vertical barbell. If mass $m_{1}$ is at radius $r_{1}$ from the center of the Earth, and mass $m_{2}$ is at radius $r_{2}$, then we are looking for a point $h$ from the center of the Earth such that the second derivative of the potential energy when rotating both masses about $h$ is zero. The trivial solution is $h = 0$. If my math is correct, the other solution is:
 * $$\frac{m_1}{r_1^2} + \frac{m_2}{r_2^2} = h\left(\frac{m_1}{r_1^3} + \frac{m_2}{r_2^3}\right).$$
 * Setting $m_{1} = m_{2}$ and $r_{1} = R$ and $r_{2} = 2R$, we have $h = 10R/9 = 1.111R$. So in this case, the stability CG is lower than both the Symon CG and the Beatty CG.
 * While the stability CG is an interesting point, it's not the same as the points described in the available references. And that's just fine, because the available references don't make any claims about stability. Melchoir (talk) 09:50, 9 November 2011 (UTC)

 * Quick explanation of the above formula: the potential energy of a particle is proportional to
 * $$\frac{m}{r}$$.
 * The first task is to determine the change in $r$ when the particle is pivoted about $h$. Some square roots are involved, but it turns out to be:
 * $$\frac{d^2}{d\theta^2} r = \frac{h(h - r)}{r}$$.
 * This only holds around $θ = 0$, but that's all we need. Then we find
 * $$\frac{d^2}{d\theta^2} \frac{m}{r} = \frac{-h(h - r)m}{r^3}$$.
 * Again this only holds at $θ = 0$, since it requires the first derivative to be zero. I may have messed up the minus signs somewhere along the way, but the formula behaves correctly: it vanishes at $h = 0$ and $h = r$, and its units are mass over length. Melchoir (talk) 03:21, 23 November 2011 (UTC)

I did give it a try. After pivoting about $h$, you could have calculated $r$ from the cosine theorem. All expressions you found for the upper $r$ and the respective potential energy can be reproduced assuming you took $θ = 0$ limit after claculating the derivatives, approximating also "pivoted" $r$ with the starting value. It does look reasonable, though I do not feel overconfident.

However, for the lower $r$, the cosine term has the opposite sign. As it is the only term that actually survives in the final expressions for potential energy second derivatives, the contributions from the upper and lower mass have opposite signs. This changes your final expression into:
 * $$\frac{m_1}{r_1^2} - \frac{m_2}{r_2^2} = h\left(\frac{m_1}{r_1^3} - \frac{m_2}{r_2^3}\right).$$.

And any solution for equal masses barbell puts $h$ below the barbell, e.g. at (6/7)$R$ in the example considered. Which is clearly unacceptable.

The problem with the "high level" considerations, such as the potential energy derivatives above the first order, is that they are less transparent to "common sense" or intuition (unless you have been doing that for decades, which I have not), so that combinations of "obvious approximations" and elementary differentiation rules can easily lead to nonsense. It was a nice try, and I am sorry I cannot suggest a constructive improvement.--Ilevanat (talk) 01:22, 25 November 2011 (UTC)

By the way, since we are in no hurry, let me ask the authors of "Young H. D., Freedman R. A., Sears and Zemansky University Physics, Addison-Wesley" how they calculated that the CG in Petronas towers in Malesia (about 450m high) was about 2 cm below their CM. Just out of curiosity.--Ilevanat (talk) 01:36, 25 November 2011 (UTC)

Stability calculations

 * Well let's get this straight. I claim that
 * $$\frac{d^2}{d\theta^2} \frac{1}{r} = \frac{-h(h - r)}{r^3}$$
 * with no restrictions on $h$. The sign changes in this formula reflect the fact that the equilibrium is unstable in the region $0 < h < r$. What's your claim? Melchoir (talk) 07:37, 25 November 2011 (UTC)

My "claim" was that this expression could be derived for $h < r$. But for $r < h$ it changes the sign, i.e. it becomes
 * $$\frac{d^2}{d\theta^2} \frac{1}{r} = \frac{h(h - r)}{r^3}$$

under the same limit procedure.--Ilevanat (talk) 11:49, 25 November 2011 (UTC)

 * Then your formula has a positive value for both $0 < h < r$ and $r < h$. Correct? Melchoir (talk) 20:18, 25 November 2011 (UTC)

Correct! Except that it is not "my" formula, it is only my attempt to reproduce your formula. And, unlike you, I believe I stated clearly how it was arrived at:

"After pivoting about $h$, you could have calculated $r$ from the cosine theorem. All expressions you found for the upper $r$ and the respective potential energy can be reproduced assuming you took $θ = 0$ limit after claculating the derivatives, approximating also "pivoted" $r$ with the starting value."

Plus:

"However, for the lower $r$, the cosine term has the opposite sign. As it is the only term that actually survives in the final expressions for potential energy second derivatives, the contributions from the upper and lower mass have opposite signs."

Is there anything unclear in these statements? Of course, you might have used some other procedure, but why would you not describe it?

However, if you only believe in the "magic power" of the "final formula", we are unlikely to reach consensus.--Ilevanat (talk) 00:03, 26 November 2011 (UTC)

 * Okay, now let's talk about the physics. If a mass at $r$ is suspended at a point $h$ where $0 < h < r$, then the equilibrium is unstable. But if $r < h$, then the equilibrium is stable. Agreed? Melchoir (talk) 04:05, 26 November 2011 (UTC)

Agreed, provided we have a rigid massless extension to the point where it is suspended.--Ilevanat (talk) 01:55, 27 November 2011 (UTC)

 * Awesome, I was afraid I'd have to draw some diagrams and upload them to Commons or something. :-) The reason I asked these questions is to establish physical plausibility. I gather that you don't see how my formula was derived, so I'll go into more detail now. I'll number the steps so that we can refer to them later if necessary.
 * Let there be a primary mass $GM = 1$ at the origin. Its gravitational field is $$-\frac{1}{r}$$
 * Let there be a secondary mass $m$, initially at a distance $r_{0}$ from the origin. We treat $r_{0}$ as a constant. Its potential energy is initially $$V_0 = -\frac{m}{r_0}$$
 * Now suppose that $m$ is rotated by an angle $θ$ about a point $h$ away from the origin, on the line connecting the origin to $m$. Then its new position vector is $$\mathbf{r} = h + (r_0 - h)e^{i\theta}$$
 * The distance from the origin is $$r = \left|\mathbf{r}\right| = \left|h + (r_0 - h)e^{i\theta}\right|$$
 * $$r = \sqrt{h^2 + (r_0 - h)^2 + 2h(r_0 - h)\cos\theta}$$
 * We start expanding around $θ = 0$: $$r = \sqrt{h^2 + (r_0 - h)^2 + 2h(r_0 - h)\left(1 - \frac{1}{2}\theta^2 + O(\theta^4)\right)}$$
 * $$r = \sqrt{r_0^2 - h(r_0 - h)\theta^2 + O(\theta^4)}$$
 * $$r = r_0 \sqrt{1 - \frac{h(r_0 - h)}{r_0^2}\theta^2 + O(\theta^4)}$$
 * $$r = r_0 \left(1 - \frac{h(r_0 - h)}{2r_0^2}\theta^2 + O(\theta^4)\right)$$
 * $$r = r_0 - \frac{h(r_0 - h)}{2r_0}\theta^2 + O(\theta^4)$$
 * Now we calculate the derivatives of the distance from the origin: $$\frac{dr}{d\theta} = -\frac{h(r_0 - h)}{r_0}\theta + O(\theta^3)$$
 * $$\frac{d^2r}{d\theta^2} = -\frac{h(r_0 - h)}{r_0} + O(\theta^2)$$
 * Now we calculate the derivatives of the potential energy: $$\frac{dV}{d\theta} = \frac{d}{d\theta}\left(-\frac{m}{r}\right)$$
 * $$\frac{dV}{d\theta} = \frac{m}{r^2}\frac{dr}{d\theta}$$
 * $$\frac{d^2V}{d\theta^2} = \frac{m}{r^2}\frac{d^2r}{d\theta^2} + \frac{-2m}{r^3}\left(\frac{dr}{d\theta}\right)^2$$
 * Substitute and simplify: $$\frac{d^2V}{d\theta^2} = \frac{m}{r^2}\left(-\frac{h(r_0 - h)}{r_0} + O(\theta^2)\right) + \frac{-2m}{r^3}\left(O(\theta)\right)^2$$
 * $$\frac{d^2V}{d\theta^2} = \frac{m}{r^2}\left(-\frac{h(r_0 - h)}{r_0} + O(\theta^2)\right) + O(\theta^2)$$
 * $$\frac{d^2V}{d\theta^2} = \left(\frac{m}{r_0^2} + O(\theta^2)\right)\left(-\frac{h(r_0 - h)}{r_0} + O(\theta^2)\right) + O(\theta^2)$$
 * $$\frac{d^2V}{d\theta^2} = -\frac{h(r_0 - h)m}{r_0^3} + O(\theta^2)$$
 * Finally we evaluate at $θ = 0$: $$\left.\frac{d^2V}{d\theta^2}\right|_{\theta=0} = -\frac{h(r_0 - h)m}{r_0^3}$$
 * Okay? Melchoir (talk) 07:35, 27 November 2011 (UTC)
 * (I just noticed a mistake in the very last term of steps 15-19. The term vanishes at the end, so the mistake doesn't affect the result, but I've corrected it inline.) Melchoir (talk) 20:12, 27 November 2011 (UTC)

I am really impressed by so many perfectly formatted lines, you must be very proficient in wiki math text editing. As for the graphical illustration, may I offer this one?

And now, back to the issue. I did not actually question the partial result you reproduced here, with the assumptions and limits used (though I was not certain about its applicability) - see my repeated statement above "After pivoting about $h$...". I drew something similar to the middle of the illustration here, and started with your line 5 (cosine th.) having in mind the upper mass at B. I left the $r$ squared on the left side, took all derivatives with all sines and cosines (knowing the first derivative of $r$ will vanish), and it took me somewhat fewer lines to get your result at the line 20 after the limit $θ = 0$.

And then I made a stupid mistake for which I appologize: looking at the lower point (A in the illustration), I only noted that the cosine term should contain the supplement angle, but overlooked the change of sides in $r - h$. So, forget about all my claims about "positive value for both $0 < h < r$ and $r < h$". Sorry that lead you to such a lengthy answer, I certainly was careless (and take all the blame), but there also was some general faillure in communication.

So, you clearly were right about the final V formula for the barbell. But I am still skeptical about its applicability, let me have another thought.--Ilevanat (talk) 02:16, 28 November 2011 (UTC)

 * Okay, thanks! I prefer to blame the medium for the failure of communication; if we were standing in a room together with a blackboard, I'm sure we would have cleared this up in a couple minutes. :-)
 * While you're thinking over the applicability of these calculations, you might also want to reconsider the right-hand side of the above diagram, File:Thin rod center of gravity.JPG. If a barbell (or rod) is allowed to rotate about its center of mass, then its potential energy is minimized in the vertical orientation and maximized in the horizontal orientation. In other words, a horizontal barbell supported at its center of mass is in an unstable equilibrium. We can define a "parallel stability CG" for rotations in the plane of the barbell, and I have some observations about this point:
 * The parallel stability CG will lie above the CM (which is the same as the Beatty CG).
 * We can also define a "perpendicular stability CG" for rotations out of the plane of the barbell; this point will coincide with the CM.
 * Without doing the math, I don't know if the parallel stability CG will coincide with the Symon CG, but given our experience with a vertical barbell, I doubt it!
 * Melchoir (talk) 03:07, 28 November 2011 (UTC)

Let me address here (still in an introductory way) my skepticism about the applicability of your result. (For your latest comments, I may need additional clarifications.) We have been discussing equilibrium stability for an extended rigid body, specifically for the simple case of a vertical barbell consisting of two point masses (connected by a massless rigid rod).

We are looking for a point of support (the point at which the body can be suspended) to achieve the indifferent equilibrium (higher/lower points of support lead to stable/unstable equilibria). The body (the barbell) is pivoted for a small angle arround the point of support in a vertical plane. In simple physical terms, it seems to me the equilibrium under consideration is the equilibrium of torques produced by the weights of the upper and the lower barbell point mass: for the indifferent equlibrium, we should have equal but opposing torques; therefore, the position of the support point must be somewhere between the upper and the lower mass. The result obtained may only depend on the details of the vanishing angle limit procedure.

To the first order differentials, the lever arms are proportional to the original distances of the respective masses from the support (pivoting) point, multiplied by the SAME pivoting angle. In the torque equation (total torque equals zero), the angle will cancel out, you do not even have to go for $θ = 0$. In the same approximation, the weights of displaced masses will be equal to the original weights, you only neglect $O(θ^2)$ order of magnitude, and again you do not have to go for $θ = 0$. Then, in the example of equal point masses at R and 2R, you get the pivoting point at 1,2R. (The weights are 4:1, so the lever arms must be 1:4.)

Now, how is it that we get different result using the potential energy derivatives? This result does not seem to be based on higher order approximations; and even if it were, the difference obtained is too big for the second order contribution.

Still, I tend to believe one should get the same result from both the torque and the energy considerations. But the first energy differential (see your lines 14 + 11) is not equivalent to infinitesimal work of the torque I described under the assumed infinitesimal pivoting. And I do not yet see the explanation.--Ilevanat (talk) 00:59, 29 November 2011 (UTC)

On the second thought, the "catch" might be in the force applied at the point of support. Such external force would be consistent with the torque analysis (it does not produce any torque around that point) and we do not consider the barbell is in free fall.--Ilevanat (talk) 01:12, 29 November 2011 (UTC)

 * I believe the faulty claim is "the lever arms are proportional to the original distances". We're interested in the first derivative of the torque, which can be expressed like so:
 * $$\frac{d\boldsymbol{\tau}}{d\theta} = \frac{d}{d\theta}\left(\mathbf{r} \times \mathbf{F}\right) = \frac{d\mathbf{r}}{d\theta} \times \mathbf{F} + \mathbf{r} \times \frac{d\mathbf{F}}{d\theta}.$$
 * The first term is certainly proportional to the distance from the pivot. However, we can't neglect the second term, which is comparable in magnitude for a strongly curved field. Melchoir (talk) 06:54, 29 November 2011 (UTC)

I do not see why one should be interested in derivatives in the torque equation. But never mind that. I decided to do the exact torque equilibrium for a simple geometry. "Our" barbell of length R is positioned so that the lower mass is at distance R from the center of the Earth and is inclined so that the position vector of the upper mass forms a small angle $θ$ with the position vector of the lower mass (the vectors intersect at the center of the Earth). The barbell is supported (suspended) along its "rod" at the distance d from the lower mass.

Equating both torques with respect to the support point gives the result:
 * $$8d\cos^3\theta = R - d.$$

Obviously, in the limit $θ = 0$ this gives your reult for CG (d=0.11R). On the other hand, if the angles between the gravitational forces are neglected (Beatty field), one gets
 * $$4d\cos\theta = R - d.$$

This gives the Beatty CG (d=0.2R) for $θ = 0$. I got the same result (section above) by effectively ignoring the small "horizontal" component of the upper weight - which was wrong, because it has a long lever arm.

In summary:
 * Indifferent equilibrium point of support for our barbell can be determined along the barbell "rod" for varying inclinations, and its position (d) is a function of $θ$.
 * Beatty result is inadequate for large bodies in the Earth field.
 * Symon result is not the equilibrium CG for a body in an external field.
 * Your result is correct for $θ = 0$. It mostly bothered me because the deravition is done for each mass separately: I still do not see how the constraint to their simultaneous motion (the rod connecting them) is incorporated. Which obviously does not matter in the limit $θ = 0$.--Ilevanat (talk) 01:46, 1 December 2011 (UTC)

 * Aha, we seem to be using conflicting notation.
 * My $θ$ is the angle through which each part of the body is rotated about the pivot point $h$. This definition of $θ$ is given in step 3 above, resulting in the equation $$\mathbf{r} = h + (r_0 - h)e^{i\theta}.$$ Since the barbell is rigid, this angle is the same for both masses. That's how the constraint is incorporated, and that's why the derivatives for each mass can be computed separately and then added.
 * Your $θ$ is a polar coordinate referred to the center of the Earth. If we adopt this convention, then $$\mathbf{r} = r_0e^{i\theta}.$$ This angle is different for each part of the body, and there is no simple relation between $θ_{1}$ and $θ_{2}$.
 * I presented an equation for the derivative of the torque because you brought up the first order differentials of the lever arms. Note that the second term of the equation is precisely the component that you had effectively ignored earlier: $r × dF$ is a small horizontal component of the weight, multiplied by a long lever arm.
 * In any case, I'm glad that we're converging on our conclusions. One loose end to tie up: If the barbell is oriented horizontally, then its indifferent-equilibrium point of support lies above its center of mass. Agreed? Melchoir (talk) 02:51, 1 December 2011 (UTC)

Agreed, intuitively, your type of pivoting. As for the rest, I was aware of differences in $θ$, but asked a different question: for a given position of the barbell (no matter how arrived at), where is the point of application for the resultant weight along the rod? But the conclusions I hastily wrote (have very little time) need reconsideration for non-zero $θ$: the point of support seems to be the point for unstable equilibrium (for greater d, the barbell will swing towards vertical position).--Ilevanat (talk) 12:22, 1 December 2011 (UTC)

 * I'm not totally clear on what you're saying here... I guess you might have the time to elaborate later?
 * Meanwhile, I'd like to explore an earlier argument in the context of the vertical barbell. We've defined four different points:
 * The center of mass at 1.5 R
 * The Symon center at 1.265 R
 * The Beatty center at 1.2 R
 * The stability center at 1.111 R
 * You've suggested that the Beatty center is less correct than the Symon center. My argument is that they're simply different points, and since neither point is particularly meaningful in terms of stability, neither point is privileged over the other. The only property that the Symon center and the Beatty center share with the stability center is that all three points, when they exist, lie on the line of action. In this sense, all three have equal claim to being a center of gravity, or even "the" center of gravity.
 * Unfortunately, I haven't seen a published source that describes the stability center as the center of gravity. But we do have sources for the Symon center and the Beatty center, so this Wikipedia article should describe both, without favoring either. Melchoir (talk) 23:46, 1 December 2011 (UTC)

Briefly (for now): Agreed about describing Symon and Beatty. However, remember, classical "textbooks such as the The Feynman Lectures on Physics characterize the center of gravity as a point about which there is no torque. In other words, the center of gravity is a point of application for the resultant force etc" (a quote from the article). For "our barbell" the equation
 * $$8d\cos^3\theta = R - d$$

gives the intersection point of the line of application of the resultant weight with the barbell rod (no approximation used), for whatever it may be worth (e.g. as a point of support for the barbell, it will ensure at least an unstable equilibrium). If we agree about that, we may consider the implications.--Ilevanat (talk) 01:25, 2 December 2011 (UTC)

 * Yes, certainly the article should also describe the CG as a general point on the line of action. I didn't mean to imply otherwise; I was just talking about which particular points on the line of action should be given special attention.
 * I haven't worked out the equation for $d$, so I'll assume your equation is correct. Actually I think I know where this is going, but I'll shut up and let you continue. :-) Melchoir (talk) 02:03, 2 December 2011 (UTC)

If you happen to know where this is going, you are ahead of me (which may not be a particular achievement, given my age and slowlyness). Anyway, let me elaborate a bit on "my equation" for $d$. The derivation begins by equating the torque from the lower weight $$\frac{m}{R^2}$$ times its lever arm with the torque from the upper weight $$\frac{m}{4R^2\cos^2\theta}$$ times its lever lever arm, which gives:

 * $$\frac{m}{R^2} d \sin2\theta = \frac{m}{4R^2\cos^2\theta}(R-d)\sin\theta$$

The result is seen by inspection. This simplifying geometry is achieved by rotating "our barbell" for $2θ$ arround the lower mass (from its vertical position). Regardless of how this barbell position is achieved, we have a unique intersection point of the resultant weight line of application with the barbell rod. Agreed?--Ilevanat (talk) 02:26, 3 December 2011 (UTC)

 * Sure! Melchoir (talk) 09:06, 3 December 2011 (UTC)

Arbitrary subsection break
Sorry for delay, I was away for the weekend. And what is worse, my schedule is getting so busy for the rest of the month that I probably will not have time for any serious consideration of our discussion. Appologies!

I would not want to make any hasty judgements prone to mistakes, as I have already been doing in the last few weeks. As I mentioned earlier, CG is not "my field" (if it is anybody's), and our brief exploration here borders with "unconfirmed research" (at an almost irrelevant and rather trivial level - this I do not say pejoratively, but just in order to illustrate the fact that I cannot raise this topic in conversation with my friends physicists working on e.g. black holes and dark matter).

Still, I believe the concept of "stability CG" is worth further consideration in the context of wikipedia (or any ecyclopedia) definition of CG, because the issues of stability in gravitational field were (as far as I can tell) the primary motivation for development of the CG concept (and of any useful - and synonymous - CM concept before the Newton laws).

Therefore, I am glad that we arrived at the same "stability CG" for a vertical barbell in the Earth field by different approaches, but I am not yet fully confident in its significance. Except for the fact that (for the same body in the same field) it is clearly different from any potential application of the Symon concept. As for the Beatty CG, I believe it is also the stability CG but for a different (parallel) field.

So much for now. I will be glad to read your thoughts and at least briefly comment.--Ilevanat (talk) 01:05, 6 December 2011 (UTC)

 * No problem! Yeah, CG doesn't seem to be anyone's field, in that there isn't a secondary source that summarizes all the primary sources we're currently citing in this Wikipedia article. If there were, then organizing the article would be much easier!

 * I agree that the stability of objects, in a broad sense, is an important motivator for the CG concept. However, it's not clear that the "stability CG" as we've defined it is precisely the concept that's in use. For the historically important example of a floating vessel, in Metacentric height, an analogy is made with a pendulum suspended from a point, which sounds promising. But the analogy isn't precise, and the whole theory assumes a uniform field. In the context of a curved gravitational field affecting both the water and the vessel, I don't know how all of the points being discussed would generalize. Maybe the stability CG would be relevant, and maybe not.
 * For the example of an artificial satellite in Earth orbit, the stability CG probably isn't relevant, since the satellite isn't being supported by a force at any particular point.

 * Back to Wikipedia's purposes, if there isn't a source that clearly states the "stability CG" definition, I don't think we can state it either. I'm really glad that you mentioned "unconfirmed research", because that's an important editorial principle (Verifiability, No original research). I've thumbed through quite a few books, in person and online, and I don't remember ever seeing this definition. Melchoir (talk) 06:03, 7 December 2011 (UTC)

Hi, I cannot but generally agree with all you said, despite my notorious obsession with the relevance of stability aspects. And yes, "CG doesn't seem to be anyone's field". I am still waiting for any answer from the authors/editors of the "Young H. D., Freedman R. A., Sears and Zemansky University Physics, Addison-Wesley"; they advertise true commitment to communication with teachers, but no response yet. Anyway, I hope to find some time for a more meaningfull correspondence during the holiday season.--Ilevanat (talk) 00:46, 13 December 2011 (UTC)

Perhaps it is time to wrap it up
I mean both CG and CM articles. I did some additional literature search on CG, but it did not help. I also did some search on the "concept of mass" history, and it did help to establish a perspective. So, let me start on a rather general level, and proceed to details in a few following days.

I still believe that both items should be kept as separate articles. In both of them it should be pointed out at the very beginning that it is quite customary to use the terms as synonyms, but that they are conceptually as well as historically distinct.

According to Merriam-Webster, the first known use of the term CM dates from 1862. Which is reasonable: the very term "mass" (in physics) was less than a century old. Newton still used the "quantity of matter", and any real understanding of the inertial and gravitational properties of matter date from about his time (see e.g. Max Jammer, his older book). The Old Greek knew only about "weight", the property of some materials (not of e.g. hot air).

According to Merriam-Webster, the first known use of the term CG dates from 1648. Webster does not date the origin of barycenter (obviously, the center of "heaviness/weight"). In contemporary usage it is often reserved for the CM of 2 celestial bodies orbiting each other (see e.g. the free online dictionary, although Webster uses it as a synonym for CM in general), but I guess the ancients may have used it for the CG they studied (though that is hardly relevant). Their CG clearly was the point of equlibrium for supporting the weight of the body. They undrestood nothing about the force of gravity, and even less about concept of mass: so, what else could it have been?

Then, if we can agree about this context, we can proceed with the discussion of each article.--Ilevanat (talk) 00:54, 3 January 2012 (UTC)

 * We agree about the language used, but we still don't agree about the scope of the concepts... Let me try an argument that's relevant to the way the Wikipedia articles should be organized. I claim that whenever anybody refers to a CG with the understanding that it's a fixed point in relation to a body, regardless of the body's orientation or position, they are talking about the CM. Agreed? Melchoir (talk) 20:19, 4 January 2012 (UTC)

 * I agree that they're synonymous, but distinguishable. I think that one article that covers both concepts should be created. Merging synonymous concepts is normally done, even if they're not perfectly synonymous, as here. The articles aren't too big to do this, and the merged article would be smaller than their sum.Teapeat (talk) 03:18, 5 January 2012 (UTC)

Agreed. Almost: I believe that some tens of thousands of my students would at least wagely recall that CG "practically" coincides with CM (though probably not many of them would remember any further details, even about CM that was extensively used). But those who did not have any physics beyond highschool are certainly in the category you described.

So, if we do not want to "perplex" that large majority, maybe a single article would be more appropriate. On the other hand, if do not want to "perplex" those who even wagely remember that there may be some differences in the physical concepts, perhaps that single article shoud have the title "Center of mass and center of gravity". Unless it is contrary to some Wikipedia policy?--Ilevanat (talk) 01:14, 5 January 2012 (UTC)

 * I am glad that other people are joining our discussion (Teapeat). Though I do not enthusiastically support formulations like "synonymous, but distinguishable", I can generally agree with the proposal. (And let me appologize for the above repeated misspelling of the word "vaguely".)

 * It may now be the time to discuss the starting sentences of this unified article. Do not take my English seriously, but would strongly prefer something with the following meaning:

 * "In common usage, CM and CG denote the same point, in which as if the entire mass and weight of a body is concentrated for purposes of various calculations in mechanics. In physics, however, CM and CG are two distinct concepts with differring history."--Ilevanat (talk) 01:14, 6 January 2012 (UTC)

 * Well, hang on, I have a more conservative proposal. I don't necessarily want to see Center of gravity merged with Center of mass. Back in October, Center of mass was very long and overrun with minutiae. I split it into a number of summary-style daughter articles: Derivation of the center of mass, Locating the center of mass, and Barycentric coordinates (astronomy). Likewise, I'd like to see the current material at Center of gravity kept intact, and even expanded, but moved to a new name: Centers of gravity in curved fields or something similar. It would be linked to from Center of mass in the appropriate section, just like the other related articles.
 * Does that make sense? If I'm not being clear, I could implement the idea very quickly for the purpose of demonstration...? Melchoir (talk) 23:34, 6 January 2012 (UTC)

But when someone types "center of gravity" in the search box, where does it lead him? Or what choices does he see? (P.S: "curved field" should rather be "non-uniform" or "non-homogenous".)--Ilevanat (talk) 01:06, 9 January 2012 (UTC)

 * In both my proposal and in the event of a merge, Center of gravity would be a redirect to Center of mass. And in both proposals, the target article would have a section that presents the idea of a center of gravity defined relative to an external field.
 * The difference is that, if we avoid a merge, then the present Center of gravity article can stay at the length of a full article, having several sections that compare the various definitions. The only real change is that the 300 articles that link to "Center of gravity" will be redirected to CM. Melchoir (talk) 01:45, 9 January 2012 (UTC)

For most of those 300 articles, CM would probably be a better link anyway. The present article on CG should perhaps be named "Center of gravity concepts in physics" (Symon CG starts out as a field source, rather than as a point in external field).

However, in the case of CG redirect to CM, I would really hate to see the article begin as it does now. We can discuss various fine details later, but I strongly believe that the first lead paragraph should contain something close to those two sentences I proposed above. If not, why bother with any additional considerations of CG (let alone articles)!--Ilevanat (talk) 23:55, 9 January 2012 (UTC)

 * Okay, I gave it a try: hopefully this is a good start. I want to avoid the word "however", which feels argumentative in the way it emphasizes one sentence over another. I'm not sure that "differing history" is justified at this point, so I left that out as well. Melchoir (talk) 11:02, 10 January 2012 (UTC)

Yes, it could be a good start. Give me a day or two for comments.--Ilevanat (talk) 01:42, 11 January 2012 (UTC)

After some consideration, my conclusion is: there are many minor points in the lead and later in the CM article that could be discussed/improved. But much of that will depend on the point of view regarding CM and CG concepts and their history.

Of course, I fully agree (or have come to agree) that the present common usage should be given priority (no matter how superficial or confusing it may be). And you seem to be willing to acknowledge the conceptual differences. So, there only remains the question of historical development, which is necessarily related to the conceptual differences.

One approach is to ignore the history, or to misrepresent it as is presently done in the CM article with the claim that Achimedes was studying CM. (Yes he was, if CM and CG are one and the same concept! But then again, if we go along with that, why bother with any other distinguishing details?) This is the most easy approach, and it may be sufficient for Wikipedia. After all, any more subtle considerations may later be reverted or re-edited by well-intended people reading casual formulations that dominate the literature.

Yet, let me for this once, outline some historical data as I see them (and, please, correct me if you have any evidence to the contrary):

The early Latin terms equivalent to "gravity" and "center of gravity" (and Greek terms "barus" and "barykentron", or something like that in Greek alphabet) denoted "heaviness/weight" and "center of weight". Achimedes was studying balancing of lever arms and static equilibrium of body weights (in present terms we would use "the torque equation") and then extended his "barykentron" concept to geometrical figures (having in mind, I would guess, bodies with homogenous distribution of weight).

The idea that matter may have a general property of gravitational attraction was first mentioned by some Arab thinkers around the year 1000 AD (primarily as attraction among the celestial bodies, I believe). But the concept was seriously considered only shortly before Newton and in his time.

The idea that matter has a property of inertia is contrary to Old Greek concepts, and became generally accepted only at Gaiileo/Newton times.

So, what about mass and CM? Newton was still using "quantity of matter", but his "matter" had both inertial and gravitational properties, and it is only from about his time that the concept of "mass" (although the term was introduced a bit later) can have any meaning, regardess of how it was called. Before that, "matter" just did not have the properties that would relate it to mass. Nor did "heaviness". Therefore, CM is indeed a post-Newtonian concept (though perhaps a bit older than Webster claims). And it must have been intended to address the inertial properties of matter: why else would anybody introduce a new term if they had in mind "center of haviness"?

You can see that some of my above arguments border with guesswork. Why? Because "the application people" use CM and CG as synonyms, and the easily accessible literature generally reflects this practice. Therefore, it would take me too much time and effort to find those rare authors who care about distinctions in conceptual and historical developments. But I doubt physics would have developed to its present level if the "real guys" were following practice of the "application people".

Anyway, in our CM and CG articles we can choose to stick mainly with the "common usage"; or, we can attempt to introduce more content that reflects the "indisputable real meaning" (yes, there is such a thing!) of the physical concepts, presently and through the history.--Ilevanat (talk) 01:12, 13 January 2012 (UTC)

 * As far as I know, you're absolutely right about the early history. Still, I don't think it forces us to choose between common usage and the truth. I think with a careful framing of the issue, we can respect both...
 * I haven't studied Archimedes, but as I understand his work:
 * He was studying gravity and weight. For him, weight was a scalar quantity and an intrinsic property of a body. Today we understand that mass is the intrinsic scalar. Weight is a vector which depends on the configuration of distant objects.
 * He understood the CG as a point which always uniquely exists and is painted onto an object. Today we understand that the CM uniquely exists and is fixed. The CG usually doesn't exist, usually can't be uniquely defined, and is not fixed.
 * He derived the law of the lever. Today we combine the law of the lever with vector arithmetic to define the CM. The CG doesn't obey any such law, and in fact, the Symon CG can lie completely outside the convex hull of the body.
 * It would appear that his CG is more like our CM than our CG, at least mathematically. On the other hand, you've already demonstrated that Archimedes' CG concept cannot simply be identified with the modern CM concept, because the foundations are separate. So what was Archimedes really studying?
 * The root problem, as I see it, is that Archimedes' treatment of the CG is inconsistent. (Many modern treatments are also inconsistent, but for other reasons.) He imagined a concept that both relates to gravity and also obeys certain convenient laws, which we now know are incompatible! His concept is too large to be governed by any precise definition. And so, like an unstable empire, the old CG has balkanized into a constellation of sub-concepts that compete to be considered its successor. There's the modern CM and the torque CG, and the latter is itself split into at least two pieces, the Symon CG and the Beatty CG. Meanwhile, non-physicists who don't care about linguistic politics are happy to nominate the CM as their preferred "center of gravity".
 * (I don't claim to have any evidence of how these splits happened historically.)
 * The practical lesson for Wikipedia is that we should be careful to avoid suggesting that any modern treatment was envisioned by Archimedes. Fortunately, that's just a matter of fine-tuning the language. For example, I just made this edit to Center of mass#History, which I hope you'll agree is a step in the right direction.
 * Meanwhile, I'd really like to sort out the article titles. I'll go ahead and move this article and redirect Center of gravity. If you, Teapeat, or anyone else feels this is premature or unwarranted, feel free to revert me or ask me to undo it myself! And we can still continue to discuss exact titles, body text, and further merges and splits on the various talk pages. Melchoir (talk) 08:46, 13 January 2012 (UTC)

For the beginning, let me de-mistify Archimedes. A number of accounts of his works can be found on the web, including a very comprehensive one by T. L. Heath from 1897. Archimedes introduces CG in the book ON THE EQUILIBRIUM OF PLANES OR THE CENTRES OF GRAVITY OF PLANES. A reasonable discussion of his assumptions can be found at http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/De%20Planorum%20%20Aeqilibriis/Intro.I.Props1-5/Intro.I.Props1-5.html#note0. Please, note his formulations (copied from that site), such as:
 * Proposition 1: Weights that incline-equally from equal lengths are equal.
 * Proposition 2: Unequal weights from equal lengths do not incline-equally, but will incline towards the larger.
 * Proposition 3: Unequal weights will incline-equally from unequal lengths and the larger weight from the smaller length.
 * etc.

He generally thinks in terms of weights (magnitudes) and lightness, and his considerations are rather incomplete. Note the comment from the above web-source:

"...Archimedes never gives a definition of ‘center of weight’, and the text has been criticized for this omission. He also never defines 'inclination’ or ‘equal-inclination’. Eutocius gives a definition, which seems inadequate. Here are two attempts at definitions, the first of which captures better what Archimedes does in Plane Equilibria and Quadrature of the Parabola, but keeping in mind the abstact nature of his works also fits the solid geometry of the Method and Floating Bodies. The second definition fits better with the account of Archimedes found in Heron' Mechanica.

1. The center of weight is a point from which a freely hanging body is stabile, no matter how it is positioned about the point.

2. The center of weight is a point such that if a body is hung freely from any point on the body, a perpendicular from the point of suspension to the horizon will go through it."

It is obvious that Archimedes has in mind the equilibrium CG, or "torque CG". Based on weight/lightness properties of bodies. Any reference to gravitational force, or to concept of mass as the property of matter, or to concepts of vectors and scalars, is entirely out of his league. After all, he was just an Old Greek.

Of course, the CG on the surface of the Earth "practically" coincides with the CM. And all his results do hold for CM, actually better than for any CG concept in a non-uniform field (and in particular his extension to geometrical figures). Still, that does not change the obvious fact that Archimedes studied the concept of the "equilibrium CG". Which could be a bit more stressed in that historical account, if you think it is worth the effort.

Therefore, I would remove the second sentence from the first paragraph of "History", and substitute CM by CG until the last sentence. And before that last sentence of the first paragraph, an explanation along the above lines could be inserted.

P.S. I do not see any conceptual relation between the law of the lever arm and "real" CM.--Ilevanat (talk) 02:00, 14 January 2012 (UTC)

Merge to center of mass
I am sure that the final merger by user: Teapeat was a mistake. The center of mass article should not cover nothing anything but the point giving the mean value of a mass distribution. Any stuff about mean values of some else distributions must constitute a separate article(s). Incnis Mrsi (talk) 10:13, 8 March 2012 (UTC)
 * Eh, I agree (to nobody's surprise). I still think a summary-style daughter article is a reasonable compromise, and hopefully it's a solution that we can eventually settle on, especially if the main article is streamlined enough in other respects. Melchoir (talk) 01:20, 12 March 2012 (UTC)
 * What is "the main article"? What means "a summary-style daughter article" and a daughter of which one? Expected value of some distribution in 3-dimensional space is more general topic, but "center of mass" is more important topic. P.S. I changed the section header to "Merge to center of mass" because the contested merger occurred not in March, but January 27. In March we actually discuss splitting, not merging. Incnis Mrsi (talk) 06:22, 12 March 2012 (UTC)
 * Oh, by the main article I mean Center of mass. In January, had a section "Gravity" with a main link to . As of today, Center of mass has a section "Centers of gravity" with the same content and no main link. Melchoir (talk) 09:24, 12 March 2012 (UTC)
 * I agree that merging was a mistake. --Steve (talk) 12:14, 12 March 2012 (UTC)

You can always unmerge them, but there's several problems you create if you do that:
 * people refer to them synonymously, and in this important sense they are the same topic (on its own this wouldn't be sufficient but:)
 * in nearly all practical cases they are exactly the same point in space within measurable accuracy (importantly this is so when there's parallel fields or spherically symmetric fields and geometries)
 * you will have split the 'center of gravity' topic across two articles
 * the combined article is currently not nearly big enough to demand splitting (it's only 13k of text)
 * all of the internal links for 'center of gravity' point to 'center of mass'
 * if you split them the new article is not linked from anywhere except the original article, and IMO is unlikely to ever be well-linked

There's certainly pros and cons either way, but on the whole it's apparent to me that there's more downsides to remerging them than upsides.Teapeat (talk) 17:58, 12 March 2012 (UTC)
 * A bunch of invalid arguments and irrelevant remarks.

> people refer to them synonymously
 * The problems of terminology are not the same than the problems with definitions. We are in Wikipedia, not Wiktionary.

>… they are exactly the same point in space within measurable accuracy
 * … like all material points in an atomic nucleus are the same point within measurable accuracy. Does this imply that the article charge radius is pointless?

> you will have split the 'center of gravity' topic
 * In most cases a lake is a body of water. But there are several, in some remote parts of the Solar Systems, filled with hydrocarbons. So, Wikipedia splits the "lake" topic. Is it really bad?

> the combined article is currently not nearly big enough
 * A stereotypical logical fallacy. A large size suggests splitting the article, but not-a-large size does not suggest non-splitting of articles.

> internal links for 'center of gravity' point to 'center of mass'
 * What? Does Teapeat mean that someone intended to link an exotic "center of gravitational force" (which differs from the CoM) notion could type [[center of mass]] assuming these are synonyms? Or Teapeat thought something nearly opposite to what he typed? Explain your thoughts carefully, not hastily. There is no rush.

> if you split them the new article is not linked from anywhere
 * If center of gravity becomes a WP: disambiguation page? If such a content will be demanded, then it will be linked. If it will not be demanded, then… it now pollutes the center of mass article, but yeah, it will be linked indeed, how cool. Incnis Mrsi (talk) 18:41, 12 March 2012 (UTC)

 * I stand by every single one of my points, as is my right, and I do not consider any of your characterizations to be accurate, nor do I find that I have been unclear. There's about a thousand links to 'center of gravity' and in nearly every case they mean 'center of mass', redirecting 'center of gravity' to a disambiguation page seems to me to be extremely undesirable.Teapeat (talk) 19:31, 12 March 2012 (UTC)
 * These two topics, while distinguishable, they are intimately related, and are, in practice, synonymous and for most (but not all) are exactly the same point. Even if we did go through and separate them, by hand, there's no guarantee that they would stay separated, people will continue to edit.Teapeat (talk) 19:31, 12 March 2012 (UTC)
 * While I like your suggestions in theory, in practice, I'm not liking it nearly so much.Teapeat (talk) 19:31, 12 March 2012 (UTC)
 * This is not "practice vs theory". These are conflicting optimizations of Wikipedia to different aims. One approach is to give more articles available just by typing some words and pressing, without thinking. Another approach is to increase clarity and do not bother much about disambiguation pages and extra clicks needed to reach a page needed, or to fix a bad link. This latter approach dominates today in scientific areas of English Wikipedia and, I hope, will be dominate for a long time in the future. So, I do not think that further attempts to join different topics based on a historically ambiguous term "centre of gravity" will succeed. Incnis Mrsi (talk) 08:45, 13 March 2012 (UTC)
 * I think the link center of gravity can and should continue to go to center of mass, which is the primary meaning of the term "center of gravity". The center of mass article can link to "Centers of gravity in non-uniform fields" as a hatnote. --Steve (talk) 15:00, 17 March 2012 (UTC)
 * Congratulations on splitting up the center of gravity article and making an article that virtually nobody will read!!!Teapeat (talk) 17:35, 29 March 2012 (UTC)

Why "gravity"?
Since the article will certainly be restored, I ask second question: which content about the gravity cannot be applied to a field of another nature? Does there exist a reason to restrict to the gravitational field only? Incnis Mrsi (talk) 16:16, 12 March 2012 (UTC)