# Talk:Centers of gravity in non-uniform fields/Archive 1. Talk Re Previous Article - Deleted

== This page is ridiculous. The author throws in the words "or other object", and that should be most of the article. Aircrft should perhaps be mentioned at most in a footnote if at all. -- Mike Hardy

Yes, this page needs to be improved. 146.6.144.109 00:54, 24 March 2006 (UTC)

OK, no problem with the addition of some stuff about aircraft, but the article should not make it look as if centers of gravity of aircraft are what the subject is primarily about. Someone experienced in physics pedagogy should add material. Michael Hardy 21:58 Jan 26, 2003 (UTC)

## Centre of Gravity -- more about the spelling than the math?

Dear Editors,

While I very much appreciate the efforts of the editors and the authors to make Wikipedia one of the best sources and references for general knowledge on the web, I would recommend not to underestimate so much the intellectual faculties of the general public as to believe that we would actually neither need nor expect at least one formula in the article about Centre of Gravity. Having dedicated special attention to the spelling of the term (see the article) the author should have at the very least given some practical information on how to calculate the Centre of Gravity say of a collection of physical points. The formula is quite simple:

Xcg = (Σ Mi*Xi) / (Σ Mi), the sums are for i=1 to N

Ycg = (Σ Mi*Yi) / (Σ Mi), the sums are for i=1 to N

where Xcg and Ycg are x and y coordinates of the Centre of Gravity;

Xi and Yi are the x and y coordinates of a physical point;

Mi is the mass of a physical point;

N is the number of physical points.

Despite the simplicity of the above formula and its wide application for a quick and rough estimation of the Centre of Gravity (especially now that computers are commonly used) I had forgotten it and was looking for a place on the internet to check it. Wikipedia is usually my starting point in such searches. I was quite disappointed that in this particular case Wikipedia offered so little practical knowledge.

I found the required information elsewhere and used it in my MS Excel calculations to solve my problem.

I suggest the following:

1) include in the article the above formula for the Centre of Gravity of a collection of physical points as it is perhaps the most simple and the most widely applied formula on the subject (if one has an arbitrary 2-D or 3-D shape simply by dividing it into regular squares or cubes one could very quickly obtain a rough estimate of its Centre of Gravity using this formula and a PC).

2) include in the article a note that for an object with symmetry the Centre of Gravity necessarily lies on the axis of symmetry (implying homogeius density of the material). And since a large number of practical objects such as rectangles and rhombs have two axes of symmetry their Centre of Gravity lies on the intersection of the two and hence is easily found.

3) include in the article a note that there is no requirement for the Centre of Gravity to lie inside the physical object. E.g. for a horse shoe it lies outside.

4) finally include in the article a bit more practical information on how to find the centre of gravity of some simple shapes (e.g. a triangle).

Kind Regards,

Plamen Grozdanov

28/Aug/2004, Bristol

P.S. Just in case someone is interested: I am a software engineer.

P.P.S. One source to use to verify the above formula could be Centre of Gravity from the University of Winnipeg

## Merging with Center of mass

I plan to move most of this article to "Center of Mass". Then this article could focus on the few objects where the center of mass is *different* from the center of gravity. (Such objects that are in a non-constant gravity field, and are non-spherical). --DavidCary 03:41, 12 Feb 2005 (UTC)

I agree with the merge. Most people are not interested in non-umiform gravitational fields. I think "Center of gravity" should redirect to "Center of mass" and what is left of this page should be called something like "Center of gravity in non-uniorm gravity" David R. Ingham 19:10, 8 April 2006 (UTC)
Agree also. The article should begin
"In uniform gravitational fields, center of gravity coincides with center of mass, therefore the two terms are synonymous in daily usage. In non-uniform gravitational fields, however, ..." —The preceding unsigned comment was added by 18.251.2.2 (talkcontribs) 04:48, 14 April 2006.
I support merging Center of gravity into Center of mass. As far as I can tell, this center of gravity is a useless concept. I'd love to see an application, but I doubt one is forthcoming. Melchoir 05:30, 17 April 2006 (UTC)
Interesting. This discussion group seems to be in favour of merging into center of mass, while the discussion group on center of mass seems generally opposed. I vote no merge, but a better clarification of the what the difference is, and why the reader probably should be in the other place Fizzybrain 23:39, 17 April 2006 (UTC)
Right. In fact, very little if any material in this article that would survive a merge, while the structure of Center of mass should not change. Really what we ought to do is clean up the discussion of centers of gravity at Center of mass and then just redirect Center of gravity to it. There will be some shuffling of interlang links, but nothing too hard. Melchoir 00:35, 18 April 2006 (UTC)

## What's the difference?

Center of inertia, center of mass, and center of gravity.

Two of them are synonyms, but no one seems to agree on which two, including Wikipedia:

• CoI = CoM: "The center of mass or center of inertia of an object is a point" - center of mass
• CoI ≠ CoM: "Similarities between center of mass and center of inertia" - center of gravity
• CoM ≠ CoG: In a uniform gravitational field (in other words, when the tidal force is insignificant), the center of mass and the center of gravity are at the same location. - center of gravity
• CoG ≠ CoM: In physics, the center of gravity (CoG) of an object is the average location of its weight. In a uniform gravitational field it coincides with the object's center of mass - center of mass

• CoM = CoG: "The center of the system of forces, in this case, is the center of gravity and is identical with the center of mass or centroid which can easily be determined by summation for a system of discrete particles or by integration for continuous masses" - The Encyclopedia of Physics, Besançon
• CoM = CoG: The center of mass, also called the centroid or center of gravity, [1]
• CoM ≠ CoG: The terms "center of mass" and "center of gravity" are used synonymously in a uniform gravity field [2]
• CoM ≠ CoG: The terms "center of mass" and "center of gravity" are interchangeable as long as their is no discernible difference in the pull of gravity from one part of the object to another. [3]
• CoM = CoG: The center of mass is the point where, if the object were changed such that all of the mass of the object were concentrated at that point, the motion of the object would be unchanged. When considering the force due to gravity, this point is called the center of gravity. [4]
• CoM ≠ CoG: In almost all cases you are likely to encounter, the center of mass and the center of gravity are the same. (The only times this wont be true is if you are studying objects that are so large that the gravitational field due to the earth is not constant over the object, like for example the moon). [5]

• CoM ≠ CoI = CoG: For a flexible body in motion, the center of mass and the center of inertia forces (aka center of gravity) do not always coincide when the body is subjected to an acceleration field that varies in space. [6]
• CoM = CoI: center of mass That point of a material body or system of bodies which moves as though the systemÕs total mass existed at the point and all external forces were applied at the point. Also known as the center of inertia. [7]
• CoM = CoI: center of mass d. Also called center of inertia. [8]

• CoI = CoG: Center of inertia (Mech.), the center of gravity of a body or system of bodies. [9]

So which is it??

I trust the sites that explicitly state that CoG and CoM are different. But which one is a synonym for CoI? I see both in the above list, and I don't know which to trust. — Omegatron 02:39, 4 November 2005 (UTC)

They may simply be used differently by different authors. -- SCZenz 02:50, 4 November 2005 (UTC)

I would assert that CoI≠CoG, although they are equivalent for almost any purpose, and that CoM is ambiguous as it does not state whether the mass referred to is gravitational (CoG) or inertial (CoI). Can't give you a reference though, and I will bow to the more knowledgeable. Physchim62 (talk·RfA) 12:57, 4 November 2005 (UTC)

So CoG ≠ CoM, CoG ≠ CoI, CoI = CoM? It seems CoI is very rarely used compared to the other two, but we should mention it when we figure it out. — Omegatron 15:15, 4 November 2005 (UTC)
I would imagine it (CoI) is mostly used in the analysis of rotational movement, but I haven't got any references handy at the moment (I'll see what I can find). We should have an article on gravitational mass as well. Physchim62 (talk·RfA) 08:54, 5 November 2005 (UTC)
Rotational movement was the reason I was looking this up, in fact... I totally got lost on this tangent and forgot about my original question.  :-) — Omegatron 18:25, 4 December 2005 (UTC)
First of all: I'm not a native English speaker, so I might be lost in translation. But here are my 2 cents: CoI is a mathematical concept, thus it applies to mathematical objects, such as lines, areas, any 3 or higher dimensional objects. The center of mass is a physical concept, which applies to physical objects, which by definition are 3dimensional. So they are by no means the same.
Um, no, that's not how physics works. Anyway, does anyone have a reliable source that attests to the usage of "center of inertia" at all? Melchoir 18:08, 17 April 2006 (UTC)

## Moved from page

144.92.172.34 20:05, 21 October 2005 (UTC)Batrob Begins

First of all, the center of gravity (CG) is not the location whre mass is considered to act!!! The CG is the location where the force of gravity is said to act on an object. Second of all, the center of mass (CM) is the average location of mass in a body!!

For example: the center of mass of an object consisting of 2 particle masses (m1 and m2) at distances x1 and x2, respectively, from the major source of gravity's center as refrence point:

Xcm = (m1*x1 + m2*x2)/(m1+m2)

Xcg = (x1*GMm1/(x1^2)+ x2*GMm2/(x2^2))/[GMm1/(x1^2)+ GMm2/(x2^2)]

So, in situations where x1 and x2 squared don't differ by much, the CM and CG are nearly identical since they'll approximately cancel out. The farther away from the origin of the gravitaional source, the more they differ.

20:05, 21 October 2005 (UTC)Batrob

## CG or CoG?

I've always seen center of gravity abbreviated CG - in aviation publications and aerodynamics texts, mostly. Which is considered 'correct'? Or are both correct? If both are, perhaps CG should be mentioned in the article on the topic. ericg 20:09, 12 December 2005 (UTC)

The lack of response is telling me no one really knows. The search 'cg "center of gravity"' returns three times more hits on google than 'cog "center of gravity"' does, and NASA appears to use CG. ericg 19:01, 28 December 2005 (UTC)

I suppose that it is a matter of convenience. I used CoG as I thought that it is unambiguous. But as we are talking about the center of gravity, even CG would be equally unambiguous. I propose that we stick to CG. If there are no objections, I'll change all CoGs to CG - Wikicheng 05:05, 29 December 2005 (UTC)

Changed - Wikicheng 04:16, 30 December 2005 (UTC)

## Err... are we sure that a center of gravity can really be said to exist? (2006-04-15)

I think we have mostly agreed that centre of mass is absolute, and that centre of gravity depends on the gravitational field. I think there's a bit of relativity which creeps in here. It's probably best described by what I was about to write...:

Existing entry: Note that the center of gravity of a body is not a point such that the gravitational field due to that body is equal to the gravitational field if all mass were concentrated there. Such a point usually does not exist. For example for two equal spherical massive bodies the center of gravity of the system is forced by symmetry, and lies midway between the centers; but gravity due to the system is not very large near that point. I was about to add: Also, a third and fourth light body placed anywhere near either of the massive bodies will each experience a greater attraction to the body that it is closest to, in accordance with Newton's law of universal gravitation; i.e. the apparent center of gravity can have multiple simultaneous positions for multiple simultaneous observers.

Can anyone tell me from which vantage point we are to assess this system's "centre of gravity" - and why? Fizzybrain

I think the "center of gravity" is supposed to be the point at which you can calculate the force exerted on the object by some given gravitational field, not the point from which the object apparently acts on other bodies, which generally will not exist. I'm not sure, because none of my own books say anything about this obscure concept, and the article doesn't actually give a definition. Melchoir 05:17, 17 April 2006 (UTC)

Well, for the purposes of the thought experiment, consider the third and fourth bodies as massive - it's pretty much the same result; i.e. the resultant direction of force/acceleration (and their perceived acting centres) will be hugely different depending on the objects' initial relative positions. I suppose though that one might still be able to give it a unique position for any given field/arrangement (i.e. it does exist). The closest we get to a definition is "the average location of its weight", and of course weight is a very vague concept on this large scale. The references seem to be more of a description than a definition (i.e. "this is where it is in this situation", rather than, "this is what it is in all situations"). Ho hum. Fizzybrain 01:38, 18 April 2006 (UTC)

## Wrong stuff

### Practical example

If you hang an object from a string, the object's CG will be directly below the string (and in line with it).

### Locating center of gravity

 Step 1: Cut an arbitrary 2D shape. Step 2: Suspend the shape from a location near an edge. Drop a plumb line and mark on the object. Step 3: Suspend the shape from another location not too close to the first. Drop a plumb line again and mark. The intersection of the two lines is the center of gravity.

(see also Center of Mass for a method of calculating center of mass in 2-D objects, which in most cases is equivalent to center of gravity)

All this talk about hanging the object only works for the center of mass, which is a rigid point within the body. The "center of gravity" moves around within the body as you subject it to different gravity fields, so if you want to determine the center of gravity, you can't swing it around. The demonstration might be appropriate at Center of mass. Melchoir 05:23, 17 April 2006 (UTC)

• But if these practical experiments (/the objects themselves) are affected by gravity then surely that makes them less useful for center of mass than center of gravity? But, center of mass is a better term, and it is essentially the same as CG in these experiments. It's tricky. Is there something to be said for each page having one method each (one theoretical, one practical)?

If you hang an object from a string, the object's CG will be directly below the string (and in line with it).

OK, the question of "below" is rather vague, but I'd say it's worth keeping, because it is familiar and easy to grasp (and of course some people will read it it to find out stuff - not just to check that we have got it right :-) ), and anyway, I'd argue that any experiment involving string is going to be assumed to be an approximation - howabout simply changing it to

If you are standing on the earth and you hang an object from a string, the object's CG will be directly below the string (and in line with it).

The experiment for Locating center of gravity seems slightly better - in this particular example the object and the measurement reference (i.e. the plumb line) are both affected by gravity, so surely the results would be consistent regardless of the gravitational field, (i.e. we don't have to imagine what "down" is)? Of course, theoretically the object and the plumb line would be affected by the gravitational field to different extents, but those differences really are going to be ridiculously slight. To give an idea of scale, when considering these "non-uniform" gravitational fields, I am thinking of experiments done showing mountains as having a significant gravitational effect - I can't remember the precise details, but essentially if you hang a plumb line next to a large mountain it doesn't point "down", compared to other plumb lines without nearby mountains.

I think it is a useful and commonly used/understood demo, and it is worth keeping somewhere, and here is the best place for it (in contrast with the purely mathematical calculation, which I had already moved to the center of mass page). The very fact that it is experimental rather than theoretical suits the "this is practically true in virtually all cases" nature of this page. It would be even less appropriate on the center of mass page, where pure physics accuracy seems to be more of a goal than familiarity.

A blanket "all these experiments assume that they are being performed in "normal" conditions - i.e. standing stationary on the surface of the Earth" would allow us to keep some nice simple and useful demos.

We run the risk of being so keen to be *absolutely precise* that we end up not having a page at all. In short, as these are practical examples which the reader is encouraged to imagine/do, it seems fair to make the same assumptions as the published texts and describe the resultant point as being the center of gravity, and to note that it is likely to be in the same position as the center of mass. Fizzybrain 01:38, 18 April 2006 (UTC)

Even a mountain doesn't produce a non-uniform gravitational field unless your measuring instrument is on the same scale as the mountain, and suffice it to say that humanity has yet to build a rigid object that large. If you're playing with everyday-scale objects while standing next to a mountain, you might notice that your "up" is different from GPS "up"; that's all. Your object will never feel any nonuniformities.
Agreed, it's not about non-uniformity, but an error can still occur. I think the inaccuracy applies to the first one, if your down-finding mechanism is not gravitational (which e.g. a "vertical" line compared with a distant one using a telescope - I think this is how the mountain experiment was done. The beauty of the second one is that it is self-correcting i.e. the down-finding mechanism is effectively calibrated. Neither of them involve the idea that CM is different from CG - this is just out of academic interest :-) - but under exceptional circumstances the first experiment could give an error. Of course, the sort of student who is going to want/need to know the subtle experimental errors will have great fun working this out for themselves.... Fizzybrain 12:13, 18 April 2006 (UTC)
I'm not saying that these experiments aren't worth keeping; I copied them to the talk page so that they can be reinserted where appropriate later. But as long as we have separate articles named Center of mass and Center of gravity, it is misleading to place them in the latter. There are really two choices of interpretation:
1. The gravitational field is essentially uniform, and the experiments locate the center of mass, which is a center of gravity during each orientation.
2. The gravitational field is essentially nonuniform, and the experiments fail to locate anything, including the center of mass and any centers of gravity, and are generally meaningless.
If I read you correctly, you agree with me that the only useful interpretation is the former.
We can discuss later whether it is a good idea to have two articles, one mathematical and one practical, about the center of mass. New articles are cheap. Melchoir 02:42, 18 April 2006 (UTC)
Yes, the only circumstances where these experiments measure anything, they measure *both* CG and CM. I'm sold. Let's merge (and let's get these experiments carefully re-inserted) Fizzybrain 12:13, 18 April 2006 (UTC)
Right, I'm working on improving Center of mass to the point where it'll be natural to talk about gravity. There are going to be some editorial questions along the way, such as how to place the series of figures and whether to split off Barycenter again, but all in due time. Melchoir 19:37, 18 April 2006 (UTC)

### More wrong stuff

Removing the following:

• In objects that are radially symmetric (in shape and density), both the center of gravity and the center of mass coincide.
• The path of an object in orbit in a true dark vacuum is subject only to gravitational forces, and therefore depends only on its centre of gravity at any moment in time.

The first item is wrong unless the ambient gravitational field is also symmetric, and the second item is handwaving nonsense. Melchoir 17:12, 17 April 2006 (UTC)

1 - agreed, as long as we continue to make it clear that in virtually all cases that anyone is going to care about (or relate to) these two centres do indeed sit at the same point on the object. Does the initial introductory paragraph say that clearly enough to the regular reader, do you think (that is not an aggressive challenge - I reckon it's OK, but maybe we can do better)?

2 - :-) I tried to patch up a dodgy definition, but you have simply excised it, hurrah! (it was even worse before....) Fizzybrain 01:38, 18 April 2006 (UTC)

## Original research

I've looked at the references given in the article, and they don't support the text. Therefore the article consists of original research. Here's what the references actually say:

• Tipler (4th edition, p.353) defines "the center of gravity" as the point at which the total weight of an object exerts the same torque as the total torque due to the weights of the particles. He draws the gravity field in question as pointing downwards everywhere, and he then defines only the "x coordinate" of the center of gravity through the formula ${\displaystyle X_{cg}W=\sum _{i}w_{i}x_{i}.}$ Immediately after this definition -- literally the next line -- he restricts his attention permanently to uniform fields and reconciles with the center of mass.
• Serway (3rd edition, p.304) says "In order to compute the torque due to the weight force, all of the weight can be considered as being concentrated at a single point called the center of gravity." He draws a similar diagram, with the weight pointing downwards everywhere, writes a similar equation for the x coordinate of the center of gravity, and in the very next sentence, guess what: he restricts his attention permanently to uniform fields and reconciles with the center of mass.

Neither of these authors spends any time discussing non-uniform gravitational fields, and they quickly start using the center of gravity as a synonym for the center of mass. Serway's questions at the end of the chapter ask "where is the center of gravity" without specifying the gravitational field. Tipler speaks (on p.359) of distributing weight towards the bottom of an object to "lower the center of gravity", even though he never defines the vertical position of the center of gravity, and his torque definition is incapable of such a distinction. These discussions make sense only because the authors have already decided that center of gravity and center of mass are the same.

Most importantly, they never say anything remotely resembling the content of this article. Melchoir 17:54, 17 April 2006 (UTC)