Talk:Center of mass

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Comments:

"The centre of mass of an object is the point through which any plane divides the mass of the object in half."

Are you sure this is true? If the center of mass is a weighted sum of all the points in an object, and the distance is part of the weighting, then the plane would not divide the mass in half, but the mass times distance. Later on in the page, the Jupiter-sun system is mentioned. This can be viewed as one object without loss of generality (think of the matter connecting them approaching zero mass in the limit The center of mass is outside of the sun, so a plane perpendicular to the line between the objects would certainly not divide the mass of the object in half! I think I will wait for some comment and then remove the sentence from the article.

Other than that, it looks like a good article. It has a lot of good examples.

Cos111 04:38 24 Jul 2003 (UTC)

I agree, I changed it. - Patrick 08:54 24 Jul 2003 (UTC)
For mass that is distributed according to a continuous, nonnegative density \rho(\mathbf x)\ge0...

We are not likely to encounter substances with negative density, but if we did, these integrals could still be evaluated and the result would be physically correct.

It seems to be a reminder that density is not negative, and also a clarification that we can integrate over the whole space, not just the masses, since we allow density to be zero. --Patrick 01:00, 23 Feb 2004 (UTC)

Also, ρ doesn't have to be continuous to be integrable. In fact, being composed of point masses (quarks and electrons) matter is never distributed continuously. More to the point, ρ is often discontinuous at the interface between two materials. A better formulation might be:

For a physical body with mass distribution \rho(\mathbf x)...
However, if you integrate over point masses, you will exclude them from your integral since you they are multiplied by a measure of 0. So, not continuous, but piecewise continuous.
The Lebesgue measure allows point masses and continous masses in one formula.--Patrick 13:38, 4 November 2005 (UTC)

If the Earth-Moon distance is rounded to one significant digit (400000 km), it's silly to come up with 4 significant digits in the answer (4877 km). I call this 'calculator blindness'. I'm not fixing it because the Earth/Moon example is duplicated in the existing article on barycenter. Since that term seems to be in use primarily in celestial mechanics, wouldn't it be more logical to move all the astronomy stuff to the barycenter page?
Herbee 19:18, 2004 Feb 22 (UTC)

Contents

[edit] Example removed from page

I removed the following example from the page:

  • To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the solar system. The influence of each is approximately proportional to the product of mass and distance. That of Jupiter is largest, its large mass more than compensates its smaller distance to the Sun than several other planets. If all the planets would align on the same side of the Sun, the combined center of mass would lie about 500,000 km outside the Sun surface.

The first sentence of this example is incredibly vague, and doesn't apparently have to do specifically with center of mass. Because of this vagueness, the second sentence is inherently confusing: the influence on center of mass is proportional to distance, while the gravitational influence is inversely proportional. The same confusion holds in the third sentence with "smaller" versus "larger". Dbenbenn 10:21, 2 Jan 2005 (UTC)

I have clarified it, but to say things accurately makes the sentences somewhat complicated. Since you seem to understand the matter, you could have improved the formulation instead of just deleting everything. You are welcome to further improve the formulation.--Patrick 00:00, Jan 3, 2005 (UTC)

[edit] Mixing up centers of mass and gravity?

The paragraph on Archimedes' discovery and the section on aeronautical significance both use the term "center of gravity", which is actually a subtly different thing from "center of mass". To my inexpert eye it looks like just replacing all the instances of "gravity" with "mass" would correct this, but I'm not certain enough at this time to just go ahead and do it. So I've tagged the problem with an HTML comment and am appealing to anyone with this article on their watchlists to give me a yea or nay before I tinker. Bryan 23:55, 18 October 2005 (UTC)

There appears to be a difference. The center of gravity weighs the mass density by the gravitational field somehow, while the center of mass just assumes a homogeneous field. For non-gravitational forces the center of gravity should thus not be used. However this is all based on disputed information. --MarSch 16:16, 4 November 2005 (UTC)
Yeah, that disputed information was quite incorrect, and the difference between "center of mass" and "center of gravity" has been greatly exaggerated. As I understand it, "center of gravity" seems to be the historically preferred term, so it is appropriate in a historical context. I'll restore the paragraph and work on explaining the difference. Melchoir 01:12, 20 April 2006 (UTC)

[edit] Mixes up center of gravity with geometric center

The center of mass of a celestial object only tends to be at the geometric center because the mass is pretty evenly distributed around the center. This configuration requires the least energy. If you consider a two-body system in which one body is much larger than the other (e.g. the Earth and the Moon), the center of mass is nowhere near the geometric center of the system. In fact, because the Earth is much more massive than the Moon, the center of gravity remains within the Earth at all times. —The preceding unsigned comment was added by 131.136.202.27 (talk • contribs) 22:48, 4 November 2005.

It depends on what you mean by geometric center of the system: the centroid, or halfway between the Earth and the Moon. The centroid is near the center of mass (not the same point because the densities differ).--Patrick 01:00, 5 November 2005 (UTC)

[edit] Distinguishes between center of mass and center of inertia

Inertia is the tendency for an object to resist an acceleration. It is proportional to the object's mass.

Gravitation is an (apparent) attraction between objects. It is proportional to the product of the masses of the objects in question and inversely proportional to the distance between them.

There are therefore two ways to determine an object's mass. The inertial mass may be determined by applying a force to the object and measuring the resulting acceleration. The gravitational mass may be determined by measuring the gravitational attraction between the object and another object. We usually measure the attraction between an object and the Earth, and call this value its weight.

In all cases, inertial and gravitational masses are identical. Much of Einstein's theory of general relativity is based on the idea that the acceleration produced by gravity is identical to that produced by application of a force. One (experimentally confirmed) consequence is that a gravitational field causes the paths of photons to be deflected, although they have no mass.

In short, the center of mass of an object is also its center of inertia, and the article should not distinguish between the two. —The preceding unsigned comment was added by 131.136.202.27 (talk • contribs) 22:48, 4 November 2005.

I am not aware of a reliable source that even defines such a thing as a "center of inertia". Melchoir 01:06, 18 April 2006 (UTC)

[edit] Definition of center of gravity not rigorous

The article states that "The center of gravity of an object is the average location of its weight." This is misleading. It is a weighted average (no pun intended), with the contribution of each part of the object being proportional to its mass. If an object's mass is not distributed uniformly, its center of gravity tends to be closer to its denser portions. For example, if a metal bar were composed of two cubes, one of them made of aluminum and one made of lead, the center of gravity would be within the lead cube, since lead is denser than aluminum. You could safely place all of the aluminum end over the edge of a desk, but not the lead end.

I would like the article to include a mathematical formula that expresses this idea. It's sure to be an integral. —The preceding unsigned comment was added by 131.136.202.27 (talk • contribs) 22:48, 4 November 2005.

"The average location of its weight" expresses that. An integral for the centroid is given, expressing this idea.--Patrick 01:07, 5 November 2005 (UTC)

I do not not like/understand the integral definition as given (common in physics ?). The result of an integration (as usually defined) is a scalar not a vector. In other words you do the integration on every coordinate and that should be indicated (and not left to context knowledge of the reader).

See http://mathworld.wolfram.com/GeometricCentroid.htm for a more clear use of the integral notation. If the integral in the article is simple a use of standard notation of an expanded integral concept, i think it should be noted/commented or otherwise replaced by a mathworld style notation. --84.132.233.173 12:30, 28 January 2006 (UTC)

That link is broken... Anyway, integration of vectors is extremely common in physics and is quite well-defined. Melchoir 01:02, 18 April 2006 (UTC)
Oh, I see: the link went here. Melchoir 01:09, 20 April 2006 (UTC)

[edit] Center of Mass vs. Center of gravity

"The path of an object in orbit depends only on its center of gravity."

As illustrated so beautifully later in the article, this is absolutely not the case. Only in an (imaginary) isolated system consisting of a single rigid body in a uniform gravitational field could this be true. The path of the moon relative to the earth alone depends only on the moon's center of gravity; however, its "absolute" path (since the earth is orbiting the sun) is a squiggly line about the barycenter of the Earth-Moon system; this barycenter depends on the masses of both the Earth and the Moon. The Earth-Moon system does not have a center of gravity, since it is comprised of multiple discrete bodies in a non-uniform gravitational field (they attract each other and are also attracted by the sun, so the gravitational field is constantly changing). Likewise, the sun is orbiting the center of the galaxy, and the path traveled depends very little on the center of gravity of the moon.

Thus, it may be more accurate to say that the path of an object in orbit depends on its center of mass, as well as on the barycenters of the system(s) in which it orbits. It would be most accurate to say that the path of an object is affected by every other mass in the universe, but most significantly by the masses of bodies with which it participates in orbital motion.

Because of this confusion, I don't think it would be a good idea to merge this article with Center of Gravity.

Pcress 07:16, 26 December 2005 (UTC)

I agree. Do not merge. --Rebroad 12:39, 22 January 2006 (UTC)
I also agree. Do not merge. The two quantities are distinct.Outofmine 15:04, 26 January 2006 (UTC)
  • Do not merge. Totally different concepts. Steve Max 21:33, 12 April 2006 (UTC)
I have removed the statement from both articles, as it accomplished the rare feat of being meaningless and yet still wrong. As for merging, it has become clear to me that there is nothing worth keeping in the Center of gravity article, so to "merge" it here would really be more like a redirect. I'll work on improving this article first to fix the various confusions. Melchoir 00:45, 18 April 2006 (UTC)

[edit] Disputed tag

There is a disputed tag on this article, but there seems not to be any point of dispute raised on the talk page. Should the tag be removed? Outofmine 15:07, 26 January 2006 (UTC)

I've relocated the tag to the most problematic section. Melchoir 00:58, 18 April 2006 (UTC)

[edit] Center of Mass vs. Center of Gravity

Well what I think is that the definition of center of mass/gravity in a way it is done: In physics, the center of gravity (CG) of an object is a point at which the object's mass can be assumed, for many purposes, to be concentrated. is a little vague.

Let's talk about center of mass (CM). For two points it is quite simple \mathbf{r^*} = \frac{m_{1}\mathbf{r}_1 + m_{2}\mathbf{r}_2}{m_{1} + m_{2}}. Generalizing for N points isn't far more complicated either:

\mathbf{r^*} = \frac{\sum_{i=1}^{N}m_{i}\mathbf{r}_{i}}{\sum_{i=1}^{N}m_{i}} (*).

Here the \mathbf{r^*} means the position vector of the CM and mi resp. \mathbf{r}_{i} is the weight resp. position of i-th point.

Now try to imagine that on each point in our N-points system acts an external force \mathbf{f}_{i}. According to Newton's third law hte i-th point will start to move with an acceleration \mathbf{a}_{i} depending on the given force and mass: \mathbf{f}_{i} = m_{i}\mathbf{a}_{i}. Summing this up for N particles gives: \mathbf{F} = \sum{\mathbf{f}_{i}} = \sum{m_{i}\mathbf{a}_{i}} (**).

After differentiating (two times) the eqution (*) we get:

M\mathbf{a^*} = \sum{m_{i}\mathbf{a}_{i}} (***),

where M = \sum{m_{i}} is the total weight of the system and \mathbf{a^*} is the acceleration of its CM.


Comparing equations (**) and (***) leads to final result:

\mathbf{F} = M\mathbf{a^*}.


OK, what is it good for? The last equation shows the physical meaning of center of mass. Simply said: Consider an N-point system under some external forces. The movement of this system (given by the set of N equations) can be simplyfied by assuming the concept of center of mass in this way: take the total weight of the system, put it into the center of mass, sum all external forces. The CM will then move like a single point (with mass M) under single force (F). So we have reduced the N-point system to only one point.

Note 1: For solid bodies everything is the same but you have to use integration instead of summation.

Note 2: The center of gravity is the point with this behavior: when we support the body in CG it will not move (better said it will be well balanced). What does it mean? Take the N-point system again. We are looking for some point where when we act with some force F (supporting force) the system will not move. The system wants to move due to gravitational forces acting on each point (for non-uniform gr. field they aren't the same size nor direction but it doesn't matter). The movement of this system can be solved easily using the CM. Put total weight into CM and act with the sum of all gravitational forces. Now it should be clear that the balancing of this system will be achieved by supporting the system in its CM. Result: Center of Mass IS THE SAME THING as Canter of Gravity. --147.175.20.101 13:25, 6 February 2006 (UTC) umer

You're right that the vagueness needs to be purged from the article, But the handwaving argument in part 2 fails to account for torques, which are what centers of gravity are all about; see Talk:Center of gravity. I intend to edit this article to make the point clear. In full generality, an object's center of mass does not need to be a center of gravity. Melchoir 00:52, 18 April 2006 (UTC)

Without reading everything, it seems to me that large gravitational field gradients are so far from common experience that "center of mass" and center of gravity should be directed to the same page. Which title the page should have depends on a choice between common usage and technical jargon. David R. Ingham 06:05, 22 March 2006 (UTC)

I'm pretty sure "center of mass" is better. It's the correct jargon, and it does pretty well on Google. Most important is to avoid confusion: the center of mass is unambiguous, and of those sources that make a distinction with a center of gravity, it's the "center of mass" concept that agrees with the content of this article and that dominates the discussion. Melchoir 00:57, 18 April 2006 (UTC)

Note: Merging the concepts of "center of mass" and "center of gravity" may be done. Keep in mind that the concept of "center of mass" pertains to mass, which is common to all physical objects. Also the concept of "center of gravity" pertains to gravity, which is specific to space near a very massive object, earth, sun, etc. Hence, the "center of mass" concept is more general where the "center of gravity" concept is a sub-concept concerning mass under the influence of gravity. J.A.T. 12:46am, 13 April 2006

[edit] Center of gravity now incorporated

I've edited the article to explain that "center of gravity" is a limited synonym for "center of mass". I've also removed the merge tag and the disputed tag, and I'll make Center of gravity a redirect to here after I sort out the interlangs. This doesn't mean that discussion about the two concepts is over; everyone, feel free to yell at me if you disagree, but please read the new article first! Melchoir 07:52, 23 April 2006 (UTC)

Oh, and Talk:Center of gravity has some material that we might want to merge into this article. Melchoir 08:16, 23 April 2006 (UTC)

[edit] Another way of looking at it

Departing from the acceptance that Newton's Laws are valid we can examine a stationary raisin loaf. For the loaf to remain stationary we deduct that the sum of all forces and moments impacting on the loaf are equal to zero, otherwise the loaf would rotate or move in some direction. For purposes of analysis any fixed reference point Pt is selected and all the forces and moments about that point can be identified and equated to zero, assuming clockwise as positive. The sum of moments of each raisin and piece of dough about the X, Y & Z axis provide three equations which each equate to zero. The sum of the forces in the direction of each of the X, Y & Z also provide three equations that equate to zero. The total mass of the loaf M exercises a moment about point Pt and per definition it is Rx, Ry, & Rz from point Pt. Analysis of the equations yield results for Rx, Ry, & Rz that are the point we are trying to locate, which in the case of a doughnut is somewhere in space.. Gregorydavid 06:52, 16 August 2006 (UTC)

[edit] Barycenter: Location within/without heavier body depends on more than just the mass ratio!

This --

In the case where one of the two objects is much larger and more massive than the other, the barycenter will be located within the larger object. Rather than appearing to orbit it will simply be seen to "wobble" slightly. This is the case for the Moon and Earth, where the barycenter is located on average 4,671 km from Earth's center, well within the planet's radius of 6,378 km. When the two bodies are of similar masses (or at least the mass ratio is less extreme), however, the barycenter will be located outside of either of them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, Jupiter and the Sun, and many binary asteroids and binary stars.

-- is a nonsense!

mMoon:mEarth = 0.0123, mJupiter:mSun = 318/333,000 ~ 0.001 - how on Earth (!) is this mass ratio "less extreme"? The Sun-Jupiter barycenter is outside the Sun only because the distance between them >> than the radius of the Sun! If Jupiter were in Mercury's orbit, the barycenter would be close to the Sun's center: r1 = 5,500 km -- much less than the Sun's radius.

--Ant 09:16, 17 August 2006 (UTC)

In the end, I was moved to completely rewrite the section. Thanks to Frankie1969 and Srleffler for catching the typos in the table! What's an order of magnitude between friends? --Ant 22:53, 18 August 2006 (UTC)

[edit] Barycenter...moves at different points in the orbit?

I don't understand this assertion:

Astronomers expect to find hundreds of Pluto-sized objects in the outer solar system. If one has a satellite that is round, and which has a certain eccentric orbit -- meaning the two objects come very close together at one point and then diverge greatly -- then the barycenter could dip inside the larger object during part of the orbit, Laughlin explained.
In such a case, the smaller object would be defined as a moon part of the time and a planet the rest.[1]

How does this make sense? Wouldn't it just end up looking like the last animation? Or is there another animation which could demonstrate this situation? —pfahlstrom 21:42, 19 August 2006 (UTC)

Try imagining each ball in that animation to be three times its current size. Sometimes it'll contain the CM, and sometimes it won't. Melchoir 22:14, 19 August 2006 (UTC)
Ah! Thanks. That does help. —pfahlstrom 21:37, 20 August 2006 (UTC)

[edit] Barycenter Independant

Was "Barycenter" origionally it's own article? I think it's a related topic, but should be it's own article. Waarmstr 00:45, 1 September 2006 (UTC)

The problem is that we don't have a reliable source that even distinguishes between the phrases "barycenter" and "center of mass". For all we know, they mean exactly the same thing and are interchangeable. Given that, it's hard to justify having separate articles. What I would support is an article on barycentric motion, which would explicitly apply the CM/barycenter concept to celestial mechanics. Melchoir 04:37, 1 September 2006 (UTC)


[edit] Barycenter inside Sun

Why does this page say that "it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body ... Note that the Sun-Jupiter system ... just fails to qualify" yet in the page for the 22nd Century, it claims "March 10, 2130: At 07:32 UTC, Sun passes through solar-system barycenter " (and links to this article)? —The preceding unsigned comment was added by 194.221.133.226 (talk • contribs) 20:04, 10 November 2006 (UTC)

It does sound like an inconsistency. Is there a source for either statement? Melchoir 20:14, 10 November 2006 UTC

[edit] Many-body problem?

The many-body problem seems to refer to quantum mechanical situations. I presume that orbital physics, centre of mass, etc, etc, is about classical mechanics, and so I've changed the link in Barycenter section to the n-body problem, which is classical. Tez 18:37, 10 May 2007 (UTC)

[edit] Symbol?

Is the symbol that looks like the BMW logo a center-of-mass symbol? (Or perhaps a center of effort symbol? I've seen it drawn on sails in books on sailboat physics.) It's common on diagrams and as markers in crash tests. I can't find the Unicode symbol for it or any information about it; the closest symbol I could find is the circled plus, ⊕. The BMW page says the logo is a roundel, but that's a broad class of symbols. —Ben FrantzDale 15:46, 25 June 2007 (UTC)

SolidWorks appears to call it the "dowel pin symbol". 155.212.242.34 17:46, 9 November 2007 (UTC)

[edit] Animations

The animations help understand the concept as it applies in astronomy. Please do not remove them. --Shkedi 02:00, 16 November 2007 (UTC)

[edit] Torque about the Center of Mass Induced by a Uniform Gravitational Field.

The gravitational torque acting on a system was cited as being Mg x R = Σmig x ri. But, isn't this the negation of torque? The convention for positive gravitational torque should satisfy "the right-hand rule". Also, the linear combination of torque for a system of particles shouldn't switch the place of r and F because the derivative in respect to time of the angular momentum is r x F (since v x mv = 0) thus the total torque for every particle in the system would be Σri x Fi. I think the switched orientation of r and F is a minor error since it is still perfectly possible to show that the center of mass is the center of gravity in a uniform gravitational field since the net gravitational torque about the center of mass is zero and the negation of zero is obviously zero, so in this case the negative of the gravitational torque about the center of mass equals the gravitational torque about the center of mass. However, for the sake of convention, the orientation should be switched so everyone agrees on sign in our "right-handed" universe. Russmoore86 (talk) 07:23, 30 January 2008 (UTC)

Fixed. Melchoir (talk) 07:34, 30 January 2008 (UTC)