# Are there any instances on Earth where it must be necessary to distinguish between center of mass and center of gravity?

Distinguishing between center-of-mass as the geometric centroid pondered by mass (the average point of the object(s) using mass as the weight of the sum) and the center-of-gravity as the centroid where you can take the total gravitational force to act on, or equivalently, where there would be no torque due to gravity:

Has there been any engineering project on Earth where it was necessary to distinguish between the two? Perhaps a building so big and asymmetrical you had to take it into account?

• Your wording is a bit confusing because you say "geometric centroid pondered by mass". "Geometric centroid" is the term used to refer to the centroid of an object due solely to its shape. Then, there's center of mass and center of gravity. The fact you mention "building so big and asymmetrical" seems to indicate you know what it is you're talking about (uneven gravitation effects on mass distributions causing the COG to be different than the COM), however, the two answers given so far seem to miss that and focus on geometric centroid vs center of mass/gravity (equal under constant G-field). Jan 3, 2022 at 1:45
• @DKNguyen This article may help to better understand "mass center". I could be wrong though. en.wikipedia.org/wiki/Center_of_mass
– r13
Jan 3, 2022 at 1:50
• @r13 That wiki only confirms what I thought. It says the difference between COG and COM is caused by a gradient in the gravitational field. That's why the OP mentions a very big building (assymetrical doesn't seem to have much to do with it however). However, all current answers do not talk about a gradient. They only talk about assymmetric mass distribution which is not necessary for COM and COG to be different. Jan 3, 2022 at 1:51
• @DKNguyen I think you have a valid point. I withdraw my answer.
– r13
Jan 3, 2022 at 2:24
• @AJN I was considering the types of situations where the distinction might become relevant, the only ones I did find on the Web were in celestial mechanics and submarines, hence the exclusion. Jan 8, 2022 at 19:45

Yes: any structure intended to be partly submerged in water, such as a ship. The submerged part is subject to a "reduced" (often reduced so much that its sign changes) value of $$g$$ that is the resultant between the actual acceleration due to gravity and the specific upthrust due to water being displaced. This can lead to the centre of mass and the centre of gravity being widely separated.

There are multiple concepts mentioned here rather than two: geometric centroid, center of mass, and center of gravity, and another "center of gravity" concept where a gravitational field apparently defines it.

Geometric centroid is the furthest off the other two. It is only the same as center of mass if either one is assuming uniform density or some balanced masses just happen to make it so.

Center of mass is a rather interesting concept that is most commonly used. Gravity and weight(balancing) happen to be an easy way to measure this but only because the gravitational field is very uniform relative to the object, but one could also leverage other phenomenon such as forces and moments experienced when rotating an object about various points.

Center of gravity is intended to be the point where the forces due to gravity from the object itself vanishes (aka cancel each other out). Isolating only the forces of gravity of an object due to itself requires bringing in some theoretical models to exclude effects of gravity from other objects. For all that we know, this happens to be the center of mass thanks to the definition of gravitational force. Maybe some day someone will theorize something where the mass and gravity relation needs to be tweaked to maintain all other non-gravity related mass physics, but until then, this center of gravity remains the synonym of the center of mass.

Finally a center of gravitational? force that the poster is calling a center of gravity. Note that by considering gravitational forces from OTHER objects, there is no guarantee that there shall exist a vanishing point of zero gravitational force applied to OUR object. Such is the case for an object of nonzero mass at rest on the ground, considering only all gravity(primarily from the earth) and ignoring the electromagnetic forces that keep it from accelerating down through the ground. The closest definition I can think of would be: points where a force (and zero moment) can be applied to cancel all gravitational forces on the object if the object is assumed to be rigid. With such a specific definition, the only answer I can think of is that it matters when it matters, and it is affected by the scale rather than the planet it is on.

If I'm considering a small object on earth in earth's reference frame, I might be able to draw a line parallel to the gravitational fields, through the center of mass to identify the points. However if I consider just a proton, I may have to compensate for gravity from other protons or neutrons as proximity overtakes mass, or maybe I just don't care about gravity from earth. Maybe my object is such that the primary effects of gravity are due to the sun despite being on earth - object as a part of earth's core (or some other) balanced subsection such that gravity due to earth cancels out.

:. For things that matter consider all forces; discrimination without a good purpose is bad.

• Your answer confuses me. If the Center of Gravity is defined to be: "the point where the forces due to gravity from the object itself vanishes (aka cancel each other out)", then why would such a point ever need to be considered when analyzing dynamics? Jan 8, 2022 at 20:04
• you could do all the math without ever relying or even identifying a center of gravity/center of mass. it simplifies a lot of math though much like a model of the solar system about the sun. It turns out that forces due to gravity are not the only thing where mass is important, and other things closely related to F=ma cancel out in a frame about that point. You could use any point anywhere, but then you would have a lot more angular acceleration terms to handle.
– Abel
Jan 8, 2022 at 22:13
• Do you have an example? If I understand correctly, there is no need to assume uniqueness. I know there is a theorem on the additivity of the center of mass when you join two systems into one, but I don't see that it happens necessarily for this center of gravity. If I join two half-rings of different mass into an "S" shape wouldn't it be possible there are two points in space with vanishing gravitational force? Jan 8, 2022 at 23:30
• en.m.wikipedia.org/wiki/Rotating_spheres. as for the two half rings, once you consider them to be one "object" you must consider the mass of both half-rings together doing so makes there be only one center of gravity.
– Abel
Jan 9, 2022 at 13:28
• OMG, the center of gravity is frame dependent, I haven't thought of that! Jan 9, 2022 at 23:44

If I understand you correctly, my answer is yes. In almost every building when we design the foundation we design it for the center of gravity, not geometrical centroid.

Consider a slab on grade supporting a square room 30m by 30m. and suppose at one corner of this room there is a piece of equipment that weighs 15 tons as compared to the rest of the room that has just 100kg/m^2 load.

We design the slab and foundation for unsymmetrical load, stronger if needed under and near the equipment. And also we make sure that the uneven load's overturning moment is compensated somehow by a wider tow or counterweight on the foundation.

## Edit

after some comments pointing out my misunderstanding:

If the question implies that small local variations in earth's gravity are cause for concern in designing large buildings the answer would be no!

The variations in the earth's gravity are due to several reasons.

• one would be the latitude of the location due to earth rotation. Closer to de equator, the smaller the effect of gravity.

• the altitude, the higher the site is the less gravity.

But these variations are smaller than, 0.07 as per this Wikipedia source article. And they happen gradually over distances longer by order of magnitude than any building's size.

I have read but forgot the source that some small variance in gravitation has been observed near a concentration of heavy metal mines! But again on a scale not consequential for building design.

• See my main comment to the OP. Jan 3, 2022 at 1:47
• Considering the edit, this is what I was looking for, but now I wonder whether we would ever actually consider it, as in some sci-fi megaproject. Thanks. Jan 8, 2022 at 20:09
• Please accept the answer so others can use it. Jan 8, 2022 at 20:13