Center of mass vs. center of gravity

[pierce15]

Are there any realistic scenarios for which center of mass is not almost exactly equivalent to the c.o.g., i.e. they must be treated separately?

 

[Simon Bridge]

Non-uniform gravity field.
http://en.wikipedia.org/wiki/Centers_of_gravity_in_non-uniform_fields

 

[BruceW]

from looking at that wiki link, I'd guess a useful application is for spherical gravitational field (like that from the Earth). So, say we were talking about the torque on a satellite (or spaceship) due to Earth's gravity, then we could use the term center of gravity in a useful way. (i.e. the center of gravity of the spaceship would be different to its center of mass).

Although, the gravitational force is going to be pretty much uniform over a normal-sized spaceship. So I guess the center of gravity would only be significantly different from the center of mass if we had a really huge spaceship. That would be cool. Would get in the way of the sun though... people might not be happy about that.

 

[Simon Bridge]

I think the main part is that the cog is defined and calculated wrt a particular gravitational field while the com just depends on the object.

thus: there would be no center of gravity wrt an external field if the field were zero at the object, but each part of the object would have a cog wrt the combined gravity of all the other bits.

I imagine there may be differences in a non-inertial frame too. but I'm too lazy right now to check properly.
The "related discussions" list below is neat reading though.

 

[BruceW]

cool, so maybe it is useful concept for a spiral galaxy, to talk about the centre of gravity of one of the arms with respect to the gravity caused by the rest of the galaxy.

 

[sophiecentaur]

A good demonstration of the dramatic possible difference between centre of mass and the centre of gravity would be to consider the cm of the Earth and Moon (that's somewhere below the surface of the Earth, as it happens)
The, if you take the Centre of Gravity as the point which attracts another nearby object, ask yourself, if you stood on the Moon, would you be attracted to the CM of the Earth Moon system? The Centre of Gravity, for someone on the Moon will be beneath their feet and 'in the Moon'.

CM is always the same. CG depends upon your reference point

 

[BruceW]

CM is always the same. CG depends upon your reference point

I don't really understand this. If it is possible to say the CG depends on reference point, then isn't it equally as valid to say the CM depends on reference point? Like for example, if we say "the CG due to only gravity caused by the moon" then we could also say "the CM due to only mass of the moon"

 

[sophiecentaur]

Centre of Mass is defined as the point in / on the object where the sum of the moments about it is Zero. It is just referenced to the object itself (and the distribution of masses within it). CG is where the force of gravity acts when in another gravitational field.* If the other field is uniform then the CM and CG happen to be in the same place but not generally. (It also applies to two uniform spheres.)


*The word 'Field' means the description of how the force on a 'unit mass' varies over a region of space. You can consider the field of the EarthMoon OR the field of the observer / object. You will get the same answer for the force and its direction whichever you choose to take as the field generator and which you choose to be the 'test object'. (The sign of the force will need to be adjusted, depending which is the attractor and which is the attracted .)

 

[BruceW]

ah right. I would have assumed that the CG is also due to the gravitational field of itself. But if that is not how CG is defined, then yeah, I can see what you mean now.