Differences between Centre of Gravity and Centre of Mass

[Shafia Zahin]

Hello,
I need help in understanding the concept of centre of gravity and centre of mass.I really get confused in the two of these.It is written in books that the centre of gravity is never changed in an object,its position is constant.But as far as I know the position of an object determines the position of the centre of gravity.I don't know whether I'm telling it right or wrong but I need a very clear concept in this topic.Please help.
Shafia.

 

[robphy]

They are often interchangeable [near a uniform gravitational field]... but they are distinct.
Center of mass is a mass-weighted average of position. http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html
Center of gravity is a weight-force-weighted average of position... e.g. there may be variations in local "g" (i.e. a different value of g for each piece of mass).
For instance, if part of the object is further away from the center of the earth, then it will have a different "g" value than the others.

 

[CWatters]

Thanks robphy. Something I'd never thought about.

So something like a very long space tether...
https://en.wikipedia.org/wiki/Space_tether
..could have the centre of mass and centre of gravity in slightly different places.

 

[Shafia Zahin]

Thank you for your help.You took the trouble to reply this thread ,that's great.But actually these things have already been discussed in my book;but I just can't figure it out from these that what centre of mass or gravity actually is and why is it that they are unchangeable? It would be great if you could give me a detailed explanation about this.Still,thank you very much once again.
With regards,
Shafia.

 

[robphy]

Consider two masses connected by a massless rigid rod.

$\frac{m_1r_1+m_2r_2}{m_1+m_2}\neq \frac{m_1g_1r_1+m_2g_2r_2}{m_1g_1+m_2g_2}=\frac{\left(\frac{GMm_1}{(R+r_1)^2}\right)r_1+\left(\frac{GMm_2}{(R+r_2)^2}\right)r_2}{\left(\frac{GMm_1}{(R+r_1)^2}\right)+\left(\frac{GMm_2}{(R+r_2)^2}\right)}$

As long as the object is rigid (i.e. two masses keep the same separation: $|\vec r_2-\vec r_1|$ is constant),
the center of mass is still in the same location relative to the two objects---you can mark an X on that spot.
However, if "the local values of g for each mass" change as the rigid object is moved, the center of gravity can move.
For example, if you rotate this rigid object so that $r_1\neq r_2$ (i.e. one higher above the Earth than the other), the center of gravity is different from the center of mass.

(In my example, I assumed the weight forces are in the same direction. I presume there will be some complications for non-parallel forces.)

 

[Paulo Zensei Heshiki]

Center of Gravity (CG ) : resulting from the application point of the forces of gravity acting on each particle of a system. the point of application of the weight force of a body.
center of mass ( CM): point where one can admit that the mass is concentrated .
In the uniform gravitational field center of gravity coincides with the center of mass

 

[Avi Mishra]

Centroid

 

[Cutter Ketch]

Perhaps an example. The far side of the moon is about 1% further away from the Earth than the near side. Due to the 1/R^2 form of gravity, particles on the near side of the moon are attracted to the Earth with around 2% more force than similar particles on the far side of the moon. Assume the moon is a uniform sphere (it's not at all, but just go with me here). If you sum up the locations of all of the material weighted by the mass of each piece the resulting center of mass would be right in the center of the sphere. If instead you want to know the center of action of the force you would add up all of the locations weighted by the force of gravity each experiences. Because the pieces on the near side feel more gravitational force the resulting center of gravity would be biased away from the center toward the earth.