Center of gravity and Center of mass different?!?!

BruceW

My definition works in a non-uniform gravitational field. My definition is that the torque around the centre of gravity is zero. So if I actively rotate the object, its centre of gravity will change such that it remains at a point around which there is zero torque. I say my definition. But it isn't mine, of course. We are all standing on the shoulders of giants.

A.T.

It is not unique in a trivial case, where it is not relevant. Only in a non-uniform fields you need to distinguish between center of mass and center of gravity.

BruceW

If I understood you correctly on the last page, your definition for the centre of gravity is:
[tex]\int \rho \vec{g} \ dV = M \vec{g}_{(cg)} [/tex]
Where I use the subscript cg to mean the gravitational field evaluated at your 'centre of gravity'. So, if we use the above equation to define the 'centre of gravity', you can see that there will generally be more than one possible value for the 'centre of gravity'. Also, this definition is not useful, because the 'centre of gravity' will not generally be a point around which there is zero torque due to gravitational forces.

I think that maybe you were thinking of the law which says that a spherically symmetric mass distribution produces a gravitational field that is the same as one where all the mass was contained at the centre of mass. But this law is completely unrelated, and does not help us with a definition of a centre of gravity.

My definition says that the centre of gravity is the line of points around which there is zero torque on the object due to gravity. The intuitive way to think about this definition is if you have an object in a non-uniform gravitational field, and you want to keep it from moving by applying a contact force at a single point, then you would have to apply the contact force somewhere along the line of points which make up the 'centre of gravity'. Otherwise, gravity will cause the object to begin to rotate.