[Physics] When the Center of Mass and Center of gravity does Not Coincide

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"The centre of mass of a system of particles is the point that moves as though (1) all of the system's mass were concentrated there and (2) all external forces were applied there." (Resnick et al, p.215)

"The gravitational force on a body effectively acts at a single point, called the centre of gravity of the body." (Resnick et al, p.330)

According to these concepts:

If we have a body large enough that we could not assume that the centre of mass and the centre of gravity doesn't coincide. Then, by definition of the centre of gravity, the gravitational force does not act on the centre of mass. This means that we have an external force which we can't consider to be acting on the centre of mass.

Q (1)-Does the centre of mass still move as if the gravitational force were applied there?

Q (2)-Aren't this concepts the best definitions of the center of mass and centre of gravity?

Q (3)-What would be the axis rotation if I apply a force to one corner of this body?

Source:
Fundamentals os physics / Robert Resnick, David Halliday, Robert Resnick – 10th edition.

Center of Mass is a mathematical concept used for ease of analysis because we are familiar with Equations of Motion for a point particle. Center of Gravity is a quite similar concept but very special to gravitational forces acting on a body.

Now, Center of Mass is the averaging of the position of masses while Center of Gravity is the averaging the position of Gravitational Forces acting on masses.

Since we observe events in the classical regime, Gravitational Field or $$g$$ value(which is just a practical representation of field - not to be confused with the Field itself) is uniform around us on Earth. So, it is likely to coincide with the center of mass.

But, the gravitational field varies as we move away from earth or as we go high. But even at the top of Mount Everest (8848 metres), the gravitational field strength is still 99.6% of its standard value. So, there will be very slight difference always w.r.t to Center of Mass and Center of Gravity which is negligible for most of the purposes.

However, if you have a varying gravitational field w.r.t position, say, $$\Phi(\vec{r})$$, then the centre of Gravity is going to be different from the centre of mass of the body or the system which we wish to study.

Q (1)-Does the centre of mass still move as if the gravitational force were applied there?

Yes, the motion along a path is the same as if net gravitational force acts on the centre of mass. But since both the centres are not the same, it will add up to a rotational motion of the body/system.

Q (2)-Aren't this concepts the best definitions of the center of mass and centre of gravity?

This question, I tried to establish the concepts above.

Q (3)-What would be the axis rotation if I apply a force to one corner of this body?

Depends on the problem, force and the body itself!