Center of gravity in wrong place?

[JohnPage]

Textbooks all say that the center of gravity of an object is the point at which all it's gravitaional effects appear to be concentrated. But that seems to depend on who's checking.

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Imagine a small mass M some distance from the Earth. That mass is attracted simultaneously by all the particles that make up the Earth. At first blush you would think that since the Earth is symetrical, they would average out and the center of gravity is at the geometric center. But that cannot be right.

The gravitaional attraction between M and each particle is an inverse square law. That means that particles that are symmetrically each side of the Earth's center do not average to the center. So as far as M sees it, the center of gravity must be closer to m than the center point, right?

I would have thought this would be a major effect on how M would orbit the Earth. Seems to me that the mass M would see the CG as a ring. The CG would follow M around a circular path around the center point.

What am I missing?

 

[ModusPwnd]

"t first blush you would think that since the Earth is symetrical, they would average out and the center of gravity is at the geometric center. "Newton proved that is the case. The proof is often located in Calculus or University Physics texts. Check it out.

 

[AlephZero]

Imagine a small mass M some distance from the Earth. That mass is attracted simultaneously by all the particles that make up the Earth. At first blush you would think that since the Earth is symetrical, they would average out and the center of gravity is at the geometric center. But that cannot be right.

Correct. But that is talking about something different from the center of gravity of the Earth on its own. You can choose to ignore the gravitational attraction of everything else in the universe (including the moon) or to include it, and it shouldn't be a surprise that the two situations are different.

The moon and Earth both orbit around their combined center of gravity position, whcih is not at the center of the earth.

If the moon was the same size as the Earth the CG of the combined bodies would be half way between the line joining them, and both orbits would follow the same curve around that point. There are system like that which can be observed, for example binary stars.