destroy this school meeting
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hmm okay
If the ring is symmetrical the centre of mass is in the middle. The centre of mass does not have to be in the material of the ring.
Note that the centre of gravity is only in the same place as the centre of mass if there is a uniform gravitational field. For small objects you can assume that the earth's gravitational field is uniform but for something like a space tether there could be a difference.
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