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The Other, Other Rope Around The Earth Problem - 2015/08/01

There's a classic problem:

If you don't know this one you really need to have a go at it.

• Upon stretching a rope around the Earth, you find that you have 6 metres excess. So you join the ends, and then go around the Earth propping up the rope equally everywhere. How high will it be?

This is also a fun question, although less elementary.

• Instead of propping it up equally everywhere, just prop it up as high as possible in one place. If it doesn't sag, how high is the prop?

• Suppose you have a rope, propped up one metre above the Earth's surface, running right around the Earth. Now suppose instead of being rope, it's steel. Release the props.
  
  What happens?

What happens?

If we consider the centres of mass of the Earth and the ring of steel, we can consider what happens when the ring is off centre. The two centres of mass will surely be attracted, so surely the ring floats back to be centred, doesn't it?

Or does it? We know that we can't always replace objects with a point mass at their centre of mass, and maybe this is one of those cases.

So what does happen?

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