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File: TheOtherOtherRopeAroundTheEarth

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Previous blog posts:
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Additionally, some earlier writings:
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!!! The /Other,/ Other Rope Around The Earth Problem - 2015/08/01

There's a classic problem:

[[[>30 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?

An alternative that's been suggested is this:

[[[>30 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?

But now Bill Mullins has asked me yet another variant:

* 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?

If it falls, where? Assuming it remains stiff (and we're in
PuzzleWorldVsCryptoWorld
so let's assume it does), that means that if it
falls in one place, that means that on the other side of the
world it rises.  Maybe it just floats there, stable. Or maybe
it does fall in one place, rises at the antipodal point, and
then remains there, off-centre, grazing the Earth's surface.

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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