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The solution to the problem of a string hung from two points in a uniform gravitational field is well known, its solution is the catenary. A solution can also be found for the problem of a spring of initially uniform mass hung between two points in a uniform gravitational field. This then leads me to wonder: is the formula for a heavy elastic material hanging from a circular rim a rotational analogue of that of the aforementioned spring problem? Presumably more tension would be exerted on any given point, but would the solution still be similar in form?

I've taken a look at some of the links mentioned here, but many of them deal with this in a far more general form, and I will admit to being at a loss at several points. Many of them also simply relate to the problem of an inelastic surface, as well.

Hence my question: Is a solution known to this problem, and if so, what is it?

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