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This problem is being discussed on the AAMT email discussion list. I have a meter long metal ruler. I push the ends together so that they're only 99cm apart, which means the ruler will bow a bit. How tall is that arc?

The problem we haven't been able to solve is 'what is the shape of the arc'?

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    You can imagine bending a much longer strip into a variety of shapes - a circle, for example, or where the two "ends" cross at right angles. You can then find a chord which is 99% of the length of the arc it cuts off (intermediate value theorem), and scale to match the problem. So a variety of shapes will be possible. What makes the difference, I think (this may be wrong), is the direction of the forces applied at the end of the arc and stuff like gravity (the answer will be a little different in a horizontal v vertical plane).2011-11-04

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Elastic materials assume shapes that are the graphs of polynomials of degree at most 3. Knowing the length of the ruler, the distance between the endpoints, and making an assumtion about symmetry with respect to the middle may give you enough data to get the equation.

Alternatively, if all you want is a bound, you could probably assume some extreme shapes, like the equal sides of an isosceles triangle, and get an answer that way.

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    @Christian, you have a point, and I currently have no answer.2011-11-04