# The Bowed Metre Stick

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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Doesn't the answer depend on the nature of the material under consideration? I suspect that the answer needs some 'physical' assumptions about the ruler (e.g., Young's modulus) and hence a purely mathematical answer may not be available. – tards Nov 4 '11 at 5:42
Why wouldn't Euler-Bernoulli apply? – J. M. Nov 4 '11 at 8:45
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). – Mark Bennet Nov 4 '11 at 9:34

I'd think that the true shape is the solution of a variational problem; e.g., minimize $\int_\gamma \kappa^2(s)\ ds$, and that polynomial graphs are just an approximation to this. – Christian Blatter Nov 4 '11 at 10:01
@Gerry Myerson: The strip doesn't know what your coordinate system is. Vanishing of the 4th derivative with respect to $x$ is no geometric invariant. – Christian Blatter Nov 4 '11 at 13:08