### 10th Putnam 1950

Problem B4

A long, light cylinder has elliptical cross-section with semi-axes a > b. It lies on the ground with its main axis horizontal and the major axes horizontal. A thin heavy wire of the same length as the cylinder is attached to the line along the top of the cylinder. [We could take the cylinder to be the surface |x| ≤ L, y2/a2 + z2/b2 = 1. Contact with the ground is along |x| ≤ L, y = 0, z = -b. The wire is along |x| ≤ L, y = 0, z = b.] For what values of b/a is the cylinder in stable equilibrium?

Solution

Take the ellipse to be x2/a2 + y2b2 = 1. We need the normal to the ellipse at a point near (0, -b) to meet the y-axis above (0, b) (because then the heavy wire will be the correct side of the vertical through the point of contact to right the cylinder).

The gradient at (a cos t, b sin t) is -b/a cot t, so the normal has slope a/b tan t, so its equation is (y - b sin t) = a/b tan t (x - a cos t). This meets the y-axis at y = b sin t - a2/b sin t. For points near (0, -b), sin t is -(1 - ε) so we need a2/b - b > b or a > b√2.