MathDB

The cross-section of a right cylinder is an ellipse, with semi-axes

a

and

b,

where

a > b.

The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of its several cross-sections. Along the upper endpoints of these minor axes lies a very heavy homogeneous wire, straight and just as long as the cylinder. The wire and the cylinder are rigidly connected. We neglect the weight of the cylinder, the breadth of the wire, and the friction of the ground.
The system described is in equilibrium, because of its symmetry. This equilibrium seems to be stable when the ratio

b/a

is very small, but unstable when this ratio comes close to

1.

Examine this assertion and find the value of the ratio

b/a

which separates the cases of stable and unstable equilibrium.

Problem Statement