Rocking by rolling: super-fast near-collisional dynamics of perturbed vertical cylinders

A slightly tilted vertical cylinder on a table-top rights itself, coming to a halt after "rocking" for a while. These rocking motions may involve collisions of the cylinder's circular bottom surface with the table-top. Here we study the more likely motions that almost but do not quite involve a face-down collision of the cylinder's bottom with the table-top. For ideal cylinders that can roll without slip or slide without friction on their circular bottoms, the typical rocking motions do not involve collisions. Instead, they involve slow inverted pendulum-like motions interrupted by infinitely fast rolling or sliding motions during which the contact point with the table shifts by a finite angle in infinitesimal time. We derive simple asymptotic formulas for this "angle of turn" of the contact point for two simple cases: sliding without friction and rolling without slip. These formulas for the angle of turn depend only on the properties of the cylinder and the assumed friction law. These results explain why not-too-tall cylinders or disks never seem to rock symmetrically on diametrically opposite ends. Also, on the other hand, very tall cylinders that appear to rock back and forth on diametrically opposite points on the bottom rim are really (often) rolling and/or sliding very quickly on the rim by a little over 180 degrees