  A beer bottle or soda can on a table, when slightly tipped and released,
falls to an upright position and then rocks up to a somewhat opposite tilt.
Superficially this rocking motion involves a collision when the flat circular
base of the container slaps the table before rocking up to the opposite tilt. A
keen eye notices that the after-slap rising tilt is not generally just
diametrically opposite the initial tilt but is veered to one side or the other.
Cushman and Duistermaat (2006) recently noticed such veering when a flat disk
with rolling boundary conditions is dropped nearly flat. Here, we generalize
these rolling disk results to arbitrary axi-symmetric bodies and to
frictionless sliding. More specifically, we study motions that almost but do
not quite involve a face-down collision of the round container's bottom with
the table-top. These motions involve a sudden rapid motion of the contact point
around the circular base. Surprisingly, like for the rolling disk, the net
angle of motion of this contact point is nearly independent of initial
conditions. This angle of turn depends simply on the geometry and mass
distribution but not on the moment of inertia about the symmetry axis. We
derive simple asymptotic formulas for this "angle of turn" of the contact point
and check the result with numerics and with simple experiments. For tall
containers (height much bigger than radius) the angle of turn is just over
$\pi$ and the sudden rolling motion superficially appears as a nearly symmetric
collision leading to leaning on an almost diametrically opposite point on the
bottom rim.

Classical Physics;Popular Physics