Grazing clearance of an impulsively struck rod (${\beta }_{\mathrm{crit}}=1/(2\pi )$)

A thin straight rod of length $l$, mass $m$, central moment of inertia $I=\beta m{l}^2$ and midpoint centre of mass starts beside a pointlike pole at that midpoint. An end impulse perpendicular to the rod gives translation $V=J/m$ and rotation $\Omega =J/(2\beta ml)$. At the first quarter-turn the trailing end is at horizontal distance $\pi \beta l-l/2$ from the pole. Strict clearance requires $\beta >1/(2\pi )$; equality is the grazing limit. The criterion is for free motion before contact, a negligibly thin rod and pole, and an impulse directed away from the pole.