= Grazing clearance of an impulsively struck rod
{title2=$\beta_{\mathrm{crit}}=1/(2\pi)$}
A thin straight rod of <length> $l$, <mass> $m$, central <moment of inertia> $I=\beta ml^2$ 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.
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