= Winding time around a circular pole
{title2=$t_{\rm hit}=L^2/(2Rv)$}
A point <mass> attached to a light inextensible string winds around a fixed smooth circular pole of radius $R$. Let the initial straight free <length> be $L$, and assume the string remains taut without slipping over the part already laid on the pole. Since its <tension> is perpendicular to the point <mass>'s <velocity>, the <speed> $v$ stays constant. If the contact point turns through $d\theta$, the free <length> $s$ obeys $ds=-R\,d\theta$, while the <involute> geometry gives $v=s\dot\theta$. Thus $s\dot s=-Rv$, or $s^2=L^2-2Rvt$. First contact of the point <mass> with the pole occurs at $s=0$, after $L^2/(2Rv)$. In the ideal model, the required <tension> $mv^2/s$ diverges at contact; finite string strength or finite <particle> size limits that idealization.
Back to article page
