= Slipping solid cylinder on an incline
{title2=$\tan\alpha>3\mu$}
For a uniform solid cylinder, take $s$ positive uphill and $\omega$ positive for uphill <rolling without slipping>. The velocity of the contact point is $w=\dot s-a\omega$. Uphill <kinetic friction> gives $\ddot s=-g(\sin\alpha-\mu\cos\alpha)$ and $a\dot\omega=-2\mu g\cos\alpha$. Starting from $\dot s(0)=a\omega(0)$, one obtains $w=-gt(\sin\alpha-3\mu\cos\alpha)$. Thus $\tan\alpha>3\mu$ makes the assumed uphill <kinetic friction> self-consistent for the entire ascent and descent. Without slip the required uphill <static friction> is $mg\sin\alpha/3$; it is available exactly when $\tan\alpha\leq3\mu$ under the same limiting coefficient.
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