= Critical pulling angle of a rolling spool
{title2=$\cos\theta_c=r/R$}
For a spool of outer radius $R$, axle radius $r$, <mass> $M$ and axial <moment of inertia> $I$, a lower-tangent pull of <tension> $T$ at upward angle $\theta$ produces horizontal <acceleration> $a=TR(R\cos\theta-r)/(I+MR^2)$ while <rolling without slipping>. This follows by combining $Ma=T\cos\theta+f$, $I\alpha=Tr+Rf$ and $a=-R\alpha$, with rightward translation and counterclockwise rotation positive. The <acceleration> reverses at $\theta_c$; zero <acceleration> means equilibrium only if initially at rest. At that angle the <static friction> limit gives $T_{\max}=\mu Mg/(\cos\theta_c+\mu\sin\theta_c)$.
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