= Rolling-cylinder loop contact threshold
{title2=$h_{\min}=11(R-a)/4$}
A solid cylinder rolling inside a circular loop has centre-path <radius> $\ell=R-a$, <kinetic energy> $3mv^2/4$, and <speed> $v^2=4g[h-\ell(1-\cos\theta)]/3$ at angle $\theta$ from the bottom. The normal contact <force> is smallest at the top. Requiring $v_t^2\ge g\ell$ gives the displayed necessary contact threshold under imposed no-slip rolling. The centre's maximum <acceleration> on the circle is at the bottom, and at the threshold it is $11g/3$ upward. This is a limiting contact condition, not a guarantee for arbitrary finite <static friction> or an unspecified approach curvature.
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