Rolling-cylinder loop contact threshold ($h_{\min }=11(R-a)/4$)

A solid cylinder rolling inside a circular loop has centre-path radius $\ell =R-a$, kinetic energy $3m{v}^2/4$, 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.

At the limiting release height for a solid cylinder, the loop normal force is $N=(7mg/3)(1+\cos \theta )$ and the required static-friction magnitude is $mg|\sin \theta |/3$. Thus $|f|/N=|\sin \theta |/[7(1+\cos \theta )]$ diverges near the top, even though both forces vanish there. A finite friction coefficient therefore requires a height above the ideal contact threshold to maintain no-slip rolling throughout. A question imposing rolling but giving no coefficient determines the ideal limiting bound, not a realistic friction-dependent minimum.