 Rolling-cylinder loop contact threshold

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.