Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B8 / b / Solution

Set $s=\sin \theta$. The maximum height becomes the concave quadratic functionUnder the stated high-angular frequency assumption, $r{\omega }^2>g$, its interior maximum isThus the optimum release lies on the upper-left quarter of the wheel for the printed clockwise rotation. The largest attained height isIf extended to $r{\omega }^2\le g$, the maximum over release positions is the limiting height $r$ at the top of the rim. The interior solution is the relevant branch for reaching a ceiling above the wheel.