 Solution

For contact loss on a smooth sphere, let $\theta$ be the angle from the upward vertical. In the limit of an infinitesimal initial displacement, conservation of energy gives $v^2=2gR(1-\cos \theta )$. The inward radial equation isContact is lost at $N=0$, so $\cos \theta =2(1-\cos \theta )$ and $\theta =\arccos (2/3)$. The distance travelled along the sphere is thereforeThe vertical drop is instead $R/3$; it is not the requested arc length. Exact rest at the unstable top would persist in the ideal mathematical solution, so the infinitesimal-displacement limit in the question is essential.