 Contact loss on a smooth sphere

A particle released from an infinitesimal displacement near the top of a smooth sphere has $v^2=2gR(1-\cos \theta )$. The normal reaction vanishes when $mg\cos \theta =m{v}^2/R$, giving $\cos \theta =2/3$. The travelled arc is $R\arccos (2/3)$, whereas the vertical drop is $R/3$.