Sphere rolling off a hemisphere ($\cos {\theta }_d=10/17$)

A uniform solid sphere released from the top of a fixed spherical surface has $v^2=10g(R+r)(1-\cos \theta )/7$ under the ideal rolling constraint. The normal force is $Mg(17\cos \theta -10)/7$, giving the title's ideal detachment angle. The required static friction is $2Mg\sin \theta /7$. Its ratio to the normal force diverges at detachment, so any finite coefficient of static friction causes prior slipping; a physical subsequent trajectory requires a sliding-friction model.