= Sphere rolling off a hemisphere
{title2=$\cos\theta_d=10/17$}
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.
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