# Hemisphere sliding down problem

by kevin0960
Tags: hemisphere, sliding
 P: 12 There is a small object, which mass is m on the top of hemisphere, with the mass of M. the size of the object is neglectable. Also, the radius of hemisphere is R. There is no friction between the hemisphere and the object, and the hemisphere and the surface. What if we slightly hit the object the object will be fall from the hemisphere. At what point the mass will completely off from the hemisphere? I attatched the the picture. --- I first tried to use energy conservation law to find the velocity of the object. $$\frac{1}{2} mv^2 + mgR(cos \theta) + \frac{1}{2}MV^2 = mgR$$ Also by conservation of momentum on x axis, we can deduct $$mv_x = MV$$ But I just stucked at here. Plz someone help me Attached Thumbnails
 P: 12 I found few more equations for that problem. First, because the object moves on the hemisphere before it falls off It must satisfy following condition $$\frac{v_y}{v_x + V} = tan \theta$$ Also, I am not quite sure about the equation but since the object is having on circular motion (at the frame of hemisphere), $$\frac{m((v_x + V)^2 + v_y^2)}{R} = mgcos\theta - N$$ and at the time the object falls from the hemisphere, N will be 0 Is this right??