Physics
posted by Chris .
A particle is at rest at the apex A of a smooth fixed hemisphere whose base is horizontal. The hemisphere has centre O and radius a. The particle is then displaced very slightly from rest and moves on the surface of the hemisphere. At the point P on the surface where angle AOP = ¦Á the particle has speed v. Find an expression for v in terms of a, g and ¦Á.

Use conservation of energy. At angle A (measured from vertical), the particles elevation has decreased by a (1  cosA).
m g (1cosA) = (1/2) m V^2
V = sqrt[2*(1cosA)/g]
This assumes that the particle remains in contact with the hemisphere. At some point it may (or may not) be fast enough to leave the surface. That is a separate problem. 
According to the analysis here, the particle does leave the sphere 1/3 of the way down:
http://www.feynmanlectures.info/solutions/particle_on_sphere_sol_1.pdf