Consider a small frictionless puck perched at the top of a xed sphere of radius R. If the puck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere?

Let the vertical height by which it descends be h. Let it acquire velocity of v.

1/2 mv² = mgh

v² = 2gh

As it leaves the surface of sphere, reaction force of surface R = 0, so

centripetal force = mg cosθ where θ is the angular displacement from the vertex.

mv² / r = mg cosθ

(m/r) x 2gh = mg cosθ

2h / r = cosθ

cosθ = (r-h) / r

2h / r = r-h / r

2h = r-h

3h = r

h = r / 3