A particle moving on the surface of a smooth sphere will remain in contact with the sphere as long as the reaction force is greater than zero. We can use this as a condition to find the point at which the particle loses contact with the sphere.
If at some point the line joining the position of the particle to the centre of the circle makes an angle %theta with the downward vertical then the component of gravity towards the centre of the circle is
Using
towards the centre of the circle gives
When
(1)
If the particle starts from rest at the top of the sphere, at the angle
above it will have fallen a vertical distance
losing
in gravitational potential; energy and gaining an equivalent amount of kinetic energy.
Substitute this into (1) to obtain