If the line from the centre of the circle to the point B makes an anglewith the upwards vertical then the force of gravity will have a componenttowards the centre of the circle.
Applyingtowards the centre of the circle gives (1)
Since the inside of the sphere is smooth, no work is done against friction, and energy is conserved in going from A to B. The potential energy at B is
Putin equation (1) and cancelfrom both sides, obtainingthenSubstitute this into (2) to obtain
Rearranging this formula forgives
Sincewe must have
The particle will leave the inside surface at some point ifThis corresponds to the particle not having enough initial kinetic energy to reach the top of the sphere.