## Motion of Particle on Inner Surface of Smooth Sphere

A particle travelling on the inside surface of a smooth sphere will just leave the surface when the reaction between the particle and the inner surface is zero.

Suppose a particle of mass at the bottom A of a sphere of radius r is given a horizontal speed It will move along the internal surface of the sphere. Suppose at some point B it has a speed  If the line from the centre of the circle to the point B makes an angle with the upwards vertical then the force of gravity will have a component towards the centre of the circle. Applying towards 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 Hence Cancelling gives (2)

Put in equation (1) and cancel from both sides, obtaining then Substitute this into (2) to obtain Rearranging this formula for gives Since we must have Obviously The particle will leave the inside surface at some point if This corresponds to the particle not having enough initial kinetic energy to reach the top of the sphere. 