Ball rolling on a rotating turntable
A rough horizontal turntable is made to rotate about a fixed vertical axis through its centre O with constant angular velocity k, where the unit vector Ωk points vertically upwards. A uniform ball of radius a can roll or skid on the turntable. Show that, in any motion of the ball, the vertical spin ω · k is conserved. If the ball rolls on the turntable, show that
V· = 2/7πk X V,
where V is the velocity of the centre of the ball viewed from a fixed reference frame. Deduce the amazing result that the path of the rolling ball must be a circle.
Suppose the ball is held at rest (relative to the turntable), with its centre a distance b from the axis {O, k}, and is then released. Given that the ball rolls, find the radius and the centre of the circular path on which it moves.