Two rigid plastic panels lie in the planes z -b and z b

Question

Two rigid plastic panels lie in the planes z = -b and z = b respectively. A rigid ball of radius b can move in the space between the panels and is gripped by them so that it does not slip. The panels are made to rotate with angular velocities ω₁k, ω₂k about fixed vertical axes that are a distance 2c apart. Show that, with a suitable choice of origin, the position vector R of the centre of the ball satisfies the equation

R^· = π X R.

where  = ½ (ω₁ + ω₂). Deduce that the ball must move in a circle and find the position of the centre of this circle.

[This arrangement is sometimes seen as a shop window display. The panels are transparent and the ball seems to be executing a circle in mid-air.]