Two uniform circular discs, each of mass m and radius a, are joined by a light (massless) rod AB of length 4a connected to the discs at their centers A, B in such a way that the planes of the discs are perpendicular to the rod and the discs can turn freely about the rod. The system (discs and rod) is placed on a horizontal table which is rough enough to prevent slipping and which is forced to rotate with variable angular velocity Ω about a vertical axis passing through the center O of AB. Initially the system is at rest.
Let i be the unit vector along OA, let k be a unit vertical vector, and let j = k x i. Let α be the angle between a fixed line on the disc with center A and a the vertical; and let φ be the angle between AB and a fixed horizontal line. Figure below shows the arrangement.
1. Determine the condition for rolling without slipping.
2. Determine the Lagrangian equations of motion with constraint force.
3. Find the angular velocity of the discs in terms of Ω.