A rigid rod of length 2a has its lower end in contact with a smooth horizontal floor. Initially the rod is at an angle α to the upward vertical when it is released from rest. The subsequent motion takes place in a vertical plane. Take as generalised coordinates x, the horizontal displacement of the centre of the rod, and θ, the angle between the rod and the upward vertical.
Obtain Lagrange's equations. Show that x remains constant in the motion and verify that the θ-equation is equivalent to the energy conservation equation. Find, in terms of the angle θ, the reaction exerted on the rod by the floor.