Calculate the moment of inertia of a uniform rod of length $2 l$ and mass $M$ about an axis through its centre and perpendicular to its length. Assuming it moves in a plane, give an expression for the kinetic energy of the rod in terms of the speed of the centre and the angle that it makes with a fixed direction.

Two such rods are freely hinged together at one end and the other two ends slide on a perfectly smooth horizontal floor. The rods are initially at rest and lie in a vertical plane, each making an angle $\alpha$ to the horizontal. The rods subsequently move under gravity. Calculate the speed with which the hinge strikes the ground.