 Two hinged rods with both ends sliding on a floor

Two identical uniform rigid bodies of rod shape, each of length $2l$ and mass $M$, form a symmetric arch with a free common hinge and both outer ends sliding without friction on a horizontal floor. At angle $\theta$, each center has speed $l|\dot{\theta }|$, giving total kinetic energy $4M{l}^2{\dot{\theta }}^2/3$. The total gravitational potential energy is $2Mgl\sin \theta$. Conservation of energy from rest at $\alpha$ gives the displayed hinge impact speed. Unlike fixing one outer endpoint, allowing both endpoints to slide keeps the arch center horizontally fixed. The floor reaction on each rod is $Mg[1+9{\sin }^2\theta -6\sin \alpha \sin \theta ]/4$, nonnegative throughout this fall.