= Two hinged rods with both ends sliding on a floor
{title2=$v_{\rm hinge}=\sqrt{6gl\sin\alpha}$}
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> $4Ml^2\dot\theta^2/3$. 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.
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