Figure $4.$ Rod on hinge attached to block sliding over ice.

A thin rod with mass $m,$ length $L,$ and uniform mass density pivots on a frictionless hinge attached to a block with mass $M,$ as shown in Fig. $5.$ Introduce $lon=\frac{m}{M}.$ The block slides on a frictionless sheet of ice. At $t=0$ the rod stands vertically, perfectly balanced. At some later time the rod is gently nudged and falls over. On its way down it briefly makes an angle of $\frac{}{4}$ with the horizontal. Assuming the rod's angular velocity is $e_3,$ what is the velocity of the block at that instant?

$()\frac{1}{2}\frac{1}{1+lon}L$

$()\frac{1}{\sqrt[\phantom{2}]{2}}\frac{lon}{1+lon}L$

$()L$