 Hinge impulse of an edge-pivoted square plate

For a uniform square plate of side $l$ hinged along an edge, $h=l/2$ and $I_H=M{l}^2/3$. An impulse $K$ normal to the plate, applied a distance $a$ below the centre, gives $v_G=3K(1+2a/l)/(4M)$ and signed hinge impulse $J_H=K(3a/(2l)-1/4)$. Therefore the hinge impulse vanishes at $a=l/6$ below the centre, a distance $2l/3$ from the hinge. The sign of $J_H$ identifies the reaction direction; its magnitude is $|J_H|$.