As shown in the figure below, an L$-$shape gate is used in a reservoir to control the level of the free

surface. The force $F$ keeps the gate closed and the hinge is frictionless. The gate has a width $b,$ a

vertical length $L_1,$ and horizontal length $L_2,$ which is half the vertical length $(L_2=\frac{L_1}{2}).$ The moment

at the hinge caused by the weight of the gate can be neglected.

Obtain an expression for $F$ that is a function of only variables defined in the figure $(\{$:$,H,b,L_1,L_2).$ Your

solution must include a free$-$body diagram of the gate and the expression for the force $F$ must be in its

simplest form.