$(30$ pts $)$ Consider the hinged gate ABC shown below. The width of the gate into the page is $1$ ft $.$ The weight of the gate is negligible. Segment $AB$ of the gate is curved forming the shape of one$-$quarter of a cylinder with radius $R=2ft.$

Segment BC of the gate is planar in the shape of a rectangle. Establish a free body diagram and solve for the magnitude of the force vec$(Q)$ required for the gate to remain in the position shown. The specific weight of water is ${}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=62\mathrm{.}4l\frac{b}{f}{t}^3.$