Rectangular plate $AB$ has uniformly distributed mass and $1m$ wide into the paper, hinged at $A,$ and resting

on a smooth bottom at B$.$ The bottom exerts no vertical force on B$..$ Neqlect air pressure. ${}_{\text{W}\text{a}\text{t}\text{e}\text{r}}=9790\frac{N}{m^3}.$

Find: $$

$(1)$ The resultant hydrostatic force on gate $AB(3$ points$)$;

$(2)$ Weight of the gate $AB$ that keeps the gate closed $(7$ points$).$ Hint: the center of pressure is at a distance

of $\frac{1}{3}\frac{p_A+2p_B}{p_A+p_B}{L}_{AB}$ from A along the plate, with $p_A,p_B$ being the hydrostatic pressure at $A,B,$ respectively,

and $L_{AB}$ is the length of $AB.$