B$2.$ There is a self$-$opening rectangular flat gate that rotates around a hinge, as shown in the figure below. The gate width is $6m.$ The gate is in equilibrium when the water level on the right side of the gate is $h_1=0\mathrm{.}8m$ and the water depth on the left side is $h_2=0\mathrm{.}5m.$

$=45,$ and the specific weight of water is $9\mathrm{.}81k\frac{N}{m^3}.$

$($a$)$ Find the forces exerted on the gate by the water from each side of the gate. $(6$ marks$)$

$($b$)$ Find the locations of the center of pressure exerted by the water from each side of the

gate. The second moment about the centroidal axis of a rectangular area is $I_{,c}=\frac{b{l}^3}{12},$

where $b$ is the width, and $l$ is the height. $y-$coordinate of center of pressure is $y_{cp}=$

$\frac{?}{\text{b}\text{a}\text{r}}(y)+\frac{I_{,c}}{(\frac{?}{\text{b}\text{a}\text{r}}(y))A},$ where $\frac{?}{\text{b}\text{a}\text{r}}(y)$ is the coordinate of the centroid, and $A$ is the area.$(6$ marks$)$

$($c$)$ If the weight of the gate and the friction of the hinge are ignored, find the range of the height of the hinge $H$ that allows the gate to open automatically, assuming that the water levels remain the same.

$(8$ marks$)$