Calculate the horizontal hydrostatic force on the vertical gate $($width $0\mathrm{.}5$ m $)$ shown in Figure $1.$

Figure

$(25$ points$)$ For pipe flow of an incompressible liquid, experimental data indicates that the flow can be classified as laminar or turbulent depending on the magnitude of the dimensionless parameter, Reynolds number $(Re=R\frac{}{}u(r)=2($:$u$:$)[1-(\frac{r}{R}{)}^2]$

$u(r)=1\mathrm{.}22($:$u$:$)[1-\frac{r}{R}{]}^{\frac{1}{7}}($:$u^2$:$)=($:$u$:${)}^2($:$u^3$:$)=($:$u$:${)}^3($:$u^n$:$)=\frac{1}{A}[u(r){]}^{n}dA$