A $4\mathrm{.}5-$m$-$long quarter$-$circular gate of radius $3$ m and of negligible weight is hinged about its upper edge $\backslash ($ A $\backslash ),$ as shown in the figure. The gate controls the flow of water over the ledge at $\backslash ($ B $\backslash ),$ where the gate is pressed by a spring. Determine the minimum spring force required to keep the gate closed when the water level rises to $\backslash ($ A $\backslash )$ at the upper edge of the gate. Take the density of water to be $1000\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$