A $4\mathrm{.}100-$m$-$long quarter$-$circular gate of radius $2$ m and of negligible weight is hinged about its upper edge $A,$ as shown in the figure. The gate controls the flow of water over the ledge at $B,$ 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 $A$ at the upper edge of the gate. Take the density of water to be $1,000k\frac{g}{m^3}.$

The minimum spring force required to keep the gate closed is

kN $.$

Find the force applied by support $BC$ to the gate $AB.$ The width of the gate and support is $3$ m and the weight of the gate is $1,600N.$ Take the density of water to be $1,000k\frac{g}{m^3}.$

The force applied by support $BC$ to gate $AB$ is

N$.$

Gate $AB(0\mathrm{.}6-m0\mathrm{.}9-m)$ is located at the bottom of a tank filled with methyl alcohol $(SG=0\mathrm{.}79)$ and hinged along its bottom edge $A.$ Knowing that the weight of the gate is $320$ N $,$ determine the minimum force that must be applied to the cable $(BCD)$ to open the gate. Take the density of water to be $1,000k\frac{g}{m^3}.$

The minimum force that must be applied to the cable to open the gate is

N$.$

An open settling tank shown in the figure contains a liquid suspension, as shown in the figure. Determine the resultant force acting on the gate and its line of action if the liquid density is $870k\frac{g}{m^3}.$

The resultant force acting on the gate is kN $.$ The line of action of the force is m from the bottom of the tank. $($Round the final answer to three decimal places.$)$

For non$-$symmetric immersed surfaces, the coordinate of the pressure center is not zero and can be either a positive or negative number. Derive an expression for $x$ coordinate of the pressure center and determine the resultant force acting on the $0\mathrm{.}7-$m$-$high and $0\mathrm{.}7-$m$-$wide triangular gate shown in the figure and its line of action. Take the density of water to be $1,000k\frac{g}{m^3}.$ The distance $d$ from the surface to the top of the gate is $0\mathrm{.}310$ m as shown in the figure.

The resultant force acting on the triangular gate is $q,$ IN$.$

The line of action of the force is Im$.($Round the final answer to three decimal places.$)$