## Water Gate

In the figure below, consider a 2 m high and 1 m wide rectangular gate is attached via a frictionless hinge at its base, and is holding back a tank of water. If water has a density of 1,000 kg/m^3, what force is required to keep the gate closed?

##
__
__**Hint**

**Hint**

$$$Pressure=\rho gh=\frac{Force}{Area}$$$

##
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__**Hint 2**

**Hint 2**

Since atmospheric pressure acts on both the non wet side of the gate, and the body of water's surface, the resultant force acting on the gate is:

$$$F_{R}=(\rho gh)sin\theta(A)$$$

where
$$\rho$$
is the density,
$$g$$
is gravity,
$$h$$
is height, and
$$A$$
is area.

Remember:

$$$Pressure=\rho gh=\frac{Force}{Area}$$$

$$$Force=\rho gh(Area)$$$

Since atmospheric pressure acts on both the non wet side of the gate, and the body of water's surface, the resultant force acting on the gate is:

$$$F_{R}=(\rho gh)sin\theta(A)$$$

where
$$\rho$$
is the density,
$$g$$
is gravity,
$$h$$
is height, and
$$A$$
is area. In our situation, the mean pressure of the fluid acting on the gate is evaluated at the mean height, and the gate is completely vertical, so
$$\theta = 90$$
degrees. Thus,

$$$F_R=(1,000\frac{kg}{m^3})(9.8\frac{m}{s^2})(\frac{2m}{2})(sin90^{\circ})(2m\times 1m)$$$

$$$F_R=(1,000\frac{kg}{m^3})(9.8\frac{m}{s^2})(1)(1m)(2m^2)=19,000\frac{kg\cdot m}{s^2}=19,000\:N$$$

Next, let's determine where the center of pressure is, which is where the resultant force is applied:

$$$y_{CP}=y_C+\frac{I_{xC}}{y_CA}$$$

where
$$y_C$$
is the gate's centroid,
$$I_{xC}$$
is the moment of inertia about the centroidal x-axis for a rectangle, and
$$A$$
is the area.

$$$I_{xC}=\frac{(base)(height)^3}{12}=\frac{(1m)(2m)^3}{12}=0.67m^4$$$

Thus,

$$$y_{CP}=(\frac{2m}{2})+\frac{0.67m^4}{(\frac{2m}{2})(2m\times1m)}=1m+\frac{0.67m^4}{2m^3}=1.33\:m$$$

The resultant force is 1.33 m from the water's surface, or 0.67 m above the hinge. Performing a moment balance about the hinge:

$$$\sum M_{hinge}=0=force\times distance$$$

$$$F_R(0.67m)-F(2m)=0$$$

$$$F=\frac{(19,000N)(0.67m)}{2m}=6,365\:N$$$

6,365 N