Forces on a Partially Submerged Gate

A gate, which is hinged at the bottom, is partially submerged under water, and a cable holds the gate closed. Use the sliders to set the angle of the gate, the weight of the gate, and the water height. Use the buttons to change the units from and ft (US customary units) to kN and m (SI units). Check the "show distance" box to display distances. The Demonstration displays the cable tension needed to support the gate. When the tension is too high (greater than 4.23 or 18.82 kN), the cable breaks.

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This Demonstration determines the cable tension necessary to support a gate submerged under water (see Figure 1). The gate is meters long, and the distance from the hinge to top of the water along the gate is meters.
The magnitude of the resultant force due to the water is found by summing the differential forces over the entire surface:
,
where is the resultant force (N), is the specific weight of water (), is the vertical distance from the top of the water to any point in the water (m), is the corresponding distance along the gate (m), is the angle of the gate (degrees), is differential area of the gate (), and is the width of the gate (m). Note that the specific weight is the specific gravity times the acceleration of gravity . The total area of the gate that is in contact with water is . This integral is from at the top of the water level to at the hinge.
Since is constant, and for a fixed value of , the resultant force becomes:
.
The integral is:
,
this is then equal to:
,
the resultant force is then:
.
The sum of the moments around the hinge is equal to the moment of the resultant force at the coordinate . Note that moment is proportional to the distance from the hinge to location of the force:
,
,
and since ,
.
That is, the resultant force is located 1/3 of the distance from the hinge to the water level along the gate. A moment balance determines the tension (kN) in the cable that is holding up the gate:
.
where Wgate is the weight of the gate, which is located at the center of the gate. The tension is then:
.
Figure 1
Figure 2
Reference
[1] B. R. Munson, T. H. Okiishi and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2009 pp. 57–60.
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