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:

,

,

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

W_{gate} is the weight of the gate, which is located at the center of the gate. The tension is then:

.

[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.