This Demonstration determines the cable tension necessary to support a gate submerged under water (see Figure 1). The gate is

feet wide at the base,

feet long diagonally and measures

feet of diagonal length.

The magnitude of the resultant force is found by summing the differential forces

over the entire surface:

,

where

is the resultant force (N),

is the specific weight of the fluid (

),

is the vertical distance from the fluid surface to the centroid center (m),

is the

coordinate of the height along the diagonal (see Figure 2 for coordinate system),

is the angle (degrees), and

is area (

).

For constant

and

, the resultant force becomes:

.

The integral

is the first moment of the area with respect to the

axis, so:

,

where

is the

coordinate of the centroid of area measured from the

axis that passes through 0.

The resultant force can be written as:

,

where

is the vertical distance from the fluid surface to the centroid of area.

The

coordinate

of the resultant force is determined by summing the moments around the

axis. Thus, the moment of the resultant force must equal the moment of the distributed pressure forces, or:

.

Therefore, since

:

.

The integral in the numerator is the second moment of the area (moment of inertia)

, with respect to an axis formed by the intersection of the plane containing the surface and the free surface (

axis):

.

Since

, where

is the second moment of the area with respect to an axis passing through its centroid and parallel to the

axis (

):

,

where

.

A moment balance determines the tension in the cable that is holding up the gate:

.

Rearranging to solve for tension:

,

where

is tension (N), and

is the weight of the gate (N).

[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.