Examples

Fluid force on a triangular plate

Find the total force on the submerged vertical plate with the following shape: Equilateral triangle, sides , top vertex at the surface, liquid is oil with density .

Solution

(1) Write the width function:

Establish coordinate system: at water line (also the vertex), and increases going down.

Method 1: Geometry of similar triangles

Top triangle with base at is similar to total triangle with base .

Therefore, corresponding parts have the same ratios.

Therefore:

Method 2: Quick linear interpolation function

Generalized “quick linear interpolation function”

Generalization:

where:

• is when comes earlier (smaller ), and if it comes later
• is created to force the quantity to equal for the given value at

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(2) Compute integral using width function:

Bounds:

• (shallow)
• (deep)

Integral formula:

Weight of water on a dam

Find the total hydrostatic force on an angled dam with the following geometric description: Tilted trapezoid. Base , Top , and vertical height . The base is tilted at an angle of .

Solution

(1) Write the width function:

Establish coordinate system: at water line (also the top edge), and increases going down.

“Quick linear interpolation function”:

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(2) Incorporate angle of incline in strip thickness:

So the area of a strip is:

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(3) Compute total force using integral formula.

Plug data into formula: