8.3 Challenging Integral Applications

Physics & Engineering Applications

Question #2 (Medium): Hydrostatic Force on a Submerged Vertical Plate

Strategy

The hydrostatic force is given by

, where is the water density,

the depth below the surface, the area of the thin th strip.

Sample Question

A vertical plate that is submerged in water has the shape shown by the diagram. Express the hydrostatic

force against one side of the place by a Riemann sum as well as an integral, and then evaluate it.

[a semi-circle with diameter of meter submerged meter deep]

Solution

Since the units are given in the Standard International metric system,

As for the th rectangular strip, the area is . If any th strip is chosen to be x distance from the

bottom of the semi-circle, then based on Pythagorean Theorem, the width (or even length) is:

. The distance the th strip is located beneath the water surface is

. Thus:

, and by the definition of definite integral, this is equal

to:

. The first integral can be solved using

trigonometric substitution, however that would be the long tedious route. This represents the

top half of a circle with radius 5, but only over interval , so only

of the circle. The area

would then be

. For the second integral, it is in chain rule form, so let ,

then , and

Aligning the interval, when , then ; when

, then . Thus:

Therefore, the hydrostatic force on the plate is .

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###### Document Summary

Question #2 (medium): hydrostatic force on a submerged vertical plate. The hydrostatic force is given by the depth below the surface, the area of the thin th strip. A vertical plate that is submerged in water has the shape shown by the diagram. Express the hydrostatic force against one side of the place by a riemann sum as well as an integral, and then evaluate it. [a semi-circle with diameter of meter submerged meter deep] Since the units are given in the standard international metric system, ( ) As for the th rectangular strip, the area is . If any th strip is chosen to be x distance from the bottom of the semi-circle, then based on pythagorean theorem, the width (or even length) is: The distance the th strip is located beneath the water surface is ( )

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