As per Figure 5.9.1(a), let the origin of a Cartesian coordinate system be at the bottom of the porthole, and let measure distance upwards. The surface of the water is then at since there are 20 ft of water above the top of the porthole.
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Figure 5.9.1(a) Porthole under hydrostatic pressure
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From Figure 5.9.1(a), the width of a horizontal strip across the porthole is , where
so the area of this strip is , or
This strip is at a depth of
The force on this strip is
The total force on the porthole is given by the integral
whose value is . At , this evaluates to lbs.
Alternatively, the area of the porthole is , and its centroid (the center) is at a depth of ft. The product of the area and the pressure at the centroid is then lbs.
(Note: There was no mathematical reason for keeping the radius of the porthole as the symbolic . It could just as well have been set immediately to for computational purposes. However, the letter typesets more compactly than the fraction .)