Chapter 5: Applications of Integration
Section 5.9: Hydrostatic Force
Under heavy seas, the top of a cruise ship's circular porthole one foot in diameter is submerged under 20 ft of water. If sea water weighs 62.5 lbs/ft3, find the total force on the porthole.
As per Figure 5.9.1(a), let the origin of a Cartesian coordinate system be at the bottom of the porthole, and let y≥0 measure distance upwards. The surface of the water is then at y=21 since there are 20 ft of water above the top of the porthole.
Figure 5.9.1(a) Porthole under hydrostatic pressure
From Figure 5.9.1(a), the width of a horizontal strip across the porthole is w=2 x, where
x2=r2−r−y2=2 r y−y2
so the area of this strip is dA=w dy, or
dA=2 2 r y−y2dy
This strip is at a depth of
The force on this strip is
P d=62.5 d dA
The total force on the porthole is given by the integral
1252∫02 r20+2 r−y 2 2 r y−y2ⅆy
whose value is 1252π r2r+20. At r=1/2, this evaluates to 512516⁢π≐1006.29 lbs.
Alternatively, the area of the porthole is π 1/22, and its centroid (the center) is at a depth of 20.5 ft. The product of the area and the pressure at the centroid is then 62.5×20.5×π/4≐1006.291397 lbs.
(Note: There was no mathematical reason for keeping the radius of the porthole as the symbolic r. It could just as well have been set immediately to 1/2 for computational purposes. However, the letter r typesets more compactly than the fraction 1/2.)
Expression palette: Definite-integral template
Write the integral giving the total hydrostatic force on the porthole. (See Figure 5.9.1(a).)
Context Panel: Simplify≻Assuming Positive
Context Panel: Evaluate at a Point≻r=1/2
Context Panel: Approximate≻10
1252∫02 r20+2 r−y 2 2 r y−y2ⅆy→assuming positive125⁢r2⁢π⁢r+202→evaluate at point5125⁢π16→at 10 digits1006.291397
(For some integrals, Maple interprets "simplify" to mean "evaluate" because the evaluated form is simpler than the unevaluated form. Because the Context Panel provides for assumptions only with the Simplify option, the integral has been so evaluated in order to apply the assumption that r>0.)
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