Chapter 5: Applications of Integration
Section 5.9: Hydrostatic Force
Pressure is defined as force per unit area. Given a pressure and an area over which is applies, the total force exerted is then the product of the pressure and the area. Such a force is called hydrostatic when the pressure is applied by a fluid such as water.
If A is the area of a plane surface parallel to the surface of a fluid weighing δ lbs/ft3, and at depth d in the fluid, the total weigh of the fluid "sitting" on the submerged surface is F=A d δ. The pressure on this surface is P=F/A=d δ lbs/ft2.
It has been verified empirically that at depth d, the pressure is the same for any direction. This observation makes it possible to calculate the total force on submerged surfaces that are not parallel to the surface of the fluid. In particular, the total hydrostatic force F on a submerged surface that is perpendicular to the surface of a fluid is F=∫P ⅆA, where dA, the element of area, is taken as a narrow horizontal strip along which the pressure is constant.
It can be shown that the total hydrostatic force on such a vertical surface is the product of the area of the surface with the pressure at the centroid of the surface.
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.
The water behind a trapezoidal dam (dimensions 25 × 15 × 15 × 15 in feet, longest edge uppermost), is to the top edge. Find the total hydrostatic force on this structure if water weighs 62.5 lbs/ft3.
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