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Unformatted text preview: FURTHER APPLICATIONS FURTHER APPLICATIONS OF INTEGRATION OF INTEGRATION 8 8.3 Applications to Physics and Engineering In this section, we will learn about: The applications of integral calculus to force due to water pressure and centers of mass. FURTHER APPLICATIONS OF INTEGRATION As with our previous applications to geometry (areas, volumes, and lengths) and to work, our strategy is: Break up the physical quantity into small parts. Approximate each small part. Add the results. Take the limit. Then, evaluate the resulting integral. APPLICATIONS TO PHYSICS AND ENGINEERING Deepsea divers realize that water pressure increases as they dive deeper. This is because the weight of the water above them increases. HYDROSTATIC FORCE AND PRESSURE Suppose that a thin plate with area A m 2 is submerged in a fluid of density ρ kg/m 3 at a depth d meters below the surface of the fluid. HYDROSTATIC FORCE AND PRESSURE The fluid directly above the plate has volume V = Ad So, its mass is: m = ρV = ρAd HYDROSTATIC FORCE AND PRESSURE Thus, the force exerted by the fluid on the plate is F = mg = ρgAd where g is the acceleration due to gravity. HYDROSTATIC FORCE The pressure P on the plate is defined to be the force per unit area: HYDROSTATIC PRESSURE F P gd A ρ = = The SI unit for measuring pressure is newtons per square meter—which is called a pascal (abbreviation: 1 N/m 2 = 1 Pa). As this is a small unit, the kilopascal (kPa) is often used. HYDROSTATIC PRESSURE For instance, since the density of water is ρ = 1000 kg/m 3 , the pressure at the bottom of a swimming pool 2 m deep is: HYDROSTATIC PRESSURE 3 2 1000kg/m 9.8m/s 2m 19,600Pa 19.6kPa P gd ρ = = × × = = An important principle of fluid pressure is the experimentally verified fact that, at any point in a liquid, the pressure is the same in all directions. This is why a diver feels the same pressure on nose and both ears. HYDROSTATIC PRESSURE Thus, the pressure in any direction at a depth d in a fluid with mass density ρ is given by: HYDROSTATIC PRESSURE P gd d ρ δ = = Equation 1 This helps us determine the hydrostatic force against a vertical plate or wall or dam in a fluid. This is not a straightforward problem. The pressure is not constant, but increases as the depth increases. HYDROSTATIC FORCE AND PRESSURE A dam has the shape of the trapezoid shown below. The height is 20 m. The width is 50 m at the top and 30 m at the bottom. HYDROSTATIC F AND P Example 1 Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam. HYDROSTATIC F AND P Example 1 We choose a vertical xaxis with origin at the surface of the water....
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 Fall '11
 AlanS.Grave
 Calculus

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