Y (b) The fluid force against the gate Figure 7. So, the depth of the water at y in feet is A convenient approach is to let the y-axis bisect the gate and place the x-axis at the surface of the water, as shown in Figure 7(b). What is the fluid force on the gate when the top of the gate is 4 feet below the surface of the water? Solution In setting up a mathematical model for this problem, you are at liberty to locate the x- and y-axes in several different ways. A vertical gate in a dam has the shape of an 8 ft 6 ft 5 ft 4 ft (a) Water gate in a dam isosceles trapezoid 8 feet across the top and 6 feet across the bottom, with a height of 5 feet, as shown in Figure 7(a). 7 Fluid Pressure and Fluid Force 507 Fluid Force on a Vertical Surface See LarsonCalculus for an interactive version of this type of example.
3 6 4 The fluid force on a horizontal metal sheet is equal to the fluid pressure times the area. Definition of Force Exerted by a Fluid The force F exerted by a fluid of constant weight-density w (per unit of volume) against a submerged vertical plane region from y = c to y = d is F = w lim ∆→ 0 ∑Ĭ h(y)L(y) dy where h(y) is the depth of the fluid at y and L(y) is the horizontal length of the region at y. Therefore, taking the limit as ∆ → 0 (n → ∞) suggests the next definition. Note that w is considered to be constant and is factored out of the summation. The force against this representative rectangle is ∆Fi = w(depth)(area) = wh(yi)L(yi)∆y.
Next, consider the representative rectangle of width ∆y and length L(yi), where yi is in the ith subinterval. To determine the total force against one side of the region from depth c to depth d, you can subdivide the interval into n subintervals, each of width ∆y. in a fluid of weight-density w (per unit of volume), as shown in Figure 7. Y Calculus methods must be used to find the fluid force on a vertical metal plate. Consider a vertical plate that is submerged x L(yi) h(yi) Δ y d c
This problem is more difficult because the pressure is not constant over the surface. Consider a surface that is submerged vertically in a fluid. In Example 1, the fact that the sheet is rectangular and horizontal means that you do not need the methods of calculus to solve the problem. The fluid force would be the same in a swimming pool or lake. This result is independent of the size of the body of water. pounds square foot (12 square feet) = 4492 pounds. Because the total area of the sheet is A = ( 3 )( 4 ) = 12 square feet, the fluid force is F = PA = ̳74. Solution Because the weight-density of water is 62 pounds per cubic foot and the sheet is submerged in 6 feet of water, the fluid pressure is P = (62)( 6 ) P = wh = 374 pounds per square foot. 506 Chapter 7 Applications of Integration Fluid Force on a Submerged Sheet Find the fluid force on a rectangular metal sheet measuring 3 feet by 4 feet that is submerged in 6 feet of water, as shown in Figure 7.
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