Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_179

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the total force exerted on the viewing window by the fresh water in the tank. The water level is even with the top of the window. Additionally, determine the distance from the water surface to the line of action of . Problem 5/179 (a) Total resultant force 1. Formula: The hydrostatic force on a plane surface is given by: where is the specific weight of the fluid, is the depth of the centroid of the area from the free surface, and is the area of the surface. 2. Substitution: For fresh water, . The window is a "stadium" shape with width and total height . It consists of a middle rectangle and two semicircular ends. Radius of semicircular ends: . Height of the rectangular part: . Area . Centroid depth : By vertical symmetry, the centroid is at the geometric center of the window. Since the top is at the surface (), . 3. Calculation: 4. Result: R h ˉ R R R=γAzˉ γzˉ A γ= 62.4 lb/ft 3 w=4 ftH= 8 ft r= = 2 w = 2 4 2 ft h = rect H−2r=8−2(2)=4 ft A=A + rect A = circle (4 ft×4 ft)+π(2 ft)= 2 16+ 4π≈ 28.566 ft 2 zˉ z=0=zˉ = 2 H = 2 8 4 ft R= (62.4 lb/ft)× 3 (4 ft)×(16+ 4 π ft) 2 R=249.6×(16+12.566) R=249.6×28.566≈7130.1 lb R≈7130 lb ● Final Conclusion: The total hydrostatic force exerted on the window is (or ). (b) Distance to the line of action 1. Formula: The depth of the center of pressure (line of action) is: where is the second moment of area about the horizontal centroidal axis. 2. Substitution: For the rectangle (from to ): .