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Exam3_2010_solution - 57:020 Mechanics of Fluids and...

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57:020 Mechanics of Fluids and Transport Fall 2010 EXAM3 Solutions Problem 1: Hydrostatic Forces on Plane Surface (Chapter 2) Information and assumptions Circular gate with diameter D = 2 m; R = D /2 = 1 m Gate orientation = 60 ° Vertical centroid location h c = 3 m from the water surface γ = 9.8 kN/m 3 for water I xc = π R 4 /4 for a circle of radius R Find Water force acting on the gate F R and the center of pressure y R . Solution (a) Water force: 𝐹 𝑅 = 𝑝 𝑐 𝐴 = 𝛾ℎ 𝑐 𝐴 (+3 points) Thus, 𝐹 𝑅 = 9.8 kN m 3 (3 m) 𝜋 4 (2 m) 2 = 92.4 kN (+2 points) (b) Center of pressure: 𝑦 𝑅 = 𝐼 𝑥𝑐 𝑦 𝑐 𝐴 + 𝑦 𝑐 where, 𝐼 𝑥𝑐 = 𝜋𝑅 4 4 = 𝜋 (1 m) 4 4 = 𝜋 4 m 4 𝑦 𝑐 = 𝑐 cos 30 = 3 m cos 30 = 3.464 m (+4 points) Hence, 𝑦 𝑅 = 𝜋 4 m 4 (3.464 m) 𝜋 4 (2 m) 2 + 3.464 m = 3.536 m (+1 point)
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