Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_160

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Fresh water in a channel is contained by the uniform 2.5-m plate freely hinged at A. If the gate is designed to open when the depth of the water reaches 0.8 m as shown in the figure, what must be the weight w (in newtons per meter of horizontal length into the paper) of the gate? Problem 5/160 (a) Weight of the gate and its moment about hinge A 1. Formula: The total weight of the plate is given by its weight per unit width multiplied by its total length . The moment of this weight about the hinge is the weight multiplied by the horizontal distance from to the center of mass. 2. Substitution: From the problem description and diagram: Total plate length Angle with the vertical 3. Calculation: Moment arm 4. Result: (counter-clockwise) ● Final Conclusion: The counter-clockwise moment due to the weight of the gate is . (b) Hydrostatic force and its moment about hinge A Ww LA A M = W W⋅ sinθ = ( 2 L )(w⋅L)⋅ sinθ( 2 L ) L=2+0.5=2.5m θ=30 ∘ M = W (w⋅2.5)⋅ sin30( 2 2.5 ∘ ) W=2.5w =xˉ1.25⋅0.5=0.625m M = W 2.5w⋅0.625=1.5625wN⋅m M = W 1.5625w 1.5625wN⋅m 1. Formula: The hydrostatic force acts on the submerged part of the gate. The pressure at depth is . The moment about is the integral of the pressure multiplied by the distance from along the plate. where is the submerged length along the plate and is the length of the plate above the water surface. 2. Substitution: , Submerged length 3. Calculation: 4. Result: (clockwise) ● Final Conclusion: