Problem 3.75 - Problem 3.75[Difficulty 2 Given Gate formed...

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Problem 3.75 [Difficulty: 2] Given: Gate formed in the shape of a circular arc has width w. Liquid is water; depth h = R Find: (a) Magnitude of the net vertical force component due to fluids acting on the gate (b) Line of action of the vertical component of the force Solution: We will apply the hydrostatics equations to this system. Governing Equations: dp dy ρ g = (Hydrostatic Pressure - y is positive downwards) F v A y p d = (Vertical Hydrostatic Force) x' F v F v x d = (Moment of vertical force) Assumptions: (1) Static fluid (2) Incompressible fluid (3) Atmospheric pressure acts at free surface of water and on outside of gate Integrating the hydrostatic pressure equation: p ρ g y = Instead of y, we use θ as our variable of integration: y R sin θ () = Therefore, dy R cos θ d θ = In addition, dA y wR sin θ d θ = Therefore, F v 0 π 2 θ ρ g R sin θ w R sin θ
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