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Problem 3.65

# Problem 3.65 - Problem 3.65 Given[Difficulty 3 Gate shown...

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Problem 3.65 [Difficulty: 3] Given: Gate shown with fixed width, bass of gate is negligible. Gate is in equilibrium. b 3 m = Find: Water depth, d Solution: We will apply the hydrostatics equations to this system. Governing Equations: dp dh ρ g = (Hydrostatic Pressure - h is positive downwards) F R p c A = (Hydrostatic Force) y' y c I xx A y c + = (Location of line of action) Σ M z 0 = (Rotational equilibrium) L y l d M h θ 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 h = F R p c A = ρ g h c A = h c d 2 = A b d sin θ ( ) = Therefore, F R ρ g b d 2 2 sin θ ( ) = To find the line of application of this force: y' y c I xx A y c + = Since I xx b l 3 12 = and A b l = it follows that y' y c b l 3 12 b l y c + = y c l 2 12 y c + = where l is the length of the gate in contact with the water (as seen in diagram)

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