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Problem 3.81 - Problem 3.81[Difficulty 3 Given Cylindrical...

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Problem 3.81 [Difficulty: 3] Given: Cylindrical log floating against dam Find: (a) Mass per unit length of the log (b) Contact force per unit length between log and dam Solution: We will apply the hydrostatics equations to this system. Governing Equations: dp dh ρ g = (Hydrostatic Pressure - h is positive downwards from free surface) dF →⎯ p dA = (Hydrostatic Force) dF H dF h θ dF V R = D /2 Assumptions: (1) Static fluid (2) Incompressible fluid (3) Atmospheric pressure acts on free surfaces and on the first quadrant of the log Integrating the hydrostatic pressure equation: p ρ g h = ρ g R 1 cos θ ( ) ( ) = Resolving the incremental force into horizontal and vertical components: dF p dA = p w R d θ = ρ g R 1 cos θ ( ) ( ) w R d θ = ρ g R 2 w 1 cos θ ( ) ( ) = dF H dF sin θ ( ) = ρ g R 2 w 1 cos θ ( ) ( ) d θ sin θ ( ) = dF v dF cos θ ( ) = ρ g R 2 w 1 cos θ ( ) ( ) d θ cos θ ( ) = Integrating the expression for the horizontal force will provide us with the contact force per unit length: F H 0 3 π 2 θ ρ g R 2 w 1 cos θ ( ) ( ) sin θ ( ) d = ρ g R 2
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