Problem 3.85[Difficulty: 3]Given:Model cross section of canoe as a parabola. Assume constant width W over entire length Lyax2⋅=a1.2 ft1−⋅=W2ft⋅=L18ft⋅=Find:Expression relating the total mass of canoe and contents to distance d. Determine maximumallowable total mass without swamping the canoe.Solution:We will apply the hydrostatics equations to this system.Governing Equations:dpdhρg⋅=(Hydrostatic Pressure - h is positive downwards fromfree surface)FvAyp⌠⎮⎮⌡d=(Vertical Hydrostatic Force)Assumptions:(1) Static fluid(2) Incompressible fluid(3) Atmospheric pressure acts on free surface of the water and innersurface of the canoe.At any value of d the weight of the canoe and its contents is balanced by the net vertical force of the water on the canoe.Integrating the hydrostatic pressure equation:pρg⋅h⋅=FvAyp⌠⎮⎮⌡d=xρg⋅h⋅L⋅⌠⎮⎮⌡d=wherehHd−()y−=To determine the upper limit of integreation we remember that x2⋅=At the surfaceyHd−=Therefore,xHd−a=and so the vertical force is:Fv20−axρg⋅
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