Problem *3.103[Difficulty:4]Given:Sphere partially immersed in a liquid of specific gravity SG.Find:(a) Formula for buoyancy force as a function of the submersion depth d(b) Plot of results over range of liquid depthSolution:We will apply the hydrostatics equations to this system.Governing Equations:Fbuoyρg⋅Vd⋅=(Buoyant force is equal to weight of displaced fluid)Assumptions:(1) Static fluid(2) Incompressible fluid(3) Atmospheric pressure acts everywhered RsinθR dmaxhWe need an expression for the displaced volume of fluid at an arbitrarydepth d. From the diagram we see that:dR 1cosθmax()−=at an arbitrary depth h:hdR 1cosθ−⋅−=rR sinθ⋅=So if we want to find the volume of the submerged portion of the sphere we calculate:Vd0θmaxhπr2⌠⎮⌡d=π0θmaxθR2sinθ2⋅R⋅sinθ⋅⌠⎮⌡d⋅=πR3⋅0θmaxθsinθ3⌠⎮⌡d⋅=Evaluating the integral we get:VdπR3⋅cosθmax33cosθmax−23+⎡⎢⎣
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This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.