Problem 3.103 - Problem*3.103[Difficulty:4 Given Sphere...

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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 depth Solution: We will apply the hydrostatics equations to this system. Governing Equations: F buoy ρ g V d = (Buoyant force is equal to weight of displaced fluid) Assumptions: (1) Static fluid (2) Incompressible fluid (3) Atmospheric pressure acts everywhere d R sin θ R d θ θ max h We need an expression for the displaced volume of fluid at an arbitrary depth d. From the diagram we see that: d R 1 cos θ max ( ) ( ) = at an arbitrary depth h: h d R 1 cos θ ( ) ( ) = r R sin θ ( ) = So if we want to find the volume of the submerged portion of the sphere we calculate: V d 0 θ max h π r 2 d = π 0 θ max θ R 2 sin θ ( ) ( ) 2 R sin θ ( ) d = π R 3 0 θ max θ sin θ ( ) ( ) 3 d = Evaluating the integral we get: V d π R 3 cos θ max ( ) ( ) 3 3 cos θ max ( ) 2 3 +
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