(Solution) - A rectangular block of height L and horizontal cross sectional a -(2025 Original AI-Free Solution)

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A rectangular block of height L and horizontal cross-sectional area A floats at the interface between two immiscible liquids, as shown below.


(a) Derive a formula for the block density, pb, in terms of the fluid densities p1 and p2, the heights h0, h1, and h2, and the cross-sectional area A. (It is not necessary that all of these variables appear in the final result).
(b) Force balances on the blocks can be calculated in two ways; (i) in terms of the weight of the block and the hydrostatic forces on the upper and lower block surfaces; and (ii) in terms of the weight of the block and the buoyant force on the block as expressed by Archimedes? principle. Prove that these two approaches areequivalent.