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Question

A uniform solid cylinder of density 0.8g/cm3 floats in equilibrium in a combination of two non-mixing liquids A and B with its axis vertical. The densities of liquids A and B are 0.7g/cm3 and 1.2g/cm3, respectively. The height of liquid A is hA=1.2cm. The length of the part of the cylinder with liquid B is hB=0.8cm and h=.2cm. If h, the length of part of the cylinder in air is x4cm. Find x.
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Solution

Let A be the cross-sectional area of the cylinder.
For floating equilibrium,
Weight of floating cylinder = Buoyant force = weight of liquid displaced.
(h+hA+hB)×A×0.8g=(hAAρAg+hBAρBg)
or (h+hA+hB)×0.8=hAρA+hBρB
(h+1.2+0.8)×0.8=1.2×0.7+0.8×1.2
0.8h+1.6=1.8 or 0.8h=0.2
=h=0.25cm

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