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Mechanical Properties Of Fluids

Question
CBSEENPH11019361

A piece of metal floats in mercury. The temperature coefficients of volume expansion of metal and mercury are γ1 and γ2 respectively. The temperature of the system is increased by θ. Find the factor by which the fraction of volume of metal submerged in mercury.

Solution
Using the law of flotation, the fraction of volume of metal submerged in the mercury is, 
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where,
ρ0 and σ0 are the densities of metal and mercury at low temperature.
When the temperature is increased by straight theta, then the fraction of volume of metal submerged in the mercury is,
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rightwards double arrow space straight f space minus space straight f subscript straight o space equals space straight f subscript straight o space left parenthesis straight gamma subscript 2 space minus space straight gamma subscript 1 right parenthesis space straight theta space

rightwards double arrow space fraction numerator space straight f space minus space straight f subscript straight o over denominator space straight f subscript straight o end fraction space equals space left parenthesis straight gamma subscript 2 space minus space straight gamma subscript 1 right parenthesis space straight theta space