A piece of metal floats on mercury .the coefficient of expansion of metal and mercury are gamma_1γ1 and gamma_2γ2 ,respectively .if the temperature of both metal and mercury increased by an amount Delta T . See description ?

by what factor does the fraction of the volume of the metal submerged in mercury changes .

1 Answer
A08
Jun 18, 2017

This is what I get.

Explanation:

Let density of metal and mercury be d_1 and d_2 respectively.

Fraction of volume of metal submerged initially =d_1/d_2 .....(1)

We know that Volume of metal will change after increase of temperature DeltaT by a factor (1+gamma_1DeltaT)
Similarly Volume of mercury will change after increase of temperature DeltaT by a factor (1+gamma_2DeltaT)

(It is assumed that gamma is coefficient of volume expansion.)

Density of metal after increase of temperature DeltaT=d_1/(1+gamma_1DeltaT)
Density of mercury after increase of temperature DeltaT=d_2/(1+gamma_1DeltaT)

Fraction of volume of metal submerged after increase of temperature =(d_1/(1+gamma_1DeltaT))/(d_2/(1+gamma_2DeltaT))
=d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT)) .....(2)

Change in fraction of volume of metal submerged =d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT))-d_1/d_2
=d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT)-1)

=d_1/d_2((1+gamma_2DeltaT)-(1+gamma_1DeltaT))/((1+gamma_1DeltaT))
=d_1/d_2((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT)

Factor of the fraction of the volume of the metal submerged in mercury changed after increase of volume=(d_1/d_2((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT))/(d_1/d_2)
=((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT)

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