A piece of metal floats on mercury .the coefficient of expansion of metal and mercury are #gamma_1# and #gamma_2# ,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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