An aluminium cube of side `20cm` floats in mercury. How much farther will the block sink when tempereture rises from `27^(@)C` to `77^(@)C` ? Density of aluminium and mercury at `27^(@)C` are `2.7` and `13.6g//cc` while the coeficient of volume expansion of mercury and linear expansion of aluminium are `1.8xx10^(-4)//^(@)C` and `23xx10^(-6)//^(@)C` respectively.

Explanation

In case of floatation
Weight=Upthrust
`W=F_(b)`
`sigmaV_(g)=pV'_(g)`
`sigmaV=pV'`
`sigma` : density of material of block
`p` : density of liquid
`V` , volume of block
`V'` : immersed volume of block
`sigmaV=pV'`
`sigmaL^(3)=pL^(2)h`
`h=(sigma)/(p)L=(2.7xx20)/(13.6)=3.97cm`
`L` : side of cube, `h` : length of cube immersed in liquid As weight of block remains same `sigmaV_(g)=p_(27)V'_(27)g=p_(27)V'_(77)g`
`p_(27)L^(2)h=p_(77)L'^(2)h'`
`L'=L(1+alpha_(s)Deltatheta),p_(77)=(p_(27))/(1+gamma_(m)Deltatheta)`
`(h')/(h)=((1+gamma_(m)Deltatheta))/((1+alpha_(s)Deltatheta)^(2))=((1+gamma_(m)Deltatheta))/((1+2alpha_(s)Deltatheta))`
`=(1+gamma_(m)Deltatheta)(1+2alphaDeltatheta)=1+(gamma_(m)-2alpha_(s))Deltatheta`
`=1+(1.8xx10^(-4)-2xx23xx10^(-6))(77-27)`
`=1.0067`
`Deltah=h-h=0.0067h=0.0067xx3.97=0.0266cm`