A cubical block of co-efficient of linear expansion `alpha_s` is submerged partially inside a liquid of co-efficient of volume expansion `gamma_l`. On increasing the temperature of the system by `DeltaT`, the height of the cube inside the liquid remains unchanged. Find the relation between `alpha_s and gamma_l`.

Explanation

Let I be side of cube at initial temperature and d the depth of cube sumerged. Then according law of floatation
Weight of solid=weight of liquid displaced
`mg=F dp g` (i)
When temperature is increased, the weight remains same, Length of side of cube increases, density of liquid decrease and depth remains unchanged (as given)
`Mg=(I')2 dp g`
But `I'=I(1+a,DeltaT)P_(I)=(P_(I))/(1+lambda_(I)DeltaT)`
Substituting these value in Eq (iii), we get
`I^(2)dp_(I)g=I^(2)(1+a, DeltaT)^(2)(P_(I))/(1+lambda_(I)DeltaT)grArr1+lambda_(I)DeltaT=(1+a,DeltaT)^(2)`
As `a, DeltaTltlt1`, using binomial theorem
`1+lambda_(I)DeltaT=1+2a,DeltaT rArrlambda_(I)=2a_(s)`