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

In equilibrium `Mg=F_(B)=pVg` (i)
When temperature is increased, depth of block inside liquid remains same.
In second case `Mg=F'_(B)=p'V'g` (ii)
Let `l` : side of cube at initial temperature
`l'` : side of cube at final temperature
`d` : depth of cube in the liquid
`p` : density of liquid at initial temperature
`p'` : density of liquid at final temperature
`pVg=p'V'g`
`pl^(2)d=p'l'^(2)d`
`pl^(2)=(p)/((1+gamma_(1)DeltaT))[l(1+alpha_(s)DeltaT)^(2)]`
`1+gamma_(1)DeltaT=(1+alpha_(s)DeltaT)^(2)`
`=1+2alphaDeltaT`
`gamma_(l)=2alpha_(s)`