A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are `gamma_1` and `gamma_2` respectively. If the temperatures of both mercury and the metal are increased by an amount `DeltaT`, the fraction of the volume of the metal submerged in mercury changes by the factor............

Explanation

In case of floatation
Weight=Upthrust
`W=F_(B)`
`sigmaV_(g)=pV'g`
`P` : density of liquid
`V` : volume of block
`V'` : volume of block inside liquid
`f=(V')/(V)=(sigma)/(p)` (i)
When temperature is increased by `DeltaT`
`f'=(sigma//(1+gamma_(1)DeltaT))/(p//(1+gamma_(2)DeltaT))=(sigma)/(p)(1+gamma_(2)DeltaT)(1+gamma_(1)DeltaT)^(-1)`
`=f(1+gamma_(2)DeltaT)(1-gamma_(1)DeltaT)`
`=f[1+(gamma_(2)-gamma_(1))DeltaT]`
{Since `gamma_(1)gamma_(2)DeltaT^(2)` is negligible}
`(f')/(f)-1=(gamma_(2)-gamma_(1))DeltaT`