A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are `gamma_1 and gamma_2`, respectively. if the temperature of both mercury and metal are increased by an amount `Delta T`, by what factor does the fraction of the volume of the metal submerged in mercury changes ?

Answer

`((1+gamma_(2)DeltaT)/(1+gamma_(1)DeltaT))`

Explanation

`((1+gamma_(2)DeltaT)/(1+gammaDeltaT))`
The condition of floating is,
Weight = Upthrust
`Vrho_(1)g=V_(f)rho_(2)g`
`(rho_(1)= " density of metal ", rho_(2)= "dnsity of mercury ")`
`therefore(V_(f))/(V)=(rho_(1))/(rho_(2))`
= fraction of volume of metal submerged in mercury = x (say)
Now, when the temperature is increased by `DeltaT`.
`rho_(1)'=(rho_(1))/(1+gamma_(1)DeltaT)andrho_(2)=(rho'_(2))/(1+gamma_(2)DeltaT)thereforeX'=((rho_(1))/(1+gamma_(1)DeltaT))((1+gamma_(2)DeltaT)/(rho_(2)))`
`=(rho_(1))/(rho_(2))((1+gamma_(2)DeltaT)/(1+gamma_(1)DeltaT))thereforeX'=X((1+gamma_(2)DeltaT)/(1+gamma_(1)DeltaT))therefore(X')/(X)=(1+gamma_(2)DeltaT)/(1+gamma_(1)DeltaT)`