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 ?

Explanation

`Vd_(Hg)g=Vd_(m)g`
Fraction of volum of metal submerged in mercury

`v/V=(d_(m))/(d_(Hg))=K_(1)` (say)
In second case `(T+/_\T)`
`v'd'H_(g)g=V'd_(m)^(')g`
`implies (v')/(V')=(d_(m)^('))/d_(Hg)^(')`
Fraction of volume of metal submerged in mercury `=K_(2)`(say)
`:.(K_(2))/(K_(1))=(d_(m)^(')xxd+(Hg))/(d_(Hg)^(')xxd_(m))=(d_(m)^(')xxd_(Hg)^(')(1+gamma_(2)/_\T))/(d_(Hg)^(')xd_(m)^(')(1+gamma_(1)/_\T))`
`=(1+gamma_(2)/_\T)/(1+gamma_(1)/_\T)`
`:. 1-(K_(2))/(K_(1))=1-(1+gamma_(2)/_\T)/(1+gamma_(1)/_\T)`
`implies(K_(1)-K_(2))/(K_(1))=(1-gamma_(1)/_\T-1-gamma_(2)/_\T)/(1-gamma_(1)/_\T)`
`(K_(1)-K_(2))/(K_(1))=((gamma_(1)+gamma_(2))/_\T)/(1-gamma_(1)/_\T)`