A piece of metal floats on mercury. The coefficient of expansion of the metal and mercury are Y _(1)and Y_(2) respectively. If the temperature of both mercury and metal are increased by DeltaT, by what factor does the fraction of the volume of the metal submerged in mercury change ?

A piece of metal floats on mercury. The coefficient of expansion of th

1.	A piece of metal floats on mercury. The coefficient of expansion of the metal and mercury are Y _(1)and Y_(2) respectively. If the temperature of both mercury and metal are increased by DeltaT, by what factor does the fraction of the volume of the metal submerged in mercury change ?
Answer» Solution :Let the toal volume of metal in air and mercury `V and V _(3)` respectivley, `rho and sigma ` be the densities of metal and mercury RESPECTIVELY . When the metla flots in equilibrium, Fraction of the volume submerged, `f _(s) = (V _(s))/( V) = (rho )/(sigma ) .....(1)` When te temperature charges, the fractionof volume submerged changes as densitied change. `(Delta f _(s))/( f _(s)) = (f _(s))/(f _(s)) - = (rho ^(1))/( sigma ^(2)) xx (sigma )/( rho ) -1""...(2)` DENSITY of LIQUID DECREASES as temperature increases `implies rho . = ( rho )/(1 + gamma Delta T .) , sigma . = (sigma )/(1 + gamma _(2) Delta T)` which on subsituation in EQN. yields `(Delta f _(s))/( f _(s)) = (1 + gamma _(2) Delta T)/( 1 + gamma _(1) Delta T) -1 = ((1 + gamma _(2) Delta T ) - (1 + gamma _(l) Delta T ))/((1 + gamma _(t ) Delta T)) = ((gamma _(2) - gamma _(1)) Delta T)/( (1 + gamma _(t) Delta T))` `= (gamma _(2) - gamma _(1)) Delta T (1 - gamma _(1) Delta T) = (gamma _(2) - gamma _(1)) Delta T` [As `(1)/( 1 + gamma _(t) Delta T) = (1 + gamma _(1) Delta T) ^(-1) = (1- gamma _(1) Delta T) ` and higher powers are neglected]