A piece of metal floats on mercury .the coefficient of expansion of metal and mercury are #gamma_1# and #gamma_2# ,respectively .if the temperature of both metal and mercury increased by an amount #Delta T# . See description ?

Question

by what factor does the fraction of the volume of the metal submerged in mercury changes .

Olivia Askew · Accepted Answer

This is what I get.

Let density of metal and mercury be #d_1 and d_2# respectively. Fraction of volume of metal submerged initially #=d_1/d_2# .....(1) We know that Volume of metal will change after increase of temperature #DeltaT# by a factor #(1+gamma_1DeltaT)# Similarly Volume of mercury will change after increase of temperature #DeltaT# by a factor #(1+gamma_2DeltaT)# (It is assumed that #gamma# is coefficient of volume expansion.) Density of metal after increase of temperature #DeltaT=d_1/(1+gamma_1DeltaT)# Density of mercury after increase of temperature #DeltaT=d_2/(1+gamma_1DeltaT)# Fraction of volume of metal submerged after increase of temperature #=(d_1/(1+gamma_1DeltaT))/(d_2/(1+gamma_2DeltaT))# #=d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT))# .....(2) Change in fraction of volume of metal submerged #=d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT))-d_1/d_2# #=d_1/d_2((1+gamma_2DeltaT)/(1+gamma_1DeltaT)-1)# #=d_1/d_2((1+gamma_2DeltaT)-(1+gamma_1DeltaT))/((1+gamma_1DeltaT))# #=d_1/d_2((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT)# Factor of the fraction of the volume of the metal submerged in mercury changed after increase of volume#=(d_1/d_2((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT))/(d_1/d_2)# #=((gamma_2-gamma_1)DeltaT)/(1+gamma_1DeltaT)#

Isabella Alvarez · Answer

undefined When both the metal and mercury increase in temperature by an amount ΔT, the metal expands by γ₁ΔT and the mercury expands by γ₂ΔT. Since the metal is floating on the mercury, for the piece of metal to continue floating, its volume increase due to expansion must be equal to the volume increase of the mercury it displaces. Therefore, the equation for this situation is γ₁ΔT = -γ₂ΔT. Solving for ΔT gives: ΔT = -γ₂/γ₁ * ΔT.