Question

A piece of metal floats on mercury. The coefficients of volume expansion of metal and mercury

Answer 1

Let density of metal and mercury be d1andd2 respectively.

Fraction of volume of metal submerged initially =d1d2 .....(1)

We know that Volume of metal will change after increase of temperature ΔT by a factor (1+γ1ΔT)

Similarly Volume of mercury will change after increase of temperature ΔT by a factor (1+γ2ΔT)

(It is assumed that γ is coefficient of volume expansion.)

Density of metal after increase of temperature ΔT=d11+γ1ΔT

Density of mercury after increase of temperature ΔT=d21+γ1ΔT

Fraction of volume of metal submerged after increase of temperature =d11+γ1ΔTd21+γ2ΔT

=d1d2(1+γ2ΔT1+γ1ΔT) .....(2)

Change in fraction of volume of metal submerged =d1d2(1+γ2ΔT1+γ1ΔT)−d1d2

=d1d2(1+γ2ΔT1+γ1ΔT−1)

=d1d2(1+γ2ΔT)−(1+γ1ΔT)(1+γ1ΔT)

=d1d2(γ2−γ1)ΔT1+γ1ΔT

Factor of the fraction of the volume of the metal submerged in mercury changed after increase of volume=d1d2(γ2−γ1)ΔT1+γ1ΔTd1d2

=(γ2−γ1)ΔT1+γ1ΔT

 i will hope that you will be satisfied from this answer ...,..... ; )