 # A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are ã1 and ã2 respectively. If the temperature of both mercury and metal are increased by an amount, Ät, the fraction of the volume of the metal submerged in mercury changes by the factor

3 years ago

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+γ1ΔT1)

=d1d2(1+γ2ΔT)(1+γ1ΔT)(1+γ1ΔT)
=d1d2(γ2γ1)ΔT1+γ1ΔT

Factor of the fraction of the volume of the metal submerged in mercury changed after increase of volume=d1d2(γ2γ1)ΔT1+γ1ΔTd1d2
=(γ2γ1)ΔT1+γ1ΔT