A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are `gamma_1` and `gamma_2` respectively. If the temperatures of both mercury and the metal are increased by an amount `DeltaT`, the fraction of the volume of the metal submerged in mercury changes by the factor............

Options

A. `(1+gamma_2DeltaT)/(1+gamma_1DeltaT)`
B. `1+gamma_2DeltaT`
C. `1+gamma_1DeltaT`
D. `(1+gamma_2DeltaT)/(1-gamma_1DeltaT)`

Answer

A

Explanation

As, `lamda_1=(DeltaV_1)/(V_1DeltaT)` and `gamma_2=(DeltaV_2)/(V_2DeltaT)`
`impliesDeltaV_1=gamma_1V_1Delta`
and `DeltaV_2=gamma_2V_2DeltaT`
Fraction of volume submerged before temperature is raised is given by `f=(rho_1)/(rho_2)`.
Fraction of volume submerged after the temperature is raised is given by `f'=(rho'_1)/(rho'_2)`.
`alpha'=((rho_1)/(1+gamma_1DeltaT))((1+gamma_2DeltaT)/(rho_2))=((rho_1)/(rho_2))((1+gamma_2DeltaT)/(1+gamma_1DeltaT))`
`implies alpha' =f((1+gamma_2DeltaT)/(1+gamma_1DeltaT))`
`(alpha')/(f)=(1+gamma_2DeltaT)/(1+gamma_1DeltaT)`