A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are ${\mathrm{\gamma }}_1\text{ and }{\mathrm{\gamma }}_2$respectively. If the temperature of both mercury and metal are increased by an amount $\mathrm{\Delta T}$, the fraction of the volume of the metal submerged in mercury changes by the factor

Let f be the fraction of the volume of metal submerged in mercury
$\therefore \mathrm{f}=\frac{\text{ density of metal }}{\text{ density of mercury }}=\frac{{\mathrm{\rho }}_{\text{metal }}}{{\mathrm{\rho }}_{\text{mercury }}}$
When temperature is increased by $\mathrm{\Delta t}$, then fraction f'
$\begin{array}{l}{\mathrm{f}}^{\mathrm{\prime }}=\frac{{\mathrm{\rho }}_{\text{metal }}\left(1-{\mathrm{\gamma }}_1\mathrm{\Delta t}\right)}{{\mathrm{\rho }}_{\text{mercury }}\left(1-{\mathrm{\gamma }}_2\mathrm{\Delta T}\right)}\\ =\mathrm{f}\left(1-{\mathrm{\gamma }}_1\mathrm{\Delta T}\right){\left(1-{\mathrm{\gamma }}_2\mathrm{\Delta T}\right)}^{-1}\\ =\mathrm{f}\left[1+\left({\mathrm{\gamma }}_2-{\mathrm{\gamma }}_1\right)\mathrm{\Delta T}\right]\end{array}$
Fractional change or fraction immersed
$\frac{{\mathrm{f}}^{\mathrm{\prime }}-\mathrm{f}}{\mathrm{f}}=\left({\mathrm{\gamma }}_2-{\mathrm{\gamma }}_1\right)\mathrm{\Delta T}$