Question

A piece of metal floating on mercury. The coefficient of volume expansion of mental and mercury are ${\gamma }_1$ and ${\gamma }_2$ respectively. If the temperature of both mercury and metal are increased by an amount $△T$, by what factor does the fraction of the volume of the metal submerged in mercury changes?

Answer 1

The fraction of metal inside mercury is given by the relation

$f=\frac{{\rho }_{metal}}{{\rho }_{mercury}}$

On increasing temperature,

${\rho }^{\prime }=\frac{\rho }{1+\gamma \Delta T}$

Hence, $f^{\prime }=\frac{{\rho }_{metal}}{{\rho }_{mercury}}\frac{1+{\gamma }_{mercury}}{1+{\gamma }_{metal}}=f\frac{1+{\gamma }_{mercury}}{1+{\gamma }_{metal}}$

Therefore, fractional change is $\frac{f-f^{\prime }}{f}\approx ({\gamma }_2-{\gamma }_1)\Delta T$