Question

A metallic bob weight 50 g in air. If it is immersed in a liquid at a temperature of ${25}^{\circ }C$, it weighs 45 g. When the temperature of the liquid is raised to ${100}^{\circ }C$, it weighs 45.1 g. Calculate the coefficient of cubical expansion of the liquid. Given that coefficient of cubical expansion of the metal is $12\times {10}^{-6}{}^{\circ }{C}^{-1}$.

Answer 1

By Archimedes principle, the weight of liquid displaced is equal to the bouyant force on the object. In this case, the the change in bouyant force is equal to $0.1g$.

Also, expression for bouyant force will be,

$F=\frac{50}{{\rho }_s}{\rho }_{l}g$

In the second case, after expansion,

$F^{\prime }=\frac{50}{\frac{{\rho }_s}{1+{\gamma }_s\Delta T}}\frac{{\rho }_l}{1+{\gamma }_l\Delta T}g=F\frac{1+{\gamma }_s\Delta T}{1+{\gamma }_l\Delta T}$

$\Delta F=F^{\prime }-F=F\frac{({\gamma }_s-{\gamma }_l)\Delta T}{1+{\gamma }_l\Delta T}$

$-0.1=45\times 75\frac{12\times {10}^{-6}-\gamma }{1+75\gamma }$

$\gamma \approx 3.1\times {10}^{-4}{/}^{0}C$

Answer is $3.1\times {10}^{-4}{/}^{0}C$