Question

A cylindrical block floats vertically in a liquid of density ${\rho }_1$ kept in a container such that the fraction of volume of the cylinder inside the liquid is $x_1$. Then some amount of another immiscible liquid of density ${\rho }_2({\rho }_2<{\rho }_1)$ is added to the liquid in the container so that the cylinder now floats just fully immersed in the liquids with $x_2$ fraction of volume of the cylinder inside the liquid of density ${\rho }_1$. The ratio ${\rho }_1/{\rho }_2$ will be:

• A. $\frac{1-x_2}{x_1-x_2}$
• B. $\frac{1-x_1}{x_1+x_2}$
• C. $\frac{x_1-x_2}{x_1+x_2}$
• D. $\frac{x_2}{x_1}-1$

Answer 1

Answer: (A)

$\frac{1-x_2}{x_1-x_2}$

As Bouyant force in both the cases are same
$\therefore {\rho }_1{x}_{1}g={\rho }_1{x}_{2}g+{\rho }_2(1-x_2)g$
On solving $\frac{{\rho }_1}{{\rho }_2}=\frac{(1-x_2)}{{x_1-x}_2}$