Question

A uniform solid cylinder of density $0.8g/c{m}^3$ floats in equilibrium in a combination of two non-mixing liquids A and B with its axis vertical. The densities of liquids A and B are $0.7g/c{m}^3$ and $1.2g/c{m}^3$, respectively. The height of liquid A is $h_A=1.2cm$. The length of the part of the cylinder with liquid B is $h_B=0.8cm$ and $h=.2cm$. If $h$, the length of part of the cylinder in air is $\frac{x}{4}cm$. Find $x$.

Answer 1

Let A be the cross-sectional area of the cylinder.
For floating equilibrium,
Weight of floating cylinder $=$ Buoyant force $=$ weight of liquid displaced.
$(h+h_A+h_B)\times A\times 0.8g=(h_{A}A{\rho }_{A}g+h_{B}A{\rho }_{B}g)$
or $(h+h_A+h_B)\times 0.8=h_A{\rho }_A+h_B{\rho }_B$
$\Rightarrow (h+1.2+0.8)\times 0.8=1.2\times 0.7+0.8\times 1.2$
$\Rightarrow 0.8h+1.6=1.8$ or $0.8h=0.2$
$=h=0.25cm$