Two non-mixing liquids of densities $\rho$ and $3\rho$ are put in a container. A solid cylinder of l

Question

Two non-mixing liquids of densities $\rho$ and $3\rho$ are put in a container. A solid cylinder of length $L$ floats vertically with its axis aligned, such that $\frac{L}{3}$ of its length is in the denser liquid ($3\rho$) and $\frac{2L}{3}$ of its length is in the lighter liquid ($\rho$). The density $d$ of the cylinder is

Professor PC Thomas Classes · Answer

(d) : Let $A$ be the cross-sectional area of the cylinder. According to the law of floatation, the weight of the cylinder is equal to the total upthrust provided by both liquids. The mass of the cylinder is $L \cdot A \cdot d$. The buoyant force from the denser liquid is $(\frac{L}{3} \cdot A) \cdot (3\rho) \cdot g$ and from the lighter liquid is $(\frac{2L}{3} \cdot A) \cdot \rho \cdot g$. Equating forces: $L \cdot A \cdot d \cdot g = (\frac{L}{3} \cdot A \cdot 3\rho \cdot g) + (\frac{2L}{3} \cdot A \cdot \rho \cdot g)$. Canceling $L A g$ yields $d = \rho + \frac{2\rho}{3} = \frac{5\rho}{3}$.