Two non-mixing liquids of densities $\rho$ and $n\rho$ ($n > 1$) are put in a container. The height of each liquid is $h$. A solid cylinder of length $L$ and density $d$ is put in this container. The cylinder floats with its axis vertical and length $pL$ ($p < 1$) in the denser liquid. The density $d$ is equal to:
(2016 - I)
Solution:
In equilibrium, the weight of the cylinder is balanced by the buoyant force from both liquids. $d \cdot A \cdot L \cdot g = \rho \cdot A \cdot (L - pL) \cdot g + n\rho \cdot A \cdot pL \cdot g$. Dividing by $A \cdot L \cdot g$, we get $d = \rho (1 - p) + n\rho p = \{1 + (n - 1)p\}\rho$.
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