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A jar is filled with two non-mixing liquids $1$ and $2$ having densities $\rho_1$ and, $\rho_2$ respectively. A solid ball, made of a material of density $\rho_3$ , is dropped in the jar. It comes to equilibrium in the position shown in the figure.Which of the following is true for $\rho_1 , \rho_2$ and $\rho_3$?
$\;{\rho _1} < {\rho _3} < {\rho _2}$
$\;{\rho _3} < {\rho _1} < {\rho _2}$
$\;{\rho _1} > {\rho _3} > {\rho _2}$
$\;{\rho _1} < {\rho _2} < {\rho _3}$
Solution
From the figure it is clear that liquid $1$ floats on liquid $2$.
The lighter liquid floats over heavier liquid. Therefore
we can conclude that ${\rho _1} < {\rho _2}$
Also ${\rho _3} < {\rho _2}$ otherwise the ball would have sink to the bottom of the jar.
Also ${\rho _3} < {\rho _1}$ otherwise the ball would have floated in liquid $1$. From the above discussion we conclude that
${\rho _1} < {\rho _3} < {\rho _2}.$
