Question

Two solid bodies $X$ and $Y$ having same volume floats in a liquid. It is observed that $X$ floats with $50\%$ of its volume immersed in liquid and $Y$ floats with $25\%$ of its volume above the liquid level. The ratio of density of $X$ to $Y$ is

• A. $3:2$
• B. $2:3$
• C. $2:5$
• D. $5:2$

Answer 1

The correct option is B $2:3$

Let $\rho$ be the density of the liquid.
In equilibrium,
Bouyant force on body $X$ by water,
$F_{bx}=\frac{\rho gV}{2}$
This will balance the weight of body $X$.
$\therefore m_{x}g={\rho }_{x}gV=F_{bx}=\frac{\rho gV}{2}$
$\Rightarrow {\rho }_x=\frac{\rho }{2}$ ...........(1)
Similarly,
Bouyant force on body $Y$ by water,
$F_{by}=\frac{3\rho gV}{4}$
This will balance the weight of body $Y$.
$\therefore m_{y}g={\rho }_{y}gV=F_{by}=\frac{3\rho gV}{4}$
$\Rightarrow {\rho }_y=\frac{3\rho }{4}$ ............(2)
So, $\frac{{\rho }_x}{{\rho }_y}=\frac{2}{3}=2:3$
[ from (1) and (2) ]