Question

A body $P$ floats in water with half its volume immersed. Another body $Q$ floats in a liquid of density $\frac{3}{4}th$ of the density of water with two-third of the volume immersed. The ratio of density of $P$ to that of $Q$ is

• A. $1:2$
• B. $1:1$
• C. $2:3$
• D. $3:4$
• E. $5:1$

Answer 1

Answer: (B)

Let, for body $P$ , volume $=V_P$
Given, Immersed volume $=\frac{V_P}{2}$
For body $Q$ volume $=V_Q$
Immersed volume $=\frac{2}{3}{V}_Q$
According to Archimedes principle,
$\because$ Weight of body $=$ weight of fluid displaced
For body $P,$
$V_p{\rho }_{p}g=\left(\frac{1}{2}{V}_p\right){\rho }_{\omega g}$
So, $\frac{{\rho }_P}{{\rho }_w}=\frac{1}{2}\dots (i)$
For body $Q$,
$V_Q{\rho }_{Q}g=\left(\frac{2}{3}{V}_Q\right)\left(\frac{3}{4}\right){\rho }_{w}g$
$(\because$ density of liquid $=\frac{3}{4}{\rho }_w)$ So, $\frac{{\rho }_Q}{{\rho }_w}=\frac{1}{2}\dots$ (ii)
From Equations (i) and (ii),
$\frac{{\rho }_P}{{\rho }_Q}=1\text{ or }{\rho }_P:{\rho }_Q=1:1$