Question

A ship floats with ${\frac{1}{3}}^{rd}$ of its volume outside. when its in water. When it floats in another liquid, ${\frac{2}{3}}^{rd}$ of its volume is outside the liquid. What is the relative density of the liquid?
[$3\text{marks}$]

Answer 1

When the ship is floating on water, ${\frac{1}{3}}^{rd}$ of its volume is outside water. This means that ${\frac{2}{3}}^{rd}$ of the ship's volume is inside water.

Let the volume of ship be '$V$'.

From archimedes' principle, the weight of the liquid displaced is equal to the weight of the ship.
$V\times {\rho }_{ship}\times g=\frac{2V}{3}\times {\rho }_w\times g$
${\rho }_w=\frac{3}{2}{\rho }_{ship}$ [$1\text{mark}$]

Now, in another liquid, ${\frac{2}{3}}^{rd}$ of the ship's volume is outside. ${\frac{1}{3}}^{rd}$ of the ship's volume is inside the liquid.

From Archimedes' principle,
$V\times {\rho }_{ship}\times g=\frac{V}{3}\times {\rho }_l\times g$
$\Rightarrow {\rho }_l=3{\rho }_{ship}$ [$1\text{mark}$]

Relative density of the liquid is given by:
R.D = $\frac{{\rho }_l}{{\rho }_w}$
R.D = $\frac{{3\rho }_{ship}}{\frac{3}{2}{\rho }_{ship}}$
R.D$=2$ [$1\text{mark1}$]