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?

• A. $2$
• B. $3$
• C. $\frac{2}{3}$
• D. $\frac{7}{3}$

Answer 1

The correct option is A $2$

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

Let the volume of ship be 'V'.

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

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,
$V{\rho }_{ship}g=\frac{V}{3}{\rho }_{l}g$
$\Rightarrow {\rho }_l=3{\rho }_{ship}$

Now relative density of the liquid is:
${\rho }_{l,w}=\frac{{\rho }_l}{{\rho }_w}$
${\rho }_{l,w}=\frac{3{\rho }_{ship}}{\frac{3}{2}{\rho }_{ship}}$
${\rho }_{l,w}=2$