Question

Two objects A and B of same dimensions are floating in the water in the same orientation. One-third of A is immersed in the water and half of B is immersed in the water. Which object has larger density?

• A. Object A
• B. Object B
• C. Density of A and B are equal
• D. Insufficient information

Answer 1

The correct option is B Object B

Qualitative approach:
The percentage of an object submerged in the water will increase with its absolute density. Since more fraction of B is immersed, it has more density than A.

Mathematical approach:
Assume:
Density of water: ${\rho }_w$,
Density of A: ${\rho }_A$,
Density of B: ${\rho }_B$,
Volume of A (equal to the volume of B): $V$

According to the Archimedes' principle, weight of displaced water by an object is equal to the buoyant force or upthrust on that object.
If body is floating the weight of object is also equal to the buoyant force.

For A,
${\rho }_w\times \frac{V}{3}={\rho }_{A}V$
${\rho }_A=\frac{{\rho }_w}{3}$ -------(1)

For B,
${\rho }_w\times \frac{V}{2}={\rho }_{B}V$
${\rho }_B=\frac{{\rho }_w}{2}$ -------(2)

$⟹\frac{{\rho }_A}{{\rho }_B}=\frac{2}{3}$
Hence, object B has comparatively higher density value than A.