Question

Body A is half immersed in the water. If another body with same dimentions is placed on body A, then what will be the density of body B (${\rho }_B$) when the system is fully immersed in the water.
The density of water is ${\rho }_w$.

• A. ${\rho }_B=\frac{5{\rho }_w}{2}$
• B. ${\rho }_B=\frac{3{\rho }_w}{2}$
• C. ${\rho }_B=\frac{{\rho }_w}{2}$
• D. ${\rho }_B=2{\rho }_w$

Answer 1

The correct option is B ${\rho }_B=\frac{3{\rho }_w}{2}$

Given:
Body A is half immersed in the water, we can conclude that the density of the body A is half of the density of water (${\rho }_w$).
${\rho }_A=\frac{{\rho }_w}{2}$
Voume of body A and B are equal,
$V_A=V_B=V$
When the system fully immerged in the water, density of the system has to be equal to the density of water.
${\rho }_w=\frac{\text{Total mass}}{\text{Total volume}}$
${\rho }_w=\frac{m_A+m_A}{V_A+V_B}$
${\rho }_w=\frac{{\rho }_A{V}_A+{\rho }_B{V}_B}{V_A+V_B}$
${\rho }_w=\frac{V\times \frac{{\rho }_w}{2}+{\rho }_{B}V}{V+V}$
${\rho }_w=\frac{\frac{{\rho }_w}{2}+{\rho }_B}{1+1}$
$2{\rho }_w=\frac{{\rho }_w+2{\rho }_B}{2}$
$4{\rho }_w={\rho }_w+2{\rho }_B$
${\rho }_B=\frac{3{\rho }_w}{2}$