Question

When a cube of wood floats in water, $60\%$ of its volume is submerged. When the same cube floats in an unknown fluid, $85\%$ of its volume is submerged. Then:

• A. Density of wood is $500{\text{kg/m}}^3$
• B. Density of wood is $600{\text{kg/m}}^3$
• C. Density of the unknown fluid is $800{\text{kg/m}}^3$
• D. Density of the unkonwn fluid is $706{\text{kg/m}}^3$

Answer 1

Answer: (B), (D)

The correct options are
B Density of wood is $600{\text{kg/m}}^3$
D Density of the unkonwn fluid is $706{\text{kg/m}}^3$

Let the volume of the cube be $V$.


When wooden cube is submerged in water:
Applying equilibrium condition along vertical direction,
$F_{bw}-mg=0$
$\Rightarrow {\rho }_{w}g{V}_f={\rho }_{b}Vg...\left(i\right)$
$\because [V_f\to \text{volume of displaced fluid}]$
$\&V_f=0.60\times V$
Hence from Eq.$\left(i\right)$,
$1000\times \left(0.6V\right)={\rho }_{b}V$
$\therefore {\rho }_b=600{\text{kg/m}}^3...\left(ii\right)$

When wooden cube is submerged in unknown fluid:
Applying equilbrium condition along vertical direction,
$F_{bf}-mg=0$
$\Rightarrow {\rho }_{f}g{V}_f={\rho }_{b}Vg$
$\because [V_f\to \text{volume of displaced fluid}]$
$\&V_f=0.85\times V$ and substituting from Eq.$\left(ii\right)$,
$\Rightarrow {\rho }_f\times 0.85V=600\times V$
$\therefore {\rho }_f\simeq 706{\text{kg/m}}^3$
Hence, options $\left(b\right)\&\left(d\right)$ are correct.