Question

A cubical metal block of edge $12\text{cm}$ floats in mercury with one fifth of the height inside the mercury. Water is poured till the surface of the block is immersed in it. Height of the water column to be poured is [specific gravity of mercury$=13.6$]

• A. $6.4\text{cm}$
• B. $10.4\text{cm}$
• C. $8.4\text{cm}$
• D. $5.4\text{cm}$

Answer 1

The correct option is B $10.4\text{cm}$

Let the height of the water column to be poured is $x$.



For initial case -
Applying equilibrium condition along vertical direction,
$F_{bm}-mg=0$ [Vertical equilibrium]
$\Rightarrow {\rho }_{m}g{V}_d-{\rho }_{b}Vg=0$
$\left[\begin{array}{c}{\rho }_m\to \text{density of mercury};V_d\to \text{Volume of displaced fluid};\\ V\to \text{Volume of block}\\ {\rho }_b\to \text{density of block}\end{array}\right]$
Here, $V_d=2.4\times 12\times 12{\text{cm}}^3$

Substituting in above equation,
$\Rightarrow \frac{{\rho }_m}{{\rho }_w}\times 2.4\times (12{)}^2-\frac{{\rho }_b}{{\rho }_w}\times (12{)}^3=0$
[dividing by ${\rho }_w$,] and ${\rho }_w=1\text{g/cc},{\rho }_m=13.6\text{g/cc}$
$\Rightarrow \frac{{\rho }_b}{{\rho }_w}=\frac{13.6\times 2.4\times {12}^2}{1\times {12}^3}$
$\Rightarrow \frac{{\rho }_b}{{\rho }_w}=2.72...\left(1\right)$

For final case, let $x$ be the height of block immersed in water.
On applying $\sum F=0$ for equilibrium along vertical direction,
$F_{bw}+F_{bm}-mg=0$
$\Rightarrow {\rho }_{w}g{V}_w+{\rho }_{m}g{V}_m-{\rho }_b(V_w+V_m)g=0....\left(2\right)$

$\text{Total volume of block}=(V_w+V_m)=\text{total volume of liquid displaced}$

Dividing Eq.$\left(2\right)$ by ${\rho }_w$,
$\Rightarrow V_w+\left(\frac{{\rho }_m}{{\rho }_w}\right)V_m-\left(\frac{{\rho }_b}{{\rho }_w}\right)(V_w+V_m)=0$
$\Rightarrow V_w+13.6V_m-2.72(V_w+V_m)=0$
From Eq.$\left(1\right)$ and substituting the values of volume,
$\Rightarrow (x\times {12}^2)+(13.6\times (12-x)\times {12}^2)-(2.72\times {12}^3)=0$
$\Rightarrow 144x+23500.8-1958.4x-4700.16=0$
$\Rightarrow 1814.4x=18800.64$
$\Rightarrow x\simeq 10.4\text{cm}$
Height of the water column to be poured is $10.4\text{cm}$