Question

A vessel contains oil (density $0.8g{\text{m/cm}}^3$ ) over mercury (density $=13.6g{\text{m/cm}}^3$ ). A homogeneous sphere floats with half of its volume immersed in mercury and the other half in oil. The density of the material of the sphere in $g{\text{m/cm}}^3$ is

• A. $3.3$
• B. $6.4$
• C. $7.2$
• D. $12.8$

Answer 1

Answer: (C)

As the sphere floats in the liquid. Therefore its weight will be equal to the upthrust force on it

$\text{Weight}\text{of}\text{sphere}=\frac{4}{3}\pi {\left(\text{R}\right)}^3\rho g\text{......}\left(\text{i}\right)$
$\text{Upthrust due to oil and mercury}=\frac{2}{3}\pi {\left(\text{R}\right)}^3\times {\left(\sigma \right)}_{\text{oil}}g+\frac{2}{3}\pi {\left(\text{R}\right)}^3{\left(\sigma \right)}_{\text{Hg}}g\text{......}\left(\text{i}\text{i}\right)$
Equating (i) and (ii)
$\frac{4}{3}\pi {\text{R}}^3\rho g=\frac{2}{3}\pi {\text{R}}^{3}0\text{.}8g+\frac{2}{3}\pi {\text{R}}^3\times 13\text{.}6g$
$\Rightarrow 2\rho =0\text{.}8+13\text{.}6=14\text{.}4$
$\Rightarrow \rho =7\text{.}2$