Question

A vessel contains oil (density $0.8{\text{g/cm}}^2$) over mercury (density $13.6{\text{g/cm}}^3$). A homogeneous sphere floats with half its volume immersed in mercury and the other half in oil. The density of the material of the sphere in ${\text{g/cm}}^3$ is

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

Answer 1

The correct option is C $7.2$

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

Weight of sphere $=\frac{4}{3}\pi R^3\rho g$
Upthrust due to oil and mercurry
$=\frac{2}{3}\pi R^3{\sigma }_{oil}g+\frac{2}{3}\pi R^3{\sigma }_{Hg}g$
Equating $\left(i\right)$ and $\left(ii\right)$
$\frac{4}{3}\pi R^3\rho g=\frac{2}{3}\pi R^3\times 0.8g+\frac{2}{3}\pi R^3\times 13.6g$
$\Rightarrow 2\rho =0.8+13.6=14.4\Rightarrow \rho =7.2$