Question

A vessel contains oil (density $=0.8 \,gm / cm ^{3}$ ) over mercury ( density $=13.6\, gm / cm ^{3}$ ). $A$ 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 $gm / cm ^{3}$ 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
Weight of sphere $=\frac{4}{3} \pi R^{3} \rho g \ldots$ (i)
Upthrust due to oil and mercury
$=\frac{2}{3} \pi R^{3} \times \sigma_{\text {oil }} g+\frac{2}{3} \pi R^{3} \sigma_{H g} g$
Equating (i) and (ii)
$\frac{4}{3} \pi R^{3} \rho g=\frac{2}{3} \pi R^{3} 0.8 g+\frac{2}{3} \pi R^{3}+13.6\, g$
$\Rightarrow 2 \rho=0.8+13.6=14.4$
$ \Rightarrow \rho=7.2$