question_answer

A vessel contains oil\[(density=0.8\text{ }gm/c{{m}^{3}})\] over mercury\[(density=13.6\text{ }gm/c{{m}^{3}})\]. 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 \[gm/c{{m}^{3}}\]is

A) 33

B) 6.4  

C) 72

D) 12.8

Correct Answer: C

Solution :

[c] As the sphere floats in the liquid. Therefore its weight will be equal to the upthrust force on it Weight of sphere \[=\frac{4}{3}\pi {{R}^{3}}\rho g\] ?(i) Upthrust due to oil and mercury \[=\frac{2}{3}\pi {{R}^{3}}\times {{\sigma }_{oil}}g+\frac{2}{3}\pi {{R}^{3}}{{\sigma }_{Hg}}g\] ?(ii) Equating (i) and (ii) \[\frac{4}{3}\pi {{R}^{3}}\rho g=\frac{2}{3}\pi {{R}^{3}}0.8g+\frac{2}{3}\pi {{R}^{3}}+13.6g\] \[\Rightarrow \,2\rho =0.8+13.6=14.4\Rightarrow \rho =7.2\]