question_answer

A vessel contains oil (density = 0.8 gm/cm3) over mercury (density = 13.6 gm/cm3). 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/cm3 is                                                                      [IIT-JEE 1988]

A)             3.3 

B)             6.4

C)             7.2 

D)             12.8

Correct Answer: C

Solution :

                                      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) \[\frac{4}{3}\pi {{R}^{3}}\rho g=\frac{2}{3}\pi {{R}^{3}}0.8g+\frac{2}{3}\pi {{R}^{3}}\times 13.6g\]\[\Rightarrow 2\rho =0.8+13.6=14.4\Rightarrow \rho =7.2\]