First slide

Pressure in a fluid-mechanical properties of fluids

Question

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

Moderate

A

3.3
B

6.4
C

7.2
D

12.8

Solution

As the sphere floats in the liquid. Therefore its weight will be equal to the upthrust force on it
Weight of sphere
$= \frac{4}{3} {πR}^{3} ρg$ …(i)
Upthrust due to oil and mercury
$= \frac{2}{3} {πR}^{3} \times σ_{oil} g + \frac{2}{3} {πR}^{3} σ_{Hg} g$ …(ii)
Equating (i) and (ii)
$\begin{array}{l} \frac{4}{3} {πR}^{3} ρg = \frac{2}{3} {πR}^{3} 0 .8 g + \frac{2}{3} {πR}^{3} \times 13 .6 g \\ \Rightarrow 2 ρ = 0 .8 + 13 .6 = 14 .4 \Rightarrow ρ = 7 .2 \end{array}$

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