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.
$Weight of sphere = \frac{4}{3} π R^{3} ρ g . . . . . . . (i)$
Upthrust due to oil and mercury
$= \frac{2}{3} π R^{3} \times σ_{o i l} g + \frac{2}{3} π R^{3} σ_{H g} g . . . . . . . (i i)$
Equating (i) and (ii)
$\frac{4}{3} π R^{3} ρ g = \frac{2}{3} π R^{3} \times 0.8 g + \frac{2}{3} π R^{3} \times 13.6 g$
$\Rightarrow 2 ρ = 0.8 + 13.6 = 14.4 \Rightarrow ρ = 7.2$

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