First slide

Pressure in a fluid-mechanical properties of fluids

Question

A hollow sphere of radius R is made of metal whose specific gravity is $ρ$ . The sphere will float in water if thickness of wall of sphere is (thickness of wall << R)

Moderate

A

$> \frac{R}{3 ρ}$
B

$= \frac{2 R}{3 ρ}$
C

$< \frac{R}{3 ρ}$
D

$< \frac{4 R}{3 ρ}$

Solution

Given
Radius of hollow sphere $= R$
Specific gravity= $ρ$
Now mass of spherical shell
$M_{1} =$ Solid volume of the sphere density of material
$M_{1} = 4 π R^{2} t ρ$
Where $M_{1}$ =Mass of the hollow sphere
t=Thickness of the hollow sphere
While the mass of water having same volume = Whole volume of the sphere $\times$ density of the water
$M_{2} = \frac{4}{3} π R^{3} ρ_{w}$
For the floatation of sphere $M_{1} \leq M_{2}$
$\Rightarrow 4 π R^{2} t ρ \leq \frac{4}{3} π R^{3} ρ_{w}$
$\Rightarrow t ρ \leq \frac{R}{3} (∵ ρ_{w} = 1 g / {cm}^{3})$
$∴$ Thickness $t \leq \frac{R}{3 ρ}$

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