First slide

Pressure in a fluid-mechanical properties of fluids

Question

A cemented sphere of radius R has a cavity of radius r which is packed with sawdust. The relative density of cement and sawdust are 2.4 and 0.3 respectively. Find the ratio of mass of cement to sawdust for which the sphere will float in water in completely submerged state.

Moderate

Solution

mass of cement   $= \frac{4}{3} π (R^{3} - r^{3}) \times 2 .4 ρ_{w}$
mass of sawdust $= \frac{4}{3} {πr}^{3} \times 0 .3 p_{w}$
for floating    $mg = {Vρ}_{w} g$
$\frac{4}{3} π (R^{3} - r^{3}) \times 2 .4 ρ_{w} + \frac{4}{3} π r^{3} \times 0 .3 ρ_{w} = \frac{4}{3} π R_{}^{3} \times ρ_{w}$
$1 .4 R^{3} = 2 .1 r^{3}$
$\frac{R^{3}}{r^{3}} = \frac{3}{2}$
   $\frac{m_{c}}{m_{s}} = \frac{\frac{4}{3} π (R^{3} - r^{3}) \times 2 .4 ρ_{w}}{\frac{4}{3} {πr}^{3} \times 0 .3 ρ_{w}}$
$\frac{m_{c}}{m_{s}} = (\frac{3}{2} - 1) \times 8 = 4$

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