A concrete sphere of radius R has a cavity of radius r which is packed with sawdust. The specific gravities of concrete and sawdust are respectively 2.4 and 0.3 for this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of sawdust will be
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Let specific gravities of concrete and saw dust are \rho_1 and $\rho_{2}$ respectively.
According to principle of floatation weight of whole sphere = up thrust on the sphere
$\frac{4}{3}\pi(R^{3}-r^{3})\rho_{1}g + \frac{4}{3}\pi r^{3}\rho_{2}g = \frac{4}{3}\pi R^{3} \times 1 \times g \\ \Rightarrow R^{3}\rho_{1} - r^{3}\rho_{1} + r^{3}\rho_{2} = R^{3} \\ \Rightarrow R^{3}(\rho_{1} - 1) = r^{3}(\rho_{1} - \rho_{2}) \Rightarrow \frac{R^{3}}{r^{3}} = \frac{\rho_{1} - \rho_{2}}{\rho_{1} - 1} \\ \Rightarrow \frac{R^{3} - r^{3}}{r^{3}} = \frac{\rho_{1} - \rho_{2} - \rho_{1} + 1}{\rho_{1} - 1} \\ \Rightarrow \frac{(R^{3} - r^{3})\rho_{1}}{r^{3}\rho_{2}} = \left(\frac{1 - \rho_{2}}{\rho_{1} - 1}\right) \frac{\rho_{1}}{\rho_{2}} \\ \Rightarrow \frac{\text{Mass of concrete}}{\text{Mass of saw dust}} = \left(\frac{1 - 0.3}{2.4 - 1}\right) \times \frac{2.4}{0.3} = 4$
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