Question

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

• A. 8
• B. 4
• C. 3
• D. Zero

Answer 1

The correct option is B

4

Let specific gravities of concrete and saw dust are ρ1and ρ2 respectively. According to principle of floatation weight of whole sphere = upthrust on the sphere

$\frac{4}{3}\pi (R^3-r^3){\rho }_{1}g+\frac{4}{3}\pi r^3{\rho }_{2}g=\frac{4}{3}\pi R^3\times 1\times g$

$\Rightarrow R^3{\rho }_1-r^3{\rho }_1+r^3{\rho }_2=R^3$

$\Rightarrow R^3({\rho }_1-1)=r^3({\rho }_1-{\rho }_2)\Rightarrow \frac{R^3}{r^3}=\frac{{\rho }_1-{\rho }_2}{{\rho }_1-1}$

$\Rightarrow \frac{R^3-r^3}{r^3}=\frac{{\rho }_1-{\rho }_2-{\rho }_1+1}{{\rho }_1-1}$

$\Rightarrow \frac{(R^3-r^3){\rho }_1}{r^3{\rho }_2}=⟮\frac{1-{\rho }_2}{{\rho }_1-1}⟯\frac{{\rho }_1}{{\rho }_2}$

$\Rightarrow \frac{\text{Mass of concrete}}{\text{Mass of saw dust}}=⟮\frac{1-0.3}{2.4-1}⟯\times \frac{2.4}{0.3}=4$