Question

A sphere of solid material of relative density 9 has a concentric spherical cavity and just floats in water. If the radius of the sphere be R, then the radius of the cavity (r) will be related to R as:

• A. $r^3=\frac{8}{9}{R}^3$
• B. $r^3=\frac{2}{3}{R}^3$
• C. $r^3=\frac{\sqrt{8}}{3}{R}^3$
• D. $r^3=\sqrt{\frac{2}{3}}{R}^3$

Answer 1

The correct option is A $r^3=\frac{8}{9}{R}^3$

According to law of floatation, a body floats if weight of the body is equal to the weight of water displaced
$\Rightarrow W_s=W_w\Rightarrow V_s\times {\rho }_s\times g=V_w\times {\rho }_w\times g\Rightarrow \frac{4}{3}\pi (R^3-r^3)\times 9{\rho }_w=\frac{4}{3}\pi R^3\times {\rho }_w\Rightarrow 9(R^3-r^3)=R^3\Rightarrow r^3=\frac{8}{9}{R}^3$