Question

A sphere of solid material of specific gravity $8$ has a concentric spherical cavity and just sinks in water. The ratio of radius of cavity to that of outer radius of the sphere must be

• A. $\frac{7^{1/3}}{2}$
• B. $\frac{5^{1/3}}{2}$
• C. $\frac{9^{1/3}}{2}$
• D. $\frac{3^{1/3}}{2}$

Answer 1

Answer: (A)

Let $\rho$ be the density of the material. ${\rho }_0$ be the density of water when the sphere has just started sinking, the weight of the sphere = weight of water displaced (approx).
$\Rightarrow \frac{4}{3}\pi \left(R^3-r^3\right)\rho g=\frac{4}{3}\pi R^3{\rho }_{0}g$
$\Rightarrow \left(R^3-r^3\right)\rho =R^3\rho 0$
$\Rightarrow \frac{\left(R^3-r^3\right)}{R^3}=\frac{{\rho }^0}{\rho }$
$\Rightarrow \frac{r}{R}=\frac{(7{)}^{1/3}}{2}$