Question

A sphere of solid material of specific gravity 8 has a concentric spherical cavity and just sinks in the water. Then, the ratio of the radius of the cavity to the outer radius of the sphere must be
A. $\dfrac{{\sqrt[3]{3}}}{2}$
B. $\dfrac{{\sqrt[3]{5}}}{2}$
C. $\dfrac{{\sqrt[3]{7}}}{2}$
D. $\dfrac{2}{{\sqrt[3]{7}}}$

Answer 1

Hint:To solve this problem, we need to use the concept of floatation. Floatation depends upon the density. If an object has density less than the density of water, it floats. This principle of floatation is given by the Archimedes. According to this principle, a body floats if its weight is equal to the weight of water displaced. Therefore, here, we will equate the weights of sphere and water displaced to determine the required answer.

Complete step by step answer:
As per the law of floatation, the sphere will float when its weight is similar to the weight of displaced water.Let ${W_s}$ is the weight of sphere, ${V_s}$ is the volume of the sphere, ${\rho _s}$ is the density of the sphere, $r$ is the radius of the cavity and $R$is the outer radius of the sphere.Also, let ${W_w}$ is the weight of water displaced, ${V_w}$ is the volume of the water displaced and ${\rho _w}$ is the density of water and $g$ is the gravitational acceleration.
\[
{W_s} = {W_s} \\
\Rightarrow {V_s}{\rho _s}g = {V_w}{\rho _w}g \\
\Rightarrow \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right) \times 8{\rho _w} = \dfrac{4}{3}\pi {R^3} \times {\rho _w} \\
\Rightarrow 8\left( {{R^3} - {r^3}} \right) = {R^3} \\
\Rightarrow 7{R^3} = 8{r^3} \\
\Rightarrow \dfrac{{{r^3}}}{{{R^3}}} = \dfrac{7}{8} \\
\therefore \dfrac{r}{R} = \dfrac{{\sqrt[3]{7}}}{2} \\
 \]
Thus, the ratio of the radius of the cavity to the outer radius of the sphere must be $\dfrac{{\sqrt[3]{7}}}{2}$.

Hence, option C is the right answer.

Note:In this question, the specific gravity of the sphere is given. Specific gravity is defined as the ratio of density of the body to the density of the water. Therefore, here, we have taken ${\rho _s} = 8{\rho _w}$ because it is given that the value of specific gravity of the sphere is 8.