Question

A sphere of radius $R$ is made of material of relative density $\sigma$ has a concentric cavity of radius $r$. It just floats, when placed in a tank full of water. The value of the ratio $R/r$ will be:

• A. ${\left(\frac{\sigma }{\sigma -1}\right)}^{1/3}$
• B. ${\left(\frac{\sigma -1}{\sigma }\right)}^{1/3}$
• C. ${\left(\frac{\sigma +1}{\sigma }\right)}^{1/3}$
• D. ${\left(\frac{\sigma -1}{\sigma +1}\right)}^{1/3}$

Answer 1

The correct option is A ${\left(\frac{\sigma }{\sigma -1}\right)}^{1/3}$

According to law of floatation, a body floats if weight of the body is equal to the weight of water displaced
$\Rightarrow W_s=W_w\Rightarrow V_s\times {\rho }_s\times g=V_w\times {\rho }_w\times g\Rightarrow \frac{4}{3}\pi (R^3-r^3)\times \sigma {\rho }_w=\frac{4}{3}\pi R^3\times {\rho }_w\Rightarrow \sigma (R^3-r^3)=R^3\Rightarrow R^3=\left(\frac{\sigma }{\sigma -1}\right)r^3\Rightarrow \frac{R}{r}={\left(\frac{\sigma }{\sigma -1}\right)}^{1/3}$