A sphere of relative density \(\sigma\) and diameter \(D\) has concentric cavity of diameter \(d\). The ratio of \(\frac{D}{d}\), if it just floats on water in a tank is _______.

Buoyant force \(\left(F_b\right)=1 \times \frac{4}{3} \pi\left(\frac{D^3}{8}\right) \cdot g\)

For Just Float \(\Rightarrow \mathrm{W}=\mathrm{F}_{\mathrm{b}}\)

\(\Rightarrow\left(\mathrm{D}^3-\mathrm{d}^3\right) \sigma=\mathrm{D}^3\)

\(\Rightarrow 1-\frac{\mathrm{d}^3}{\mathrm{D}^3}=\frac{1}{\sigma}\)

\(\Rightarrow 1-\frac{1}{\sigma}=\left(\frac{\mathrm{d}}{\mathrm{D}}\right)^3\)

\(\Rightarrow\left(\frac{\sigma}{\sigma-1}\right)^{\frac{1}{3}}=\left(\frac{\mathrm{D}}{\mathrm{d}}\right)\)