$\Rightarrow W_{s}=W_{w}$

$\Rightarrow V_{s} \times \rho_{s} \times g=V_{w} \times \rho_{w} \times g$

$\Rightarrow \frac{4}{3} \pi\left(R^{3}-r^{3}\right) \times \sigma \rho_{w}=\frac{4}{3} \pi R^{3} \times \rho_{w}$

$\Rightarrow \sigma\left(R^{3}-r^{3}\right)=R^{3}$

$\Rightarrow R^{3}=\left(\frac{\sigma}{\sigma-1}\right) r^{3}$

$\Rightarrow \frac{R}{r}=\left(\frac{\sigma}{\sigma-1}\right)^{1 / 3}$