Answer
Outer diameter = 6.2 cm, Inner diameter = 5.27 cm
Work Step by Step
When the system is at equilibrium. We can write
Buoyant force = Weight of the shell
$F_{B}=mg$
According to the principle of Archimedes, we can rewrite the above equation.
$V\rho_{w}g=mg=> \frac{4}{3}\pi R_{o}^{3}\rho_{w}=m$
$R_{o}=\sqrt[3] {\frac{3m}{4\pi\rho_{w}}}=\sqrt[3] {\frac{3\times1\space kg}{4\pi(1000\space kg/m^{3})}}=6.2\space cm$ = Outer diameter
We can write,
Mass of the shell $m= (V-V_{h})\rho_{s}$
$m=[\frac{4}{3}\pi R_{o}^{3}-\frac{4}{3}\pi R_{i}^{3}]\rho_{s}$
$\frac{3m}{4\pi \rho_{s}}=R_{o}^{3}-R_{i}^{3}=>R_{i}^{3}=R_{o}^{3}-\frac{3m}{4\pi \rho_{s}}$
$R_{i}^{3}=(6.2\times10^{-2}m)^{3}-(\frac{3(1\space kg)}{4\pi(2600\space kg/m^{3})})$
$R_{i}=5.27\space cm$ = Inner diameter
