Question

The ratio of radii of gyration of a spherical shell and solid sphere

Answer 1

Let $R_{1} ,R_{2}$ represent radii of spherical shell.

$R_{1} >R_{2}$

Radii of spherical shell= $R_{1} -R_{2}$

Volume of spherical shell= $\frac{4}{3}\pi [(R_{1})^{3}-(R_{2})^{3}]$

Formula for radii of gyration of Spherical shell=

                            $\frac {1}\text { Volume of spherical shell}\times [\int_{R_{1}}}^{R_{2}}}4\times\pi\times r^{4} dr]$

As,    $\int r^{4} dr=\frac{r^{5}}{5}$

Radii of gyration of spherical shell=$\frac{3}{5}\times \frac {R_{2}^{5}-R_{1}^{5}}{R_{2}^{3}-R_{1}^{3}}$

Volume of solid sphere of radius R is $\frac{4}{3}\pi R^{3}$

Ratio of radii of gyration of a spherical shell and solid sphere=

                            $\frac{9}{20}\times \frac{1}{\pi\times R^{3}}\times \frac {R_{2}^{5}-R_{1}^{5}}{R_{2}^{3}-R_{1}^{3}}$