7.9 Spherical shell

2.1 Pure translational motion
2.2 Pure rotational motion
2.3 Plane motion

3.1 Angular velocity $\omega $
3.2 Angular acceleration $\left( \alpha \right)$
3.3 Kinematics equation for rotational motion
3.4 Analogy between translational motion & rotational motion

6.1 Parallel-axis theorem
6.2 Perpendicular axis theorem

7.1 Thin rod
7.2 Rectangular lamina
7.3 Circular ring
7.4 Circular disc
7.5 Solid cylinder
7.6 Cylindrical shell
7.7 Solid sphere
7.8 Hollow sphere
7.9 Spherical shell
7.10 Solid cone
7.11 Hollow cone
7.12 Hollow hemisphere
7.13 Parallelopiped
7.14 List of moment of inertia $(I)$ and radius of gyration $(K)$ of different bodies

Consider a hollow sphere of outer radius $R_2$, inner radius $R_1$ and mass $M$

Volume of the hollow sphere is, $(V) = \frac{4}{3}\pi \left( {R_2^3 - R_1^3} \right)$

Mass per unit volume, $\left( {{\lambda _V}} \right) = \frac{M}{V} = \frac{{3M}}{{4\pi \left( {R_2^3 - R_1^3} \right)}}$

Consider an infinitesimally small section in the shape of a hollow sphere of raidus $r$, thickness $dr$ and mass $dm$

Area of the hollow sphere of radius $r$, $A = 4\pi {r^2}$

Volume of the infinitesimally small section of mass, $dV = 4\pi {r^2}dr$

Mass of an infinitesimally small section, $dm = {\lambda _V}dV$ $$\begin{equation} \begin{aligned} dm = \left( {\frac{{3M}}{{4\pi \left( {R_2^3 - R_1^3} \right)}}} \right)\left( {4\pi {r^2}dr} \right) \\ dm = \frac{{3M}}{{\left( {R_2^3 - R_1^3} \right)}}{r^2}dr\quad ...(i) \\\end{aligned} \end{equation} $$

Moment of inertia of an axis about the diameter of a hollow sphere of radius $r$ and mass $dm$ passing through its $COM$ is, $$dI = \frac{{2(dm){r^2}}}{3}\quad ...(ii)$$

From equation $(i)$ & $(ii)$ we get, $$\begin{equation} \begin{aligned} dI = \left( {\frac{{2{r^2}}}{3}} \right)\left( {\frac{{3M}}{{\left( {R_2^3 - R_1^3} \right)}}{r^2}dr} \right) \\ dI = \left( {\frac{{2M}}{{\left( {R_2^3 - R_1^3} \right)}}{r^4}dr} \right) \\\end{aligned} \end{equation} $$

Integrating the above equation for the spherical shell of outer radius $R_2$ & inner radius $R_1$ is, $$\begin{equation} \begin{aligned} \int\limits_0^{{I_{COM}}} {dI} = \int\limits_{{R_1}}^{{R_2}} {\frac{{2M}}{{5\left( {R_2^3 - R_1^3} \right)}}{r^4}dr} \\ \left[ I \right]_0^{{I_{COM}}} = \frac{{2M}}{{5\left( {R_2^3 - R_1^3} \right)}}\left[ {\frac{{{r^5}}}{5}} \right]_{{R_1}}^{{R_2}} \\ ({I_{COM}} - 0) = \frac{{2M\left( {R_2^5 - R_1^5} \right)}}{{5\left( {R_2^3 - R_1^3} \right)}} \\ {I_{COM}} = \frac{{2M\left( {{R_2} - {R_1}} \right)\left( {R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_2^3 + R_1^4} \right)}}{{5\left( {{R_2} - {R_1}} \right)\left( {R_2^2 + {R_2}{R_1} + R_1^2} \right)}} \\ {I_{COM}} = \frac{{2M}}{5}\left( {\frac{{R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_1^3 + R_1^4}}{{R_2^2 + {R_2}{R_1} + R_1^2}}} \right)\quad ...(iii) \\\end{aligned} \end{equation} $$

Radius of gyration $(K)$ is given as, $$\begin{equation} \begin{aligned} {I_{COM}} = MK_{COM}^2 \\ \frac{{2M}}{5}\left( {\frac{{R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_2^3 + R_1^4}}{{R_2^2 + {R_2}{R_1} + R_1^2}}} \right) = MK_{COM}^2 \\ {K_{COM}} = \sqrt {\frac{2}{5}\left( {\frac{{R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_2^3 + R_1^4}}{{R_2^2 + {R_2}{R_1} + R_1^2}}} \right)} \\\end{aligned} \end{equation} $$

For hollow sphere, $${R_1} = {R_2} = R\quad ...(iv)$$

From equation $(iii)$ & $(iv)$ we get, $$\begin{equation} \begin{aligned} I = \frac{{2M}}{5}\left[ {\frac{{5{R^4}}}{{3{R^2}}}} \right] \\ I = \frac{{2M{R^2}}}{3} \\\end{aligned} \end{equation} $$

For solid sphere, \[\left. \begin{gathered} {R_2} = R \hspace{1em} \\ {R_1} = 0 \hspace{1em} \\ \end{gathered} \right\}\quad ...(v)\]

From equation $(iii)$ & $(v)$ we get, $$\begin{equation} \begin{aligned} I = \frac{{2M}}{5}\left[ {\frac{{{R^4}}}{{{R^2}}}} \right] \\ I = \frac{{2M{R^2}}}{5} \\\end{aligned} \end{equation} $$