RotationalInertiaOfASolidSphere 2006-03-27 21:02:11 2006-03-27 21:02:11 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The rotational inertia or moment of inertia of a solid sphere rotating about a diameter is \begin{equation} I = \frac{2}{5}M R^2 \end{equation} This can be shown in many different ways, but here we have chosen integration in spherical coordinates to give the reader practice in this coordinate system. If we choose an axis such as the z axis, then we just have one moment of inertia given by \begin{equation} I = \int {z^2 dm} \end{equation} It is important to understand this distinction and the more general case about an arbitrary axis is handled by the inertia tensor. Since we have chosen z as our axis of rotation, then z in formula (2) is the distance from dm/dV to the z axis. In the figure below this is shown as the purple line. Then from spherical coordiantes we obtain z through $$ z = r \sin \theta $$ leaving us with the integral $$ I = \int {r^2 \sin^2 \theta} $$ \begin{figure} \includegraphics[scale=.6]{InertiaSphere.eps} \caption{Rotational inertia of a solid sphere rotating about a diameter, z} \end{figure}