Talk:PlanetPhysics/Moment of Inertia of a Circular Disk

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: moment of inertia of a circular disk %%% Primary Category Code: 45.40.-f %%% Filename: MomentOfInertiaOfACircularDisk.tex %%% Version: 3 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Here we look at two cases for the \htmladdnormallink{moment of inertia}{http://planetphysics.us/encyclopedia/MomentOfInertia.html} of a homogeneous circular disk

(a) about its geometrical axis,

(b) about one of the elements of its lateral surface.

Let \emph{m} be the \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html}, \emph{a} the radius, \emph{l} the thickness, and $\tau$ the density of the disk. Then choosing a circular ring for the element of mass we have

$$dm = \tau \cdot l \cdot 2\pi r \cdot dr$$

where $r$ is the radius of the ring and $dr$ its thickness.

\begin{figure} \includegraphics[scale=.6]{Fig87.eps} \vspace{20 pt} \end{figure}

Therfore the moment of inertia about the axis of the disk is

$$I = 2\pi l \tau \int_0^a r^3 dr$$ $$I = \frac{\tau l \pi a^4}{2}$$ $$I = \frac{ma^2}{2}$$

The moment of inertia about the element is obtained easily by the help of \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} II. Thus

$$I' = I + ma^2$$ $$I' = \frac{3}{2}ma^2$$

It will be noticed that the thickness of the disk does not enter into the expressions for $I$ and $I'$ except through the mass of the disk. Therefore these expressions hold good whether the disk is thick enough to be called a cylinder or thin enough to be called a circular lamina.

\subsection{References}

This article is a derivative of the public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} book, "Analytical \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}" by Haroutune M. Dadourian, 1913. Made available by the \htmladdnormallink{internet archive}{http://www.archive.org/index.php}

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