3.6.1: Tables of geometries
- Page ID
- 656
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following tables present several moment of inertias of commonly used geometries.
Table 3.1 Moments of Inertia for various plane surfaces about their center of gravity (full shapes).
Shape Name | Picture Description | \(\mathbf{x_c}\), \(\mathbf{x_c}\) | \(\mathbf{A}\) | \(\mathbf{I_{xx}}\) |
Rectangle | ![]() |
\(\dfrac{a}{2}\,;\dfrac{b}{2}\) | \(a\,b\) | \(\dfrac{a\,b^{3}}{12}\) |
Triangle | ![]() |
\(\dfrac{a}{3}\) | \(\dfrac{a\,b}{3}\) | \(\dfrac{a\,b^{3}}{36}\) |
Circle | ![]() |
\(\dfrac{b}{2}\) | \(\dfrac{\pi\, b^2}{4}\) | \(\dfrac{\pi b^4}{64}\) |
Ellipse | ![]() |
\(\dfrac{a}{2}\; \dfrac{b}{2}\) | \(\dfrac{\pi\, ab}{4}\) | \(\dfrac{ab^2}{64}\) |
\(y=\alpha\,x^2\) Parabola |
![]() |
\(\dfrac{3\,\alpha\,b}{15\,\alpha-5}\) | \(\dfrac{6\alpha -2}{3} \, \left( \dfrac{b}{\alpha}\right)^{\dfrac{3}{2}}\) | \(\dfrac{\sqrt{b}\,\left( 20\,{b}^{3}-14\,{b}^{2}\right) }{35\,\sqrt{\alpha}}\) |
Quadrant of Circle |
![]() |
\(\dfrac{4\,r}{3\,\pi}\) | \(\dfrac{\pi\,r^2}{4}\) | \(r^4\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\) |
Ellipsoidal Quadrant |
![]() |
\(\dfrac{4\,b}{3\,\pi}\) | \(\dfrac{\pi\,a\,b}{4}\) | \(a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\) |
half of of Ellipse |
![]() |
\(\dfrac{4\,b}{3\,\pi}\) | \(\dfrac{\pi\,a\,b}{4}\) | \(a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\) |
Circular Sector |
![]() |
\(0\) | \(2{\alpha}\,r^2\) | \(\dfrac{r^4}{4} \left(\alpha - \dfrac{1}{2}\sin2\alpha\right)\) |
Circular Sector |
![]() |
\(\dfrac{2}{3}\dfrac{r\,\sin\alpha}{\alpha}\) | \(2{\alpha}\,r^2\) |
\(I_{x^{'}x^{'}} =\)\(\dfrac{r^4}{4} \left(\alpha + \dfrac{1}{2}\sin2\alpha\right)\) |
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.