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3.6.1: Tables of geometries

  • Page ID
    656
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    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 rectangle.png \(\dfrac{a}{2}\,;\dfrac{b}{2}\) \(a\,b\) \(\dfrac{a\,b^{3}}{12}\)
    Triangle triangle.png \(\dfrac{a}{3}\) \(\dfrac{a\,b}{3}\) \(\dfrac{a\,b^{3}}{36}\)
    Circle circle.png \(\dfrac{b}{2}\) \(\dfrac{\pi\, b^2}{4}\) \(\dfrac{\pi b^4}{64}\)
    Ellipse ellipse.png \(\dfrac{a}{2}\; \dfrac{b}{2}\) \(\dfrac{\pi\, ab}{4}\) \(\dfrac{ab^2}{64}\)
    \(y=\alpha\,x^2\)
    Parabola
    parabola.png \(\dfrac{3\,\alpha\,b}{15\,\alpha-5}\) \(\dfrac{6\alpha -2}{3} \, \left( \dfrac{b}{\alpha}\right)^{\dfrac{3}{2}}\) \(\dfrac{\sqrt{b}\,\left( 20\,{b}^{3}-14\,{b}^{2}\right) }{35\,\sqrt{\alpha}}\)
    Quadrant
    of Circle
    qCircle.png \(\dfrac{4\,r}{3\,\pi}\) \(\dfrac{\pi\,r^2}{4}\) \(r^4\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\)
    Ellipsoidal
    Quadrant
    qelliptic.png \(\dfrac{4\,b}{3\,\pi}\) \(\dfrac{\pi\,a\,b}{4}\) \(a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\)
    half of
    of Ellipse
    hElliptic.png \(\dfrac{4\,b}{3\,\pi}\) \(\dfrac{\pi\,a\,b}{4}\) \(a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)\)
    Circular
    Sector
    cSector.png \(0\) \(2{\alpha}\,r^2\) \(\dfrac{r^4}{4} \left(\alpha - \dfrac{1}{2}\sin2\alpha\right)\)
    Circular
    Sector
    lcSector.png \(\dfrac{2}{3}\dfrac{r\,\sin\alpha}{\alpha}\) \(2{\alpha}\,r^2\)

    \(I_{x^{'}x^{'}} =\)\(\dfrac{r^4}{4} \left(\alpha + \dfrac{1}{2}\sin2\alpha\right)\)

    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.


    This page titled 3.6.1: Tables of geometries is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


    This page titled 3.6.1: Tables of geometries is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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