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3.4.2.1: General Discussion

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    649
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    For body with thickness, \(t\) and uniform density the following can be written \[I_{xxm} = \int_{m} r^{2} dm = \rho t \quad \int_{A} r^{2} dA \] The moment of inertia about axis is \(x\) can be defined as

    Moment of Inertia

    \[I_{xx} = \int_{A} r^{2} dA = \frac{I_{xxm}}{\rho t} \]

    where \(r\) is distance of \(dA\) from the axis \(x\) and \(t\) is the thickness.

    parallelTheorem.png

    Fig. 3.4. The schematic that explains the summation of moment of inertia.

    Any point distance can be calculated from axis \(x\) as \[x = \sqrt{y^{2} + z^{2}}\] Thus, equation 18 can be written as \[I_{xx} = \int_{A} \left(y^{2} + z^{2}\right)dA \] In the same fashion for other two coordinates as \[I_{yy} = \int_{A} \left(x^{2} + z^{2}\right)dA \] \[I_{zz} = \int_{A} \left(x^{2} + y^{2}\right)dA \]

    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 3.4.2.1: General Discussion 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 3.4.2.1: General Discussion 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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