3.4.2.1: General Discussion

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For body with thickness, $$t$$ and uniform density the following can be written $I_{xxm} = \int_{m} r^{2} dm = \rho t \quad \int_{A} r^{2} dA \tag{17}$ The moment of inertia about axis is $$x$$ can be defined as

Moment of Inertia

$I_{xx} = \int_{A} r^{2} dA = \frac{I_{xxm}}{\rho t} \tag{18}$

where $$r$$ is distance of $$dA$$ from the axis $$x$$ and $$t$$ is the thickness.

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

Any point distance can be calculated from axis $$x$$ as $x = \sqrt{y^{2} + z^{2}}\tag{19}$ Thus, equation 18 can be written as $I_{xx} = \int_{A} \left(y^{2} + z^{2}\right)dA \tag{20}$ In the same fashion for other two coordinates as $I_{yy} = \int_{A} \left(x^{2} + z^{2}\right)dA \tag{21}$ $I_{zz} = \int_{A} \left(x^{2} + y^{2}\right)dA \tag{22}$

Contributors

• 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.