3.4.2.1: General Discussion

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.

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 $I_{yy} = \int_{A} \left(x^{2} + z^{2}\right)dA$ $I_{zz} = \int_{A} \left(x^{2} + y^{2}\right)dA$