# 3.4.2.1: General Discussion

- Page ID
- 649

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.