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3.4.2.2: The Parallel Axis Theorem

The moment of inertial can be calculated for any axis. The knowledge about one axis can help calculating the moment of inertia for a parallel axis. Let \(I_{xx}\) the moment of inertia about axis \(xx\) which is at the center of mass/area. The moment of inertia for axis \(x'\) is \[I_{x'x'} = \int_{A} r'^{2} dA = \int_{A}\left(y'^{2} + z'^{2}\right)dA = \int_{A}\left[\left(y+\Delta y\right)^{2} + \left(z + \Delta z\right)^{2}\right]dA\tag{23}\] equation 23 can be expanded as \[I_{x'x'} = \int_{A}\left(y^{2} + z^{2}\right)dA + 2\int_{A}\left(y\Delta y + z \Delta z\right)dA + \int_{A} \left(\left(\Delta y\right)^{2} + \left(\Delta z\right)^{2}\right)dA \tag{24}\] The first term in equation 24 on the right hand side is the moment of inertia about axis \(x\) and the second them is zero. The second therm is zero because it integral of center about center thus is zero. The third term is a new term and can be written as \[\int_{A}\left(\left(\Delta y\right)^{2} + \left(\Delta z\right)^{2}\right)dA = \left(\left(\Delta y\right)^{2} + \left(\Delta z\right)\right)\int_{A}^{2} dA = r^{2} A \tag{25}\] Hence, the relationship between the moment of inertia at \(xx\) and parallel axis \(x'x'\) is 

Parallel Axis Equation

\[I_{x'x'} = I_{xx} + r^{2}A\tag{26}\]

 

Fig. 3.5. The schematic to explain the summation of moment of inertia.

The moment of inertia of several areas is the sum of moment inertia of each area see Figure 3.5 and therefore, \[I_{xx} = \sum_{i=1}^{n} I_{xxi}\tag{27}\] If the same areas are similar thus \[I_{xx} = \sum_{i=1}^{n} I_{xxi} = nI_{xxi}\tag{28}\]

Cylinder Moment of Inertia

Fig. 3.6. Cylinder with an element for calculation moment of inertia.

Equation 28 is very useful in the calculation of the moment of inertia utilizing the moment of inertia of known bodies. For example, the moment of inertial of half a circle is half of whole circle for axis a the center of circle. The moment of inertia can then move the center of area. of the