# Does anybody know where this calculation for the mass moment of inertia for an eighth of a circle comes from?

These attached photos describe flaps which are a portion of a circle and equations are attached as the mass moment of inertia, but there are no descriptions. I am using this for part of my research, but I do not know if this was something calculated through an integral or not, but I am very unaware of this being anything related to what the mass moment of inertia would be. The equations I am given are:

$$l=2R \frac{\sin{\alpha}}{\alpha} \\ \alpha=\pi/8 \\ I_3 = m_3 k_{y'}^2 \\ k_{y'}^2 = \frac{1}{72 \alpha} [18 \alpha^2 + 9\alpha \sin{2\alpha} - 32 (1-\cos^2{\alpha})]$$

• what is exactly the question? Are you interested in the derivation or an explanation?
– NMech
Feb 26, 2021 at 17:41
• @NMech Ah okay, sorry for the photo confusion, these are the same objects. One is a top view and one is a side view. The flap rotates along the y axis. I am not sure where they are getting this equation for the moment of inertia. I know that the moment of inertia for even part of a circle is much simpler and I tried plugging things in, but it doesn't make sense to me especially since there is no R present in the last equation. This equation may also be completely wrong, but I have no reference to prove or disprove that as I am not sure what they were able to do. Feb 26, 2021 at 17:47