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For total angular momentum of a disk traveling around a fixed point(pivot) at angular velocity $\Omega$, is the rotation of the disk around its own axis ($\omega$) relevant in the equation?

No need for a complicated explanation :)

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Yes, you can write

$$\vec{L}_{Tot}=\vec{L}_{CM}+\vec{L}^*$$

where $\vec{L}_{CM}$ is the angular momentum of the center of mass and $\vec{L}^*$ is the angular momentum viewed from the center of mass.

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