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Say we have a disc with its center at the origin. The disc has a mass M and a radius R, the density distribution of the disc is constant and is spinning as shown. The moment of inertia with respect to the rotation axis of the disc passing through the origin is 0.5*MR^2, but this is something very specific. Is there a thing like a moment of inertia of this disc with respect to point P just like you can choose any point in space to define an angular momentum, or does moment of inertia needs to be with respect to the rotational axis of the object?

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  • $\begingroup$ If you're interested in a more general moment of inertia that applies to any axis then the thing you want is the moment of inertia tensor. $\endgroup$
    – M. Enns
    Feb 8, 2022 at 2:07

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Provided the rotating object is made to rotate around another axis, then yes. You always need an axis of rotation for moment of inertia to make sense.

You can define the moment of inertia about any axis, and not just the axis of rotation of the object, using the parallel axis theorem which states that if a body has a moment of inertia $I$, and if the object now rotates about another axis (parallel to the original axis) then $$I'=I+ml^2$$ where $l$ is the perpendicular distance between the two axes, and $I'$ is the moment of inertia about the new axis.

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  • $\begingroup$ The disc doesn't rotate around the point P though. Point P is an arbitrary point in space, the disc still rotates around its center. The question is that even though P is arbitrary and is not on an axis of rotation, is defining a moment of inertia with respect to P a thing? $\endgroup$ Feb 8, 2022 at 1:23
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    $\begingroup$ I see what you mean. No, because about P there is no rotation, so it doesn’t really makes sense to define an axis of rotation. Cheers $\endgroup$
    – joseph h
    Feb 8, 2022 at 1:49
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Yes, the moment of inertia is only defined with respect to some axis of rotation see for example https://en.wikipedia.org/wiki/Moment_of_inertia#Point_mass. Calculating the moment of inertia with respect to some axis, shifted parallel to the centre of gravity (in your case shifting from the origin to point P) can be done by using the parallel axis theorem https://en.wikipedia.org/wiki/Parallel_axis_theorem

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