# Does moment of inertia has to be with respect to a rotational axis?

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?

• 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. Commented Feb 8, 2022 at 2:07

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.