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In the book Feynman's Lectures on physics volume 1 chapter 19, He explains prallel axis theorem as follows.

Suppose we have an object, and we want to find its moment of inertia around some axis. That means we want the inertia needed to carry it by rotation about that axis. Now if we support the object on pivots at the center of mass, so that the object does not turn as it rotates about the axis (because there is no torque on it from inertial effects, and therefore it will not turn when we start moving it), then the forces needed to swing it around are the same as though all the mass were concentrated at the center of mass, and the moment of inertia would be simply $I_{1}=MR_{cm}^2$, where $ R_{cm}$ is the distance from the axis to the center of mass. But of course that is not the right formula for the moment of inertia of an object which is really being rotated as it revolves, because not only is the center of it moving in a circle, which would contribute an amount $I_{1}$ to the moment of inertia, but also we must turn it about its center of mass. So it is not unreasonable that we must add to $I_{1}$ the moment of inertia $I_{c}$ about the center of mass.

I could see the first half of his explanation, but I could not understand last half of it. Why the forces needed to swing it are the same as if all the mass were concentrated at the center of mass and why we should turn the object about its center of mass?

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If the object is not spinning, then moving it will take the same effort whether the mass is spread out, or all at one point.

Imagine that you are on a roundabout, but you are sitting on a disk that it itself able to spin. Initially you are facing North, and you are exactly in the middle of your disk (although it is off axis of the roundabout). Now the roundabout starts spinning. Your center of mass starts going around in circles, but you continue to look North.

After a while, the roundabout has reached a steady speed. You get bored of looking North - you want to talk to the person next to you on the roundabout. In order to keep facing her, you have to spin your little disk until it has the same angular velocity as the roundabout.

Net result: to get you in that state, first the roundabout moved your center of mass; next, you set yourself spinning about the axis of your little disk.

Not sure if this is clear without a diagram - I am struggling to think how to come up with a good one that doesn't involve creating an animation (which I am really really bad at).

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