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Τ = I*α

contradicts my intuition. Suppose that you take a part of a body in the radius r from the rotational axis and exert a force on that.

It will then get a linear acceleration. We can then find the angular acceleration using:

a = r*α

Which means that the angular acceleration is bigger the closer we are to the axis of rotation. I mean that if we take 2 identical rotating disk, and measure the linear acceleration of a point on the first disk to be a at r=1m, and then find for the second disc, that the linear acceleration is a at r=2m, then obviously the first disk must have a higher angular acceleration... But then this suggests that a force exerted closer to the rotational centre means a higher angular acceleration than one exerted further away, and that is not implied by the rotational analog of Newtons second law. What am I doing wrong in my assumptions?