I'm trying to understand the derivation of the torque equation →r×→F=Iα. My textbook derives this easily enough from Newton's 2nd Law for a single point with mass m and radial distance r, with the force applied at the same distance r, as (if we drop the vector notation for simplicity) F=ma=m(rα), so rF=mr2α=Iα. (Note, there are both 'ay's and 'alpha's there; they look similar.)
The textbook stops there and concludes that the equation holds for any rotating body. But then I tried this derivation for two points positioned along a rigid, massless rod (which points perpendicular to an axis about which it rotates). If there's a point-mass m1 a distance r1 from the axis of rotation and another point-mass m2 at a distance r2, and the force is applied at (for example) distance r2, I get F=m1a1+m2a2=m1r1α+m2r2α, so r2F=(m1r1r2+m2r22)α. But m1r1r2+m2r22 isn't the correct value for I here. What am I missing?
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