For a single point mass : τ=Ftr=matr=(mr2)α=Iα
For multiple point masses bound together : ∑τi=(mir2i)α=Iα
But how do we go from that to Iα=Ftrapp (for the multiple masses case), where rapp is the point where force is applied? That is saying that the entire sum of torques of individual masses ∑τi is physically equivalent to (i.e. produces the same angular acceleration of the bound rigid object as) a single torque Ftrapp applied to the point rapp. (Or putting it another way, that applying a single force Ftrapp will result in many Ft_i forces on each mass element such that ∑Ft_iri will be equal to Iα.)
Every place I've looked for a derivation just jumps from ∑τi=Iα, to τ=Iα, as if they are the same thing.
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