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Suppose I consider a massless rod of several discrete masses with an external force being applied to one of the masses. Following the derivation here, the net torque on the system depends only on the external forces. Namely, the text states that all internal forces are along the rod and thus cancel when taking their respective cross products with $r_\alpha-r_\beta$.

But let us consider any one of the masses not experiencing an external force. The net force on this mass is simply the sum of the internal forces acting on it due to the other masses, which once again are along the direction of the rod as stated from the text. But if I were to calculate the net force on this mass strictly based on its motion, I would write $F=ma=mr\alpha$ which is a tangential force! How then could this be understood as the sum of internal forces acting on it if they're all along the direction of the rod?

Update: Apologies, it appears that I've misread the text. Namely $r_\alpha-r_\beta$ or the direction along the vector connecting $m_\alpha$ and $m_\beta$ need not be perpendicular to the direction of rotation. This makes sense, since lattice interactions need not be directly along the rod. Apologies for this!

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Internal forces are forces which do not cause any change in the acceleration of center of mass. That doesn't mean they can't accelerate an individual mass with respect to the center of mass. Thus, they can be perpendicular to the rod.

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