Why can't we prove that sum of internal torques always sum to zero from Newton's laws?

Question

We can argue that internal forces on a body add up to zero by saying that forces are created in pairs (third law) inside the system and hence the net sum must be zero. Similarly we by the third law of rotational dynamics, we should be able to argue the same for rotations.

However, it is written in the book of Kleppmer and Kolenkow: introduction to mechanics, that it is not possible to prove that internal torques sum to zero using newton's laws and we must accept it as an experimental fact. Why exactly does my previously stated argument fail?

Thanks to @Rosnaik I figured out my argument was indeed correct. However, I do wish to know what Kleppner was trying to say here.

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Reference page-260, under dynamics and fixed axis rotation

Answer 1

Suppose you have two particles 1 and 2 interacting with each other. Force on 1 by 2 is $F_{12}$ and force on 2 by 1 is $F_{21}$. From Newton's third law it follows that $F_{12} = - F_{21}$. Now let's calculate the torque of this two particle system. $$\tau_{internal} = \mathbf{r_1} \times \mathbf{F_{12}} + \mathbf{r_2} \times \mathbf{F_{21}}$$ $$ = (\mathbf{r_1} - \mathbf{r_2}) \times \mathbf{F_{12}}$$ When does this torque vanish? It vanishes when $\mathbf{r_1} - \mathbf{r_2}$ is parallel to $\mathbf{F_{12}}$. In other words, the net internal torque in a system is zero only when the internal forces are central, i.e., if they point along the line connecting the two particles.

Remember Newton's laws do not require this. Newton's third law only says the forces have to be equal and opposite. It does not say that they also have point along the connecting vector. That depends on the nature of forces. You have to do some extra experiments to find that out. If, experimentally you find that there are non central forces between particles, then internal torque cannot be zero.

Answer 2

This is just a small addition to the other answers...

Like the other answers state, just from Newton's 3 laws, it is impossible to predict that net internal torque is zero.

But it is fair enough to assume that the interactions between the particles are electrostatic. Since electrostatic force is a central force, we can conclude that the net internal torque is zero.

Answer 3

If the sum of the internal torques was not zero, then the system could undergo spontaneous angular acceleration in violation of conservation of angular momentum (and energy).