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

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.

Reference page-260, under dynamics and fixed axis rotation

• Commented Jun 12, 2021 at 0:09

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.