The Angular Momentum of a System of Particles — Question 4

Question 4

Internal central forces cancel torque pair by pair

Two particles exert equal and opposite internal forces $\vec F_{12}$ and $-\vec F_{12}$, where $\vec F_{12}$ acts on particle 1 and is parallel to $\vec r_1-\vec r_2$.

1. Show that their combined internal torque about any common origin is zero.

2. Would equal and opposite forces alone be sufficient if they were not collinear with the separation?

Question 4 - Solution

Internal central forces cancel torque pair by pair

(a) Derivation and calculation

The combined internal torque is $$\vec\tau_{12}=\vec r_1\times\vec F_{12}+\vec r_2\times(-\vec F_{12}) =(\vec r_1-\vec r_2)\times\vec F_{12}.$$ For a central pair, $\vec F_{12}=\lambda(\vec r_1-\vec r_2)$, giving $$\boxed{\vec\tau_{12}=\vec0}.$$

(b) Result and interpretation

Equal opposite forces alone do not suffice. For example, $$\vec r_1-\vec r_2=2a\hat x,\qquad\vec F_{12}=F\hat y$$ give $$\vec\tau_{12}=2aF\hat z\ne0.$$ They form a couple. A complete physical model must then include the angular momentum of any omitted internal rotations or mediating objects; particle translation alone is not a closed description.

Physical check

The separation vector is independent of origin, so the cancellation for a central pair holds about every common origin.