# A question regarding Newton's Third Law for torques

I was trying to prove why, if two objects are rotating around some axis, and are interacting, then the sum of the torques produced on both objects via the mutual interactions will be zero. However, I am having a confusion as to how the forces are being applied. I will use a diagram to make thinks clear.

Consider the following diagram:

Here , there are two particles, $A$ and $B$, which are rotating about an axis, which is the center of this circle. Let $B$ exert a force on $A$, which I have denoted by a vector. The torque produced on $A$ is thus magnitude of the force times the perpendicular distance from the axis, which again I have shown. By Newton's third law, an equal and opposite force acts on $B$, which I have denoted by a vector, equal in magnitude but opposite in direction. Now here is my confusion: If Newton's law says that there is an equal and opposite force along the same line of action, then the torques cancel each other. But, if that is the case, then $B$ isn't lying on the line of action, so how would it experience the force? If I draw an equal and opposite vector through $B$, then the torques don't cancel each other.

Note that the force vector on $A$ is any general force, not necessarily lying on the line segment joining $A$ and $B$. What is wrong here?

## 1 Answer

You're right; the torques only cancel out if and only if the forces are applied along the line connecting the particles. Here's the proof: Let $\vec{F}_{AB}$ be the force on $A$ due to $B$, and define $\vec{F}_{BA}$ similarly. The torque on $A$ due to $B$ is then $$\vec{\tau}_{AB} = \vec{r}_A \times \vec{F}_{AB}$$ and the torque on $B$ due to $A$ is then $$\vec{\tau}_{BA} = \vec{r}_B \times \vec{F}_{BA}.$$ The net torque is then $$\vec{\tau}_\text{net} = \vec{r}_A \times \vec{F}_{AB} + \vec{r}_B \times \vec{F}_{BA}.$$ But by Newton's Third Law, $\vec{F}_{BA} = - \vec{F}_{AB}$, which allows us to simplify this to $$\vec{\tau}_\text{net} = (\vec{r}_A - \vec{r}_B) \times \vec{F}_{AB}.$$ Note that $\vec{r}_A - \vec{r}_B$ is a vector pointing from $B$ to $A$; note also the cross product of two non-zero vectors vanishes if and only if they are parallel. We conclude that $\vec{\tau}_\text{net} = 0$ if and only if the force between the two particles acts along the line connecting them.

What this means is that in your diagram, if $A$ and $B$ are really exerting forces on each other in this way, then the net torque they exert on each other will not in fact be zero. The net torque will only be zero if the particles interact with each other via a central force, which is a fancy way of saying that the force is parallel to the line connecting the particles.

• Thnx a lot. Sorry I was a lil late in accepting the answer. Commented Aug 31, 2017 at 13:53
• @codetalker: No worries about the "lateness"; just yesterday I had an answer from May 2015 accepted, so by that standard you're doing fine. Commented Aug 31, 2017 at 14:28