# Every action has an equal and opposite reaction. Is is true for torques as well?

A torque is exactly the same as two equal and opposite forces acting at different points on a body.

Each force has an equal and opposite reaction force, and the reaction forces are the same as an equal and opposite torque.

Yes, but the opposite reaction torque is not always useful.

If you defined torque as $\tau = r \times F$, then the force $F$ always follows Newton's third law. For any contact force, $r$ is the same. Since $r \times (-F) = -(r \times F)$, it follows that the "third law force" delivers a "third law torque".

For non-contact forces, $r$ is different, but the math will always work out. Otherwise, angular momentum would not be conserved for the closed system of both objects.

All of that said, this "third law torque" can be deceptive. The original torque is defined with respect to some axis - wherever $r$ is measured from. Unless you are interested in how both of the interacting objects are rotating around the same axis, the interaction torque is useless. If you are unsure, you can always find the underlying force and use that third law pair.

Something to keep in mind: torques are not as fundamental as forces. I say this for two reasons. First, torques are defined in terms of forces. Second, the torque produced by a force depends on our subjective point of reference. With that being said, if you have confusions about torques, the best place to start is to just think about forces instead.

So let's do that. Let's say I apply a tangential force of magnitude $F$ to the edge of a wheel of radius $R$ with my hand. Well then by Newton's third law the wheel applies a force to my hand of equal magnitude and opposite direction as the force I applied to the wheel. Both forces act at the same point in space: the point of contact between my hand and the wheel.

Let's look at the torque of these forces about some point, say the center of the wheel. Then the torque of my force is $$\tau_{\text{me}}=FR$$ and by Newton's third law the torque of the wheel's force is $$\tau_{\text w}=-FR=-\tau_{\text {me}}$$ So we do in fact get an "equal but opposite torque".

It is worth pointing out that just like for forces this does not mean both torques act on the wheel. My torque acts on the wheel. The wheel's torque acts on me.

I'll also point out that just like how Newton's third law for forces gives us linear momentum conservation, this version of the third law for torques gives us angular momentum conservation.