How Do Equal and Opposite Torques Work with Different Moment Arms?

[BrainSalad]

Angular momentum is conserved, which means that the change in angular momentum of object A must be exactly canceled by the change in angular momentum of the object exerting a torque on object A. So, the objects, A and B, exert equal and opposite torques on each other. But, the contact forces between the objects must also be equal (Newton's 3rd). Torque= F x r, so what if the moment arm (r) of object A is larger than object B? If torques are equal and forces are equal, moment arms must also be equal, but this is certainly not always the case. What's going on?

 

[jbriggs444]

"angular momentum" encompasses more than just the rotation of rigid objects around their centers of mass. A moving, non-rotating object also has angular momentum around any specified point. Before you can even have a well-defined angular momentum, you have to specify that point.

Say you have specified that point and you have two objects that are interacting with a contact force -- they bump into each other. By Newton's third law, the forces are equal and opposite [as you have understood]. By the definition of angular momentum, the moment arms are equal -- it is a contact force so both forces act at the same point and both moment arms originate at the same point. It follows that the two torques are equal and opposite.

If you have a force-at-a-distance, things are a little messier, but it works out that the cross product of force times moment arm is still equal and opposite.

 

[A.T.]

If torques are equal and forces are equal, moment arms must also be equal, but this is certainly not always the case.

The equal but opposite torques must be both computed around the same point.

 

[BrainSalad]

Thanks to both of you, that clears things up. Can't wait to learn this stuff formally.