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.

 

[PJC01]

Thank you for your question. I can provide an explanation for the concept of equal and opposite torques.

First, let's define torque. Torque is the measure of the twisting force that causes an object to rotate. It is calculated by multiplying the force applied to an object by the distance from the point of rotation to the point where the force is applied. This distance is known as the moment arm.

Now, let's consider the scenario you presented where objects A and B are exerting equal and opposite torques on each other. This is possible because of the principle of conservation of angular momentum. This principle states that the total angular momentum of a system remains constant unless acted upon by an external torque. In other words, the change in angular momentum of one object must be exactly canceled by the change in angular momentum of the other object.

So, in the case of objects A and B, if they are exerting equal and opposite torques on each other, it means that the change in their angular momentum is equal and opposite. This can only happen if the forces and moment arms are also equal and opposite.

Now, let's address the concern about the moment arms being different. You are correct that in some cases, the moment arm of object A may be larger than the moment arm of object B. However, this does not necessarily mean that the torques are not equal and opposite. The key factor here is the force applied to each object. If the force applied to object A is larger than the force applied to object B, then the moment arm of object A can be larger and still result in equal and opposite torques.

To summarize, the principle of conservation of angular momentum ensures that equal and opposite torques are exerted on objects A and B. This is possible because the forces and moment arms are also equal and opposite. The difference in moment arms can be compensated by a difference in the forces applied to each object. I hope this explanation helps to clarify the concept of equal and opposite torques.