Net Torque and Conservation of Angular Momentum

[SprucerMoose]

G'day all,

I have been reading that if there are no external torques on a system, then angular momentum will be conserved. Can a system be defined as anything or only a single rigid object. Could I define the "system" as 2 disks rotating side by side? If so then the total angular momentum of this system would be the sum of momentum of the 2 disks. What if these these disks had each edge pushed into each other by an equal force on either side so the net force on the system remained zero? Assuming friction is present each disk would exert and equal and opposite force on the other. If each disk has a different radius, wouldn't this create a non-zero net internal torque and cause a change in angular momentum of a closed system? This contradicts my first statement, so could someone please point out the error in my reasoning.


The only way I can see Newtons 3rd law and the conservation of angular momentum holding would be if the above circumstance results in a net torque on the two disks together causing them to rotate together (about each other), while exerting a torque on each other. I have no proof for this it is just my intuition. The reason I'm not sure is because today I was also told that you cannot define a net torque on a system of several objects like this, but to me this leads to a breakdown in the conservation of angular momentum, or do I have it wrong?

 

[Filip Larsen]

I have been reading that if there are no external torques on a system, then angular momentum will be conserved. Can a system be defined as anything or only a single rigid object.

This is valid for any system, it does not have to be a rigid body.

Could I define the "system" as 2 disks rotating side by side? If so then the total angular momentum of this system would be the sum of momentum of the 2 disks. What if these these disks had each edge pushed into each other by an equal force on either side so the net force on the system remained zero? Assuming friction is present each disk would exert and equal and opposite force on the other. If each disk has a different radius, wouldn't this create a non-zero net internal torque and cause a change in angular momentum of a closed system? This contradicts my first statement, so could someone please point out the error in my reasoning.

Yes, you can easily have a system of two spinning objects with a net angular momentum that combine without any external forces and produce a "single" object that still has the same angular momentum as before. If the combined object is rigid then it has to rotate around its center of mass in order to have this angular momentum.

The only way I can see Newtons 3rd law and the conservation of angular momentum holding would be if the above circumstance results in a net torque on the two disks together causing them to rotate together (about each other), while exerting a torque on each other. I have no proof for this it is just my intuition. The reason I'm not sure is because today I was also told that you cannot define a net torque on a system of several objects like this, but to me this leads to a breakdown in the conservation of angular momentum, or do I have it wrong?

There surely is no breakdown in conservation of angular momentum in your example. And the net torque on an isolated system is zero.

While I understand that you want to understand what happens at the level of forces and torques that makes conservation of linear and angular momentum "happen", you may find it easier to accept the "opposite view", that is, regard the conservation laws to be more fundamental and use that to explain the (net) exchange of forces and torques.

On the other hand, if you sat down and really made the proper calculations of forces and torques in your example, I'm sure you will arrive at the conclusion that angular momentum is conserved.

 

[kewljerk]

I could not get what actually your doubt is. You are yourself solving all your doubts.
Initially angular momentum was non-zero. When you brought them in contact to each other, net force on the system is zero but, still there is a net torque on the system as the direction of torque is same. As torque is different therefore, there would be a change in angular momentum.
Do tell if still there is a doubt.

 

[SprucerMoose]

I could not get what actually your doubt is. You are yourself solving all your doubts.
Initially angular momentum was non-zero.

My assumptions are not solving any doubts. I am assuming things to be true based on intuition and not a rigorous proof, hence the reason I am asking in a forum where someone who understands, based on proof, may be lurking.

When you brought them in contact to each other, net force on the system is zero but, still there is a net torque on the system as the direction of torque is same. As torque is different therefore, there would be a change in angular momentum.
Do tell if still there is a doubt.

A change in angular momentum breaks the law of conservation of angular momentum for a closed system as far as I understand it.

 

[Filip Larsen]

My assumptions are not solving any doubts. I am assuming things to be true based on intuition and not a rigorous proof, hence the reason I am asking in a forum where someone who understands, based on proof, may be lurking.

What is it you want proof of? The laws of mechanics, like Newtons and Eulers laws, are defined for conservations laws to hold. For instance, for a system of particles Newtons law says dp/dt = f, where p is the linear momentum vector, f is total external force, and d()/dt is time derivative in an inertial reference frame. As you can see, this law directly says that as long as f is zero there can be no change in total linear momentum for the system. Same goes for Eulers law, i.e. for the rotational state of the system.

 

[SprucerMoose]

I do not know euler's law. I set up the earlier scenerio to see where the error was in my original reasoning. Is the net torque on a system the sum of all torques within that system (even if the system is made of non-attached bodies)? Is this net torque, if it is defined that way, always zero in the absence of an extnernal torque to the system? Do torques occur in action reaction pairs or have an equivilent, explicit conservation law?

As I said, I was told earlier today that a net torque on a system is NOT defined this way. I asked a lecturer the earlier example and was told that an idea of NET torque on a System of multiple bodies does not make sense.

My idea for how this 'may' be resolved is just that, an idea. I don't know if that is what would occur in such a situation and that is why I am asking. Must these two disks rotate around each other for momentum to be conserved?

 

[AlephZero]

If only makes sense to define "angular momentum of a system" based on the rotatiion of everything in the system about a single point. And it is easiest if that single point is the CG of the entire system.

If you have two spinning rigid disks, the angular momentum is of the whole system is NOT equal to the angular momentum of each disk about its own CG, because the two disks may also be rotating about their combined CG.

Conservation of angular momentum and conservation of linear momentum both follow from the fact that the internal forces (actions and reactions) in a system are always equal and opposite.

Angular momentum can be transferred from one part of the system to another, just like linear momentum can.

I don't understand what your Lecturer was trying to say, but you only told us the answer and not the exact question that you asked, so it's impossible to decide whether he/she was "right" or not.

 

[Filip Larsen]

I do not know euler's law.

Euler's law is to rotation what Newtons law is to linear motion for a particular system and says that time rate change in total momentum of the system equals the net torque applied to that system, all relative to some reference frame and reference point. In lack of a better online reference you can see a brief description on http://en.wikipedia.org/wiki/Euler's_laws_of_motion

Is the net torque on a system the sum of all torques within that system (even if the system is made of non-attached bodies)?

No, the sum of all torques "between" bodies or parts internal to the system cancels due to Newtons third law. If you have a system of two spinning discs that exchange angular momentum between them, then the sum of all the forces and the sum of all the torques between the two discs are zero.

Is this net torque, if it is defined that way, always zero in the absence of an extnernal torque to the system?

Yes.

Do torques occur in action reaction pairs or have an equivilent, explicit conservation law?

Yes, every torque has a corresponding reaction torque relative to the given reference point.

As I said, I was told earlier today that a net torque on a system is NOT defined this way. I asked a lecturer the earlier example and was told that an idea of NET torque on a System of multiple bodies does not make sense.

My idea for how this 'may' be resolved is just that, an idea. I don't know if that is what would occur in such a situation and that is why I am asking. Must these two disks rotate around each other for momentum to be conserved?

If you have two rotating discs whose center of mass (CM) is at rest relative to each other and at rest relative to an inertial reference frame (zero linear momentum), and the discs then combine to one rigid object, then the CM for this object is still at rest relative to inertial space (with a position somewhere between the CM of the two discs). The angular momentum of the combined object (e.g. around the common CM) will be equal to the angular momentum of each of the original rotating two discs around the same reference point (e.g. common CM). If this angular momentum is non-zero, the combined discs will be rotating as a rigid body around the common CM.

Note, as AlephZero also said, that you cannot just add the angular momentum vectors for two objects unless they both use the same reference point, but there are equations to translate the angular momentum from one reference point to another so you can add angular momentum of parts of system to get the total angular momentum of the system.

 

[rcgldr]

Getting back to the OP's example, you'd need some type of frame work or gravity on the system in order to push the two rotating disks together edge to edge. Assuming that both disks initially rotate in the same direction and the framework or gravity causes the disks to be pushed together. When the disks make contact the linear friction force at the point of contact will result in angular acceleration of the entire system (the disks may also separate in the case of gravity), so as the disks slow down, the entire system speeds up and angular momentum is conserved.

 

[Filip Larsen]

... so as the disks slow down, the entire system speeds up ...

The actual rotational speed of the combined object can be less, equal or more than the rotational speed of the original discs depending on how mass is distributed when they combine, so "speeds up" is a term that to me sounds misleading. I would merely say that the combined rigid object rotates with a rotational speed that conserves angular momentum.

And to the OP:

The reason conservation laws "work" is not so much because the various forces and torques are as the are (since force and torque are defined as the time rate change of linear and angular momentum), but because all micro interactions satisfy the conservation laws. Any interaction between two objects (whether we consider them as particles or as continuously extended objects) satisfy the conservation laws in isolation so they are also satisfied no matter what "sequence" of interactions you line up to get a macroscopic mechanical effect.

The only way you would be able to "break" the conservation laws is if you can measure an interaction between two object that break the laws. Note that I said measure, and not calculate, since all "legal" calculations in mechanics already satisfy the conservation laws, so if you can calculate a break in conservation this is only an indication that you have made a calculation or modelling error.