Cons. Angular Momentum & Reference Frames

[withchemicals]

Hi! I'm currently a student taking a classical mechanics course.

Finals are coming up, and I've come to realize that I seem to have a firm grasp of most of the material (energy, forces, etc...) but not momentum. I know this because I was flabbergasted by a problem on my last midterm that asked about angular momentum with respect to the pivot of a pivoting rod and at the point of collision (a ball hits the rod at the end of it).

I was under the impression that momentum is conserved regardless of reference frame (it certainly seemed so for linear momentum, but I may by wrong).

1. How do different reference frames affect angular momentum and conservation of angular momentum? Is this only for the specific case of a pivoting rod hit by a ball at one end (as opposed to a rod floating in space that is hit by a ball at one end)?
2. Also, how can a non-rotating ball that travels linearly have angular momentum?
3. Lastly, why is there a net external torque at the pivot -again- depending on reference frame?

I'm afraid that these questions might be simple to a seasoned person, so bear with my silly questions! I really wish to nail angular momentum, but my textbook simply has 2-3 pages on the subject. In addition, my textbook doesn't seem to have any questions to address these situations, so I don't have any resources to work with. I tried Google-ing, but I couldn't come across anything that talks about these situations, so references would be great too!

 

[Doc Al]

1. How do different reference frames affect angular momentum and conservation of angular momentum? Is this only for the specific case of a pivoting rod hit by a ball at one end (as opposed to a rod floating in space that is hit by a ball at one end)?

Please describe this ball & pivoting rod problem exactly so we can understand what the issue is. What different 'reference frames' are you talking about?

If a ball strikes a rod that is constrained by a frictionless pivot at one end, then a natural axis to use is the pivot, since there will be no net torque about that point.

2. Also, how can a non-rotating ball that travels linearly have angular momentum?

It will have an angular momentum with respect to some external point, not an angular momentum about its center of mass. The angular momentum of a point particle is defined to be L = r X p, where p is the particle's linear momentum.

3. Lastly, why is there a net external torque at the pivot -again- depending on reference frame?

Again, you need to specify the problem exactly. Unless there's friction at the pivot, I don't see why you'd think that there's a net torque about the pivot.

 

[withchemicals]

Whoops! Sorry.

Pretty much the rod was vertically hanging from a ceiling and pivoted at one end. A ball would have a horizontal velocity and hit it directly at the opposite tip of the rod (not pivoted and hanging down). This collision is inelastic. The question asked about angular momentum conservation with respect to the pivot and with respect to the other end where the ball hits.

I don't have the specific question because my professor doesn't return exams (stupid), but this should be enough!

 

[Doc Al]

The question asked about angular momentum conservation with respect to the pivot and with respect to the other end where the ball hits.

OK. To start with, do you agree that angular momentum would be conserved about the pivot, since there's no net torque about that point?

As far as using the other end as your reference, realize that its accelerating. And that the pivot, since it exerts a force on the rod, would exert a torque about that point.

 

[withchemicals]

OK. To start with, do you agree that angular momentum would be conserved about the pivot, since there's no net torque about that point?

As far as using the other end as your reference, realize that its accelerating. And that the pivot, since it exerts a force on the rod, would exert a torque about that point.

Hmm... I'm trying to reason and so far, I understand that there is an angular momentum at the "end of the stick" as viewed from the pivot. Am I correct to say that there is an angular momentum at the pivot as viewed from the "end of the stick"? This is because the pivot exerts a torque which makes it seem as if the pivot caused a rotation rather than the colliding ball. Are these statements true?

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Also, in order to talk about conservation of angular momentum, must we define the point at which we are considering conservation of angular momentum and the reference frame?

If so, from the pivot reference frame, the pivot should have angular momentum conserved because there is no net torque. The "end of the stick" should have angular momentum conserved w.r.t. the pivot because the angular momentum of the ball the collides participates and causes a torque on the pivot.

On the other hand, with respect to the "end of the stick", the pivot does not conserve momentum because from that reference frame, it would seem as if the pivot end is rotating spontaneously(?). In addition, from the "end of the stick" frame, the ball would seem to be colliding but causing no net torque (from the reference frame), so conservation of momentum would be violated there too(?).

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My apologies my potentially dumb questions, and thanks for your time!

 

[Doc Al]

Hmm... I'm trying to reason and so far, I understand that there is an angular momentum at the "end of the stick" as viewed from the pivot.

Here's how I would express it: Before the collision, the system (stick + ball) has a non-zero angular momentum with respect to the pivot.

Am I correct to say that there is an angular momentum at the pivot as viewed from the "end of the stick"?

Before the collision, the system has zero angular momentum about the impact point. The stick isn't moving, and the angular momentum of the ball is zero about that point.

This is because the pivot exerts a torque which makes it seem as if the pivot caused a rotation rather than the colliding ball.

Viewed using the impact point as the reference, the pivot exerts a torque.

Also, in order to talk about conservation of angular momentum, must we define the point at which we are considering conservation of angular momentum and the reference frame?

Yes.

If so, from the pivot reference frame, the pivot should have angular momentum conserved because there is no net torque. The "end of the stick" should have angular momentum conserved w.r.t. the pivot because the angular momentum of the ball the collides participates and causes a torque on the pivot.

Angular momentum of the system is conserved with respect to the pivot since no net torque acts about that point. The forces between ball and stick exert equal and opposite torques, so they cancel out; the force exerted at the pivot exerts no torque about that point.

On the other hand, with respect to the "end of the stick", the pivot does not conserve momentum because from that reference frame, it would seem as if the pivot end is rotating spontaneously(?). In addition, from the "end of the stick" frame, the ball would seem to be colliding but causing no net torque (from the reference frame), so conservation of momentum would be violated there too(?).

Yes. Using the impact point as the reference, angular momentum of the system is not conserved.

My apologies my potentially dumb questions, and thanks for your time!

Not dumb at all! Angular momentum (and reference frames) is tricky stuff.