[David Cooper (OP)]

If an ice skater pulls his/her arms in while rotating, the rotation rate can increase dramatically. Everything I'm finding by searching for information about this describes this case as conservation of momentum with zero external torque. My interest in this came out of a conversation with someone on Quora who claimed to have disproved the conservation of angular momentum, but his attempt to do so was wayward, but I think he has found an genuine error with the idea of conservation of angular momentum and I've identified a mechanism by which external torque appears to be being missed.

Picture a ball on a string flying round in circles. If you shorten the string, the ball is forced to move round a smaller circle, but its speed of movement through space increases. Clearly energy is being added to it by the pull on the string. I've seen a video of a lecture with the lecturer laying down the law about how there cannot be any external torque involved in this because the angle of the pull of the string is at 90° to the direction the ball's moving it, but he's clearly wrong: pulling the string northwards while the ball is moving eastwards will add a northward component of movement to it, and quarter of a lap later, that vector will line up with the changing tangential movement of the ball and add to its tangential speed.

That's the mechanism that I've identified for this, but I can't find any other description of this mechanism from anywhere else, so I'm wondering if it's a new discovery or if it's just old hat. I obviously expect the latter, but my searches aren't finding anything other than people asserting that there cannot be any external torque involved. The person who claimed to have disproved angular momentum is sure that people's measurements are faulty and the increase in rotation speed isn't as big as is predicted by angular momentum, and he's telling me that my mechanism is pseudoscience, so I'd like to find this mechanism spelt out in the existing literature so that I can point him to it, assuming that it isn't a new discovery.

Here's what I think happens when a skater pulls his/her arms in: (s)he can only do this by adding energy released from muscles, and that's the source of the extra energy of the rotation. When the arms are moved back out, energy is lost as heat rather than being maintained in the rotation. In the ball-on-a-string case, you could pull the string in using a motor, and the closer in the ball gets and the faster it moves, the more extreme the pull has to become to be able to move it closer. When letting the string out again, it can do work by generating power with a dynamo, in a lossless system recovering all the energy that was put in when pulling the string.

If you let the string out in one go without holding onto it until the string goes tight at maximum length, the ball will shoot outwards and then apply a huge force to the string as it stops, losing that energy as heat rather than maintaining it in the speed of movement of the ball, so either way that energy is necessarily lost from the tangential movement of the ball. When you let out the string by any method, if we look at the same point in the lap as before, you reduce the northward acceleration at the point of the lap when that acceleration is strongest, so it can't maintain its previous speed and has to lose kinetic energy.

This can't really be a new discovery, so does anyone know where existing documentation of it can be found?

[alancalverd]

The energy of a skater's rotation (½Iω²) is conserved, along with his angular momentum Iω. Arms in or out alters the moment of inertia I (= ∫mr²) and thus rotational speed ω.

[Bored chemist]

The energy of a skater's rotation (½Iω²) is conserved, along with his angular momentum Iω. Arms in or out alters the moment of inertia I (= ∫mr²) and thus rotational speed ω.

Great.
And now, apply the same logic here
https://www.thenakedscientists.com/forum/index.php?topic=80136.0

[Halc]

If an ice skater pulls his/her arms in while rotating, the rotation rate can increase dramatically. Everything I'm finding by searching for information about this describes this case as conservation of momentum with zero external torque.

That information sounds correct to me. With the skater, there's always some friction with the ice, so the momentum is slowly drained away. She can only keep it up so long.

My interest in this came out of a conversation with someone on Quora who claimed to have disproved the conservation of angular momentum, but his attempt to do so was wayward, but I think he has found an genuine error with the idea of conservation of angular momentum and I've identified a mechanism by which external torque appears to be being missed.

Quora doesn't have a policy of promoting good answers. I ignore the site for any meaningful question. Go to physics stack exchange which has standards.

Picture a ball on a string flying round in circles. If you shorten the string, the ball is forced to move round a smaller circle, but its speed of movement through space increases. Clearly energy is being added to it by the pull on the string

The speed of movement through space increases  since as it spirals in, the force on the ball is not entirely tangential. A spiral path is not perpendicular to that string. This is fine. Whatever is reeling in that string is performing work, adding energy to the system.

I've seen a video of a lecture with the lecturer laying down the law about how there cannot be any external torque involved in this because the angle of the pull of the string is at 90° to the direction the ball's moving it

That's wrong. It is 90 only while moving at a fixed radius, not while being reeled in.

but he's clearly wrong: pulling the string northwards while the ball is moving eastwards will add a northward component of movement to it, and quarter of a lap later, that vector will line up with the changing tangential movement of the ball and add to its tangential speed.

You're describing tangential force which can only change the velocity, not the speed.

Here's what I think happens when a skater pulls his/her arms in: (s)he can only do this by adding energy released from muscles, and that's the source of the extra energy of the rotation. When the arms are moved back out, energy is lost as heat rather than being maintained in the rotation.

In an ideal case, negative work is done moving the arms back out. No energy loss as heat. Angular speed resumes prior rate.
I agree that angular momentum must be preserved in this case, in which case the energy expended to draw in the arms goes into increased speed of the component, preserving angular momentum. For a closed system, the body (central mass) is not fixed and has to be taken into account. It applies torque to the skater arms and slows them down, thus preserving the angular momentum.  With the ball and the string, that torque might not be there, so the situation may not be analogous.

[David Cooper (OP)]

The energy of a skater's rotation (½Iω²) is conserved, along with his angular momentum Iω. Arms in or out alters the moment of inertia I (= ∫mr²) and thus rotational speed ω.

The point is that the energy is different: energy has to be added when the skater's arms are in and is lost when they are extended. If you think about the case with a ball on a string where all the mass can be thought of as being in a single point, let's say one metre from the centre. If there's also a hook on the end of a 10cm rod sticking out from a post 90cm out from the centre and we have a cutter on this post, we can suddenly stop the ball going round the 1m radius circle and go round a 10cm radius circle instead, and it's speed of movement is unchanged - just going in a tighter circle doesn't speed it up. In this case the energy in the system is unchanged. If we shorten the string to 10cm instead, we end up with the ball moving in tight circles with a 10cm radius again, but at a much higher speed because of the energy added by pulling the string. The skater is like this second case with the string being shortened, so there is more energy in the rotation with the arms in than with them out. Conservation of angular momentum only happens because there's an addition of or removal of some of the energy when the distance from the centre of rotation to the mass changes.

[David Cooper (OP)]

If an ice skater pulls his/her arms in while rotating, the rotation rate can increase dramatically. Everything I'm finding by searching for information about this describes this case as conservation of momentum with zero external torque.

That information sounds correct to me. With the skater, there's always some friction with the ice, so the momentum is slowly drained away. She can only keep it up so long.

But I'm referring to cases where the distance from centre of rotation to the mass changes. In those cases, there is external torque.

The speed of movement through space increases  since as it spirals in, the force on the ball is not entirely tangential. A spiral path is not perpendicular to that string. This is fine. Whatever is reeling in that string is performing work, adding energy to the system.

That's the kind of thing he calls unscientific wishful thinking bull****, which is why I want to find something in the scientific literature about it. Unfortunately, what my searches are turning up are descriptions that are incorrect.

I've seen a video of a lecture with the lecturer laying down the law about how there cannot be any external torque involved in this because the angle of the pull of the string is at 90° to the direction the ball's moving it

That's wrong. It is 90 only while moving at a fixed radius, not while being reeled in.[/quote]

Here's the video in question: https://www.youtube.com/watch?v=kJyI7IFamK0&lc=Ugz0J8n1-2VqM_CJVOJ4AaABAg.8eeMTzJe1yH9BsNV9vr7xN - jump to 26 and a half minutes in to get to the relevant point.

In an ideal case, negative work is done moving the arms back out. No energy loss as heat. Angular speed resumes prior rate.
I agree that angular momentum must be preserved in this case, in which case the energy expended to draw in the arms goes into increased speed of the component, preserving angular momentum. For a closed system, the body (central mass) is not fixed and has to be taken into account. It applies torque to the skater arms and slows them down, thus preserving the angular momentum.  With the ball and the string, that torque might not be there, so the situation may not be analogous.

With the skater, the energy expended when pulling in the arms should be considered  as external torque, even though it's coming from inside the system. Here's the thing: it looks as if at least some experts have an incorrect understanding of this, but you've immediately homed in on what's actually going on, as I expected. There must be other experts who do understand it correctly, and I want to find something in official literature that spells out these mechanisms. I'll try physics stack exchange, but I posted here first because I was hoping to get your opinion on it.

[Bored chemist]

With the skater, the energy expended when pulling in the arms should be considered  as external torque, even though it's coming from inside the system.

No it shouldn't.
Because that would violate the conservation of angular momentum.

If you aren't careful, you end up thinking none of the conservation laws are valid and you end up in the same mess as Alan has here
https://www.thenakedscientists.com/forum/index.php?topic=80136.0

[David Cooper (OP)]

With the skater, the energy expended when pulling in the arms should be considered  as external torque, even though it's coming from inside the system.

No it shouldn't.
Because that would violate the conservation of angular momentum.

It's only that injection of energy that gives you the conservation of angular momentum. Think about the ball on the string again. It's moving round on the end of a 1m string. If you put something in its path it will punch it. Repeat the experiment, but pull the string in first to 50cm, and then when you put something in the path of the ball it will punch it much harder. Energy has been added.

[Bored chemist]

It's only that injection of energy that gives you the conservation of angular momentum.

No.
It's this "invariance with respect to rotation gives the law of conservation of angular momentum;"
From here
https://en.wikipedia.org/wiki/Noether%27s_theorem

"Is conservation of angular momentum properly understood?"
Plainly not by everyone.

[Jaaanosik]

With the skater, the energy expended when pulling in the arms should be considered  as external torque, even though it's coming from inside the system.

No it shouldn't.
Because that would violate the conservation of angular momentum.

It's only that injection of energy that gives you the conservation of angular momentum. Think about the ball on the string again. It's moving round on the end of a 1m string. If you put something in its path it will punch it. Repeat the experiment, but pull the string in first to 50cm, and then when you put something in the path of the ball it will punch it much harder. Energy has been added.

David,
When the ball is reeled in from 1m to 0.5m there is a work done. The potential energy of the ball increases.
This costs energy. That's the reason why the ball has more kinetic energy when released.

Having said that a good question is what increased the ball velocity?
The answer is Coriolis force.
Coriolis force is consider fictitious force as gravitational force.
Is the Coriolis force that 'external torque'?
This is going to be a nice thread.
Jano

MOD EDIT: Coriolis is not the answer here, please read subsequent posts for the correct discussion.

[Halc]

But I'm referring to cases where the distance from centre of rotation to the mass changes. In those cases, there is external torque.

This can take place in space. There can be no external torque there since there is nothing again which a reaction force can be applied. You pull your arms in, which yes, requires work being done, and the angular velocity increases, all without external torque. External torque is only needed to change the angular momentum, and that is not changed at all in this scenario. It cannot be changed since the system in space is isolated and there is nothing else to which momentum can be transferred.

The speed of movement through space increases  since as it spirals in, the force on the ball is not entirely tangential. A spiral path is not perpendicular to that string. This is fine. Whatever is reeling in that string is performing work, adding [kinetic] energy to the system.

That's the kind of thing he calls unscientific wishful thinking bull****, which is why I want to find something in the scientific literature about it. Unfortunately, what my searches are turning up are descriptions that are incorrect.

Who calls the thinking bull?  What part is wrong then? You say yourself that it requires energy for the skater to pull her arms in. It can be recaptured when she puts them back out.  If it's bull, then he has to explain the vector arithmetic. I found the best way to do it was to have the ball on a string yanked in fast to half the radius. That must increase the linear speed, all without any change to angular momentum.

Ball is at top of circle (radius R), moving at +v to the right.  We put enough brief 'yank' force on the string (down) to bend that path 60° downward. 1/cos(60) = new velocity of 2v 60° down of right. String goes slack and ball travels in straight line for distance √3R/2 at which point its motion is perpendicular to the center again at half radius and double the linear speed (and thus 4x the angular speed).  The string takes up the tension again at this point and ball now circles at half the radius.
The angular momentum formula demands this since the angular momentum would have halved had the speed of the object on the string stayed the same as it gets reeled in.
The skater spins furiously. The crowd cheers, except they don't because they all do that and the crowd has seen it before. Imagine the first time this was demonstrated though...

Here's the video in question: https://www.youtube.com/watch?v=kJyI7IFamK0&lc=Ugz0J8n1-2VqM_CJVOJ4AaABAg.8eeMTzJe1yH9BsNV9vr7xN - jump to 26 and a half minutes in to get to the relevant point.

Yea, he's correct when he says there's no net torque to the system (or at least there wouldn't be if he was in space), but wrong when he says the string pulls perpendicular to the motion, which is easily shown as I did above using simple trig on the spiral path of the weight. The string is not perpendicular to the motion, so there's a component of the tension adding speed to the weight.
It's sad to see a lecturer not know his geometry, and also to not have a student point this out.
Were the force from the string to be tangential, the speed of the bob would stay the same and the angular momentum would decrease as it is reeled in, in violation of conservation laws. Surely the professor should see this.

With the skater, the energy expended when pulling in the arms should be considered  as external torque even though it's coming from inside the system.

That's a contradiction.  It's internal torque since it is applied between arms and rest-of-skater. External torque comes from the ice and air, both of which can only slowly rob the skater of momentum.  If in space, there cannot be external torque.
Maybe you have a different definition of 'external torque' that may help shed light on what you mean by those words.  You say:

It's only that injection of energy that gives you the conservation of angular momentum.

The energy comes from inside the system, and thus energy is preserved. The skater doesn't have her partner prodding her with a stick to get her spinning like that. No energy is injected. Some kind of muscle energy is translated into kinetic energy, but no net injection of energy takes place.

Here's the thing: it looks as if at least some experts have an incorrect understanding of this, but you've immediately homed in on what's actually going on, as I expected. There must be other experts who do understand it correctly, and I want to find something in official literature that spells out these mechanisms. I'll try physics stack exchange, but I posted here first because I was hoping to get your opinion on it.

Well you got it, complete with numbers.

[David Cooper (OP)]

Right - I see what's going on now. Thanks for all those answers, everyone. I'd misunderstood what counts as external torque. The fact that energy has to be used to get the ball in is irrelevant to that, and what shows it to be completely irrelevant to that is that you can't change the speed the ball goes round at any fixed distance by expending any amount of energy - that requires a push against something external, and there isn't any such push.

[Bored chemist]

Right - I see what's going on now. Thanks for all those answers, everyone. I'd misunderstood what counts as external torque. The fact that energy has to be used to get the ball in is irrelevant to that, and what shows it to be completely irrelevant to that is that you can't change the speed the ball goes round at any fixed distance by expending any amount of energy - that requires a push against something external, and there isn't any such push.

Yes, that's about it.
Now, can you explain it to Alan?

[David Cooper (OP)]

Yes, that's about it.
Now, can you explain it to Alan?

Well, he must be wrong, but he had a good reason for thinking what he did. No amount of energy created by the air moving relative to the ground should be able to change the angular momentum without an external input. If we imagine a planet like the Earth but with no heat differences to make the air move about, would the air just have to sit in place and go round with it without any relative movement? I think it would have to. Alan said something about the Coriolis force causing wind, and that wind would then cause drag, but without an external input to that, it can't slow the Earth down, so that would become a perpetual energy machine generating heat if it was a real mechanism. The conclusion that we're forced to reach is that the Coriolis force can't drive the wind. If the air's already moving though, it can influence how it moves, and that's the part that's misleading: it must need the air to be moving relative to the Earth to begin with.

[David Cooper (OP)]

By the way, seeing as some interesting physics has been clarified in this thread after it's move out of the physics section where it may have interested people who will now never see its resolution and therefore won't benefit from that clarification, please try to find ways to get these improved understandings to reappear in the physics section at some point in the future when they fit a thread there.

[Colin2B]

By the way, seeing as some interesting physics has been clarified in this thread after it's move out of the physics section

David
I moved it because @Jaaanosik was dragging the thread into new theories territory with Coriolis forces, which was the last post at the time. I was dealing with a lot of spam at the time and didn’t have time to split the topic intending to go back and sort it out.
I’m putting it back as the topic has moved in the right direction.

[Jaaanosik]

OK, question.
1kg weight is moving in a uniform circular motion on 1m string with tangential velocity 1m/s.
The weight is reeled in to 0.5m radius.
What is the tangential velocity at 0.5m radius?
Thanks,
Jano

[Jaaanosik]

OK, question.
1kg weight is moving in a uniform circular motion on 1m string with tangential velocity 1m/s.
The weight is reeled in to 0.5m radius.
What is the tangential velocity at 0.5m radius?
Thanks,
Jano

Post 10 explained all that, including an example with exactly that question. Do the math yourself if it's still unclear. At what tangential velocity is angular momentum conserved? It's really simple, and doesn't require asking the group and sorting between the inconsistent answers.

Of course, easy, right?
2m/s

What acceleration made the weight to gain the tangential velocity?
Jano

[Jaaanosik]

Post 10 explained all that, including an example with exactly that question

Of course, easy, right?
2m/s

What acceleration made the weight to gain the tangential velocity?
Jano

Post 10 explained all that, including an example with exactly that question

So consider instead a tether-ball setup.  That ball also starts at maximum radius and spirals in, except this time there is no work being done to reel in the line. The pole expends no energy as the radius decreases on its own.  What is the speed of the ball when the radius is halved?

I am lost in your post #10. I do not see what you are trying to describe.

As you can see in my example, the angular momentum is conserved, yet the weight has bigger tangential velocity.
So what increased the tangential velocity?
Jano

[Momentus]

I had not seen this post when I posted on Angular momentum.https://www.thenakedscientists.com/forum/index.php?topic=80399.msg612026#msg612026
Do wish it had been referred to.

This is another instance of Dark Motion. Angular Momentum is not conserved.

For angular momentum to be conserved the tangential velocity must remain constant. There can only be Sir Isaac Newton’s centripetal acceleration present. The logic that follows from this is that espoused in the lecture, this is how it must behave, in order to fulfil the conditions for momentum to be conserved, there fore we assume that it behaves in this way. Nothing to see here, move along. I think the students were wise not to call him on that one. The object of education is to get a degree, not to challenge orthodoxy

As several posts point out, there is a tangential force in addition to the centripetal force when a mass is moved on a spiral path. That force is reacted by change in the magnitude of the velocity and hence a change in momentum.

I have drawn some diagrams to illustrate the point.

 In fig 1 a mass is moving along a spiral path. The graphic shows how the force acting on the mass can be resolved into two vectors. One vector changes the direction of the momentum, acting at right angles to the path, the other vectored force changes the magnitude of the momentum by acting along the line of action.

Fig 2 shows the same system with two equal and opposite masses with opposed forces. This shows a closed system, with no external force applied. All the forces and vectors are equal and opposite, yet the momentum of the system is changed. As there is no outside force, that change occurs within a closed system.

 Momentum can be created or Destroyed. It is not Conserved.

Fig 3 is a closer view of the forces involved.

The mass is accelerated along its line of travel by force B, work is done. When spiralling inward,  the tangential speed is increased. When spiralling outward the tangential speed is reduced. Energy is conserved.

hat vector will line up with the changing tangential movement of the ball and add to its tangential speed.

A spiral path is not perpendicular to that string.

That's wrong. It is 90 only while moving at a fixed radius, not while being reeled in

So what happens now?

This forum has a stated purpose to inform the general Public on science. Is there any reciprocal arrangement?