Conservation of Angular/Linear momentum in a system?

[mfactor]

Consider a woman standing on a turntable, which can turn without any friction. Both are initially at REST. A woman (m kg) starts to walk around the rim of turntable clockwise. she walks at a constant velocity with respect to the earth. Obviously, the turntable (I kgm^2) starts to rotate counterclockwise.

In any closed system, Angular monetum and linear momentum are conserved.

So, angular momentum of the woman with respect to the center of the table has the same magnitude as the angular momentum of the turntable, but both vectors cancel each other out, creating zero-sum. (since both are initially at rest.)

Also, the woman gains linear momentum. But I cannot think of an opposing linear momentum that would cancel out the linear momentum of the woman, when I consider the woman-turntable system.

So here comes my question. Where is the other linear momentum that opposes that of the woman? Should the Earth be affected (very very slightly) so that the change in linear momentum of Earth is the same as the change in linear momentum of the woman?

 

[Clausius2]

I don't see where is the linear momentum of the woman. Her velocity is zero from the laboratory frame.

And also, why is she a "she" instead of a "he"??

 

[masudr]

Clausius2 replies correctly.

I could also add that if we only consider the particle of the turntable that makes contact with the woman's foot, then that particle, at point of contact is moving in the opposite direction at the same velocity as the woman's foot.

 

[Doc Al]

Consider a woman standing on a turntable, which can turn without any friction. Both are initially at REST. A woman (m kg) starts to walk around the rim of turntable clockwise. she walks at a constant velocity with respect to the earth. Obviously, the turntable (I kgm^2) starts to rotate counterclockwise.

I presume you mean that the woman walks at a constant speed (not velocity) with respect to the earth.

In any closed system, Angular monetum and linear momentum are conserved.

Right. But this is not a closed system. The axis of the turntable is presumably fixed to the earth. Since the turntable has no friction, angular momentum will be conserved. But linear momentum will not be conserved.

So, angular momentum of the woman with respect to the center of the table has the same magnitude as the angular momentum of the turntable, but both vectors cancel each other out, creating zero-sum. (since both are initially at rest.)

OK.

Also, the woman gains linear momentum. But I cannot think of an opposing linear momentum that would cancel out the linear momentum of the woman, when I consider the woman-turntable system.

The turntable is restrained from moving due to its fixed axis.

So here comes my question. Where is the other linear momentum that opposes that of the woman? Should the Earth be affected (very very slightly) so that the change in linear momentum of Earth is the same as the change in linear momentum of the woman?

Yes, the Earth is affected ever so slightly. The turntable+woman+earth system is an isolated system; linear momentum is conserved.

 

[WhyIsItSo]

I might get laughed at, but...

I see the problem this way. The woman and the turntable each have inertia. Since the turntable has zero friction, consideration of the Earth is not an issue for this problem. There is the factor of her effects from leverage on the pole supporting the system, but that does not factor into the conservation of momentum issue.

There are TWO conditions to consider.

1. The time she is acclerating.

2. The time she is at a constant speed.

While she is accelerating, she is buying momentum from the inertia of the turntable. The turntable buys momentum in the opposite direction from the woman's inertia. Her momentum is linear (sort of), and the turntable's is angular.

Once she reaches her "cruise speed", she and the turntable continue with constant speed, and equal but opposite momentum.

As far as I can see, it is that simple.

 

[Doc Al]

I might get laughed at, but...

I see the problem this way. The woman and the turntable each have inertia. Since the turntable has zero friction, consideration of the Earth is not an issue for this problem. There is the factor of her effects from leverage on the pole supporting the system, but that does not factor into the conservation of momentum issue.

The original post asked about both linear and angular momentum. If all you care about is angular momentum, then it's true that the Earth is not an issue.

Once she reaches her "cruise speed", she and the turntable continue with constant speed, and equal but opposite momentum.

As far as I can see, it is that simple.

They have equal and opposite angular momenta. But you've ignored the issue of why linear momentum is not conserved.

 

[WhyIsItSo]

Doc Al,

If you insist on taking the OP exactly as written, then:

she walks at a constant velocity with respect to the earth.

is not going to happen. As someone has already pointed out, her speed may be constant, but her velocity must change.

Now, I may not know whether to call her momentum angular, linear, or loop-the-loops, but whatever her momentum is, the turntable will have equal and opposite momentum.

If you want to take the true scenario in which the woman and turntable are not the system, but include the earth, then yes, her acceleration required to alter her velocity such that she remains on the turntable is countered by the Earth via the pole the turntable is mounted on (as the OP stated). However, you seem to be confused on this point:

Right. But this is not a closed system. The axis of the turntable is presumably fixed to the earth. Since the turntable has no friction, angular momentum will be conserved. But linear momentum will not be conserved.

You are commiting an error of logic here. Either the Earth is part of the system, and linear momentum is concerned, or we take the hypothetical scenario where the woman and turntable are the closed system, in which case we must ignore linear momentum. You can't argue half one case and half the other.

Stating this is not really a closed system is realistic, but given that fact, everything is conserved nicely.

Where's the problem?

 

[Bystander]

C'mon, folks --- where's the center of mass of the turntable, woman, Earth system? Does it move? Where're the COMs of the components of the system? What are their movements?

 

[Doc Al]

If you insist on taking the OP exactly as written, then:
is not going to happen. As someone has already pointed out, her speed may be constant, but her velocity must change.

I think I realize that, since that "someone" was me.

Now, I may not know whether to call her momentum angular, linear, or loop-the-loops, but whatever her momentum is, the turntable will have equal and opposite momentum.

Basically you admit to not knowing the difference between angular and linear momentum, yet feel obliged to share your opinion nonetheless.

If you want to take the true scenario in which the woman and turntable are not the system, but include the earth, then yes, her acceleration required to alter her velocity such that she remains on the turntable is countered by the Earth via the pole the turntable is mounted on (as the OP stated). However, you seem to be confused on this point:

You are commiting an error of logic here. Either the Earth is part of the system, and linear momentum is concerned, or we take the hypothetical scenario where the woman and turntable are the closed system, in which case we must ignore linear momentum. You can't argue half one case and half the other.

Perhaps you should read the original post once more. The question was why linear momentum was apparently not conserved.

Stating this is not really a closed system is realistic, but given that fact, everything is conserved nicely.

Again, you seem to want to say that "everything" is conserved for the turntable+woman system. Obviously that's not true. (If you include the Earth as part of the system, then both angular and linear momentum are conserved.)

Where's the problem?

Beats me! It was a simple question that deserved a simple (and accurate) answer, such as the one I gave.

 

[wysard]

Simple and straightforward

Assuming, for the sake of argument, the friction on the axis of the turntable is zero here is what happens.

As the woman begins to walk, the turntable turns underneath her by the friction transmitted through her feet to accelerate up to walking speed. In a perfect system what happens is that the woman gains linear momentum (walking) which is transmitted to the turntable which moves in an equal and opposite direction according to the distance from the centre of rotation, the relative mass of the turntable to the woman and how fast she walks. In a simple system if the turntable weighs the same as the woman as she walks she appears not to move from an external frame. In essence she begins to walk, and the table turns.

Conservation of both angular and linear momentum are conserved.

There is no linear action which can affect her unless the frame is changed. Wind, hitting her with a rock, or anything else requires the the frame to expand to include the effect and it's source that interacts with her. Including, for instance, a rocket she brought with her...lol

 

[Doc Al]

Assuming, for the sake of argument, the friction on the axis of the turntable is zero here is what happens.

As the woman begins to walk, the turntable turns underneath her by the friction transmitted through her feet to accelerate up to walking speed. In a perfect system what happens is that the woman gains linear momentum (walking) which is transmitted to the turntable which moves in an equal and opposite direction according to the distance from the centre of rotation, the relative mass of the turntable to the woman and how fast she walks. In a simple system if the turntable weighs the same as the woman as she walks she appears not to move from an external frame. In essence she begins to walk, and the table turns.

Not exactly. The woman and turntable exert equal and opposite forces (and thus torques) on each other. From an external frame, the woman moves clockwise (say) with some angular momentum, while the turntable moves with the opposite angular momentum. Regardless of the relative mass of woman and turntable, she will move clockwise with respect to an external frame.

Conservation of both angular and linear momentum are conserved.

As explained earlier, only angular momentum is conserved, not linear momentum. (The fixed axle exerts a force on the turntable, preventing it from sliding back as the woman goes forward.)

 

[FredGarvin]

Resurrection of a year old, solved thread?

 

[masudr]

Resurrection of a year old, solved thread?

Slightly bizarre.

 

[TVP45]

In any closed system, Angular monetum and linear momentum are conserved.QUOTE]

In reality, neither are conserved. If you use a person walking (rather than a single contact point rotator), you get a very non-conservative system. The human provides nonconservative work.

Since we walk in straight lines, the first step involves a rotation of the standing foot and a push off, followed by a lesser force strike. The process then follows that pattern except the inner step is always
shorter.

Initially, the turntable goes backwards WRT the walking person. If that persists, the person falls off. Most people develop a walking strategy that is somewhat like a child "pumping" a swing - they adopt a gait closer to a decerebrate one and carefully time their steps. They then walk forward. I have seen one person who could even sort of jog around a turntable.

Even this requires a certain moment of inertia for the turntable, i.e., one that is within a factor of 10 of that of the person. If you posit a massless turntable, you cannot walk on it. If you have a very large moment, you always walk forward quite easily and the turntable may not really turn at all, save for slight oscillations.

You could, of course, assume an infinitely large radius with a mass approaching zero, and that might work. But that would not really be a turntable.

 

[cesiumfrog]

In reality, neither [linear nor angular momentum] are conserved. [..] The human provides nonconservative work.

That is very wrong. The friction beneath your feet may dissipate your work as thermal energy, but it is only by forgetting the Earth that you have concluded momentum is ever not conserved. (Could a person spinning through empty space halt their spin nor redirect their velocity?)

 

[rcgldr]

Change the situation to one where the woman is pedaling a bicycle at the edge of the turntable ... or maybe a unicycle with a cone shaped wheel, so that the path is truly circular.

Or maybe it's a robot programmed to "walk" so that it's center of mass follows a circular path at a constant rate.

 

[TVP45]

That is very wrong. The friction beneath your feet may dissipate your work as thermal energy, but it is only by forgetting the Earth that you have concluded momentum is ever not conserved. (Could a person spinning through empty space halt their spin nor redirect their velocity?)

Well, I may indeed be incorrect in thinking that conservation requires conservative forces. I'll have to puzzle over that.

But, yes, a person spinning through space can alter their spin. You might watch the high divers at the next Olympics and note how they correct when they over-rotate.

 

[cesiumfrog]

But, yes, a person spinning through space can alter their spin. You might watch the high divers at the next Olympics and note how they correct when they over-rotate.

Watch closer! By bending their body (altering rotational moment of inertia despite that the angular momentum stays constant) they can alter their speed of rotation (choosing to somersault quickly at first, then slowing the rotation at the right point so as to enter the water head-first), but if they were to over-rotate (into a back-flop position) it isn't possible for them to reverse their rotation to correct it.

 

[TVP45]

Yes. I went back and found the JAP reference on cat spin. I need to think through all that again. Thanks.