Linear momentum conservation puzzle

[OceanBreeze]

My only spoiler is to remind of that stated back in #1. Unless an axial impulse (more explicitly, an impulse profile) exists in the common shaft, the system experiences a net shift in COM - towards the charged-up battery. Or whatever other energy storage (or dissipation) arrangement that might have been chosen.
Same question to answer as always; is there such an axial impulse? If so, how exactly does it arise?

Axial forces do exist in motor-generators due to the magnetic center when running under load is not the same as the mechanical center when at rest.
I had considered this earlier because I have experience with thrust bearings and thrust washers, having seen one pop ten feet into the air when the end plate was removed on a large electric motor!
However, since you stated in the OP a condition the “power transmission is via pure torsion in a rotating drive shaft” I assumed that ruled out any consideration of an axial impulse.
But, even if there is an axial impulse, we are dealing with a closed system and any internal impulse force will be met by a counter force and the COM should not move. Right?
I am beginning to suspect that there may be something wrong with the question that is being asked here.
It is a given, In any totally isolated system, mass is neither entering or leaving and there are no external forces acting on it, therefore there cannot be any change in the center of mass or the linear momentum due to internal forces.
But what if the isolated system is started off or forced into state where the masses and forces are unbalanced? In that case, the internal forces will act to towards equilibrium and shift some mass around.
In the problem under consideration, the system is isolated but it is not starting off at equilibrium. The unbalanced mass will shift from one battery to the other as it seeks equilibrium. For linear momentum and center of mass to be conserved, we are to assume that “something else” must provide for translational symmetry; finding that “something else” is the problem.
My thought is, if you start or force an isolated system into an unbalanced state that requires an external force and external energy. In that case, because an external force acted on the system the COM can move as it goes towards equilibrium with no violation of conservation. There is no “something else” to be found. Once the isolated system achieves equilibrium (both batteries are equally charged) then COM will ensure there is no further shifting of the COM due to any internal forces.
I wish you will provide your answer soon so I can stop thinking about this!

 

[The God]

The source battery supplies voltage and current (say power) to the induction motor. So if there is any axial impulse in the transmission shaft, it has to come from change in V / I. No one disputes such change on discharge of battery.

 

[Q-reeus]

Axial forces do exist in motor-generators due to the magnetic center when running under load is not the same as the mechanical center when at rest.
I had considered this earlier because I have experience with thrust bearings and thrust washers, having seen one pop ten feet into the air when the end plate was removed on a large electric motor!
However, since you stated in the OP a condition the “power transmission is via pure torsion in a rotating drive shaft” I assumed that ruled out any consideration of an axial impulse.
But, even if there is an axial impulse, we are dealing with a closed system and any internal impulse force will be met by a counter force and the COM should not move. Right?
I am beginning to suspect that there may be something wrong with the question that is being asked here.
It is a given, In any totally isolated system, mass is neither entering or leaving and there are no external forces acting on it, therefore there cannot be any change in the center of mass or the linear momentum due to internal forces.
But what if the isolated system is started off or forced into state where the masses and forces are unbalanced? In that case, the internal forces will act to towards equilibrium and shift some mass around.
In the problem under consideration, the system is isolated but it is not starting off at equilibrium. The unbalanced mass will shift from one battery to the other as it seeks equilibrium. For linear momentum and center of mass to be conserved, we are to assume that “something else” must provide for translational symmetry; finding that “something else” is the problem.
My thought is, if you start or force an isolated system into an unbalanced state that requires an external force and external energy. In that case, because an external force acted on the system the COM can move as it goes towards equilibrium with no violation of conservation. There is no “something else” to be found. Once the isolated system achieves equilibrium (both batteries are equally charged) then COM will ensure there is no further shifting of the COM due to any internal forces.
I wish you will provide your answer soon so I can stop thinking about this!

Re that in red above: no it didn't. Power transmission is always via pure torsion in the shaft, but that doesn't per se exclude the possibility of an axial force or rather force profile.
The rest of your attempts there are commendable, but have not come down with any real answer. Keep trying - if you like.

 

[Q-reeus]

The source battery supplies voltage and current (say power) to the induction motor. So if there is any axial impulse in the transmission shaft, it has to come from change in V / I. No one disputes such change on discharge of battery.

Still too vague. Is there an axial impulse? How could it arise with only pure torsional power input and output?

 

[The God]

Still too vague. Is there an axial impulse? How could it arise with only pure torsional power input and output?

Axial or torsional, whatever, can only come through v and I. No amount of magic is going to impart any impulse to motor, and no amount of charging of second battery is going to kick the generating set.

This is as far as ac supply is concerned (Battery to inverter to motor), so to create some impulse you need to pump in some DC or does it get dissipated in inverter braking system or filtering capacitors.

 

[Confused2]

I'd like to discuss the proposition that the centre of mass stays in the same place. Defining 'in the same place' as a test inertial mass and the centre of mass of the system proposed in the OP will not move relative to each other as the experiment progresses. There are two (or more) analogues for this:
1/ Walking forwards along a surfboard of negligible mass where the force exerted by the feet causes the surfboard to move and the surfer stays in the same place. Keyword force
2/ A bucket of water at some height feeds water into a lower bucket on the left. If the system is mounted on a frictionless surface do the buckets move to keep the centre of mass in the same place? I am fairly (but not absolutely) sure the bucket system doesn't move.

A very good poser from Q-rees. Are we looking for "how the centre of mass stays in the same place " when it doesn't stay in the same place?

 

[OceanBreeze]

Re that in red above: no it didn't. Power transmission is always via pure torsion in the shaft, but that doesn't per se exclude the possibility of an axial force or rather force profile.
The rest of your attempts there are commendable, but have not come down with any real answer. Keep trying - if you like.

At the risk of being the class idiot, I will express some more thoughts:

"The unbalanced energy-mass will shift from one battery to the other as it seeks equilibrium. For linear momentum and center of mass to be conserved, we are to assume that “something else” must provide for translational symmetry; finding that “something else” is the problem".

If “something else” does come back in the opposite direction of the energy-mass (that represents battery charge) is moving, then where is that something else when the process is completed and the battery charges are equalized?
Whatever comes back the opposite way to restore/maintain linear momentum must still exist in some form, somewhere in the isolated system because it IS an isolated system, like the Hotel California, it can never leave.
Ultimately, whatever it is, when the process completes, it remains as heat; the entire isolated system must be slightly hotter at the end then at the beginning, and it cannot shed that heat, only distribute it evenly.
So, ultimately, we have some shift in battery charge countered by some increase in heat and that will be the new static equilibrium of the isolated system. That must be true.
The dynamic situation, while the charge is being transferred, is still unclear. I can’t see thermal energy travelling axially along the shaft, being the office holder of the counter momentum that is required to maintain linear momentum and COM dynamically.
However, there will be a counter-emf (cemf) generated in the armature of the motor due to loading from the generator.
That cemf is the only agent I can think of, that comes back across going the other way, dynamically. That is my final, apparently not-so-educated, guess.

 

[Q-reeus]

Axial or torsional, whatever, can only come through v and I. No amount of magic is going to impart any impulse to motor, and no amount of charging of second battery is going to kick the generating set.

This is as far as ac supply is concerned (Battery to inverter to motor), so to create some impulse you need to pump in some DC or does it get dissipated in inverter braking system or filtering capacitors.

Why do you keep adding unnecessary and irrelevant complications? At one end, a power supply - say a battery - electrically feeds energy to an adjacent motor. Via the common shaft, that same energy transmits mechanically to a generator/alternator at the other end. Which in turn electrically transfers energy to an adjacent energy reservoir - say a battery. Keep it simple.

 

[Q-reeus]

I'd like to discuss the proposition that the centre of mass stays in the same place. Defining 'in the same place' as a test inertial mass and the centre of mass of the system proposed in the OP will not move relative to each other as the experiment progresses.

You will have to clarify further. Would someone with a very sensitive theodolite detect any relative motion between the test mass and a given marker on the system?

There are two (or more) analogues for this:
1/ Walking forwards along a surfboard of negligible mass where the force exerted by the feet causes the surfboard to move and the surfer stays in the same place. Keyword force

If the walker exerts a lateral force, there must by Newton's 3rd law be a back reaction propelling the walker forward. If the surfboard becomes 'infinitely light', no force can be exerted on it (obviously friction is excluded here). Hence none back on the walker. Magnitudes of relative accelerations will be inversely proportional to mass ratio.

2/ A bucket of water at some height feeds water into a lower bucket on the left. If the system is mounted on a frictionless surface do the buckets move to keep the centre of mass in the same place? I am fairly (but not absolutely) sure the bucket system doesn't move.

If the buckets are both laterally constrained by a common rigid frame (but upper one free to tilt or whatever is needed), then no motion of the frame occurs, again by Newton's 3rd law. If each bucket is free to move laterally, there will be relative motion between them but so as to keep the combined COM constant.

A very good poser from Q-rees.

Thank you.

Are we looking for "how the centre of mass stays in the same place " when it doesn't stay in the same place?

LOL. Once you answer my first question above, i will give some further clues. Enough for maybe one or two here at SF to figure the rest out in detail.

 

[Q-reeus]

At the risk of being the class idiot, I will express some more thoughts:

"The unbalanced energy-mass will shift from one battery to the other as it seeks equilibrium. For linear momentum and center of mass to be conserved, we are to assume that “something else” must provide for translational symmetry; finding that “something else” is the problem".

If “something else” does come back in the opposite direction of the energy-mass (that represents battery charge) is moving, then where is that something else when the process is completed and the battery charges are equalized?
Whatever comes back the opposite way to restore/maintain linear momentum must still exist in some form, somewhere in the isolated system because it IS an isolated system, like the Hotel California, it can never leave.

Ultimately, whatever it is, when the process completes, it remains as heat; the entire isolated system must be slightly hotter at the end then at the beginning, and it cannot shed that heat, only distribute it evenly.

So, ultimately, we have some shift in battery charge countered by some increase in heat and that will be the new static equilibrium of the isolated system. That must be true.

We are perfectly entitled to idealize to a lossless energy conversion/transfer. Neater that way. And it's what I clarified in #30.

The dynamic situation, while the charge is being transferred, is still unclear. I can’t see thermal energy travelling axially along the shaft, being the office holder of the counter momentum that is required to maintain linear momentum and COM dynamically.
However, there will be a counter-emf (cemf) generated in the armature of the motor due to loading from the generator.
That cemf is the only agent I can think of, that comes back across going the other way, dynamically. That is my final, apparently not-so-educated, guess.

Well....there is no room in #1 scenario for emf's existing outside of the electrical pathways coupling between battery & motor one end, and generator & battery the other. No emf exists in and axially along the shaft. So no resulting current exists there. But I can understand the frustration - where is it? Or even is it? Answers are coming!

 

[hansda]

For those not familiar with what a typical common-shaft motor-generator set entails: http://www.electrical4u.com/motor-generator-set-m-g-set/
Suppose an initially fully charged battery supplies the motor, a similar but initially depleted battery being charged up by the generator. Each battery adjacent to the respective motor, generator. Energy E thus a mass equivalent E/c^2 ends up being transferred from one location to another. But assuming the entire system is isolated, conservation of momentum requires the centre of mass is not changed by the transfer of mass-energy from battery to battery. At some stage then, one expects an axial force exists along the common shaft (assume a continuous one-piece shaft). Providing just enough impulse to keep the overall centre of mass fixed.

The problem is how, or even if, that is possible given power transmission is via pure torsion in a rotating drive shaft. In more familiar cases; e.g. tandem-shaft belt or chain or gear transmission, or electrical transfer via wires, compensating forces are (relatively) easily identified. Not here.

Looked at the issue many years ago, and iirc it took a good hour or so to arrive at an answer I was reasonably satisfied with. Instead of providing my answer(s) here and now, leaving it as is for ~ 24 hrs. Maybe somewhat more. A teaser challenge. Let's see who here can offer an answer with clear, credible justification, within that rough time frame.

Just knowing that F = dp/dt will or should suggest what specific physical condition in above setup needs focusing on. Actually, there are two convenient limiting cases worth analyzing. The only hint offered is there are just two relevant variables to work with. Angular velocity and torque.
Hand-wavy suggestions won't cut it. An acceptable answer(s) should ideally be quantitative not just qualitative.

If anyone manages to or thinks they have found an essentially direct answer from a textbook or online article etc., be honest enough to admit it and reference to such. Good luck - back later.

Here the battery connected with motor will supply current to the motor. This current is due movement of an electron. This electron will comeback to the same battery. So linear momentum associated with this electron is conserved.

Similarly the battery associated with the generator; an electron will travel in that close loop.

There is no mass(or electron) transfer from one battery to another battery. Its only energy conversion from EM to Mechanical and then again from Mechanical to EM.

 

[QuarkHead]

I assume that mass-energy lost in turning the motor is equal to that lost in turning the generator.
I further assume a feedback such that as the charges in the two batteries equalize, the shaft slows and finally stops.
Now, since there exists no such thing as a perfectly rigid shaft, some of the source battery's mass-energy will be stored in the shaft as torsional mass-energy, and this will be greatest at the end of the shaft furthest from the motor.
As the feedback kicks in and the mass-energy of the two batteries equalize, so the torsional mass-energy at the distal end of the shaft decreases and becomes zero when the two batteries are equally charged.
Therefore the COM does not shift

 

[Confused2]

Q-Reeus mentions a theodolite...

You will have to clarify [my definition of mass] further. Would someone with a very sensitive theodolite detect any relative motion between the test mass and a given marker on the system?

Take a balance beam. Place a charged battery on the left and an uncharged battery on the right. Move the balance point until the beam is horizontal. The balance point is by definition 'The Centre of Mass' (?). Now connect the charged battery to the uncharged battery by any means whatsoever. The Charged battery gets lighter as it gives up energy, the uncharged battery gets heavier as it gains energy. The balance point (centre of mass) shifts towards the uncharged battery as it gains charge. Yes/no?

 

[Q-reeus]

Here the battery connected with motor will supply current to the motor. This current is due movement of an electron. This electron will comeback to the same battery. So linear momentum associated with this electron is conserved.

Similarly the battery associated with the generator; an electron will travel in that close loop.

Yes to both above.

There is no mass(or electron) transfer from one battery to another battery. Its only energy conversion from EM to Mechanical and then again from Mechanical to EM.

And? You do not accept that mass/energy in one battery has been exchanged with mass/energy in another such battery? There is no issue with an evident shift in COM to address?

 

[Q-reeus]

I assume that mass-energy lost in turning the motor is equal to that lost in turning the generator.
I further assume a feedback such that as the charges in the two batteries equalize, the shaft slows and finally stops.
Now, since there exists no such thing as a perfectly rigid shaft, some of the source battery's mass-energy will be stored in the shaft as torsional mass-energy, and this will be greatest at the end of the shaft furthest from the motor.
As the feedback kicks in and the mass-energy of the two batteries equalize, so the torsional mass-energy at the distal end of the shaft decreases and becomes zero when the two batteries are equally charged.
Therefore the COM does not shift

I will give you credit for being innovative there. But it's wrong. For a number of reasons but one in particular. Any elastic energy stored in the shaft will locally have an energy density 0.5σ²/G, where σ is the shear stress amplitude, and G the shear modulus: https://en.wikipedia.org/wiki/Shear_modulus
The latter is entirely material dependent. Hence not fundamental to the physics of OP scenario. Further, any realistic calculation will show the possible elastic energy contribution for a typical MGS will be entirely negligible wrt the electrochemical energy exchanges. Finally, shaft elastic energy is transitory and does not figure in a final before vs after balance, whereas the electrochemical exchanges do.

 

[Q-reeus]

Q-Reeus mentions a theodolite...

Take a balance beam. Place a charged battery on the left and an uncharged battery on the right. Move the balance point until the beam is horizontal. The balance point is by definition 'The Centre of Mass' (?). Now connect the charged battery to the uncharged battery by any means whatsoever. The Charged battery gets lighter as it gives up energy, the uncharged battery gets heavier as it gains energy. The balance point (centre of mass) shifts towards the uncharged battery as it gains charge. Yes/no?

Again, idealize to lossless connecting wires. Newton's 3rd law will hold at every instant, so a transfer of mass/energy from battery to battery is compensated by an emf acting on the wire lattice charges, directly and indirectly through conduction electrons. You have to be careful though. If the batteries are on a swing, the heavier i.e. charged-up battery will tilt down wrt the lighter battery. The reaction force via the wires is essentially horizontal and taken up via the swing pivot.

Which is all a distraction because the key conundrum is to do with energy transfer via pure torsional means. Hmm.....I may play this out a while longer.

 

[OceanBreeze]

Finally, shaft elastic energy is transitory and does not figure in a final before vs after balance, whereas the electrochemical exchanges do.

That is the real conundrum; how is the final balance the same as the initial balance. I can see the torsional transfer of energy-mass being met by opposing forces during the transfer time (counter emf or back emf does exist and the magnetic force in the stator is opposed by the magnetic force in the rotor, etc) but it is difficult to see how the final arrangement of mass is counter-balanced.

Uh oh; I hope this isn’t going to end up being: the increased mass of the charging battery results in a greater gravitational force with the MGS, pulling it closer to the MGS and reducing its moment, thereby maintaining the center of mass.

 

[OceanBreeze]

Q-Reeus mentions a theodolite...

Take a balance beam. Place a charged battery on the left and an uncharged battery on the right. Move the balance point until the beam is horizontal. The balance point is by definition 'The Centre of Mass' (?). Now connect the charged battery to the uncharged battery by any means whatsoever. The Charged battery gets lighter as it gives up energy, the uncharged battery gets heavier as it gains energy. The balance point (centre of mass) shifts towards the uncharged battery as it gains charge. Yes/no?

Nitpick: I think it is more correct to say the balance point is the center of gravity, but as long as the gravity field is uniform, it is also the center of mass.

 

[Q-reeus]

That is the real conundrum; how is the final balance the same as the initial balance. I can see the torsional transfer of energy-mass being met by opposing forces during the transfer time (counter emf or back emf does exist and the magnetic force in the stator is opposed by the magnetic force in the rotor, etc) but it is difficult to see how the final arrangement of mass is counter-balanced.

Uh oh; I hope this isn’t going to end up being: the increased mass of the charging battery results in a greater gravitational force with the MGS, pulling it closer to the MGS and reducing its moment, thereby maintaining the center of mass.

Nope. Gravity not involved. Especially not linked to a bootstrap scheme.

 

[Q-reeus]

Nitpick: I think it is more correct to say the balance point is the center of gravity, but as long as the gravity field is uniform, it is also the center of mass.

Agreed. Not much to do with OP though.