Poss. Symmetry Break?

[Wubbly]

If one car's stationary and the other has twice the speed, the relative velocity between them is equal to both cars closing at half that speed, so why isn't the energy?

Maybe there is a base reference frame that determines the maximum energy, and all other reference frames are arbitrary? Maybe the base reference frame is anchored about the center of mass of the overall system, and moves according to the movement of the center of mass of the system? The car approaching the brick wall at 100 is approaching the system center of mass at 100. The two cars approaching each other at relative 100 are approaching the center of mass of the system at relative 50?

[ME]

Maybe the base reference frame is anchored about the center of mass of the overall system

Sounds good to me.

--
Perhaps a slightly different view:
I think the (unmovable) wall acts like a mirror (the center of mass) and is not actually part of the system - as it is unmovable one could give it enormous mass. Two cars are also mirrored and their center acts like a wall. A car with double speed approaching a stationary car has a moving center of mass (or mirror).

[MrVibrating]

If one car's stationary and the other has twice the speed, the relative velocity between them is equal to both cars closing at half that speed, so why isn't the energy?

Maybe there is a base reference frame that determines the maximum energy, and all other reference frames are arbitrary? Maybe the base reference frame is anchored about the center of mass of the overall system, and moves according to the movement of the center of mass of the system? The car approaching the brick wall at 100 is approaching the system center of mass at 100. The two cars approaching each other at relative 100 are approaching the center of mass of the system at relative 50?

This seems like a powerful insight..! Momentum would be divided symmetrically either side of the center of mass, providing an emergent frame from which to balance the energy..

Will chew it over some more, but this is cool. I'd been stumped up to now..

So the next question is, how might we leverage this frame in a system of vertically-rotating and/or counter-rotating masses? As an emergent frame, not tied to any particular body, perhaps this has benefits - maybe additional mobility or trasportability or immunity from gravitation or N3 or something.. maybe gravity's inertial equivalence could come to bear (Grr!) here - the pseudo 'everything's accelerating upwards' frame..

Yes i'm flailing blind, but the point is that here we have a situation in which the net energy of a system might vary by 200% as a function of this center of mass frame. It's a potential weak spot, a single point of failure and possibly wide open to attack, so worth scoping out..

Such an exploit would also be consistent with Wolff's conclusion that the weights somehow gained additional energy while falling..

Think about it - if we could chuck 5 J CW while 5 J fell CCW, the difference between a net total of 10 J or 20 could be decided by something itself massless & ephemeral... like poltergeists..

[MrVibrating]

Perhaps the compression of a car can be replaced by a spring.

1 car going 50 against a wall compresses one spring.
2 cars heads-on at 50 compresses two springs - where their common mid-point remains stationary on collision.
1 car going 100 against a stationary car will compress its own and the others.
The spring doesn't care from which side it's been compressed, it just absorbs the energy (in these cases each one absorbs the same amount); although an actual spring would give it back.

This is basically what i was thinking of simming; trying to load more or less energy into a spring from equal velocities in different frames. But i was confounded by the simple 2-body elastic scenario... trying to find a preferential frame between contra-rotating discs (?), and sundry other brainstorms..

Like you say, though, stick the spring on either mass and for its part, it's entirely FoR-agnostic. It shouldn't care whether each mass is at 10 m/s or one is at 20 with the other stopped. If motion is simply relative then the net KE of a simple two-body system could be double or half itself, depending on how you look at it..

One day, one of these little conundrums - or at least, someone's ingenious resolution of one - is gonna bear out. (Grr!)

[MrVibrating]

Distilling all these ideas down, consider the following proposal:

Two identical counter-rotating discs.

The system's center of mass frame is now invariant, and replaced by a center of angular momentum - the axis.

But whether this axis is itself rotating or not seems fairly arbitrary.. if it's co-rotating with one disc, then from its' POV the other disc has all the momentum, and thus the system KE is twice as high.

And presumably, simply spinning up a spring - without loading it - needn't require a fraction of that energy difference..

In summary, it would seem that for a given relative angular velocity, the reference division of angular momentum can be accelerated independently of the respective angular inertias, hence the input energy of accelerating this reference frame is thermodynamically decoupled from the energy gain resulting from the effective redistribution of momentum and velocity..!

For example, spend 10 J on a CW momentum, and another 10 J on an identical CCW momentum; net energy = 20 J, relative to a static rotary spring sandwiched between them.

But the spring can have a low mass and MoI, and might only cost 2 J to spin up to the same speed as either disc... at which point, from its' FoR the static disc has no energy, while the rotating one has 40 J.

This should be easy to test in a sim..

[MrVibrating]

Two 6 meter radius discs of WM2D's 'standard' material, each with a mass of 113.097 kg

One disc spinning, the other pinned stationary to earth:

Initial momentum = 244300 kg-m^2-°/s

Initial relative velocity = 120°/s

Initial KE = 4464.904 J

A rotational spring then connects the two discs, with a 10 N-m/° constant. The spinning disc is thus braked by the spring against the earth's inertia, loading all of its energy into the spring.

As the de-spun disc halts, the static disc is unpinned and the spring thus allowed to spin both up, dividing that stored energy equally between them.


Final momentum, velocity and KE:

172700 kg-m^2-°/s

84.852°/s

2232.45 J

..for each disc.

Final relative velocity thus (2 * 84.852) = 169.704°/s

Net energy conserved

Net momentum is not conserved - having increased by 101100 kg-m^2-°/s, from our starting conditions of 244300 kg-m^2-°/s, rising to (2 * 172700) = 345400 kg-m^2-°/s, a ratio of 1.414/1

Relative velocity increased by 49.704°/s, again a ratio of 1.414/1

This ratio being equal to the square root of two..


So, a number of interesting results here: presumably this 2 sqrt corresponds to the 1/2 square of KE divided by twice the inertia as the second disc is accelerated along with the first...? Not exactly clear on this yet but that ratio is a common constant, equal to the ratio of the hypotenuse of an isoceles right triangle to either leg, and the secant and cosecant of a 45° angle, amongst other things.

But more interestingly still is the fact that net momentum has increased at all! By 1.414:1... How curious! That was unexpected, and likewise for relative velocity - up by 1.414:1, from 120°/s to 169.704°/s.

Suffice to say, if that 169.704°/s resided on one disc only, it would have an energy of 8929.636 J... precisely twice our starting energy.

For good measure i'll run the same experiment backwards - beginning with both discs spinning in opposite directions, braking against each other as they load the spring. Then, at the moment they halt, i'll pin one stationary and let the other be spun up by the unwinding spring...

Presumably the results will mirror those here - net momentum and velocity should be reduced by the same 1.414:1 ratio they've risen by here..




sim attached (set to 200 Hz for quick perusal, but originally run at a 5kHz crawl for the above measures)

[MrVibrating]

...the weird part is, i expected the available momentum to be equally divided between the two inertias.

And it basically is... there's just more of it.

Ditto the velocity.

So that's where my predicates fell apart.. but this unexpected outcome may yet present its own opportunities..

Surprising how the simplest things can be so unintuitive, innit?

[Mark]

For your consideration...

In the world of electronics, a ratio of 1.414:1 is associated with the Root Mean Square [RMS] of a sine wave.

For a cyclically alternating electric current, RMS is equal to the value of the direct current that would produce the same power dissipation in a resistive load.

https://en.wikipedia.org/wiki/Root_mean_square

I find your results, as Laugh-in's Wolfgang would say, "Verrrry interesting".

[MrVibrating]

Soon as i saw the figure it rang bells, but that's another interesting correlation, for what is after all a kind of 'power ratio' for division of energy into momentum and velocity.. much more interesting than the 50:50 split i'd anticipated.

Still wanna do the previous test i mentioned, spinning up the spring to equal speed with one disc before the transfer commences, just for thoroughness... but i'm thinking that the inverse results of the last test are a foregone conclusion - P and V will be reduced by a factor of 1.414 as expected. A far more compelling issue would be the possibilty to transfer the relative velocity between two opposing momentums, into an absolute velocity of a discrete momentum!

I wonder if this can be done - ie. two 1 kg masses on a collision course, each with a 10 m/s ground velocity, and hence a 20 m/s relative velocity... so, could we somehow convert or transfer that relative velocity into a standalone ground velocity - one 1 kg mass at 20 m/s?

In short, converting relative velocity into 'real' velocity could double our KE... so getting to the root of this 1.414:1 transfer ratio seems a good start.. Will have to repeat this experiment in a linear system to see if the same conversion factor arises..

[MrVibrating]

Hmm, a scissorjack could convert two opposing horizontal momentums into a single vertical momentum...

Something to try maybe. Just won't be too surprised if the total's (1 / 1.414) = 70% of whatever we started with...

[Mark]

Correction to my previous post --
I originally stated that "a ratio of 1.414:1 is commonly known as the Root Mean Square [RMS] of a sine wave", and that is incorrect. ( Old-guy brain fart :-)

The RMS [average power] of a sine wave is the reciprocal value: .7071 times the peak.
The peak power of a pure sine wave is 1.414 times the average.

What I found interesting is that the sim program shows the interaction of the discs and the spring to be sinusoidal, not linear. Or, I now wonder, is the resultant ratio purely coincidental, making this assumption incorrect?

And MrV, would you please verify that Blue disc goes to 0.0 velocity before the Red disc is unpinned? I ran Disc1.gif in Gimp and, stepping through it, it doesn't show the transition point. I figure that you are very diligent about such things, but I am curious nonetheless. Thanks.

[MrVibrating]

The spring constant's linear - a straight 10 N-m/° throughout its range, so not squaring up per the usual Hooke's law for springs (not a deliberate design point, just didn't see a point in changing it)... so momentum is gained or lost at a fairly steady rate, resulting in a more or less sinusoidal velocity plot (see revised sim below).

The flaky-looking release triggering seems to be due to a confluence of the reduced timing accuracy of the anim (which was at 200 Hz, as opposed to the original 5 kHz, just to make the .gif smaller and watchable) and the arbitrarily-high masses and energies - i usually begin with a 6 meter radius wheel for every design as it has plenty of divisors, but in retrospect the resulting magnitudes are pretty high.

The actual moment the pin is released is at 2.96 seconds, coinciding with the exact frame of the sim (2960 @ 1kHz) where the blue disc KE reaches its minimum of "1.770e-005 J" = 0.0000177 J or 17.7 microjoules.. as close to stationary as possible at this timing resolution. Raising the timing resolution further will allow us to zoom in on ever-finer slices of time, adding ever-more significant digits, but to all intents and purposes the blue disc grinds to a halt at 2.96 seconds, to within a few microseconds.

Below is the same sim at 1 kHz, with added velocity plots, a pause at the trigger release, and a short continue after the second pause (monster .gif!)...



You can still see one or two glitches as the sim works through, but they don't seem to matter - we basically have kJ loads with microjoule error margins.

[Mark]

OK, now i get it. When I first read "momentum is gained or lost at a fairly steady rate, resulting in a more or less sinusoidal velocity plot", I was like, say what?! Then I looked at the new gif, and I actually said to myself "oooh, duh!".

Thanks for helping me see it, MrV. Much appreciated.

[Fletcher]

I took the liberty to make a few small changes to your sim Mr V.

Simply to make it easier to change inputs at the push of a button and compare outputs e.g. changed velocity of disks from degs per sec to rpm etc (in numbers).

Don't know if the changes have any relevance to your findings or method here ?

P.S. if the pause doesn't work just open it up and retype the Input[14] and save it again.

[MrVibrating]

Regarding this 1.414:1 / .7071 constant; the scissorjack scenario i mentioned above might provide an angle on it - since the resultant of a vector sum is proportionate to the hypotenuse of its coordinates (per Py's theorem, a^2 + b^2 = c^2), so maybe we should expect a 30% variation in net momentum.. will check this when running the linear test later. Still seems weird tho since torque moments are always aligned, unlike linear vectors...