Poss. Symmetry Break?

[MrVibrating]

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.

I was unsure how to describe it at first too - the acceleration curve is kinda linear, velocity curves near sinusoidal but with linear sides..

Still unsure what's happening though... as far as angular velocity and momentum are concerned, it appears that 1 / 2 = 0.7.. a surprisingly high value of 'half', but there it is..

[Fletcher]

I get the two AM's added to 1.41 or about 70.5% of initial.

I think you'll find it in the AM formulas based on MOI which is about 70% displacement for a solid disk IIRC. Just a guess.

The Joules add correctly though it seems.

Here is the sim again except I have taken out the Red disk initial velocity of 20 rpm and made it zero.

That means only the Blue disk has 20 rpm initial velocity at start.

[MrVibrating]

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.

Really cool, thanks mate.. i still haven't tried adding controls to values yet, will defo be copying your technique next time i need to play with input variables. Here, though, all are fairly arbitrary - i began with a 6 meter radius of 'standard' material just from habit, then raised the spring constant to reduce the cycle period to a reasonable time... i switched from RPM to °/sec so that the momentum was in more coherent units (to me, anyway).

All we've done is drop an arbitrarily-spinning mass onto an identical static mass, via a spring.

The strange part is that if we drop the spinning mass directly onto the static one, the momentum and velocity are indeed split 50/50 - the sums of the two final P's & V's equal the inital P & V (i just double-checked this, to be sure).

But when the spring's there, the sums of final peak P's & V's are both 1.414 times our starting values... with energy nonetheless still conserved...

If energy's constant, but momentum and velocity have increased by 70%, then mass (or, in this case, angular inertia) must've decreased by 30%... or something.. maybe the spring is doing something non-physical?

I'll shortly re-do the test with a more natural exponential spring constant and lower masses / inertias, to see if the anomaly persists...

[MrVibrating]

I get the two AM's added to 1.41 or about 70.5% of initial.

I think you'll find it in the AM formulas based on MOI which is about 70% displacement for a solid disk IIRC. Just a guess.

The Joules add correctly though it seems.

Here is the sim again except I have taken out the Red disk initial velocity of 20 rpm and made it zero.

That means only the Blue disk has 20 rpm initial velocity at start.

Oops that was an error of course - although cancelled out by being pinned at startup - but yep, shouldn't've been there.

I too presumed it had to be a valid, albeit unexpected result, due to the reassuringly-conserved energy... but then why's it only arise when there's a spring involved, not in a direct elastic collision?

Will be investigating this further, seems a trifle dodgy for now...

[MrVibrating]

Ok, this time the spring constant is changed to 1 N-m per degree squared - a more natural behaviour - with everything else left at previous defaults (altho i moved the pin to the center of the red disc to try stabilise the glitching):




..results this time are:

Energy conserved

Momentum conserved

Net velocity rises by *1.414


So this seems more realistic... Still, you have to wonder whether the previous linear spring constant is impossible, or just unusual but perfectly do-able. If it's the latter, then we may have discovered a way of causing a net rise in system momentum, which seems a fascinating and potentially useful discovery in its own right.. as previously noted, apparently providing another means of varying the effective MoI.

But with an old-fashioned exponential spring constant, that anomaly disappears - net momentum's constant throughout the interaction, along with net energy, although we still have this 70% rise in relative velocity.- which again, is an interesting finding and may yet prove useful..

I'm stacking up unresolved questions though, need to slow down a bit - my objective is still to try to convert a relative velocity into an objective one - to try and access this doubling of energy from the rest frame of an equal opposing momentum... ('Mythbusters' be damned)..

[MrVibrating]

..so quick realisation - the previous rise in net system momentum was a kind of integration error - with a traditional spring we get rounder sinusoids for the V & A plots, but the slab-sided plots from the linear spring have streched integrals along the displacement axis..

But when i say a kind of 'integration error', that doesn't necessarily imply it's an inaccurate result, or non-physical..

...and if we can create momentum... things could get very interesting indeed..!

Serendipitous oversights, eh? The very backbone of unintuitive discoveries, where would we be without 'em.



"Fundamental research is what i'm doing when i've no idea what i'm doing" - Wernher Von Braun

[MrVibrating]

So, offhand, i think linear spring constants are a fairly common engineering requirement, no? Many systems try to optimise spring designs to keep responses within a linear range. An exponential slope is just a practical consequence of material properties (flexibility / ductability coefficients etc.).


All we would need is a spring operating within a narrow linear range, determined by the relative forces and momentums - IOW instead of needing 270° or whatever of travel, we could accomplish the same interaction at higher force within a shorter arc segment & period; ie. net system momentum should rise in direct proportion to the linearity of the spring's response through an abitrarily-narrow angle.

On reflection, making momentum seems to be of dubious value in and of itself, if it's simply trading inertia for velocity at a constant energy. Perhaps it's trivial, tho it still seems novel to me (the implied MoI variation especially), and either way, seems worth a closer look on the off-chance it exposes some other potential weakness..

[MrVibrating]

...cool followup would be to try to transfer two opposing horizontal momentums, into a single vertical one... via a linear spring...!

Or vice-versa. The implication here is that the net momentum should again vary by up to 70%...

[MrVibrating]

..further thoughts...

With the exponential spring constant, net velocity still rises by *1.414, but independently of net momentum.

This implies that the inertial component of that conserved momentum (the 'm' in P=mV) has decreased in proportion to the rise in velocity... ie. we have an effective MoI variation, for a conserved P and KE.

But with the linear spring constant, Occam's razor would suggest that momentum rises because velocity does - presumably for the same reason as in the first case - except this time, the effective MoI does not decrease in proportion to the greater net velocity - hence that invariant MoI is dragged up by the velocity rise, by the same factor of 1.414, and we thus have a net rise in momentum.

But here things get even more perplexing - energy is also function of MoI and velocity, yet net energy remains constant..!?

If momentum has increased because MoI was invariant as net velocity increased, then why hasn't the net energy, which is a function of the same terms?

In short, if net momentum's increased, but net energy hasn't... ie. if mass/inertia is constant and net velocity increases, independently of energy.. this seems to imply some kind of time paradox, or reference frame distortion? The two results seem mutually contradictory... how can we have a rise in momentum - via a rise in velocity - without a corresponding energy rise?

This doesn't seem to make the slightest bit of sense...?

[MrVibrating]

Could WM2D be making an assumption of CoE, in the top-down manner suggested by Fletcher et al, and thus in spite of the apparent momentum rise?

A gain in momentum w/o a KE rise seems mutually irreconcilable..

[MrVibrating]

All we've done is drop a spinning disc onto a static one, via a spring... it would seem ludicrous to suppose that such a simple interaction could produce OU, so the possibility that WM might be enforcing unity is moot.

Yet how can net momentum rise, but not net KE, if they're calculated from inter-dependent terms (inertia and velocity)?

Maybe it's a reference frame discrepancy, and momentum and KE are being calculated in different frames?

Why would the discrepancy only arise when using a linear spring constant, but not when using a non-linear one?

If net energy is conserved, then what's paid for the net momentum rise?

[MrVibrating]

Linear springs - linear forces generally, as applied to accelerating bodies - can produce momentum anomalies, as i've shown in previous findings, by effectively circumventing the usual practicalities of Newton's 3rd law.

But i'm not sure that's what's happening here..

The 70% figure also arose earlier, when considering how MoI varies with radial distribution (MoI for a point mass = mass times radius squared) - so halving the radius causes a fourfold drop in MoI and 4* rise in RKE; if we wanted to simply halve MoI and double RKE, we'd need to reduce radius by 70%.

So there's three examples of that constant which could potentially cross-reference with the velocity / momentum gain here.


1) net velocity rises by 70% regardless of which type of spring is used, with P and KE still conserved

2) net velocity and momentum rise by 70% when a linear force spring is applied (but paradoxically, not net energy)

3) net velocity and KE both double when MoI halves, conserving momentum and corresponding to a drop in radius of 70%

Maybe there's a thread of continuity weft through there, but i can't piece it together yet..

Gonna stay off-piste with this until i get a resolution. Can't afford to be turning our backs on any apparent paradox..

[MrVibrating]

Going back over that last sim, discs1.2, i've found i mis-read the results:

It began with 488600 kg-m^2-°/s, but then peak momentum between the two discs was 2 * 34550, so 691000 in total.

691000 / 488600 = 1.414 (alternatively, the other way round with input P / output P = 0.707)...

Again.


So it isn't an effect particular to linear springs! Duh, dumb error, sent me off on one for 24 hrs... typical.

So net momentum rises by the sqrt 2 when any spring is used this way.


For comparison, i just tried something similar - same two identical discs, one moving, the other static as before, and this time meshing them directly together, via a 1:1 chain and sprocket set, that magically appears after 500 ms. This time, momentum IS conserved, and divided equally between the two discs. Their velocity is thus half the original, and their energy quarters - so the sum of the final energy is half what we started with (the other half having been 'destroyed' by the division into a second MoI).

Logically, then, if we can do the exact opposite - transfer the momentum of two discs into a single disc.. the net energy should be doubled..!?

Something to try next...


Still baffled by that 1.414 thing, but at least it's more consistent now. Still, why's it only arise when using springs to mediate the momentum, but not when the discs are meshed directly? The only substantive difference is that the former depends upon a stator, against which to load the spring, and then unload it again into the two discs... whereas the latter only needs a stator to support the discs, but not to leverage torque against.

So it seems likely that the extra momentum has come from earth, somehow. Yet the energy doesn't rise - and of course why should it in a simple elastic collision - despite the net rise in momentum.

I just double checked again, and the discs have the right amount of energy, which divided by their velocity gives their correct MoI, so energy balances both bottom up - calculated form the outcome results - and top-down, divided from the initial conditions.. so there's evidently no anomaly. Just 1.414 times more momentum than we began with. But only when springs are used, not drive belts / chains / gears or clutches etc. Seems weird, but must be trivial... dunno...

Moving swiftly on, then... is my most recent suggestion plausible - transfer the momentum of two discs into one, gaining energy the same way it's destroyed when transfering in the opposite direction..?

Will have a play and report back later..

[ME]

Still baffled by that 1.414 thing

Perhaps you could do the same but linear test with two blocks in zero G.
If you'll get 2 than such could indicate the difference between linear and rotational stuff?

[MrVibrating]

Absolutely - it's actually one of the suggestions i noted earlier, like i say i'm getting a bit ahead of myself... the linear spring constant thing was a wasted diversion, momentum isn't conserved in either case whatever type of spring is used. A direct copy of the test in a linear system would be a good cross-reference - load a regular coil spring with energy from a collision between a mass and earth, then unload the spring between a pair of the same masses, to compare the balance of input to net output momentums...


In the meantime, here's the sim i mentioned earlier - same basic interaction, but now the two discs are side-by-side, and directly connected by a chain and sprocket set.

So initially the blue disc begins with a 120°/sec (20 RPM) starting velocity, then after coasting for 500 ms, the transmission is engaged and its momentum shared with its identical twin red disc:




..as we can see, momentum and velocity are equally divided, but each disc only has a quarter the inital system energy, so half of it has disappeared.

And i do mean "disappeared" - it's been divided into double the initial MoI, so velocity and momentum have halved... and KE quartered (reduced by half the square of the change in velocity).

So momentum conserved, relative velocity conserved, but energy semi-annihilated.

Clearly, if we ran this same sequence of actions in reverse - give equal energy to two identical rotors, then transfer the momentum from one to the other.... our net energy will have doubled!

How effing sweet is that? What's the catch, i hear you ask..? Well, i've absolutely no idea how to do it, for starters... hoping you wonderful people might assist here..

Cards on the table, here's a clear-cut path to a genuine energy gain... it looks within reach, and it definitely works in reverse... How might we invert it?