PS to everyone else, James has just shown greater clarity of thought on this issue than me or anyone so far, so my guess is he's either about to kill it dead, or blow it wide open...
I'm on the edge of my seat..
Decoupling RKE from GPE, for fun and profit
Moderator: scott
re: Decoupling RKE from GPE, for fun and profit
Lets change the force from one newton to gravitational force in the following paragraph.
A newton is the quantity of force that causes a one kilogram mass to accelerate to one meter per second after the force is applied for one second. 1 N causes 1 kg to (a) accelerate by 1 m/sec/sec. After one second it is moving one meter per second; after 2 seconds it is 2 m/sec, after 3 seconds it is 3 m/sec, after the application of one newton for 400 seconds the one kilogram is moving 400 meters per second.
The gravitational force of 9.81 newtons is the quantity of force that causes a one kilogram mass to accelerate to 9.81 meters per second after the force is applied for one second. 9.81 N causes 1 kg to (a) accelerate by 9.81 m/sec/sec. After one second it is moving 9.81 meter per second; after 2 seconds it is 19.62 m/sec, after 3 seconds it is 29.43 m/sec, after the application of 9.81 newtons for 400 seconds the one kilogram is moving 3924 meters per second.
After three seconds the one kilogram has fallen 44.145 meters. From the distance formula d =1/2 v²/a: or d =1/2at²
So; 1 kilogram moving 29.43 m/sec is 29.43 units of momentum (mv). We need 29.43 units of momentum to rise 44.145 meters.
Now lets take a 9 kilogram rim mass vertical fly wheel and drape a string over its circumference. Lets place one kilogram on the end of the string. The acceleration will be 1/10th gravitational acceleration or .981m/sec/sec. How long does it take to make 29.43 units of momentum?
Well the final velocity will have to be 2.943 m/sec for all 10 kilograms. From d =1/2 v²/a we get (a = .981 m/sec/sec) 4.4145 meters of drop for the one kilogram.
The yo-yo despin device will throw all the motion into one kilogram. So the same one kilogram can be dropped 4.4145 meters for a rise of 44.145 meters. And we did not even us a fusee.
Are these scarey numbers: ten to one? Wait until you use a fusee.
My data confirms these Newtonian concepts.
A newton is the quantity of force that causes a one kilogram mass to accelerate to one meter per second after the force is applied for one second. 1 N causes 1 kg to (a) accelerate by 1 m/sec/sec. After one second it is moving one meter per second; after 2 seconds it is 2 m/sec, after 3 seconds it is 3 m/sec, after the application of one newton for 400 seconds the one kilogram is moving 400 meters per second.
The gravitational force of 9.81 newtons is the quantity of force that causes a one kilogram mass to accelerate to 9.81 meters per second after the force is applied for one second. 9.81 N causes 1 kg to (a) accelerate by 9.81 m/sec/sec. After one second it is moving 9.81 meter per second; after 2 seconds it is 19.62 m/sec, after 3 seconds it is 29.43 m/sec, after the application of 9.81 newtons for 400 seconds the one kilogram is moving 3924 meters per second.
After three seconds the one kilogram has fallen 44.145 meters. From the distance formula d =1/2 v²/a: or d =1/2at²
So; 1 kilogram moving 29.43 m/sec is 29.43 units of momentum (mv). We need 29.43 units of momentum to rise 44.145 meters.
Now lets take a 9 kilogram rim mass vertical fly wheel and drape a string over its circumference. Lets place one kilogram on the end of the string. The acceleration will be 1/10th gravitational acceleration or .981m/sec/sec. How long does it take to make 29.43 units of momentum?
Well the final velocity will have to be 2.943 m/sec for all 10 kilograms. From d =1/2 v²/a we get (a = .981 m/sec/sec) 4.4145 meters of drop for the one kilogram.
The yo-yo despin device will throw all the motion into one kilogram. So the same one kilogram can be dropped 4.4145 meters for a rise of 44.145 meters. And we did not even us a fusee.
Are these scarey numbers: ten to one? Wait until you use a fusee.
My data confirms these Newtonian concepts.
re: Decoupling RKE from GPE, for fun and profit
Keep it simple. Design the simplest fusee experiment that will give you a result and go from there.
My opinion: a fusee at any point is just like a particular gear ratio. My experiments with gear ratios have shown me inertia is energy dependant. Or I=r^2×g^2 (g=gear ratio). A fusee will not decouple a reference frame.
I also suspect decoupling reference frame will not give you a gain in energy, but have not experimented with that.
My opinion: a fusee at any point is just like a particular gear ratio. My experiments with gear ratios have shown me inertia is energy dependant. Or I=r^2×g^2 (g=gear ratio). A fusee will not decouple a reference frame.
I also suspect decoupling reference frame will not give you a gain in energy, but have not experimented with that.
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Trevor Lyn Whatford
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re: Decoupling RKE from GPE, for fun and profit
Hi Mr V,
the sad thing is, a Fusee is used in spring clocks to drop the power train down a gear while the springs power is winding down, so I have to agree with T 79 its just a gear thing.
Fusee is not needed if you you keep the spring wound with GKE (drop weights) and that is where one of the answers will be found, not on the drop but on the lift (reset).
the sad thing is, a Fusee is used in spring clocks to drop the power train down a gear while the springs power is winding down, so I have to agree with T 79 its just a gear thing.
Fusee is not needed if you you keep the spring wound with GKE (drop weights) and that is where one of the answers will be found, not on the drop but on the lift (reset).
I have been wrong before!
I have been right before!
Hindsight will tell us!
I have been right before!
Hindsight will tell us!
London south bank?MrVibrating wrote:...
Bumped into an old physicist buddy on the South Bank earlier and ran it by him. He was stumped. Initially insisted that inertia must be velocity-dependent, LOL... basically he couldn't fault it.
Who is she that cometh forth as the morning rising, fair as the moon, bright as the sun, terribilis ut castrorum acies ordinata?
Re: re: Decoupling RKE from GPE, for fun and profit
I first became familiar with fusees by looking through the glass side windows of the antique grandmother clock that stood on our mantlepiece when I was a boy.Trevor Lyn Whatford wrote:Hi Mr V,
the sad thing is, a Fusee is used in spring clocks to drop the power train down a gear while the springs power is winding down, so I have to agree with T 79 its just a gear thing.
Fusee is not needed if you you keep the spring wound with GKE (drop weights) and that is where one of the answers will be found, not on the drop but on the lift (reset).
I managed to wreck the chimes by winding the hands backwards. :-(
Who is she that cometh forth as the morning rising, fair as the moon, bright as the sun, terribilis ut castrorum acies ordinata?
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MrVibrating
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@James...
Struggling to follow you now - a 10-fold rise in GPE? I read your post back umpteen times, making notes, but still not following. An asymmetry only arises if we invoke a new reference frame; the quasi-static frame of the accelerating mass. It's a KE asymmetry, not a gravitational one, and dependent upon this privileged reference frame.
An accelerating mass normally has two frames - one relative to a stator (us and the Earth), and its own on-board frame, in which it is stationary and the Earth is acceletaring. No energy asymmetry is possible between them, due largely to Newton's 3rd law.
So my little twist on this - and the singular point behind this thread - is the suggestion that we might consider a third reference frame that is "quasi-static" relative to the accelerating mass. This special frame would arise naturally if we could violate Newton's third law and keep accelerating our mass by its own bootstraps, or equivalently, by freely catching up with it after every acceleration. Hence each successive stroke of input work would be subject to the same conditions as when the mass is at rest in the Earth's frame. 1J will keep accelerating 1kg by 1.414m/s, because the mass is always stationary relative to the frame in which energy is being applied and measured.
But N3 is inviolable, so we're denied that reference frame via this route. We can't freely run after a mass, nor apply unilateral force to it, such as by making it accelerate itself via on-board collisions.
It's important to grasp this state of affairs, since this is the setup for the logic that follows. It's the inspiration for the whole rationale..
_____________________________
It's this next bit that really matters...
The masses in the above status quo are merely the relative locations of the respective inertias.
In other words, these locations are incidental to the energy equations, which in their terms are location-agnostic - the "mass" in our work-energy equivalence is, more fundamentally, just inertia; and its displacement, dimensionally subjective.
For a linear acceleration, the locations of the masses and their inertias are hard-coupled and so equivalent. To perform consistent work upon a source of inertia we have to chase after its mass, incurring energy symmetry due to N3.
But in a rotating system, things get switched around: the location of the mass never changes; its position is stationary relative to its stator and Earth, regardless of how fast it spins.
But its angular inertia is in motion relative to the stator & Earth. And it's this inertia we want to perform work upon.
So, the dumb solution would be to keep accelerating the stator relative to the Earth, so that it's always stationary relative to the rotor's inertia. Then we'd always get our consistent 1.414m/s acceleration per input Joule, but while still incurring energy symmetry from the work done on a "stator" that's really just more rotor. This approach just loops the two ends of the linear problem together, incurring the same limitations. We're still treating the inertia as if its distance is increasing in one dimension, when rotation is of course equal motion in two dimenions, and hence from the third dimension the inertia only has angular displacement..
So, can we simulate the same net result as physically chasing the inertia, without having to do so? Perhaps we can take advantage of the fact that it's just spinning in front of us.
The desired effect of running after it, as far as the energy terms are concerned, is maintaining a resting-state F*d input work integral - if the distance component is increasing, then so are successive input work integrals for a consistent yield of acceleration, resulting in the half-square exponent that we're trying to dodge.
Alternatively we could raise the force component of the input F*d integral, while reducing the displacement, keeping input energy equal and maintaining the static inertial load, but for an ever-diminishing displacement per cycle.
A single fusee just trades increasing angular displacement of a rotor for decreasing force, per unit of input energy.
If however we drive the rotor fusee from a second fusee from which we lower a weight, then we can offset decreasing displacement and rising force per unit of input energy, against increasing displacement with decreasing force of output angular acceleration.
The objective result is a slow, constant rate of drop - minimising the kinetic energy of the falling weight, while stabilising the GPE to MoI conversion.
The bottom line is the hypothesis that we might be able to balance the falling weight against the accelerating MoI - whereas normally our focus would be on converting GPE to RKE, here the RKE is considered incidental, and our focus is on MoI - we want the weight to 'feel' a constant angular inertia from the flywheel throughout its descent, despite the flywheel's acceleration, such that each Joule of input energy is converted to an equal torque * angle of accelerated rotor mass.
If i'm simply invoking a miracle then my mistake must be here in the power train. The intended cheat is to effectively chase a mass's inertia without physically chasing the mass itself - ie., simulating the results of an N3 break without actually needing one.
If inverted fusees don't cut it, perhaps something else will. The core concept is simply that a rotating system may offer some way of compensating or sidestepping the increasing distance between a source of inertia and the reaction mass against which it is being accelerated, dodging the square rise in requisite input energy for a consistent rate of acceleration.
Alternatively, any means of emulating the effects of an N3 break - an effectively-reactionless collision, or whatever - will provide access to the same asymmetry - input energy will scale linearly within the accelerating frame, while output squares up in the stationary frame.
But without some means of ducking our 3rd law overheads there can be no freeloading reference frame and hence no asymmetry. And while we could trade a KE gain for a GPE rise, a GPE rise from first principles without explicitly invoking a symmetry break must be in error..
Struggling to follow you now - a 10-fold rise in GPE? I read your post back umpteen times, making notes, but still not following. An asymmetry only arises if we invoke a new reference frame; the quasi-static frame of the accelerating mass. It's a KE asymmetry, not a gravitational one, and dependent upon this privileged reference frame.
An accelerating mass normally has two frames - one relative to a stator (us and the Earth), and its own on-board frame, in which it is stationary and the Earth is acceletaring. No energy asymmetry is possible between them, due largely to Newton's 3rd law.
So my little twist on this - and the singular point behind this thread - is the suggestion that we might consider a third reference frame that is "quasi-static" relative to the accelerating mass. This special frame would arise naturally if we could violate Newton's third law and keep accelerating our mass by its own bootstraps, or equivalently, by freely catching up with it after every acceleration. Hence each successive stroke of input work would be subject to the same conditions as when the mass is at rest in the Earth's frame. 1J will keep accelerating 1kg by 1.414m/s, because the mass is always stationary relative to the frame in which energy is being applied and measured.
But N3 is inviolable, so we're denied that reference frame via this route. We can't freely run after a mass, nor apply unilateral force to it, such as by making it accelerate itself via on-board collisions.
It's important to grasp this state of affairs, since this is the setup for the logic that follows. It's the inspiration for the whole rationale..
_____________________________
It's this next bit that really matters...
The masses in the above status quo are merely the relative locations of the respective inertias.
In other words, these locations are incidental to the energy equations, which in their terms are location-agnostic - the "mass" in our work-energy equivalence is, more fundamentally, just inertia; and its displacement, dimensionally subjective.
For a linear acceleration, the locations of the masses and their inertias are hard-coupled and so equivalent. To perform consistent work upon a source of inertia we have to chase after its mass, incurring energy symmetry due to N3.
But in a rotating system, things get switched around: the location of the mass never changes; its position is stationary relative to its stator and Earth, regardless of how fast it spins.
But its angular inertia is in motion relative to the stator & Earth. And it's this inertia we want to perform work upon.
So, the dumb solution would be to keep accelerating the stator relative to the Earth, so that it's always stationary relative to the rotor's inertia. Then we'd always get our consistent 1.414m/s acceleration per input Joule, but while still incurring energy symmetry from the work done on a "stator" that's really just more rotor. This approach just loops the two ends of the linear problem together, incurring the same limitations. We're still treating the inertia as if its distance is increasing in one dimension, when rotation is of course equal motion in two dimenions, and hence from the third dimension the inertia only has angular displacement..
So, can we simulate the same net result as physically chasing the inertia, without having to do so? Perhaps we can take advantage of the fact that it's just spinning in front of us.
The desired effect of running after it, as far as the energy terms are concerned, is maintaining a resting-state F*d input work integral - if the distance component is increasing, then so are successive input work integrals for a consistent yield of acceleration, resulting in the half-square exponent that we're trying to dodge.
Alternatively we could raise the force component of the input F*d integral, while reducing the displacement, keeping input energy equal and maintaining the static inertial load, but for an ever-diminishing displacement per cycle.
A single fusee just trades increasing angular displacement of a rotor for decreasing force, per unit of input energy.
If however we drive the rotor fusee from a second fusee from which we lower a weight, then we can offset decreasing displacement and rising force per unit of input energy, against increasing displacement with decreasing force of output angular acceleration.
The objective result is a slow, constant rate of drop - minimising the kinetic energy of the falling weight, while stabilising the GPE to MoI conversion.
The bottom line is the hypothesis that we might be able to balance the falling weight against the accelerating MoI - whereas normally our focus would be on converting GPE to RKE, here the RKE is considered incidental, and our focus is on MoI - we want the weight to 'feel' a constant angular inertia from the flywheel throughout its descent, despite the flywheel's acceleration, such that each Joule of input energy is converted to an equal torque * angle of accelerated rotor mass.
If i'm simply invoking a miracle then my mistake must be here in the power train. The intended cheat is to effectively chase a mass's inertia without physically chasing the mass itself - ie., simulating the results of an N3 break without actually needing one.
If inverted fusees don't cut it, perhaps something else will. The core concept is simply that a rotating system may offer some way of compensating or sidestepping the increasing distance between a source of inertia and the reaction mass against which it is being accelerated, dodging the square rise in requisite input energy for a consistent rate of acceleration.
Alternatively, any means of emulating the effects of an N3 break - an effectively-reactionless collision, or whatever - will provide access to the same asymmetry - input energy will scale linearly within the accelerating frame, while output squares up in the stationary frame.
But without some means of ducking our 3rd law overheads there can be no freeloading reference frame and hence no asymmetry. And while we could trade a KE gain for a GPE rise, a GPE rise from first principles without explicitly invoking a symmetry break must be in error..
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MrVibrating
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Ideally we want minimal mass acceleration (minimum KE of the drop mass), with maximal acceleration of the wheel (maximal as in OU, via an emulated symmetry break).Kirk wrote:as you accelerate the mass you decelerate the wheel.
A single fusee can't work, no.Tarsier79 wrote:Keep it simple. Design the simplest fusee experiment that will give you a result and go from there.
My opinion: a fusee at any point is just like a particular gear ratio. My experiments with gear ratios have shown me inertia is energy dependant. Or I=r^2×g^2 (g=gear ratio). A fusee will not decouple a reference frame.
I also suspect decoupling reference frame will not give you a gain in energy, but have not experimented with that.
The simulated reference frame acceleration here affords a linear rise in input energy for an exponential rise in output energy, diverging after two Joules.
Still think dual inverted fusees may be worth testing, but a well-presented rebuttal would save a lot of time and effort..
Again, a single fusee can't work, since the objective is to convert a consistent amount of GPE to a consistent amount of inertial acceleration. A pair of fusees may work, or if not, some other approach might have more success. The hypothesis is simply that 1 Joule might accelerate 1 kg by 1.414 m/s repeatedly, in a linear, cummulative sum, while its KE in the stationary frame increases by half the square of its momentum.Trevor Lyn Whatford wrote:Hi Mr V,
the sad thing is, a Fusee is used in spring clocks to drop the power train down a gear while the springs power is winding down, so I have to agree with T 79 its just a gear thing.
Fusee is not needed if you you keep the spring wound with GKE (drop weights) and that is where one of the answers will be found, not on the drop but on the lift (reset).
Yep, Upper Ground, ITV studios. I do a lot of work for film and TV, which isn't anywhere near as glamorous as it sounds.Grimer wrote: London south bank?
Still, top marks for trying to cheat time. That's just the sort of initiative we need..Grimer wrote:I first became familiar with fusees by looking through the glass side windows of the antique grandmother clock that stood on our mantlepiece when I was a boy.
I managed to wreck the chimes by winding the hands backwards. :-(
re: Decoupling RKE from GPE, for fun and profit
So presumably W3 in your profile refers to London W3, where I went to school (Gunnersbury Grammar). I did wonder.MrVibrating wrote:Grimer wrote:
London south bank?
Yep, Upper Ground, ITV studios. I do a lot of work for film and TV, which isn't anywhere near as glamorous as it sounds.
Who is she that cometh forth as the morning rising, fair as the moon, bright as the sun, terribilis ut castrorum acies ordinata?
re: Decoupling RKE from GPE, for fun and profit
I cheated time a long time ago....:-)MrVibrating wrote: ... top marks for trying to cheat time. That's just the sort of initiative we need..
Who is she that cometh forth as the morning rising, fair as the moon, bright as the sun, terribilis ut castrorum acies ordinata?
re: Decoupling RKE from GPE, for fun and profit
Why would more than 1 fusee work when 1 doesnt. The fusee is just a variable gear ratio. 2 variable gear ratios magically create energy?
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MrVibrating
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Maybe if one is cursed by a gipsy?
The concept is to try to maintain a linear rise in input work, for a regular half-squared output RKE.
The proposed solution is to try to keep the weight of the drop mass balanced against the MoI of the flywheel across an accelerating RPM range, to try and achieve a consistent, constant linear rate of flywheel acceleration for a given increment of input GPE.
To my addled thinking this requires two inverted fusees, at least one of which should be under some kind of supernatural charm.
A single fusee would have to be doubly-magic, which as I keep saying is above my wizard grade, but that's just mechanics 101.
I do wish people would listen...
The concept is to try to maintain a linear rise in input work, for a regular half-squared output RKE.
The proposed solution is to try to keep the weight of the drop mass balanced against the MoI of the flywheel across an accelerating RPM range, to try and achieve a consistent, constant linear rate of flywheel acceleration for a given increment of input GPE.
To my addled thinking this requires two inverted fusees, at least one of which should be under some kind of supernatural charm.
A single fusee would have to be doubly-magic, which as I keep saying is above my wizard grade, but that's just mechanics 101.
I do wish people would listen...
re: Decoupling RKE from GPE, for fun and profit
Let's see if this will help Kaine. There are some similarities IMO.
In my thread I was investigating whether horizontal load masses could be accelerated by a falling driver mass which might lead to a complete (or near complete) transfer of momentum from driver to load. If so this would mean a gain in KE above GPE lost.
I was using the sim program and both geared pulley systems and then latterly the storksbill arrangements.
In the pulley systems the the driver' velocity increased downwards (albeit slowly increasing) and the loads velocity also increased by a fixed factor of the gearing ratio. The accelerations were 'fixed' and did not change because the gearing was fixed.
In the storksbill systems the gearing was variable. This allowed the driver to slow and reverse its acceleration while descending whilst the loads acceleration initially increased dramatically then began to reduce. Overall the load velocity continued to increase as the drivers velocity decreased eventually to zero. In sim world there was no excess KE apparently.
.............................
In Mr V's case he is using a double fusee (one up and one down) to create a variable gearing also, like a CVT. The hanging driver attached to the bottom fusee should be able to be slowed right down so it has very little KE, and the RKE of the 'disk' load (attached at the end of the top fusee) should be able to be sped up dramatically because of this constant gearing change brought about by the double fusee action.
Since angular momentum is a different beast entirely from linear momentum he is wanting to investigate whether a faux N3 break is created which could lead in principle to a gain in load energy (RKE) above driver GPE lost.
If I have that wrong in any way I'm sure Mr V will correct me with the detail.
Sorry for butting in Mr V.
In my thread I was investigating whether horizontal load masses could be accelerated by a falling driver mass which might lead to a complete (or near complete) transfer of momentum from driver to load. If so this would mean a gain in KE above GPE lost.
I was using the sim program and both geared pulley systems and then latterly the storksbill arrangements.
In the pulley systems the the driver' velocity increased downwards (albeit slowly increasing) and the loads velocity also increased by a fixed factor of the gearing ratio. The accelerations were 'fixed' and did not change because the gearing was fixed.
In the storksbill systems the gearing was variable. This allowed the driver to slow and reverse its acceleration while descending whilst the loads acceleration initially increased dramatically then began to reduce. Overall the load velocity continued to increase as the drivers velocity decreased eventually to zero. In sim world there was no excess KE apparently.
.............................
In Mr V's case he is using a double fusee (one up and one down) to create a variable gearing also, like a CVT. The hanging driver attached to the bottom fusee should be able to be slowed right down so it has very little KE, and the RKE of the 'disk' load (attached at the end of the top fusee) should be able to be sped up dramatically because of this constant gearing change brought about by the double fusee action.
Since angular momentum is a different beast entirely from linear momentum he is wanting to investigate whether a faux N3 break is created which could lead in principle to a gain in load energy (RKE) above driver GPE lost.
If I have that wrong in any way I'm sure Mr V will correct me with the detail.
Sorry for butting in Mr V.
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MrVibrating
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OK, non-snarky version:
Single fusee
--------------
- As the weight winds off from the thick end of the fusee towards the thinner end, we DO get the increasing RPM, and if the MoI and drop weight are well matched to the fusee then we can neatly perch the weight against the MoI for the full drop across the full flywheel acceleration.
- However, as the radius of the fusee winding decreases, so does the distance the weight drops each cycle, meaning the per-cycle input energy decreases.
- The proposed asymmetry can only work if the per-cycle input energy remains constant - ie. the drop distance has to remain constant per-cycle.
Dual fusee
------------
So if we add a second fusee, winding from its thin end towards its thick end, and drop the weight from this one while using its rotation to drive the other one accelerating the flywheel, then its increasing girth compensates the decreasing girth of the other one, maintaining a consistent drop distance and thus input energy per cycle across an accelerating RPM range.
That would create energy. Quite a lot of it, in fact.
If it worked.
I suspect i may be asking for conflicting requirements here, and that such a power train is mathematically impossible... but at the same time the asymmetry depends upon "creating" energy by effectively "destroying" the displacement that otherwise develops between mutually accelerating masses... so there may be a fine line between making a dumb error and actually describing a viable asymmetry; either way we need to discount half the square of the conventional input energy.
Again, the sole meditation behind all this is the question "why does KE square up?" as found by s'Gravesande - as opposed to following the linear m*V Aristotelian presumption..
If it squares up for truly fundamental reasons, then they're likely mechanically insurmountable. Any attempt around them will just end up trading force for displacement, and the whole premise is a fool's errand.
But if it only arises due to practical engineering challenges, then there may be a way to work around it, or compensate for it.
Because KE squares up, each successive unit of input energy accomplishes progressively less work. To keep the work done per unit of input energy constant, we need to either sustain the properties of the initial conditions throughout the acceleration, or else freely reset them between each additional unit of input energy.
If we could do that then each unit of input energy will accelerate the same amount of mass by the same change in velocity. Hence ten consecutive units of input energy will cause ten equal accelerations, the last accelerating the mass up to ten times the velocity of the first.
So i'm suggesting a way to attempt this step-wise sequence in a continuous process, trading each unit of dropped GPE for an equal unit of accelerated mass.
Such a 1:1 conversion of work may seem perfectly conservative and reasonable, until you recall that KE is still squaring up from the stationary perspective. Hence if input energy only increases linearly, while output energy increases by a half-square more, then the two values diverge after two units of input energy, with output KE peaking at between fifty and a hundred times input energy, depending on how bad your maths are.
So it's just a hypothetical asymmetry, only accessible if we can accomplish this 1:1 linear work ratio for a progressively accelerating mass.
Fusees, or constantly-variable transmissions more generally, are one attempt at a solution.
Maybe there's others.
Or maybe the whole scheme's walled in by symmetries at every turn, and forever closed off to us at a very fundamental level.
The only question that matters here is:
Why does KE square up, and does it have to (as in really, really have to)?
Single fusee
--------------
- As the weight winds off from the thick end of the fusee towards the thinner end, we DO get the increasing RPM, and if the MoI and drop weight are well matched to the fusee then we can neatly perch the weight against the MoI for the full drop across the full flywheel acceleration.
- However, as the radius of the fusee winding decreases, so does the distance the weight drops each cycle, meaning the per-cycle input energy decreases.
- The proposed asymmetry can only work if the per-cycle input energy remains constant - ie. the drop distance has to remain constant per-cycle.
Dual fusee
------------
So if we add a second fusee, winding from its thin end towards its thick end, and drop the weight from this one while using its rotation to drive the other one accelerating the flywheel, then its increasing girth compensates the decreasing girth of the other one, maintaining a consistent drop distance and thus input energy per cycle across an accelerating RPM range.
That would create energy. Quite a lot of it, in fact.
If it worked.
I suspect i may be asking for conflicting requirements here, and that such a power train is mathematically impossible... but at the same time the asymmetry depends upon "creating" energy by effectively "destroying" the displacement that otherwise develops between mutually accelerating masses... so there may be a fine line between making a dumb error and actually describing a viable asymmetry; either way we need to discount half the square of the conventional input energy.
Again, the sole meditation behind all this is the question "why does KE square up?" as found by s'Gravesande - as opposed to following the linear m*V Aristotelian presumption..
If it squares up for truly fundamental reasons, then they're likely mechanically insurmountable. Any attempt around them will just end up trading force for displacement, and the whole premise is a fool's errand.
But if it only arises due to practical engineering challenges, then there may be a way to work around it, or compensate for it.
Because KE squares up, each successive unit of input energy accomplishes progressively less work. To keep the work done per unit of input energy constant, we need to either sustain the properties of the initial conditions throughout the acceleration, or else freely reset them between each additional unit of input energy.
If we could do that then each unit of input energy will accelerate the same amount of mass by the same change in velocity. Hence ten consecutive units of input energy will cause ten equal accelerations, the last accelerating the mass up to ten times the velocity of the first.
So i'm suggesting a way to attempt this step-wise sequence in a continuous process, trading each unit of dropped GPE for an equal unit of accelerated mass.
Such a 1:1 conversion of work may seem perfectly conservative and reasonable, until you recall that KE is still squaring up from the stationary perspective. Hence if input energy only increases linearly, while output energy increases by a half-square more, then the two values diverge after two units of input energy, with output KE peaking at between fifty and a hundred times input energy, depending on how bad your maths are.
So it's just a hypothetical asymmetry, only accessible if we can accomplish this 1:1 linear work ratio for a progressively accelerating mass.
Fusees, or constantly-variable transmissions more generally, are one attempt at a solution.
Maybe there's others.
Or maybe the whole scheme's walled in by symmetries at every turn, and forever closed off to us at a very fundamental level.
The only question that matters here is:
Why does KE square up, and does it have to (as in really, really have to)?