Rotational Kinetic Energy and Conservation of Momentum

[Dadface]

Some of the energy transfer systems being suggested here suffer from their own inherent losses.An idealised spring system can store only fifty percent of the energy needed to distort it and an idealised capacitor charging system can store only fifty percent of the energy needed to charge it.
The idea of elastic transmissions by collision is interesting and although it can work for linear motion(perfectly for some collisions with microscopic objects eg identical gas atoms under certain conditions) it would be interesting to see how it can be achieved,in practise,with rotational motion and macroscopic structures like flywheels.
If the energy is transferred by some sort of gear/clutch linkage then,assuming an idealised system,the maximum energy of the whole assembley,as a result of linkage is given by:

Final Energy=Initial Energy*I/(I+Ix)

I= moment of inertia of original flywheel.

Ix equals effective moment of inertia of everything else( second wheel plus clutch/gears etc)

Although the equation applies to a system which has been massively simplified I think it illustrates that considerable energy losses are inevitable.

 

[cmb]

Some of the energy transfer systems being suggested here suffer from their own inherent losses.An idealised spring system can store only fifty percent of the energy needed to distort it and an idealised capacitor charging system can store only fifty percent of the energy needed to charge it.

AFAIK - that's not right. You're saying that a capacitor being repeatedly charged then discharged would heat up by the total power that the capacitor is handling? (because it is causing as much loss as power that it passes?!)

Or you are saying that if I have a 10 kg mass I attach on a spring, and after letting go the spring stretches 1 m with the mass on the end of it, that the spring will heat up by 50J!?

By these means you can calculate how many times a spring can bounce - each bounce would reduce the energy stored in the spring by 50%. Not many bounces, then? So are you saying a spring in simple harmonic motion would never have a displacement in excess of 70% of the previous displacement!?

I think you are confusing the issue of this thread, as well as purveying some errors.

 

[Ken G]

Yes, I think a factor of 1/2 slipped in on you there Dadface-- springs and capacitors can in principle store 100% of the work done on them.

 

[Dadface]

AFAIK - that's not right. You're saying that a capacitor being repeatedly charged then discharged would heat up by the total power that the capacitor is handling? (because it is causing as much loss as power that it passes?!)

Or you are saying that if I have a 10 kg mass I attach on a spring, and after letting go the spring stretches 1 m with the mass on the end of it, that the spring will heat up by 50J!?

By these means you can calculate how many times a spring can bounce - each bounce would reduce the energy stored in the spring by 50%. Not many bounces, then? So are you saying a spring in simple harmonic motion would never have a displacement in excess of 70% of the previous displacement!?

I think you are confusing the issue of this thread, as well as purveying some errors.

1.Consider ,for example,a helical spring with a mass M added resulting in a final static extension of x:

PE lost =Mgx

Energy stored=0.5Mgx (average force times extension)

2.Similarly the energy stored in a charged capacitor=0.5QV(Q= charge V=volts)

In the case of the spring if a mass is added and released the system overshoots the equilibrium position and vibrates with damped harmonic motion before coming to rest.In the case of the capacitor energy is lost by Joule heating as the charging current flows through the connecting wires.Mainly heat losses in both cases.

 

[cmb]

1.Consider ,for example,a helical spring with a mass M added resulting in a final static extension of x:

PE lost =Mgx

Energy stored=0.5Mgx (average force times extension)

eh? Why is the 'average force' not Mg!? Did its mass change somewhere during the fall?

 

[cmb]

2.Similarly the energy stored in a charged capacitor=0.5QV(Q= charge V=volts)

So this is true so long as the charge stays the same but the capacitance changes!

The thing which stays the same is the capacitance, C, so the energy is 1/2[CV^2].

 

[jimgram]

Delta-Kilo: You have expressed the fundamental question I originally posited except that the momentum/energy transfer is through an IVT which varies ratio over a specific time such that the outcome will be flywheel A gives up momentum/energy to flywheel B and the satellite (flywheel C). Depending of course on the relative moments of inertia and the time and the beginning and ending ratio's of the IVT, the outcome can be anything - but the goal is that flywheel A slows some percentage - say, from ω to ω/2 - and flywheel B accelerates from zero to some speed which will give it the energy and momentum that was given up by flywheel A. Of course the torque applied to the satellite will be equal to the torque applied to (-) FW A +FW B, and therefore momentum is conserved. Having done all of this, however, you will find the ending kinetic energy is significantly different that beginning kinetic energy

 

[Dadface]

eh? Why is the 'average force' not Mg!? Did its mass change somewhere during the fall?

Average because provided that the elastic limit is not passed the extension is directly proportional to the load,in other words during stretching the force in the spring increases from zero to a maximum.The energy can also be found by integration which is equivelent to finding the area under the force extension graph.

 

[A.T.]

A slows some percentage - say, from ω to ω/2 - and flywheel B accelerates from zero to some speed which will give it the energy and momentum that was given up by flywheel A. Of course the torque applied to the satellite will be equal to the torque applied to (-) FW A +FW B, and therefore momentum is conserved. Having done all of this, however, you will find the ending kinetic energy is significantly different that beginning kinetic energy

Yes, and the reason for this change in KE is directly related to your requirement of the final ω. If you want KE conserved you are not free to choose the final ω.

This is the same as in the linear collision case, where you cannot choose the final velocities of the balls, if you want kinetic energy to be conserved.

See my comments here:
https://www.physicsforums.com/showpost.php?p=3486833&postcount=26

 

[jimgram]

I am not choosing a final ω. As I stated in my original post, I have two flywheels with specified inertia and a foundation (I.E. satellite, earth, moon, flywheel 'c', etc.). I have specified an inital velocity of FW A at 10,000 rpm and an initial velocity of FW B of zero. They are coupled with an IVT (e.g. Torotrak) which provides ratio's from zero (no output) to 0.075 (13.333:1). I will vary the ratio from zero to max. in 14 seconds.


My objective is to calculate the results. I can conserve momentum by using the torque applied to the transmission housing, assuming some (preferably large) inertai for my foundation (resulting in a very low ω). It's simple to arrive at the ending ω of FW A, B, C and to show the conservation of momentum. But, unless I calculate the work done decelerating FW A and accelerating FW B and subtracting this work from the total ending kinetic energy, kinetic energy is not conserved. This has nothing to do with elastic collisions which will result in FW A stopping and reversing which does not occur.

 

[Dadface]

So this is true so long as the charge stays the same but the capacitance changes!

The thing which stays the same is the capacitance, C, so the energy is 1/2[CV^2].

No,my previous comments applied to the charging of a fixed capacitor where when Q increases then so does V in direct proportion(Q=CV).As far as the energy equation is concerned make your choice:

E=0.5CV^2

E=0.5QV

E=0.5(Q^2/C)

Note the 0.5 factor in all three equations.

Charging a capacitor can be considered as analogous to stretching a spring.The energy stored in a stretched spring is the area under a force extension graph and the energy stored in a charged capacitor is the area under a charge voltage graph.

 

[Delta Kilo]

The angular velocity of the flywheel as a function of time (and thus a function of n (the ivt ratio from zero to ne) is equal to the initial momentum of the flywheel (Ifwa*ω fwai) divided by the total inertia as seen by the flywheel (Ifwa+(n2*Ifwb):

ω fwa(t) = (Ifwa*ω fwai)/(Ifwa+(n2*Ifwb)

How did you arrive at this formula? It seems wrong. It can be rewritten as I_{ab} \omega_a(t) = I_a \omega_a(0) where I_{ab} = I_a + n^2 I_b is the "total" of "effective" moment of inertia. In this form it looks very much like momentum conservation law, but:
a) Angular momentum of the flywheels is not conserved due to torques on the gearbox mounts and
b) Effective moment of inertia I_{ab} is a kind of surrogate with limited applicability. It can be plugged into \tau = I \frac{d\omega}{dt} and E=\frac{1}{2}I\omega^2 (in fact it is derived from it) but not much else. It is easy to see that it fails for the angular momentum.

 

[cmb]

Average because provided that the elastic limit is not passed the extension is directly proportional to the load,in other words during stretching the force in the spring increases from zero to a maximum...

...the maximum being 2Mg when it is decelerated to rest at the bottom of the SHM spring articulation!

Doh!

 

[cmb]

No,my previous comments applied to the charging of a fixed capacitor where when Q increases then so does V in direct proportion(Q=CV).As far as the energy equation is concerned make your choice:

E=0.5CV^2

E=0.5QV

E=0.5(Q^2/C)

Note the 0.5 factor in all three equations.

Charging a capacitor can be considered as analogous to stretching a spring.The energy stored in a stretched spring is the area under a force extension graph and the energy stored in a charged capacitor is the area under a charge voltage graph.

This line of discussion should be disregarded by anyone looking into this thread for guidance. It is wholly misleading.

Springs and capacitors can act as stores of energies approaching 100% of input energy.

 

[jimgram]

Delta-Kilo: Originally this solution was posted by K^2 (https://www.physicsforums.com/showthread.php?t=443644&highlight=flywheel+momentum+exchange" ) I have refined the result and used an inverted ratio since it's easier to use '0:.075' than 'inf.:13.33'. Using Mathcad, I can show angular velocity results, torque results, and momentum results that all 'work' correctly. I'm not understanding the kinetic energy relations, however. Since this situation will be reduced to practice, it's difficult without a clear understanding.

 

[RonL]

Delta-Kilo: You have expressed the fundamental question I originally posited except that the momentum/energy transfer is through an IVT which varies ratio over a specific time such that the outcome will be flywheel A gives up momentum/energy to flywheel B and the satellite (flywheel C). Depending of course on the relative moments of inertia and the time and the beginning and ending ratio's of the IVT, the outcome can be anything - but the goal is that flywheel A slows some percentage - say, from ω to ω/2 - and flywheel B accelerates from zero to some speed which will give it the energy and momentum that was given up by flywheel A. Of course the torque applied to the satellite will be equal to the torque applied to (-) FW A +FW B, and therefore momentum is conserved. Having done all of this, however, you will find the ending kinetic energy is significantly different that beginning kinetic energy

If I'm understanding correctly, the rate of exchange is the confusing part.
Look at the charging of flywheel A, at start, low speed and high torque increases to high speed and low torque. Now using that high end, energy, (high speed, low torque) to start energizing flywheel B at the most demanding point, plus the loss to the satellite, the imbalance of exchange should become obvious.

Also, just for me to be clear, is the goal to eliminate spin to the satellite ?
my mind does not think in much other than, action/reaction pairs.

Ron

 

[Dadface]

...the maximum being 2Mg when it is decelerated to rest at the bottom of the SHM spring articulation!

Doh!

What do you mean by Doh !Is that supposed to represent a condescending utterance.Try searching and may I suggest that you start on this site by searching for:
"Where does the missing energy go?"

 

[Dadface]

This line of discussion should be disregarded by anyone looking into this thread for guidance. It is wholly misleading.

Springs and capacitors can act as stores of energies approaching 100% of input energy.

It would be more constructive if you showed how you the 100 percent efficiency can be approached.Try a search on this topic as well and a good start would be to go to the "hyperphysics" site and then go to the section entitled "Energy stored in a capacitor".

 

[cmb]

What do you mean by Doh !Is that supposed to represent a condescending utterance.

Yes, sorry, it was. If a mass passes through 'x' displacing a spring until it reaces zero acceleration (thus, the mass reaches a point where it is finally acting on the spring with Mg of force) then it means it is at its centre of SHM. In other words, it is moving. And, guess what, indeed the energy in the spring at the time will be 0.5Mgx, just as the kinetic energy of the mass with be the other 0.5Mgx. That kinetic energy will keep the mass falling, extending the spring to '2x' (in your dimensional terms), thus storing a total of 2Mgx of energy in the spring once the mass decelerates to zero velocity at the 'bottom' of its SHM - just before it gets pulled back up by the energy stored [NOT lost] in the spring.

It's all elementary SHM.

 

[cmb]

It would be more constructive if you showed how you the 100 percent efficiency can be approached.Try a search on this topic as well and a good start would be to go to the "hyperphysics" site and then go to the section entitled "Energy stored in a capacitor".

If you were to connect a capacitor, of C Farads, to a battery of V volts through a resistor of R ohms, then the time it takes to charge up to 99% of the voltage is ~4RC. This is very well known. Similarly, if you then connect a capacitor charged to V volts to ground via the resistor, the time it takes to get to 1% of V is ~4RC.

If what you were saying was right, it'd take 8RC to charge up, and 4RC to discharge. It doesn't. See http://www.electronics-tutorials.ws/rc/rc_1.html.

Sorry, your discussion is very misleading and I implore folks to ignore your posts here.

 

[Dadface]

Yes, sorry, it was. If a mass passes through 'x' displacing a spring until it reaces zero acceleration (thus, the mass reaches a point where it is finally acting on the spring with Mg of force) then it means it is at its centre of SHM. In other words, it is moving. And, guess what, indeed the energy in the spring at the time will be 0.5Mgx, just as the kinetic energy of the mass with be the other 0.5Mgx. That kinetic energy will keep the mass falling, extending the spring to '2x' (in your dimensional terms), thus storing a total of 2Mgx of energy in the spring once the mass decelerates to zero velocity at the 'bottom' of its SHM - just before it gets pulled back up by the energy stored [NOT lost] in the spring.

It's all elementary SHM.

Did you actually read my posts?tThe energy value I quoted was when the mass reached its final static extension this being after the vibrations die down due to damping.

 

[Dadface]

If you were to connect a capacitor, of C Farads, to a battery of V volts through a resistor of R ohms, then the time it takes to charge up to 99% of the voltage is ~4RC. This is very well known. Similarly, if you then connect a capacitor charged to V volts to ground via the resistor, the time it takes to get to 1% of V is ~4RC.

If what you were saying was right, it'd take 8RC to charge up, and 4RC to discharge. It doesn't. See http://www.electronics-tutorials.ws/rc/rc_1.html.

Sorry, your discussion is very misleading and I implore folks to ignore your posts here.

Go to
1.hyperphysics
2.capacitor
3.capacitance concepts
4.energy storage device

 

[cmb]

Did you actually read my posts?tThe energy value I quoted was when the mass reached its final static extension this being after the vibrations die down due to damping.

What's that got to do with how much energy a spring can store? When the spring gets to the bottom of its extention, then it may be storing, say, 99% of the energy that the falling mass has just dissipated. Say each bounce after that it loses 1% [just an example figure] of the stored energy, which, if it carried on losing 1% per bounce, would give it a ~70 bounce 'half-life' (that is, it'd drop to half the original energy after 70 bounces).

The relationship between the final rest position of an oscillating mass and the rate of energy loss during oscillations (which is a function of the efficiency of energy storage) is... none at all...

This is really very confusingly diverging the point of this post. You are wrong and off topic, I'll add no more because I'd be adding to the off-topic-ness ... Moderator, please...

 

[Dadface]

What's that got to do with how much energy a spring can store? When the spring gets to the bottom of its extention, then it may be storing, say, 99% of the energy that the falling mass has just dissipated. Say each bounce after that it loses 1% [just an example figure] of the stored energy, which, if it carried on losing 1% per bounce, would give it a ~70 bounce 'half-life' (that is, it'd drop to half the original energy after 70 bounces).

The relationship between the final rest position of an oscillating mass and the rate of energy loss during oscillations (which is a function of the efficiency of energy storage) is... none at all...

This is really very confusingly diverging the point of this post. You are wrong and off topic, I'll add no more because I'd be adding to the off-topic-ness ... Moderator, please...

Any vibrations are transient only ,they die down.The vibrational energy can't be stored,in other words kept for use as needed.Also,stretching methods can be used where there will be no vibrations at all but whatever method is used the energy stored is the same.
Did you do the searches I recommended?If not please read the following quote taken from the hyperphysics site:

from hyperphysics "Storing Energy in a Capacitor"

The title makes it obvious what the section is about and at the end of the section the following note is added:

Note that the total energy stored QV/2 is exactly half the energy QV which is supplied by the battery,independent of R(R referring to the resistance)

Check it for yourself.

Please note that in my first post I made a brief response to the energy storage systems referred to here and then moved on in an attempt to get involved in the main thrust of the discussion referred to by the OP.You were the one who caused this discussion to go on by suggesting that I was wrong.

 

[cmb]

Note that the total energy stored QV/2 is exactly half the energy QV which is supplied by the battery,independent of R(R referring to the resistance)

'QV' isn't the energy supplied by the battery. When the volts are low, the 'QV' value is lower (because the volts are low). Let's say that some capacitor has two Coulombs of charge on it. The first Coulomb took less energy to put it on the capacitor than the second - because the second was pushed up to a higher potential.

Imagine a kg mass on each step of your 10 step stair case, each step of height h. The first kilo only takes a little energy, but by the time you've filled up the staircase, the one on the last step takes 10 times as much energy to walk it up the stairs than the last. If you are claiming that QV is the energy being put onto a capacitor (and that 'V' is the highest voltage) then it is like you are calculating the potential energy of all those 1kg masses on your stairs as 10 x the height of the highest one. Clearly it isn't 100gh. It is 50gh, because it is a triangular number with some having been put on the stair case when they didn't have to be put so high up it. Same with charge into a capacitor.

Check it for yourself.

Yes, it is not at all as clear as hypherphysics usual standard. But if it has taken you to the point of thinking that the energy it takes for a battery feeding Q Coulombs into a capacitor that reaches V volts is QV, then your understanding has been mislead. A battery delivers [1/2].QV of energy to push Q of charge into a capacitor with a charged voltage V, because the battery does not need to push ALL of the electrons up to a potential of V, only the very last one!

Sorry, I said I'd not thread-drift and did so, but that misunderstanding really needed to be addressed here.

 

[Dadface]

So hyperphysics and others got it wrong did they?I wouldn't mind betting that next you are going to announce that the equation defining the volt in terms of energy and charge is incorrect.
If you really think there's a way of making the energy storage efficiency of either system(spring or capacitor)approach 100 percent then show how.At least that would be interesting and productive.

 

[Delta Kilo]

Delta-Kilo: Originally this solution was posted by K^2 (https://www.physicsforums.com/showthread.php?t=443644&highlight=flywheel+momentum+exchange" ) I have refined the result and used an inverted ratio since it's easier to use '0:.075' than 'inf.:13.33'.

I see. In the derivation you refer to, n is a constant and so it can be moved outside of the integral. In your case n = n(t) is a function of time and so the trick no longer works.

What it means you just cannot change the ratio of an ideal (perfectly rigid) gearbox at whim because the moment you try you get unlimited torques. In other words the ratios \frac{\omega_1}{\omega_2} = n and \frac{\tau_2}{\tau_1} = n cannot both be true while n is changing, something has to give. For example, you can have an elastic element somewhere in your gearbox in which case the ratio of angular velocities will be violated briefly as the elastic element first compresses and accumulates energy and then expands and releases energy. Or you can have slipping clutch that allows angular velocity ratio to be violated and converts energy to heat, or a combination of both. Or you might have perfectly rigid gearbox but moving the gear handle would actually require force and would produce some work which will be added/removed to/from the balance of kinetic energies of the flywheels.

At the end of the day you select what you want to be conserved: energy, torque ratio or angular velocity ratio, and write the equations accordingly.

 

[cmb]

So hyperphysics and others got it wrong did they?

Hyperphysics says

From the definition of voltage as the energy per unit charge, one might expect that the energy stored on this ideal capacitor would be just QV. That is, all the work done on the charge in moving it from one plate to the other would appear as energy stored. But in fact, the expression above shows that just half of that work appears as energy stored in the capacitor.

Yes, that is very misleading, because all the work done on pushing a charge into a capacitor DOES (or, at least, in principle does, notwithstanding fractional percentage inefficiencies) appear as stored energy. I am not sure it is what the writer intended to say, but on the basis that it can be interpreted as you've written it then, yes, it is wrong.

Just think for yourself for a moment and think of the example I gave of masses on a series of steps. When you charge up a capacitor, it does not START at the voltage you are aiming to charge it up to, so the charge for, say, the first dV is q.dV. But once it is already charged to V', then the energy for the next dV above that is q.[V'+dV]. You can't just take the voltage of the finally charged capacitor and say 'ah, now I multiply ALL of the charge I put in by the final voltage'! How did the battery know to push the first few electrons up by the voltage of the finally charged capacitor!?

Think for yourself! Work it out. You have to multiply the charge by the AVERAGE voltage during the charging period, and as that is from 0 to V so the total energy is the AVERAGE voltage during capactor charging, times the total final charge, so it is Q.[V/2]

 

[Dadface]

Hyperphysics says

Yes, that is very misleading, because all the work done on pushing a charge into a capacitor DOES (or, at least, in principle does, notwithstanding fractional percentage inefficiencies) appear as stored energy. I am not sure it is what the writer intended to say, but on the basis that it can be interpreted as you've written it then, yes, it is wrong.

The quote below comes from the textbook:

Nelkon & Parker Advanced Level Physics Fourth Edition

"The energy comes from the battery.This supplies an amount of energy equal to QV during the charging process.Half of this energy goes to the capacitor.The other half is transferred to heat in the circuit resistance.If it is a high resistance the transfer is made quickly;if it is a low resistance the transfer is made slowly.In both cases,however,the total amount of heat produced is the same,0.5QV"


In your post above you seem to have forgotten that work is done not just to charge the capacitor but also to drive the charging current through the circuit resistance.The energy losses due to heating are significant and cannot be ignored.

 

[Ken G]

The problem is actually the false analogy between springs, and capacitors attached to a battery. A battery may be assumed to always maintain the voltage V-- note that if we had a different kind of battery that gradually ramped up to V, we could charge the capacitor to V gradually, losing no energy to dissipation, and getting the full QV/2 stored energy. But if the voltage is at V the whole time, then no matter how small the initial resistance, an energy QV must appear somewhere, as that is the work done by a battery that is always at V. So Dadface is partially correct, but the logic is skewed-- it's not that charging capacitors always wastes 50% of the energy, it is that capacitors always store QV/2 energy (that's the integral of V(Q)dQ ), so if you charge them in a way that does more work than that, heat will have to get made.

The same could be said of springs-- they store kx²/2 because that's the integral of kx dx, so if you do more work than that, you'll have to make heat. But usually you don't do more work than that-- you usually balance the spring force while you stretch it, analogous to a battery slowly reaching V. It all depends on the context, there is no law that says conservative systems are only 50% efficient in general.