# The weight of flywheel

by jartsa
Tags: flywheel, weight
 P: 440 A spinning flywheel is lowered into a gravity well using an elevator. Energy flows from the flywheel into the elevator motor (which functions as a generator in this case). The energy is red shifted by gravity, and equivalently by centrifugal force. To be more specific the force part of work = force * distance is reduced. So rotational energy always weighs less than other kinds of energies. Any objections?
Mentor
P: 10,809
What do you mean by "Energy is red shifted by gravity"? Or "equivalently by centrifugal force"?
Photons can be red-shifted. Energy as a general concept does not have any frequency which could be shifted.

 So rotational energy always weighs less than other kinds of energies.
How is that related to the other parts of your post, and why do you think this would be true?

 Any objections?
I don't see anything which would make sense in physics, therefore: No.
 P: 440 Let us consider two identical light bulbs in a spinning carousel, one at the middle, the other near the rim, powered by a battery at the rim of the carousel. Obviously the inner light bulb emits less energy. We have here a centrifugal pump that decreases the energy of the energy that flows in the "wrong" direction through it. Now I postulate the general red- or blue shift of energy: A photon that climbs up from a gravity well loses energy. Any energy that climbs up from a gravity well loses energy. And that is what I mean by red shift of energy.
Mentor
P: 10,809

## The weight of flywheel

Your carousel is not a gravity well.
The light bulb near the rim, as seen from the center, will emit less photons per time (due to time dilation), but these have a higher average energy (due to the movement of the light bulb). While the second effect is a conversion of rotational energy to light, time dilation will give you a longer lifetime of the battery (again as seen from the center).

I can use 1eV to store an electron 1eV away from some surface, bring this construction to a different height and get this 1eV back. Of course, the work I need for lifting the system is increased if I do this charging.
 P: 440 Let us consider the energy and the angular momentum of the flywheel. We almost agree here that a free falling object does not gain energy: http://www.physicsforums.com/showthread.php?t=588937 If energy stays the same in free fall, then using an elevator should cause a decrease of energy. The mass-energy of our flywheel in the elevator decreases, while its angular momentum stays the same. An observer in the elevator will say that the rotational energy of the flywheel increases, when the elevator moves down.
Emeritus
P: 7,437
Unfortunately, angular momentum in GR is rather tricky and technical. The good news is one can find treatments of the topic, for instance Jeffrey Wincour's paper "Angular Momentum in General Relativity", published in Held, "General Relativity and Gravitation, one hundred years after the birth of Albert Einstein".

The bad news is that it's not particularly clear to me, even after reading this paper, how to handle the simple physical experiment of lowering a gyroscope into a gravity well.

For asymptotically flat space-times, angular momentum can be defined at spatial infinity and/or null infinity, but apparently it isn't even clear in general how to be able to compare the two results. I.e, from the previously cited reference:

 Quote by Held Pulsars, rotating black holes, binary stars, quasars, and galactic centers are typical of the physical systems for which angular momentum provides a useful tool of analysis and for which relativistic effects might be expected. Fortunately, it is an extremely good approximation to treat such systems as if they were completely isolated from the rest of the universe and immersed in their own asymptotically flat space-time. As described herein, the total angular momentum can then be formulated as a two-dimensional integral over a sphere at infinity analogous to the Gaussian integrals for charge in electromagnetic theory. However, there are complications that can effect either the convergence or the uniqueness of this integral. One important choice in this construction is between the asymptotic limits of null infinity or spatial infinity. In fact, there is not even any clear procedure for checking the agreement between these two choices of limit, except in special cases. In one such case, rotational symmetry, there is a unique procedure for constructing the component of angular momentum about the symmetry axis.
Perhaps some other poster has some insight or some other paper which might handle the interesting issue of lowering a gyroscope into a gravity well - at this point, I can mostly say that the OP's treatment is unlikely to be meaningful, and that the problem appears to be rather difficult.

Using quasi-Newtonian definitions of angular momentum is more or less meaningless, unless one can compare them to more rigorous notions. And it appears there are some definitoinal issues here regarding how to take the appropriate limit..
 Physics Sci Advisor PF Gold P: 5,507 [Edit: deleted post, got the analysis backwards.]
Physics
PF Gold
P: 5,507
 Quote by jartsa The mass-energy of our flywheel in the elevator decreases, while its angular momentum stays the same.
Not if energy is being transferred out of the flywheel to the elevator motor, as the OP specifies. That means the flywheel is spinning more slowly, so its angular momentum decreases.
P: 3,967
 Quote by pervect The bad news is that it's not particularly clear to me, even after reading this paper, how to handle the simple physical experiment of lowering a gyroscope into a gravity well.
You are right that this is the place to start. We need to understand the simplest scenario first before we have any hope of understanding the more complex situation expressed in the OP.

I would suggest that an efficient flywheel effectively acts as simple primitive clock so that as we lower it into a gravity well in slows down relative to coordinate time but appears to rotate at the same speed relative to a local clock.

In a slightly more elaborate setup imagine we have a massive body that is rotating sufficiently slowly that we do not need Kerr coordinates and approximates to SC coordinates. We lower a flywheel and a comoving observer slowly towards a pole. The comoving observer would not notice any change in the speed of the flywheel, but as he got nearer the pole the rotation speed of the massive rotating body would appear to be speeding up relative to his clocks and flywheel. Does that sound reasonable?

It might be helpful to note that the angular momentum per unit rest mass (L) of a particle spiralling in towards a black hole remains constant when measured in terms of proper time. i.e.

$$L = r^2 \frac{d\theta}{d\tau}$$

Similarly the energy per unit mass (E) remains constant for an in spiralling particle in terms of proper time with the relation:

$$E^2 = \left(1-\frac{2m}{r}\right) \left(1-\frac{L^2}{r^2}\right) \left(\frac{dr}{d\tau}\right)^2$$

Ref http://www.fourmilab.ch/gravitation/orbits/

I believe the above formulas are somehow obtained from the Lagrangian of the test particle, but I do not understand that myself.

Hope that helps.
 P: 440 Let's say a spinning flywheel with a quite large diameter lies flat on the floor of a descending elevator. Let's say there is a brake where the braking energy goes, when the elevator moves down. The brake happens to be a thing with a quite small diameter, somewhere near the flywheel axis. I'm quite sure there is a change of spinning rate in this shape changing system. (Mass-energy flows into the brake from somewhere else) Most probably the change is the increase of the spinning rate of the flywheel, according to all observers. As this flywheel behaves in this odd way, it is quite unlikely that it would exert a force on a scale in a normal way. Sorry guys for being all Newtonian. But it seems simple: shape change -> other changes
P: 3,967
 Quote by jartsa Let's say a spinning flywheel with a quite large diameter lies flat on the floor of a descending elevator. Let's say there is a brake where the braking energy goes, when the elevator moves down. The brake happens to be a thing with a quite small diameter, somewhere near the flywheel axis. I'm quite sure there is a change of spinning rate in this shape changing system. (Mass-energy flows into the brake from somewhere else) Most probably the change is the increase of the spinning rate of the flywheel, according to all observers. As this flywheel behaves in this odd way, it is quite unlikely that it would exert a force on a scale in a normal way. Sorry guys for being all Newtonian. But it seems simple: shape change -> other changes
Could you describe this experiment in more detail? What is changing shape? If the brake now a friction type like the brakes on a car or are you using an electric generator for the brake? In the OP you mentioned an elevator motor but did not mention if it is the elevator or remains at the top powering a pulley and cable system. You also mentioned work = force * distance without mentioning what distance you had in mind. You assume that the force part of the equation is reduced, but if the distance you have in mind is the distance the elevator drops the the coordinate distance is less than the ruler distance and so the force does not necessarily change.

Are you suggesting a scheme where the kinetic energy of the falling elevator and flywheel system is used to accelerate the flywheel via a gearing system to bring the elevator to rest lower down? Clearly in such an artificial case, the flywheel will speed up as observed by a local observer even if the flywheel is not initially rotating. This is just a conversion of potential energy to kinetic energy of the falling elevator to rotation energy of the flywheel and there is nothing unusual about that that. Alternatively are you talking about using the rotation energy initially stored in the spinning flywheel to bring the falling elevator to rest?

Your statement in the OP that ".. rotational energy always weighs less than other kinds of energies." is almost certainly wrong.

Here is what happens to a free spinning flywheel in a lowered elevator in the simplest case, without any motors, gears, brakes or pulley etc attached to the flywheel. If the elevator is brought to rest at a lower altitude, the speed of the flywheel will appear the same as when it was higher up according to an observer in the elevator. To an observer that remains at the top, the flywheel will appear to have slowed down by the gravitational redshift factor. The top observer also calculates that the inertial mass of the flywheel has increased by the same factor and the angular momentum of the flywheel is unchanged. The top observer also measures the stored energy of the flywheel to be less lower down, while the observer inside the elevator does not see any change in the stored energy of the flywheel. If the flywheel is resting on some weight scales, then when the elevator is at rest its weight is GMm/r^2*gamma where gamma = 1/sqrt(1-2GM/(rc^2)) which increases faster than the Newtonian expectation, as r gets smaller. No sign of our flywheel weighing less here.
P: 440
 Quote by yuiop Could you describe this experiment in more detail? What is changing shape? If the brake now a friction type like the brakes on a car or are you using an electric generator for the brake? In the OP you mentioned an elevator motor but did not mention if it is the elevator or remains at the top powering a pulley and cable system. You also mentioned work = force * distance without mentioning what distance you had in mind. You assume that the force part of the equation is reduced, but if the distance you have in mind is the distance the elevator drops the the coordinate distance is less than the ruler distance and so the force does not necessarily change. Are you suggesting a scheme where the kinetic energy of the falling elevator and flywheel system is used to accelerate the flywheel via a gearing system to bring the elevator to rest lower down? Clearly in such an artificial case, the flywheel will speed up as observed by a local observer even if the flywheel is not initially rotating. This is just a conversion of potential energy to kinetic energy of the falling elevator to rotation energy of the flywheel and there is nothing unusual about that that. Alternatively are you talking about using the rotation energy initially stored in the spinning flywheel to bring the falling elevator to rest? Your statement in the OP that ".. rotational energy always weighs less than other kinds of energies." is almost certainly wrong. Here is what happens to a free spinning flywheel in a lowered elevator in the simplest case, without any motors, gears, brakes or pulley etc attached to the flywheel. If the elevator is brought to rest at a lower altitude, the speed of the flywheel will appear the same as when it was higher up according to an observer in the elevator. To an observer that remains at the top, the flywheel will appear to have slowed down by the gravitational redshift factor. The top observer also calculates that the inertial mass of the flywheel has increased by the same factor and the angular momentum of the flywheel is unchanged. The top observer also measures the stored energy of the flywheel to be less lower down, while the observer inside the elevator does not see any change in the stored energy of the flywheel. If the flywheel is resting on some weight scales, then when the elevator is at rest its weight is GMm/r^2*gamma where gamma = 1/sqrt(1-2GM/(rc^2)) which increases faster than the Newtonian expectation, as r gets smaller. No sign of our flywheel weighing less here.

Let's forget the OP, for a while.

Let's say we build a system of water pipes, where water from the earth's equator area is pumped to the polar region, where the water cools and then returns back to the equator.

Running this system will cause a speeding up of earth's rotation.

Let's say exactly at the north pole there is a iceberg. We extract some gravitational potential energy from this iceberg. If we store the energy at the equator, then the spinning of the earth will slow down. If we store the energy at the pole in a store in which the shape of the energy is such that the rotational inertia of the energy is smaller than it was before we started this energy extraction, then the spinning of the earth will speed up.

Shape of energy, that's funny. If we convert some wind energy, which has a shape of a hollow sphere, into geothermal energy which has a shape of solid ball, the the spinning of the earth will speed up.

EDIT:
It would be good and appropriate if this had something to do with previous discussion. So I add: When storing the energy that we extracted from the iceberg, it does not matter if we store the energy high above the north pole or deep below the north pole. Also the type of the energy doesn't matter.

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