Very Fast Flywheel

[losfomot]

Say we were to make a 10 meter (arbitrary number) diameter flywheel in a vacuum and, with a series of gears, constantly increase the rpm of that wheel. There will be a huge difference in speed between the hub and the outer edge of the wheel...

 

Relativistically, what would happen to the wheel?

 

(apart from flying apart from the force of the spin.... let's assume it is made from a VERY strong material)

[Fuzzwood]

Erm, it spins?

[insane_alien]

well, when the perimiter is approaching the speed of light the torque required to keep constant acceleration would increase.

 

i suppose there would be a limit for its rotational velocity.

[Klaynos]

It's non-inertia so there'd probably be some GR effects...

[insane_alien]

It's non-inertia so there'd probably be some GR effects...

 

in laymans: "Damn thats f****d up"

[losfomot]

The perimeter is moving near the speed of light, but the hub is not anywhere near that speed...

 

wouldn't time dilation and length contraction come into play? More so the closer to the perimeter?

[swansont]

IIRC it was this kind of issue that led Einstein to realize that in this accelerating system, the coordinate system would have to be curved, owing to the effect of tangential length contraction.

[Sisyphus]

I have a hard time picturing what one would actually see in this situation, say, from the hub looking out or the rim looking in.

[alan2here]

Wouldn't the atomic bonds between the atoms pritty much hold everything together? The only odd thing would be that if it was really big then it could only rotate slowly even if it was really light (as in not much mass)? I may be wrong.

[Norman Albers]

People are currently building high-kinetic energy storage flywheels, perhaps to 10,000 rpm, for vehicle or fixed energy storage. Yes they fly apart from the extreme centrifugal forces so folks are working out the engineering here. When you get to a significant fraction of the SOL yes, things really get out of hand.

[losfomot]

People are currently building high-kinetic energy storage flywheels, perhaps to 10,000 rpm, for vehicle or fixed energy storage. Yes they fly apart from the extreme centrifugal forces so folks are working out the engineering here.

 

10,000 rpm does not sound particularly fast... how big are they?

 

When you get to a significant fraction of the SOL yes, things really get out of hand.

 

Do you mean just because of the centrifugal forces, or a relativistic effect? Can you elaborate on that?

[Norman Albers]

losfomot, you calculate the centrifugal force on a kilogram of even SPIDERMAN SILK at the radius and speed we are talking of. To get us started in the relativistic discussion, the differential expression in coordinates riding along with the rotation is: [math]ds^2=(1-\frac{\omega^2 r^2)}{c^2}) c^2dt^2 - [dr^2+(1-\frac{\omega^2r^2}{c^2})^{-1}r^2d\phi^2 + dz^2] [/math].

[losfomot]

at the radius and speed we are talking of.

 

Ummm... what is the radius we are talking about? That was kinda my question. Are you referring to my original post which brought up a 10 meter diameter? Or are you assuming that I already know the radius of the real-world flywheels that you just brought up?

 

In all honesty, I am not interested in the centrifugal forces acting on the flywheel, I am more interested in the relativistic effects. I am sure that, as you get into relativistic speeds, the centrifugal forces become extremely large. But let's assume that the flywheel is made of nanotubes or lets even skip ahead to a material that is strong enough to withstand any centrifugal force that we might reach for the duration of this thread.

 

To get us started in the relativistic discussion, the differential expression in coordinates riding along with the rotation is: [math]ds^2=(1-\frac{\omega^2 r^2)}{c^2}) c^2dt^2 - [dr^2+(1-\frac{\omega^2r^2}{c^2})^{-1}r^2d\phi^2 + dz^2] [/math].

 

 

 

I hope we don't have to use that equation.

 

I thought we should be able to simply take the length of the perimeter along with the rpm, and we will have the speed at which the rim of our flywheel is traveling.

 

For example... using your 10,000rpm with my original 10 meter diameter flywheel, the rim would be only moving at 5236 m/s or 1.746% of the speed of light.

 

What I am really interested in is what happens when the rim travels closer to the speed of light? What would time dilation and length contraction do to the flywheel? As Sisyphus put it... what would you see from the hub looking out or the rim looking in? What would you see as an outside observer watching the flywheel turn? (aside from a blur) Would the fly wheel hold together? (again, disregarding centrifugal (or centripetal) forces)

 

For example, from the point of view of the rim, I would imagine that the diameter of the flywheel would have to be significantly smaller than if measured from the point of view of the hub. If I am right, are these just differences in measurement or would these effects have a physical impact on the flywheel?

[Norman Albers]

If you look at the metric expression, it tells you that radial changes are the same; there is no coefficient in front of the [math]dr^2[/math] term. On the other hand, there is time dilation and Lorentz contraction of the circumference. Also the first term implies relativistic mass increase. I guess it's sort of the opposite of the Schwarzschild scenario, where as you near a horizon the circumference is less than [math]2\pi r[/math]. Here there is similar behavior but applying to the circumferential measures, so the result is opposite. In the example of 10,000 rpm, I figure an acceleration of one million meters/sec/sec, or a hundred thousand g's for each meter of radius in the nonrelativistic regime. I've not thought enough here yet to describe the cinema. You have created another ultimate challenge to SPIDERMAN....Thwwppp.

[someguy]

that's a good question, when i think of this scenario, I can only imagine that the center would just tear out at some point, if we assume no matter how distorted it gets we could continue to spin it faster.

 

you could even just think of an extra long super light rod, one end on a spindle and you at the other end. if it spins fast enough, just the extra mass of the center part of the rod would cause, at some point, your rod to bend, assuming you are continuously accelerating it.

[losfomot]

I guess it's sort of the opposite of the Schwarzschild scenario, where as you near a horizon the circumference is less than [math]2\pi r[/math]. Here there is similar behavior but applying to the circumferential measures, so the result is opposite.

 

The result is opposite? I thought it would be the same, only your looking at it from the opposite direction. With a black hole your approaching the horizon from the outside... With our flywheel, your approaching the horizon (the rim) from the inside... in both cases, the circumference of the flywheel (or event horizon) gets smaller, the closer you get to the rim.

 

In the example of 10,000 rpm, I figure an acceleration of one million meters/sec/sec, or a hundred thousand g's for each meter of radius in the nonrelativistic regime.

 

Assuming you are correct, a 300 meter radius would give an acceleration (at the rim) equal to the speed of light (sort of a reverse black hole). And the rim would have a rotational speed of only (about) 10% of the speed of light.

 

Hmmm... We are going to need lower rpms and a bigger flywheel in order to get the rim moving faster, because no amount of spiderman silk or nanotubes or anything else will hold up to that kind of centrifugal force.

[Norman Albers]

I'm so sorry I wasted my time. The physics which demands to be seen is just relativity: an energy density which is convinced to circulate is indistinguishable from mass. The relativistic mass of the DOOMSDAY WHEEL increases arbitrarily, depending upon your energy input budget. (I keep searching for my thread on the Thermonuclear Box of Manure.) Too bad if you don't know what I mean. Too much has been erased.

[losfomot]

The physics which demands to be seen is just relativity

 

Isn't that what we were talking about?

 

an energy density which is convinced to circulate is indistinguishable from mass.

 

I don't know why it has to circulate, but yes, mass is just a stable form of very dense energy... I agree, others probably don't.

 

The relativistic mass of the DOOMSDAY WHEEL increases arbitrarily, depending upon your energy input budget. (I keep searching for my thread on the Thermonuclear Box of Manure.) Too bad if you don't know what I mean. Too much has been erased.

 

Yes, an outside observer would see the wheel increasing in mass with an increase in speed... moreso closer to the rim of the flywheel. This is how the wheel stores energy to compensate for the speed of light being constant in all frames.

 

(Now, if you were to throw an indestructible stick into the spokes of our relativistically fast, mass-increased flywheel... stopping it all at once, I would have to agree with the term DOOMSDAY WHEEL)

 

I'm so sorry I wasted my time.

 

What is so disappointing? What were you expecting?

[Spyman]

The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.

 

In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. But the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.

 

[Note that a cylinder was considered in order to circumvent the possibility of a disc "dishing" out of its plane of rotation and trivially satisfying C<2πR. Subsequently when a rotating disc is substituted it is assumed that this distortion possibility is also excluded.]

 

The paradox has been deepened further by later reasoning that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR.

 

The Ehrenfest paradox may be the most basic phenomenon in relativity that has a long history marked by controversy and which still gets different interpretations published in peer-reviewed journals.

http://en.wikipedia.org/wiki/Ehrenfest_paradox

[alan2here]

losfomot,you calculate the centrifugal force on a kilogram of even SPIDERMAN SILK at the radius and speed we are talking of.

 

can I recommend arogal, is is less dense that spiderman silk

 

more would fit around the circumference, which would thus measure greater than 2πR.

 

Are you saying the edge would get bigger?

[Spyman]

Are you saying the edge would get bigger?

It is a paradox, the circumference gets both larger and smaller, but the radius is unchanged.

(2πR does no longer hold true.)

[alan2here]

I'm having difficulty imagining how this can be, like it can be thought of as being bigger and smaller like a photon can be thought of as a particle and a wave? Are you talking about relative to an observer on the disk and an observer outside the disk?

 

Ah you mean that the paradox is theoretical and probably created by broken maths?

[Spyman]

I'm having difficulty imagining how this can be

Lorentz length contraction is a real physical effect.

 

I think the solution is in the curving of spacetime:

 

In the frame of a observer outside the disk, spacetime is warped by speed and no longer flat.

 

The disc is bendt into the shape of a cone, but without changing size in the three spatial dimensions.

(It still has the same height and radius, (width & length), but the circumference changes.)

 

Since we only can view 3D it still looks like a disc, but we can measure if C=2πR or not.

 

I am sorry, but I don't have enough knowledge of GR to explain it better/properly for you...

[alan2here]

lol, ty for trying

[Country Boy]

its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. But the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.

The first part of this is true- since the radius is perpendicular to the direction of motion, the radius will not change. But you do NOT gety a contradiction" that R= r and R< r. What happens is that C= 2 pi r is no longer true- the geometry is no longer Euclidean.

 

[Note that a cylinder was considered in order to circumvent the possibility of a disc "dishing" out of its plane of rotation and trivially satisfying C<2πR. Subsequently when a rotating disc is substituted it is assumed that this distortion possibility is also excluded.]

 

The paradox has been deepened further by later reasoning that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR.

Absolutely not. If you mean measuring rods moving with the circumference of the disk, from the point of view of a person in the "lab frame", in which the disk is spinning, both circumference and measuring rods, moving at the same rate, will be reduced by same amount and so the same number of rods will cover the circumference. From the point of view of a person ON the circumference of the disk, the neither rods nor circumference will have changed. From the point of view of a person on the disk but closer to the center, both circumference and rods will have reduced by the same amount but less than observed from the laboratory frame- he will see a non-Euclidean geometry but closer to Euclidean geometry than the laboratory frame. Of course all that change in "geometry" is because rotation is an acceleration, producing "fictitious forces".