Relativistic angular velocity

  • #26
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The contracted length of the wire is equal to ##2\pi r##, yes. Thus when it slows down the wire will be longer than the rim of the disc and will fall off. This is true even assuming (unrealistically) that the disc does not undergo elastic expansion when spinning.

If you imagine making many radial cuts into the disc (so it looks like a jet turbine), you would find that these cuts open up as the disc accelerates, since each blade of the "turbine" would length contract in the tangential direction. If you don't make cuts, the whole rim of the disc must "want" to length contract - but is unable to do so due to the symmetry of the situation. So you will find elastic stresses in the tangential direction.

Presumably a spinning hoop would try to decrease its radius due to Lorentz contraction (massively overwhelmed by elastic expansion and outright disintegration in practice, I should imagine).
It is very interesting! Does physical realization of the disk affects its behavior?

The spool can be made in the form of a ring without an axle and spokes. Will the wire stretch after stop and the rim will not?

What if the disc was winding a wire for many years and is now actually a spiral of Archimedes? What will happen after the stop? Which layers will separate from the axle and which will not?

Let’s assume that the spool does not contract in radial direction and the very long wire has been wound around the spool. The spool moves with velocity ##v=R\Omega## in the frame of unwound wire.

In the frame of the spool the rim of the spool cannot move faster than ##c##, the angular velocity ##\Omega## of the spool cannot be higher that ##c/R##, if linear velocity of the rim approaches ##c##

In the frame of the unwound wire, as speed of the spool approaches ##c##, rotation should generally cease due to time dilation.

But the wire cannot be whole in one frame of reference and torn in another.
 
  • #27
Ibix
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It is very interesting! Does physical realization of the disk affects its behavior?
I've been assuming throughout that the disc is sufficiently stiff that it doesn't expand under centrifugal force. Any actual material will disintegrate long before relativistic effects become the slightest bit significant, so the whole conversation is somewhat unrealistic.

A thin ring of some material with enough tensile strength for elastic effects to be negligible beside relativistic effects would have to decrease in radius when spun up, because the material would length contract and internal stresses would cause it to shrink. A disc can't do this, of course, because the material inside any elementary ring gets in the way. So the material becomes stressed over and above the obvious centripetal forces. Exactly what happens depends on the details of the material, I think.
What if the disc was winding a wire for many years and is now actually a spiral of Archimedes? What will happen after the stop? Which layers will separate from the axle and which will not?
All of the wire is moving at relativistic speed and is not under any tension as it is fed in. So it acts like a thin-walled cylinder (i.e., the thin ring I described above) and expands when it is slowed (assuming it's made of this same ridiculously strong material and can somehow be persuaded to bend around a spool). What happens to the spool depends on whether it was another thin-walled cylinder (that shrank when accelerated and expands when it decelerates) or a solid cylinder (that couldn't shrink so doesn't expand). But either all of the wire falls off or none of it does.

I should add that I haven't finished reading Gron's paper, linked by Laurie K above, and if he contradicts me he's probably the one who's right.
 

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