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So, the linear velocity of a body is limited to speed of light (some may argue its not a "limit", but...). Are there any such limits for angular velocity of a body? The thing that bothers me is that tangential velocity of an outer rim of of wheel is $\omega \times r$. This means if you can rotate an arm of size 100m at 300K rad/s and you can achieve speed of light on the tangent. Does that put limit how fast you can rotate an object?

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The answer to your question is yes, it puts an upper bound on how fast you can rotate an object.

Now, I suppose someone's next questions is likely to be "well, what's the fastest angular speed at which an object can rotate?" The answer to this is that it depends on the size of the object. According to the most basic classical/relativistic analysis, no point on the surface of a rotating object can move tangentially faster than the speed of light. In other words, the fastest angular speed of a rotating object is:

$$\frac{c}{R} = \omega$$

While we're on this subject, this very argument was the one of the first clues that the electron's "spin" was not in virtue of any literal spinning.

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  • $\begingroup$ At the risk of stating the obvious, it's perhaps worth adding that if the reference point/axis is not taken through the centre-of-mass of the body, then by taking the reference point/axis far away the angular momentum can be made arbitrarily large. $\endgroup$ – diracula Jun 26 '17 at 8:18
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There appear to be no restrictions on the angular velocity of rotation, if we allow the object to contract freely. For simplicity, we can consider a hollow coil (or ring) with a wire wrapped around it. Imagine that we begin to pull the end of the wire, and the wire in the reference frame of the coil moves at a speed close to the speed of light. The wire Lorentz – contracts then.

It would be rather strange to assume that the unwound wire undergoes relativistic shrinkage, but the wire on the coil does not. Thus, the rotating ring undergoes relativistic shrinkage as the speed of rotation increases. Thus, the ring can be rotated with an infinitely large angular velocity.

Kinematics of rotating wheel that described in the Relativistic Trolley Paradox (reference below) demonstrates that a wheel rolling on the rail, if allowed to contract freely, should reduce in diameter and the angular velocity of rotation of the wheel will tend to an infinitely large value. Otherwise, this leads to a contradiction.

If, due to some reasons, we do not allow the spool to contract freely, then proper length of the rim must increase and angular velocity will be limited.

It should be noted, that if a wire on the spool and the spool are toothed (for example, like a bicycle chain on a gearwheel), only Lorentz – contracted gearwheel will allow the chain to run without gritting.

Related reference: https://oda.hioa.no/nb/a-relativistic-trolley-paradox

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