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I've seen ideas and previously asked questions for reaching the speed of light through a long bar rotating at some high RPM for the tip to reach $c$. However it seems impossible, so I'm concerned is it impossible still to reach like $c/2$ and if not what fraction through this method is realistic?

Qmechanic
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identiconnnn
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    It says you are a new contributor, so where are your previously asked questions ? This question is unclear, you can accelerate electrons in a betatron to c/2 – trula Sep 17 '23 at 10:11
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    As in the rotating bar example, a centrifuge is limited by the speed of sound. I have some related info here: https://physics.stackexchange.com/a/686011/123208 – PM 2Ring Sep 17 '23 at 11:54
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    To be clear, it isn't possible to spin a centrifuge with a mechanical arm faster than sound. But particle accelerators work differently. A charged particle moving in a uniform magnetic field travels in a circle. Using this, the LHC spins particles at $> 99.9999$% c. See Accelerator Science: Circular vs. Linear for more. – mmesser314 Sep 17 '23 at 15:05
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    To be clear, faster than the speed of sound in the material that is spinning. Supersonic centrifuges going faster than the speed of sound in air are just fine. – Jon Custer Sep 17 '23 at 16:03
  • I'm sorry by what I meant by a centrifuge, I mean at least 100 km pipe in space spinning at 10,000 RPM with a lightweight solar sail attached to it at the end to be released tangentially into the void for space exploration, would this be possible? How shorter would the pipe have to be and what kind of speed are we looking at here? Thanks for your time. – identiconnnn Sep 18 '23 at 10:13
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    Could someone please explain how any of the answers to the claimed duplicate "How is length contraction on rigid bodies possible in special relativity since definition of rigid body states they are not deformable?" are relevant to this question. I'd understand if this question was closed because of @PM-2Ring's nice answer to "Does a spinning disk weigh more than when it is stationary?", but I don't understand how any of the answers to the length contraction question answer this question. – David Bailey Sep 19 '23 at 13:24
  • @identiconnnn A pipe that long isn't rigid. Consider: a guitar string is made of steel. Your 100 km pipe is going to be floppier than a strand of spider web. The speed of light is roughly a million times the speed of sound in air. The speed of sound in solids is faster than in air, but you can only reach a tiny fraction of c by spinning some kind of arm around. – PM 2Ring Sep 19 '23 at 15:16
  • “ I've seen ideas and previously asked questions for reaching the speed of light through a long bar rotating at some high RPM for the tip to reach c”. Where have you seen this? Which questions? – ZeroTheHero Sep 20 '23 at 21:52
  • @trula I think he means he's seen questions that have been "previously-asked", not that he's been the one who asked them – Señor O Sep 24 '23 at 18:55

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The real answer to your question is to direct you to the concept of specific strength which is the concept that is directly applicable to your question. This is the limit where a bar (or strand) of material will break under its own mass. It matters little how lightweight your payload is (even just a sail), because it's the bar's own limits that matter.

Many commenters mentioned that the limit for this is when the tangential velocity exceeds the speed of sound in the material. This was used to resolve the Ehnrenfest paradox in relativity. Long story short, when you approach a tangential speed equal to the speed of sound in that medium, you must also have centrifugal pressures that exceed the specific strength of the material and it rips itself to shreds.

So what's realistic? That's not really an easy question to answer, because we have to question what level of technology you consider "realistic." The most directly applicable answer would be to point to SpinLaunch, a space launch company whose entire business model is basically to implement your idea as best as can be done with modern technology. At the present, they reach 2.1 km/s. So if you want "realistic using modern technology," the state of the art is 0.000007c.

You are welcome to play games with the materials. SpinLaunch uses carbon fiber. I do not know what carbon fiber they use (it is likely proprietary), but Wikipedia's page pegs the best carbon fiber on their list at a specific strength of 7GPa. Its theorized that carbon nanotubes could get up to 300GPa, so in theory one could shave a few zeros off of of that -- almost 2 zeros to be precise.

There is an ultimate limit to specific strength, thanks to relativity. Interestingly enough, if you treat magnetic field lines as "tethers," they achieve this ultimate limit. So maybe the more realistic approach is to ditch the bar entirely, and just rely on electromagnetic attractions to do the trick.

I didn't do the math myself, but based on what I saw, I have suspicions that this ultimate limit would indeed be c, so there would be no fundamental force preventing any fraction of c using electromagnetic tethers. Of course there will be practical limits, but nothing fundamental.

Cort Ammon
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