15

Imagine a bar

spinning like a helicopter propeller,

At $\omega$ rad/s because the extremes of the bar goes at speed

$$V = \omega * r$$

then we can reach near $c$ (speed of light) applying some finite amount of energy just doing

$$\omega = V / r$$

The bar should be long, low density, strong to minimize the amount of energy needed

For example a $2000\,\mathrm{m}$ bar

$$\omega = 300 000 \frac{\mathrm{rad}}{\mathrm{s}} = 2864789\,\mathrm{rpm}$$

(a dental drill can commonly rotate at $400000\,\mathrm{rpm}$)

$V$ (with dental drill) = 14% of speed of light.

Then I say this experiment can be really made and bar extremes could approach $c$.

What do you say?

EDIT:

Our planet is orbiting at sun and it's orbiting milky way, and who knows what else, then any Earth point have a speed of 500 km/s or more agains CMB.

I wonder if we are orbiting something at that speed then there would be detectable relativist effect in different direction of measurements, simply extending a long bar or any directional mass in different galactic directions we should measure mass change due to relativity, simply because $V = \omega * r$

What do you think?

Qmechanic
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6 Answers6

31

Imagine a rock on a rope. As you rotate the rope faster and faster, you need to pull stronger and stronger to provide centripetal force that keeps the stone on the orbit. The increasing tension in the rope would eventually break the it. The very same thing would happen with bar (just replace the rock with the bar's center of mass). And naturally, all of this would happen at speeds far below the speed of light.

Even if you imagined that there exists a material that could sustain the tension at relativistic speeds you'd need to take into account that signal can't travel faster than at the speed of light. This means that the bar can't be rigid. It would bend and the far end would trail around. So it's hard to even talk about rotation at these speeds. One thing that is certain is that strange things would happen. But to describe this fully you'd need a relativistic model of solid matter.

People often propose arguments similar to yours to show Special Relativity fails. In reality what fails is our intuition about materials, which is completely classical.

Marek
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16

There is a real object with relativistic speed of surface - millisecond pulsar. The swiftest spinning pulsar currently known, spinning 716 times a second. Surface speed of such pulsar with radius 16 km is about $7*10^7$ m/s or 24% speed of light.

It is calculated that pulsars would break apart if they spun at a rate of more than 1500 rotations per second.

voix
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6

In your calculations you assume that your propeller is a rigid body.

You cannot use that assumption, when your speeds are not much smaller than the speed of light. Because "there are no rigid bodies in relativity".

Kostya
  • 20,288
4

remember that in a three-dimensional description of special relativity the impulse of an object is given by

$$\mathbf{p} = \gamma m \mathbf{v}$$ with the so-called Lorentz-factor $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

Now, do you think you can accelerate the masses within the slab to a speed greater than light or do you think that something is wrong with your physical model of the system?
Sincerely

Robert

Robert Filter
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4

I say no. Assuming all the practicalities work, you can get arbitrarily close to c. But not reach c. You can see this easily from the relativstic formula for kinetic energy:

$E_k = mc^2(\frac{1}{\sqrt{1-v^2/c^2}}-1)$

As $v$ approaches $c$, the energy you need to supply to a particle at the end of the bar tends to infinity.

Mateen Ulhaq
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TROLLHUNTER
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3

Dear Hernan, as the distant parts of the bar are approaching the speed of light, they become heavier, so it becomes harder to accelerate them: you can never reach (or surpass) the speed of light. It doesn't matter whether you try to accelerate the "final segments" of the bar by jets or by their attachment to the rest of the bar that is being pushed in the middle: the speed of light can never be reached.

If you want to speak in terms of torques and moments of inertia (of the bar), the moment of inertia goes to infinity - much like the mass itself - when the velocity of some points on the bar approaches the speed of light. So much like you have to modify the formulae for masses of moving objects by relativistic effects, you need to modify the formulae for the moments of inertia.

So your statement that you need a finite energy to get to the speed of light is invalid. You would need an infinite energy. For speeds below the speed of light, the total energy that you need can simply be calculated as the sum of the kinetic energies of all the segments of the bar.

Luboš Motl
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