Pole through Earth, faster than c?

[coktail]

Here's a question from a coworker:

Based on the idea that the tips of fan blades move faster the base relative to the center of the fan, if you were to somehow put a very, very (impossibly) long pole through the spinning Earth, could the tips of the pole reach c or faster given that the pole was long enough?

I know the answer is "no," and that it involves time dilation, length contraction, and possibly the relativity of simultaneity, but I wasn't sure how to explain it in this scenario.

This question could also be asked in more abstract terms, but my coworker likes the Earth/pole thing, so I went with that.

I'd really appreciate an explanation from the perspective of an observer on Earth measuring the relative velocity of the tip of the pole, as well as from the perspective of of an observer on the tip of the pole measuring the relative velocity of Earth.

As always, thank you!

 

[harrylin]

Here's a question from a coworker:

Based on the idea that the tips of fan blades move faster the base relative to the center of the fan, if you were to somehow put a very, very (impossibly) long pole through the spinning Earth, could the tips of the pole reach c or faster given that the pole was long enough?

I know the answer is "no," and that it involves time dilation, length contraction, and possibly the relativity of simultaneity, but I wasn't sure how to explain it in this scenario.

This question could also be asked in more abstract terms, but my coworker likes the Earth/pole thing, so I went with that.

I'd really appreciate an explanation from the perspective of an observer on Earth measuring the relative velocity of the tip of the pole, as well as from the perspective of of an observer on the tip of the pole measuring the relative velocity of Earth.

As always, thank you!

From the Earth, you would need to put in infinite energy to get the ends of that pole moving at c.

I'll leave to others (if anyone volunteers) to go through the motions to Lorentz transform it to one of the other frames... But note that the tip of the pole is only an infinitely short time at rest in an inertial frame.

 

[coktail]

Thank you, harrylin.

I think my coworker's idea was that Earth is already spinning, so no additional energy would be required to power the pole's movement.

As to your note, do you mean that the tip of the pole would be at rest relative to someone on Earth standing at the base of the pole and looking up at the tip? If so, I was thinking along those lines as well. It's similar to how the top of a skyscraper doesn't seem to be moving if you're standing at the base of it.

 

[HallsofIvy]

This is just a variation on the puzzle where you have a perfectly rigid "pole" several light years long, extending from Earth to a planet in another start system. Push the pole a little bit and the other end immediately moves so that by using, say, short and long pushes, and 'Morse Code', we could communicate much faster than the speed of light.

What that really tells us is that, assuming relativity, you cannot have "perfectly rigid" things, even in theory. Instead the "push" would move through the pole at the speed of sound in the material, which must be less than c.

Similarly, with a rotating pole, or fan, no part can move, locally, faster than the speed of light. If points near the center of rotation were moving at a large percentage of the speed of light, the pole (or fan, etc.) must "warp", bending backwards so that its speed is not larger than the speed of light.

 

[russ_watters]

No, it is a completely different problem than infinite speed communication:

The "coworker" is wrong about the lack of additional energy required: just like with a figure skater's arms, extending a pole from Earth would slow its rate of rotation without additional energy input.

So the issue is the energy requirement, not the rigidity. Even if then pole were perfectly rigid, the required input energy would still be infinite.

 

[coktail]

I like how you put "coworker" in quotes. It really is a coworker, I swear! But now I'm curious as well.

Why would the pole slow down Earth's rotation? There's no atmosphere in space to drag on the pole, so what force would act against it? Does it help if it's a super-light pole?

 

[harrylin]

Thank you, harrylin.

I think my coworker's idea was that Earth is already spinning, so no additional energy would be required to power the pole's movement.

Of course the Earth is already spinning; the problem is to get the pole spinning at c.

As to your note, do you mean that the tip of the pole would be at rest relative to someone on Earth standing at the base of the pole and looking up at the tip? If so, I was thinking along those lines as well. It's similar to how the top of a skyscraper doesn't seem to be moving if you're standing at the base of it.

No. I meant that your colleague will have to fabricate such a pole and try to get the ends moving fast. It's the ice skater pirouette thing inversed: extend your arms and you (as well as your arms) rotate slower. Conservation of energy, you know?

 

[coktail]

Of course the Earth is already spinning; the problem is to get the pole spinning at c.

How about if we build the pole segment by segment so it's always spinning with Earth as it gets built? Then we never need to get it spinning because it will be built on Earth while Earth is spinning. I imagine it would still slow down like someone extending their arms while spinning in the pirouette analogy, though I don't understand why that happens.

I'll look up the conservation of energy, but any help is always appreciated.

 

[coktail]

Found this here: http://physics.bu.edu/~duffy/py105/AngularMo.html

"Angular momentum is proportional to the moment of inertia, which depends on not just the mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation. This leads to some interesting effects, in terms of the conservation of angular momentum.

A good example is a spinning figure skater. Consider a figure skater who starts to spin with their arms extended. When the arms are pulled in close to the body, the skater spins faster because of conservation of angular momentum. Pulling the arms in close to the body lowers the moment of inertia of the skater, so the angular velocity must increase to keep the angular momentum constant."

 

[russ_watters]

You got it!

 

[harrylin]

Found this here: http://physics.bu.edu/~duffy/py105/AngularMo.html

"Angular momentum is proportional to the moment of inertia, which depends on not just the mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation. This leads to some interesting effects, in terms of the conservation of angular momentum.

A good example is a spinning figure skater. Consider a figure skater who starts to spin with their arms extended. When the arms are pulled in close to the body, the skater spins faster because of conservation of angular momentum. Pulling the arms in close to the body lowers the moment of inertia of the skater, so the angular velocity must increase to keep the angular momentum constant."

Yes indeed - it was somewhat a slip of the pen that I wrote energy as an ice skater basically conserves momentum. However I was really thinking energy. It is perhaps possible with a trick to maintain the kinetic energy so as to increase the momentum of the extending pole while inducing a counter momentum of the earth; fact is that the speed remains limited as you can't get free energy, as far as we know!

 

[PAllen]

Here is how to pull it off :

Obtain a supply of un-obtainium, which can remain, if not rigid, at least hold together no matter how low the density (we can allow a steady state to be reached - need not happen spin up all at once). Build out your rotor such the the density of each additional meter is inversely proportional to γ(v)*r^2, with v proportional to r (of course). The very tip is massless unobtainium - must move at c. Total mass, energy, and angular momentum can remain finite and as small as you like, by virtue of the decreasing mass per meter of rotor.

The tip will be about 25 billion km. from earth. (Of course, this achieves c without slowing down the Earth appreciably; not faster than c).

 

[lightarrow]

I think HallsofIvy is right. The problem is similar to the one involving the effective length of a circumference of a spinning disk. The pole would probably be bent because of relativistic effects.