# Homework Help: A pole, 2 lightyears in length - conceptual question

1. Jul 16, 2009

### vorcil

A pole in zero gravity, two light years in length

you push it 1m,

does it move instantly at the other end?
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its in my physics textbook but dosen't give an explanation as to why it dosen't move?

i know nothing can move faster the speed of light

2. Jul 16, 2009

### sylas

No, it does not move instantly at the other end. Your final answer is the explanation; nothing moves faster than the speed of light. The "pole" is mostly empty space; a collection of atoms held together by electromagnetic forces. It takes time for movement of one atom to result in a force at another.

3. Jul 16, 2009

### vorcil

my reasoning for nothing going faster than the speed of light...

gamma(from special relativity) = 1/(squareroot ( 1-(v^2/c^2)))

if v > c, 1-X which would give a negative number
the negative number becomes a complex number upon square rooting it

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and it's not really reasonable to have a complex number

is that why nothing can go faster than the speed of light?

4. Jul 16, 2009

### queenofbabes

This "gamma" is used to calculate accelerations and times, we can't have them being imaginary >.<

As an object approaches the speed of light, it requires an infinite force to accelerate it any further, so it's impossible to go faster than that.

5. Jul 16, 2009

### Mentallic

Even if v=c, the denominator becomes 0 which isn't allowed either.

However, I wouldn't word it exactly in this way. I mean, the equation was "created" (for lack of a better word) to describe the effects of special relativity.

Sure, you could argue that the mathematics of this equation has been supported by experimental evidence, but the equation itself was derived with the intentions of relativity in mind.

I guess what I'm saying is that for e.g. it's not fair to say it's a square because its area is described as s2 where s is a side length. The square came first, not the equation. It can be used as evidence to support it though.

6. Jul 16, 2009

### vorcil

can you link me to some math of this? i looked around wikipedia but couldn't find anything, ended up reading about schrodingers cat lol

7. Jul 16, 2009

### queenofbabes

8. Jul 16, 2009

### PhanthomJay

The speed of light has nothing to do with the 'speed' of the force transmission through the pole (other than that the speed of light would be the absolute maximum limiting speed). The 'speed' of the force through the pole is more or less the speed of sound through that medium (material) , for reasons noted in sylas' earlier post. If the pole was made of steel, I expect it would take over 100,000 years for the movement to occur at the other end (relatively speaking ).

9. Jul 16, 2009

### turin

This is irrelevant, and misses the point of Special Relativity. Special Relativity makes no claim whatsoever about the existence of such objects (i.e. atoms, and rods composed of atoms), and furthermore applies regardless of the composition of the rod. That is, while Special Relativity originated from electromagnetism, it is not limited thereto.

Suppose a perfectly uniform and continuous "mathematical rod" of length 2-lyrs. This is a continuum (wave) mechanics problem that applies to any hypothetical material. Special Relativity restricts the equations of motion (i.e. the wave equation). Then you must describe the "push" on one end in a covariant way (i.e. promote spatial vectors to Lorentz vectors). Assuming a longitudinal "push", you can do this in a single spatial dimension, so that you can draw a 2-D spacetime diagram of this process. From this, you can put a lower limit on the time delay between the "push" and the resulting motion at the other end.

10. Jul 16, 2009

### negitron

That was easily the most ridiculous explanation for the mechanics involved I've ever seen. Sorry, but mathematically-correct or not, that post is virtually meaningless and lacking in any sort of educational value for the OP. Posts #2 and #8 had it.

11. Jul 16, 2009

### sylas

I'll defer to #8 as my favourite, for a simple explanation at an appropriate level.

12. Jul 16, 2009

### negitron

Agreed. However, I wanted to acknowledge your contribution, as well.

13. Jul 16, 2009

### Mentallic

ok they're both unrelated. But if we were wanting to make the transmission speed of the force through this pole move faster, what could we change? e.g. Composition of the pole; amount of force applied;??

14. Jul 16, 2009

### negitron

Change the composition, specifically, the property sometimes called stiffness which is quantified by Young's modulus.

15. Jul 16, 2009

### Mentallic

and what would be the limit of the transmission speed if this stiffness were increased indefinitely? (Nearing perfect rigidity)

16. Jul 16, 2009

### negitron

Presumably c, although this is wandering well outside of my expertise. And I suspect the molar mass of the material in question puts a smaller upper bound on it.

17. Jul 16, 2009

### PhanthomJay

Since the molecules of the material make contact with each other at the speed of sound when the force is applied, and since the speed of sound in a solid medium is roughly equal to the sq rt of E/p, where E is the elasticity modulus of the material through which the force is transmitted, and p is its density, then the speed of sound, and the speed of the molecular collisions, would be fastest when E is maximized and p is minimized. E and p are properties of the material. Steel has a high elasticity modulus (doesn't deform very easily), so even though it is rather dense,the speed of sound thru steel is quite high. Aluminum is 1/3 less dense than steel, but also 1/3 less stiff, so the speed of sound in alum is on the same order of magnitude as steel. Now diamond will transmit sound the fastest amongst most common materials, but who can afford it, especially in the 2 light year length? So only by changing the composition of the material can you increase the force transmission speed. Increasing the force will incease the acceleration of the cm of the pole, but I do not believe it will affect the time it takes for the far end to start moving.

18. Jul 16, 2009

### PhanthomJay

Agree. Try one of Einstein's rigid measuring rods.

19. Jul 16, 2009

### negitron

Look, I'm not touching Einstein's rigid rod.

20. Jul 16, 2009

### PhanthomJay

And I'm not touching that response. :rofl:

21. Jul 16, 2009

### Mentallic

lol :rofl:

PhanthomJay, thanks, that was a very concise response.
And is E - the elasticity modulus - the same as Young's modulus that negitron mentioned?

22. Jul 17, 2009

### PhanthomJay

Yes, it is.

23. Jul 17, 2009

### turin

negitron, your response to my post was quite rude, and, since you claim in a subsequent post that the content of my post is well oustide your area of expertise, I don't understand how you qualify it.

Anyway, I don't want to hijack this thread with a some little dispute, so I would like to hear your explanation for E<ρc, where E is the elastic modulus of the rod, ρ is the density of the rod, and c is the speed of light.

My main point was the irrelevance of a discrete lattice for the explanation. Explanations that involve the speed of sound tend to lead the eager young minds to a misunderstanding about Special Relativity. (Of course, I will admit, I am simply assuming that the question is posed in the context of Special Relativity, and it may very well be a solid-state physics question for all I know.)

24. Jul 17, 2009

### PhanthomJay

If I may interrupt again, the OP needs to clarify the point of the question. I did not view it as a question on Special Relativity, but rather, a question about how long it takes for the far end of a long pole to move when a force is applied to the near end. Maybe I misinterperted it. Nonetheless, I would argue in the following manner: If the pole was say 6000 meters long and made of steel, and the speed of sound in steel is 6000m/s, and an axial pushing force is applied to one end, then it would take at least one second to see any movement at the other end. Movement would certainly not be instantaneous. I would further argue that if the material were completely rigid (that is, undeformable, with infinite stiffness (E modulus), a hypothetical case for sure), then although non relativistic formulas would yield the speed of sound as infinite for this case, implying instantaneous movement at the far end, that in reality, since nothing can go faster than light, the speed of sound would be limited to the speed of light in this hypothetical case. Now if the pole was 2 light years in length, the question of 'how long does it take the far end to move' may get into SR theory, and I'm no expert on that. But I'm guessing, without doing the math, a couple of hundred of thousand years for me to see it as measured by my clock, if steel; and if ideally rigid, 4 years (2 years for the force vibrations to get there and 2 more years for me to see the event thru my powerful telescope). What do you think? And what does Vorcil say about the original question??

25. Jul 17, 2009

### turin

It seems to be a contradiction, then. If E=∞, then doesn't that result in an infinite sound speed? Imagine a very simplified model: a lattice of a single spring connecting two masses at a distance of 2 lyrs. Wave propagation in the lattice is based on Newton's laws:

Law 3: Push on mass #1 and then the spring pushes back.

Laws 1 and 2: But the spring only pushes back because the spring itself is connected to another mass, which presents inertia (requires force to be accelerated).

Law 1: (only applicable for many lattice points) The lattice can sustain a wave because, after a given mass in the lattice is pushed, a force is required to return it back to some eq position.

Basically, the claim that "nothing can go faster than light" is simply vacuous, and this kind of thinking leads to confusion about what Special Relativity really says. For example, the concept of simultaneity "goes faster than light", meaning that the two ends of the rod can be said to exist at the same moment in time, even though they are separated by some distance. Granted, there is no single massive object moving faster than light w.r.t. any other massive object; however, note that the issue involves the temporal relation between the spatial position of two spatially separated objects (namely the two ends of the rod).

Yes. Here is the distinction that I suggest. If the purpose of the question is to demonstrate material properties, such as the finite propagation of an impules in one part of a real material to another part of that material, then sure, there will be a delay. However, 2 lyrs is a rather exotic length for any material, and so I assumed that the this question regarded some distant future (in which humankind could actually construct such a rod). That being the case, I did not want to limit the consideration to known materials with such low stiffness-to-density ratios. In other words, if we're allowed to imagine the existence of a 2-lyr rod, then I think we should imagine a perfectly rigid material.

Just a comment on this: if you set your watch to t=0 at the moment you push your end of the rod, and you know that the rod is 2 lyrs to the other end, then you would automatically know to subtract the 2-yr delay, and this is the idea of simultaneity. So, you would be able to say (assuming that Special Relativity is correct) that the other end of the rod moved 2 yrs after you pushed your end. Then again, if the speed of sound in the rod is infinite, then you would still have to wait 2 yrs to see the other end move. And that's the point. Simultaneity. And the other point is that simultaneity is relative. The problem with the infinite speed of sound is not some specific material property, it is a fundamental property of space and time. You have to ask yourself, "What would an observer in a boosted frame see?" The answer, according to Special Relativity, is that,

if a perfectly rigid rod (E=∞) is pushed at one end, then a boosted observer could actually see the other end of the rod move before the push.