Action and reaction when pushing an extremely long stick

[Nantes]

There is a classical thought experiment on trying to exceed the speed of light: if you push a light-year-long rod, on whose end is a button that will be pressed by the rod, wouldn't you be vastly exceeding the speed of light (which is the speed of cause & effect) by pressing the button instantly? The answer is no, the reason why is summarized in this answer I pulled from a reddit user:

"Mechanical disturbances travel through materials at the speed of sound, which is much slower than the speed of light. Any time you push an object, you are not actually pushing the whole object. You are pushing the end near you, which then gets compressed, bounces back from the compression, causing a part of the object a little ways down to get compressed, and so forth, until the compression wave travels through the whole object to the other side. For small rigid objects, this happens so fast, that it seems like the entire object is moving at once. But for large objects, you simply can't ignore this process."

Ok, but then I started thinking: if such a rod was in space, and you pushed it, you would be pushed back due to Newton's third law. But what would happen to the rod? Would it stay completely immobile until the shockwave traveling at the speed of sound reached the other end (hundreds of thousands of years later), at which point it would finally budge forward?
Or would it somehow start moving forward immediately after the push? But how, if the atoms at a given point in the rod can't go forward because the atoms just ahead of them are blocking the way?

Note: I'm assuming the rod would be made of an impossibly light material, otherwise its total mass would be impossible to budge with human strength.

Thanks

 

[andrewkirk]

A compression wave will go through the rod.
What will happen is that your end of the rod, held by your hand, will move forward a bit and consequently the molecules in the part of the rod you are holding, and just for a little way behind that, will compress together slightly (move closer together). That pattern of compressed molecules will travel through the rod as a wave until it reaches the other end. No part of the rod will move until the wave reaches it. When the wave reaches a part of the rod, that part starts to move towards the button, and continues to move in that direction until the wave has passed it. Once a part of the rod has been passed by the wave, it is closer to the button than it was before.
When this hits the button, the button moves in, to the switched 'on' position.

The person holding the rod is pushed back as soon as they push their end of the rod - they don't have to wait for the wave to reach the button. It is the forces between molecules in the rod, resisting the compression, that creates the push-back force.

 

[Nantes]

The person holding the rod is pushed back as soon as they push their end of the rod - they don't have to wait for the wave to reach the button. It is the forces between molecules in the rod, resisting the compression, that creates the push-back force.

Yup, that much I had figured :P

If I understand correctly, it will move forward only as much as the length reduction due to the compression shockwave, which I figure is a very small amount. Then, when the wave finally reaches the end and the tip of the rod starts moving forward by inertia, the start of the rod (close to your hand) won't "know" the far-away tip started moving until the information travels all the way back to the start, right? So it'll actually take at least twice as long for the start of the rod to begin moving forward?

 

[Dale]

which I figure is a very small amount.

It can be arbitrarily large.

the start of the rod (close to your hand) won't "know" the far-away tip started moving until the information travels all the way back to the start, right? So it'll actually take at least twice as long for the start of the rod to begin moving forward?

It would take that long for the start of the rod to decompress from a small displacement, but it could start moving immediately regardless of the state of the other end.

 

[Nantes]

It would take that long for the start of the rod to decompress from a small displacement, but it could start moving immediately regardless of the state of the other end.

How? The rod is a solid, its molecules have strong bonds between them. When the molecules at the very tip start moving forward, they also drag along the molecules behind them, which drag along the molecules behind them and so on, and this process cannot be faster than light. How do both tips start moving at once?

 

[Dale]

How do both tips start moving at once

They do not. The near tip starts moving immediately, the far tip starts moving once the compression wave arrives.

 

[Vanadium 50]

Consider your rod to be a stiff spring (because it is). That will help you visualize what's going on.

 

[Nantes]

Considering that 't = 0' is the moment the force is applied:

How would the frozen start of the rod know the compression wave has reached the other end and suddenly begin moving, if that knowledge cannot be transmitted faster than the speed of light?

 

[jbriggs444]

The end of the rod that is pushed only compresses as long as a force is being applied. If the person pushing does so for one second then the compression is only present for one second. If the pressure is then released and if the speed of the compression wave is 5000 meters per second then this will result in a compressed region 5 kilometers in length. That compressed region will move down the rod. There is no "freezing" going on.

 

[Nantes]

The end of the rod that is pushed only compresses as long as a force is being applied. If the person pushing does so for one second then the compression is only present for one second. If the pressure is then released and if the speed of the compression wave is 5000 meters per second then this will result in a compressed region 5 kilometers in length. That compressed region will move down the rod. There is no "freezing" going on.

That makes more sense, thanks! That's one half of the situation explained. But what about when the whole thing starts moving due to the astronaut's force applying an acceleration to the rod? How such an extremely long rod move in unison, if information about the state of each part of the rod cannot be transmitted faster than c ?

 

[Dale]

How wo frozen start of the rod know the compression wave has reached the other end and suddenly begin moving, if that knowledge cannot be transmitted faster than the speed of light?

It doesn't. In this scenario the end of the rod would unfreeze as soon as the astronaut stops pushing on it.

 

[Dale]

How such an extremely long rod move in unison, if information about the state of each part of the rod cannot be transmitted faster than c ?

It doesn't move in unison. At those scales no material is rigid, it flops around like Jello. Think about tapping a piece of Jello on a frictionless plate and you will have a reasonable intuition about how the rod moves.

 

[jbriggs444]

If the compressed region reaches the other end of the rod and meets no resistance then it will reflect from the far end in inverse phase -- as an expanded region which will propagate back up the rod to the starting end. An ideal rod will "ring" like this forever. In a real rod, the vibrations will eventually damp out, leaving only a uniformly moving (and slightly warmer) rod.

 

[russ_watters]

What you have here is a reeeeaaaaaallly low frequency tuning fork (hit on the end). It'll mostly just vibrate.

 

[Nantes]

I see, thank you very much guys!