Speed of information in classical mechanics

[louk]

Speed of information – effects of on classical mechanics.

(This is not a homework or coursework question. Even if the problem stated is simple, the answer is maybe not.)

Problem: A mass is hanging vertically in a not heavy, quite inelastic and hard to compress cable which is fixed in the top end. The top cable end is then released. When will the mass at the bottom part of the cable start to move downwards?

a) Immediately. In this case information is transferred faster than light. (There are no moving objects and just one observer – say placed halfway down).

b) The time corresponding to speed of light over the length of the cable. For this alternative one have to ask what is the mechanism for transferring the information with the speed of light. Compare also to case c) below.

c) The time corresponding to speed of sound in the cable.


It turns out that c) is the standard – or at least the most frequent - answer. But at a closer look, this answer raises a lot of problems.

If one strictly states that information is propagated with speed of sound, then it is quite easy to see that a number of unexpected (and easily observable) effects should occur. Those are first listed and then discussed below.


1. The top part of the cable has to start to move well head before the bottom part.

2. If there are two masses – one suspended in a steel cable and one in a rod made of cork and both top ends are released the same time, then the mass suspended in the cork rod will start to move substantially later. (The speed of sound in cork is about ten times slower than in steel.)

3. The classical spring/mass oscillation equations have to be adjusted.


1. The top part of the cable has to start move before to bottom part.

If the information is transferred with the speed of sound, then there is an event-horizon moving downward with the speed of sound. Below the event-horizon nothing can happen - no information has reached that part of the cable. Therefore, the cable part below the event-horizon has to be in rest. The top part of the cable above the event-horizon should be accelerated downwards, there is a downward force but no upward force (assuming that the mass of cable itself is low). At the event-horizon there must somehow (how?) be an upward force balancing the downward force so the lower part of the cable does not move.

When the event-horizon reaches the mass at the bottom, the time elapsed is Length_cable/V_sound_cable. For a 1000 meter long cable in steel – and using the speed of sound in the steel which is about 5000 m/s - this time will be 0.2 seconds. The top part should then have moved downwards:

time^2*g/2 – that is 0.196 meter.


An effect that the top part starts falling well before the bottom part should also be quite easy to demonstrate in an experiment.

The first thing that comes up to explain such an effect is that the cable was elongated by the mass and is just contracting when the top end is released. But this explanation does not hold fully. The elongation due the tension is dependent on the mass of the weight but the distance moved by the top part when released depends only on the speed of sound and the length of the cable. One can therefore place a smaller weight at the bottom end so that the elongation due to the tension will be much smaller than 0.196 meters (also assuming that the speed of sound does not vary too much with the tension of the cable).

One may try to explain the effect that the top part of the cable is moving while the bottom part stands still, as the cable elongation is first contracted to the length it had without any mass and then becomes compressed. But the compression part of the explanation does not really work out – from where comes the force needed for such a compression? The only force available is a very small one caused by the acceleration of the mass of the top part of the cable caused by the spring contraction.



2. Two suspended masses in cables with the same lengths but of different materials and released the same time will reach the ground at different times.

If one mass is suspended in a steel cable of say 10 meters and another mass is suspended in a rod of cork of the same length, then, if both top ends are released at the same time the masses will start moving at different times. The information that the top end has been released will propagate with 5000 m/s in the steel cable and 500 m/s in the cork rod. The mass cannot start to move until the information has reached the masses and this will take 2 milliseconds for the mass suspended in the steel cable and 20 milliseconds for the mass suspended in the cork rod.


3. The classical equations for spring and mass oscillations needs adjustments.

The standard spring and mass problem is a mass, attached to a spring, which is first displaced by force and then released. The force from the spring displacement causes the mass to oscillate in some way. One physical relation in this problem is F=k*d, where F is the force of the spring, k is Hookes spring constant and d is the displacement of the mass and thus also the elongation of the spring.

However, when the mass is released, the full spring cannot actively pull the mass directly since the information that the mass have been released, first has to propagate over the full spring. The full force of the spring cannot be accounted for immediately. The k*d expression has therefore to be multiplied with a factor increasing linearly from zero to k over the time needed for the event-horizon to propagate over the full spring length with the speed of sound for the material of the spring. (If the spring is spiralled, then the length should be the full curled up length.) The net result will probably be some kind of phase or maybe even a frequency adjustment in the oscillation.


The objective is not to find an exact modified formula for the spring/mass oscillation – but to conclude that such modifications have to be included - if the speed of information really is the same as the speed of sound.


The above three examples illustrates that: If the speed of information is the same as the speed of sound in mechanical systems, then a lot of classical mechanics seems to need adjustments!


All comments are appreciated. (I am just trying to sort out those questions)

Thanks in advance.

 

[russ_watters]

If the information is transferred with the speed of sound, then there is an event-horizon moving downward with the speed of sound. Below the event-horizon nothing can happen - no information has reached that part of the cable. Therefore, the cable part below the event-horizon has to be in rest. The top part of the cable above the event-horizon should be accelerated downwards, there is a downward force but no upward force (assuming that the mass of cable itself is low). At the event-horizon there must somehow (how?) be an upward force balancing the downward force so the lower part of the cable does not move.

The propagation of the loss of that force is this "event horizon" you speak of. What actually happens as the shock wave propagates is that the top of the cable gets accelerated down toward the hanging weight. The magnitude of that acceleration and the speed of the propagation of the "event horizon" are related and are a function of the elasticity of the material.

The top part should then have moved downwards:

time^2*g/2 – that is 0.196 meter.

Actually no. As what I said above implies, the elasticity will pull the cable toward the weight much, much faster than that.

An effect that the top part starts falling well before the bottom part should also be quite easy to demonstrate in an experiment.

Yes. Try it with a stapler and a bunch of rubber bands tied together (I'm in my office...).

The first thing that comes up to explain such an effect is that the cable was elongated by the mass and is just contracting when the top end is released.

Yes, but in contracting, it also acceleratese the top of the cable faster than (before) the bottom.

But this explanation does not hold fully. The elongation due the tension is dependent on the mass of the weight but the distance moved by the top part when released depends only on the speed of sound and the length of the cable. One can therefore place a smaller weight at the bottom end so that the elongation due to the tension will be much smaller than 0.196 meters (also assuming that the speed of sound does not vary too much with the tension of the cable).

As said above, the effects are additive: the extra distance traveled by the top part will be greater than .196 - by how much depends on the weight.

One may try to explain the effect that the top part of the cable is moving while the bottom part stands still, as the cable elongation is first contracted to the length it had without any mass and then becomes compressed. But the compression part of the explanation does not really work out – from where comes the force needed for such a compression? The only force available is a very small one caused by the acceleration of the mass of the top part of the cable caused by the spring contraction.

If this were a rod, it would oscillate like a spring, in tension and compression. Since it is a cable, it won't compress, it'll just coil itself up. Cables don't do compression.

2. Two suspended masses in cables with the same lengths but of different materials and released the same time will reach the ground at different times.

Yes.

3. The classical equations for spring and mass oscillations needs adjustments.

The standard spring and mass problem is a mass, attached to a spring, which is first displaced by force and then released. The force from the spring displacement causes the mass to oscillate in some way. One physical relation in this problem is F=k*d, where F is the force of the spring, k is Hookes spring constant and d is the displacement of the mass and thus also the elongation of the spring.

However, when the mass is released, the full spring cannot actively pull the mass directly since the information that the mass have been released, first has to propagate over the full spring. The full force of the spring cannot be accounted for immediately. The k*d expression has therefore to be multiplied with a factor increasing linearly from zero to k over the time needed for the event-horizon to propagate over the full spring length with the speed of sound for the material of the spring. (If the spring is spiralled, then the length should be the full curled up length.) The net result will probably be some kind of phase or maybe even a frequency adjustment in the oscillation.

You are correct about the issue, but it isn't a flaw in the spring-mass equation, just a domain of applicability issue. The classical spring-mass equation simply doesn't address the signal propagation issue. It only applies in situations like you described, where the elastic modulus is extremely high and distances long enough for signal propagation to have an impact.

The above three examples illustrates that: If the speed of information is the same as the speed of sound in mechanical systems, then a lot of classical mechanics seems to need adjustments!

There aren't a whole lot of situations where this particular issue would be relevant, but rest assured, an engineer designing a device where this mattered would take it into account. However, I can't, off the top of my head, think of any real-world situation where this would be an issue. Perhaps in a vibration analysis of some sort (though I haven't done much of that - that's structural engineering and I'm mechanical).

 

[Danger]

However, I can't, off the top of my head, think of any real-world situation where this would be an issue. Perhaps in a vibration analysis of some sort (though I haven't done much of that - that's structural engineering and I'm mechanical).

You're mechanical? Hmm... I always thought, I guess due to your career, that your specialty was fluidics, which I thought was part of Aero.
Anyhow, Aero is where I can think that the speed of sound in the material might be important. That can apply to either the reaction of materials at trans- or supersonic speeds regarding the airframe, or equally to the speed of turbine or compressor blades.

 

[russ_watters]

I started out Aero, didn't do so hot, and finished up mechanical.

 

[Danger]

Ahhh... the 'Peter Principle'?

 

[AlephZero]

There aren't a whole lot of situations where this particular issue would be relevant, but rest assured, an engineer designing a device where this mattered would take it into account. However, I can't, off the top of my head, think of any real-world situation where this would be an issue. Perhaps in a vibration analysis of some sort (though I haven't done much of that - that's structural engineering and I'm mechanical).

Actually, it matters in just about every transient dynamics or vibration analysis, except when analysing the "steady state" dynamics response - and even for steady state dynamics, a vibration mode or a standing wave is equivalent to two traveling waves moving in opposite directions at the local "speed of sound", and it's often convenient to think about it in that way.

In general the "speed of sound" in a solid depends on the type of wave (axial, transverse, etc), the shape of the solid the wave is traveling through, and the stress field in the structure, as well as on the material properties. The actual speed of wave propagation through the solid can be orders of magnitude different from the conventional definition of "the speed of sound in a solid" = sqrt(Youngs modulus/density).

All the "corrections to classical mechanics" the OP mentions are already part of classical mechanics - though they are probably not taught in a first dynamics course which deals with idealized mechanical components like point masses and massless springs.

 

[louk]

Thank you very much for all comments and the time you spent for this. The comments are very enlightened. (Sorry - I was not aware that the time delays due to the signal propagations are indeed included in the more advanced classical mechanics)

But I still have an unresolved puzzle with this (which actually was the reason to start thinking about this) and that is it hard to balance the forces in the beginning when releasing the cable or rod in the top end. This has to do that the cable weight above the shockwave can be very small in the beginning and the acceleration of that small weight has to produce an upward force equal to the gravitational force of the heavy mass in the bottom (which is constant) – in order for the bottom part to stay in rest.

First, I have no problem with the reverse situation – when you take the rod in top end, lift it and hang it in a fixed point above ground. All forces balances nicely and I see no problems for signal speeds in the same range as the speed of sound in the material. It is the release of the rod in top end which creates problems.

If I understand Russ correctly, he agrees that there is an upward force at the “event-horizon” (where the shockwave is) - after the top end has been released - that exactly balances the downward force of the mass below, so it can stay in rest. Even the slightest movement of the bottom end mass can be interpreted as transferred information.

I have problems to understand how the upward force in the beginning can be large enough. The problem is that the mass to be accelerated in the very beginning is a very small part of the top of the cable with very little mass. Simplified - the acceleration needed as a function of time will be: a = constant/time – so the acceleration needed is infinite to start with – and huge when a distance a little bit over the atomic scale has been reached.

I am grateful for comments on this.