I Dropping an extended Slinky -- Why does the bottom of the Slinky not fall?

[Orodruin]

Microscopically, approximating a slinky with a set of mass-and-springs subsystems is just as wrong as using a continuum approximation. Neither are good descriptions if you go to the fundamental level. The question becomes: is either a better approximation? Assuming a sufficient number of sub-masses and springs, the predictions of the models will be indistinguishable. The ”better” model becomes the one where making the predictions is easier, which is arguably the continuum model. (I at least prefer solving a rather simple PDE over diagonalizing a $10^{20}\times 10^{20}$ matrix - no matter how sparse …)

 

[DaTario]

Microscopically, approximating a slinky with a set of mass-and-springs subsystems is just as wrong as using a continuum approximation. Neither are good descriptions if you go to the fundamental level. The question becomes: is either a better approximation? Assuming a sufficient number of sub-masses and springs, the predictions of the models will be indistinguishable. The ”better” model becomes the one where making the predictions is easier, which is arguably the continuum model. (I at least prefer solving a rather simple PDE over diagonalizing a $10^{20}\times 10^{20}$ matrix - no matter how sparse …)

Hi, Orodruin. I have never seen the PDE associated with this problem. It seems that it must be somewhat hard to solve, as relevant properties of the medium, including its size, are changing in time and differently in each location. The matrix I would leave to quantum computers,

 

[Orodruin]

Hi, Orodruin. I have never seen the PDE associated with this problem. It seems that it must be somewhat hard to solve, as relevant properties of the medium, including its size, are changing in time and differently in each location. The matrix I would leave to quantum computers,

The PDE is just the wave equation (written in the correct variables), but the pure wave equation is not really describing the case completely as it results in a solution where the top falls past the other parts of the slinky before the bottom moves. In a real slinky, compressions are not actually possible and therefore there will be collisions of the slinky windings. These could be modelled separately, but even the wave equation (that is sufficient to describe the lower part where collisions still have not occured) is sufficient to conclude the bottom does not move until some time has passed. The wave equation solution is relatively easy and outlined and illustrated here in a thread from yesterday:
https://www.physicsforums.com/threa...inearly-elastic-string-under-gravity.1061144/

 

[DaTario]

The PDE is just the wave equation (written in the correct variables), but the pure wave equation is not really describing the case completely as it results in a solution where the top falls past the other parts of the slinky before the bottom moves. In a real slinky, compressions are not actually possible and therefore there will be collisions of the slinky windings. These could be modelled separately, but even the wave equation (that is sufficient to describe the lower part where collisions still have not occured) is sufficient to conclude the bottom does not move until some time has passed. The wave equation solution is relatively easy and outlined and illustrated here in a thread from yesterday:
https://www.physicsforums.com/threa...inearly-elastic-string-under-gravity.1061144/

It seems to be a nice work, in deed. Considered as one spring, the Slinky has mass and its natural length in the horizontal is such that it does not admit compression. But one consequence of its having mass is that when it is put in the vertical the new natural length, accompanied by a pattern of distensions that are not homogeneously distributed, is such that it admits a certain degree of compression.

But my point is basically that the wave argument is somewhat opaque for those who study this problem in general don't know how to associate the concept of information with the fundamental interactions which are present in this context. The tentative explanation (or the outline of an explanation) I provided in post #60 has a bit more transparency (IMO), showing dinamical reasons for the observed rest of the loop which is closest to the ground.

 

[Orodruin]

But my point is basically that the wave argument is somewhat opaque.

I disagree with this. The wave argument is very transparent. A disturbance to any shape of the slinky cannot travel faster than the wave speed as long as the wave equation holds (which it admittedly does not once you start getting hit by the upper part of the slinky, but that too takes time). As long as the shape at the bottom is not disturbed, the bottom won't move because it is subject to exactly the same forces it was subjected to when the full slinky was hanging still.

 

[DaTario]

I disagree with this. The wave argument is very transparent. A disturbance to any shape of the slinky cannot travel faster than the wave speed as long as the wave equation holds (which it admittedly does not once you start getting hit by the upper part of the slinky, but that too takes time). As long as the shape at the bottom is not disturbed, the bottom won't move because it is subject to exactly the same forces it was subjected to when the full slinky was hanging still.

Ok, but what about the more newtonian argument that the rest is caused by the fact that the decrease of the elastic force in time is compensated by the appearance of accelerated masses in the system? Do you think it makes some sense?

 

[Orodruin]

Ok, but what about the more newtonian argument that the rest is caused by the fact that the decrease of the elastic force in time is compensated by the appearance of accelerated masses in the system? Do you think it makes some sense?

On the level of the end of the slinky, no. The end of the slinky is subject to exactly the same forces as before. If you consider the full slinky, obviously the average acceleration is g downwards. As the bottom is not accelerating, upper parts must be accelerating faster than g. The distribution of the acceleration is determined by the internal tension in the slinky.

 

[DaTario]

Is the example of the man climbing the ladder well stated in your opinion?

 

[DaTario]

On the level of the end of the slinky, no. The end of the slinky is subject to exactly the same forces as before. If you consider the full slinky, obviously the average acceleration is g downwards. As the bottom is not accelerating, upper parts must be accelerating faster than g. The distribution of the acceleration is determined by the internal tension in the slinky.

the last turn is subject to a stack of subsystems. As time passes, the superior subsystems lose the elastic forces that kept them in balance and in return, masses begin to accelerate. The logic of this exchange seems to allow the maintenance of balance in this stack of subsystems.

 

[A.T.]

Thank you A.T., for your contribution, but regarding your comment on relativity, this is a Classical Physics' discussion, isn't it?

Yes, and in Newtonian physics, when appropriate, we use the approximation of extended bodies as perfectly rigid objects (infinite propagation speed of mechanical disturbances). But this approximation definitely not appropriate for a slinky.

 

[DaTario]

Yes, and in Newtonian physics, when appropriate, we use the approximation of extended bodies as perfectly rigid objects (infinite propagation speed of mechanical disturbances). But this approximation definitely not appropriate for a slinky.

But who is proposing the slinky as a rigid body?

 

[A.T.]

But who is proposing the slinky as a rigid body?

To have instantiations motion at the bottom, you would need infinite propagation speed of mechanical disturbances, which implies a perfectly rigid body.

 

[DaTario]

My present position (and proposition) was described in post #60. Initially I proposed that the fall of the first loop would produce a local departure from equilibrium that would lead to a rapid chain reaction producing some disequilibrium effect in the lowest loop. However, I later realized that throughout the process, masses were leaving rest and starting to form an accelerated subsystem. This accelerated subsystem of increasing mass can provide the necessary support to maintain the balance of the lower turns.

I would like to know if you agree with this line of reasoning.

 

[A.T.]

This accelerated subsystem of increasing mass can provide the necessary support to maintain the balance of the lower turns.

The balance of forces on the lower turns depends on the force and thus deformation of the turn right above them. It does not depend of whether some turns way above them are accelerating or not.

 

[DaTario]

The balance of forces on the lower turns depends on the force and thus deformation of the turn right above them. It does not depend of whether some turns way above them are accelerating or not.

It seems we have a chain of interdependent subsystems, haven't we?

 

[PeroK]

Let me try another analogy. Imagine there is an earthquake under the ocean that sets off a tsunami. The water wave moves at a finite speed and could take an hour or more to reach the shore. The whole ocean of water does not move uniformly at once. Or, look at the time sound waves take to move through the air. They are not instantaneous either.

Initially I proposed that the fall of the first loop would produce a local departure from equilibrium that would lead to a rapid chain reaction producing some disequilibrium effect in the lowest loop. However, I later realized that throughout the process, masses were leaving rest and starting to form an accelerated subsystem. This accelerated subsystem of increasing mass can provide the necessary support to maintain the balance of the lower turns.

I would like to know if you agree with this line of reasoning.

By "a local departure from equilibrium that would lead to a rapid chain reaction producing some disequilibrium effect in the lowest loop", I guess you mean a wave. It's nothing more and nothing less. Waves in different objects move at different speeds. The slinky is a visual representation of this. It's not like you are being asked to accept something abstract. What is the point of this thread?

 

[DaTario]

Let me try another analogy. Imagine there is an earthquake under the ocean that sets off a tsunami. The water wave moves at a finite speed and could take an hour or more to reach the shore. The whole ocean of water does not move uniformly at once. Or, look at the time sound waves take to move through the air. They are not instantaneous either.



By "a local departure from equilibrium that would lead to a rapid chain reaction producing some disequilibrium effect in the lowest loop", I guess you mean a wave. It's nothing more and nothing less. Waves in different objects move at different speeds. The slinky is a visual representation of this. It's not like you are being asked to accept something abstract. What is the point of this thread?

I apologize to everyone for having used user Notinuse's post to ask anyone interested in this topic if a more Newtonian explanation (with elastic forces decreasing and masses becoming accelerated) could be accepted. Apparently no one that contributed here found the idea interesting. Maybe it's a good time to stop. I don't want to make anyone sad.

 

[Dale]

if a more Newtonian explanation (with elastic forces decreasing and masses becoming accelerated) could be accepted. Apparently no one that contributed here found the idea interesting.

For my part, that is not what I objected to. This is a scenario that is well within the scope of Newtonian mechanics. And discrete element models are well studied with known uses and limitations.

I personally objected to

1) the claim that such a discrete element model is more fundamental than a continuum explanation

2) the claim that a discrete explanation would result in instantaneous movement at the bottom

The first objection is a matter of opinion, but the second objection is not. A correct Newtonian discrete element explanation leads to waves and has the same feature that the bottom doesn’t start moving immediately.

 

[DaTario]

For my part, that is not what I objected to. I personally objected to

1) the claim that such a lumped element model is more fundamental than a continuum explanation

2) the claim that a discrete explanation would result in instantaneous movement at the bottom

The first objection is a matter of opinion, but the second objection is not. A correct Newtonian discretized explanation leads to discretized waves.

Hi,

Regarding your point number 1, if what you call continuum explanation is the well known wave equation, I would say that a 'lumped element model' is likely to be free from aproximations tipically used in the first approach such as small-amplitude, continuous medium, uniformity and isotropy, neglecting of damping and dispersion, quasi-static approximation, linear approximation and homogeneous boundary conditions.

Furthermore, the slinky, falling, that is, accelerated, vertically, with both ends free, does not exactly constitute an example of a familiar and safe application of the wave equation and the notions pertinent to this context (wave physics).

Regarding the point number 2, I abandoned this view when in post #60 I exposed the idea that along with the systematic decrease, over time, of the elastic forces between the upper turns, there was an increase in the number of particles having acceleration in the system, and that these two processes together would explain ( or would probably explain) the maintenance of the equilibrium condition of the lower turns.

 

[Orodruin]

Regarding your point number 1, if what you call continuum explanation is the well known wave equation, I would say that a 'lumped element model' is likely to be free from aproximations tipically used in the first approach such as

Let us consider these one at a time:

small-amplitude,

… is typical for transversal waves, not longitudinal waves. Furthermore, the discrete model suffers similarly from possible non-linearities for large displacements in an actual material if you create a model with masses and springs.

continuous medium,

… is no more of an approximation than a discrete model with a chain of masses and springs is. If you really want to make a molecular level approximation, a spring is far from a single chain of molecules.

uniformity and isotropy,

… is not better handled by a discrete model.

neglecting of damping and dispersion,

… is again not better handled by a discrete model. It is as easy to introduce in a discrete model as it is in a continuous one so that is nothing particular either.

quasi-static approximation,

… is not in either model.

linear approximation

… is in both models. Can be taken out of either but then leads to a harder problem to solve.

and homogeneous boundary conditions.

… in in both models (or neither, depending on the actual boundary conditions).

To me, your distinction simply does not hold up to scrutiny.

 

[DaTario]

To me, your distinction simply does not hold up to scrutiny.

Hi, Orodruin. I think you show signs that you are happy with the explanations you give about this system. I was looking to see if there are people out here who aren't happy with this explanation. The thesis that the two paths of explanation are not distinct seems unlikely to me and I do not think it has been sufficiently well defended.

 

[Orodruin]

Hi, Orodruin. I think you show signs that you are happy with the explanations you give about this system. I was looking to see if there are people out here who aren't happy with this explanation. The thesis that the two paths of explanation are not distinct seems unlikely to me and I do not think it has been sufficiently well defended.

I mean, the effect is well known and it is quite clear that regardless of the description you use you end up with the same result: Making a free body diagram of the lower part of the slinky will have it hanging still until what is just above starts contracting. There really is nothing more to it and it works the same irrespective of the model you use.

If you disagree with this then I believe you simply have not worked enough with the discrete model. In both models you have a propagation of the disturbance “down-spring” and the bottom doesn’t move until that disturbance reaches the end.

 

[DaTario]

I mean, the effect is well known and it is quite clear that regardless of the description you use you end up with the same result: Making a free body diagram of the lower part of the slinky will have it hanging still until what is just above starts contracting. There really is nothing more to it and it works the same irrespective of the model you use.

If you disagree with this then I believe you simply have not worked enough with the discrete model. In both models you have a propagation of the disturbance “down-spring” and the bottom doesn’t move until that disturbance reaches the end.

Believe me when I say I'm not getting off topic. But how do you explain the electric field produced by a uniformly charged plane?

 

[Orodruin]

Believe me when I say I'm not getting off topic. But how do you explain the electric field produced by a uniformly charged plane?

The answer is still that the continuous model prediction is just as good as the discrete model prediction here. Besides, you are missing the point of the above discussion, which is that both models - continuous and discrete - have the same fundamental explanation of why the bottom does not fall until some time has passed. Just as the uniformly charged plane has an electric field produced by non-zero charge - whether that charge be continuously or discretely distributed. Changing from a continuous to a discrete model does not affect the actual physics in the limit where the discrete model goes to the continuum limit.

Also in the case of the electric field, both models are wrong to some extent.

 

[Orodruin]

To add to that, there is actually quite a set of literature on this subject and the predictions of the models - continuous and discrete alike - closely reproduce the behaviour of a real falling slinky.

 

[Orodruin]

Waves in different objects move at different speeds. The slinky is a visual representation of this.

It should be pointed out though, that what we have in the slinky drop experiment is not your typical wave moving at the wave speed of the medium. It is moving faster than the wave speed of the medium along with an abrupt change in the coil density. These are characteristics of a shock wave rather than your normal wave propagating at a speed set by the medium.

 

[DaTario]

Just as the uniformly charged plane has an electric field produced by non-zero charge - whether that charge be continuously or discretely distributed. Changing from a continuous to a discrete model does not affect the actual physics in the limit where the discrete model goes to the continuum limit.

I did not invoke the problem of the electric field produced by a uniformly charged plane, to discuss discrete or continuous approaches. Sorry if I gave you this impression. What I want to exemplify is taking advantage of an opportunity to observe something unique that both contexts seem to offer. Regarding the uniformly charged infinite plane, let's see the following:

We know that considering any point charge, when we move away from it the field in our position decreases in magnitude. When we take a step back, moving away from a plane, we are at the same time moving away from all the area elements of this plane. The total field is the vector sum of the fields produced by all elements of area in the plane. What keeps the total field constant regardless of the distance from the plane is a perfect compensation between the decrease in the modulus of the electric field vectors originating from each charge element and the geometric effect of greater alignment between these vectors. Thus, if in a position further away from the plane the vectors have a smaller module, they are also more aligned, that is, they are closer to parallelism, so that this geometric condition perfectly compensates for the effect of distance.

In the slinky problem I have the impression that something similar happens. Elastic forces start decreasing from the upper part of the slinky but at the same time we see masses being accelerated so that balance is maintained for the remaining part of the spring. It is more or less like we set initially $y$ as a vertical coordinate for a mass and spring system, and write Newton's second law:
$$m \frac{d^2y}{dt^2} = - k \Delta y - mg.$$
Now we write it in the form:
$$m \frac{d^2y}{dt^2} + k \Delta y = - mg.$$
so that we can see in the first term an informal suggestion of the compensatory mechanism to which I refer. (* Informal mainly because the slinky is not such a simple mass and spring system *). If the elastic tension decrease but some of the masses start accelerating, then we can still have the equilbrium.

But I recognize that I don't have a closed method to express clearly and consistently this compensation, so I decide not to proceed with this debate. I thank you all.

 

[Orodruin]

Sorry, but I do not see the analogy here. There are no forces from the upper part of the string on the lower end. All of the forces are contact forces and the force on the lower part is the force from the part immediately above it. In a discrete model, the only elongation that matters for the bottom mass is the very last spring. It is irrelevant what the elongations of the other springs are.

You can of course do a harmonic series expansion in terms of standing waves, but that also won’t give you a good description because, as has been discussed elsewhere, what occurs in the slinky drop experiment is a shock wave rather than a wave propagating at the characteristic wave speed of the slinky.

 

[A.T.]

It seems we have a chain of interdependent subsystems, haven't we?

The instantaneous balance of forces on each chain element depends only on the current deformation of the neighboring elements. It does not depend on what other elements further up are currently doing.

Explaining the local force balance at the bottom by acceleration of the elements at the top make no sense. It's only relevant for the global momentum conservation, as the center of mass must accelerate down at g.

 

[DaTario]

Orodruin and A.T.

I understand your points and they seem ok to me. As I said, my point is weakly supported, I will no longer defend it here. Thank you very much for this debate. I would like to point out some minor divergencies I have relatvely to what you said in these last posts:

A.T. said:

"The instantaneous balance of forces on each chain element depends only on the current deformation of the neighboring elements."

I would note that, besides gravity, at each time there is an element of mass in the slinky which is on the moving interface between those accelerated masses (above) and those static masses (below). For this particular mass element (at this particular time), it seems that the balance or unbalance of forces depends also on the acceleration of neighboring elements.

Orodruin said:

1) "All of the forces are contact forces"

I do not agree with this. As I said, there is gravity (field force) and there is inertia, which reponds by the term $m y''(t)$.

2) "You can of course do a harmonic series expansion in terms of standing waves"

I would find it enoumously anti intuitive or non pratical, to try harmonic series expansion, as the spring is changing size which makes its harmonic family to change continuously over time. I would not say it is impossible, but it seems not to be a good idea.