Of falling chain ladders

In the video linked below, the chain ladder on the left (against my expectations) fell faster. I couldn’t understand this as the chain between the rungs is essentially slack after the rung below it hits the table.

Then it dawned on me that the angled rungs might play a part, acting somewhat as a fulcrum or lever. This wouldn’t happen if the rungs were parallel and horizontal, would it?

So, I’m assuming the table plays a part in this as well but I’m not sure why. Wouldn’t the ladder hitting the ground have that same lever thing happening?

All I can guess is that it had something to do with tension waves moving through the chains… but I’m not sure about any of the details.

That, or the table didn’t actually make any difference, and there was actually a slight difference in how they were dropped (out of frame, where we couldn’t see it).

Yeah, that’s possible. I was giving the team of physicists at Cornell University the benefit of the doubt. Maybe I shouldn’t have?

ETA: I see the video I linked to doesn’t explain who did the experiment - the first video I watched, did. Sorry.

It helps to think of a simplified free-body diagram (FBD) of the chain falling freely (Ladder A) versus the one hitting the table (Ladder B). Neglecting air resistance (which is virtually negligible at these speeds and would apply each run independently) Ladder A is in an equilibrium state and may be treated essentially as a rigid body as all of the chains between segments remained tension, and so every part falls at the same rate, and the momentum state remains constant. The chains on Ladder B, on the other hand, lose tension as each rung strikes the table, which is a net reaction force (and thus acceleration) acting upon the ladder. As the force acts, it transfers some of the momentum from impact back up the attached section of chain, (which is far from a purely elastic because the chains between rungs cannot transfer any compressive load to the next run) but is sufficient to slightly unload the chain in the same way that the acceleration that you would measure in an ascending elevator car would be slightly more than one that was stationary. You can see the cumulative effect near the end of the video as the chains on the upper segments start to unload even before the run below the impacts the table.

I don’t think that the rungs being angled has any effect on this and it is just done to alternate the unloading of the chain on Ladder B so that you don’t see any whipping motion in/out of the plane of view. If they were aligned level then you might get a wave that whips up each side at the same rate, whereas with the alternating lengths any tendency for each segment of chain to whip will be out of phase with its complementary segment, so the only way for a wave to effectively propagate is in torsion. You might see a tendency for that at the very end of the video, as the upper rungs all turnt he same counterclockwise direction but that may just be incidental in how those last few rungs landed, too. I’ll have to think about that some more to see if there is any particular effects of uneven lengths of the chain.

Stranger

That can’t be it, first, because tension in the chains of a falling ladder should be negligible, and second, because the one that hits the table is actually the one that falls faster.

I found this:

Which seems to imply the angled rungs seem to propel the left side down faster, but I still don’t understand why the same principal wouldn’t apply the rungs on the right hitting the floor.

It doesn’t have anything to do with the tension in the chains—which you are correct should be effectively negligible—but the transfer in momentum of the impacting rungs onto the table propagating back up to the chain above. In fact, upon reviewing the video, you can see that the left side chain above the fourth link violently whips up past the rung above it, and by the time the ninth or tenth rung hits the table you can clearly see that the chain in the rungs above is clearly going somewhat slack, and even the runs themselves are moving in response. This response isn’t as pronounced as it would be with, say, a coil spring because the spring would be a continuous elastic element in both tension and compression, while the ladder is virtually incapable of bearing any compressive load but as a ‘body’ it still has to respond to the force applied to it as it strikes the table, and some small portion of that is conveyed in accelerating the ladder in the upward direction; in effect, increasing the net acceleration.

The explanation given in that article is kind of confusing, and characterizing the impacting link of chain as “pulling” on the rest of the chain is conceptually fraught. Think of what you would see dropping two unloaded coil springs; the spring that was freely falling would stay the same length, while the spring that impacted would compress as the net force of impact was applied to it. The same thing is occurring to the ladder except it isn’t as obvious because the ladder is not a continuous linear elastic system.

Stranger

Well, that part’s easy: Because it’s not hitting the floor. The floor is enough below the table that it doesn’t hit until after the video is over.

Your previous explanation was in terms of the tension in the chains, so you were in fact saying that it has something to do with the tension in the chains.

I was just observing that the chains were visibly taut (no visible slack) in the video prior to Ladder B impacting the table. The amount of strain (elastic) energy in them from any tension prior to being dropped is negligible and is not a consideration in the phenomenon in question. If it helps, just replace the word “tension” with “taut” in the first post for clarity.

Stranger

Think about the experiment with two weights on springs instead.
Hang them, the springs extend.
Let go, two things happen at once. the entire system begins free fall, and the spring contract as in free fall there is no gravity on them or the weights pulling the lower end apart from the top end.

The center of gravity of both systems falls at the same rate. the top of each spring closes downward in addition to the falling center of gravity (so faster than g while the bottom contracts upward (so slower than g ).

Depending on the construct of the spring, it may have time to overshoot compression and rebound during the trip down, but let’s pretend the fall is over before that.

So the question is - then what? Chains consist of links that like any metal are slightly deformed by the weight below, and as the system falls, the links will rebound to unloaded shape, shortening the chain overall. As the one chain hits the table, does the remaining airborne segment(s) rebound/contract faster, since the rebound consists of middle links “pulling in” the links at the extremities for a length of chain?

The acceleration of the ladder that lands on the table is because of the geometry of the rungs: when one end of the rung hits the table, the center of mass of the rung continues to fall at almost the same rate, which requires the elevated end of the rung to swing down faster, pulling the rest of the ladder down slightly to do so. The free-falling ladder does not even begin to hit the ground until after it has fallen completely past the table.

A ladder with horizontal rungs, or chains of the appropriate shape, will not experience this extra acceleration. By ‘appropriate shape’ I mean a chain that doesn’t interact with the links above it once it has hit the table. You can imagine chains where the links have long aspect ratios and have to topple over once they hit the table, thus exerting lateral forces up the line of links. That could speed up or slow down the fall of the whole chain, depending on the geometry.

That mechanism might be true for the first rung that strikes the table but it clearly isn’t the case for any of the rungs that hit after as you can observe both sides of the chain above the impacting rung are slack and never become taut, and yet the Ladder B continues to accelerate faster than Ladder A. In any case, simply converting linear momentum to angular momentum and then back alone cannot result in net acceleration per conservation of momentum.

Stranger

This isn’t entirely true. Since the ladder is in freefall, none of the links are ever under any tension - except for a very brief moment as each rung’s lower end hits the pile of rungs on the table. During that short period, that rung’s upper end gives a brief tug to the still-falling portion of the ladder. It doesn’t seem to happen consistently, but only because the irregular pile of rungs on the table sometimes prevents rotation of the rung that’s going through it’s landing process, and also become sometimes there’s excess slack in the strings that the landing/rotating rung is unable to completely take up (and so unable to exert any meaningful tension on the still-airborne portion of the ladder).

It took me all of five minutes to find the site where the guy who did the experiment shows how and why it was done, with an added bonus explaining it all. Enjoy.

http://ruina.tam.cornell.edu/research/topics/fallingchains/Falling_Chain_experiments.html

Plus he even wrote a paper about it:
Grewal, A., Johnson, P., & Ruina, A. (2011). A chain that speeds up, rather than slows, due to collisions: How compression can cause tension. American Journal of Physics , 79 (7), 723-729.

I’m still not understanding his logic why a rung where one ends hits the table, the other end will accelerate faster than gravity. The force making the rung rotate around impact A is relative to g*cos(Θ) which will always be less than g and so less than free fall from gravity.

Some of the momentum of the falling rung is turned into rotational energy, but that subtracts from the total velocity of the center of gravity of the rung. Is this enough to exceed just gravity?

Plus we have to remember things falling are accelerating, it’s not a constant velocity situation.

Because the end that strikes the table first is now accelerating in the opposite direction, not stopped dead. The rung is not rotating around impact A, it’s rotating around its centre.

Factoring in all sorts of things about degree of elasticity of the impact etc. of course.

Torque and angular momentum follow a set of rules that are equivalent of the Newtons 3 laws and conservation of momentum, and energy … eg a body will remain at a constant angular velocity, possibly zero, unless acted on by a net external torque.

eg look at the static force when the low part of the rung is touching the table… gravity can be seen to be a force acting on the centre of the gravity of the rung… but the axis of rotation is at the impact site… so there is a torque from gravity … and the impact of the rung is creating a force up, at the extreme away from the CoG , with a complication that the force over time is more of an impulse, a huge force acting for a very short time, but anyway its an impluse of force and torque… so there is a torque on the rung, adding to gravity, making it rotate … transmitting as an impluse through the string… And hence that is why looking at the tension in the string "over most of the time " is wrong, the string may transmit a significant impulse in a very short time, and then be slack for most of the time…

All that can be said is that the gravity torque and the impact torque impulse can only speed the falling, not slow it, because the strings of the chain ladder only transmit tension (for the most part… only << 1% … negligible compressive forces … )

I would content that even horizontal and flat shaped rungs could show increase velocity downward , in a gas, due to the sloped surface of the air its squashing out of the way… and more if the rungs were round or wedge shaped and the impact tends to through them outward rather than bouncing them straight back up…

You could minimise the effect of shape using a flat ,eg domino, rung in a vacuum … then do the experiment again with the horizontal and sloped rungs… tune the experiment so that the horizontal rungs could be seen to stack neatly and not affect the speed of the ladder above, but then make the rungs sloped, and they will again affect the speed of the ladder falling.

I have an idea here - let’s say you have a stick moving at a constant velocity, angled up as in the video, and one end (end A) hits a solid object, and the materials are such that end A doesn’t just stop with a thud, but bounces back the opposite direction.

In this case, will end B, immediately after A makes its impact, continue at the same speed, slow down, or speed up? I haven’t thought this through completely, but it seems possible that it might speed up due to the stick now having some angular movement about its center.

If this is the case, it explains why the impacting ladder would speed up. The B end of each rung pulls down on its string connected to the rung above, thus giving a little more downward acceleration to the whole thing.