# Does the attached mass go up or down in the spring paradox?

Take a look at this video.

It shows that when 2 springs in series are transformed into 2 springs in parallel the attached mass goes up. But I'm interested in what happens in that split-second after cutting the blue cord but before the green and red ropes become taut (i.e. when the springs are pulling their respective ropes but the latter are still slack). Does the attached mass go up during that time? Down? Or does it maybe depend on the actual springs used (their pulling strength or something)?

### Summary

We do not know if the hanging weight will have a an instant of going downward or not

It depends on the mass and spring constant of the lower spring and mass of the hanging weight.

### Logic

We cut the rope. There is gravitational force on the weight. The springs begin to contract rapidly and contract until the strings are tight. (John Hunter’s answer assumes the mass has to descend to make the strings taut, but the contracting springs do most of the moving to tighten the strings).

Assume the strings have no mass.

After cutting but before tight strings, the [lower spring / hanging weight] is a free system not affected by the upper spring or weightless un-taut strings. Let’s consider this system during that time:

The system as a whole is only acted on by gravity so as mentioned by alephzero it will fall - it’s center of mass will accelerate downward at F/M = g, the acceleration of gravity, where M = m_spr + m_hw (spring and hanging weight).

In addition to that, the spring (which has mass) will contract. As alephzero said, if the hanging weight has a much much higher mass than the lower spring, the mass will go down as this system falls (until the strings are tight).

But what if the spring doesn’t have a mass that is negligible compared to the hanging weight? There are cases where the spring constant and/or spring mass are high enough that the mass never (even instantaneously) has a downward acceleration of velocity. The spring contracts with enough inertial force that it’s instantaneous force on the weight is greater than m_hw g.

To be sure there are such cases, take an extreme case where the mass of the hanging weight is negligible compared to the spring’s mass and the spring constant is high. The spring will begin to fall under its own weight while contracting rapidly about its center and its lower end will go up (hence the hanging weight will)

• This seems to be the most complete answer. I'm glad you decided to write it after all. :) Just to extend it further - assuming the ropes do have mass (but lower density than either the spring or the hanging weight) does the green rope affect the [lower spring / hanging weight] system and actually slows its fall down (ever so slightly)?
– NPS
Jul 30 at 15:34
• Thanks. The density isnt the main parameter because they have different thickness/size/shape. Yes I guess we would have to consider a very small portion of the string. Not the whole thing because it is loose. If the string was hanging down off the weight we would just count its whole mass as added. That is an extremely complex problem with deformation forces on the string (it is unbending which does not take zero force). At first we can think of it as a J connected to the spring and the weight the spring bears equals the short end of the J. If things happen fast enough thats good enough. Jul 30 at 15:43
• I do wish i had been slightly less obnoxious about it though. Commenting everywhere about who was wrong, and deleting and and re-commenting. Why only this question got me so obsessed have no idea. Lmao those engineering debates.. look out! Probably this crappy med Im taking for another two days. (Tmi 😉) Jul 30 at 16:12
• @ANPS This doesn't seem right, it was said "John Hunter’s answer assumes the mass has to descend to make the strings taut, but the contracting springs do most of the moving to tighten the strings", but if that were the case the contracting springs provide less force to support the weight (the extended spring previously balanced the weight), so the weight would drop briefly. Jul 30 at 16:24
• @Al Brown, P.s. no worries, a good question has a way of getting people obsessed! Jul 30 at 16:26

Consider the bottom spring and mass as a single system.

When the blue string is cut, there is a net external force on the bottom spring+mass equal to their weight. Therefore the center of mass of the bottom spring+mass will start to move down with acceleration $$g$$.

However there are also the internal forces in the spring which are no longer balanced by the force from the blue string.

Assuming the mass of the spring is small compared with the weight, this will cause the top of the spring move downwards with an acceleration much bigger than $$g$$, and the mass to move downwards with an acceleration slightly smaller than $$g$$.

Similarly the bottom of the top spring will move up, with an acceleration much bigger than $$g$$.

When the green and/or red strings become tight, the situation changes because of the tension in those strings.

In the experiment where the blue string is cut, all this happens so fast that you can't see the weight's small movement down before it starts to move up. A high speed video would show it, if there was also a fixed vertical scale visible to measure the position of the weight accurately.

• Yeah the thats exactly right. I answered adding qualitative analysis of the case where mass of lower spring is not negligible compared to mass of hanging weight... the spring could weigh much more than the mass and the mass will only go up Jul 30 at 15:28
• @AlBrown But if the mass of the spring is not negligible, the tension in the spring is not uniform, and a proper analysis is more complicated. The limiting case of a heavy spring with no added mass at the bottom would be a good place to start. This gets interesting because the initial extension of the spring, and its acceleration after you cut the string, both depend on the mass and stiffness of the spring. Jul 30 at 18:22

It's a good question.

The answer seems to be about what happens between 55 and 56 seconds on the video.

At 55 seconds, it shows the red string like the left image below.

But surprisingly the string is only about 0.5cm longer than the right image - you can test this by using a piece of string about 30cm long.

So, the weight does drop, but only by about 0.5cm until the string becomes tight and bounces the weight back up.

The video doesn't show it, (it seems only possible to stop it at 1 second intervals), but how long would it take?

Using the equation of motion $$s=\frac{gt^2}{2}$$ where $$s$$ is the drop distance of 0.5cm gives a time of $$0.03$$ seconds, and the video simply doesn't show it, even frame by frame.

When the blue cord is cut, several things happen very rapidly. At the instant the blue cord is cut, the two springs in series become unloaded, and they rapidly contract up until the red and green cords become taut. At that point, the springs become loaded again and their rate of contraction would change in a complicated way based on what the weight does.

While the springs are contracting and before the red and blue cords become taut, the weight is free falling, so it is moving down under the acceleration of gravity. When the red and blue cords become taut, the springs "catch" the weight that is falling. Because the time interval for this catch is so short, the weight has a very low downward velocity, and this velocity must still be stopped and reversed to pull the weight upward. Because the weight has inertia, this means that for an instant, the upward contraction of the springs is reversed by a very small amount. After that instant, the two springs will continue contracting upward at a rate that is somewhat slower than when the weight was in free fall.

Note that it would take high speed video (e.g., 100,000 frames per second) to observe these changes in the cords, springs, and weight, so these changes are not observable in the posted video.

• I answered adding qualitative analysis of the case where mass of lower spring is not negligible compared to mass of hanging weight... the spring could weigh much more than the mass and the mass will only go up. (Consider the case where the mass of the hanging weight is negligible compared to the mass of the spring to see this) Jul 30 at 15:30
• @AlBrown, I answered based on what I observed in the video. Jul 30 at 16:59
• Good point David. Clearly the springs were light in the video Jul 30 at 17:03

This is an excellent question, and its real answer is all about how the springs contract.

As a matter of fact, to answer this question, we cannot think about the springs in an idealized way, we must consider that the springs are extended objects with mass and tension. As such, when the blue rope is cut, the relaxation of the springs happen in the form of a relaxation shock wave that takes time to travel from one end of the spring to the other. This relaxation shock wave has the property that the spring coils in front of it remain perfectly taught until they are reached by the shock front. To these parts of the spring it is as if the rope had not been cut at all. As the shock front passes a spring coil, it fully relaxes, turning its tension energy into kinetic energy. Once it is fully relaxed, it's moving at constant speed along with the rest of the already relaxed spring coils. The process looks a bit like this:

\/\/\/\/\/\/\/\/\/\/ fully extended, no movement
\/\/\/\/\/\/\/\/\/V
\/\/\/\/\/\/\/\/VV   relaxation shock wave traveling left
\/\/\/\/\/\/\/VVV
\/\/\/\/\/\/VVVV
\/\/\/\/\/VVVVV
\/\/\/\/VVVVVV
\/\/\/VVVVVVV
\/\/VVVVVVVV
\/VVVVVVVVV
VVVVVVVVVV          fully relaxed, moving left
NVVVVVVVV
NNVVVVVV            compression shock wave traveling right
NNNVVVV
NNNNVV
NNNNN               fully compressed, not moving


The key insight is, that while the springs relax and become shorter, the far end of the springs remain utterly ignorant of the relaxation wave that's moving closer. The consequence is, that the long ropes become taught before the weight at the far end of the lower spring realizes what's going on.

With these insight, we can now fully describe what will be happening:

1. When the rope is cut, the weight remains suspended beneath the lower spring.

2. As the relaxation shock wave moves up the upper spring, the red rope becomes taught. The added force along the red rope causes the weight to accelerate upwards.

3. The sudden onset of force from the red and green ropes causes a second shock wave to start traveling across the springs. This time, the shock wave stops the movement of the springs, turning their kinetic energy back into tension.

4. On each spring, a wave of a relaxed portion is moving towards the outer ends, both ropes are taught with enough tension to hold the weight by themselves, and the weight is accelerating upwards at $$1g$$.

5. The relaxation shock waves hit the ends of the springs and reflect off it, becoming compression shock waves in the process.

6. While the compression shock wave moves up the lower spring and the following tension wave has not reached its lower end, the lower spring pushes the weight downwards. This cancels the force from the red rope, making the weight enter free fall.

7. The quickly following tension shock wave restores the force on the weight, allowing it to continue its acceleration upwards.

8. As the waves travel up and down the springs, they become less sharp and dissipate their excess energy, making the springs more smoothly tensioned. At the same time, the weight enters an oscillation around its final position.

9. All these oscillations dissipate their remaining energy and die away until the system enters its new equilibrium.

TL;DR:
Due to the retardation of the signal about the cut wire along the lower spring, the weight first gets the signal of the taught red rope and starts its movement with accelerating upwards.

• Awesome picture! :D But anyway, I'm not sure I understand you correctly. Are you saying it's impossible (in any case) for the hanging mass to start moving downwards immediately after the blue rope gets cut?
– NPS
Aug 6 at 21:23
• @NPS Yes. The red rope starts tugging the weight upwards before the lower end of the lower spring releases its force on the weight. Unless you change the material of the ropes in such a way that they delay the tension wave more than the springs delay the relaxation wave, the first acceleration of the weight is upwards. Aug 6 at 21:42
• I may be way off here but I'm not sure that's true. If we had just a single spring attached at the top and stretched only by gravity and the top end was then released then perhaps the lower end would remain in the same spot until the spring fully relaxed. But if there's a mass attached to the spring's lower end wouldn't that mass pull it downwards (before the ropes became taut)?
– NPS
Aug 6 at 21:59
• @NPS The point is that the acceleration of the springs mass provides the reaction force that keeps the lower part of the spring taught. I know that this sounds a bit unintuitive, but it's the way things work. As an analogy, think of air inside a 300m long, narrow tube that's coiled up. It acts like a long spring under compression (air pressure). Now, if you speak into one end of the tube, you'll hear your words come out of the other end with a delay of one second. Same thing if you suddenly connect one end to an evacuated chamber: It takes a second before the BANG is heard at the other end. Aug 7 at 7:04

In the video he shows what happens when he releases the blue rope slowly. You can see that until the red and green cords become taut, the springs are still in series and the weight goes down.

• I think that's a different situation as here the springs are still connected and still in series whereas when you cut the blue rope the springs are not attached to one another anymore and nothing's pulling them from one side until their respective ropes become taut.
– NPS
Jul 30 at 15:01