Slinky Drop

David Beckham was not known for excellence at Physics. But his intuition about where a ball would go when he kicked it was astonishing. And though our skills are not quite so exceptional, we all rely on our intuition about the how the world is going to behave in unexpected situations.

But  sometimes our intuition fails us completely.

I recently experienced such a failure. As the video shows, if a slinky spring is suspended from one end, hanging under its own weight, and then the support is removed, the slinky falls in an unexpected (to me) manner. The top of the spring moves down, but initially the bottom of the spring does not move at all until the top collides with it.

Slinky

Illustration of the stages of falling of a ‘slinky spring’. The top falls first and the bottom remains completely motionless until the top hits it. This wasn’t what I expected.

I don’t intend to try to explain this to you. But I did spend some time trying to explain it to myself. I couldn’t think how to model a slinky spring, so instead I modelled a similar situation – a set of masses connected by springs, initially suspended and then dropped.

Slinky 2

A system of weights and springs. Initially the weights (1 kg) were separated by springs at their normal length (30 cm). When the system was suspended, each spring stretches by a different amount. When dropped, this system should display the same effect as the slinky but be easier to model.

I was able to write a short program to solve all the sums for how the springs and masses behaved, using Newton’s laws of motion thousands of times to see how the masses moved. I then worked out the extension of each spring over time and results are shown in two graphs below. The key understanding is that each mass responds only to gravity and the two springs above it and below it in the chain.

The first graph shows how the springs first stretched, with springs near the top extending more until they settle down, and then after 3 seconds the top spring is removed.

Slinky Graph 1

Graph showing the extension of each spring in the model versus time. The spring at the top extends more because it has to support the weight of the entire ‘chain’ of masses. The graph below shows in detail what happened in the red dotted section – after the top spring was ‘cut’.  Click for Larger Version

The graph below shows detail from the first graph.  When released, the top springs reduce their extension quickly, but the bottom springs didn’t change their extension at all until the the upper weights made them move. If one draws a line on the graph its slope corresponds to the speed of a ‘sound wave’ propagating down the chain of masses and springs.

Slinky Graph 2

Graph showing the extension of each spring in the model versus time after the top spring was ‘cut’.  Springs at the bottom are not affected until the disturbance reaches them. Click for Larger Version

So now I feel I understand what is happening. But each time I see the video, I am still amazed. You can see the beautiful video made by the ever enthusiastic Veritasium here. I hope it inspires you as much as it inspired me.
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The model I solved considered 10 x 1 kg masses separated by springs with a regular length of 30 cm and a spring constant of 500 newtons per metre. The expected speed of sound is approximately 6.7 metres per second, and the simulation gives a result close to this.

7 Responses to “Slinky Drop”

  1. teddnet Says:

    I love counter-intuitive observations. Strangely though, this wasn’t counter-intuitive to me. After all, the only thing that happened to the spring was that an upward force on the top end was removed. No extra upward force was applied to any part of the spring. So no part of it should accelerate upwards. The only way the bottom of the spring should move upwards on release is if it was constrained from moving up in the first place, but of course it was not. It was, to use a technical term, dangling.

    • protonsforbreakfast Says:

      If that wasn’t counter intuitive then you are even cleverer and more clear-thinking than I had thought! I think the reason this looks so odd (to me at least) is because of the truly exceptional properties of the slinky. It is very compliant resulting in massive extensions and a very slow speed of ‘sound’. M

  2. Nestor Patrikios Says:

    Very nice. In the Veritasium video you can see the bottom end, although amazingly still, does start to unwind almost immediately – the twisting impulse propagating more rapidly than the longitudinal wave.

    • protonsforbreakfast Says:

      Yes, its very striking. I have wondered about the significance of that. I guess it tells the same story. After releasing the end the torsion of the slinky is only released after a time at which a torsional wave can reach a particular point on the spring. Veritasium’s video is so beautiful and wherever he shoots it is always sunny. All the best. M

  3. Physics is more than applied mathematics | Protons for Breakfast Blog Says:

    […] don’t mind if you don’t read the article, but do enjoy the […]

  4. Eusa Says:

    The explanation of The General Relativity is simple:

    The slinky accelerates upwards and the tension gradient in the slinky is equivalent to its proper acceleration. When the dragging releases, all the tensions in the slinky must be released in order from the release point. The duration of levitation of the bottom part is dependent of the mass energy (increase of time) of the slinky and the stiffness (decrease of time) of the slinky spring.

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