Impossible Slinky

What happens when you drop a Slinky? Specifically, what happens if you let it dangle down motionless, then let go of the top end? It falls, right? And maybe bounces around a bit as it does so?

Here’s a slow-motion video. I came across it the other week. Unless you’ve either seen it before or carefully worked out the physics, I think you’ll be quite startled.

It looks utterly impossible. Most of the Slinky apparently levitates, remaining almost completely stationary for most of the time, while the top part falls. YouTube being YouTube, last time I looked some of the comments were from people refusing point-blank to believe the explanation in the video. I’m not sure what they wanted the explanation to be—maybe some kind of impossible magic more in keeping with the impossible-looking behaviour? I don’t know.

The video talks in terms of information about what’s happened at the top of the Slinky taking time to get to the bottom. Although that’s accurate, I found a different way of looking at it more helpful. I need to explain a bit of very basic mechanics, though. Just one of Newton’s Laws of Motion.

Before Newton, a popular idea was that in order for something to move, you had to apply a force to it. Give it a push or a pull. This seems reasonable from everyday experience: something resting on the table will stay there until some force makes it move, and to drag a heavy object across the floor you have to keep on pulling all the time it’s moving.

That’s not actually the case, though. Everyday heavy objects need a force to keep them moving because there’s another force, friction, resisting the movement. What forces do is to accelerate objects: that is, to change their velocity. Start something moving in empty space and it’ll just keep on moving in a straight line unless you apply some force to change the motion. How much force? Newton’s Second Law answers that: force equals mass times acceleration. That is, to accelerate something weighing 2 kg by as much as something weighing 1 kg, you need to push twice as hard.

“Yes but what about gravity?” I hear you saying (whether you are or not). “That’s there all the time, and I can feel it pulling me down into my chair, but I’m not accelerating at all. I’m just sitting here. I’m not moving and I’m not starting to move either. And all the things on the table are just sitting there too.”

The answer, of course, is that there are two forces acting on you: gravity pulling you down, and the chair pushing you up. They cancel out so you don’t accelerate.

“Yes but why do they cancel out? Why is the force from the chair just the right size?” That bothered me when I first learnt basic mechanics. One answer is simply to cite Newton’s Third Law: “To every action there is an equal and opposite reaction”. The chair obeys this and reacts to gravity pulling you down onto it by pushing up against you. The Third Law says the forces are equal and so they are. But that doesn’t seem much like an answer. It amounts to “The forces balance simply because they do, and we’ve got a name for it: Newton’s Third Law of Motion.”

OK then: what would happen if the chair didn’t push quite hard enough? It wouldn’t quite cancel out the gravitational force on you, so you’d accelerate downwards. Into the chair. Squashing the upholstery and thereby making it push a bit harder, so you accelerated less . . . you’d overshoot a bit and bounce up again . . . and after a few bounces you’d end up stationary, at exactly the right point for the two forces to cancel out. The forces have to be equal because if they weren’t, they’d adjust your position until they were.

If you put a mug of tea on the table, the same explanation applies, but on a smaller scale. The outer electrons of the atoms of the two objects are repelling each other. Push them closer together and they repel more. Instead of macroscopic bouncy upholstery we’ve got a microscopic bouncy electric field. The mug rests in just the right place for  the repulsion to cancel out the gravity trying to pull it through the table.

The key thing is: stationary objects around us stay stationary because the net force acting on them is zero. They accelerate when the forces are out of balance.

How does all this relate to the Slinky? And why, when you let go of the top, doesn’t the force holding it up disappear so gravity wins and it accelerates downwards? Why does it “levitate” so counterintuitively? Surely it’s disobeying everything I just described?

Consider just one small piece of the Slinky. Imagine holding it in your hand, gently pulling one turn of it away from the next. To stretch it further, you have to pull a little harder. A given amount of stretch requires a precise strength of pull; a given strength of pull produces a precise amount of stretch.

Now consider the dangling, stationary Slinky. Think about just one turn of the spring. It has three forces acting on it:

• The turn above is pulling upwards on it. How hard depends solely on how far apart the two turns are.
• The turn below is pulling downwards on it. Again, how hard depends just on how far apart the two turns are.  (They’ll have stretched just enough to support the part dangling below.)
• Its own weight is also pulling downwards.

Before the Slinky is dropped, the three forces are in balance, so the turn we’re looking at remains stationary. It has to; moving would require accelerating. That would require at least one of the forces to change, so they didn’t balance any more. They won’t change unless one of these changes:

• The weight of this turn of the Slinky—which is constant.
• The distance between this turn and the one above. This requires the turn above to move (this one won’t until the forces change).
• The distance between this turn and the one below–i.e. the turn below must move.

In other words, the only way for a turn within the Slinky to start moving is for an adjacent one to move first.

The top turn is supported not by another turn above, but by an upward force from your fingers. When you let go of the Slinky, you simply remove this force. Now the only forces acting on the top turn are its own weight and the pull of the turn below. So it accelerates downwards, becoming the first one to move. This moves it closer to the turn below. The reduced stretch means the two turns don’t pull on each other quite so hard, so the one below no longer has quite enough upwards pull on it to keep it stationary, and it too starts accelerating.

But further down the Slinky, where nothing has moved yet, the turns are spaced  just as before. Since the pull between them depends only on their spacing, it is unchanged. Neither has the pull of gravity changed. So the forces remain in exactly the same balance, happily continuing to cancel each other out. There’s nothing to start those turns moving; they have to remain stationary.

You know from playing with such things that if you waggle one end of a Slinky or similar, a wave of movement travels along it. The same happens here: it takes time for the pulling together of adjacent turns to travel down the Slinky. And since none of the turns can start falling until the wave reaches them, they remain suspended in that impossible-looking way.

Yes but—it still seems wrong. What’s holding the bottom part up now you’ve let go? How can it hang there with nothing to hang from?!

Imagine doing pull-ups: you pull yourself up towards the bar by pulling down on it. Similarly, as the bottom part of the Slinky hangs down from the top part, it supports itself by pulling down on the top part. Once you let go, it continues doing exactly the same thing. It’s still hanging from it. The only difference is that the downward which previously went into resisting the upward pull from your hand now goes into making the top section fall faster than it would under gravity alone. It’s being forcibly accelerated downwards,  and the reaction to that accelerating force is what supports the stationary section.

The Slinky is simply behaving the way it has to.

And yet, it still looks impossible. And it took more words to explain than I expected.