Why the Mythbusters Newton’s Cradle failed

10 December 2011

What with the Mythbusters having a bit of a mishap this week (a vast understatement by the way, and thank goodness no one was injured), folks have likely forgotten about their most elaborate and ambitious project of the current season: the supersized Newton’s Cradle. The thing was enormous, consisting of giant orbs hung from steel girders suspended over an empty drydock. It was an awesome concept.

But it was also a dismal failure.

Why? Well, there’s the inherent difficulty of precisely aligning such a massive structure such that the balls are in a perfectly straight line and a minimum of energy is lost to sideways motion. This was what much of Adam’s and Jamie’s fine-tuning addressed—but try as they might, they couldn’t get the giant clack-clacking effect they’d hoped for.

An in-depth analysis on Wired goes into the physics of a Newton’s Cradle, and what might have gone wrong, but ultimately punts a definitive conclusion by stating that “the camera angle wasn’t the best for analysis.” Now, I am not a professional physicist, but I think a hint at the real problem may be summed up in one comment in that Wired article: “It seems that these balls are not elastic.”

Right. See, the balls they used were not the solid steel balls of an ordinary Newton’s Cradle, scaled up, which apparently would have been prohibitively expensive to acquire. Instead they were homemade: spherical steel casings, each with a thick steel disk at the equator and both hemispheres filled with concrete.

As I said, I am not a physicist, so what follows might be off-base. But my impression of the impact event in a normal Newton’s Cradle goes like this:

  • When one ball strikes another, the first ball’s momentum is transferred as a force acting on a single point (ideally, that is) on the surface of the second ball.
  • That force of impact radiates in all directions through the second ball. The energy can’t escape from the ball (except for the bits that become heat, and that clacking noise), so as it crosses through the interior of the ball, the energy that reaches the surface is reflected (or refracted?) back into the interior.
  • Ultimately all that energy converges back at a single point on the surface of the ball, exactly antipodal to the impact point.
  • That energy convergence causes the ball to react and move—and if there’s another ball touching that convergence point, the energy is transferred into that next ball, and the Newton’s Cradle does its thing.

So far, so good. Here’s the problem: as I said, the interior of the Mythbusters balls were mostly concrete, not steel. Therefore most of the energy entering each impacted ball was muddled, diffused, slowed as it moved through that medium. Only the energy passing through the steel equatorial disk—a small fraction of the whole—was transferred efficiently into the next ball. The result was as seen on TV: powerful action, anemic reaction.

I believe that, had the Mythbusters used enormous, solid, hardened steel balls for their giant Newton’s Cradle, they might have come up with the amazing visual they—and we—were all hoping to see.

  1. February 15th, 2012 at 05:12 | #1

    It’s actually a little bit simpler than that. You don’t need to care so much about “a single point ideally” or “energy converges back” or anything like that. It’s true that there must be dissipation, which means that kinetic energy isn’t being conserved. So let’s just go through the motions:

    *Conservation of energy* states that energy is a stuff which cannot be created or destroyed, but can only be transmitted — and at least kinetic energy is given by the formula K = ½ m v².

    *Conservation of momentum* states that unidirectional momentum is also a stuff which cannot be created or destroyed, but can only be transmitted — and momentum has the formula p = m v.

    Now if you just handle the case of a Newton’s cradle of two identical balls: one comes in at speed s, and one is at rest, and afterwards they have velocities u and v, which might be positive or negative. Our equations say:

    s = u + v (conservation of momentum)
    s² = u² + v² (conservation of kinetic energy)

    But a little algebra tells you that (u + v)² = u² + v² + 2 u v. So for both of these equations to be true together, it must be the case that 2 u v = 0, so that one of the balls must always remain stationary, if kinetic energy is conserved.

    But now I claim I’ve solved this for one ball crashing into an entire Newton’s cradle: because that is just a bunch of individual ball-crashes, and each individual ball-crash must work the above way. (And they only tested one ball crashing, so there.)

    Now, conservation of momentum in their experiment is pretty much *inescapable* because the only thing that the balls can really push on, other than each other, is the air — and wind resistance is not going to be particularly important here. So it *has* to be that the energy is not getting transmitted to the next ball, but is getting wasted.

    If energy is wasted then we have:

    s = u + v (conservation of momentum)
    s² > u² + v² (incoming kinetic energy is greater)

    s² > s² – 2 u v
    u v > 0

    This means that both of the balls must end up with positive velocity, for us to lose energy. And this is exactly what you see in the video: the whole Newton’s cradle starts moving forward, no part of it really stops.

    The nice thing about this analysis is that the wrong words are clear. You don’t say “energy entering each impacted ball was muddled, diffused, slowed” — it was simply lost. It doesn’t matter how it was lost; it was lost.

    The question is just where the energy goes (since we do still have a conservation of energy law), and that’s not quite so clear. When you watch the video there is a loud “crack” of sound, and sound carries energy, so that’s a little of it. Another thing which can happen right at the impact site is that the impact could set up a vibration in the whole sphere. Such a vibration would eventually turn into heat. I *think* this is what you mean when you say that the energy is “muddled”. By itself, shifting to steel wouldn’t necessarily help with this, and I’m inclined to say that it’s not my first suspect. The reason why is that the speed of sound in solids is much, much faster than that ball was going.

    If you have a Slinky, you might like to grab it by one end and hang it out in front of you, and try to excite vibrations in it. You’ll notice that there is one frequency, the “resonant frequency,” where you’re pushing just as the thing wants to be pushed and pulling just as the thing wants to be pulled. With a very tiny force, you can build this Slinky into moving very high. The same happens with children on swing sets, who move just their legs a little and can nonetheless drive the swing to go very high — until the chain starts to slack at the top of their flight, which (a) makes them feel endangered and kills their motivation, and (b) ups the energy losses quite a bit.

    What I want you to notice about the Slinky, though, is the two opposite regimes. The first one I want you to try is to vibrate your hand up and down at a rate much faster than the resonant frequency. You’ll notice that the Slinky does seem to vibrate, but the bottom actually really doesn’t move anywhere at all! The thing doesn’t really absorb the energy, and if you stop vibrating then the thing quickly quiets down.

    The next one is the regime that a Newton’s cradle is in: try to move the Slinky up and down very slowly, compared to its resonance frequency, and you’ll notice that the bottom more or less just follows the top, and it again doesn’t absorb much energy.

    You might be indirectly right, though: if the deforming steel ball is somehow pulverising the concrete at the steel-concrete interface, that would be a pretty serious energy loss. A good test here, to keep the cheapness of the original design, would be to just use hollow steel balls. There’s no reason why you can’t use a hollow ball in a Newton’s cradle; it might have odd new vibrational modes, but if it’s thick enough then those modes shouldn’t matter much. So it would be a nice test to see if the concrete is really the shock-absorber material that you’re claiming it is. ^_^

    There’s another interesting energy loss shown off in the videos which might be important: the wrecking balls, especially the ones in the middle, actually start to *rotate* back and forth. That’s pretty significant, because when you scale up a ball, its mass scales up like r^3, but its moment of inertia scales up like r^5, so rotation will scale differently as you double the size of the experiment. I haven’t figured out roughly how much energy was contained in these rotations relative to the collisions, though.

  2. February 15th, 2012 at 12:28 | #2

    Thank you for the extensive analysis. As I recall, they did try using hollow balls in early experiments, and found they were mostly good for making dents in each other. Your comments about resonance frequencies makes me wonder: since each ball, hanging from a cable, is acting as a pendulum, is it possible that the length of the cable affects the responsiveness of the ball when struck? Perhaps the best Newton’s Cradles work well because their dimensions are somewhere in a “sweet spot” of ball size and weight, cable length, etc.

  3. C.F.Reed
    September 18th, 2013 at 04:29 | #3

    Following up on the use of concrete to fill the balls: the density of good quality construction concrete has a density of 2.5 – 2.7 gm/cc, approximating that of aluminum/aluminium, and nowhere the density of steel at 7.8 gm/cc. Therefore, I think that scaling, or lack of scaling is also a contributing factor. I do agree that there would have been damping of the energy in the mass of the concrete. Also, the concrete was only allowed to cure for 14 days, whereas construction concrete, such as that that appeared to be used on the program, typically reaches working strength at 28 days (I have worked in a concrete testing laboratory), and under favourable conditions will continue to gain strength almost indefinitely.

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