# How are these marbles being accelerated?

This question refers to an effect visible starting at around 5m45s in this video1. (The question will make little sense if one has not first watched the clip.)

The observation

At around 5m45s we see 5 marbles beginning to move, single-file, in a straight line, slanting down from right to left, at an approximately 45º angle. During this diagonal traverse the marbles seem to be almost floating through mid-air. Their motion does not look like what one would expect of marbles falling under gravity. (For one thing, the trajectory is rectilinear, not parabolic.)

This is pretty cool, but it's what happens next that really puzzles me.

Once each marble reaches the horizontal channel-like piece at the bottom of its diagonal traverse, it shoots to the left with much greater speed than I would have expected. IOW, there is an apparent discontinuous jump in kinetic energy that I'm having a hard time accounting for.

The easy part

OK, first, here's the explanation for the odd-looking diagonal motion: the marbles are moving between two rigid transparent plates. These plates are positioned relative to each other at a slight angular deviation from a parallel configuration, so that at any horizontal position, cannot fit below a certain vertical position: it gets stuck there. The locus of all these points where the marble just fits between the plates determines the diagonal trajectory observed in the video.

The not-so-easy part

But what about the subsequent motion along the horizontal channel?

Let $\frac{1}{2}mv_-^2$ be the kinetic energies of, say, the first marble right before it touches the horizontal channel, and let $\frac{1}{2}mv_+^2$ be its kinetic energy shortly after the marble starts moving along this channel.

If when I first saw this sequence the video had been stopped right before the first marble starts moving along the horizontal channel, I would have expected that that $\frac{1}{2}mv_+^2 \approx \frac{1}{2}mv_-^2$, and therefore $v_+ \approx v_-$.2

But this naive expectation appears to be wrong: it sure looks to me like $v_+ \gg v_-$, and therefore $\frac{1}{2}mv_+^2 \gg \frac{1}{2}mv_-^2$.

(It is possible that what I'm calling a "horizontal channel" is not so, and that its leftmost end is lower than its rightmost one, but any effect due to gravity that such an inclination could have would not be apparent right at the beginning of the marble's motion along the channel. IOW, it could not account for the visible jump from $v_-$ to $v_+$. It is also possible that the marbles are getting accelerated through some hidden trickery, but this would be very much against the style and the spirit of these educational videos.)

The only candidate for an explanation that I can come up with is that during the diagonal traverse between the clear plates each marble picks up (somehow) a great deal of angular momentum. IOW, it gets "revved up", as it were. Once it touches the horizontal channel, the friction between the two results in the conversion of some of this high angular momentum into a fast leftward rolling by the marble.

One objection to this hypothesis is that a light smooth marble is more likely to dissipate most of its angular momentum through sliding friction, spinning in place.

But putting aside this objection, how exactly would the marble pick up so much angular momentum during the diagonal traverse?

Alternatively, is there some other explanation for the apparent jump in kinetic energy?

1 The video is a collection of short clips from the Japanese educational children's TV show Pitagora Suitchi (ピタゴラスイッチ or ピタゴラ装置), which means literally "Pythagorean device(s)". (In the US such contraptions are usually referred to as "Rube Goldberg machines".) The phenomenon that motivated this question happens in the clip that begins at around 5m35s. FWIW, this clip is titled "5 marbles" (５つのビー玉), and it dates from 2003. All these clips were produced by the group of Satō Masahiko at Keiō University.

2 Here, I am using $v_-$ and $v_+$ to represent scalar "speeds" rather than vectorial "velocities".

• I'm amazed that even after one puts as much effort as I did in writing this question, the first vote it gets is a downvote... I really don't get what's with you people. – kjo Dec 12 '15 at 18:29
• I downvoted because you linked to a 9+ minute long video and said that the question won't make sense without watching it (you are probably correct, I stopped reading after a few more sentences). It would be much better if you could make this self contained (i.e., no need to watch the video). – Kyle Kanos Dec 12 '15 at 18:30
• You did not read what I wrote. I gave the exact timepoint where the phenomenon I'm asking about takes place. It's trivial to cue the video up to that point, and the phenomenon in question takes only a few seconds. Furthermore, I gave additional details in a footnote. You didn't bother to read and understand any of this, and decided to just downvote somebody. Just like that. Wow. Shame on you. – kjo Dec 12 '15 at 18:34
• Not to worry, kjo. I up-voted you because it is an interesting question. Your linking to the video is no problem with me since, like you said, you did specify the time-index to look at. However, it would help to keep the question shorter and more focussed since a lot of people are not going to want to read through a long description like the one above. – user93237 Dec 12 '15 at 18:38
• If you want to try placing shame on me, feel free. Even with a time stamp, telling someone that it won't make sense without watching any part of the video does not provide for any sort of useful or good question in my books. – Kyle Kanos Dec 12 '15 at 18:40

OK, I get it.

What I had to realize is that if the marbles had rolled down a normal inclined plane with the same vertical drop, their speed along the horizontal channel would be comparable to what's seen in the video. IOW, the kinetic energy at the end is consistent with the potential energy at the beginning.

What made the motion seen in the video confusing to me is that during the diagonal traverse a much greater fraction of the marble's initial potential energy gets converted into rotational kinetic energy, if compared to the more familiar case of a marble rolling down an incline, in which only 2/7 of the potential energy goes into rotational kinetic energy, the rest going into linear kinetic energy. This is why, in the video, the linear motion during the diagonal traverse looks so unnaturally slow.

In fairness to myself, as I said in my original post, I hypothesized that the marbles probably had picked up a substantial angular velocity by the time they touched the horizontal channel.

Many of the comments in essence reiterated this, which I sort of knew already.

My question was in fact asking why this gain in angular velocity.

The following thought experiment was what finally clarified the situation for me.

Imagine first a sphere $\mathcal{S}$ of radius $R$, and a cylinder $\mathcal{C}$ of radius $r < R$ and length $L > 2 R$. IOW, the cylinder $\mathcal{C}$ is both skinnier and longer than the sphere $\mathcal{S}$.

Now imagine the rigid solid $\mathcal{S}^\prime$ obtained by superposing $\mathcal{C}$ and $\mathcal{S}$ so that their centers of mass coincide. IOW, $\mathcal{S}^\prime$ is similar to a sphere of radius $R$, but it has a couple of cylindrical bits sticking out antipodally along one of its axes.

Finally imagine two parallel rigid rails, that are far enough apart that the cylindrical bits of $\mathcal{S}^\prime$ can be rested on them, and $\mathcal{S}^\prime$ can be rolled along the two rails while hanging, as it where, between them.

The whole set-up should look something like this:

If the two rails are tilted by some angle, say $0 \lt \theta \lt \pi/4$, so that they describe an inclined plane, then $\mathcal{S}^\prime$ will roll downhill on the rails, as its potential energy gets converted to kinetic energy.

Key assumption: all rolling happens without slipping.

Let let $KE_l(t)$ and $KE_r(t)$ be the linear and rotational kinetic energies of $\mathcal{S}^\prime$ as it rolls down the inclined rails.

The fraction of the total kinetic energy coming from the linear motion is approximately1

$$\frac{KE_l(t)}{KE_l(t) + KE_r(t)} \approx \frac{\frac{1}{2}m(r\omega(t))^2}{\frac{1}{2}m(r\omega(t))^2 + \frac{1}{5}m(R\omega(t))^2}=\frac{5 \left(\frac{r}{R}\right)^2}{5 \left(\frac{r}{R}\right)^2 + 2}.$$

...where $m$ and $\omega(t)$ are the mass and angular velocity of $\mathcal{S}^\prime$, respectively.

Similarly, the fraction of the total kinetic energy coming from the rotational motion is approximately

$$\frac{KE_r(t)}{KE_l(t) + KE_r(t)} \approx \frac{2}{5 \left(\frac{r}{R}\right)^2 + 2}.$$

IOW, for a fixed $R$, the relative contributions of the linear and rotational motions to the total kinetic rapidly approach 0% and 100%, respectively, as $r \to 0$.

The phenomenon shown in the video can be thought of as the limiting case where $r \to 0$. The points of contact between the marbles and the clear plates play the role of $\mathcal{S}^\prime$'s two cylindrical bits rolling down the inclined rails.

BTW, I must say that I think this video segment is the prettiest demonstration I recall ever seeing of the principle of energy conservation.

1 The expression for $KE_r$ above is actually the rotational kinetic energy for the original sphere $\mathcal{S}$; this approximation becomes better as $r/R$ becomes smaller.

• This really isn't much of an answer to your question, in the SE sense of the term, more a long comment. You don't explain e.g. how when the marbles roll on the diverging rails, translational energy is converted to rotational energy. Or how that acquired rotational energy causes the marbles to re-accelerate in the next stage of their travels. – Gert Dec 13 '15 at 2:37