15 uncoupled simple pendulums of increasing lengths dance together | Hacker News Reader

15 uncoupled simple pendulums of increasing lengths dance together

sciencedemonstrations.fas.harvard.edu

258 pointsSandB0x25 comments

This made curious what patterns emerge when the objects move along circles instead. So I made this 10 minute hack to simulate it:

http://www.gibney.org/spiral_clock

Very pretty. It's a lot less crowdy if you change line 105 in your script to

  var speed =(nr_elements - i)/1000;

Yeah, then the inner objects move faster and the outer objects move slower. I find it hard to say, which version I prefer.

Physics prefers that change, at least when gravity drives the motions.

If you adjust the periods as distance^1/3, you get a simple model for a solar system.

Interesting visual side-effect: watch the spiraling to completion, then flip back to HN and watch the text swirl.

This version is nice because it's even clearer what's going on at integer-divisor points in the process, with different numbers of "arms" on the pattern.

Wow, it is beautiful. I'm trying to wrap my mind around the scheduling and the scale. It looks like it's calibrated to run an entire cycle in an hour, so if you see 6 arms (say) at the center, it's been running for 1/6 of an hour.

I'm trying to grok how the outward propagation works. It seems that a structure at the core propagates outward, while the core reorganizes itself into the next integral division (say 1/5). So you can get 5 arms at the core and 6 arms near the outer edges at the same time.

Check out this capture image. 4-way symmetry at the core, 5-way symmetry in the clusters halfway out, 6-way symmetry (faint but recognizable) at the outermost edge. http://www.dos486.com/misc/spiral-clock.gif

You also get recognizable structure for non-integral divisions, say 2/5. These structures are shorter-lived because each element is passing by an element two spins away instead of one so they converge and diverge faster.

it seems as if there is another interpretation lurking here... that you can also explain this phenomenon as a single wave of increasing frequency in time observed at 15 points. Because of the discrete nature of the points, there is an aliasing effect as the wavelength of the wave gets shorter. For instance, once the wavelength is equal to the spacing between pendulums, all pendulums will line up. When the wavelength is twice that of the pendulums, they will seem to alternate, etc.

This explains the awesome visual effect of this experiment. Though I'm not exactly sure why adjusting the lengths of the pendulums would mimic a wave of traveling with ever increasing frequency. Maybe it has something to do with dispersion (we see an overlap of waves that travel at speeds proportional to their frequency/length)?

A great experiment would be to have two of these pendulum systems side by side, except one is made of 30 pendulums at half the spacing. Then when the 15 pendulum system is lined up, the 30 system should be alternating,etc.

You wish for it, HTML5 delivers. http://pepsdev.com/pendulums/

A slight tweak to draw the pendulums too makes the simulation look much nicer:

      function draw (Ls, t, h) {
         // only for in the program, needed?
        for (var i = 0; i < Ls.length; i++) {
          ...
          ctx.lineTo(h * L, 0)
          ctx.arc(h * L, 0, 6, 0, 360)
          ctx.stroke()
          ctx.fill()
          ...
        }
      }

to do it right you really need to start another path, but this works fine in Chrome (looks little wonky in Firefox)

thanks, that's much nicer indeed.

awesome! at 60 seconds my theory is confirmed! thanks

Don't miss the moiré patterns in the wires.

My friend's version: http://sorryrobot.com/pendula/

The hard part about understanding this is modeling the position of each pendulum easily. If you want something that cycles from 0 to 1 and back try: Absolute value of (x mod 2 + 1) note:drop the abs to see direction. If they all cycle in 60 seconds it's easy to see that any cycle will start over. At 1/2 the full cycle time they will split into left and right sides etc. Now, work backward after the fastest ball does 4 cycles you can see something that looks nothing like a sine wave.

However, you now need to substitute that position as time into the pendulum equation. At which point you see the sine wave behavior of sticking near the edges as the ball picks up momentum followed by a fast transverse through the center. Still, it's not really a sine wave but it's fairly close. To get a true sine wave all the pendulums would need to be the same length and the balls dropped the same distance but at slightly different times. However, in the experiment they are close to the same length and distance fallen so it looks about right.

Related: http://wheelof.com/whitney/index.php?var=v6 Warning: Flash + sound

The mathematics behind this are pretty simple. It's just that the human mind is great at picking up patterns.

That is an awesome site. Thanks! I've never seen anything like that before.

Hypnotic video.

Only vaguely related, but this reminds me of a story a couple of years back involving a machine learning system that was able to derive the laws of Newtonian physics from observing the motion of a hinged pendulum.

http://worldsciencefestival.com/videos/robot_learning_rebuil...

There's a lot of information buried in the motion of these objects :)

The Exploratorium in SF has one of these you can play with. Show up at their monthly "After Dark" and you won't have to compete with the kiddos.

You can see a physical example of this at Technorama (in Switzerland) for those of you with kids that enjoy a good day out.

Here's my attempt (requires WebGL, realistically only works in Chrome):

http://www.glowscript.org/#/user/David/folder/Public/program...

Here's an exposition on the math: http://hippomath.blogspot.com/2011/06/pendulum-waves-mathema...

I've seen this setup in the Mathematicum which is a museum dedicated to math, in Gießen, Germany.

I really didn't know this was a phenomenon..

It looks like a sorting algorithm visualization.

s/vinostroj/vlnostroj

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