From order to chaos and back again
To Do and Notice
What do you see?
Notice that there are 9 pendulums.
The length of each pendulum is different and yet the pendulum bobs are all at the same height.
Lift the large lever to pull all the pendulums to one side. Drop the lever quickly to release them at the same time.
Notice that the pendulums move together at first, then some begin
to change position relative to their neighbors, creating the
eponymous snake pattern!
Eventually, all apparent pattern is lost until suddenly, every other
pendulum is moving opposite its neighbors.
Then chaos returns again.
Eventually, all the pendulums return and move together in one
line.
Notice that the pendulums are numbered, the longest says 15 swings in 30 seconds, the next 16 in 30 seconds, and so on to the shortest which says 24 swings.
You can study individual pendulums by moving the ones you do not want so that the pendulum is caught behind the plastic shield. (The shortest pendulum can be wrapped around its support structure.)
Take a stopwatch and verify the timing of the longest pendulum, which is claimed to make 15 swings in 30 seconds.
Get the pendulum swinging and start the stopwatch when the pendulum is closest to you at a count of zero. It is important to start interval measurement with a count of zero not one. Count each time the pendulum is nearest to you. At the 15'th count stop the watch.(For more on counting from zero go here.)
Notice that after 30 seconds the first pendulum will have made 15 swings and the second exactly 16, thus they return to swing together. In fact note that each pendulum will have made exactly one swing more than its longer neighbor.
Even and Odd
What happens after 15 seconds? The first will have made 7.5 swings and the second will have made 8, they will be half a swing different and so at opposite ends of their swings, one near you, one far away. In fact after 15 seconds all of the even number pendulums will have finished an integer number of swings while all of the odd ones will have completed a halfaswing. This creates the pattern in which each pendulum is moving opposite to its neighbor.
Math Roots
Differences
Look at just two pendulums, start with number 15 and 16.
Notice that when they start together, after 30 seconds they return to swing together only once at the end of the interval. Notice also that the difference between 15 and 16 is 1.
Now look at pendulums 16 and 18, also 15 and 17. Both pairs swing together twice in 30 seconds. The even number pair match up half way through the 30 second interval. Half way through they have made an integer number of swings, half of 16 is 8, half of 18 is 9. The odd number pendulums also match half way through, except each matches on a half swing. Half of 15 is 7.5, half of 17 is 8.5.
Notice also that the difference between the numbers of swings in 30 seconds of each pair is 2.
Between pendulums 15 and 18 the difference is 3, notice that they swing together 3 times in 30 seconds. They synchronize after 10 seconds, 20 seconds and again at 30 seconds. In 10 seconds the 15 pendulum has completed 5 swings and the 18 pendulum has completed 6 swings. The pattern here is that pendulums which make a number of swings that have a greatest common divisor like 3, will synchronize 3 times in 30 seconds, and will have made an integer number of swings.
Pendulums 16 and 19 also synchronize 3 times in 30 seconds, however, after a third of the time, after 10 seconds, the 16 pendulum will have completed 16/3 or 5 1/3 swings, the 19 pendulum will have completed 6 1/3 swings so they will be together, but not after an integer number of swings.
So the difference in number of swings per 30 seconds between pendulums gives the number of times they repeat their pattern in 30 seconds. However, only pendulums with a greatest common divisor will repeat after an integer number of swings.
Pendulums 16 and 20 have a gcd of 4 and so synchronize 4 times in 30 seconds.
What happens with pendulums 18 and 24? They have a gcd of 6. How about pendulums 15 and 20?
The lengths of the pendulums.
Recall that the frequency of a pendulum, F, in oscillations per second is inversely proportional to the square root of the length of the pendulum, L.
F a 1/(L)0.5
Thus if we number the pendulums by their frequency, e.g. F = 1 per 30 seconds, F = n per 30 seconds etc. then the lengths of pendulum number n is L a 1/F^{2 }a 1/n^{2}
and the series of lengths of pendulum number n goes down inversely proportional to n squared.
The exact equation is F = 1/(2p)(g/L)^{0.5} Go to Pendulums to do a complete exploration.
Choosing the timing
Why did we choose the number of swings in 30 seconds instead of in a minute? The number of swings of the pendulums per minute would be 30,32,34...48. By choosing the number of swings to be per 30 second interval the number of swings becomes different by consecutive integers: 15,16,17...
Scientific Explorations with Paul Doherty 

2 August 2002 