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# Pendulum Waves (page 2)

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Author: Christopher Crockett

### Why?

The lengths of the pendulums are designed so that all of them complete a different whole number of swings every 30 seconds. The first (longest) pendulum swings 25 times in 30 seconds, the next one 26 times, the next one 27, and so on; the final (shortest) pendulum completes 33 swings in the same interval. This means that every 30 seconds, all the pendulums will swing to one side together.

Everything that happens in the middle of this interval is a stunning display of many pendulums, each with a slightly shorter period than the previous one, moving in and out of phase with one another. As the shorter pendulums start getting ahead of the longer ones, they slightly “lead” the ones next to them and create a wave effect along the meter stick. At 15 seconds—halfway through the 30-second cycle—every other pendulum (starting with the second longest) will have completed a whole number of cycles, while the remaining pendulums are all synced together at a “half cycle”. When this happens, half the pendulums are all grouped together on one side, with the remaining pendulums grouped together on the other side.

### Going Further

You can modify the apparatus to add more pendulums or create a longer cycle than 30 seconds. The trick is to first decide how long you want the overall cycle to be (we used 30 seconds in this design, but it can be as long as you like). Then, decide how many times you want the longest pendulum to swing back and forth in that interval. With that in mind, the period of each successive pendulum has to be set so that it swings one more time than the previous one in the same interval.

Remember that the period of an ideal pendulum depends only on its length:

• where g = 9.8 m/s is the acceleration due to gravity,
• L is the pendulum length (in meters), and
• T is the period in seconds.

The number of times (N) a pendulum will swing in the overall cycle is represented by

• where Tmax is the overall cycle length in seconds.

Alternatively, you can use the following shortcut to design your own pendulum wave apparatus:

…which tells you the length, L(n), of the n-th pendulum, where k is the number of cycles the whole apparatus goes through before repeating the pattern.

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