Robin asked me to put up some of the final videos of the bowling ball pendulum after tuning. They are big files, so you may have to let them load for a bit or lower your youtube resolution settings if you have a slow connection.
This one, from the side, is in slow motion, and it shows a beautiful corkscrew motion as it goes:
Throughout the project, I was impressed by the idea that the patterns we see are equally in the math and in our minds. They are real patterns, but they come about from the relationship of separate objects.
A number of people have asked about the math that governs this project. Here's a run-down:
- The period (length of time for one back and forth) of a pendulum is governed only by the distance from the support to the center of mass of the pendulum. This is nearly the center of the ball in this case, but not quite, as the center of mass is shifted upwards by the mass of the hook-eye and cable. The mass of a pendulum doesn't affect the period.
- The equation that governs the period is Time=2*pi*(length/acceleration due to gravity)^.5
- Once we set the longest pendulum's length, we could figure out how long it took to go 48 times back and forth. This is about 2 minutes, 40 seconds. We called that a "full cycle."
- From there, we could take that "full cycle" time and back-calculate the necessary lengths so the next pendulum would go 49 times in the same full cycle time, the following one 50 times per cycle, etc.
- You'll see that at the half-cycle point (about 1 minute, 20 seconds) there is a moment when every other ball is in line at the "to" and "fro" points; a very fast and brave superman could fly down the middle at this moment.
- After that mid point, the patterns reverse, more or less, to reform a line that starts the original curve again at 2:40.
Though the math is very clean, Matt Tibbits, Theo and Eric Witherspoon, and many others and I spent a lot of time tuning this and trying to get the real thing right. It wasn't so easy, and it never was perfect. However, it got good enough for us to feel satisfied, and we learned some of the limitations of our materials. Here are some things that complicate the tuning:
- While the a ball's weight doesn't affect the period, it does affect the amplitude over time. That is, a heavier ball will start with more potential energy and thus will have more energy to work with in overcoming the friction in the hooks and air resistance. So, for example, the very light red-orange ball doesn't swing out as far as the other balls over time. I considered putting jam on the hooks for the heavy balls and olive oil on the light balls' hooks, but that started to sound messy.
- The lighter the ball, the more a given amount of cable and eye hook weight would throw the center of mass off from the center of the ball.
- The beam from which the balls were suspended flexed a little side to side, and we suspect this changed the balls' motions in some small and mind-boggling ways. We added a stiffener, but still there was some transfer of energy from ball to ball through the beam.
- The hooks started getting loose over time and added some wiggle to our already wiggly re-calculations.
If you're still reading this, you are as much of a nerd as I, and you might be interested in the following equation that predicts where any two balls in the sequence will line up. This came up because we were using a ball-ball-comparison to tune the pendulums and started to see patterns. I include it mostly so I'll have a record of this for when Margot and I rebuild this in the woods below our house this summer:
- let x = number of swings per full cycle of a longer ball in any pair we're comparing.
- let S = any integer or half integer representing a 0 velocity point (swing out, E = .5, back for the first period, E= 1, out for the second swing, E = 1.5 etc.) where the two balls might match up.
- let n = the number of the comparison ball, if counting the first ball as 1 (i.e. n=2 for the neighboring ball, 3 for the third in the line, etc.)
- let q= an integer representing the number of full periods ahead the second ball is that allows it to match the first
and anything that satisfies this equation is a match. If S is a half-integer, the match it on the out-swing, a full integer, the match is back at the starting side.
For those of you who haven't seen it, here is another video from Bev Hill, our bowling ball procurement expert:
Pendulum with Ground Level Observers