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Tubular Balls

Can Gravity Choreograph a Ballet? A demonstration using pendulums reveals an awe-inspiring harmonic quality apparent in standard Earth gravity.

The demonstration. This is a computer generated demonstration which simulates real-world gravity. Two views run simultaneously. The view on the left shows a side-view of the pendula and fulcrum. The right-hand view shows the same gismo end-on. The right-hand view has an optimum angle to observe multiple wave forms.

The word pendulum is from the Latin pendulus; meaning hanging. A simple pendulum is a weighted (bob) suspended from a pivot so that it can swing freely. The pendulums used in this demontration have different lengths. The amount of time a pendulum takes to swing from end-to-end is called its period. The length, or angle of a pendulum's swing is called its amplitude.

What you see. The demonstration starts as 15 pendula are set in motion at the same time - with equal force. Care is taken that they do not jostle one another. As soon as they begin to swing freely the pendula quickly move out of phase. It is these harmonic phase shifts which create the mesmerizing dance. The dancing bobs will display both traveling and standing waves, plus beating and random motion as they pass from chaos to perfect harmony in the blink of an eye. The demo ends when the pendula again match phases, at which point the dance repeats. Play the movie as many times as you like.

How it works. Our movie is 50-seconds long. The shortest pendulum's period is set to 1.0. The next to 1.1 and so on. The longest pendulum's period is 2.4. The period of a simple pendulum is based on length and can be approximated using Formula (1) (shown below) where g is standard gravity for Earth, or the nominal acceleration of an object in a vacuum near the surface of the Earth. The length of a simple pendulum can be determined using Formula (2). Formula (3) expresses Earth standard gravity.

In our movie the pendula have no friction or air-resistance. The rods are massless and the bobs have identical mass and never slow down. The amplitudes of all our pendulums are set to 40-degrees. While amplitude changes do effect the period of a real pendulum - higher amplitude increases the period - it is ignored in this demonstration.

Formulas

About the demonstration. The inspiration for our Pendulum Wave demonstration came from Nils Sorrensen and the folks at Harvard University, who claim they were inspired by Richard Berg at the University of Maryland, who says he learned about it from Moscow State University, who claim they saw it first in America. It is not yet known who came up with the demonstration first.

Tubular Balls Movie:  ©2010 A.A. Azevedo
Music: Tubular Bells, Mike Oldfield, ©1973

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