Swinging science

I recently saw this video over at Phil Plaitt's Bad Astronomy website.  It's a neat demonstration of 15 pendula (plural of pendulum) with regularly increasing string lengths that produce standing waves, travelling waves, beating, and seemingly random motions over time.  Order leads to apparent chaos which leads back to order again.  Watch it - it's awesome!




The design of this apparatus was based on a 20-year-old paper from the American Journal of Physics.  The period of the longest pendulum is 51 cycles per minute.  Each adjacent pendulum adds one more cycle per minute and the shortest of the 15 pendula therefore has a period of 65 cycles per minute.

The entire dance of the pendula takes 60 seconds to go through one cycle.  If you watch carefully, you'll see the release is around 0:27, half the pendula are at opposite ends of their swings around 0:57 and they come back to their intial starting points around 1:27 (although much reduced in amplitude).

How do you adjust the period (the time it takes for one complete back and forth swing) of a pendulum?  Many people would guess that the mass at the end of the string would determine the period but that doesn't matter.  The period of a pendulum is independent of the mass.

As an approximation (the solution can only be approximated by an infinite series - see this Wikipedia entry for more details).

Where T is the period of the pendulum in seconds, L is the length of the string in meters, and g is the acceleration of gravity (9.8 m/s²).  As you can see, the only thing that you can change is L, the length of the string.  So, if you want a pendulum with a period of 1 second, the length of the string would have to be:

     1 s = 2 p (L / g)^1/2

     1 s = 2 p (L)^1/2 / (9.8 m/s²)^1/2

     (1 s) (9.8 m/s²)^1/2 / 2 p = (L)^1/2

     0.498 m = (L)^1/2

     L = 0.248 m » 25 cm

So, a pendulum with a 25 cm string (about 9.8 inches), will have a back and forth period of 1 second.  Doesn't matter if you hang a 1 gram weight on it or a 100 kg professor.  That's why pendulums are so useful for timekeeping.

Pendula were first studied and used for timing in 1602 by Galileo (presumably inspired by watching a chandelier swing in the Pisa Cathedral during a boring church service).  In 1656, the Dutch polymath Christiaan Huygens (1629-1695) built the first pendulum clock which greatly increased the accuracy of existing mechanical clocks in use at the time in Europe.

Another interesting characteristic of pendula is that their back and forth motion stays in the same plane regardless of the movement of the pivot.  In other words, if the pendulum is oscillating back and forth in a north-south direction, and you rotate the frame to which the pivot is attached, the pendulum will stay in it's north-south orientation.  In 1851, French physicist Leon Foucault (1819-1868) used this property of a pendulum to illustrate the rotation of the Earth on its axis.

Going back to the formula for the period of pendula, we see that the acceleration of gravity (g) is also important.  Well it turns out that the gravitational attraction on Earth varies by about 0.5% from place to place due to variations in the density of subsurface materials.

Since the period of pendula changes with changes in g, they can be used as gravimeters (and once were up until better techniques were developed).  There's actually an entire field of study called gravitational geophysics which studies gravity anomalies on the Earth's surface (shown in the image at left).  One practical application is prospecting for oil and mineral deposits.

Science is not just cool, it's incredibly useful and valuable to society.  It all started with an observant man whose mind wandered during a boring mass in the Pisa Cathedral.