| MadSci Network: Physics |
Hi Julian,
Interesting question, and I've been pondering it for quite a bit. You see, the motion of pendulums is one of the first things modern physics "discovered"-so we had better have a good answer!
A pendulum is in essence a weight attached to a string (or rod), that is allowed to swing through the air. What basically happens is the following. When we lift the mass in a pendulum, we put energy in it. We can think of this energy as being "stored" in the position of the mass. If we release it, the mass will fall, releasing the energy, but because of the string, it can only fall in one direction. At the lowest point, all the extra height energy (we call this "potential energy" is gone, but the mass has gained speed. In effect, the potential energy is converted in a different kind of energy, kinetic energy. When the pendulum swings up again, its speed decreases, and the energy is re-converted to potential energy. This cycle continues until friction has removed all the energy, which for a well-designed pendulum can take a very long time.
We can make two interesting observations about swings: firstly, that the time it takes them to swing is independent of the attached mass, and secondly, that is independent of how far they swing. Both discoveries have been integral to the discovery of the theories of gravity.
To "prove" the fact that the motion of a pendulum is independent of the mass, consider the moving of one pendulum. This pendulum is swinging, and it takes a certain amount of time to swing. Now, imagine I take another pendulum, which is exactly identical and hence swings just as fast. Now, imagine is use a tiny, interconnecting rod to connect them together. Would they now suddenly start moving faster (or slower)? Doesn't seem logical, right? Of course, doing this, I can put as many small weights together as I want, and this proves (by induction) that the motion of a pendulum doesn't depend on the weight that's on it. Furthermore, it also proves that objects fall with a speed that does not depend on their mass (that little fact took mankind a long time to discover).
The second bit is a bit more difficult. Imagine you pull your little weight twice as high. This would mean the pendulum has to travel further to get from one side to the other. However, the energy stored in it is also twice as large, meaning it moves faster. It turns out these effects (almost) exactly cancel. A rather detailed description is available in [1].
These discoveries, which were made by Galileo, were used by Christiaan Huygens to build a clock. Having something (in this case a pendulum), of which the time of oscillation does not depend on the size of the swing, is of course very convenient for time keeping-all you need to do is count! The only thing our swinging time depends on are gravity and the length of the pendulum.
The effect of gravity is small but interesting. You see, gravity isn't exactly constant on the Earth [2], so a clock in, say, Inquitos, Peru (where you live), gives a slightly different time than one in Amsterdam, the Netherlands (where I live). Using the formulas in [2] (for the computation of the constant of gravity in Inquitos, [3], for the location of Inquitos, and [1] for the formulas describing the motion of the pendulum, we find that the difference in gravity is 0.33%, leading to a 0.166% difference in time. In other words, if I had a clock that would work fine in Amsterdam, it would be slow by 5 seconds per our in Inquitos. This is 2 minutes per day-quite a bit!
Anyhow, back to the original question. The time it takes for a pendulum to move, provided they are all in the same place, only depends on the length of the pendulum. This means that if all the clocks in the store have pendulums of the same length, and we all release them at the same time, they all move the same (we call this "in phase") and will continue to do so. If the lengths of their pendulums are different, they will of course all move with different speeds. This does mean that if we wait long enough, a few of them will through the lowest point in their motion at the same time. In fact, if we keep waiting, any number, or even all of them, will go through the lowest point at the same time. Because humans are very good at recognizing patterns, you will see and notice this if you are in the store. Of course, they won't stay like this; after a few ticks, the differences in pendulum length will cause them to start run out of phase again.
As for forces acting on the pendulums, the only physical force that would "lock" the pendulums in one motion is gravity. Interestingly, at cosmic scales, this does happen: the Moon, for instance, always has the same side pointing towards the Earth due to gravity. However, for a pendulum, this effect is much (by a several dozen orders of magnitude) too weak to have any measurable impact.
In conclusion, I think the explanation is the fact that if you look long enough, a set of pendulums, all moving at different speeds, will eventually move through the same point in their motions at the same time. There are no significant physical forces that can change this behavior.
Regards,
Bart Broks
If you read the details of the three sources you will see that synchronization depends on some kind of forcing between the several motions, and that the actual type of synchronization (anti- or otherwise) depends on the relative amplitude of the forcing. I have tried to obtain confirmation, or not, of the phenomenon of pendulum clocks being sychronized in a clock shop, but none of the shops I contacted replied. It would make for a very interesting experiment!
John Link, MadSci Physicist ]
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