Preview of a Homemade Chaotic (Magnetic) Pendulum

About a year ago while I was at a "curiosity" shop in the Singapore Science Centre, I came across an interesting table-top display sold under a hefty price tag. It was marketed with a catchy name "Random Oscillating Magnetic Pendulum" or ROMP and so I thought, hey, since the mechanism is essentially a perturbed pendulum, so why not make one myself that is worthy of display?

Sometime during mid April, I went to my father's workshop for its carpentry. I covered the base of the pendulum with white Formica laminated under a sheet of stainless steel which enable permanent magnets to stick on (picture above). The magnets provide perturbative force which disturbs the oscillation of a rigid pendulum (with ferrite magnetic tip) hanging from a static boom (arm). The designated poles of the magnet does not matter as we are only interested in its interaction between the magnetic tip and the magnets on the base - either repulsive or attractive. So, three bobs of different length (305, 311 and 314 mm respectively) was cut from a single piece of square wooden rod and after gluing a magnet to its tip, I aesthetically finished it with stained lacquer. 

The picture above (I digitally "stitched" two photos together) shows free-oscillations of two geometrical axis relative to the base of the pendulum using long-exposure photography. I gummy tacked a blue L.E.D to the tip of the bob and connected its electrodes to a battery with a fine copper wire from a stripped cable to minimize erroneous dampening and it is essentially a photographic technique called "light painting". 

The pendulum motion from this apparatus can never be a harmonic oscillator. This is due to natural dampening sources which eventually causes the pendulum to stop: in addition to friction between the hooks at the top of the bob (check video at the end of this article), air resistance and torsional force, we also need to note some residual "attractive" forces exist between the tip magnet and the stainless steel plate below the base affects the periodic motion. 

When a single magnet with same pole as the one on the pendulum tip was introduced at the center of its (where the pendulum tip will be when rest) oscillating path, the resultant trajectories can be drastically modified. This demonstration is included in the video while photographic example exhibits similar motion in 30 second exposure as the deflected path painted a chrysanthemum flower pattern around the central base magnet. 

These examples shows obvious changes in the pendulum's motion when natural oscillation under the influence of gravity was disturbed by the opposing magnetic force as it approaches the base magnet. However, the interesting part was that the geometry (square cut) of the magnets actually provide "sites of lowest energy" or metastable spots as the tip of the bob shows repeated, predictable, "dying out" oscillation when the system's energy is dissipating away.

Even with one base magnet, it begin to show this system exhibits chaos as the trajectory evolution of the pendulum depends highly on its initial conditions. We will soon see why it is almost impossible to get the exactly-same "chrysanthemum pattern" even though we tried our best to start the oscillation at the same spot. 

See, a chaotic system is characterized by having exponential growth (Lyapunov exponent) in error or more popularly known as the butterfly effect. Lets say if we have an instrument which allow us to measure initial velocity and position of the pendulum with extreme precision. Therefore, we should in theory be able to predict the trajectory outcome of the pendulum based on known physical laws like Newtonian mechanics and electromagnetism even though the analysis is extremely tedious. 

The catch is, there is a limit on how accurate our measurements can be done (for example, we cannot measure 0.00001 mm with a ruler). In fact, quantum mechanics dictate it is fundamentally impossible to know exactly a particle's position and momentum at a same time. Thus, our measurements are bound to have a slight uncertainty no matter how good the instrument is. As we mentioned earlier about the dynamics of the butterfly effect, these initial uncertainty from our measurements, although small, will grow exponentially as the we let the pendulum swing under influence of the magnet. Every single moving moment of the pendulum constitute a larger uncertainty to its motion and eventually it grows so large we are not able to predict which direction the next move will be. 

In the picture above, I used a red laser pointer to trace the path of oscillation (long-exposure photography) when 8 more magnets are introduced, all under "repulsion" with the magnetic tip. Tiny changes of either the position of the base magnet(s) or starting pendulum position will always produce a completely different movement. This is the essence of a chaotic system: given the knowledge of all initial positions, it is still difficult to predict the outcome even though the natural laws governing this system are fully deterministic.

So theories aside, I have not really quantified all the parameters which allows me to calculate the Lyapunov exponent. After all, the aim of this project was simply to prove "display-worthy" demonstrations can cost much lesser compared to similar products on the shelf. However, given free time in the future, with decent computer programming skills, it might just be possible to consider a mathematical modelling of this system to calculate its exponent value. 

On a side-note, I find it amusing that even chaos can be classified into subtle or complex types. Mathematical physicists are still working out if there is a fractal pattern called strange attractors emerging from the chaos generated by noise of certain systems. Strange isn't it?   

The occurrence of chaos in nature is not rare by all means. The weather is a good example. It is one of the reason why scientist find it tremendously difficult to accurately predict due to chaotic nature of air-flows, wind currents, temperature variations et cetera. Dynamic phenomena such as the pattern of Saturn's rings, SARS outbreak and the trigger event of heart attack are other examples of chaotic systems.

So, after discussing about the occurrence of chaos, it is not surprising to get one thinking (especially geophysicists) about the so called butterfly effect - "If a kupu-kupu (butterfly) flaps its wing in Taman Tasik Perdana, will it dramatically change the weather pattern in San Francisco?"

Pendulum Waves (Not So)

Before I start, I would like to thank my ever-supportive father, who spend so much effort for making this beautiful wooden frame. He is a skilled craftsman (carpenter) who inspired me partly to become an experimental physicist.

Now, about this video:

As I have mentioned, it is a demonstration of a so-called pendulum wave. I was very much impressed by the pattern of repeating oscillations shown to me by a friend not too long ago, so one day I decided: why not build one? After all, you never truly know something until you design and test it yourself - it is the heart and reason why we do experiments in science.

So I bought 20 fishing leads as bobs, they are graded (as casted on their surface) "3". I am not entirely sure what it meant, but I presume it is an index of weight. However, simple mechanics dictates that the period of oscillation in a pendulum is independent of the bob's mass. Hence, slight difference of mass between the bobs should not matter in a synced-oscillations as it would not affect its "swinging frequency".

I used a series of plastic zip-strips as pivots for the pendulums. 16 bobs are arranged in successive 0.5 cm increment, sharing pivots with another.

Comparing with available pendulum waves demonstration, this video appears to exhibit a failed example. Initially, the collective oscillation shows pattern resembles a wave-like motion, but because there is a small difference in frequency between the successive bobs which results in de-phasing after a few cycles of oscillation, which collectively shows a quick decay from a wave-like pattern into seemingly random movements.

However, it is important to note that this "failure" was not unpredicted for three reasons:

1. The bobs are sharing the same pivot with each other. i.e. there is only one continuous string that ties from one-end to another end of the wooden frame. As such, the component mechanical forces acting on each bob during their individual oscillation could perturb the adjacent bob - creating a dynamic system.

2. Successive bobs are arranged in a linear increase of length, that is the height difference between successive bobs are constant at 0.5 cm. If the frequency of swinging is critically dependent on the bob length, i.e. tiny difference such as 0.5001cm from one bob to another, then we can expect the synced-movement to decay rapidly as time goes on.

3. The string that is holding the bobs is very thin. At times, some bobs twist while swinging (we call torsion) which dampens the oscillation for each pendulum.

Below is a video of a simulated "pendulum wave" if we vary the bob lengths linearly. A good friend of mine used MATLAB(TM) to simulate this 19-bobs movement, and we can see obvious de-phasing of the synced-oscillation after the first mode at about 00:30 video time. Note the pattern of the mode decay is similar to my video earlier. 

Therefore, we speculate that the supposedly "ordered" displacements between the bobs is critically dependent on the linear increment in frequency of each successive bobs instead of the length.

My Essay (Part IV) - Clockwork Universe

The moon - Like any other celestial object are bounded by gravity. Gravitational attractive force is pulling moon closer to earth but this force is countered and balanced by the outward rotational (centrifugal) force which results an orbit.

Because of the three immutable laws, Newton is able to synthesise a new equation that describes the motions of planet. This is what we affectionately call “Newton’s Universal Law of Gravitation”. This law is powerful because it is so short – barely five characters in the equation. Enough to predict how much force it needed to propel a rocket to space, how much force pulls us on the surface of the earth, how much force the moon need to stay in orbit. In fact, Since the discovery, the equation is still being used now by all space agencies to launch anything into orbit! Here’s the nasty bit:


Here’s how it works. The F which means the attractive force between two bodies in space is equal to the product of mass 1 (object A in space) and mass 2 (object B in space) divided by the square of distance between them. Do not mind the G. It’s a constant which has a fixed value. What this means is that for any two objects in space, now there’s a formula to apply and we get the value for the forces between them. The motions of everything in this universe became calculable.

Newton’s law stated that gravitation is always attractive. That is bodies in space will always attract each other no matter what they are made of. This has a logical flaw which is pointed out by Richard Bentley, another brilliant thinker at his time, sending a letter to Newton. That innocent looking letter packed with such a load that blows Newton back thinking about this: “If all things in this universe are attractive in nature, shouldn’t the stars all come onto each other crashing? The universe is unstable.”

In reply towards Bentley’s letter at 1691, Newton argued that such case will only happen if there are finite number of stars and finite volume of space between them. So, he proposed that in the universe, there should be infinite stars separating from each other in equal distances and space is infinite in volume. Also, God has to intervene once in a while to nudge the stars so that they will not move towards one another destroying the stability of the universe.

Newton’s law is elegant, in the sense that the law is simple yet it could predict the motions of stuff in our experiences. Take a stone and throw for example. Using Newton’s law (knowing the mass of the stone) we could predict how high the stone will fly and with a computer, we can draw a path of its precise trajectory. Ironically, it was precision that render Newton’s law at describing some astronomical observations, wrong.

My Essay (Part III) - Galileo to Newton

Newton's Principia, with his actual pen-strokes on correction.

Galileo not just only discovered there are other moons on other planet. He also proposed that all things move due to acceleration of gravity. This was demonstrated by Galileo, as rumour has it, was done at the top of the Pisa tower when he dropped two balls, one ball made of iron and the other made of wood. The result is that both balls of the same size dropped at the same time landed on the floor together at the same time suggest that different weight does not affect the speed of the balls. Another experiment was he put a ball on an incline plane and let it roll from the top to the bottom and he found that the ball is moving faster and faster. We call it accelerating. So the question is what drives the ball to accelerate?

The legacy of Galileo was indeed profound and it made an impact as how science is done. As quoted from physicist Leon Lederman’s book: The God Particle

“For our purposes, he (Galileo) was a physicist, and a great one, far beyond his advocacy of Copernicanism. He broke new grounds in many fields. He blended experiments and mathematic thinking. When an object moves, he said it’s important to quantify its motion with mathematical equation. He always asked “how do things move? How? How?” But he didn’t ask “Why? Why is the ball falling?” He was aware that he was just describing motion, a difficult enough task for his time.”

Galileo didn’t mention that why the ball is falling, and that needed another great mind to solve the puzzle. Enter Isaac Newton. Sir Isaac Newton is perhaps one of the greatest minds of all time. He revolutionised the way we see how objects move and he literally created a new branch of mathematics for it. In 1687, Newton published the celebrated “Philosophiae Naturalis Principia Mathematica” which is perhaps the single most important work ever published in physical sciences.

In the seminal masterpiece, Newton described how everything in this universe worked – Force. In the papers were three immutable laws that describe all motions in the universe. Suddenly, the universe became a measureable, definite clockwork. The first law was the concept of inertia, which is a fancy word that simply means: an object does not move or moves in a constant velocity until an external force acts on it. This is particularly useful in your life when you tried to get your ketchup conveniently by hitting at the bottom end of the bottle repeatedly, the sudden force acting on the bottle propel the sauce out from its bottle. The second law is quite simply F = MA. It’s a simple mathematical relation. Let’s read it out loud; “Eff equals to Emm Ayy”. What that means was the force is equal to the product of the mass and the acceleration of the object. The third law is better. It says that when a force is exerted on an object, the object exerts the same force back on the opposite direction. (Every action there’s a reaction). So next time if someone bitch-slapped you, don’t bother. Because your face already slapped that person’s hand on that painful moment! (You felt pain more due to the more pain receptors on the surface of your cheek than the slapping palm)