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?"

A Chinese Saying

I remember when I was much younger, my father used to keep me quiet in an argument whenever I came out with an unanswerable question or a fact with a ubiquitous statement among oriental families. It goes something like this: "我食鹽多過你食米" which literally means "I ate salt much more than you ate rice". 

It was a figurative statement referring to his experience in life being much more longer and richer than mine gives him a right for authority and I, as a younger generation should learn to respect it. 

There goes a while each time this happens, I would stay quiet and let him have his way. Now, I was curious to find out if that aged-old "dictum" is even theoretically possible. 

So, before we start out with the math, we have to make 4 assumptions to ease the calculations:

1. An average person eats about 200 grams of rice (as one bowl) every-day.

2. An average Asian, start feeding on rice at the age of 5. 

3. My father, a savoury food lover, consume 3 times more salt than the recommended diet by Scientific Advisory Committee Nutrition (SACN) in 2003. 

4. He will start consuming salt in amount stated in assumption (3) since his birth. 

Based on assumption (1) and (2) we say the child would consume 73 kg of rice per year starting at the age of 5. 

The SACN recommended table salt intake per day is 4 grams per day. Following assumption (3) would translate into 4.38 kg of salt per year. 

Now, lets follow this simple timeline diagram, we can conclude an algebraic expression for the age my father have to give birth to me for the time he claimed "我食鹽多過你食米" to be valid. 

Naturally, we can say the collective amount of rice the child ate would be 73n and collective salt intake for father would be 4.38m. From the diagram, we can also see clearly that in terms of years, m=x+5+n. 

For the amount of salt needed to be equal to the amount of rice consumed, we simply equate the two rates of consumption to find the ratio of time:

73n = 4.38m

73n = 4.38(x+5+n)

which simplifies to:

15.67n-5 = x , when n=1, child at 6 years old, n=2, child at 7 years old, ...

Which means, if I were to be 7 years old, having ate two years' worth of rice, my father would have to consume salt for 26 years before bringing me into this world. Sounds perfectly reasonable isn't it? 

Extend that to 3 years more, you will realize this is not possible. Say if he claimed that statement while I was 10 years old, I would have to be born when he is 73 years old for the statement to be valid. 

*  *  *

Truth is, these days he no longer use such statement on me since I was in middle school not because I argued with him using this simple proof; but time itself and the things he had shown to the family are the elements which strengthens the father-and-son bond to such an extend where statement like this becomes irrelevant. 

This post is merely a muse of my childhood, brought back and checked using simple mathematics. 

Solutions: Roots of a Quadratic Equation

In elementary mathematics, one will come across a polynomial equation of the second degree which are called quadratic equations.Here's the general form of a quadratic equation:

Where a ≠ 0 (The equation will become linear instead)

In high-schools, students are generally given the quadratic formula to solve the equation. The answer for the solved equation are called the roots of the quadratic equation. The solutions can be real or complex and there are two solutions to each quadratic equation.

The formula is given as:

Here, we will go through the derivations to the formula.

We begin as the standard form of a quadratic equation:

Moving the last term to the opposite side and dividing the terms with a;

By applying completion of squares;

rearranging the equation we finally get;

A little thing about waves

Adapted from Wikimedia. Translated and edited by myself.

The red curvy line is a representation of wave.

All the other components are noted down.

The period of the wave (wave period) is usually denoted as the symbol T.

The wavelength is usually denoted with the greek letter lambda (λ).

The frequency of a wave is given by the formula:

Where the symbol f is the frequency.

*Note that frequency is the reciprocal of the wave period.*

The unit for frequency is Hertz. (Hz)

In short, the shorter the wavelength means higher frequency (more waves in a second) and if the wavelength is longer, it means it has less waves in a second.

For musicians,they see frequency as pitch. Everytime when they play a tone, it reprisents a note with a specific frequency. For example:

Start with the middle C in piano, and it goes C, D, E, F, G, A, B, C. This is called an octave. As you press the keys starting from the middle C to the next adjacent keys, you are actually listening to the a note with a higher frequency each time you press it.

So, each time you press the adjacent keys, all you hear is the note with a higher pitch.

Click here for a more details about waves. (high school level)