The Pendulum
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"One, two, three... Infinity."
George Gamow

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This is the third article… after this we have infinity.
Please, dear Reader, forgive me for the mathematics in this article. Since I shall pass on this message also to those who understand mathematics, I have no other choice. Please, dear reader and friend, accept what is explained below. I promise to use as little as possible mathematical jargon, and I shall repeatedly get back to the analysis of the physical phenomenon proper, for this is what really matters. For more than ten years, I have been carrying a pendulum in my pockets.
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This has nothing to do with any superstition whatsoever, and at times it may be difficult to explain why it is there. Even so, I do not get tired watching it like a kid contemplating its favorite toy. To me, it is the gate to Hyperphysics. No reason to rush.
Look at Figure 1.
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When applying the second of Newton´s Laws and after appropriate simplifications, we will have:
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Which should be read ‘theta double dot, plus g over l, sine theta is equal to zero’. Normally, this equation is solved by approximating sine theta by angle theta. The error induced for angles of up to 10 degrees is less than 0.2%, and for angles of up to 20 degrees it can be as much as 0,8%. Therefore it is a good approximation. The differential equation shown above is no more than a special, though simplified, case of a more general expression of the formula below:
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the characteristic equation of which is:
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Have a look at this latter equation.
Yes, it is a second-degree equation.
Do you remember the problem with the hen house?
What happens when the delta is negative and the equation has complex roots?
It is calling ‘the fourth dimension’!
This should be enough mathematics for this article, I think. For those who may be interested: motion of the pendulum is called “Second-order differential linear equation”. The worst is that even good texts do not always contemplate complex solutions. Upon having a closer look at the problem, interested readers will easily see the complex solution come forth, which is exactly what describes the oscillating motion. Having honored the obligation, let us talk about the physical phenomenon we are interested in.
See Figure 2.
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This is the pendulum I have been keeping in my pocket for years; mysteriously. I know two ways to make it oscillate. One goes like the pendulum of a clock, the other is a circular motion, like in the picture. The pendulum motion in a pendulum clock is in two dimensions, in a plane. The circular motion is in three dimensions, in a cone. In order not to leave mathematics behind, let us put origin O at the resting point. So, quite naturally, axis X shall remain in the direction of the pendulum’s motion.
The expression
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describes precisely the oscillating motion of the pendulum. I would like to talk a bit about sines and cosines. To those who are good at electronics, AC is nearly a synonym of sinusoidal current. The mathematical form as used by electronics professionals is exactly the expression described above. Letter a refers to maximum amplitude, Omega is the function of the frequency, and b represents the phase.
Some curiosities now.
The cosine function is exactly equal to the sine function with a 90º phase advance. Now this is exactly why you will never hear electronics pros talk about a ‘cosinusoidal function’. For them, cosines virtually do not exist. To them, sine is enough. However, putting it straight: sine and the phase. Using an oscilloscope, It is quite easy for them to measure the phase difference of two sinusoidal signals.
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What is an oscilloscope?
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The oscilloscope is an electronic instrument that measures an electric signal as a function of time, or, if it is double beam, one signal as a function of the other one. In other words, the oscilloscope is sort of a small TV set (in fact, it is the ancestor of television). The horizontal axis is X and the vertical axis is Y, or vice-versa. Using it this way, we can compare two sinusoidal signals. .........................................................................................
What appears on the screen?
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What appears on the screen has a name. It is called ‘figure of Lissajous’. .........................................................................................
These figures, when they are stable, look like ellipses, circles, the well-known little fish and other interesting figures, provided that the frequency of one signal is a multiple of the other.
And what happens if we put in the same signal twice? Obviously, a straight line inclined 45º (the X = Y function).
So, this means that the figure may be a simple straight line.
Another curiosity.
Function
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is well known. It is called exponential function.
This function appears very frequently when solving differential equations. The very simple reason is that the derivative of the exponential function is the exponential function proper.
In the differential equation we mix the function with its different derivatives. Should one expect to encounter exponential functions upon solving differential equations because they generate derivatives that are similar to themselves?
This is exactly what happens.
Look, how interesting.
If z is a complex number, z = s+it, then, according to Euler:
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Now look how the sine appears!
The derivative of the sine is minus cosine; a curiosity as well.
Since the cosine is the sine itself with a phase difference, we can see that the sine function also has the interesting exponential function property to auto-generate after a mathematical derivation.
Let´s get back to our pendulum.
The answer of the differential equation describing the oscillatory motion of the pendulum may present a complex solution.
The complex solution leads us to the sines and cosines that characterize oscillatory motions
. The figure has been created projecting sunlight onto a white screen, with the help of a mirror.
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What shape should the mirror have, if we want the focus to be round?
Just for fun, I used an elliptic mirror, and got a circle.
The figure shows very clearly that the motion of the pendulum is not reciprocating. The pendulum is clearly moving in a circle.
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My very dear reader, when we are limited by our three-dimensional space, we have no way whatsoever to know which is the “ACTUAL” motion of the pendulum. It is like a person who can only look at a white screen. He or she will only see the oscillatory motion and will not be able to give an account about the tree-dimensional motion. .........................................................................................
And what is the practical application of all this?
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The application may be thrilling!
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Just wait. .........................................................................................
Among those few toys I had when I was a child, there was a whistling top (read the resume of my biography). Whistling tops are made of metal sheet and make a whistling noise when spinning. And it fascinated me to the greatest degree.
What impressed me most was when the top would spin madly, stabilize for a short while and then get unbalanced again.
It looked like as if the top would hide its motion.
Play with a top and you will see what I mean.
Besides the rotating motion, which we all know, there are the precession and nutation phenomena.
These are the strange motions.
The reader may guess the complexity of the equations describing these motions. Complex solutions are obvious.
Although people say they are useless, well-done simulation will show these motions. There will be no way to avoid the complex part of the solutions, which means the multidimensional models, in other words, the Hyperspace models.
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There is no way to ignore Hyperspace!
What is this so important for?
Which is the biggest top we can put our hands on?
Think...
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The answer is : Earth !
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Has anyone ever conducted a good mechanical computer simulation to find out what may happen due to glacier thawing?
Nobody does really care about this!
So, Earth may anytime begin to spin madly?
The answer, unfortunately, is: yes!
Earth is a nearly spherical ball, flattened off at the poles, where we have a strong concentration of mass. The concise explanation by a mechanical engineer would be: a ball rotating quite unbalancedly, and distribution of continents and oceans permanently disturbed by tides!
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One more mechanical engineer´s opinion: to change the mass distribution of this ball may be reckless!
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When I was in college I asked my teacher questions about recession and nutation. I wanted to know more about my top. The teacher tergiversated, of course. .........................................................................................
What may happen : in a very near future, let´s say years, Brazil may become a new Antarctic and today´s Antactic may become an equatorial continent!
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What is the mechanical explanation of the fact that the axis of the Earth has an inclination of approximately 23 degrees? Superficial-minded people may say there may have been a collision with a celestial body. Where is the scar? Is it not quite strange that the axis of Uranus has an inclination of 82.5 degrees and that part of its surface lies in darkness (night, or gelid winter), for periods of 42 years?
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A collision with a celestial body, would be an ostrichian explanation. .........................................................................................
I am talking about facts, about what has happened in our solar system. Why would it not be dynamic lack of balance like that of a toy top?
Because, if there were , there would be an unseen danger, hidden in differential equations with complex solutions, models that are conceivable only in Hyperspace, hidden from us, unwise and incautious humans: the threat of a cataclysm!
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I reserve the right not to speculate. .........................................................................................
The importance of using new, clean energy sources may be greater than we realize! I love my pendulum. I do imagine Galileu Galilei, 400 years ago, watching the chandelier of a cathedral and seeing in the balancing motion of the chandelier an instrument to measure time. In my pendulum, I see my three-dimensional limitation, and ask myself: what may be, at this very moment, the REAL motion of my little toy? Never shall I know the answer. The question however, fascinates me!
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