INTRODUCTION: What do you do when you are trying to grasp the fundamentals of a well-known physics equation, an equation you had not been formally taught it in school? Wikipedia, of course! But what happens if what Wikipedia has to say on the subject confuses you even more? You do your best to reason out the fundamental elements that comprise the equation on your own recognizance. You hope that what you come up with will somehow miraculously match up with what the academic textbooks have to say on the subject.
The process of discovery can occasionally lead to surprising conclusions, especially when you get around to comparing notes with what the priesthood of physics has to say on the subject. You might discover the fact that while your version of the equation seems to posses fundamental differences when compared to what is formally laid out in the textbooks, what you came up with nevertheless seems to explain the phenomenon in exactly the same way. Not only that you can use your own equation to make the exact same predictions. This recently happened to me while trying to grok a well know algebraic formula, the Conservation of Angular Momentum, or CoAM. It is intimately related to my on-going study of Celestial Mechanics through the use of computer simulation. Here's one of my prior posts pertaining to personal research I've done in the field made back in March of 2012: http://www.mail-archive.com/vortex-l@eskimo.com/msg64010.html While continuing my research I eventually realized I needed to understand the fundamentals of CoAM because I came to realize that the equation is an essential part of the physics that helps explain how Celestial Mechanics (CM) behaves. CoAM helps explain why a satellite orbiting a gravitational mass, like a planetary body, typically assumes the path of an ellipse where one of the foci is located at the center of the planetary body. Why does the velocity of an orbiting satellite as it swoops away from the planetary mass slow down? CoAM explains it. Why does the satellite's velocity speed up dramatically during the return phase. Again CoAM explains the reason why. What is even more astonishing is why does the speeding satellite after it has made its nearest approach break away? How can that possibly happen? Why doesn't it crash into the planetary body since the gravitational influence being felt would be at its greatest strength? Again, CoAM explains why that doesn't happen. I would conjecture that exactly how CoAM constantly comes to the rescue is not necessarily that well groked by most folks, including physicists. I certainly didn't understand nor appreciate the incredible dance of physics that is involved, not until I started taking a long hard look. It is my hope that how I finally learned to grok CoAM might help others who may also occasionally feel disenfranchised from what traditional physics books might have to say on similar subjects. The experience lead me to a belief that there may turn out to be many roads that lead to the Grand City of Rome. Not only that, sometimes traveling down a less beaten path can have its own unique surprises and rewards. I suspect Andrea Rossi is a perfect example of such an individual who found his own unique pathway to the City of Rome. I suspect he chose a road rarely travelled by others. The path he chose could possibly end up turning the world of physics upside down - assuming his eCats really do work, and perhaps most important of all, he gets the chance to sell them en masse to the world. MY SEARCH FOR COAM BEGINS: Initially I tried reading what Wikipedia had to say on the subject. The authors weren't of much help to me. See: http://en.wikipedia.org/wiki/Conservation_of_angular_momentum#Conservation_o f_angular_momentum http://tinyurl.com/yf28c7l Something was missing. Nowhere in the all of the turgid mathematical equations that had been written down was there the slightest hint of a squared value. That bothered me. It bothered me because of my own extensive computer simulation research into Celestial Mechanics, of how orbital bodies are attracted to a central gravitational mass. I was also acutely aware of Kepler's most famous law concerning planetary motion, his 2nd law which states: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time Kepler's 2nd law introduces a constant that manifests in our three-dimensional universe in the form of a flat and fixed 2-dimensional piece of area. No matter what shape that flat patch of area assumes the amount of area remains constant. First a qwik refresher course on "area". An area, such as a rectangle, is determined by multiplying two 1 dimensional lengths held at 90 degrees to each other. It is often expressed as: area = x * y. If, as sometimes happens, x = y, representing a square, then you can simplify the rectangular equation to: area = x^2. There was the squared value! Based on my own experience of working with computer simulations of orbiting bodies I strongly suspected that a squared value was also probably an intimate part of the CoAM equation. The most obvious culprit that would possess this mysterious squared value would be r, the radius. r would be the distance of where the rotating body is located from the center of the axis. At this point, while armed with the suspicion that a squared value had to exist somewhere, I had to take matters into my own hands by trying to construct the essence of the equation for CoAM on my own reconnaissance. I had to use the most elementary tools that I was familiar with and hope to god that at the end of my construction project something actually made sense! There are never guarantees that one can achieve success when embarking on these kinds of journeys. But then, it's the journey itself that often turns out to be the most valuable part of the entire learning experience. Ironically it can turn out to be irrelevant whether one achieves success, or not, because failure often turns out to be the final conclusion. Therefore, take time to smell the roses along the way. You might miss something earth shattering! MY PERSONAL CONSTRUCTION OF COAM BEGINS: * I know that 360 degrees comprises a complete circle. I know many in the scientific community prefer to use radians when working with trigonometric values, but screw that! Most laymen implicitly understand that 360 degrees makes up a complete circle. * I know that if you were to spin a weight at the end of at string while making pains not introduce any form of torque, the string will continue to sweep a full circle in equal intervals of time. * This also means that within each and every single degree for which the rotating weight passes through in its circuitous journey an equal slice of fractional time occurs. Every single degree possesses the same amount of fractional time. What this means is that we have established a constant value with the orbiting weight. * Perhaps less appreciated is the fact that this also means that the amount of area that the weight is continuously sweeping through would also turn out to be the equivalent of an constant. Let me repeat that: The area being swept through is also a constant. The constant is actually a squared value comprised of x * y, or in simplified terms x². . . . Ok, at this point some may have recognized the fact that there seems to exist a profound similarity with a fundamental law that explains how CoAM behaves and Kepler's 2nd law which states that: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. We need to test this suspicion by plugging in a different value for the radius of the spinning weight that is attached to a string and see whether the velocity does change accordingly. Through simple observation everybody intuits the fact that when a spinning ice skater pulls in her arms and legs her entire body begins spinning at significantly faster rates. The action dramatically reveals the law of Conservation of Angular Momentum (CoAM) in action. Because as the ice skater pulls in her extremities and her body begins spinning faster, this suggests that the most likely equation that will explain the phenomenon will manifest in two fundamental forms: (1) x/r or (2) x/r² where x remains a constant value. In this case x is determined to be fixed value of time. As the value of r, or radius, gets smaller, the equation causes the rate of spin (or rotation) to increase for the same slice of time. By doing so the conservation of the value of Angular Momentum. is maintained. So, which of the above two equations is most likely to be the correct one? Let's assume I have a weight attached to a central axis of a string which is spinning at a constant rate. Let's assume that the length of that string, the radius, r, is ten feet. Feet??? Not meters? Humor me! Let's assume the weight makes a complete 360 degree sweep every four seconds. How much time transpires while the string makes a sweep 30 degrees? That can be approximated by the following formula: [30 degrees * 4 seconds] / 360 degrees = 0.33333 seconds. We can also determine the amount of approximate area swept for every 0.33333 seconds because we know that the area of a circle is determined by the equation: p*r². In our case we would substitute the following values: [30*pi*10feet²] / 360 degrees = 26.2 square feet. CoAM dictates that, in our case, for every 0.3333 seconds for which the weight is rotating through an area of 26.2 square feet accumulates. In order for CoAM to be conserved a constant area of 26.2 square feet must always be maintained no matter what the current radius might be. So... which equation most likely one explains the spinning ice skater? Is it x/r or is it x/r²? I finally stumbled across the answer to my conundrum when I discovered another web site, KHANAcademy.org, where the Conservation of angular momentum was explained to me in a whimsical way by an instructor who did not take himself too seriously as he scribbling down a bunch of equations on an electronic blackboard. See: http://www.khanacademy.org/science/physics/v/conservation-of-angular-momemtu m The answer was finally revealed to me about 5 minutes into the instructors webinar. There, on the screen he had scribbled out the following equation: M*W*r^2 = constant where: M: Mass of the spinning object. W: How fast the object is spinning: r: the radius (distance) from the center of the spinning axis. The value of r, the value of radius, is squared. It seemed to me that the instructor almost snuck the power of "2" onto the end of radius r. It seemed to me that he really didn't spend any time at all on explaining why the value r needs to be squared. But no matter. I got my answer. Granted, one should always take what one gets from the internet with a grain of salt. However, I suspect that in this case it's probably safe to take the instructor's formula as the right one...until proven otherwise. Assuming radius, r, needs to be squared that means that in my case if the string, the radius of our spinning weight, is slowly reeled in from an initial distance of 10 feet to 5 feet CoAM dictates that the number of degrees for which the weight will have to rotate through (for the same fixed period of time) can be determined by using the following formula: [30deg *pi*10feet²] / 360deg = [xdeg *pi*5feet²] / 360deg We can systematically simplify the sequence of transformations as follows: [30deg *pi*10feet²] = [xdeg *pi*5feet²] [30deg *pi*10feet²] / [pi*5feet²] = x [30deg * 10feet²] / [5feet²] = xdeg 30deg * 4 = xdeg xdeg = 120 degrees This means that when the radius of the string is shortened to 5 feet, in order to maintain constant angular momentum, and incidentally the same constant area of space, the number of degrees that one must sweep through for the same period of time, has to adjusted to 120 degrees. 120 degrees makes sense as the value is the equivalent of mimicking the law of inverse square of the distance, which ironically most of us should be quite familiar: x/r². The same equation explains the effects of gravity as well as lots of other phenomenon, like how the strength of electromagnetic radiation (light and radiation) changes depending on the distance from the source. CONCLUSIONS: What this exercise brought home to me was an unexpected bonus of insight. My little journey of logic taught me the fact that the law of Conservation of Angular Momentum appears to be virtually indistinguishable from the Inverse Square of the Distance law. While many might accept such a revelation or self-evident, so what's the big deal, what might not be as obvious is the fact that Kepler's 2nd law which states, a line joining a planet and the Sun sweeps out equal areas during equal intervals of time, literally seems to play as much of an intimate part of explaining how CoAM works as it does in explaining how the behavior of rotating planets behave. This was brought home to me when during one of my prior analysis periods while studying the effects of the 2nd law. I noticed that the 2nd law works just as well with a satellite in a hyperbolic trajectory racing past a gravitational body. Not only that, the 2nd law works just as well with a satellite traveling past a body that exerts NO gravitational influences at all! Armed with these revelations, it was not that much of a stretch for me to conjecture that Kepler's 2nd law would probably also help explain how CoAM behaves. At first glance making such a revelation might not make sense to some because if you attempt to incorporate both the effects of gravity x/r² with the effect of CoAM which also contains the same x/r² equation then wouldn't both of these identical equations when combined tend to cancel each other out? Wouldn't they cancel each other because the attractive forces of gravity would be working in exact opposite to the centripetal forces generated from the effects of CoAM? Miraculously, a cancellation of influences does not happen! Why and how that doesn't happen is a fascinating tale in its own right. It's a tale I have already touched on within the Vort Collective, back around March of 2012. Some may recall that I posted a thread where I mentioned the fact that some of my computer simulations appeared to mimic the effects of Celestial Mechanics pertaining to the distance a satellite would be predicted to be at if one charted the distance the satellite as the value of y while simultaneously fixing a constant slice of time as the value of x. In order to make everything line up the computer simulation has to incorporate the following elements, where the first element mimics the forces of gravity and the second equation mimics centripetal forces: y = x/r^2 - x/r^3 Again, I refer readers to the following post: http://www.mail-archive.com/vortex-l@eskimo.com/msg64010.html Generating such a plot tends to produce a top heavy bouncing ball kind of sine wave. Nevertheless, it mimics the exact distance from the gravitation body a satellite will be at, for any specific slice of time. Woah! How did the negative element, x/r³, get into the equation? I suspect a few smart Vorts, perhaps a few afflicted with the ADD gene, might be able to figure that one out in due course. It is one of many goals I have set for myself, to hopefully explain why x/r³ might help explain the nature and shape of the elliptical orbit, such as a planet orbiting a star. Hopefully, I'll be able to get around to the explanation after I overhaul my web site. "Small steps, Sparks. Small steps." Finally, my little journey has given me a greater appreciation for the adventures that Andrea Rossi might have gone through as he accidentally burnt his hand and in the process discovered an anomalous source of heat from one of his experiments. I sometimes suspect that had Rossi been more academically trained in matters of nuclear physics and/or chemistry it is possible that the spurious heat source he stumbled across might have been dismissed as nothing more than a careless mistake on his part. He might have been more inclined to dismiss what his senses were telling him, what his burned hand told him. His formal training would have "taught" him... drummed out of him an almost naive-like willingness to entertain the possibility that the affect he stumbled across could possibly be worth a second look. It might also might help explain why many with impressive credentials with formal academic training in physics and chemistry often seem to come up empty-handed. They come up dry because formal training increases the chances that they will continue to blind themselves. An occasional observation hinting of an insignificant little anomaly that spuriously crops up in their data would automatically be perceived nothing more than a mistake that would best be ignored. A revelation for me was the realization that being afflicted with just a tiny-tiny bit if ignorance can occasionally... just occasionally be a very good thing! Occasionally being somewhat ignorant of what is currently taught in physics and chemistry can turn out to be the harbinger of unexpected surprises and discoveries. You might be more inclined to forge your own unique path to the City of Rome. You would be so inclined because you would literally have no other choice but to forge your own path. By doing so you may eventually stumble across observations that turn out to be revelatory, observations and insights which mainstream academia had never seriously considered. It all comes out of the simple fact that there would be more of an innate willingness to smell each and every single plant one encounters in a way where formal education would have paid less attention to. You would be less inclined to dismiss some of the more subtle fragrances as nothing more than another weed the textbooks had taught us long ago to ignore. DISCLAIMER: Final note. I don't mean to imply that my personal journey, my "discovery" has not already been observed by many within academic and scientific research fields. It's quite possible that what I posted here has already been written up in a series of obscure text books that are currently collecting dust in a basement of a university library. Nevertheless, what I hope I was able to convey here was the joy that can be experienced in the pursuit of a discovery, which in the greater scheme of things may actually be nothing more than a re-discovery of a discovery. Nevertheless, that doesn't take away from the fact that embarking on such journeys can occasionally produce unexpected surprises - some with earth shattering consequences. Regards Steven Vincent Johnson www.OrionWorks.com www.zazzle.com/orionworks
