Bravo Steven, it sounds like you have been having a good time in your path of discovery. I find myself in the same boat on occasions where I discover something that is new to me but then I find that it has been documented by others and recently wikipedia has been my worst foe. At least you and I have the satisfaction of knowing that we are venturing into the unknown. I like to think of my new concepts as suggesting that I could have been there first, but was not. Regardless of the timing, it is mind opening to venture into these types of studies.
Much is learned about the natural world by actually exploring interesting concepts. When the pieces fall together we get a better view of the entire picture that tends to remain in our memories far longer than one can recall a equation. And as you suggest, unusual modifications from the original well known paths lead to interesting observations that might not make sense at the first look. I have a tendency to think differently than others about natural phenomenon. You appear to exhibit that curse as well! This type of reasoning has served me well in the past when I have solved extremely complicated problems that have been unresolved for months until an unorthodox idea appears. The more we think about things in a different manner, the more likely it is to stumble upon these wondrous ideas. Keep stressing your mind as it is extremely good for you. A number of years ago I played chess with some very good players and I suspect that you may have done the same. That should be a requirement for guys and gals that want to enter into the engineering or science fields in the future. The deep planning necessary to play chess at a reasonable level is greatly advantageous for problem solving. I find the CoE as a very useful guide as well. That along with the CoM, coupled with a observation reference change can make many problems become much more transparent. I wish you good fortune my friend. Dave -----Original Message----- From: OrionWorks - Steven Vincent Johnson <orionwo...@charter.net> To: vortex-l <vortex-l@eskimo.com> Sent: Sun, Jun 24, 2012 6:25 pm Subject: [Vo]:Groking CoAM, Kepler and Rossi INTRODUCTION: What do you do when you are trying to grasp the fundamentals of a well-known hysics equation, an equation you had not been formally taught it in school? ikipedia, of course! But what happens if what Wikipedia has to say on the ubject confuses you even more? You do your best to reason out the undamental elements that comprise the equation on your own recognizance. ou hope that what you come up with will somehow miraculously match up with hat the academic textbooks have to say on the subject. The process of discovery can occasionally lead to surprising conclusions, specially when you get around to comparing notes with what the priesthood f physics has to say on the subject. You might discover the fact that while our version of the equation seems to posses fundamental differences when ompared to what is formally laid out in the textbooks, what you came up ith nevertheless seems to explain the phenomenon in exactly the same way. ot only that you can use your own equation to make the exact same redictions. This recently happened to me while trying to grok a well know algebraic ormula, the Conservation of Angular Momentum, or CoAM. It is intimately elated to my on-going study of Celestial Mechanics through the use of omputer simulation. Here's one of my prior posts pertaining to personal esearch 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 he fundamentals of CoAM because I came to realize that the equation is an ssential part of the physics that helps explain how Celestial Mechanics CM) behaves. CoAM helps explain why a satellite orbiting a gravitational ass, like a planetary body, typically assumes the path of an ellipse where ne of the foci is located at the center of the planetary body. Why does the elocity of an orbiting satellite as it swoops away from the planetary mass low down? CoAM explains it. Why does the satellite's velocity speed up ramatically during the return phase. Again CoAM explains the reason why. hat is even more astonishing is why does the speeding satellite after it as made its nearest approach break away? How can that possibly happen? Why oesn't it crash into the planetary body since the gravitational influence eing felt would be at its greatest strength? Again, CoAM explains why that oesn't happen. I would conjecture that exactly how CoAM constantly comes to he rescue is not necessarily that well groked by most folks, including hysicists. I certainly didn't understand nor appreciate the incredible ance of physics that is involved, not until I started taking a long hard ook. It is my hope that how I finally learned to grok CoAM might help others who ay also occasionally feel disenfranchised from what traditional physics ooks might have to say on similar subjects. The experience lead me to a elief that there may turn out to be many roads that lead to the Grand City f Rome. Not only that, sometimes traveling down a less beaten path can have ts own unique surprises and rewards. I suspect Andrea Rossi is a perfect xample of such an individual who found his own unique pathway to the City f Rome. I suspect he chose a road rarely travelled by others. The path he hose could possibly end up turning the world of physics upside down - ssuming his eCats really do work, and perhaps most important of all, he ets 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 uthors weren't of much help to me. See: http://en.wikipedia.org/wiki/Conservation_of_angular_momentum#Conservation_o _angular_momentum http://tinyurl.com/yf28c7l Something was missing. Nowhere in the all of the turgid mathematical quations that had been written down was there the slightest hint of a quared value. That bothered me. It bothered me because of my own extensive omputer simulation research into Celestial Mechanics, of how orbital bodies re attracted to a central gravitational mass. I was also acutely aware of epler's most famous law concerning planetary motion, his 2nd law which tates: A line joining a planet and the Sun sweeps out equal areas during equal ntervals of time Kepler's 2nd law introduces a constant that manifests in our hree-dimensional universe in the form of a flat and fixed 2-dimensional iece of area. No matter what shape that flat patch of area assumes the mount of area remains constant. First a qwik refresher course on "area". An rea, such as a rectangle, is determined by multiplying two 1 dimensional engths 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 implify the rectangular equation to: area = x^2. There was the squared value! Based on my own experience of working with omputer simulations of orbiting bodies I strongly suspected that a squared alue was also probably an intimate part of the CoAM equation. The most bvious culprit that would possess this mysterious squared value would be r, he radius. r would be the distance of where the rotating body is located rom the center of the axis. At this point, while armed with the suspicion that a squared value had to xist somewhere, I had to take matters into my own hands by trying to onstruct the essence of the equation for CoAM on my own reconnaissance. I ad to use the most elementary tools that I was familiar with and hope to od that at the end of my construction project something actually made ense! There are never guarantees that one can achieve success when mbarking on these kinds of journeys. But then, it's the journey itself that ften turns out to be the most valuable part of the entire learning xperience. Ironically it can turn out to be irrelevant whether one achieves uccess, or not, because failure often turns out to be the final conclusion. herefore, take time to smell the roses along the way. You might miss omething earth shattering! MY PERSONAL CONSTRUCTION OF COAM BEGINS: * I know that 360 degrees comprises a complete circle. I know many in the cientific community prefer to use radians when working with trigonometric alues, but screw that! Most laymen implicitly understand that 360 degrees akes up a complete circle. * I know that if you were to spin a weight at the end of at string while aking pains not introduce any form of torque, the string will continue to weep a full circle in equal intervals of time. * This also means that within each and every single degree for which the otating weight passes through in its circuitous journey an equal slice of ractional time occurs. Every single degree possesses the same amount of ractional time. What this means is that we have established a constant alue with the orbiting weight. * Perhaps less appreciated is the fact that this also means that the amount f area that the weight is continuously sweeping through would also turn out o be the equivalent of an constant. Let me repeat that: The area being wept through is also a constant. The constant is actually a squared value omprised of x * y, or in simplified terms x². . . . Ok, at this point some may have recognized the fact that there seems to xist a profound similarity with a fundamental law that explains how CoAM ehaves and Kepler's 2nd law which states that: A line joining a planet and he Sun sweeps out equal areas during equal intervals of time. We need to est this suspicion by plugging in a different value for the radius of the pinning weight that is attached to a string and see whether the velocity oes change accordingly. Through simple observation everybody intuits the fact that when a spinning ce skater pulls in her arms and legs her entire body begins spinning at ignificantly faster rates. The action dramatically reveals the law of onservation of Angular Momentum (CoAM) in action. Because as the ice skater ulls in her extremities and her body begins spinning faster, this suggests hat the most likely equation that will explain the phenomenon will manifest n two fundamental forms: (1) x/r or (2) x/r² where x remains a constant alue. In this case x is determined to be fixed value of time. As the value f r, or radius, gets smaller, the equation causes the rate of spin (or otation) to increase for the same slice of time. By doing so the onservation of the value of Angular Momentum. is maintained. So, which of the above two equations is most likely to be the correct one? et's assume I have a weight attached to a central axis of a string which is pinning at a constant rate. Let's assume that the length of that string, he radius, r, is ten feet. Feet??? Not meters? Humor me! Let's assume the weight makes a complete 360 degree sweep every four econds. How much time transpires while the string makes a sweep 30 degrees? hat 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 econds because we know that the area of a circle is determined by the quation: 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 eight is rotating through an area of 26.2 square feet accumulates. In order or CoAM to be conserved a constant area of 26.2 square feet must always be aintained no matter what the current radius might be. So... which equation most likely one explains the spinning ice skater? Is it /r or is it x/r²? I finally stumbled across the answer to my conundrum when discovered another web site, KHANAcademy.org, where the Conservation of ngular momentum was explained to me in a whimsical way by an instructor who id not take himself too seriously as he scribbling down a bunch of quations on an electronic blackboard. See: http://www.khanacademy.org/science/physics/v/conservation-of-angular-momemtu The answer was finally revealed to me about 5 minutes into the instructors ebinar. There, on the screen he had scribbled out the following equation: M*W*r^2 = constant where: M: Mass of the spinning object. : How fast the object is spinning: : 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 nstructor almost snuck the power of "2" onto the end of radius r. It seemed o me that he really didn't spend any time at all on explaining why the alue r needs to be squared. But no matter. I got my answer. Granted, one hould always take what one gets from the internet with a grain of salt. owever, I suspect that in this case it's probably safe to take the nstructor's formula as the right one...until proven otherwise. Assuming radius, r, needs to be squared that means that in my case if the tring, the radius of our spinning weight, is slowly reeled in from an nitial distance of 10 feet to 5 feet CoAM dictates that the number of egrees for which the weight will have to rotate through (for the same fixed eriod 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 rder to maintain constant angular momentum, and incidentally the same onstant area of space, the number of degrees that one must sweep through or 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 f inverse square of the distance, which ironically most of us should be uite familiar: x/r². The same equation explains the effects of gravity as ell as lots of other phenomenon, like how the strength of electromagnetic adiation (light and radiation) changes depending on the distance from the ource. CONCLUSIONS: What this exercise brought home to me was an unexpected bonus of insight. My ittle journey of logic taught me the fact that the law of Conservation of ngular Momentum appears to be virtually indistinguishable from the Inverse quare of the Distance law. While many might accept such a revelation or elf-evident, so what's the big deal, what might not be as obvious is the act that Kepler's 2nd law which states, a line joining a planet and the Sun weeps out equal areas during equal intervals of time, literally seems to lay as much of an intimate part of explaining how CoAM works as it does in xplaining how the behavior of rotating planets behave. This was brought ome to me when during one of my prior analysis periods while studying the ffects of the 2nd law. I noticed that the 2nd law works just as well with a atellite in a hyperbolic trajectory racing past a gravitational body. Not nly that, the 2nd law works just as well with a satellite traveling past a ody that exerts NO gravitational influences at all! Armed with these evelations, it was not that much of a stretch for me to conjecture that epler's 2nd law would probably also help explain how CoAM behaves. At first glance making such a revelation might not make sense to some ecause if you attempt to incorporate both the effects of gravity x/r² with he effect of CoAM which also contains the same x/r² equation then wouldn't oth of these identical equations when combined tend to cancel each other ut? Wouldn't they cancel each other because the attractive forces of ravity would be working in exact opposite to the centripetal forces enerated from the effects of CoAM? Miraculously, a cancellation of nfluences does not happen! Why and how that doesn't happen is a fascinating ale in its own right. It's a tale I have already touched on within the Vort ollective, back around March of 2012. Some may recall that I posted a hread where I mentioned the fact that some of my computer simulations ppeared to mimic the effects of Celestial Mechanics pertaining to the istance a satellite would be predicted to be at if one charted the distance he satellite as the value of y while simultaneously fixing a constant slice f time as the value of x. In order to make everything line up the computer imulation has to incorporate the following elements, where the first lement mimics the forces of gravity and the second equation mimics entripetal 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 ine wave. Nevertheless, it mimics the exact distance from the gravitation ody 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 ew smart Vorts, perhaps a few afflicted with the ADD gene, might be able to igure that one out in due course. It is one of many goals I have set for yself, to hopefully explain why x/r³ might help explain the nature and hape of the elliptical orbit, such as a planet orbiting a star. Hopefully, '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 dventures that Andrea Rossi might have gone through as he accidentally urnt his hand and in the process discovered an anomalous source of heat rom one of his experiments. I sometimes suspect that had Rossi been more cademically trained in matters of nuclear physics and/or chemistry it is ossible that the spurious heat source he stumbled across might have been ismissed as nothing more than a careless mistake on his part. He might have een more inclined to dismiss what his senses were telling him, what his urned hand told him. His formal training would have "taught" him... drummed ut of him an almost naive-like willingness to entertain the possibility hat the affect he stumbled across could possibly be worth a second look. It might also might help explain why many with impressive credentials with ormal academic training in physics and chemistry often seem to come up mpty-handed. They come up dry because formal training increases the chances hat they will continue to blind themselves. An occasional observation inting of an insignificant little anomaly that spuriously crops up in their ata would automatically be perceived nothing more than a mistake that would est be ignored. A revelation for me was the realization that being afflicted with just a iny-tiny bit if ignorance can occasionally... just occasionally be a very ood thing! Occasionally being somewhat ignorant of what is currently taught n physics and chemistry can turn out to be the harbinger of unexpected urprises and discoveries. You might be more inclined to forge your own nique path to the City of Rome. You would be so inclined because you would iterally have no other choice but to forge your own path. By doing so you ay eventually stumble across observations that turn out to be revelatory, bservations and insights which mainstream academia had never seriously onsidered. It all comes out of the simple fact that there would be more of n innate willingness to smell each and every single plant one encounters in way where formal education would have paid less attention to. You would be ess inclined to dismiss some of the more subtle fragrances as nothing more han 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" as not already been observed by many within academic and scientific esearch fields. It's quite possible that what I posted here has already een written up in a series of obscure text books that are currently ollecting dust in a basement of a university library. Nevertheless, what I ope I was able to convey here was the joy that can be experienced in the ursuit of a discovery, which in the greater scheme of things may actually e nothing more than a re-discovery of a discovery. Nevertheless, that oesn't take away from the fact that embarking on such journeys can ccasionally produce unexpected surprises - some with earth shattering onsequences. Regards teven Vincent Johnson ww.OrionWorks.com ww.zazzle.com/orionworks
