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Given the discussion that generated your question, you are (I think) asking about biological evolution. Every technical field has its own set of jargon, words and phrases that have a special meaning in that field. By way of analogy, consider the word "weight". Legally and colloquially, weight is a synonym for mass. Weight in physics is something different. In elementary physics, weight is the force caused by gravitation. In more advanced physics, weight is the net force of everything but gravitation. Regardless of which physics definition of weight one uses, one thing is certain: It is not a synonym for mass. Biological evolution is about how species change. It pertains to life, not pre-life.

As you've already been told, probability and statistics are a part of mathematics. Some rather advanced mathematics, measure theory, is needed to properly understand probability. Secondly, there's a whole lot more to mathematics than just numbers and formulae. Many professional mathematicians never deal with numbers and formulae except in the abstract. Finally, modern weather forecasting is extremely math-based. A hugely complex computational fluid dynamics model lies at the heart of a modern weather forecasting system. The model has to be propagated over time and space according to the laws of physics. Along the way it is updated with measurements of various degrees of fidelity and resolution. It's math as far as the eye can see. Analog computers also are (or rather, were; see below) highly math based. The first step in building an analog computer is to develop the differential equations (math!) that describe the system to be modeled. The next step is to build an electric circuit that follows the same set of differential equations (more math!) as the system to be modeled. The mathematics of the electric circuit are dictated by the differential equations that describe each of the components in the circuit and the differential equations that describe the wiring between components. The key problem with analog computers is that solving a new problem involved creating a brand new analog computer. "Programming" analog computers is a painstaking, time-consuming process. With digital computers, different problems to be solved merely require different programs. The computer itself doesn't change. That's a huge advantage over analog computing and it is the key reason that analog computers pretty much bit the dust half a century ago with the advent of digital computers. There is no such thing as physics without math. It is just pre-scientific story telling. The real problem here is that western cultures have taken on this attitude that being at all adept at mathematics is a subject for nerds and geeks only, and if you are good at mathematics, you must be a nerd or a geek. Could you imagine a congress critter (or your country's equivalent) admitting that they can only read at the sixth grade level? Why is it acceptable for them to only be able to do math at the sixth grade level? It's not only acceptable, it's chic.

Addendum to my last post: Even simpler than using forward Euler for that initial guess is to simply use the initial state.

It was debunked here, too. He just cannot acknowledge it. Bell's theorem kills the idea.

No. You should treat those as simultaneous equations. That means you need an initial guess for [imath]y_1(n+1)[/imath] and for [imath]y_2(n+1)[/imath]. It also means that once you have a [imath]y_1(n+1), y_2(n+1)[/imath] pair you use that pair simultaneously to obtain the next iteration on [imath]y_1(n+1), y_2(n+1)[/imath]. Since forward Euler depends on initial value rather than end values, you can use forward Euler to obtain those initial guesses for the final values. Once you have those initial estimates for the end values, iterate with backward Euler until [imath]y_1(n+1), y_2(n+1)[/imath] have both converged to a stationary value.

You need an initial guess. One approach is to use forward Euler for that first guess. With this you obtain the simplest of the predictor/corrector class of integration techniques.

I would read what Swansont wrote as he wrote it. Most physicists agree with Max Tegemark on why mathematics is so unreasonably effective in the natural sciences. It's quite simple: The physical world is completely mathematical. The key job of a theoretical physicist is to uncover that mathematics.

Are you saying , that if or while the angular velocity is increasing, there WILL be a centrafugal force ? "Centrafugal" is not a word. Did you mean centrifugal, centripetal, or something else? Swansont said nothing about what happens while angular velocity is increasing. He talked about what happens if speed is constant. If speed is constant, so is angular velocity. If the speed isn't constant, the acceleration will have a tangential component as well as a radial component. The radial component is still directed toward the center: centripetal. The tangential component is neither centripetal nor centrifugal. It's tangential.

Of course he is not using math. This is a popularization of science book. Such books tend not to be best sellers; the lay audience is a bit hostile to science. The target audience is someone who perhaps can do math at the ninth grade level and who was quite confused by that low level of mathematical sophistication. One way authors can guarantee their pop sci books won't sell well at all is to sneak some advanced math like [imath]F=ma[/imath] (or heaven forbid, [imath]\vec F = \frac {d^2 \vec x} {dt^2}[/imath]) into their text. It's very hard to find a pop sci book that uses any math whatsoever. The language of physics and astronomy is math. It has been since before Newton's time, and the interconnectedness has only grown stronger since then. There is no escaping this fact

How, exactly, do you see that? He had a germ of an idea. That's a rather vague concept. What he did say is that he saw a conflict between Newtonian mechanics and Maxwell's electrodynamics. Do you seriously think no one else saw that conflict? That was the central problem of physics in the latter part of the 19th century. The 16 year old Einstein didn't have anything to offer because the stupid, moribund physics establishment was neither stupid nor moribund. This problem attracted the best minds of the time. I'm not quite sure what you're saying here. I dispute that they couldn't have figured it out. What if there had been no Einstein? The core conflict would still have been resolved; Lorentz and Poincare were hot on the trail of solving this problem. For a short time physicists would have had Lorentz Ether Theory as a solution. To kristalris: Lorentz Ether Theory predated Einstein's special relativity and is mathematically indistinguishable from special relativity? So why don't physicists teach it? The answer is simple: LET makes the Lorentz transformation axiomatic and it postulates an unknowable aether frame. LET is an ugly theory. In comparison, special relativity is a beautiful theory. It's axioms are simple and physically testable. Nonetheless, that LET does predate special relativity is one of the reasons Einstein was not given the Nobel prize for his work on special relativity. Back to the "what if" thought experiment: LET wouldn't have lasted for long. Quantum mechanics was just around the corner. Photons don't need a medium. They have no problem traveling through a vacuum. The quantum mechanics description of electromagnetic radiation obviates the need for an aether. Someone else would have seen Einstein's solution. It might have taken a bit longer, but it would have happened. In the end, physics almost certainly would still have arrived at special relativity with its simple, elegant hypotheses. I never said that, so please do stop putting words in my mouth. Of course unknowns have made contributions, huge contributions, to science. Every great scientist was an unknown at some point. The best way to move from the "unknown" category to the "widely known" category is to make some great discovery and publish it. Most widely known scientists made this transition when they are young adults. Young adult scientists, that is. Einstein was not an outsider, uneducated in the field. Feynman was not an outsider, either. Nor was Newton. Nor was Hubble. There's a huge, huge gap between outsider and unknown. The only outsider who I can think of who made a significant contribution to science in the last hundred plus years is Alfred Wegener, and even that one is dubious. Wegener's continental drift was superseded by plate tectonics, and plate tectonics differs from continental drift in a number of ways. That's one possible exception, and the exception proves the rule: Outsiders do not make meaningful contributions to science.

There's nothing in that citation that supports your claim. Here's what you claimed: Here's what your link, http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/origins_pathway/, contains: There's nothing in that regarding the 16 year old Einstein reporting his thought experiment to friends. That's pure fabrication on your part. There's nothing in this about length contraction, either. That is also a fabrication on your part. Einstein did not conceive of length contraction in 1895 as one of the key tools needed to resolve the conflict between Newtonian mechanics and Maxwell's electrodynamics. There's not one inkling in that quote regarding length contraction. Length contraction wasn't even Einstein's idea. It's called the Lorentz contraction, not the Einstein contraction. How much Einstein knew in 1905 of others' work on resolving this conflict has long been debated. He obviously was aware of at least some part of that work because his 1905 paper discusses and uses the Lorentz transformation. You are jumping to conclusions and are guilty of hindsight bias in claiming that special relativity could and should have been realized ten years earlier had only the stupid and moribund physics establishment talked to the 16 year old Einstein. At best the 16 year old Einstein independently found something that that stupid, moribund establishment was already aware of. Your claim is unsubstantiated nonsense.

Baloney, and not just run of the mill baloney. This is 100% pure baloney stuff. Pure BS, if you will. Stop making stuff up. This claim of yours is ridiculous and unfounded. You need to prove it. In the words of Randall Munroe, Einstein himself is the source of that story about Einstein at 16 year old. He wrote of this more than half a century later in his autobiography. The description is short and vague, and who knows how true it is given that half a century passed between the occurrence and Einstein's writing of it. You have embellished that story beyond anything claimed by Einstein. At best, the 16 year old Einstein independently found something of which the physics establishment was already extremely well aware, that Newtonian mechanics and Maxwell's electrodynamics don't jibe with one another. Nothing would have transpired even if physicists had talked to the 16 year old Einstein. At that time he had nothing to offer that physicists didn't already know. He only did have something to offer after he had received the requisite knowledge. Aside #1: It's amusing how the crackpot community treats Einstein. Some parts venerate him as a god, other parts think of him as the devil incarnate. Aside #2: Studiot is correct. Einstein does not count as an outsider. He was a part of the physics establishment when he wrote his 1905 papers.

Both, actually. Math is the bedrock of many of the sciences, and in physics, ALL CAPS is the way to say this. If you aren't using math you aren't doing physics. You're at best explaining physics to your grandma by means of (inevitably poor) analogies. As far as not getting to the bedrock, science doesn't do that. It at best tries to create an ever improving model of what can be observed. Science provides models of behaviors, and it is inherently limited by what can be observed. If you want bedrock, you want religion. That religious bedrock might not be right (and oftentimes it is at odds with observable reality), but it is comforting. That's part of the appeal of pseudoscience. Just like religion, pseodoscience purportedly offers simple explanations that claim to be deeper than those offered by the sciences. That it is counterfactual is irrelevant. Don't be lulled by pseudoscientific mumbo jumbo such as that put forth by Robin Pike.

Mod edit: post and responses moved from discussion on centrifugal force that went on a tangent http://www.scienceforums.net/topic/72515-coriolis-and-centrifugal-force-visualized-in-a-rotating-massloop/page-3#entry745809 —————————— No, Mike. Please don't go there. You will not find the answer to the meaning of life, the universe, and everything in pseudoscience. All you will find is charlatan nonsense. If you want to understand physics you need to understand those "math descriptors". If you don't want to learn those, then all we can offer ultimately is "because I said so". In fact, even if you learn the very hairiest of those "math descriptors", there's still going to be a "because we said so" brick wall that you hit. Science does not answer "why" questions. That's what charlatans and crackpots pretend to do. It's so satisfying to have those ultimate answers. Unfortunately, those ultimate answers are inevitably wrong. They explain nothing.

Baloney. I've said baloney a number of times. Do I specifically have to say baloney to your nonsense concept that Einstein developed special relativity when he was sixteen? Okay then. I will say exactly that, in bold and in a big font so you cannot possibly miss it. Einstein did not develop the theory of special relativity when he was sixteen. You are greatly inflating what Einstein may or may not have thought when he was but sixteen. Whether or not the sixteen year old Einstein was aware of it, professional physicists were deeply aware of the conflict between Maxwell's equations and Newtonian mechanics. Whether or not the sixteen year old Einstein was aware of the Michelson-Morley experiment, professional physicists were extremely aware of it at that time. The conflict between Newtonian mechanics and Maxwell's electrodynamics was one of the key problems in physics in the latter half of the 19th century. At best, Einstein independently discovered this already known conflict when he was sixteen. You are 100% wrong here. Did you even read the wiki article Swansont posted? Hindsight is absolutely wonderful. It tells us what we should have done a year ago, a decade ago, or even longer in the past had we only known then what we know now.

I'm using math because that is the language of physics. If you can't understand that language you are at a bit of a disadvantage. Rhetorical question: Can you push a rope and expect a force to be transmitted to the other end? The answer is of course not. Ropes are flexible. A rope under tension can transmit a force, but only in one direction. So what kind of force does one of those hammer throwers experience? First, look at the kinematics of the throw. The thrower and hammer are connected to one another via a rope or cable, and both are rotating about their common center of mass. The force on the hammer is directed toward the thrower, and hence toward the thrower/hammer center of mass. This is a centripetal force. The force on the thrower is directed toward the hammer, and hence toward the thrower/hammer center of mass. This also is a centripetal force. There is no centrifugal force here.

Nonsense. While they were unorthodox thinkers, they were not uneducated and they were not crackpots. Copernicus was self-educated, but he was born and died prior to the age of science. Galileo was educated at and taught at the University of Pisa. Newton was educated at and taught at Cambridge. Darwin was also educated at Cambridge. You are also comparing a different age. Galileo and Newton represent the start of science, Darwin the start of biology as a science. Nowadays scientists relish new concepts and experiments that turn everything upside down. That too is nonsense. Einstein did not have the theory of special relativity when he was 16. He had perhaps the germ of the idea, but nothing specific. He had this idea in part because the school he was attending at this time encouraged creative thinking. His educators did exactly the right thing by fostering that creativity and by teaching him what they knew at the time. More nonsense. Hindsight is so wonderful. Finally, something that isn't nonsense. This is spot on.

That is complete baloney. You are singing the siren song of the crackpot. Trained scientists, engineers, and mathematicians do not owe a crackpot one scintilla of their time with regard to the crackpot's purported theory. There is no "should" here. What trained scientists, engineers, and mathematicians do owe society at large is education. Most PhDs don't particularly like teaching physics 101, introductory calculus, etc. They nonetheless do it willingly because training that next generation is their duty. The crackpot has a duty as well, which is to to receive that education. A crackpot who intentionally remains uneducated in some field deserves zero time from the experts in that field.

This is the crux of your problem. For some reason you cannot accept the fact that acceleration does not necessarily change radial distance. Acceleration is a vector, not a scalar. The acceleration vector represents the rate at which the velocity vector is changing. If some object is undergoing uniform circular motion the acceleration vector always points toward the center of the circle along which the object is traveling. This is a mathematical necessity. Now let's apply Newton's second law. Given an object in uniform circular motion, the net force on that is necessarily centripetal. No centrifugal force is needed. No, you didn't. Look at your diagram. Draw a free body diagram. What centrifugal force do you think you measured? What you measured was a centripetal force.

No, for three reasons. 1. What you wrote doesn't quite make sense. What, exactly do you mean by "gravitational relative mass"? 2. Granting for the sake of argument that this could be wrangled into something make sense, the answer is still no for the simple reason that there are multiple planets. They interact. This is the N-body problem. Yes, a solution theoretically exists, but there's no practical way to find it or to use it. 3. Granting for the sake of argument that a solution could be formulated, it's still not possible. An exact solution requires exact knowledge of position and momentum. That is something we *know* cannot exist.

The platform in that diagram is depicted as spinning, so it is presumably drawn from the perspective of an inertial observer. From that perspective, there is *no* centrifugal force acting on any of the people in that diagram. Regardless of the frame of the observer, not one of the people in that diagram feel an outward force. What they do feel is an inward (centripetal) force. The centrifugal force is a fictional device used to extend Newton's second law from the inertial frames in which it is valid to rotating frames (where strictly speaking, Newton's second law is not valid). You cannot feel the fictitious centrifugal force because it's a fiction. A very useful fiction, but still a fiction.

That makes no sense. It's just a mishmash of buzzwords. Try putting some math behind those buzzwords. Even better, how about some predictions of experimental outcomes that will differentiate your buzzword physics from the current understanding of gravitation?

There is no need to bring general relativity into the picture when Mike doesn't even understand Newtonian mechanics. There's even less of a reason to mix up the two concepts. One key difference between Newtonian mechanics and general relativity is that gravity is a real force in Newtonian mechanics but it isn't in general relativity. You mentioned both the force of a gravitating mass and geodesics. That is a very bad thing to do. Newtonian mechanics and general relativity are profoundly different ways of looking at gravitation. It is best not to mix the two. Surely not. There is no balancing force, and no balancing force is needed to explain an orbit from the perspective of an inertial frame in Newtonian mechanics. All that is needed is gravity. Draw a picture of a small body orbiting a much more massive object. The standard picture has the small body moving along a curved path (a circle, ellipse, parabola or hyperbola) about the larger, central object. There's no balancing force here. There can't be. That orbiting body is following a curved path. That means it's velocity vector isn't constant, which in turn means it is accelerating. That acceleration vector always points toward the central body. The only force needed to explain this acceleration is gravity. There is no other force. Surely not again. There's a number of things that you are confused about here. One is that you don't feel the force of gravity. The general relativistic explanation is quite easy. Gravity isn't a real force in general relativity, and the only things you can feel are real forces. That's going counter to my own advice not to invoke general relativity, so I'd best explain why you can't feel gravity from the perspective of Newtonian mechanics. Gravity, from a Newtonian perspective, acts so close to uniformly across every part of your body that you can't feel it. Aside: You truly would feel gravity were the Earth millions of times more massive. Now gravity would not be close to uniform across your body. The tidal forces would act to pull your body apart. This is called "spaghettification." That effect still does exist for our puny little Earth, but it's so ridiculously small that you cannot feel it. So what do you feel as "weight"? The answer is that you feel everything but gravity. You aren't feeling gravity when you are standing in that elevator car. You are instead feeling the elevator floor pushing up on your feet. You feel that upward push by the floor, not the downward pull by gravity. That upward force propagates throughout your body, and you feel that too. You do not feel gravity. An astronaut on the International Space Station feels weightless not because gravity is non-existent (it's about 89% of Earth surface gravity) but rather because the only force acting on that astronaut is gravity. Suppose you wake up, rather woozy, and find yourself in what looks like an elevator car. You try to remember how you got there. You remember a party last night, but then what? Suppose that unbeknownst to you, that party was hosted by Bug Eyed Monsters. Yep. You've been abducted by aliens and are now far, far away from the Earth in a spaceship accelerating at 1g. The aliens have merely secured you in a room that looks like an elevator car; they'll leave you there until they have time to experiment. You're not anywhere close to a gravity field, yet you feel normal weight. That's because of the spacecraft's acceleration. What you feel as weight isn't gravity. It's everything but gravity. Now let's switch gears to another thought experiment. Imagine taking a slow ride all the way up a space elevator. The top end of the space elevator is well beyond geosynchronous orbit . We'll ride all the way to the top. You feel your normal weight as you start the ascent (it's a slow climb). Eventually you'll notice that you are feeling lighter and lighter. By the time you reach geosynchronous altitude you feel completely weightless. Once you pass geosynchronous altitude you start feeling pulled toward the ceiling. You'll need to stand on what previously was the ceiling to continue the ride to the top. Your feeling of weight now start to increase as you climb ever higher. During the first part of the climb that force from the elevator floor is upward, so by definition it's a centrifugal force. This is just a kinematic classification of a real force, in this case, it's the normal force that keeps your body from falling into the floor during the first part of the climb. Note that this normal force becomes a centripetal force once you climb past geosynchronous altitude. There is no need to invoke a fictitious balancing force to explain your motion, or what you feel. From the perspective of an inertial observer, the only forces acting on you are that of gravity and the normal force. Note that the normal force vanishes when you reach geosynchronous altitude. At this point, the only real force acting on you is gravity. Period. What about from the perspective of an Earth-fixed observer? Yes, now you can explain a geosynchronous orbit using the concept of the fictitious centrifugal force. However, from this perspective, a geosynchronous satellite isn't moving. It's standing still. Invoking the fictitious centrifugal force to explain orbits is wrong for a number of reasons. One is that the object isn't "orbiting". It's standing still. How is that an "orbit"? Another is that this explanation fails in the more general case of an elliptical orbit (or a parabolic or hyperbolic trajectory). There's no need to invoke the fictitious centrifugal force, and doing so obscures the true picture of what "orbit" means.

Gravitational potential energy is negative semidefinite, meaning that it is always negative or zero. It is zero in the limit of distance tending to infinity, negative for all finite distances. Kinetic energy is positive semidefinite, meaning that it is always positive or zero. Specific orbital energy is the sum of gravitational potential energy and kinetic energy, divided by mass. It is zero for a parabolic trajectory. There's no balancing act here. The object is escaping. Positive energy means a hyperbolic trajectory. All bound orbits, circular or elliptical, have total energy less than zero. There's no balancing act. How can there be? An orbiting body is accelerating.

Wrong. You guys across the pond are the ones who invented the concept of common law over a thousand years ago. You should learn how your own legal system works. The US learned a lesson from that success and then notched things up a bit. Judges in the US have the power to declare laws unconstitutional. Australia in turn learned a lesson from the US and notched things up even more. The Australian High Court is even more supreme than is the US's Supreme Court. Judicial review is a very important part of American and Australian law. This, too, is wrong. Argument from authority is not always a fallacy. In addition to reading up on common law, I also suggest that you read up on appeal to authority. In particular, when the Supreme Court of the US writes a majority opinion, that opinion is fact, at least until it gets overturned. (And that doesn't happen very often.)