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This looks a lot like homework, and we have a policy here of not doing your homework for you. How would you describe this problem mathematically? What are the relevant equations?

Equality is the best symbol for this expression, not approximately equals. This gets at the heart of the definition of the real numbers. Almost all real numbers (i.e., the irrationals) are defined by limits. In the problem at hand, there is no number between [math] \lim_{x \rightarrow 2 }\left(\frac{x^2-4}{x-2} \right)[/math] and 4: They are exactly equal to one another. The expression [math]\frac{x^2-4}{x-2}[/math] is of course undefined at [math]x=2[/math], but the limit is not. So this problem also gets at the heart of the definition of continuity. A function is continuous at a point if the value of the function is defined at that point and the limit of the function as the argument approaches the point in question from any direction exists and is equal to the function value.

The full title of the book is "Occult Ether Physics: Tesla's Hidden Space Propulsion System and the Conspiracy to Conceal It". For the utter amusement of the members of ScienceForums, the referenced book is on the web at http://www.bibliotecapleyades.net/tesla/occultether/occultether.htm. There is absolutely nothing to this. First indicator: There is no mathematics. Mathematics is the heart and sole of physics. Without mathematics, the author is doing philosophy, and is doing so very badly. Second indicator: The author posits a vast conspiracy that spans 90 years to keep the true nature of universe hidden. The only way two people can keep a secret is if one of them is dead. In that 90 year span somebody would have talked. Either all of the tens (hundreds?) of thousands of physicists alive today are part of this vast conspiracy (somebody would have talked!) or the people behind this conspiracy have managed to hide the real laws of physics from the normal run-of-the-mill idiot physicists. Either the conspiracy is world-wide and involves high-level officials in the governments of all of the advanced nations for the past 90 years, or the real laws of the universe are somehow hidden from all but those in the know in the CIA and NASA (which the author calls NAZIA). Think about it this way: For the first 70 years of this conspiracy, the Soviet Union was the mortal enemy of the USA. What kept the Soviet Union from discovering this secret and using it to beat the US to the Moon? Learn the mathematics. This will take some honest effort.

You have a misunderstanding of what "theory" means in science. On the extremely rare occasion where someone posts a theory in the speculations forum, the poster is informed that the post is a known result and the post is moved to the appropriate science forum. Most people who post stuff in the speculations forum post absolute garbage. Occasionally someone will post a WAG - a wild assed guess. Every once in a while someone will post a real speculation, and sometimes people post what could actually be called a hypothesis. Scientific theories are not wild assed guesses. They are backed up with *lots* of evidence, are preferably supported by logic, and they most certainly have been vetted by the peer review process. Wild-assed guesses, conjectures, and hypotheses are not scientific theories. It takes a lot of work by the proponent to move a concept from "hypothesis" to "theory". The speculations forum exists for a couple of reasons (at least). Some fora aggressively delete all posts that propound non-standard concepts. Other fora go to the other extreme and embrace non-standard concepts. ScienceForums has taken a middle-of-the-road approach. We don't delete conjectural posts because that is just too mean-spirited. We don't out-and-out embrace them, either; that is just stupid. There is a world of difference between being open-minded and playing the patsy. Another reason is that, just perhaps, someone will post something of value. Skeptic Lance in his post talked briefly about plate tectonics. This concept has its roots in Alfred Wegener's idea of continental drift. Geologists ignored this idea for half a century or so in part because Wegener was not a geologist. Maybe, just maybe, some outsider will post an idea in our speculations forum that is worthwhile. ScienceForums will have made an incredible contribution to the body of science if ScienceForums helps this worthwhile idea take root a bit faster than the half a century it took for continental drift to take hold. That's all speculation, of course, because to date no useful new idea has been proposed in the SF speculations forum.

That is a hypothesis (maybe a theory) but it is not a fact. The brain has some aspects of a computer, fact. The brain IS a computer on the other hand is a very strong statement. That the brain IS a computer is a widely held but by no means universal view. John Searle definitely holds an opposing view. For example, see http://users.ecs.soton.ac.uk/harnad/Papers/Py104/searle.comp.html

Couldn't resist ...

Just to add: Both of these relative forces, F_N-F_C and F_F-F_C, are directed away from the Earth. ======================== The high tides typically do not occur when the Moon is directly overhead (or directly underfoot). The ocean tides are fairly complex. They vary *a lot* based on terrain (e.g., the Bay of Fundy). Some places have diurnal tides, other places have just daily tides, and yet others have something in between (two high tides, but one is much higher). On average, the tidal bulge leads the Moon by a bit because the lunar day is 24 hours and 50 minutes long. This phase difference in turn is what causes the Earth rotation rate to slow down a bit (one solar day is now 86400.002 seconds long; it was 86400 seconds long in 1820 or so) and what causes the Moon to retreat from the Earth.

That is flat-out wrong. Even the ancients knew the world was round and made some fairly accurate assessments of its size. Science per se started in the 1600s or so, or about 1800 a couple thousand years after Eratosthenes. What passed for science before the scientific revolution is not really science in the sense of using the scientific method. There aren't all that many scientific theories that have been utterly dismissed. Newtonian mechanics -- most engineering is applied Newtonian mechanics. Darwin's theory of evolution didn't have a causal mechanism and relied too much on gradualism. Modern synthesis, which predates the discovery of DNA, is a meld of Darwin's theories and genetics. Evolutionary theories have progressed considerably since then. That said, schools still teach discuss Darwin's work. A couple of examples of theories that have been utterly falsified are the caloric theory of heat and the theory of a luminiferous aether. These utterly falsified ("never right") theories are exceptions.

The drift rate of electrons in a typical wire driven by a typical battery is very, very slow -- on the order of millimeters per second. The signal (e.g., closing the circuit by connecting both ends of a wire to a battery) travels very quickly. This signal velocity is a significant fraction of the speed of light (1/3 c, IIRC) as compared to the information velocity, or about 1011 millimeters per second. The free electrons in a metal act a lot like a gas. One way to envision what is happening is to construct a mechanical analog of an electronic circuit. Envision a simple electronic circuit comprising a battery, a variable resistor, and some pieces of wire connecting the battery and resistor to form a simple closed circuit. A mechanical analog uses a pressurized gas tank (just pressurized air) in lieu of the battery, an adjustable valve in lieu of the variable resistor, and plumbing in lieu of the wires. Rather than closing the circuit the valve in this mechanical analog simply vents gas to the air. With the valve initially closed, the gas in the plumbing will be at the tank pressure. When the valve is opened slightly, gas molecules will start leaving the valve and other gas molecules will leave the tank and start moving toward the valve. These molecules will move through the plumbing much, much slower than the signal velocity, which is the speed of sound in this case.

That is some fairly simple math. In short, the tidal force is Inversely proportional to the cube of the distance (rather than the square of the distance, as is the case with gravity), which is why the Moon has a greater tidal effect than does the Sun, Directed away from the center of the Earth at the two points on the Earth's surface intersected by the line connecting the Earth and Moon centers of mass, Directed toward the center of the Earth anywhere on the great circle defined by the plane that contains the center of the Earth and normal to the Earth-Moon line. Kind of like that, but in reverse. Think of it as attaching strings to two opposite points on a rubber ball. Those opposite points will define a great circle on the ball -- the set of all points halfway between the two points. If you pull on the strings, and squeeze on the great circle you will get a picture of what the Moon is doing to the Earth.

The Earth as a whole is accelerating toward the Moon: [math]\boldsymbol{a}_{e\to m} = -\,\frac {GM_m}{||\boldsymbol{r}_{e\to m}||^3}\boldsymbol{r}_{e\to m}[/math] The acceleration at some point pon the surface of the Earth is slightly different from this: [math]\boldsymbol{a}_{p\to m} = -\,\frac {GM_m}{||\boldsymbol{r}_{p\to m}||^3}\boldsymbol{r}_{p\to m}[/math] The tidal acceleration at that point is the difference between the gravitational acceleration toward the Moon at that point and the gravitational acceleration of the Earth as a whole toward that point. The acceleration at the point on the Earth's surface directly in line with the Moon's and Earth's center of mass is directed toward the Moon with magnitude approximately [math]a_{tidal} \approx 2\frac {GM_m}{||\boldsymbol{r}_{p\to m}||^3}r_e[/math] The tidal acceleration at the point on the opposite side of the Earth has nearly the same magnitude and is directed away from the Moon. Hence the bulges.

You read it incorrectly. The jurors in question were grand jurors, not petit trial jurors. Cheney and Gonzalez didn't have to appear in court; this was a pretrial dismissal hearing. The defense attorneys did not have to present the defense case; doing so is inappropriate at a dismissal hearing. I suspect that this was the opening salvo of a series of requests for dismissal. The next shot would surely have been a request for dismissal on the grounds that the prosecutor had no case. The defense did not have to make this second request because they hit the target with the first shot.

In that case, the runner obviously performs more work. Since there is no friction, all of the energy expended by the runner/walker goes into changing the person's kinetic energy. Who crosses the finish line with more kinetic energy?

The work in both cases is zero (assuming the person comes back to rest in each case, that is).

No! First off, that is conflating mathematics with physical reality. Secondly, the whole of mathematics cannot be proven true. Gödel's incompleteness theorems get in the way.

Sysco, You are trying to ascribe physical reality to mathematics. That is a bad idea, even with something as basic as counting. One drop of water added to one drop of water makes one (bigger) drop of water, one proton colliding with one antiproton results in zero protons, and one bottle colliding with one bottle makes a lot of shards of glass. None of these invalidate 1+1=2. Finding that space itself is quantized will not invalidate the mathematical concept of a point. Finding that the universe is non-Euclidean (if that happens) will not invalidate mathematical concept of Euclidean geometry. Euclid's geometry will still be as true then as it is now and as it was in Euclid's time. That final point about Euclidean geometry illustrates a key difference between mathematics and science. Mathematical theorems, once proved true, remain true forever. Scientific theories are at best provisionally true. You are conflating mathematics with the use of mathematics as a tool by scientists, economists, etc. Don't do that! ======================= Daniel S. Freed, "Higher Algebraic Structures and Quantization", preprint at http://xxx.lanl.gov/abs/hep-th/9212115 Louis Crane, "Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC", preprint at http://xxx.lanl.gov/abs/gr-qc/9504038 John Baez wrote an extended "Tale of n-Categories" beginning in http://math.ucr.edu/home/baez/week73.html and ending in http://math.ucr.edu/home/baez/week100.html

I disagree: Quantum dots do not exist mathematically. They exist physically, and you physicists have developed a mathematical model of them that is consonant with their theoretic and experimental behavior as far as you can tell. Mathematics is something distinct from science. A mathematical truth remains a mathematical truth forever. Absolute truth doesn't exist in science. Some young whippersnapper will someday find a new feature of quantum dots that shows the existing models of quantum dots are but an approximation. The mathematics you physicists and electrical engineers currently use to describe quantum dots may be of reduced interest because of their reduced physical validity, but they will remain valid mathematically.

IQ tests are good at measuring the lack of intelligence (i.e., those in need of remedial help), which is not surprising as (a) that is what they were originally designed to do, and (b) defining what constitutes lack of intelligence is a much easier task than defining what constitutes an abundance of intelligence. Scores above 130 are more-or-less meaningless (indistinguishable) on most standard intelligence tests. I would argue that the degree above which IQ scores have relevance is even lower: Say about 120 or so. Evidence: Nobel laureate Richard Feynman's IQ = 124 IQ = 125

Word salad. The nature of the universe does not dictate anything with regard to mathematics. Mathematicians can (and do) create a complete valid mathematical constructs that have absolutely nothing to do with physical reality. The expansion of the universe has nothing to do with the nature of a point as a point is a mathematical construct.

Not again! Can someone lock this thread?

Translational states and rotational states decouple in an inertial frame. This is not the case in a rotating frame. Moreover, the equations of motion are vastly simpler in an inertial frame. Finally, your inertial navigation system measures state derivatives (acceleration and angular rate) with respect to inertial. Just because you want to control state in the NED frame doesn't mean you have to do physics in the NED frame. I disagree. Attitude rate measurements are with respect to inertial but are expressed in INS case frame. Since the transformation from the vehicle body frame to the INS case frame is a fixed (non-varying) transformation, it is just as valid to say that attitude rate measurements are respect to inertial but are expressed in body frame coordinates. This is the natural frame for propagating rotational state. It's a rotating frame, so you will need to incorporate the inertial torque [math]-\boldsymbol{\omega} \times ({\mathbf I} \boldsymbol{\omega})[/math] into your rotational equations of motion. Note well: That is the only pseudo-torque you need to incorporate into your equations of motion if you represent rotational state with respect to inertial. Choose some other rotating frame as the basis and you get an absolute mess, and you get the added disadvantage of coupled translational and rotational states.

Power isn't conserved a conserved quantity. Why should it be? Power is a process variable. An object doesn't hold a certain amount of power. An object holds stored energy. A more powerful object can release more stored energy per unit time than can a less powerful object. Let's use your example of two robots, one of which is more powerful than the other. The more powerful robot simply lifts 100 kilogram crates directly. The less powerful robot uses mechanical advantage to overcome the lack of power. Both robots perform the same amount of work (and thus use the same amount of energy) in lifting a 100 kilogram weight to some specified height.

You did undeed "put that much work in when using a lever to hoist up something heavy". What you put in was a reduced force, but at the expense of having to apply that force over a larger distance. Work is, in layman's terms, force times distance. To lift a 100 kg object 10 centimeters directly without the aid of a simple machine means you have to apply a 980.665+ Newton force as you lift the object the requisite 10 centimeters. When you have lifted that 100 kg object the 10 centimeters off the ground you will have performed 98.0665 joules of work. Suppose instead you use a lever with a 10:1 mechanical advantage. Now you can raise the same object by applying a 98.0665+ Newton force, but you have to apply that force over a 1 meter span of distance. When you have lifted that 100 kg object 10 centimeters off the ground you will have still performed 98.0665 joules of work.

You have two rotations here: The NED frame with respect to the Earth-fixed frame, and the Earth-fixed frame with respect to inertial. Your rate gyros (if you have them) measure rotation with respect to inertial, of course. The angular velocity with respect to NED is somewhat easy because angular velocity vectors are additive. Angular acceleration is a horse of a different color. The angular velocity of the plane with respect to inertial and the plane with respect to NED are related vectorially. Denoting, for short, I as the inertial frame, E as the earth-fixed frame, L as the local NED frame, and B as the plane's body frame, [math]\boldsymbol{\omega}_{I \to B:B} = \boldsymbol{\omega}_{I \to E:B} + \boldsymbol{\omega}_{E \to L:B} + \boldsymbol{\omega}_{L \to B:B}[/math] The plane body frame is not the natural frame for expressing either the earth rotation rate ([math]\boldsymbol{\omega}_{I \to E}[/math]) or the NED frame rotation rate ([math]\boldsymbol{\omega}_{E \to L}[/math]). With appropriate representations, the above becomes [math]\boldsymbol{\omega}_{I \to B:B} = \boldsymbol{\omega}_{L \to B:B} + \mathbf T_{L \to B}( \boldsymbol{\omega}_{E \to L:L} + \mathbf T_{E \to L}\, \boldsymbol{\omega}_{I \to E:E}) [/math] Differentiating with respect to time, [math]\dot{\boldsymbol{\omega}}_{I \to B:B} = \dot{\boldsymbol{\omega}}_{L \to B:B} + \dot{\mathbf T}_{L \to B}( \boldsymbol{\omega}_{E \to L:L} + \mathbf T_{E \to L}\, \boldsymbol{\omega}_{I \to E:E}) + \mathbf T_{L \to B}( \dot{\boldsymbol{\omega}}_{E \to L:L} + \dot{\mathbf T}_{E \to L}\, \boldsymbol{\omega}_{I \to E:E}) [/math] Note: I intentionally omitted the earth's angular acceleration from the above, as this term is very, very small. The time derivative of a transformation matrix is given by [math]\frac{d\mathbf T_{A \to B}}{dt} = -\,\mathbf X(\boldsymbol{\omega}_{A \to B:B})\mathbf T_{A \to B}[/math] where [math]\mathbf T_{A \to B}[/math] is the transformation matrix from frame A to frame B, [math]\boldsymbol{\omega}_{A \to B:B}[/math] is the angular velocity of frame B with respect to frame A expressed in frame B coordinates, and [math]\mathbf X(\mathbf u)[/math] is the skew symmetric cross product matrix generated from the vector u. With this, [math]\dot{\boldsymbol{\omega}}_{I \to B:B} = \dot{\boldsymbol{\omega}}_{L \to B:B} - \boldsymbol{\omega}_{L \to B:B} \times (\mathbf T_{L \to B}( \boldsymbol{\omega}_{E \to L:L} + \mathbf T_{E \to L}\, \boldsymbol{\omega}_{I \to E:E})) +[/math][math] \mathbf T_{L \to B}( \dot{\boldsymbol{\omega}}_{E \to L:L} - \boldsymbol{\omega}_{E \to L:L}\times (\mathbf T_{E \to L}\,\boldsymbol{\omega}_{I \to E:E}))[/math] And with this result, perhaps you might want to rethink whether you should be doing your calculations in the inertial frame and transforming to NED when finished.

Inertial frames in general relativity do not have infinite extent like they do in Newtonian mechanics. Suppose you have a tiny mass orbiting the Sun, far from Pluto. A cartesian reference frame with non-rotating set of axes and with its origin collocated with the tiny mass will form a local inertial frame. A nearby object also in orbit about the Sun will not appear to be undergoing any acceleration in that frame. Pluto will appear to be undergoing an acceleration, but that is because your frame is no longer inertial there. Gravitation is a pseudo force in general relativity. You can always find an inertial frame in which there is no gravitational acceleration.