D H

That's too simple a calculation. You are implicitly assuming that the iridium earth will be of constant density. Iridium, like any other material, compresses under pressure. The center of the iridium earth will have a density significantly greater than 22,560 kg/m3 because of the gravitational pressure resulting from all of the material above it.

Mass and charge are fundamentally different. The burden is upon you to show some kind of evidence that they are the same. Handwaving and just saying they are the same is not evidence. Your model has to be able to explain how the top quark, which has 2/3 of the charge of an electron, has a heck of a lot more mass than an electron. Your model has to be able to explain everything from electron and quarks, to apples falling on peoples' heads, to galaxies, and beyond. No, it is not.

I'll explain it, bit by bit. Along the way I'll correct some bad grammar in my post. That should be self-explanatory. It's such a neat set it has a name. I should have been a bit more explicit there. The process in question is repeatedly removing the middle third of the remaining parts of the set. Start with the interval on the real number line the contains all points between 0 and 1, inclusive. This a closed set, which for a simple set such as this means the end points of the set are in the set. The set of all points exclusively between 0 and 1 is an open set. The end points of the set are still 0 and 1, but this time the end points are not themselves in the set. The closed interval is denoted [0,1] while the open interval is denoted, (0,1). Removing the middle third is a bit ambiguous: Do we take 1/3 and 2/3 out, or leave them in? If you take them out the result is a pair of sets that is neither open nor closed. Leaving those points in makes things cleaner, so leave them in. The result after the first iteration is the union of the closed intervals [0,1/3] and [2/3, 1]. The second iteration involves removing the open intervals (1/9,2/9) and (7/9, 8/9), leaving four intervals [0,1/9], [2/9,1/3], [2/3, 7/9] and [8/9, 1]. The third iteration removes four middle thirds, leaving eight intervals, and so on. What's left will be a lot clearer by working in base 3. The base 3 representations of 3, 9, 27, 81 are 10, 100, 1000, and 10000. The first iteration removes (1/103, 2/103). The second iteration removes (1/1003, 2/1003) and (21/1003, 22/1003), and so on. Some background: we write the fraction 1/2 as 0.5 in base 10 (or 0,5 if you are from Europe), and 1/3 as 0.333... (the ... means the 3s continue forever). In base 2, 1/2 is 0.1 and 1/3 is 0.0101.... In base 3, 1/2 is 0.111... and 1/3 is 0.1. Back to the problem at hand: Except for 1/3 = 0.1 in base 3, the first iteration throws out all numbers whose first ternary digit is 1. Except for 1/9=0.013 and 8/9=0.213, the second iteration throws out all numbers whose second ternary digit is 1. The nth iteration throws out all numbers whose nth ternary digit is 1 (except for those nasty endpoints). Those endpoints look problematic. You may have heard the argument that 0.999... = 1. This is also true for lots of other numbers: 0.1999... = 0.2, for example. Terminating decimals such as 1.0, 0.2, and 0.125 have an equivalent non-terminating form: Just subtract 1 from the terminal digit and tack on an infinite sequence of 9s. I'll call 0.999... the non-terminating decimal expansion of 1. This same concept applies to other bases. For example, 1/3 in base 3 is written compactly as 0.1[/sub]3[/sub], but can also be written as 0.0222...3. If we do that to the endpoints in the construction of the Cantor set, the nth iteration of the construction can be viewed as throwing out all numbers whose nth ternary digit is 1. The end result is the set of all numbers between zero and one inclusive whose non-terminating ternary expansion does not contain a 1. So, how "big" is this set? Change all of the twos in the non-terminating ternary expansions of the members of the set to ones. The result looks like a set of numbers between 0 and 1 written in binary. So, view this set as a set of binary numbers. The result is the set of all numbers between 0 and 1 (inclusive) written in binary. Now take the non-terminating binary expansion of the real numbers between 0 and 1 (inclusive) and change all of the ones into twos. The result, when viewed in base 3, is the Cantor set. This little trick maps the Cantor set to all of the the real numbers between 0 and 1 (inclusive) one-to-one and onto. Since the set of all real numbers between zero and one is uncountably infinite, so is the Cantor set. While the Cantor set maps one-to-one and onto with the set of all points between zero and one, the Cantor set obviously is not the same as the set of all points between zero and one. The Cantor set is all broken up while the set of all points between zero and one is nice and continuous. The Hausdorff dimension is a measure of the dimensionality of some set. For example, the set [0,1] has a Hausdorff dimension of 1 (as does the entire real number line), and the unit square (or even the 2D plane) has Hausdorff dimension of 1. The Cantor set truly is "smaller" than the set [0,1] when viewed in terms of this dimensionality metric. That's embarassing. The Cantor set is a set of measure zero, not has set measure zero. Think of measure as an extension of the concept of length. The set [0,1] has a length of 1. The first iteration of the construction whacks a 1/3 of the total length, leaving 2/3. On the second iteration, 4/9. On the third, 8/27. On the nth, all that is left is (2/3)n of the original length of 1. This remaining length vanishes as the number of iterations increases. In the limit, there is no "length" left: All that is left is a bunch of dust. Each stage of the construction of the Cantor set entails removing an open set from a closed set. The result of doing this is a closed set. (The endpoints are in the set.) When the process is complete, the result is still a closed set. A very funky closed set, but a closed set nonetheless. The rationals are everywhere dense in the reals. What this means is that given any real number, a neighborhood of that number, no matter how small, will always contain at least one rational number. The contrary is true for the Cantor set. Members of the Cantor set are isolated points ("dust"), and any real number in (0,1) that is not in the Cantor set is part of a non-zero open interval that has no points in common with the Cantor set.

No, you haven't. Have you taken this quiz?

The proton is about 1836 times as massive as an electron, but has exactly the same charge as an electron (sans a sign change). The top quark is about 338,000 times as massive as an electron, but has only 2/3 the charge of an electron. The neutron and neutrino have zero charge, yet neither has zero mass (and the two have vastly different masses). Mass and charge are fundamentally different.

There are four fundamental forces. Electromagnetism is only one of them. No, it can't, for several reasons: What you wrote is not even a well-formed expression. Is the QQ at the end part of the denominator or the numerator? The only way to rescue this "equation" is to put that QQ in the numerator (I assume you meant the square of the elementary charge), so I'll do that: [math]G=\kappa \,(M_p+M_e) M_e Q^2 / 8[/math] But even that doesn't make sense. It has the wrong units. The gravitational constant has units of length3/mass/time2. The right-hand side of your "equation" has units of length3×mass3/time2. The expression can be made dimensionally correct by putting the mass terms in the numerator rather than the denominator: [math]G=\kappa \frac{Q^2}{8(M_p+M_e) M_e}[/math] So now the right- and left-hand sides of the equation agree dimensionally. How about numerically? G is of course 6.67300×10-11 m3kg-1s-2 in SI units. The right-hand side, on the other hand, evaluates to 1.8917×1028 m3kg-1s-2. Oops. Even if the result was close (and the above does not qualify as close), you can't just throw a bunch of physical constants and numbers together and call it a "theory". What is the rationale for your equation? Without a rationale, all you have is numerology, and very, very bad numerology at that.

Moderation note: This thread was split from this discussion on gravity.

Moderation note: The off-topic posts about "Dot wave theory" were moved to a separate thread.

The problem is that this is not technically correct: The opening paragraph of the Wikipedia article: Technically correct, poorly written. The problem is that the reference system for sidereal time is not quite inertial. How to communicate this without jargon? A star will return to nearly the same overhead position in one sidereal day. The reason the star doesn't return to exactly the same position is because the Earth's rotation has a little wobble to it, and sidereal time is measured with respect to this wobbling reference. It takes about 26,000 years for the Earth's rotation axis to trace out a cone with about a 23 degree half angle, making this wobble very small.

I agree with iNow. Wikipedia is, in general, quite accurate. One just has take what it says with a grain of salt. Ahhh. You are lazy. Nothing wrong with that per se, but when you get to more advanced topics you will not so readily find someone around to pester. Learning how to understand what you are reading is an important skill, and it won't magically pop into existence. Better to read wikipedia then ask specific questions about difficulties or clarifications. Klaynos was prescient. Read the first paragraph of the Wikipedia article on sidereal time to see why.

The Shuttle has released a long cable with a satellite on the end, twice. The goal was to generate electrical power, not change the orbit. TSS-1 flew on STS-46. Mechanical problems arose after releasing only 256 meters of cable. The engineers went back to the drawing board. TSS-1R flew on STS-75. This time they managed to release 19.7 kilometers of cable before the cable broke. Two strikes and you're out. NASA has not flown a tether experiment on the Shuttle since.

People who have to do navigation, or geodesy, or surveying, or cartography, or ... for a living generally don't use spherical coordinates. The Earth is a spheroid, not a sphere, so professionals (and any maps you have strewn about your car or house) use geodetic coordinates -- i.e., an ellipsoidal reference system. Latitude, for example, is not the angle subtended by a line to the center of the Earth and the Earth's equatorial plane. Rather, it is the angle subtended by a line normal to the surface (the reference ellipsoid, actually) and the Earth's equatorial plane. The difference between geocentric and geodetic latitude at 45 degrees north is about 12 arcminutes, or 22+ kilometers. That's akin to saying that a heading of 1 milliarcsecond north of due east is northward. We live in a three dimensional universe (at least that is how it appears to be), so another dimension is needed to describe position (or velocity, or whateve) in three space. That other dimension is of course down (or up, depending on whether you are talking to an airplane pilot or a gunnery seargent). Working with angles is a royal pain, so people use locally cartesian coordinate system instead. The two standard systems are north-east-down and east-north-up. Unless qualified with geocentric, north is geodetic north and up/down is normal to the reference ellipsoid.

The proposed amendment is still in the greedy thief honorable citizen signature gather phase. No, they are asking for a bailout from the uberrich people who live in the state, and the moderately rich people who bail out from the state. Note well: The Hasta La Vista tax is a tax on wealth, not income. The solution is obvious to some on the left. Donald Trump's 1999 proposal one-time tax on wealth (not income) is regaining a lot of steam as a solution to the deficit. Confiscate Tax enough of those nasty uberrich's accumulated wealth and the deficit will vanish.

One obvious way to "cure" the debt and the deficit is to tax the bejesus out of the uberrich. As goes California, so goes the nation. From http://www.sos.ca.gov/elections/elections_j.htm#1344,

Google is your friend. Here is a rather fruitful search string: cyanobacterial fuel cells.

The next bubble will be theft by proxy, aka wealth redistribution, in the wake of the coming Democratic hegemony. The bubble will burst when the Democrats truly do bankrupt the country. Fortunately for them, they will find some way to blame the Bush.

What you described is Cantor's tenerary set, or the Cantor set for short. The result is an uncountably infinite set of points that has Hausdorff dimension of about 0.63, has set zero measure, is closed, and is nowhere dense.

North never points up. It is always horizontal to the surface (horizontal to the reference ellipsoid, to be exact) -- except at the North Pole of course, where "north" is not defined. Any direction horizontal to the surface at the North Pole is south.

Association fallacy. Just because economic models didn't work in this particular instance does not mean all economic models are wrong, and the failure of economic models says absolutely nothing about other kinds of models.

Well, so what? It is not the government's job to decide which jobs are more important to society, or how many hours I must work to fulfill my obligation to society. That is essentially what you are advocating here. The government is already far too invasive as it is. Letting them know how much I make on an annual basis already gives the government more insight into my personal business than I prefer. If I could make $1000/hour, I would gladly cut my work week down to half a day per week or so. (I would need to make considerably more than I do now to keep myself entertained with all of that extra free time.) It is not the government's job to tell people who can pull off such a stunt that they owe society a minimum of 20 hours per week. Suppose your proposal becomes law. If I were one of the ones who could pull off the incredible stunt of making $1000/hour without working a 60-80 hour week, I would work a extra few hours a week and donate that extra money to political candidates far and wide to unseat those who pass Mr. Skeptic's proposal.

Problem #1: What makes this "wrong"? People who makes $1000/hour tend to work 60 hour weeks, or more. Actually, people who effectively make $1000/hour are not paid by the hour. They are paid a salary. So, here is one problem right off the bat: Problem #2: How do you convert a salary to an hourly wage? Is the government going to tell employers that all exempt employees are now required to work a 40 hour (or 50 hour) week? That $1000/hour corresponds roughly to a salary of $2 million/year. People who make even one twentieth of that salary typically do some work for free. For example, I occasionally charge my time to special charge codes designated "uncompensated overtime". Some people are paid different rates for different jobs. Problem #3: How do you figure in the time where someone works but is not paid? Some people are paid different rates for different jobs. Problem #4: How do you figure the hourly tax rate when the hourly rate varies? Suppose one spouse makes the lion's share of a couple's income. The other spouse works 10 hours a week at $20/hour to augment the income a bit and spends another 60 hours per week raising two kiddos. Problem #5: What if someone works fewer hours per week than the government deems "necessary"? Does 100% of their income (or more) go to taxes? Bottom line: How could the government possibly tax at the hourly wage? A libertarian would say that the best way the government can do this is by not doing very much at all. Because we have a progressive tax system. Suppose you are paid 26 times per year (i.e., every other week). The formulae for computing taxes for a given pay period assume your annual income is 26 times your pay for the pay period in question. This coupled with your tax status and the number of deductions your claim determines your tax rate. This rate is applied to your pay. Suppose on the other hand they computed a rate based on your nominal pay and suppose you work 50 hours of overtime for 48 weeks. You will have underpaid your taxes by a considerable amount because we have a progressive tax system.

Payroll taxes aren't taxes, at least not in the eyes of Democrats. That money (at least the share paid by taxpayers) is going into some trust fund, a "lock box" kept just for me, isn't it? That's not a tax, its a retirement program. Except of course it isn't. These "tax cuts" for people who aren't paying any income taxes will only further the outcry against Social Security. There is no "lock box", and there is no trust fund. These tax cuts are indeed a hidden form of welfare.

OK then. The integrals are to be taken over one full period, here, zero to 2*pi. However, the function in question is identically zero over the latter half of the interval. The obvious (and correct) thing to do is to just take the integral from 0 to pi (but obviously keep the original scale factor): [math] \aligned a_n &= \frac 1 {\pi} \int_0^{\pi}\sin^2\theta \cos n\theta \,d\theta \\ b_n &= \frac 1 {\pi} \int_0^{\pi}\sin^2\theta \sin n\theta \,d\theta \endaligned [/math] Note: There is no [math]\theta^{\prime}[/math] stuff in the integrals. I believe that is the source of your confusion. All of the parameters in the integral are the dummy variable [math]\theta[/math]. The an and bn are just numbers derived from the function f. They are not functions, at least not in this case. What you need to do is to use some trig substitutions to evaluate those integrals.

When dealing with physics, it is best to develop mathematical models for an idea. WTF are logical mathematical metaphors? Non sequitur. The OP needs to learn what the conservation laws truly entail, as opposed to his parodies of the conservation laws. This is basic physics. One cannot learn advanced physics without a thorough understanding of the basis. In other words, the OP needs to learn to walk before he learns to run.

Hi, mooey. Could you write the definition of the function again? You gave the same function for 0 to pi and pi and 2*pi, and that doesn't quite make sense. Why not just say [math]f(\theta)=\sin^2 \theta[/math] and be done with it? BTW, to write a series of equations so they line up nicely: [math] f(\theta) = \Bigl{\lbrace} \aligned \phantom{-}\sin^2 \theta &\quad \text{when $\theta \in [ 0,\pi ]$} \\ -\sin^2 \theta &\quad \text{when $\theta \in ( \pi,2\pi ]$} \endaligned [/math] [math] f(\theta) = \Bigl{\lbrace} \aligned \phantom{-}\sin^2 \theta &\quad \text{when $\theta \in [ 0,\pi ]$} \\ -\sin^2 \theta &\quad \text{when $\theta \in ( \pi,2\pi ]$} \endaligned [/math] Note: "Real" LaTeX, as opposed to the scaled-down version available here, gives a lot more capability.