timo

Hm, ok. No, spin is not a physical motion of something in space.

Example particle collision: A + B -> C + D. Let's say A and B are electron and positron at rest (meaning their energy is E=mc² = 511 keV for each) and C and D are photons. Due to conservation of energy, the sum of the energies of C and D must equal 2*511 keV. Due to conservation of momentum, they must have the same energy (it's not too hard to show, but the reason doesn't really matter here). Therefore, each of the photons must have an energy of 511 keV which -due to photons being massless- must be kinetic energy. So what happens is that an electron and a positron at rest annihilate into two photons with an energy of 511 keV that fly away in opposite directions (directions coming from conservation of momentum). You might claim that this process released 1022 keV of energy. Spin did not come into that calculation at any point. I might have misunderstood you, but you seemed to ask whether spin determines the energy balance of a reaction. The answer is: The energy balance is always E(after) = E(before). So the energy after the collision is completely determined by the energy before the collision (because it is the same). Whether the spins affect the inital energy depends on the situation (more precisely on the scenario you set up and the simplifications you use). In my example above, the energy was independent of the spin orientation.

Energy released is purely determined by conservation of energy. The sum of the energies of the particles after collision will equal the sum of the energies of the particles before collision (same statement goes for momentum). What you call "energy released" is just some mass (to be considered something like condensed energy or potential energy for this case) of the initial particles converted into kinetic energy of the final particles. Spin does not play a role, except that in some cases spin orientation can contribute a term to energy (e.g. for a bound electron in a magnetic field).

It's actually not true that they necessarily annihilate into purely photons (try that with an up-quark and an anti-down quark). Constraints come from conserved quantum numbers (most notably energy and momentum) and for practical purposes also from the probability of the interaction.

... and I refrained from commenting on "if the universe is expanding in all directions over a period of time t, the size is going to be at least 2t".

When the term under the the square root increases, the lower result (for x) must decrease and the higher result must increase. Therefore, these values (or your previous ones) cannot be correct. Simply redo the calculation starting from plugging values into the quadratic forumula and don't try to patch the erronous calculation. Assuming with "it" you mean my comment about the range of arcsin: Completely independently of your homework: sin(0°) = 0, sin(180°) = 0. So what does arcsin(0) equal to? 0° or 180°? Point to be learned: [math]\sin^{-1}[/math] is not the inverse of sin, except for special cases that do not hold true here. The problem is that what I said already was on the borderline of what I consider appropriate for helping with homework. Finding that more than one (the one you get from arcsin(x)) solution exists and finding the other solutions imho is exactly the point of the problem that requires a bit of creativity and understanding of mathematics beyond rearranging equations. Because of this and to avoid possibly unnecessary confusion (my idea is probably not the only way, it's just the approach that popped up into my mind, first) I will not give further advice on that point. Just to make one important point clear and to reduce confusion: You understand that the way you approached the problem is correct and that if you don't screw up the numbers when applying the quadratic formula you will get 1 correct solution, are you? So you have solved at least part of the problem.

- On b² you forgot to square and put 5 instead of 5²=25. - I hope you understand why your calculator gives an error for [math]\sin^{-1} 1.197 [/math]. - [math]\sin^{-1} x [/math] does not map on the whole range -180° <= phi <= 180°. You might (you do) miss solutions. Not exactly sure how you get the 2nd solution (you did make a sketch/plot to see that there's two solutions, did you?). I'd try by reexpressing x=cos(phi - 90°) rather than x=sin(phi). Sidenote: - The meaning of "quadrant" you probably meant is incompatible with my understanding of the term. Quadrant often refers to the four areas divided by the axes of a plot. Incidently, the 1st and 2nd quadrant are those where the value of the function is/would be positive

Try being a bit more specific, e.g.: - What is a "real concept" ? - Why do want to know ? - Are you speaking about the most general/abstract definition of waves or of specific waves? In either case: Why do you think waves belong into the category "Modern and Theoretical physics" ?

For a relatively (compared to a typical forum post) long explanation, the people on WP already put quite some effort into giving a good answer for the non-further specified question "what is X". So you'd best look up the term there and then ask further questions here (further questions on WP itself are generally not welcome except when they serve the purpose to improve the respective article).

Off-topic: I think what you are refering to might be the de-coupling of matter and radiation and the time before that. I do not think that the statement "there was very little stuff emitting photons" is correct, at least I wouldn't know why this should be the case. Going backwards in time: With decreasing time, temperature of matter increases. At some point (stated as 3000 K in the two sources I looked into), most (or practically all, just pump up the temp a bit) of the matter will be in ionized form. These ions and electrons strongly interact with the radiation, which is referred to as the universe being opaque to radiation. This also means that whatever information about previous events was coded in the radiation, it would be spoiled by repeated interaction of the radiation with the matter. Going forwards in time from there: Around the mentioned T=3000 K (something like time = O(100000 y), I think) matter starts to form electrically neutral atoms, which then have less interaction with the radiation. If you take this to the extreme and claim that this less basically equals none at all, then the state of the radiation at this time (a thermal equilibrium with the matter, hence the thermal distribution of the cmb) would be pretty much conserved, except for some changes due to expansion. Bottom-line: The inability to see beyond a certain point in cosmo history is not due to the fact that little radiation was emitted back then, but actually because a lot of stuff interacted with the radiation, then. Disclaimer: Above is mostly based on vague memories, fetching a book from my shelf to read it up and verify some keywords on WP, so don't take it as an expert statement. EDIT: On 2nd thought, you might have simply referred to that no stars had formed, yet. That would then cover later stages of the universe. The cmb which I basically talked about is pre-stellar.

Looks fine. A few comments: - I personally disagree with "only one solution for a given sphere", but that's really just semantics. For me, two different orientations of the drilling are two different solutions, but of course they will result in the same remaining volume. - The edges not only "not necessarily" move closer in terms of absolute distance, they don't do that at all -> the distance is always 6 inches, of course. - You forgot a factor of pi in the term for the area of a circle, meaning your final result is off exactly by this factor. EDIT: @ post #13. No, it was pretty clear what you meant. I just thought that writing "you're off by a factor of pi" was too little for a post

For example that you cannot drill a hole with a diameter of 12 inch into a sphere with a diameter of 6 inch. EDIT: A for a picture visualizing a sphere and a sphere with a cylindrical hole in it: Left greyish object is a sphere, the right one is a sphere with a cylindrical hole in it.

Few comments: - The question is a typical 1st semester physics homework, i.e. not some super-complicated problem requiring advanced math. - It is not obvious (to me) that there is a unique answer. But thinking about the problem a bit, it's also not obvious that there shouldn't be one. Making up examples with spheres with a diameter less than 6 inches just looks like an attempt to cover the own inability by blaming someone else of not being sufficiently specific. If three different people (the book as original source also counted) tell you there is a unique answer, then perhaps that simply is the case. -- As a matter of fact, if you take "there is a unique answer regardless of the sphere size" as granted (you shouldn't, you'll miss the point of the exercise), then you can trivially get the answer (which is 36pi inch³, btw) from taking a sphere with a radius of 3 inch (because the hole then has -asymptotically, if you want- a volume of 0). - The point of the problem is that two of the unknowns (diameter of the hole and radius of the sphere) cancel in such a way, that the result is just a plain number. If you just hope for that and straightforwardly calculate the volume, then the problem is solved by 1 sketch, two simple relations between the appearing variables and a 5-line calculation. Hardly unsolvable compared to that typical topics on sfn are about quantum gravity or climate models.

It wouldn't be too surprising if chemicists did more labs than physicists, so I'd guess the relative numbers will be roughly the same everywhere. For the the quality and requirements of the physics labs, from own experience (having studied at two different universities) I can say that they can vary tremendously between different universities.

The zero refers to the extension of the mass-distribution in coordinate space and probably (not exactly sure about this) the measure of the volume occupied by the mass. The non-zero Schwarzschild radius belongs to a standard coordiante system (Schwarzschild coordinates) and defines a spherical shell at which this coordinate system breaks down. The non-zero Schwarzschild-radius is a description of the gravitational field generated by a mass, not a description of its spacial extent (e.g., the SR of the sun is ~3 km, that certainly is not the size of the sun - sidenote: In the case of the sun, the simple Schwarzschild solution fails at the edge of the mass-distribution, already).

I don't get your question. What kind of calculations are you doing with the Schwarzschild radius? The formula for the radius itself, R=2Gm/c², already is very simple - there's little to simplify or to get wrong, there.

I remember having read articles (newspaper style, not research articles) claiming that left-handed people are over-represented in a lot of "extreme" groups, e.g. high intelligence, low intelligence, number of smokers, criminals, successful sportsmen (especially sports like fencing), ... No guarantee on any of the mentioned groups, I just wanted to visualize that the overrepresentation is not only in "desirable" groups but also in "undesirable" ones. I suppose a lot of studies have been done on the statistical representation, I've never heard of a reason for it (with the exception of directly competitive sports, where the reason seems kind of obvious).

The key is the question "what is the speed of time". The term is not a commonly-used one and hence has no generally-understood meaning. Furthermore, it is not a constructed term whose meaning would be obvious. In other words, you are asking for the meaning of a term that you basically made up yourself. That depends on what "time" means (obviously). Usually, it can either refer to: - Coordinate time. That's the one you meant. I'd personally prefer saying "it's a coordinate" over "it's a dimension", mainly because making it a coordinate you can assign a quantity to it. That's just personal taste, though. - Eigentime. That is not a dimension or a coordinate but losely speaking a property of an object (which changes under travelling through spacetime). You could well define something like the speed of eigentime via the differential increase of eigentime (of an object) with coordinate time (of the chosen coordinate system). The speed then would be [math]1/\gamma[/math].

At least from the theory side, QM has nothing to do with this impossibility.

Assuming this was not meant in the Terry Pratchett sense of "does a falling tree make a noise if there's no one who can hear it?" (he also gives a very natural scientific answer to the question: "Who cares?") then I tend to disagree with that statement. Especially when later on you seem to say that a system "in a superposition" was not in a definite state. A "superpostion of states" is just a vector in the vector-space of states (the Hilbert space) that is not a basis vector of the currently-chosen basis (degenerancy left aside, here). Barely exciting, but contradicting what I think you said: A non-basis vector is just as valid and definite as a basis vector, not some vague and undecided thing. What you are probably referring to is that under the process of measurement, the current (definite!) state is projected on one of the basis vectors (or a subspace spanned by more than one, in the case of degenerancy) of the basis associated to the measured property (QM name: "Observable"). This leads to a state which will have a definite value for the observable because the mentioned basis vectors are associated to certain but possibly different values of the observable (QM terminoligy: Are eigenstates of the observable). In short: While a "superposition of states" (I don't particularly like that term, but afaik above is a common usage for it) does not necessarily reflect a state with a definite value for a chosen observable, it is a clearly-defined and unique state that is just as valid as a "pure state". Might sound like nitpicking to you, but I think by not taking non-eigenstates of the current observable serious you close for yourself the doors to a more elegant way of understanding QM (via linear algebra). I'm not too familiar with modern cosmology, but I don't think "big bang theory" (whatever that shall be, I think it's just standard cosmology) suggests so. The simplemost model simply states that when approaching a certain finite value of the coordinate called time (this value being usually shifted to t=0), the distance between any two points in space approaches zero. That does certainly does not sound like a big ball to me. What about an internal observer? I am not really sure to what extent the concept of an observer is really understood. Historically? I claim that the rock-particle was known long before the rock-wave . I think your statement is true only for light, in which case you still might have to make the restriction that science started around the time of Maxwell (dunno what the view on light was before that time).

So what exactly are you going to do, meaning where are you going to do your civil service?

The big difference I see is that \lambda, ... are TeX codes that a lot of people are familiar with while [ math], [/ math] are SFN-only tags. Other pages use different tags for enclosing TeX-code (e.g. [tex] on PF, <math> on WP). I often get the wrong tag, first.

Combining ajb's post #4 above with the information that in the framework of relativity the length of the path is the time experienced by the traveler on that path, http://www.scienceforums.net/forum/showpost.php?p=336818&postcount=4 might be helpful.

I do . Took some time to get the mathematical proof of convergence working, though.

In that case I'm pretty sure I'm right about the value, too. Didn't want to explicitely write it down, because the main challenge seems to be guessing the pattern you had in mind, and it would have become pretty obvious if I had given the exact value. Would be interesting to see the full proof, though. In addition to some other potential problem I see, I am not sure that the existance of a fix-point is sufficient. EDIT: No, the number wasn't correct; I got a sign-error in the pq-formula. Should be ~0.62.