title: Wavefunction in quantum mechanics and locality

text: Every wavefunction of a form $\Psi(x)$ can be described as a superposition of multiple free particle solutions. We can see the following Fourier transform: $$ \psi(x) = \int e^{ik\cdot x} \psi(k) dk $$ We can make it time dependent by replacing $e^{ik\cdot x}$ with $e^{ik\cdot x - i E_k t}$ where $E_k = \sqrt{k^2 + m^2}$. The result gives the positive energy time evolution of any initial wavefunction data. The question is, A. This may violate locality, as $E_k$ can be set as we want to, as $k$ can be set as we want to. Is this correct? B. Why is this related to requiring a multi-particle theory to fix formalism in Klein-Gordon equation? (Context: Complete set and Klein-Gordon equation) C. How is adding negative energy preserve locality?


title: Saddle point and transition state of a molecule

text: During a transition of a molecule from one state to another, it pass through a transition state. This transition state, mathematically speaking, is a saddle point. That means, the second derivative along all the coordinates, except one, is greater than zero. I have two questions: 1) Is it possible for a molecule to have a transition state at a maximum if there is no saddle point on the potential energy surface? 2) In the case of a diatomic molecule, there is no saddle point on the potential energy surface. So the transition state should be at a maximum right?


title: Fluid in a rotating cylinder

text: I have been wondering why a fluid in a rotating container has a parabola shape? Is it possible to prove this mathematically?


title: Possibility of stable muonic structures?

text: In an analogy to the neutron, which decays rapidly as a free particle, but when bound in a nucleus it is stable, would it be possible to crease a structure that permits the stability of muons - be it muonic atoms, some sort of muonic lattice, etc.? Granted, the energetic benefit of the neutron $\beta$ decay is only about 0.08% of its rest mass, while the muon is roughly 200 times as massive as the electron, thus having much more to gain from decaying - making this proposition seem somewhat unlikely.


title: More forces at low energy?

text: Just a quick question: In high energy experiments, the fundamental forces are thought to merge into a single force. My question is, in very low energy experiments (very low), do the forces we know about split up even more/is this possible or likely? Thanks


title: Pressure derivative of bulk modulus

text: Hi all what is the definition of pressure derivative of bulk modulus if it is a pressure derivative of bulk modulus at zero pressure. if the pressure is zero how it is derivative by pressure?


title: Is there electromagnetic abrasion / erosion?

text: When a sensor registers electromagnetic radiation, such as that used for the transmission of communication like light or radio waves, does this radiation "abrade" or "erode" the material of the sensor? Are there any traces of the received radiation that can be detected, even if only theoretically? An example from a biological background would be that high energy light (such as that in the UV range) destroys the pigments in the eyes, which is why animals that live longer than a few years cannot see in the UV range. Maybe something similar happens with man made sensors for different wavelengths of electromagnetic radiation.


title: Why do physicists use LHC?

text: 1. My question is why are we colliding particles in LHC to produce new ones?    2. And these particles that they sometimes say live for a fraction of a second, how in space they exists then?    3. In space all these particle exist without smashing to each other, why do we need to smash them to produce them?


title: How to calculate the 2-point function of gravitons?

text: I'm curious about how to calculate the 2-point function of graviton, but there are no textbooks of general relativity covering this problem. So how to calculate it? In which book can I find the answer?


title: Writing down an entanglement in bra-ket notation

text: I have a relatively complex (for me anyhow) situation which I would like to capture in bra-ket notation, rather than in words. However, I've not been able to find a source which will help me understand better how to do so, and I was hoping that one of you might help me out. I'll first describe exactly what I want to express, so that it's a little more clear what I mean. Note that this is not a homework question or something, it has to do with an experiment that I came up with, so chances are that it might be somewhat impossible to actually write it out the way I want to. I have a system that lives on the Bloch sphere, like this one, but with $\left|\uparrow\right>$ instead of $\left|0\right>$ and $\left|\downarrow\right>$ instead of $\left|1\right>$ ![Bloch Sphere](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Bloch_Sphere.svg/256px- Bloch_Sphere.svg.png) Now, initially I am able to fully initialize something, say a spin, into 1 state, being $\left|0\right>$. Then, I rotate the state by pi/2 around the x-axis, and I wait a time t. In this time t, some more complex stuff happens. Depending on the state of a second spin, stuff will happen. This second spin can be in either of three states, $\left|-1\right>$, $\left|0\right>$ and $\left|1\right>$. Now, during this time $t$, the state of the first spin will precess about the Bloch sphere depending on the state of the second spin. If the second spin is in state $\left|-1\right>$ the first state will rotate around the z-axis clock wise, if the second spin is in state $\left|0\right>$ the first spin will not change, and if the second spin is in state $\left|1\right>$ the first spin will rotate about the $z$-axis counterclockwise. So, after this time $t$, I will rotate the first spin again, but this time by $\pi/2$ around the $y$-axis. This means that if I choose t such that it is exactly a quarter of the period in which the spin makes a full circle, the two spin states will be entangled. I can use this to establish what the second spin state is by reading out the first spin state: If $\left|-1\right>$, then I will readout $\left|\uparrow\right>$ with $P=0$, if $\left|0\right>$, then I will readout $\left|\uparrow\right>$ with $P=0.5$, and if $\left|1\right>$, then I will readout $\left|\uparrow\right>$ with $P=1$. I understand that this probably might be a little confusing, which is of course also the primary reason why I would just want to write this in nice, clean bra-ket notation. If there's anything in particular that's not clear, please let me know. And if someone could help me get started (possibly by pointing me to a similar example) I'd be very grateful. **Edit:** Alright, I've done a little bit of reading on what I could find, and this is how far I get now Initially: $\left|\psi\right>$ = $\left|\psi_1\right> \otimes \left|\psi_2\right> = \left|\uparrow\right> \otimes \frac{1}{\sqrt{3}}\left( \left|-1\right> \+ \left|0\right>\+ \left|1\right> \right)$ Rotate first spin by $\pi /2$ around x-axis $\left|\psi\right>$ = $R_x (\frac{\pi}{2}) \left|\psi_1\right> \otimes \left|\psi_2\right> = \frac{1}{\sqrt{2}} \left( \left|\uparrow\right> -i \left|\downarrow\right> \right) \otimes \frac{1}{\sqrt{3}}\left( \left|-1\right> \+ \left|0\right>\+ \left|1\right> \right)$ But here I get stuck again, as it is here that the rotation conditional on the second spin has to happen, and that I don't know how to do.


title: electrical potential energy stored in vacumm for a single point charge?

text: I have come to know the electrostatic potential energy in vacuum is given by $${\frac{1}{2}} \epsilon_0\int d^3x {E^2} $$ and this energy is due to the mutual electrostatic coulomb potential energy. So only a single point charge in space also contributes to some electrostatic potential energy according to $${\frac{1}{2}} \epsilon_0 \int d^3x {E^2} $$ as due to a single point charge also we have electric field at every point in space. Now I am not getting it physically why should there be any Coulomb energy in this case as there is no another charge to provide the electrostatic coulomb potential energy?


title: Fusion of two neutrinos

text: What are the results of fusing an electron neutrino and a muon neutrino? If you have an answer, then why? I looked this up and searched for stuff on Google, Wikipedia and a few other sites, but all the results say are stuff on nuclear fusion and neutrinos. **BONUS QUESTION:**   What happens if you put another muon neutrino into the mix?


title: Understanding Counter - EMF?

text: When an coil rotor is moving around a magnetic field there is -V induced to resist the input V. Let's take an example, a 12 V DC motor induces -10V, and the actual running voltage is 2V. If there we're no -V in the process, the motor would preform a lot better? What is the advantages and disadvantages of counter EMF? Is it wasting energy? I understand the concept, but I can not fathom how initially, the motor runs at 12V then drops to 2V(actual) and still preforms the same? Are all current motors today lose 70%+ of their input voltage due to counter EMF?


title: Causality in QFT from vanishing commutator and the EPR paradox

text: The question relates to this post. As shown in Peskin and Schroeder's introduction to quantum field theory p. 28., $$[\phi(x),\phi(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0$$, which implies the measurements between two spacelike places do not affect each other. However, in the EPR paradox, even two experiments are spacelike separated, once experimentalist $A$ obtained $+$ result of $S_z$, $B$ has to obtain $-$. Though at this stage of EPR, no information is delivered. It still seems to me, actually, that the two measurements affect each other. Is there any further reasoning to reconcile this inconsistency?


title: Cosmological horizon, energy and information

text: If something crossed the horizon of the observable Universe, does this mean that all information about it is lost forever? If yes (or no), how this fits with the information-cannot-be-lost law? Also, does it mean that the energy of our observable Universe is constantly decreasing? If yes, what are the consequences of that?


title: Poynting Vectors and Feigel Effects

text: There is a surprising number of papers seriously discussing the "feigel effect" This has been linked to the Abraham-Minkowski controversy also here. Although there are good discussions picking apart the feigel effect calculations, there does not appear to be a conclusive refutation of the effect under certain experimental circumstances. Much of this is linked to when the Poynting vector vanishes in the vacuum. Is there a legitimate possibility for seeing the Feigel effect? What would that mean about the nature of the poynting vector in physics? The Feigel Process: Lorentz invariance, regularization, and experimental feasibility Final Report Lorentz-invariant description of the Feigel Process for the Extraction of Momentum from a vacuum Final Report Alternative derivation of the Feigel effect and call for its experimental verification


title: Why did decoherence start in the first place?

text: We learn that the quantum wave function $\Psi$ collapses when it interacts with a classic object (measurement). **My question is:**   Why are there _classic_ objects after all, how did it all start? In a _quantum universe_ (and I understand we live in one) there should only be _superposition_ , how did these _classic_ beings came into existence (which "decohere" the others) in the first place? Is it a statistical necessity like _entropy_ \- or doesn't, quite simply, anybody know the answer to that riddle?


title: Is the Abraham-Minkowski controversy resolved?

text: A paper was published in 2010 claiming to resolve the Abraham-Minkowski controversy. Is this paper viewed as definitive by physicists?


title: Example of inviscid flow with drag

text: This was my engineering viva question and I couldn't think of any example, for I am totally convinced that there cannot be drag in a inviscid flow. It also bothers me whether it is possible, a tricky scenario in fluid mechanics or gas dynamics.


title: Extremizing a Hamilton-Jacobi Equation

text: How can one make sense of the idea of extremizing a Hamilton-Jacobi equation? In Schrödinger's paper "Quantisation as a Problem of Proper Values I" ( _Annalen der Physik_ (4), vol. 79, 1926, p. 1, available e.g. in _Collected papers on wave mechanics_ , AMS publishing) he begins with the classical time- independent Hamilton-Jacobi equation $$H(q,\frac{\partial S}{\partial q}) = E$$ and instead of assuming an additive solution for separation of variables he assumes a multiplicative solution thus defining $S = K\ln(\psi)$ to get $$H(q,\frac{K}{\psi} \frac{\partial \psi}{\partial q}) = E.$$ Instead of solving it he talks about how this last equation "can always be transformed so as to become a quadratic form (of $\psi$ & its first derivatives) equated to zero", neglecting the relativistic variation of mass. (I'm assuming that transforming this to a quadratic form means a Taylor expansion in powers of $\psi$, though I'm not sure about that). Then he seeks "a function $\psi$, such that for any arbitrary variation of it the integral of the said quadratic form, taken over the whole of coordinate space (I am aware this formulation is not entirely unambiguous), is stationary, $\psi$ being everywhere real, single-valued, finite, and continuously differentiable up to the second order." In other words, he basically extremizes $$I = \iiint\left(\frac{K^2}{2m}\left[\left(\frac{\partial \psi}{\partial x}\right)^2 \ + \ \left(\frac{\partial \psi}{\partial y}\right)^2 \ + \ \left(\frac{\partial \psi}{\partial z}\right)^2\right] + (V - E)\psi^2\right)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ and ends up with the time independent Schrödinger equation, but the issue is extremizing the Hamilton-Jacobi equation. Where is the justification for this? I can only find a reference to it here (in the ps.gz link), so it does seem to be something more than one person has done, but what is the theory behind this? Does it function as another method of solving Hamilton's equations? If so, why?


The ability to manipulate light spectra (as in mercury lamps, sodium lamps, infra red) is within the ambit of contemporary technology. Stars are said to be unique in the spectrum they emit ; http://skyserver.sdss.org/dr1/en/proj/basic/spectraltypes/ writes to say > The best tool we have for studying a star's light is the star's spectrum. A > spectrum (the plural is "spectra") of a star is like the star's fingerprint. > Just like each person has unique fingerprints, each star has a unique > spectrum. with my limited understanding of physics, I may have understood wrong!! Can contemporary technology simulate the spectrum of a star?


title: Valence band and conduction band, trying to get a clear picture!

text: I am trying to get a clear picture of the valence band, conduction band, and the band gap. Now I've been researching it for a little while now and understand most of what's going on. I'm still a little confused about some specifics though, so if you could please address the following, that would be great! My favorite description thus far has been _"The most important aspect of semiconductor bandstructure may be summarised as follows; at absolute zero the highest completely ﬁlled band (the valence band) is separated from the lowest empty band (the conduction band) by an energy gap or band gap Eg of forbidden states. Therefore the material does not conduct electricity at T = 0."_ So from this description I have a pretty clear idea of what the valence band is (with the supplement of a few diagrams). Also, I know the conduction band is above the valence band in terms of energy, and is where electrons are free to roam the material as conduction electrons. My problem is where this "conduction band" is physically. It seems like there should be some allowed state, where if you just excite one electron you pass into that new state. From the above description, it seems to say some of these states are "forbidden", and that is my main question resides. Where is this conduction band, what determines its range (an example would be fantastic), and how do we determine what is necessarily a "forbidden" state? Thanks =)


title: Derivation of Bloch's theorem

text: I'm having a problem following a derivation of Bloch's theorem, looking at a one dimensional lattice with $N$ nodes and spacing a, we impose periodic boundary conditions, meaning that the wave-function of a single electron $\psi(x)$ can be decomposed in the following modes: $$\psi(x)=\sum_{q}\phi(q)\text{e}^{iqx},$$ where $$q = \frac{2\pi m}{N a}\; m\in \mathbb{Z}.$$ Otherwise it would be multivalued. Schrodinger's equation gives:$$(\epsilon - \frac{\hbar^2q^2}{2m})\phi(q)=\sum_{k \in B}\tilde{V}(k)\phi(q+k)$$ where $\epsilon$ is the energy eigenvalue of $\psi$, $B$ is the reciprocal Bravais lattice ($k=\frac{2\pi n}{a}$ with $n\in \mathbb{Z}$), $\tilde{V}$ is the Fourier transform of the potential. So the $\phi$ modes separated by a vector belonging to the reciprocal Bravais lattice are coupled but I don't see how it implies that only $q$ vectors that are in this class of equivalence (whose difference belongs to $B$) should appear in the first sum (the expansion of $\psi$ in modes).


title: If batteries are a source of energy, would not lower-valued resistors cause a violation of the conservation of energy?

text: Here's the conundrum I have been facing. Yesterday I asked the question: how can batteries constantly motivate electrons to complete a circuit? (What maintains constant voltage in a battery?) After some thought I realized that the concept is that of an energy source. The batteries are a source of energy, and regardless of the mechanism, they will constantly 'push' for a goal. My solution finally came when I realized that speaking about someone's 'motivation' was identical to the constant push of the battery. Here's the issue: My sister wanted to get me coffee yesterday for my birthday. She was attempting to give it to me before my class started. Unfortunately, she had difficulty with traffic, and so was able to deliver only after my class started. I realized when she told me that she tried as hard as she could, and that what really mattered was her motivation. The push she gave was constant regardless of the resistance she encountered. The parallel here to the constant push of a battery was strong; voltage didn't matter in respect to resistance for a given battery. Now I have a different issue. The voltage drop is described as the per-capita potential energy difference of electric charges at points a and b right? The total energy would then be the amount of 'capitas' multiplied by this per capita potential energy difference. More capitas mean more total energy. Here's the problem. If the motivation of a battery is described by voltage, then if we look at a circuit with a resistor and compare it to another circuit with an equal-length resistor but with a wider surface area, the potential difference is supposed to stay the same in both cases. However, in taking the charge from one surface to the other, more energy is required for the resistor with larger surface area! The amount of energy supplied is different in either case, yet from the same 'constant' energy source. This seems so wrong to me right now. Yet I know that Power=Voltage*(Current)^2, so this actually makes sense in terms of the equation. I wonder what is lacking in my fundamental understanding of the potential difference in comparison to the idea of motivation/constant energy source. I'm really fundamentally misunderstanding something complex here that I don't quite have my finger put on yet. Please help clear up my view of a battery as a constant pusher to line up with this changing energy ouput a battery can give.


title: Optics: Derivation of $\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$

text: I have been given this formula from optics here, with no background: $$\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$$ Where $n$ is the refractive index and $\hat{u}$ is a unit vector tangent to the path $s$ that light takes inside a medium. Does anyone know if this formula has a name? I am looking specifically for a derivation of it. I have looked through the optics book by Hecht with no luck - I assume it comes from fermats principle of least time in some form. Any help greatly appreciated.


title: Eyes open under water

text: Yesterday I looked underwater with my eyes open (and no goggles) and I realized I can't see anything clearly. Everything looks very, very blurry. My guess is that the eye needs direct contact with air in order to work properly. With water, the refraction index is different, and the eye lens are not able to compensate for correct focalization on the retina. Am I right ? If so, what lenses should one wear in order to see clearly while under water ?


title: What is the role of the vacuum expectation value in symmetry breaking and the generation of mass?

text: Consider a theory of one complex scalar field with the following Lagrangian. $$ \mathcal{L}=\partial _\mu \phi ^*\partial ^\mu \phi +\mu ^2\phi ^*\phi -\frac{\lambda}{2}(\phi ^*\phi )^2. $$ The potential is $$ V(\phi )=-\mu ^2|\phi |^2+\frac{\lambda}{2}|\phi|^4. $$ The classic stable minimum of this potential is given by $\phi =\frac{\mu}{\sqrt{\lambda}}e^{i\theta}=:v$ for any $\theta \in \mathbb{R}$. We then define a new field $\psi: =\phi -|v|$, rewrite the Lagrangian in terms of the new field $\psi$, et voilà: out pops the mass term $$ 2|v|^2\lambda \psi ^*\psi . $$ People usually explain that this shows the existence of a field whose quanta are particles of mass $|v|\sqrt{2\lambda}$. In this sense, the original field has $\phi$ acquired a mass. I don't buy it. I don't see where this argument has actually used the fact that $v$ is the vacuum expectation value of the field. Yes, it is natural to expand about the classical minima, but why not do something stupid instead and define a new field $\psi :=\phi -7$. Once again, after rewriting the Lagrangian in terms of the new field, you should find that it has indeed acquired has a mass term. Playing this trick over and over, by picking a number different than $7$, you should be able to find a mass term with any mass you like. Obviously, this doesn't make any physical sense. There is something special about the substitution $\psi :=\phi -|v|$. There must be more to it than a tedious algebraic manipulation, but what is it? Why does nature possess particles of the mass that this substitution yields, as opposed to any other substitution? On another note, what exactly does this have to do with the global $U(1)$ symmetry present. It seems that the only thing that has played a role so far is that the vacuum expectation value of the field is non-zero, but yet I've always seen this mass generation presented alongside symmetry breaking. What precisely is the relationship between the two?


title: Empty universe in the past, non-empty in the future

text: My question is the following. Are there solutions to the Einstein field equations, which have the property that there is a hypersurface of constant time and to the past of that surface space is empty (Minkowski space-time) and to the future it is not (non-vanishing stress-energy tensor)? At first thought it seems strange to have nothing and suddenly something. On the other hand define $g_{\mu\nu}(x)$ to be the Minkowski metric for the past of the surface and any functions (sufficiently smooth) that match it on the surface and are different than the Minkowski for the future. There are many ways (infinitely many) to do this. Then one can define the stress-energy tensor using Einstein's equations. This would seem to work, but there are certain conditions on the tensor, for example positivity, or other restrictions for physical reasons that I am unaware of, so it may be there are no such functions. So the questions is: does anyone know of an explicit example or a more convincing argument than "there are so many functions, there has got to be some that work"? Of course a reason why it doesn't work, if that is the case, would be good too. Thanks.


title: Is it possible to break bulletproof glass with your voice?

text: In _The Adventures of Tintin_, an opera singer (the Milanese Nightingale) broke a bulletproof glass case using her voice. Is that scientifically possible? From the _Wikipedia_ page, a typical bulletproof glass > is usually made from a combination of two or more types of glass, one hard > and one soft. The softer layer makes the glass more elastic, so it can flex > instead of shatter. When they did it on Mythbusters, the singer had to stand really close to the glass that is so thin for it to break. Any thoughts on whether the movie scene is even humanly possible?


title: what co-operates with a Super Symmetry Transform

text: The Super-Symmetry operators reduce or increase helicity by a half unit. And this result in an algebra that where the fermions and bosons vacuum energy sums to zero. And it seems to generate gravity, when the upgraded to a local symmetry. But as yet, physicist seem to have found no good way to break super- symmetry, i.e. Supergravity. What other transformations can one do at the same time as applying a super-symmetry operator? And still keep the good properties of super-symmetry. At first view, I think one could swap left and right handed particles/fields, and swap sets of particles from one group containing both fermion and boson to another. I.e. switch to the other one of a direct product group. Does mirror- symmetry commute and/or co-operate with super-symmetry?


title: A formula in Sung-Sik Lee's paper

text: I want to ask if anyone has gone through the derivation of the second equality in the following formula ![enter image description here](http://i.stack.imgur.com/okF7P.jpg) which comes from http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.165102.


title: The Uncertainty Principle and Energy Nonconservation

text: The uncertainty principle is listed in most textbooks and articles as $$ \Delta E \Delta t \geq \frac{\hbar}{2}.$$ This can be derived in many ways in many different settings, most of them involving commutation relations with appropriate operators. This is often interpreted such that $\Delta E$ is the amount of energy that can be "borrowed" and $\Delta t$ is the time for which it can be borrowed. The uncertainty principle is then used to argue that if $\Delta t$ is large (that is, if the energy is borrowed for long times), then $$ \Delta E \sim \hbar/2\Delta t$$ is small, making this effect negligible on large time scales. However, the uncertainty relation is an inequality. If $\Delta E$ is actually the amount of energy we can borrow, and if $\Delta t$ is the time for which it can be borrowed, then if $\Delta t$ is large, $\hbar/2\Delta t$ is small, and $$\Delta E \geq \hbar/2\Delta t.$$ This puts no upper limit on $\Delta E$ at all, and in fact gives it a lower limit. $\Delta E$ must be _at least_ $\hbar/2\Delta t$, but it could also be orders of magnitude larger and still satisfy the uncertainty principle. In fact, $\Delta E$ and $\Delta t$ can both be infinite and satisfy the uncertainty principle. My question is, then, since this argument is so frequently cited in textbooks, what justifications are used for interpreting $\Delta E$ and $\Delta t$ as relating to energy nonconservation? Am I missing something? Is there a reason for the switch from $\geq$ to $\sim$ in the relation? Why don't we observe the infinite violations of conservation of energy (in time and energy) that are predicted by this interpretation of the uncertainty principle? Why are only the minimum-uncertainty cases used in the literature? I'm assuming there is a reason, and trying to figure it out. Thanks in advance for clarification!


title: How is it that angular velocities are vectors, while rotations aren't?

text: Does anyone have an intuitive explanation of why this is the case?


title: Why do people with bigger hands/wrists have bigger torque advantages in arm wrestling?

text: I have arm wrestled people with less strength than me who win simply because they have thicker wrists, or larger hands. It has nary to do with having better technique either, because their wrist size alone seems to be a big advantage. I want to know why, as explained by physics, why the torque they produce is so much larger, even with less muscular force by the axis than with me having greater strength, but smaller wrists & hands? For example, I have 7-inch wrists, and 7.2-inch hands. My opponent has 8-inch wrists, and 8.2-inch hands. They have been tested to have less physical power in every muscle group used in arm wrestling compared to me, yet, they win, and they have no technical advantage over me in the sport either. I have noticed that this advantage goes beyond arm wrestling, but to ANY angular/torque-like position of some grasping another's hand/wrist outwards in front of their bodies: ![enter image description here](http://i.stack.imgur.com/s0Fls.jpg) ![enter image description here](http://i.stack.imgur.com/aqkBc.jpg) Please tell me why this is. EXTRA: This question, or a question of this nature, has never been asked on this site, so please cut me some slack here! I know the question is poor at best, but some insight would be nice!


title: Why some nuclei with "magic" numbers of neutrons have a half-life less than their neighbor isotopes?

text: It's easy to find the "magic" numbers of neutrons on the diagrams of alpha- decay energy: 82, 126, 152, 162. Such "magic" nuclei should be more stable than their neighbors. ![](http://elementy.ru/blogs/u/i/2598.gif) ![](http://elementy.ru/blogs/u/i/2599.gif) But why some nuclei with "magic" numbers of neutrons have a half-life less than their neighbor isotopes with odd numbers of neutrons? Examples for "magic" number 126: A half-life of "magic" Po-210 is **138 days** , whereas a half-life of neighbor isotope Po-209 is **102 years**. A half-life of "magic" Ra-214 is **2.46 sec** , whereas a half-life of neighbor isotope Ra-213 is **2.74 min**. Examples for "magic" number 152: A half-life of "magic" Cm-248 is **348 thousand years** , whereas a half-life of neighbor isotope Cm-247 is **16 million years**! A half-life of "magic" Cf-250 is **13 years** , whereas a half-life of neighbor isotope Cf-249 is **351 years**. P.S. Source of the diagrams data


title: How is it possible to move something without completely lifting it?

text: For example, let's assume a chair here: It can be "slid" across the force if we use minimal upwards force, but not enough to actually "lift" the chair. Why should it move? Here's a better example: I have seen some strong people lift and slide the back of cars, however, the back tires of the car never elevate completely off the ground. With regards to friction, though, how is it possible to actually "move" something without lifting it completely? How does this work is a better question. This image should illustrate the car example: ![enter image description here](http://i.stack.imgur.com/foRJg.jpg) Is there enough space within it to allow the angular force to make it "slid" a pinch, or what else would explain its ability to move without being completely suspended in the air that would, say, differ from this with a human?


title: Was Joule's experiment able to show: thermal energy = $mgh$

text: Is my understanding of this experiment correct? There is the quantity $MC\delta \theta$ associated to the heat which I think is not an absolute value to be compared with $mgh$ since the $\delta \theta$ could be defined arbitrarily. So this experiment does not prove : thermal energy = $mgh$ it could however show a linear relation between $\delta \theta$ and $h$.


title: How do astronomers measure the distance of the planets from its star?

text: I know they have accurate means of measuring the distance of the planets in the Solar System from the Sun. I'm skeptical how can they use the same tools or techniques for other systems so far away as 22 light years. According to this link the planet GJ667Cc resides in the "habitable zone". How are they able to measure the distance of something from something while observing it from a distance of 22 light years?


title: Centre of mass and objects tipping

text: ![enter image description here](http://i.stack.imgur.com/0dOhU.jpg) In this picture if the only parts of an object in contact with a surface are the lines AB and CD am I right in saying that the centre of mass has to be in the area enclosed by ABCD (as shown). I think this is correct as if the mass lies outside this region taking moments about e.g.an axis AC would lead to a resultant moment. If this is true please can you also give me a more generalised rule. thanks


title: Are there known turbulent nonlinear equations where the cascade is a thermal gradient?

text: In a recent answer (here: The equipartition theorem in momentum space ), I suggested that if you have an appropriate first order equation (in the answer I used a second order equation, but it is more likely to be true for a first order equation) of the form: $$ i\partial_t \phi = (\nabla^2)^N \phi + \lambda \phi^3 + f + D $$ (EDIT: I first had $|\phi|^2\phi$, which is the nonlinear Schrodinger equation, but this has the same frequency shifting property as the relativistic version--- the frequencies are not completely perturbatively decoupled even with a resonance condition. The only reason I went to first order was to fix this annoying problem, and there is no condition of being physical. Although this will not conserve the total $|\phi|^2$, it has a conserved energy.) Where f is a driving force with only long-wavlength components, and D is a damping force which only affects short wavelength components. The field $\phi$ is a complex scalar. Then if N is large and $\lambda$ is sufficiently small (by scaling, $\phi$'s magnitude can substitute for $\lambda$, so the limit of small $\lambda$ is really the limit of small $|\phi|$), the cascade due to the nonlinear term is a thermal gradient draining energy from the hot low frequency modes to the cold high-frequency modes. I don't know if this model is in the literature, of if the thermal analysis is accurate (I am pretty sure it is correct, because the resonance matching does lead to local flow of energy in |k|, but I am not sure if there isn't a block to thermalization due to some problem with mixing in each |k| modes separately). In the original answer, I used a second order equation, which has frequencies of both signs. This allows some extra mixing between distant modes, which shifts the frequency of low modes based on the occupation of high modes. While I don't think this changes the cascade significantly, I wanted a clean example, so I used a first order equation here. (EDIT: The clean example does not use the absolute value, so that the frequencies have to strictly additively combine.) Is this model analyzed in the literature? Does it thermalize into a thermal gradient cascade?


title: Why whirlpool causes force towards its center?

text: Higher velocity implies low pressure. So in whirlpool the velocity of water have to decrease with radius in order to have force towards its center. My question is how is explained whirlpool formation in simple words.


title: Solve a problem of work and energy

text: > A man pulling sled of his daughter by a massless rope, climbing a snowy hill > whose slope is equal to 15 °. Considering that the mass of the sled is > $4Kg$, the girl's $26Kg$ and $\mu _c = 0,25$, calculate the work done by the > tension of the string after walking with constant speed a distance of $130m > \Delta x$ So, I know: $\varphi=15°; \alpha =30°; m_d=26Kg; m_s=4Kg; \mu _c =0,25; d=130m$ So, the equations will be: Weight in Y: $P_y=P.\cos\varphi $ Weight in X: $P_x=P.\sin\varphi$ Tension in Y: $T_y=T.\sin\alpha$ Tension in X: $T_x=T.\cos\alpha$ Normal force: $P_y-N-T_y=0 \implies N = T_y-P_y$ Newton's 2nd law: $\sum F = m_t.a \implies T_x-F_r-P_x=m_t.a$ Frictional force: $F_r=\mu _c.N$ work definition: $W=m_t.a.d$ Total mass: $m_t=m_d+m_s \implies m_t =30Kg$ I want to get the acceleration of the object and then replacing $a$ I can get the string tension, so (...) $a={{T_x-F_r-P_x}\over m_t}$ Now, ${T_x-F_r-P_x} = \sum F \implies {T_x-F_r-P_x} = W =m_t.a.d$ But again, If such replacement, the unknown $a$ appears again. ¿How can I solve this?


title: Definition of electric charge and proper explanation

text: Is there a definition of electric charge and proper explanation of it? It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter." How is it though that matter can get charged? Defining charge as the property of feeling a force with other charged matter seems circular. What is charge? Is there a non-circular definition / explanation?


title: One more relation with spherical spinors

text: Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\\ \sqrt{l \mp m +\frac{1}{2}}Y_{l, m + \frac{1}{2}} \end{pmatrix}. $$ How to prove the relation $$ (\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -r\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}? $$ I tried to rewrite the $(\hat {\sigma} \cdot \mathbf r )$ as $$ (\hat {\sigma} \cdot \mathbf r ) = r\begin{pmatrix} \cos(\theta ) & \sin(\theta )\,e^{-i\varphi } \\\ \sin(\theta )\, e^{i \varphi } & -\cos(\theta ) \end{pmatrix}, $$ but it's hard to use recurrence relations for associated Legendre functions after that. Is there some hint which simplifies the proof? Maybe, I can first show that $(\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -ra\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}$, because the parity of the left side is opposite to the parity of $\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}}$, and $(\hat {\sigma} \cdot \mathbf r )$ commutes with rotation operator, because refers to the scalar pdoduct?


title: What star catalog is suitable for a programmer wanting to model observable stars in 3D?

text: I have searched around the web and looked at Wiki entries and I am overcome by the number of catalogs available. What I want is a catalog that includes all objects you might conceivably see with equipment under $500k (I hope that captures all backyard astronomers). Apart from that the main requirements are that it is up to date, machine readable (CSV is relatively easy. SQL might be too much to hope for?) and free to use. Can anyone say which catalog would be best for that purpose and where I can get it? (Is there one numbering system used for stars? I'm seeing HDE, SAO and Gliese numbers, Bayer and Flamsteed designations and proper names e.g. Rigol. If there isn't a universally identified system there must be some kind of cross reference? Thinking like a programmer - what's my primary key?) While Physics isn't my first choice for this question the SE Astronomy site didn't make it through beta, and its page now directs users here.


title: A problem with $E = mc^2$

text: (Rest) mass of proton = (rest) mass of neutron (approximately) = $1.67 \times 10^{-27}$ kg. let $m$ be the mass of nitrogen atom then $m = 2.32 \times 10^{-26}$ kg. $c = 3 \times 10^8$ m/s (approx). Energy $E = mc^2$. $E = 2.32 \times 10^{-26} \times 3 \times 10^8 \times 3 \times 10^8$ Joules. $E = 20.88 \times 10^{-26+16}$ Joules. $E = 20.88 \times 10^{-10}$ Joules. Energy required to propel a nitrogen atom at the speed of light is very less. Is my calculation wrong? Because even if we take a cluster of 10000 atoms of nitrogen we can make them cross the speed of light easily.


title: The Holstein-Primakoff Representation (approximation)

text: I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S - n_j} a_j \\\ S_j^- = a^\dagger_j\sqrt{2S - n_j} \\\ S^z_j = S - n_j $$ Where $a_j$ and $n_j$ are operators. When we derive the magnon dispersion relation, we make the assumption that $$ \frac{\langle n_j \rangle}{\langle S^z \rangle}\ll 1 $$ which is fine. As far as I understand this only means that we assume that most of the spins are pointing along the z-direction. However, when we go further in the derivation, we do a series expansion in $n_j/S$, so that i.e. $$ S_j^+ = \sqrt{2S}\sqrt{1- \frac{n_j}{2S}+...} \approx \sqrt{2S}\sqrt{1- \frac{n_j}{2S}} $$ This is where I don't understand. Say that we are in a spin $1/2$-system. In that case we can for one site have at most 1 magnon excitation and $S=\pm 1/2$, so for each individual site $n_j/2S$ is not much smaller than one. My question is then, how can we justify that the series expansion makes sense in the operators at each individual site? Will the contributions from each site when summed up not contribute because $$ \frac{\langle n_j \rangle}{\langle S^z \rangle}\ll 1 $$ or have I misunderstood something? Any thoughts would be much appreciated.


title: Terminology for line integral of magnetic field

text: One of the quantities appearing in the integral form of Maxwell's Equations is the line integral of the magnetic field around a closed loop. (The relevant equation states that this is equal to the current through any surface bounded by that loop, plus the displacement current in the case of changing electric fields, with some constant coefficients possibly thrown in, depending on how you manage your units. This is usually called Ampère's Law, with Maxwell's correction for the displacement current.) Is there a name for this quantity? More generally, is there a name for the line integral of the magnetic field along an arbitrary non-closed curve? (Then this is not equal to any other named quantity or sum of quantities.) Ideally, I'd want a name for the integral of the H-field (rather than the B-field, when one distinguishes them) in amperes (or the equivalent in non-SI units), but I am not really picky about those details. It seems that every other quantity in the integral form of Maxwell's Equations has a name (magnetic flux, electric flux, charge, etc), so I'd hope that this one does too. Of course, the term for the line integral of the electric field (electromotive force) is somewhat of a historical oddity, so maybe this quantity is too obscure to have a name. Still, you'd think that somebody would have given it one sometime! ETA: A theoretical way to measure this quantity is to place a conductive tube (as conductive as possible, with as small diameter as possible) around the curve in question. The magnetic flux through the tube will induce a current running around (not along) the inside of the tube, which will in turn serve to cancel (or shield) the source of the magnetizing field, in accordance with Lenz's Law (in the broad sense). The current so induced, serving to completely cancel the magnetic field near the curve in question, should equal this quantity, if I'm thinking about this correctly. (Well, I'm having a little trouble getting the sign straight, but one way or the other it should work!)


title: Am I faster than my shadow?

text: I guess that everything that happens to me (or I do), happens before my shadow "records" the information. Is that right? Is it the same for any observer?


title: Bullet energy loss in solid materials

text: Does someone know simple model of energy loss in solid materials for a solid bullet? For example: I want to estimate momentum transferred to a thin plastic plate given by a small bullet. No matter how difficult model is. But it should depend only on plate/projectile parameters? **EDIT:** No precise solution is needed. It is interesting to know order of energy transfer. One simple model that i suggested was model of exponential suppression: $$ {\delta E}(r) \approx E_{0} e^{ - \frac{r}{d} } $$ where $d$ is a "terminal length" - after crossing this length bullet loses $e$ units of energy. So, knowing max distance in material one can find parameter $d$ It is not obvious that this formula is close to reality, but it is first step. Also it is not clear if one can use this formula for thin plates. I have no experience solving such problems, so i asked if someone can point me (links, books, ideas).


title: Pressure and Buoyancy

text: It is well known that you can trap water in a drinking straw by placing the tip of your finger over the top while the straw is in the water, and then lifting it out. The inner diameter of the straw is 0.5 cm; the height of the trapped water column inside the straw is 5.0 cm. Find the pressure in the air pocket between your finger and the water.


title: How to use accelaration data of moving object to calculate distance?

text: I read couple of similar question on this forum and few blogs on web, though I am still confused,I am determined to calculate object displacement using accelerometer data. So, I tried using www.freescale.com/files/sensors/doc/app_note/AN3397.pdf , which gives nearly 20-50% error. I also tried this equation on data and got some better results.               V1 = V0 + ((a1) * t)     X2 = abs((V1 * t) + (((a1)/2)*t*t))     X2 = X0 + X2     V0 = V1     X0 = X2      Here t is the time differance between two samples. X0 is the final displacement. V1,a1 current values. V0 previous value. Here is my graphs for the movement from Android sensors readings. 1)From Linear Acceleration Z axis graph, how do I calculate the distance? 2)Which values should I take and which should not? how many data samples I should use for calculation. For positive acceleration only or for an entire positive negative cycle of acceleration data? 3) What the Linear acceleration graph means. Please explain its values based on movement of object.


title: Measure maximum volume of liquid of a water tap

text: I need to know the consumption of a water tap in Liters. I'm able to know the cross-sectional area of the pipe and how long the tap is open. With these data, how can I know the consumption/volume of water that pass through the water tap? It can be rough estimation. I can assume a maximum volume of water... Thanks in advance!


title: Research on galaxy "stiffness"?

text: > **Possible Duplicate:**   >  Are galactic stars spiraling inwards? Has there been any research on the effects of interstellar gravitational attraction that would make collective star motion in a galaxy behave more like a solid disk? I understand rotational velocity has been observed to be higher than expected for stars farther from galaxy centers. This doesn't agree with Newtonian physics in which Pluto goes around the Sun more slowly than Mercury. An explanation for the faster star motion observed in galaxies was created and called dark matter. This seems to not comply with Occam's Razor, where the most simple possibility is probably right. I visualize the attraction between stars as stiffening the collection, or increasing the viscosity of the rotational behavior which could explain faster rotation towards the outside of a galaxy. The rotation would behave somewhere between atoms locked together in a frisbee, and a fluid circling the drain. Has any research been done to prove this wrong?


title: How would water drain out of a sealed pipe?

text: While hiking through the Grand Canyon, I started wondering. Say we have a pipe for the purpose of transporting water across a canyon, with the bottom submerged in a pool of water. Like the blue line in this terribly drawn image: ![enter image description here](http://i.stack.imgur.com/7iVLE.png) Now let's seal the two ends of the pipe, which is full of water. What would happen if the pipe broke at the bottom of the U? Note: It shouldn't matter whether the pipe is a U or a vertical line, I simply thought about this with a U-shaped pipe. * * * If the ends of the pipe weren't sealed, the water inside the pipe would leak out until the height of the water outside the pipe was the same as the height of the water inside the pipe. But the ends are sealed. A vacuum can support 10m of water, so perhaps the water will drain out of the pipe until it is 10m above the height of the water outside of the pipe, leaving a big near-vacuum behind. This would explain why the Grand Canyon's pipe has such a large number of valves. On the other hand, we learn from http://what-if.xkcd.com/6/ that water boils into a vacuum. Does this leave a significant amount of water vapour behind?


title: Relationship between local and global scaling (Weyl) symmetry

text: Theorem 5.1 on page 80 of this paper says that _Assuming that the matter fields satisfy their equations of motion, the matter field action is locally Weyl invariant if and only if the corresponding energy-momentum tensor is traceless._ If I have understood this right, Weyl invariance reduces to scale invariance in flat spacetime, which is basically what I am up to at the moment. My question now is: If one has Weyl or scale invariance in accordance with this theorem, does this mean that global Weyl or scale invariance (such that the scaling parameter does not depend on position) is guaranteed too? If not why not?


title: What is the importance of Joule's experiment?

text: I was reading about the experiment of Joule (Italian wiki page). I'm not sure how it's called in English, since there is only an Italian and French version. In any case, in the page it is stated that (my translation): > Through this experiment, Joule determined the mechanical equivalent of heat > to be equal to 4,186 J/cal, value of extraordinary precision for the time. Is the importance of this experiment "only" ascribable to the fact that the evaluation was really precise and exact for the time? Or are there other reasons? I don't need a whole list of reasons, just the most exemplary/important1 fact(s) will suffice. * * * 1: Objectively considered as such by most researchers, experts, etc.


title: Microgravity: how powerful is an astronaut

text: In 2008, NASA astronaut Clayton Anderson tossed a 635-kg tank of ammonia coolant overboard from the International Space Station (ISS). Subsequently the tank burned up in earth atmosphere as planned. Obviously in the 1-g environment on Earth's surface, no human could have lifted the tank let alone tossed it. The astronaut was only able to do so because the tank weighed much less in the microgravity environment of the orbiting (free-falling) ISS. Because I was asleep in much of physics class in high school, I am unable to work out a formula for how much ~~weight~~ mass a human can chuck overboard from the ISS unassisted. In particular, I have only the vaguest notions of inertia and action/reaction. (What I mean is, I can look up Newton's laws of motion but I am unable to use them to solve my question.) In the picture within the article that I linked to in my first paragraph above, we see that the astronaut appears to have his feet strapped into a footrest at the end of a robotic arm. Am I right that this is vitally important because otherwise he would not have a backstop to push off from? Anyway, can you give me a way to calculate, using simple arithmetic, what is the heaviest object that an average-strength human can toss overboard from the ISS? Also, why is not all trash discarded from the ISS by being thrown overboard manually or using a device instead of being packed into a space-craft that will depart from ISS and burn up in the atmosphere? _(A day later:)_ Thank you to all three who took the time to reply so far, I upvoted each of your Answers. I could not for the life of me single out one for "acceptance" so no green checkmark from me this time :)


title: Vector transformation in special relativity

text: Please note that I am very new on this website so have some difficulties in writings as required here but trying really hard to learn quickly. La-Tex is the main problem but please understand me that I am serious. How do Vectors transform from one inertial reference frame to another inertial reference frame in [special relativity]. A bound vector in an inertial reference frame ($x$,$ct$) has its line of action as one of the space axis in that frame and is described by $x$* _i_ *,then what would it be in form of new base vectors ( **a** ) and ( **b** ) in a different inertial system ($x`$,$ct`$) moving with respect to the former inertial system with $v$* _i_ * velocity.Let ( **i** ) and ( **j** ) be the two bounded unit vectors with the line of action as co-ordinate axis($x$) and($ct$) respectively and senses in the positive side of co-ordinates and similarly ( **a** ) and ( **b** ) are defined for co-ordinates ($x`$) and ($ct`$) respectively.


title: Are there good home experiments to get a feel for the behavior of yield-stress liquids?

text: What common household liquids / foodstuffs show Herschel-Bulkley fluid behavior? I assume tomato soup to be one, since when sloshing it around lightly on a spoon the visible surface moves as one (as can be seen by following the dots of black pepper that don't change relative position) - is this observation basically correct? What are (other) easy and sure ways to identify yield-stress behavior (vs. other non-Newton flows)? How can common states of flow (through a pipe/ open channel, discharge from a small source into a large standing volume of the fluid, mixing) be observed - not necessarily quantitatively? Are there easy to produce transparent yield stress fluids?


title: Ohm's Law Intuition

text: When we derive Ohm's Law using the Drude Model, we assume at one point of time that $E=V/L$, when is fact, $E=dV/dL$, unless $E$ is constant, in which case the assumption $E=V/L$ is true. But I don't understand why the electric field in a conductor must be constant as current flows. Is there a convincing explanation that is perhaps related to the way atoms behave and orient themselves? Also, if the assumption $V=E\cdot L$ makes sense, I can understand why Ohm's Law should work for a homogeneous electric circuit. However, I don't understand why it should work for a heterogeneous circuit - perhaps one with two different resistors connected in series. And please don't use the traffic jam analogy. Surely there must a more theoretical way to explain this (using Classical Physics).


title: Why isn't general relativity the obvious thing to try after special relativity?

text: To preface my question, I ask this as a mathematics student, so I don't have a very good sense of how physicists think. Here is the historical context I'm imagining (in particular taking into account the development of differential geometry in the 19th century): Classical mechanics is about Lagrangians of matter fields on $\mathbb{R}^3$ (with the flat metric) Sometime in the 1820s Gauss speculated about replacing the flat metric on $\mathbb{R}^3$ by one with non-vanishing curvature Special relativity is about Lagrangians of matter fields on $\mathbb{R}^{3,1}$ (with the flat metric). I'm taking this jump for granted since as I understand it, it was arrived at from experimental results on electromagnetism at the end of the 19th century. Now in line with Gauss, it seems like it would be extremely natural to speculate about replacing the flat metric on $\mathbb{R}^{3,1}$ by one with non-vanishing curvature (and in the same spirit also considering more exotic topologies for the underlying manifold). Given that then we would have to ask exactly which metric we are looking for, it seems natural to say that there should be a Lagrangian term corresponding to the metric. The Einstein-Hilbert functional is probably the simplest one to try. And so we get the Einstein equations. Alternatively (as I heard from someone is the actual history) you could observe that the energy-momentum tensor is of course a 2-tensor, and so for an Euler-Lagrange equation the most natural metric-dependent expression would be $\operatorname{Ric}=T$. Since $T$ is always divergence free it would be natural to replace $\operatorname{Ric}$ by $\operatorname{Ric}-\frac{1}{2}Rg$ just from taking the contracted second Bianchi identity into account. And so again we get the Einstein equations. I've often heard it said that if Einstein had not come up with special relativity, someone else probably would have in the next five or ten years. However if he had not come up with general relativity, it would have taken much longer to discover. **Why is this?** I feel that I must be missing something here.


title: Should physicists abandon Popperian falsification criterion?

text: Many of the most brilliant ideas of modern physics are very much unpopperian. In view of this, it has become more and more difficult to hold an orthodox Popperian attitude in physics. It simply looks like more of a philosophical dogma nowadays. My question is, should physicists abandon this criterion at least partially for good? If so, what criterion should replace it?


title: What is the angular momentum spectrum of an sp${}^3$ electron?

text: So, one thing has been annoying me ever since I learned about orbital hybridization: you explain the shape of molecules by postulating that the orbitals of multi-electron atoms are linear combinations of the orbitals of the hydrogen atom. Fine. Here's what gets me, and I've asked several chemistry professors about this, and I haven't gotten an answer. If I measure the orbital angular momentum of an electron in a hydrogen atom, I get the value $L = \hbar \sqrt{\ell(\ell +1)}$, so an s orbital has angular momentum $0$ and a p orbital has angular momentum $\hbar$. So, what happens when I measure the orbital angular momentum of a sp${}^{3}$ electron? Do I get $\frac{3\hbar}{4}$ with probability 1? Or do I get $\hbar$ with probability 0.75, and $0$ with probability 0.25? Are the original orbitals really the fundamental thing, and the hybridized orbitals fundamentally just quantum superpositions of these things? Or does the multi- electron potential create a state that just looks like a linear combination of one s and 3 p orbitals? Or is the actual answer more complicated and this a heuristic picture that I'm taking too seriously?


title: Is our universe a 2+1D superconformal field theory?

text: If our universe comes from a flux compactification of string theory over 6 dimensions with a nonzero flux, then it can't be continually deformed to another compactification with zero flux as the total flux is a topological invariant. The cosmological constant of our universe is positive, which in string theory means it is a metastable state. The stable vacuum for a flux compactification in string theory is a BPS compactification with negative cosmological constant. What if our metastable observable universe is actually embedded in a larger stable anti de Sitter background? An inflating metastable or slowly rolling or chaotically inflating patch of this asymptotically AdS universe inflates to our observable universe. By the AdS/CFT correspondence, our universe can be described by a superconformal field theory in 2+1D with no gravity. Is there any flaw in my reasoning?


title: Higgs Boson Mass

text: The WIKI Higgs boson site has an interesting diagram illustrating likely Higgs mass intervals that experiments (LEP & Tevatron) or indirect measurements have determined with a 95% confidence level. It posits that the feasible mass ranges are in the intervals of about 115-158 GeV or 175-185 GeV. The Tevatron confirmed the existance of the top quark (172 GeV) in 1995, so it had the energy and time to detect the Higgs in the 115-158 GeV ranges, but could not. Given LHC's greater energy and luminosity, why haven't there been anouncements of the Higgs discovery? Do we understand enough about potential particle creation to definitively know they are Higgs decays?


title: Why is the sum over string worldsheets of varying genuses unitary?

text: It can be shown the sum over all Feynman diagrams in quantum field theory leads to a unitary S-matrix. Where can I find a proof that the sum over string worldsheets with varying genuses also leads to a unitary S-matrix?


title: Uniqueness of quasiclassical consistent histories

text: The current zeitgeist here is on interpretations of quantum mechanics, so let me add my own two cents here. As you may know, consistent histories is an alternative interpretation proposed in a series of papers 1990-1994 by Gell- Mann and Hartle. In a shocking article, Dowker and Kent proved there are many distinct consistent histories for most quantum systems. Even if we constrain ourselves to quasiclassical histories, i.e. those histories which are close to classical, being quasiclassical now does not guarantee being quasiclassical in the future, or in the past. This is disturbing. Do histories which are quasiclassical for all times exist? If they do, are they unique? Is the nightmare scenario of two everywhere quasiclassical consistent histories which are mutually incompatible possible?


There're some easy concepts which I don't get exactly in my mind. I've been studying these things for two years of my engineering but they still bother me. Battery is one of them. Can someone explain?   * Whether a battery store energy or simply it makes energy? I have searched it on Google and Yahoo but didn't find any helpful thing there (for me). So I will be glad if I got my problem solved here. P.S. I hope that the question would not again be an off-topic, as it always does and moreover people don't suggest then where to go. It's a real sad thing.


title: Weyl Large Number Coincidence

text: I've just learned of the Weyl Large Number Coincidence on the Wikipedia. It looks interesting. Here is my interpretation of it. What is the smallest "quantum" of energy in the Universe? $$E_{min} = \hbar H$$ where $H$ is the Hubble parameter. This make sense as the Hubble parameter is roughly the inverse of the age of the Universe ($H=2\times10^{-18}s^{-1}$). You can't define a non-zero frequency smaller than that. Now consider the gravitational self-energy of a particle, $E_g$, given by $$E_g = \frac{Gm^2}{r}$$ where $m$, $r$ is the mass,size of the particle. The smallest possible particle must have a gravitational self-energy of the size of $E_{min}$ therefore we have $$\frac{Gm^2}{r} = \hbar H$$ We can link the mass $m$ and size $r$ of a particle by using the Compton wavelength quantum relation $$m c r = \hbar$$ Putting the two equations together and eliminating $r$ we find $$m^3 = \frac{\hbar^2H}{Gc}$$ If we plug values into this expression we get a mass given by $$m \approx 10^{-28}kg$$ This is nearish to the proton mass. Interesting or just a coincidence?


title: What is the strange event in this simulation of a galactic collision?

text: I was watching this video on YouTube: 2 Spiral Galaxies w/Supermassive Black Holes Collide Around half way, and again almost at the end, the black holes seem to suddenly give off some sort of force which blasts away a lot of the matter around them. What is this effect known as or caused by?


title: Taking length/area element when trying to find E.resistance

text: I am trying to find out which 'element to take' (and why), When trying to find the resistance of some material with non-uniform resistivity $\rho$. _I'll give an example_ : Say we have a co-axial cable with length $L$, inner radius $a$ and outer radius $b$ with resistivity $\rho(r)$   * If we look at the cable's resistance when the current is parallel to it's symmetry axis, we take the element $dA=2\pi r dr$ and the length as a (constant) function $\ell (r) = L$.   * If we look at the cable's resistance when the current is radial, we take the element $d\ell = dr$ and $A$ as a function : $A(r)=2\pi L r $ if the resistivity would be $\rho (z)$ - where $z$ is the axis of symmetry - would we switch that? I mean, in the second case would we take $\ell (r) =r\Rightarrow \ell_{total} =b-a$ and $dA= 2\pi(b-a) dz $ My reasoning is that we take $dA$ or $d\ell$ in respect to where the resistance grows: parallel to the current - we take $d\ell; $ otherwise - $dA $ I would be glad to hear if I'm right or wrong, and be corrected if needed. * * * Edit: This isn't really a homework question, it's tagged as one because it is homework related. I want to know how I should get the resistance in any way, and to conceptually understand why. a textbook refrence will be great too. * * * Thanks in advance, Daniel.


title: D-branes wrapping divisors and/or cycles

text: What is the difference between a divisor and a homology cycle? What is the difference between a D-brane wrapped around a divisor and a D-brane wrapped around a cycle? Thanks.


title: Radio Transmission distance (FM)

text: I am trying to find a formula (if one exists, and this is the right place) to try and calculate how far a radio station can transmit, under ideal conditions. I have searched Google, and I can not find a solid answer.   * Broadcast Frequency: 89.7 MHz    * Effective Radiated Power: 910 Watts   * Transmission Power: 936 Watts   * Transmitter Height: ~39m   * Terrain: Flat Thanks for the help.


title: Observers in (Schwarzschild-) de Sitter spacetime

text: In (pure) de Sitter spacetime, the cosmological horizon is said to be ‘observer dependent’. I imagine that as the observer always being in the center of that horizon. Another (spacelike separated) observer also has spherical horizon, but a different one. Consider a test mass in the shared region of both observers. Can both observers reach consensus on anything, such as its position, momentum or energy? I would know how to answer this if the test particle was in the center of the cosmological horizon, since then the Schwarzschild-de Sitter metric could be used. However, an observer in that metric is also always in the center (right?). Therefore, a more general phrasing of my question is: what does the Schwarzschild-de Sitter metric look like for an observer not at the center?


title: How is it possible for objects to travel faster than the speed of sound when particles interact at the speed of sound

text: First of all, I am sorry if this is a stupid question but: I've heard that atoms interact with each other at the speed of sound (when you for instance push a chair, the atoms collide with each other in a chain reaction at the speed of sound, making the chair move). How can then airplanes fly faster than the speed of sound without something going crazy?


title: Ballistic Pendulum Demo Problem

text: I have a question about the following problem: > ![enter image description here](http://i.stack.imgur.com/nOtw4.png) I got the solution $v=\frac{M+m}{m} \sqrt{2gh}$. But my real question is in the following picture: > ![enter image description here](http://i.stack.imgur.com/gVtk4.png) In the above slide, how can you derive the solution $v=(\frac{M+m}{m})d\sqrt{\frac{g}{L}}$ and thus, $v=3d$? Also, it seems that $v=3d$ is not correct since their unit is not matching... Can somebody explain this???


title: How fast would a spaceship have to go to reach Alpha Centauri within a person's lifetime?

text: Given the recent news about the discovery of an "Earthlike" planet orbiting Alpha Centauri (our nearest stellar neighbour) it got me wondering just how fast would spaceship have to travel to be able to reach Alpha Centauri within a person's lifetime (say 60 years)? The reasoning is that even if we sent an unmanned probe - assuming that it would be a one way trip - the journey time would have to be sufficiently short to keep people's interest in the mission so that when it actually arrived and sent back data there would be somebody home to receive the data. The added complication is that the craft would have to slow down sufficiently so that it could at least enter orbit around the star (I'm not going to suggest that it manages to orbit one of the planets!). I suppose what I'm really interested in how this speed compares to the speeds we've managed to attain so far and thus get an idea of how much technology must advance for us to be able to even think of achieving this. I know about relativistic effects due to travelling at a high percentage of the speed of light - but I'm not really interested in that here.


title: Retarded field interaction

text: I was doing the following thought experiment: imagine a particle that moves at a certain velocity. Imagine also that the particle generates a field that propagates at velocity $v_f$. Well, if the particle's velocity is higher than $v_f$ then it will interact with its own field if for some reason the particle decelerates (imagine a plane that goes Mach 2 and then "brakes" to zero, it will hear its own sound after a while). But what if the particles's velocity is exactly $v_f$? My guess is that it will interact with its own field... is this correct?


title: Schrödinger's equation, time reversal, negative energy and antimatter

text: You know how there are no antiparticles for the Schrödinger equation, I've been pushing around the equation and have found a solution that seems to indicate there are - I've probably missed something obvious, so please read on and tell me the error of my ways... Schrödinger's equation from Princeton Guide to Advanced Physics p200, write $\hbar$ = 1, then for free particle $$i \psi \frac{\partial T}{\partial t} = \frac{1}{2m}\frac{\partial ^2\psi }{\partial x^2}T$$ rearrange $$i \frac{1}{T} \frac{\partial T}{\partial t} = \frac{i^2}{2m}\frac{1}{\psi }\frac{\partial ^2\psi }{\partial x^2}$$ this is true iff both sides equal $\alpha$ it can be shown there is a general solution (1) $$\psi (x,t) \text{:=} \psi (x) e^{-i E t}$$ But if I break time into two sets, past -t and future +t and allow energy to have only negative values for -t, and positive values for +t, then the above general solution can be written as (2) $$\psi (x,t) \text{:=} \psi (x) e^{-i (-E) (-t)}$$ and it can be seen that (2) is the same as (1), diagrammatically ![energy time diagram](http://i.stack.imgur.com/0JBc2.jpg) And now if I describe the time as monotonically decreasing for t < 0, it appears as if matter(read antimatter) is moving backwards in time. Its as if matter and antimatter are created at time zero (read the rest frame) which matches an interpretation of the Dirac equation. This violates Hamilton's principle that energy can never be negative, however, I think I can get round that by suggesting we never see the negative states, only the consequences of antimatter scattering light which moves forward in time to our frame of reference. In other words the information from the four-vector of the antiparticle is rotated to our frame of reference. Now I've never seen this before, so I'm guessing I've missed something obvious - many apologies in advance, I'm not trying to prove something just confused.


title: Distribution of charge on a hollow metal sphere

text: A hollow metal sphere is electrically neutral (no excess charge). A small amount of negative charge is suddenly placed at one point P on this metal sphere. If we check on this excess negative charge a few seconds later we will find one of the following possibilities: (a) All of the excess charge remains right around P. (b) The excess charge has distributed itself evenly over the outside surface of the sphere. (c) The excess charge is evenly distributed over the inside and outside surface. (d) Most of the charge is still at point P, but some will have spread over the sphere. (e) There will be no excess charge left. Which one is correct and why? I guess it is some kind of electrostatic induction - phenomena going on. Am I right? I understand that excess charge is distributed over hollow sphere and that negative and positive charges are distributed opposite sides, but don't know which one positive or negative go to inside surface.


title: Time crystals : fake or revolution?

text: This article about "crystals of time" just appeared on the PRL website. Viewpoint: Crystals of Time (http://physics.aps.org/articles/v5/116) The authors (including famous Frank Wilczek) claim that some systems in their ground state are time periodic and this is related to some spontaneous breaking of some "time" symmetry. From my point of view such systems were known before this claim (persistent currents, bright solitons among them)...So I want to know what is special about these new "crystals of time"?


title: Are there isotopes that ONLY decay through beta decay?

text: Are there isotopes that only go through beta decay and nothing else to become stable ?


title: Why is the value of spin +/- 1/2?

text: I understand how spin is defined in analogy with orbital angular momentum. But why must electron spin have magnetic quantum numbers $m_s=\pm \frac{1}{2}$ ? Sure, it has to have _two_ values in accordance with the Stern-Gerlach experiment, but why precisely those values?


title: Tension in a chain fountain

text: I was reading the following paper: http://arxiv.org/pdf/1310.4056v2.pdf There were a few things I couldn't follow: 1)Equation (0.1) > $T_C/r=\lambda v^2/r$ I understand this takes the form of Newton's 2nd law, where $T_C$ is the force and $v^2/r$ is the acceleration. I don't understand why the tension is divided by $r$. I'm guessing it has to do with the dimensions of $\lambda$ (mass/length), but why is the radius of curvature the correct length to divide by? Is the sum of the tensions across the string a meaningful quantity? 2)Equations (0.2) and (0.3), and Figure 1 Do the tensions $T_T,T_C,T_F$ describe the tension at any point in their respective regions, or just the points at which they are shown in the figure? I.e., Is the tension uniform in those 3 respective regions or is it constantly varying along the chain? What I'm trying to understand is, could any point in (for example) the curved region be chosen to represent $T_C$? If not, why aren't the chosen points arbitrary? Sorry for the abundance of questions. I'm guessing there's something fundamentally wrong with my understanding of tension, and that an answer correcting my misconception could potentially answer most or all of the questions simultaneously.


title: On the shape of magnetic and electric fields in an electromagnetic wave

text: Electromagnetic waves are generally depicted like this: ![enter image description here](http://i.stack.imgur.com/Gn4wv.png) Where the electric fields and magnetic fields exist in the planes perpendicular to the direction of propagation. I also realize that as the electric field changes while the wave is propagating, a magnetic field is induced and vice versa (by faraday's and maxwell's laws of induction). But, those laws predict that the fields will be circular. So, won't the electric and magnetic fields look different? Won't they be circles along arrows that are drawn in the figure? I haven't seen anything written about this anywhere.


title: Coulomb force in SI and cgs

text: Coulomb force in SI is $ F = \frac{Q1*Q2}{4\pi\varepsilon R^{2}} $ while in CGS $ F = \frac{Q1*Q2}{R^{2}} $ why is it? I mean doesn't it any make difference in dimension? since $ \varepsilon $ itself has dimention. And similarly for magnetic force also.


title: Why is there a factor of $4\pi$ in certain force equations?

text: I mean to ask why there is $4\pi$ present in force equations governing electricity? Though all objects in universe are not spherical and circular, the constant of proportionality in both equations contain $4\pi$. Why?


title: Uncertainty of permittivity of vacuum

text: **Question** : The value of permittivity of vacuum, $\epsilon_0$, is given with absolutely no uncertainty in NIST Why is this the case? * * * **More details** : The permeability of vacuum can be given by $$\mu_0=\frac{1}{\epsilon_0 c^2}$$ which comes from the definition of a magnetic field in special relativity, where we solve the problem of a wire with electrons flowing in, and we calculate the force exerted on an external charge moving with some velocity (for details, refer to the book Electricity and Magnetism, E. M. Purcell), and we define a new field called "magnetic field" with the form of Lorentz force, where the magnetic field is $$B=\frac{I}{2\pi \epsilon_0 c^2 r}$$ where $I$ is the current due to the flow of electrons in the wire, and $r$ is the distance of the external charge from the wire. And there we get the definition of $\mu_0$ that makes $B$: $$B=\frac{\mu_0 I}{2\pi r}$$ and there starts the concept "magnetism". Why am I giving this detailed example? Because I wouldn't like to get the answer that $\mu_0$ has no error, and that's why $\epsilon_0$ has no error, and then we fall into circular logic. So I expect a reason which is independent of $\mu_0$. Thank you in advance.


Why do we take the value of the constant in Coulomb's law as $\frac{1}{4\pi\varepsilon_0}$?


title: Coulomb's Law: why is $k = \dfrac{1}{4\pi\epsilon_0}$

text: This was supposed to be a long question but something went wrong and everything I typed was lost. Here goes.   1. Why is $k = \dfrac{1}{4\pi\epsilon_0}$ in Coulomb's law?    2. Is this an experimental fact?    3. If not, what is the significance of this definition?


title: Is there energy stored when iron is magnetized?

text: When a piece of iron is magnetized, and the domains are aligned, Is there energy stored? If so, how much energy is stored? If there is an attraction between that same iron and the source of the exterior magnetic field,where work is done, and there is energy that is transferred. Is the energy equal to that of which is stored in the alignment of the domains? **And how much energy is stored or needed to align the domains?**


title: How to calculate the number of glass sheets that will be broken by a falling object?

text: In season 1, episode 7 of King of the nerds the contestants are asked to calculate how many sheets of glass will be broken by a falling object. They are shown 1 example case and then asked to calculate the result for the fallowing 3 test, for which they are given the data. After watching this I wanted to know how to calculate the answer. A schematic drawing of the experiment (drawn by me): ![enter image description here](http://i.stack.imgur.com/mk6GC.png) Screen captures of the experiment from the show: ![enter image description here](http://i.stack.imgur.com/NDC57.png) ![enter image description here](http://i.stack.imgur.com/CcWN4.png) A screen capture with all the data given: ![enter image description here](http://i.stack.imgur.com/FIszw.png) I've translated the data to normal (metric) units: _**Ball drop data_**   * The ball weighed 5.44 kg   * The ball diameter was 0.1905 m   * The distance from the bottom of the ball to the ground was 4.26 m   * The distance from the bottom of the ball to the first sheet of glass was 2.38 m   * The sheets of glass were 0.6096 m x 0.6096 m x 0.00635 m thick   * The sheets of glass were spaced 0.0762 m apart   * There were 20 sheets of glass   * The weight of one sheet of glass was 5.21 kg   * 4 sheets of glass broke _**Challenge Variations_** **(A)**   * Ball weight 2.72 kg   * Ball diameter 0.1524 m   * Glass thickness 0.003175 m   * Glass weight, one sheet 2.72 kg   * 26 sheets of glass, spaced 0.0762 m apart   * First sheet 1.9304 m from bottom of ball **(B)**   * Ball weight 3.63 kg   * Ball diameter 0.17145 m   * Glass thickness 0.00635 m   * Glass weight, one sheet 5.21 kg   * 13 sheets of glass, spaced 0.1524 m apart   * First sheet 1.9304 m from bottom of ball **(C)**   * Pig weight 6.804 kg   * Glass thickness 0.00635 m   * Glass weight, one sheet 5.21 kg   * 26 sheets of glass, spaced 0.0762 m apart   * First sheet 1.9304 m from bottom of ball The results of the experiments were:   * A: 5 sheets of glass were broken   * B: 2 sheets of glass were broken   * C: 5 sheets of glass were broken I'm interested in the correct way to calculate the analytically solution to the problem, and what physics is behind it.


Total number of subatomic particles in the universe. Are they finite ? assuming any of GR or QM or even ST.


title: Velocity and acceleration question

text: > 1.The magnitude of the total force acting on a ball rolling without slipping > down a ramp is greater than the magnitude of the total force acting on the > same ball if it slides down the ramp without friction. True or false? I chose true for this one, as I figured that if our force has to oppose friction, then it must be greater than that force without friction. However the correct answer is false. > 2.The magnitude of the velocity of an object must change if the magnitude of > its acceleration is a constant. True or false? I chose true for this one too. If acceleration is constant, then velocity is linear. This turned out to be false as well. Can anyone explain to me why I was wrong for both of these?


title: how to calculate gibbs free energy per unit mass, per unit volume, and per mole?

text: I've ran into conflicting information on how to calculate the Gibbs free energy of fuels during combustion per unit mass, volume and mole. A sample solution for hydrogen would be really appreciated!


title: What is required to create water droplets of a consistent size and shape over a distance of 3 to 10 feet?

text: What would I need to do to create water droplets of a consistent size and shape over a distance of 3 to 10 feet? Are there any special requirements for the nozzle? Can I use pure water or should I have some kind of additive? Any special requirements on the valve, etc, that is produced?


title: SU(2) critical point and volume dependence

text: I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their head whether the value of SU(2)'s critical point on the phase diagram has any volume dependence? I have tried to find out in the literature but so far I only found two papers on SU(2) critical phenomena and I haven't been able to sort out that fact yet. Any help would be appreciated. **EDIT:** sorry, I mean in the $\beta_{\frac{1}{2}}$ and $\beta_{1}$ ("beta fundamental and beta adjoint") plane, with a Wilson action $\beta_{1/2}(\chi_{1/2}-2)+\ldots$ keeping only the first three terms in the expansion (up to 3/2). $j$ is the representation. I am doing Migdal-Kadanoff recursion and plotting the different $\beta$s against each other (this is the flow of the $\beta$s throughout the recursion). There is a first order phase transition line in the $\beta_{1/2}$vs $\beta_1$ plane that ends at a critical point, I would like to know if that line (and the critical point) moves around depending on the volume.


title: Can the Euler-Lagrange equations be derived from an infinitesimal Principle of Least Action?

text: The Euler-Lagrange equations can be derived from the Principle of Least Action using integration by parts and the fact that the variation is zero at the end points. This has a mystical air about it, with the system somehow "sniffing" out all possible paths in the distant future and choosing the one that minimizes the action. An infinitesimal version would make the method less mystical in this sense, and so I would like to ask: Can the Euler-Lagrange equations be derived from an infinitesimal principle of least action?


title: What really is a thermodynamic system?

text: I'm starting to study Thermodynamics and I'm pretty confused about what a thermodynamic system really is. When studying mechanics we focus our attention on systems of particles. So when we talk about systems of particles we know what is it all about, a collection of some particles in space subject (or not) to a collection of constraints. When we study fluids, for example, the fluid is still a system of particles, each particle being a molecule of the fluid. But now this description is too messy to work with, so we use the continuum approximation. Anyway, again we have a system of particles in some region in space. Now, in Thermodynamics, what is a thermodynamic system? Texts usually talk about "pressure" and "volume" like if it were a fluid, but certainly Thermodynamics is much more general than just fluids. Wikipedia's page says the following: > A thermodynamic system is a macroscopic volume in space, the adventures of > which are to be studied according to the principles of thermodynamics, along > with its walls and surroundings. Not just any physical system is a > thermodynamic system, but only those that can be adequately described by > thermodynamic variables, such as temperature, entropy, internal energy, and > pressure. So a thermodynamic system is just a system of lots of molecules of some substance with the continuum assumptions we make in fluid mechanics? What really is the definition of a thermodynamic system and what's the motivation behind the definition?


title: Why does a magnetic field generate clearly visible separation?

text: I have noticed that when particles of metal are captured by a magnetic field. They separate into clearly visible patterns of lines. Why don't the particles form a more even distributed pattern? Do these gaps/lines exist in the magnetic field itself? Can someone explain why this happens? As illustrated by this images. ![Magnetic field example](http://i.stack.imgur.com/4Je0i.jpg) ![Magnetic field example](http://i.stack.imgur.com/q7xoX.png) ![Magnetic field example](http://i.stack.imgur.com/UAfTr.jpg)


title: What would happen in the final days of the universe?

text: I would like to know the stages of how the universe would end and what would happen and what the possible scenarios are. I understand that eventually all the stars would burn out and that would introduce a big freeze that would make it harder to sustain life. Forgive me if this is incorrect as I do not have any expertise in this area, I would appreciate any information on this.


title: Internal energy and photon absorption

text: I just wish to confirm whether my understanding is correct. I know that photon absorption/emission brings about quantised changes in electron energy levels. Photons (infrared) also interact with chemical bonds to bring about quantised changes in rotational and vibrational kinetic energy in greenhouse gases for example. My understanding of internal energy is that it comprises vibrational, rotational, translational and potential energy. What constituent of internal energy does an electron excitation represent? Is it only lower frequency photons that interact with chemical bonds?


title: Why do stickers curl?

text: When peeling a sticker off its base, the immediate reaction is that it curls; why is this? I am having trouble finding an answer to this. Could it be that the glued side expands upon contact with the air? And, on a somewhat related note, is this a similar reason for why a ribbon curls when glided over with a blade?


title: A moving inclined plane with block problem

text: I'm trying to solve for a challenging problem : a moving inclined plane with a block ![picture and free body diagrams](http://i.stack.imgur.com/OUAsX.jpg) I want to solve for the acceleration components for the block and the plane and the normal force acting on the block. Let $O=(0,0)$ be an external origin, $h$ be the upper left height of the inclined plane, $x_1$ be the x-position of the center of gravity of the inclined plane, $x_2$ be the x-postion of the center of gravity of the block, $y$ be the y-position of the center of gravity of the block, $m_1$ be the mass of the plane. Let m2 be the mass of the block, $\mu_1$ be the coeffiction of kinetic friction between the bottom of the inclined plane and the level surface, $\mu_2$ be the coeffiction of kinetic friction between the block and the upper surface of the inclined plane, $\theta$ be the angle of the plane with the horizontal and $F_p$ a force applied to the inclined plane. With those defined variables, I make two separable free body diagrams for the block and for the inclined plane, indicating all of the external forces acting on each. It then comes those two vectorial equations : Block : $$ m_2\vec{a_2}=\vec{W}_{weight-of-block}+\vec{F}_{plan-acting-on- block}+\vec{F}_{friction-from-plan-to-block}+\vec{N}_{normal-from-plan-to- block} $$ Plane : $$ m_1\vec{a_1}=\vec{W}_{weight-of-plane}+\vec{F}_{pushing- force}+\vec{F}_{block-acting-on-plane}+\vec{F}_{friction-from-level-to- plan}+\vec{N}_{normal-from-level-to-plane}+\vec{F}_{friction-from-block-to- plane}+\vec{N}_{normal-from-block-to-plane} $$ I am quite not sure whether I should include the $$\vec{F}_{friction-from- block-to-plane}$$ and the $$\vec{N}_{normal-from-block}$$ to plane into the plane's acceleration calculation. Am I right ? I notice that from the geometry of the figure, I can write down the relation : $$tan(\theta)=\frac{h-y}{x_2-x_1}$$ This implies the relation : $$-a_{2y}=tan(\theta)(a_{2x}-a_{1x}) \tag{1}$$ Writing down the equations for the x- and y- components of the accelerations of the block and of the plane , this yields : $$ m_2a_{2x}=m_1 \sqrt{a_{1x}^2+a_{1y}^2} cos(\theta) - \mu_2 N_2 sin(\theta) +N_2 cos(\theta)\tag{2} $$ $$m_2a_{2y}= m_2 g+m_1 \sqrt{a_{1x}^2+a_{1y}^2} sin(\theta) + \mu_2 N_2 cos(\theta) +N_2 sin(\theta)\tag{3}$$ $$m_1a_{1x}=F_p - m_2 \sqrt{a_{2x}^2+a_{2y}^2} sin(\theta) - \mu_2 N_2 cos(\theta) - N_2 sin(\theta)\tag{4}$$ $$m_1a_{1y}=-m_1 g - m_2 \sqrt{a_{2x}^2+a_{2y}^2} cos(\theta) - \mu_1 N_1 + N_1 + \mu_2 N_2 sin(\theta) - N_2 cos(\theta)\tag{5}$$ Since $$N_1=m_1 g$$, equation 5 becomes : $$m_1a_{1y}= - m_2 \sqrt{a_{2x}^2+a_{2y}^2} cos(\theta) - \mu_1 N_1 + \mu_2 N_2 sin(\theta) - N_2 cos(\theta)$$ I am confused at this stage because it should be $$a_{1y}=0$$, that is to say, the plane remains at the ground level surface. Where am I wrong ? Does this comes from my previous question ? I want to solve this problem with Maple and plot the solutions.


title: Topology needed for Differential Geometry

text: I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from the Internet on topological spaces, connectedness, compactness, metric, quotient Hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge. Could you please also suggest a good differential geometry books that covers diff. geom. needed in physics in sufficient detail, but not too mathematical? I heard some names such as Nakahara, Fecko, Spivak. How are these?


title: Non-relativistic spinors

text: Even in a non-relativistic theory Spinors can arise as irreducible representations of the rotation subgroup of the symmetries of the theory. Why do people then put so much emphasize on the role of Relativity and the Dirac's equation in the topic? Is it just for historical reasons?


title: Would a fast inter-stellar spaceship benefit from an aerodynamic shape?

text: Some (generous) assumptions:   * We have a spaceship that can reach a reasonable fraction of light speed.   * The ship is able to withstand the high energies of matter impacting at that speed. Given the amount of matter in inter-stellar space, at high speed, would it encounter enough of it and frequently enough that an aerodynamic shape would significantly reduce its drag (and thus save fuel)?


title: Normal force on varying mass

text: > A cart filled with a mass $m_0$ of sand leaks it at a constant rate of $\mu > \ \mathrm{kg/s}$ through a hole at the bottom of the cart. Suppose that the > cart is moving on rails. Find the normal reaction force from the rails as a > function of time. Ignore the mass of the empty cart. Does the function $\mathbf{N}(t) = (m_0 - \mu t) g\hat{\mathbf{j}} $ really work? Doesn't the sand leaving through the hole give an upthrust to the cart, to reduce the effective normal weight? Note that the sand gains momentum in the $y$-axis. I'm feeling confused at this point.


title: How to apply Guass Law to Voltages

text: So, I know $\oint E\centerdot dA = 4\pi Q_{enc}$ I'm trying to solve for a TEM mode with two concentric (infinite) cylindrical wave guides of radius a and b, $a<b$. I know that for TEM modes, I can solve by assuming that the outside and inside are at two different potentials, $\pm V$. I'm told the solution is $\vec E=a\sqrt {\mu/\epsilon}H_0\hat r/r$. It seems to me that the solutions found $\vec H$ first, and then found $\vec E$. I should be able to do the reverse, and end up with the same answer. So, my question is, how can I apply Gauss' Law in this situation? Or, is there simply a better way to solve for $\vec E $ and $\vec H$?


title: Presence Of an Another Universe

text: > **Possible Duplicate:**   >  Experimental evidence for parallel universes Is there an universe similar to ours somewhere else, I mean I had heard from Einsteins theories that,actually when I am typing this question here, there is an another world somewhere else where I would be typing this question simultaneously. Is this true, if so any evidences.


title: Rainbow around Sun

text: From the perspective of a person, a rainbow is formed when the Sun is behind the person, and there is a critical angle made by the rainbow. However, on several occasions, usually at noon when the Sun is higher, I saw a ring around the Sun made of the colors of the rainbow. Is that a rainbow? Is within the definition of a rainbow? And how is it possible?


title: Why does a system try to minimize potential energy?

text: In mechanics problems, especially one-dimensional ones, we talk about how a particle goes in a direction to minimize potential energy. This is easy to see when we use cartesian coordinates: For example, $-\frac{dU}{dx}=F$ (or in the multidimensional case, the gradient), so the force will go in the direction of minimizing the potential energy. However, it becomes less clear in other cases. For example, I read a problem that involved a ball attached to a pivot, so it could only rotate. It was then claimed that the ball would rotate towards minimal potential energy, however $-\frac{dU}{d\theta} \neq F$! I think in this case it might be equal to torque, which would make their reasoning correct, but it seems like regardless of the degrees of freedom of the problem, it is always assumed that the forces act in a way such that the potential energy is minimized. Could someone give a good explanation for why this is? Edit: I should note that I typed this in google and found this page. where it states that minimizing potential energy and increasing heat increases entropy. For one, this isn't really an explanation because it doesn't state why it increases entropy. Also, if possible, I would like an explanation that doesn't involve entropy. But if it is impossible to make a rigorous argument that doesn't involve entropy then using entropy is fine. As a side note, how does this relate to Hamilton's Principle?


title: Measuring the mass using angular velocities

text: I have 2 objects which are intially connected together, $O_1$ and $O_2$. When they are connected together, they have a rotation rate about their center of mass of $w_1$. $O_2$ is cleanly released from the connected system, and $O_1$ is now rotating at a rate of $w_2$. Given that any mass properties of the system can be measured, and that the mass of $O_2$ is known, how can one find the mass of $O_1$, given that this is the one property of the system which cannot be directly measured? Also, what will happen to $O_2$ after release? One can assume that there is absolutely no friction/resistance.


title: Velocity distribution in Plummer's models and others mass distributions

text: The Plummer's sphere is an model for the mass density in a globular cluster of stars. For an $N$-body simulation I have initialized the position of $N$ masses with a Monte-Carlo technique but cannot find a way of initializing the velocity initial conditions. Is there a simple function that given a position in a Plummers sphere assigns a velocity to a given mass? Lots of sites list the velocity for a circular orbit but is this a good approximation to a globular cluster and how should it be treated off the $x$-$y$ plane?


title: Why is work not related to velocity?

text: A very simple question; why does it cost me more energy to **very slowly** lift a mass $m$ over my head compared to very fast? The definition of work does not state anything about velocity, only the distance travelled, but I definitely feel more exhausted in the first case.


title: Will a rocket produce more thrust if fired in air, rather than vacuum?

text: I've been in an argument with a friend. He claims that when a rocket engine is fired in air, it get significantly more thrust due to the rocket pushing gas into the atmosphere, and the atmosphere expands (due to there suddenly being more "air" in the same area of space), and that expansion helps to propel the rocket forwards (or upwards, whatever). I claim that the only force that acts as thrust to the rocket is the response force from firing a large mass of gas out of the back, and so the gas exerts the same force on the opposite direction (namely, forwards of upwards), which propels the rocket. * * * He further claims that a theoretical rocket in honey, would fly much better than it would in vaccum. I claim that the resistance would nullify and theoretical benefit to thrust. Which of us is closer to "the truth"? Do you get


title: Universe in the magnetic spectrum

text: Can we scan the universe in magnetic spectrum and then separate the paramagnetic , diamagnetic and ferromagnetic objects and get a new picture of the cosmos.


title: QFT's that have no action

text: What does it mean to have a QFT that can not be encoded by an action. What is by far the most powerful approach of study in such a case. What is the best studied physical theory that falls into this category. What is QFT? I think it is a set of rules that facilitate descriptions of things we can observe, but what is the most mathematically accurate way of capturing what it is? I have read of cases where the path integral approach fails what does this mean? Is there any geometric structure whose properties describe the space of all posible QFT's? What happens when you can't use an action?