title: The pressure in a water spout and Bernoulli's equation

text: This is a conceptual question about the application of Bernoulli's equation to a water spout. There is a classical problem found in many physics texts which goes something like "you have a garden hose with a nozzle which flares inward so the radius is smaller at the end. How high does the water shoot into the air?" So there are obviously details for the exact problem (like what is the angle of the hose, pressure or velocity, etc) but I am specifically interested in the applicability of Bernoulli's equation to the water which has left the hose. It would seem to be that after the water has left the hose, it is no longer satisfying the conditions for Bernoulli's equation. I can't quite put my finger on why; I can't exactly see the the flow is not laminar but since the pressure outside the flow (the air) is certainly not at the same pressure as the water, it's certainly doesn't seem to be steady. In my understanding, you would treat the water in the nozzle with Bernoulli's equation (or continuity, depending on exactly conditions) and then simply treat the water as droplets acted on by gravity. If that's true, can someone clarify exactly what conditions of Bernoulli's equation are being violated? Alternatively, if I am wrong, can you convince me that the water can still be considered a "fluid" for the purposes of applying Bernoulli's equation? EDIT: A specific example in response to a comment. This shows that assuming Bernoulli still applies is equivalent to assuming the pressure of the stream is the same as the atmospheric pressure. Vertical oil pipe of height $h_1$, oil (gauge) pressure at its base of $P$ and fluid velocity $v$. How high into the air down the oil shoot? Solution 1) Bernoulli fails when the fluid exits the pipe. The pressure when the fluid leaves is equal to 0 (gauge) so we have Bernoulli's equation at the top of the pipe $$P+\frac{1}{2}\rho v^2=\frac{1}{2}\rho v'^2+\rho g h_1\longrightarrow v'^2=v^2+\frac{2P}{\rho}-2gh_1$$ Then using energy conservation we have $$mgh_1+\frac{1}{2}mv'^2=mgh_2\longrightarrow h_2=h_1+\frac{v'^2}{2g}$$ $$h_2=h_1+\frac{1}{2g}\left( v^2+\frac{2P}{\rho}-2gh_1 \right)=\frac{v^2}{2g}+\frac{P}{g\rho}$$. Solution 2) Bernoulli holds throughout the motion. The (gauge) pressure at the very top (and throughout the entire spout) is zero, so Bernoulli gives us $$P+\frac{1}{2}\rho v^2=\rho g h_2\longrightarrow h_2=\frac{P}{g\rho}+\frac{v^2}{2g}$$


title: Laws of Motion: Acceleration to be applied on a free falling object to reduce velocity to 0

text: Assume that an object of mass 'm' is falling to the earth.      The force acting on the same would be F = m*g = 10m (assuming g = 10m/s^2).   In this case the velocity of the object at time 't' = 10 seconds will be 100 m/s (v = u + at).   At this point, if an opposing force of F' = -10m is applied on the falling object, how long would it take for the velocity of the falling object to reduce to 0?      Would it be valid to say that the opposing force of F' = -10m has to be applied for time 't' = 10 seconds to reduce the velocity of the falling object to 0?


title: Abstract, generic derivations of energy

text: How generic can be derivation of energy? In a system with gravity and masses – it is potential energy and kinetic energy. What if a constraint would be specified that no mass and velocity should be allowed (or possible) to use? For example, suppose there is a system of massless particles. Is there a way to define and use energy in such system? If yes, then how? In general, I am dealing with abstract systems, where no typical gravity and mass occur. So it is hard for me to use potential and kinetic energy. Yet, I think it is possible to define an energy in such system, a kind of intensity of a process. Is there such thing defined in physics? What are most abstract derivations of energy?


title: Why is Gleason's Theorem not enough to obtain Born Rule in Many Worlds Interpretation?

text: The Many Worlds interpretation suffer from at least 2 "wounds", the preferred basis issue and perhaps the most notorious probability issue. How do you make sense of probability in a model where everything happens? There are all these elaborate attempts at deriving Born Rule by Wallace et al. and at least 20 different people at arXiv. And there are some very good papers that provide criticism for these attempts, but my question is why isn't Gleason's Theorem enough?


title: Acceleration: Value Disparity?

text: If we consider a ball moving at an acceleration of $5ms^{-2}$, over a time of 4 seconds, the distance covered by the ball in the first second is $5m$. In the 2nd second will $5 + 5 = 10m$. In the third second it will cover a distance of $5 + 5 + 5 = 15m$ and so on and so forth. Now, when we substitute this answer in the equations of motion derived from the area under velocity- time and distance-time graphs, we see a variation: $$s = 1/2at^2$$ $$ = 1/2*5*4^2$$ $$ = 1/2*5*16$$ = 40 metres is the distance covered. Now if we go back to our initial description of acceleration we that in the 1st sec = 5m 2nd sec = 10 m 3rd sec = 15 sec 4 sec = 20 sec. Total distance covered in this case is 5 + 10 + 15 + 20 = 50 metres? 40 != 50? Why this disparity between the values? Can someone please explain?!


title: Is there a limit to frames per second to be captured?

text: Today we see ~100,000 fps cameras for hyper slo-mo. Is there a physical limit for frames per second for a camera? Speed of light maybe? How?


title: Simple circuit with resistances

text: I have a question about a circuit that I made in class! So, we had a circuit with 2 resistors, one of $R_1 = 330 \Omega$ and one of $R_2 = 470 \Omega$. The tension applied to the circuit should be $10V$ but we applied $9.99V(V_f)$. When measuring the resistances with a multimeter the value for the $R_1$ was $328 \Omega$ and for the $R_2$ was $470 \Omega$. The value for both $R_1$ and $R_2$ in series was $R_1 + R_2 = 799 \Omega$. The tension measured in $R_1$ was $4.02V(V_{r1})$ and in $R_2$ was $5.74V(V_{r2})$ The intensity of the current ($I$) was $12.34 mA$ With that said, we were asked to calculate $R_1$, $R_2$ and $R_1+R_2$ theoretically using the values of $V_{r1}$, $V_{r2}$ and $V_f$. I used Ohm Law: $R = \frac VI$ $R_1 = 326 \Omega$ and $R_2 = 465 \Omega$ **My question is: should be $R_1+R_2$ be calculate using $V_f (9.99/0.01234)$ as it was asked or calculating an equivalent resistance? By the first method the result is $810$ and be the second is $791$. The ammeter's resistance is $11 \Omega$ at the used scale $(20mA)$.** **I'm asking cause I found it weird that $810$ is bigger than $799$(experimental value) given that the experimental should be bigger because of ammeter's resistance.** Sorry if the question is dumb, but I'm not a Physics student though I attend Physics classes.


title: Can the mass of a SUSY particle depend on the process it participates in?

text: I believe that mass is property of every particle,as well as spin etc.Now I'm interested in SUSY particles in cMSSM model.Can it be,that mass of a SUSY particle (at one point in five parameter space) depend on process it is participating in,or parents particles? I doubt it.


title: Action principle for the Electromagnetism and Gravity

text: Here is the formula for the stress energy tensor: $$ T_{\mu\nu} = - {2\over\sqrt{ |\det g| }}{\delta S_{EM}\over \delta g^{\mu\nu}} $$ (This follows from varying the total action $S = S_H + S_{EM}$, where $S_H={c^4\over 16\pi G} \int R \sqrt{ |\det g_{\mu\nu}| } d^4 x$ is the Hilbert action and it gives the Einstein's equations, and $S_{EM}$ are other terms in the Lagrangian, that contribute to the right hand side of the Einstein's equations in form of the $T_{\mu\nu}$ above.) The Lagrangian of the electromagnetic field is: $$ S_{EM1} = -\int {1\over 4\mu_0} F_{\alpha\beta} F^{\alpha\beta} \sqrt{ |\det g| } d^4 x $$ and using the formula above, we get for the stress energy tensor: $$ T_{\mu\nu} = {1\over \mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right) $$ which is the correct elmag. stress energy tensor. However, the interaction part of the elmag. Lagrangian is $$ S_{EM2} = -\int j_\mu A^\mu \sqrt{ |\det g| } d^4 x $$ and if we interpret this as a function of $g^{\mu\nu}$ it would also contribute to the stress energy tensor like this: $$ \delta S_{EM2} = -\delta\int j_\mu A^\mu \sqrt{ |\det g| } d^4 x =-\int \delta (g^{\mu\nu} j_\mu A_\nu \sqrt{ |\det g| }) d^4 x = $$ $$ =-\int (\delta g^{\mu\nu}) (j_\mu \sqrt{ |\det g| })) A_\nu d^4 x $$ where the $j_\mu \sqrt{ |\det g| }$ is treated as the current _density_ (and thus not depending on $g^{\mu\nu}$ when varying), however, clearly the stress energy tensor corresponding to this would be: $$ T_{\mu\nu} = 2 j_\mu A_\nu $$ (Possibly only the symmetric part contributes, because the antisymmetric part cancels with $g^{\mu\nu}$, so we would get $T_{\mu\nu} = j_\mu A_\nu + j_\nu A_\mu$.) In either case, such terms should then appear on the right hand side of the Einstein's equations. However, I don't think that this is correct. Does anybody know what is wrong here?


title: Guessing what a simple partial differential equation is describing physically

text: Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this: $\partial_tA=C_3\partial^2_{xx}A+C_2\partial_x\left(vA\right)+C_1A+C_0$ From inspecting it and doing some searching, it seemed like it might be trying to describe a phenomena that had advection and diffusion occurring at the same time. Is this a correct interpretation? Note that $C_3>0$, $C_2<0$, $C_1<0$, $C_0>0$ and $v(x,t)$ is velocity. I'm not sure, but it seems reasonable to assume velocity can be either negative or positive. Also, $A(x,t)$ should be positive. Note that I'm not looking for a solution to the equation, just the physical interpretation of what kind of physics would be involved.


title: Effect of rotation on turbulence threshold for Reynolds number?

text: If the significance of the Reynolds number is: ![Total Momentum Transfer / Molecular Momentum Transfer](http://upload.wikimedia.org/wikipedia/en/math/c/9/a/c9a9472f2f2cc9e56b7d0b295734882e.png) Then what is the effect of angular momentum on the transition from laminar to turbulent as in a convective vortex? Waterspouts, in particular, seem to be obvious cases where a very large Rayleigh Number would be assigned, yet the systems are admitted by observational experts to be dominated by laminar flow!


title: What is the physical sense of the transition dipole moment?

text: So if the states are the same we achieve the expectation value of the dipole moment for a given state. I mean $ \langle \mathbf{\mu} \rangle = \langle \psi \vert \hat{\mathbf{\mu}} \vert \psi \rangle$ But I don't feel the physical sense in the case of transition dipole moment when psi-functions on both sides are different $\langle \psi_{1} \vert \hat{\mathbf{\mu}} \vert \psi_{2} \rangle$ Help me to understand, please.


What's the meaning of negative accuracy for measurements of physical quantities? Can measured values of a physical quantity ever have a negative accuracy? I read some materials about accuracy and am still confused. The Wikipedia article http://en.wikipedia.org/wiki/Accuracy_and_precision explains the accuracy as defined for interpreting observed values of a random variable which has certain probability distribution. I am not sure how much the interpretation is applicable to measurement of physical quantities as probability isn't necessarily a well-defined physical quantity.


title: What makes now special?

text: I recently came across the paper Relation between the psychological and thermodynamic arrows of time (arXiv). Their argument makes sense to me, however their concept doesn't seem to address what 'now' is. If each person exists in an unchanging state in the block universe, and the flow of time is an illusion caused by the way memory functions, then how exactly does this illusory sensation of 'now' work? Are there an infinite number of me's at every point in time, all experiencing their own now? Or is there one me that 'travels' though the block universe, experiencing the illusion of time? And what is it that is experiencing this illusion of movement?


title: Electromotive force

text: EMF is defined as: **"The potential difference across the terminals of a battery or dry cell when it is giving no current to the circuit."** So, if current starts to flow from the battery, will the potential difference across the ends of the battery not called EMF....?


title: Angular momentum confusion

text: Could somebody please explain what is going on here? We have a system of two indistinguishable spin-1 bosons. We shall adopt the center of mass frame. Let $S$ = total spin   $L$ = relative orbital angular momentum   $J$ = $L+S$ = total angular momentum Prove that $J = 2m$ where $m$ is an integer. If given that $J=1$, what are the permissible $(L,S)$ pairs? I am lost with this. I have managed to show that for the states $S=0, S=2$ interchanging particles $1,2$ is symmetric, whereas it is antisymmetric for $S=1$. However, I don't know how to use this here. What is the significance of the CoM frame choice?


title: What interaction does a Fayet-Iliopoulos coupling parameterize the strength of?

text: Couplings give the strength of interactions. They appear in a Lagrangian as a coefficient to the two (or more) interacting fields. Unfortunately, I'm struggling to find a source that describes what it is that the Fayet- Iliopoulos coupling actually couples. As far as I know, it is simply the coefficient (given by the greek letter 'xi') in the Fayet-Iliopoulos D-term. The D-term is just one of the terms that appears in the expansion of a superfield and doesn't describe an interaction. My apologies if the answer is obvious.


title: Meaning of the word "Moment"?

text: This question is more of a question about the origin of a physical term moment used in many contexts. My question is about the linguistic or historical meaning of the word "moment". Please don't provide the mathematical equations for the moment of inertia or the electric of magnetic dipole moment. My question is about the essence of the word. I'm asking this question because as a non-native English speaker, I find it confusing as how did this word got to be used in the different contexts.


title: What is the cause of strain hardening and strain softening?

text: Why is it, that - if displayed in a strain/stress-diagram - the curve will at first be linear, and then change into a "curved shape", where it will first decrease again, and only later increase again? Is there any tendency, for which materials this holds true and for which it doesn't? Can I identify these by certain characteristics. As an example graph, please consider this graph: ![sample stress-strain curve](http://i.stack.imgur.com/3aBj6.gif)


title: Can we safely assume $\Psi(x,t) = \psi(x)e^{-i\omega t}$ always in QM?

text: In the particle in a box, harmonic oscillator and in Hydrogen Atom, we can safely assume $$\Psi(x,t) = \psi(x)e^{-i\omega t}.$$ So why not make it a postulate to consider the wave function to be always in the form $$\psi(x)e^{-i\omega t}$$ We can still explain all the foundational experiments, so my question is why not make this amendment to the theory so that we can avoid a lot of nightmares and some peace of mind. Why unnecessarily consider so general thing when we can do away with simple things most of the time? Oh my actual question is where do we run into problems (practical physical situations/experiments) if we consider such a thing? If we run into problem anywhere, I hope to make any other simple changes to fix them but without going back on this one.


title: Multimeter has a resistance

text: If a multimeter has a resistance (1M ohm, say) when measuring voltages how do I take that into account in my error?


title: Computer-Generated Holograms: I'm completely lost. How are they physically implemented?

text: I have been reading about holography, and I think I understand the general concept, but one thing that has me completely lost is how computer generated holography works in practice. I think I get the basic idea behind how CGHs work. If we were to take a 3D object, like a Utah teapot, we could emulate the behaviour of an actual laser beam bouncing off the teapot and interfering with itself, thus forming the hologram. Now, here's where I'm confused: I've read about printing holograms (as in, with a regular printer), recording actual holograms on CCDs, patterning a holographic plate with the fringes using an LCD, and even holographic displays. What I don't get at all is how this is even vaguely possible? Aren't the interference fringes which make up the hologram _much_ smaller than wavelength of light? Even if we had LCDs with massive resolution, wouldn't the diffraction limit prevent the using of them to pattern the plate, in the same way that visible light photolithography is nearing its physical limitations in the microfabrication? Basically, I've never seen a straight forward explanation of how computer holograms are actually transferred to the physical recording medium. As far as I know, it is possible, because there are companies currently doing it (such as Zebra Imaging). However, reading over patents and other papers in the literature yielded no clear understanding of how this really works, most authors seems to gloss over the implementation, and often seemingly contradict themselves. It was my understanding that one needed an electron microscope to actually make out the fringes because they are so small. If this is the case, why does one not need an electron microscope to etch the fringes?


title: Defining Measurement in Quantum Mechanics

text: I should begin by saying that _I am a total newbie when it comes to Quantum Mechanics._ Therefore my question might sound metaphysical to people who know their stuff. So please forgive. What I am trying to understand is the basic system of an electron in a box. Suppose I place a detector at a fixed point inside and try to detect the presence of an electron. There are two possibilities: (1) The detector detects an electron. The Copenhagen interpretation explains this by saying that the act of measurement forced the wave function to collapse at that point. (2) The detector did not detect anything. Would this "not detecting an electron" qualify as a true measurement in the Copenhagen sense? I mean, if this qualified as a measurement would it change the wave function? What led me to this confusion is the explanation in _Griffiths' Quantum Mechanics._ He states that the same experiment when done soon after possibility (1) gives us the same result i.e, observe the electron (which does not confuse me.) But what happens when you do the experiment soon after possibility (2)? Would I still not observe the electron? What if I extend this to the two-slit experiment and place the detector at only one slit. If I didn't observe the electron, it means that it passed through the other hole, which doesn't lead to the pattern. So again, does this make "not detecting the electron" a true observation which collapses the wave function?


title: Quantum tunneling is faster than light travel?

text: Quantum tunneling is faster than light travel ? My reasoning is that the particle cannot be detected inside the tunnel so if it travels from A to B it must be instantly going from A to B , hence faster than light travel ? This seems legit for the particle interpretation. And also for waves imho. Some people mention the uncertainty principle but I do not see how that explains it.


title: Confused about the Photoelectric Effect?

text: If an incident photon with frequency **_f_** hits a metal with a work force of **_w_** : How do I find the color of the incident photon and the threshold frequency of the metal? I missed out on the days in my physics class where they covered work force and stopping voltage, so what equations are necessary to find the color and threshold frequency? I know this is probably an easy question, but if you could please show me the steps with an explanation, that would really help me prepare for my test on Monday.


title: Discretization of action in path integral

text: I am reading Peskin and Schroeder (path integrals) and it states that discretising the classical action gives: $$S~=~\int \left(\frac{m}{2}\dot{x}^{2}-V(x)\right) dt ~\rightarrow~ \sum \left[\frac{m}{2}\frac{(x_{k+1}-x_{k})^{2}}{\epsilon}-\epsilon V(\frac{x_{k+1}+x_{k}}{2})\right] $$ Is it convention that $$\dot{x} ~\rightarrow~ (x_{k+1}-x_{k})/2$$ and $$x~\rightarrow~ (x_{k+1}+x_{k})/2~?$$ I understand that $\epsilon$ is effectively $dt$, but I am not sure why $x$ behaves like this.


title: Color of an incident photon?

text: If the incident light at 360nm causes photoemission of electrons, wouldn't the color be ultraviolet? I know that really isn't a color, but that's what my chart of the light spectrum says. Unless I am misinterpreting it, which I might be. What's also mentioned in the question was that it has a stoping voltage, although I don't think that affects it.


title: Hamiltonian matrix propertu

text: A professor made an statement to prove the variational theorem: Because the Hamiltonian (H operator of quantum physics) is diagonal in its own eigenfunction, the terms in $\left \langle \Phi _{m} \left |\widehat{ H} \right |\Phi _{n}\right \rangle$ reduce to $E_{n}\delta _{mn}$ given $\widehat{H}|\Phi _{n}>=E_{n}|\Phi _{n}>$ . This is not obvious to me, does anyone have a proof that the matrix of the hamiltonian is diagonal? Please assume that I do not even know about properties of a Hermitian matrix (beginner). Thank you for your help.


title: Will a rotating object traveling along a linear path continue to rotate forever in space?

text: Suppose engineers built a cylinder-shaped space ship which rotates to create artificial gravity:                 __________________      /<-b \             \     /      \             \  -----> a     \      /             /      \_b->/_____________/      The space ship uses a rocket to travel through space in the direction of `a`. Smaller nozzles ensure that the space ship rotates in in the direction indicated by `b`.   * If the space ship wants to travel in a straight line, without stopping, would the astronauts need to frequently turn on the smaller nozzles to keep the space ship rotating at the proper speed or would the space ship constantly rotate at the same speed forever?   * If the space ship was constantly rotated counter-clockwise to create artificial gravity, would this rotation cause the space ship to drift off course?


title: Is scattering cross section determined by mass?

text: This is just for visible light/photon scattering. Let's say I have small spherical particles in water, if I keep the weight concentration of glass the same but change the diameter of glass sphere from 0.5$\mu$m to 1$\mu$m, how will the scattering power, or the total scattering cross section change for visible light, e.g., at 632nm? I was told as long as the mass doesn't change, the scattering cross section won't change, the size of diameter only modifies the phase function of the scatter. is that true?


title: If a space ship accelerated constantly, would its astronauts constantly feel the forward movement?

text: I know that if a space ship suddenly traveled very fast, its astronauts would be fly against the back wall, potentially getting hurt. If the space ship suddenly stopped, they would also fly against the front wall. Suppose that a space ship could accelerate at a constant rate, e.g.:               Time    | Speed     ------------------     1:00 pm | stopped     1:10 pm | 100 Kph     1:20 pm | 200 Kph     1:30 pm | 300 Kph        * As the space ship is accelerating at a constant rate, would the astronauts constantly feel the forward motion of the craft for the duration of their journey?   * Would the astronauts begin to feel that the motion is becoming faster and faster?   * If the space ship continued to accelerate in this manner for many hours, would the astronauts eventually be injured by the great speed?


title: Pareto efficiency and Standard Model parameters

text: Pareto Efficiency is a well understood concept in economics, which basically is a condition where no one actor could be made better off without some one being made worse off. This condition allows one to define a Pareto frontier of distributions that are Pareto Efficient of Pareto Optimal. The question I have is whether there are similar measures or statements about the efficiency of Standard Model parameters? What references might be available? Update: If we say we are saying the parameters are optimized for life, how is that term proposed to be defined? My thoughts this morning were that at the heart of the anthropic principle there really is a statement of information preservation. So if a change in parameters leads to a loss of global information, it seems that might be a good place to start. If there are reasons why this is infeasible or nonsensical, those would be good answers.


title: What is the relationship between the radius of a ring in a rotating space station and the strength of the artificial gravity generated?

text: Suppose engineers built a rotating space station similar to Space Station V from the film _2011: A Space Odyssey_ , but with multiple concentric rings where astronauts can live.                   ____________        /  ____c___  \       /  /  __b_  \  \      /  /  /  a \  \  \     /  /  /      \  \  \     \  \  \      /  /  /      \  \  \____/  /  /       \  \________/  /        \____________/      Ring `a` is 10 meters from the center, ring `b` is 20 meters from the center, and ring `c` is 30 meters from the center.   * If the station is set to rotate at a speed where the gravity within ring `b` is identical to Earth's gravity and the rings rotated together, would the gravity in rings `a` and `c` be significantly stronger or weaker?   * If the concentric rings could rotate freely, would rings `a` and `c` need to rotate faster or slower than ring `b` to ensure that the gravity at each ring had identical strength?


title: Thought Experiment: Force on magnets in a Stern Gerlach Experiment

text: Background: In the SG experiment, an inhomogenous magnetic field affects a force on particles passing between two magnets. "Measurement" takes place when a screen is placed on one end, blocking one of the states. The final measurement we perform (spin state of particles exiting) is heavily dependent on whether we perform a measurement or not, whether by blocking particles exiting the device or by blocking particles by setting up a screen within a modified SG device ![Modified SG device](http://i.stack.imgur.com/motIz.png) The interaction between the particles and magnets should also exert a force on the magnets, causing them to move slightly. The motion of these magnets seems to not affect the outcome of the experiment significantly, yet that would imply that the motion of the magnets could be measured without affecting the outcome of the experiment. This seems at odds with Heisenberg's uncertainty principle, that we are getting information for free. My Professor tells me that the force exerted on the magnets by the particles is so minuscule that it cannot be measured (I assume it is similar to how we can't measure particles' momentum and position simultaneously, because any photons we shoot at the particle will end up imparting change?) What's going on here? I wanted to double check here and see where I went wrong with my assumptions.


title: Young and freedman vs resnick halliday

text: I am a grade 12 student with a very basic knowledge of calculus. Which is a better book - fundamentals of physics by resnick halliday or university physics by young and freedman? For understanding the concepts well and also for the quality of problems and proper tips to solve them


title: Do I need to take elastic potential energy into account?

text: > Let's say I have a vertical spring with end points $A$ and $B$ and length > $a$ and a mass attached to the endpoint $B$. The mass is dropped from the > point $A$ and I need to find the kinetic energy of the mass at a point $ak$ > where $k>1$. Do I need to include the Elastic Potential Energy to find the kinetic energy? If yes, why can't I just use the Kinetic Energy and Gravitational Potential Energy? I know $\text{Work Done}=mgka$ by the particle and the change in GPE is $mg(ka-a)$ but I don't get what else am I missing?


title: Laser beam shaping

text: Hi I like to build a laser pointer that reflects different shapes for a project I am working on. I know this can be done with Diffractive Optical Elements. Are there other options? DOEs are expensive. I see all these laser pointers with different shapes (arrow, hand etc), I am suspecting there is a lower cost but lower performance way. Any pointers?


title: Is it possible to reproduce Double-slit experiment by myself at home?

text: I want to reproduce this experiment by myself. What I need for this. What parameters of slits and laser/another light source it needs? Is it possible to make DIY-detector?


title: Why is an electric and magnetic field present in all universe?

text: the Earth has its fields due to the motion of molten metals in the core but how does this whole universe has its fields without any discontinuity? And we can say for sure that fields are present in whole universe as we can see light traveling thousands of light year and as light is an EM wave it can't travel without the presence of fields.


title: Is there more to an electron/photon, other than the equations that describe their measurable properties?

text: Could non-linear models of quantum physics add more to our understanding?


title: Field energy of/from virtual Photons

text: I have a slightly out-of line question: Consider a single electron (or it's current if you please) The STATIC ELECTROMAGNETIC field surrounding it will (no doubt) have a field energy (T) to go with. The standard description of interaction is by exchanging a virtual photon. (For simplicity suppose, that only one virtual photon (of arbitrary momentum) can be exchanged) Question: Is there any way to express the field energy "in terms of" virtual photons??? Line A: Every virtual photon has energy-momentum. The EM field energy-momentum is some weighted sum over all virtual photons. Line B: A photons energy-momentum is very different from the EM field energy, because $T \simeq F_{\mu\nu}F^{\mu\nu}$. But what can be said about the EM T of a single photon? Line C: Your choice NOTE: Please note that I own a PhD in physics, so you can answer on any level you like.


title: Observations used to derive orbital and timing information required for GPS satellites

text: Although I have studied GPS receiver position finding procedure in some detail, I am struggling to find enough information to allow me to deduce how the essential highly accurate orbital and time information required for each GPS satellite is actually obtained. I can only surmise that the information is somehow slowly and iteratively derived, for each satellite, from all the satellite signals received (probably receiver time-tagged) by the earthbound GPS monitoring stations, that the position information is resolved in an earth-centred inertial frame-of-reference (ECIF) and that the time information (in the ECIF) is synchronized to that from the US Navy master time clock value referred to the earth's geoid. I realise that relativistic corrections are also required at the receiver for orbit ellipticity and to refer the satellite positions to the earth-centred, earth-fixed (ECEF) WGS84 frame of reference. I am particularly wondering whether any other types of observations of the GPS satellites are also involved and I'm hoping someone could kindly give me a rigorous-descriptive rather than mathematical explanation.


title: Is there a simple model explaining Faraday effect?

text: I find magneto-optical effects fascinating, and especially the Faraday effect. But most sources only give a phenomenological description, while I want a deeper explanation of its mechanism. Is there a simple model that can explain the formula $\beta = \mathcal{V}Bd$? No need for a precise calculation of the Verdet constant.


title: Cosmological redshift interpretation

text: Can the cosmological redshift be interpreted as atomic frequencies increasing by the scale factor as the Universe expands? This explanation seems closer to the truth than the popular idea that a photon's wavelength somehow expands while it travels to us from a distant galaxy. Metric expansion only occurs with proper distances (between events at the same cosmological time). I think people take the standard derivation of the cosmological redshift to imply that the photon's oscillation period increases with the scale factor. But this again is equivalent to assuming its wavelength increases.


title: Invariance of a tensor under coordinate transformation

text: I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of obvious. So does the invariance of a tensor under coordinate transformation mean what I stated above or does it mean that under a set of particular transformation the representation of a particular tensor also doesn't change. Quoted from Wikipedia: > _A vector is invariant under any change of basis, so if coordinates > transform according to a transformation matrix $L$, the bases transform > according to the matrix inverse $L^{−1}$, and conversely if the coordinates > transform according to inverse $L^{−1}$, the bases transform according to > the matrix $ L$._ Can someone please shed some light on this?


title: Incompatibility Between Relativity and Quantum Mechanics

text: Why does Gravity distort space and time while the electromagnetic, strong, and weak forces do not? Does this have to do with why Quantum Mechanics and Relativity are incompatible?


title: Why do we need 12 atoms to store 1 bit of data?

text: **Recent research at IBM** has found a way to store 1 bit of data in 12 atoms. While that is a big accomplishment compared to what we have today, it does seem like a waste to a non-physics eye like me. From this figure on the same page: ![enter image description here](http://i.stack.imgur.com/8KTon.png) it looks like we can determine 1 and 0 based on the alignment of magnetic properties of 12 atoms. **But why is a smaller unit, like just one atom not good enough?**


title: Does the Banach-Tarski paradox contradict our understanding of nature?

text: Since the Banach-Tarski paradox makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this: If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics. In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right? Does this mean that the mathematical foundation of physics is flawed? (I hope this question is not too provocative!)


title: Is there any idea why the electric charges of electron and muon are equal?

text: Is there any idea explaining why the electric charges of electron and muon are equal? Edit: The total charge of a particle is proportional to the integral of its own electric field flow through the sphere of a big radius surrounding the particle at rest. The free Dirac equation describes charged fermion. It contains the mass term $m$. If $m$ tends to zero, Dirac equation tends to the pair of Weil equations that describe electrically neutral particles. Does it mean that charge somehow depends on mass? If yes, why do the electron and muon (both described by Dirac equation, but with different mass terms) have the same electric charges?


title: Statics software for structural engineering

text: I'm attempting to expand my knowledge of engineering software. I've found comsol and ansys for acoustics and thermodynamics/fluid dynamics (not necessarily in that order), now I'd like to see if I can find something for architectural statics/load calculations. Which packages are in common industry use?


title: Friction acting on a particle on a rough place

text: If a particle on a rough inclined is attached to a spring so that the spring is parallel to the inclined place, when the particle is equilbrium just because of the weight of the particle extends the spring somewhat. Does friction need to be taken into account (because it is not moving), will it be the maximum friction $\mu R$ (coefficient of friction times the reaction)? and will it be acting up or down the slope?


title: Cosmological constant (positive, dS) as SUSY/supertring issue

text: The "almost" de Sitter expansion of current cosmological model implies a puzzle for supersymmetry (SUSY) and also for string theory. A positive value of the cosmological constant implies a non-null vacuum energy that is, as far as I know, inconsistent with SUSY which has to be broken in order to get a tiny cosmological constant. Moreover, interacting theories in de Sitter spacetimes are hard to handle in current string theory due to some issues concerning the S-matrix and that field strengths become imaginary in those spacetimes. My question is: Is the currently accepted LCDM model of cosmology a challenge for SUSY/string theory? Wouldn't it be saying that SUSY and string theory are "wrong" or at least incomplete (or they will be, since we will approach a pure dS spacetime in the cosmological future it nothing changes)? Afterall, to my knowledge, de Sitter spacetimes are not well behaved in interacting field theories and thus, even in superstring M-theory. Remark: AdS/CFT is remarkably more "fit" to the SUSY/superstring idea, but it seems that the Universe has chosen the "wrong" sign for $\Lambda$!


title: Calculate magnitude of force acting on some area by falling object

text: I have simple question: is it possible to calculate magnitude of force acting on some area by falling object? Let's say I have an object with mass $5\text{ kg}$. I drop that object in height 1 meter. It potential energy was $$\begin{align} E_p &= hmg \\\ E_p &= 5(9.81) \\\ E_p &= 49.05 \text{ J} \end{align}$$ Near the ground, kinetic energy of object is $$E_k = E_p$$ So the velocity is $$\begin{align} E_k &= \sqrt{2\frac{E_p}{5}} \\\ E_k &= \sqrt {19.62} = 4.429 \frac{\mathrm{m}}{\mathrm{s}} \end{align}$$ Is there any way how to calculate force of object against ground when it falls on it? If not, what _something_ (against ground) can be calculated for this problem? **EDIT:** Let's say that ground is from glass, thickness $10\text{ cm}$. What properties would be affected?


title: Deriving work done on a solid

text: For a homework problem, I've been asked to show that the work done by increasing pressure isothermally on a solid of mass m is: $$W \approx -\frac{m\bar{\kappa}}{2\bar{\rho}}(P_f^2 - P_i^2)$$ where $\bar{\kappa}$ and $\bar{\rho}$ are, respectively, the mean compressibility and density (unchanged from $P_i$ to $P_f$). Starting with the definition of both the compressibility ($\kappa = -\frac{1}{V}(\frac{\partial V}{\partial P})_T$) and the density ($\rho = \frac{m}{V}$), you can work backwards to get: $$W \approx -\frac{m}{2}\left(-\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T \right)\left(\frac{V}{m}\right)(P_f^2 - P_i^2)$$ $$W \approx\frac{1}{2}\left(\frac{\partial V}{\partial P}\right)_T (P_f^2 - P_i^2)$$ The part about it being a solid confuses me because I'm not sure if the ideal gas law can be utilized. Is it enough to simply use the definition of work to get you the rest of the way from there: $$W = \int P\,dV = \int \frac{nRT}{V}\,dV$$ I think the multivariable calculus is getting in the way. I'm not sure how to interpret the partial derivative. Can someone help me connect the dots?


title: Newton's third law of motion, and the collapse of objects under heavier weight

text: Suppose a table is placed on earth. The weight of the table exerts a force on the earth say $10N$, then the earth will also exert a force of $10N$ on the table. If we place a truck on the table weighing $4000N$, then the total force exerted by the table and the truck is $4010N$ on the earth. The earth would also exert a force of $4010N$ on the table. Therefore, the net force on the table is $10N$ and the table colapses. Is this what happens to a table scientifically when it gets broken by something having a relatively a much greater weight? Whereas, my textbook says: > Consider a book of weight $W$ which is placed on a horizontal table, it > pushes the table with a force in downward direction in response to this > action the table also pushes the book in the upward direction the resultant > force acting the book must be zero that is, > > $F = 0 => F = 0$ > > Two forces are acting on the book, > > (a) the force of gravity acting downward which we call the weight of book. > > (b) the other force is the push of table on the book in the upward > direction. > > These two forces are equal and opposite to keep the book in the stationary > position. If this is so, then the truck should also have remained in the stationary position. However, what we normally see is that the truck will crush the table.


title: Can deterministic world view be denied by anything other than quantum mechanics

text: If we ignored quantum mechanics and looked at the world with a deterministic Newtonian view. Does not that mean that there is no randomness and that if all the information of the state of the universe during the big bang is accessible one can predict the state of the universe at any period of time and predict that I am writing this question right now. Of course something like that denies the free will but I am asking if there is any thing other than quantum mechanics that denies the deterministic world view.


How can I create hindrances to radio waves?


title: What is the direction of an accelerated particle due to a variation of magnetic field?

text: Let an ion of charge $q$ and mass $m$ traverse a circular orbit of radius $R$ under the influence of a uniform magnetic field $\mathbf{B}=B(r)\,\hat{\boldsymbol{z}}$, where $r$ is the distance from the symmetry axis and $z$ the 'height' or 'vertical' axis ( **basically a betatron** ). I'm a bit confused, I've managed to show that when the magnetic field is increased in magnitude, the induced electric field is $$E=-\frac{R}{2}\frac{d\langle{B}\rangle}{dt}\hspace{0.5in}\text{where}\hspace{0.5in}\langle{B}\rangle=\frac{1}{\pi{R}^2}\int_0^R{B}(r)\,2\pi{r}\,dr$$ i.e. $\langle{B}\rangle$ is the spatial average of $B(r)$. And hence, that the electric field introduces a force $\mathbf{F}=q\mathbf{E}$ on the ion that accelerates it as $$\mathbf{a}=-\frac{qR}{2m}\frac{d\langle{B}\rangle}{dt}\,\hat{\boldsymbol{\theta}}$$ where $\theta$ is the azimuthal coordinate. **The thing I'm having trouble with is the minus sign** , because I know that in a betatron an ion accelerates (positively, not as 'desaccelerate') when the magnetic field is increased. I thought I'd find out what the direction of the velocity of the ion was when the field did not change, and found from Lorentz force that in general for any (positive or negative) charge $q$, $$\mathbf{F}=-qvB\,\hat{\mathbf{r}}\;\Longrightarrow\;\mathbf{v}=-\mathrm{sgn}(q)v\,\hat{\boldsymbol{\theta}}$$ Now I don't know if this last expression is true, I haven't found it anywhere and the sign function seems rare; honestly I just managed, after various attempts, to make the direction of both the velocity (at unchanging $\mathbf{B}$) and the acceleration (at changing $\mathbf{B}$) fit so that the ion would increase or decrease its speed upon an increase or a decrease, respectively, of magnetic field. I'd like to know whether this is true and/or if there's a simpler or more straightforward way of showing this. Thank you in advance.


title: Proposed model for existence of bar in galaxies

text: What are the current explanations and models for the formation of a central bar in some galaxies, and what is the difference that can produce a barred galaxy instead of a spiral one ?


title: Are waves on water an example of gauge invariance?

text: So: Is the close similarity of small waves crossing water of varying depths ("depth potentials") an example of an approximate gauge invariance? If so, do other "only the surface dynamics matter" visualizations exist for more complicated gauge theories, ones that might help students understand them better? Finally and more speculatively, would the depth-dependent forms of large waves (think surfing) have any useful tutorial parallels to broken symmetries? (This question is mostly a sanity check. I like good tutorial visualizations, but only if they remain true to the original physics.)


title: Why are they called "cyclic" coordinates?

text: In Lagrangian formalism, when $\frac{\partial L}{\partial q} = 0$, the coordinate $q$ is called _cyclic_ and a corresponding conserved quantity exists. But why is it called _cyclic_?


title: What are the limitations of Smoothed-Particle Hydrodynamics?

text: I've been excited by some of the possibilities of Smoothed-Particle Hydrodynamics (SPH). I have seen some very exciting demonstrations of their use in 3D graphics, but I am wondering how well the formulas stack up to actual fluids. What is left out? How would I measure this objectively?


title: photoelectric effect

text: A 1.0 mW laser ($\lambda$ = 590 nm) shines on a cesium photocathode (ϕ = 1.95 eV). Assume an efficiency of $10^{-5}$ for producing photoelectrons (that is, 1 photoelectron is produced for every 10^5 incident photons) and determine the photoelectric current. So the way I tackled this problem was the following: we know $$KE_{max} = hf - \phi$$ $$KE_{max} = 0.1180 eV$$ now I dont know how to finish? I know the current is a rate of time but there is no time given here? Also what is the stopping potential? I did a little bit of research online and found this: $$V_{0} = \frac{W}{q}$$ and $$KE_{max} = eV_{0} $$ I tried equation them but I dont know what to solve for (i solved for e which was the wrong answer).


title: Why does a Lorentz scalar field transform as $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$?

text: This problem is from Srednicki page 19. Why $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$? Can anyone derive this? $\phi$ is a scalar and $\Lambda$ Lorentz transformation.


title: What is the equation for the scale factor of the universe, a(t), for the best fit of data to the $\Lambda CDM$ Model of Cosmology?

text: Ideally I like a single formula or multiple formulas for different time ranges that would cover the time from the end of inflation through 100+ billion years after the big bang using the $\Lambda CDM$ Model. I know that from the end of inflation back to the time of the big bang would be much more speculative, but some wild estimate would be appreciated for that time range also! The Friedmann–Lemaître–Robertson–Walker metric defines the scale factor, a(t), from the metric: $$-c^2d\tau^2 = -c^2dt^2+a(t)^2d\Sigma^2$$ where $d\Sigma^2$ ranges over the 3 dimensional space of the universe and does not depend on time. Usually the scale of the scale factor is set by defining $a(t_{now}) = a(13.78 B yr) \equiv 1$


title: How much of the sky is visible from a particular location?

text: From a particular point how much of the sky can be observed. For simplicity sake let us assume the particular point is the head of a 6 foot tall man floating in the middle of the ocean with no visible barriers between him and the horizon. If he stays in exactly the same location being able to turn 360 degrees how much of the sky is visible? Would we calculate the circumference of the circle with the distance to the horizon as the radius? The surface area of a portion of a sphere? Taking it further what percentage of the total sky can be seen at any given time. The answer above divided by the total surface area of earths atmosphere?


title: Symmetric eigenfunctions?

text: So a symmetric eigenfunction / wavefunction is defined as: $$P_{ij} ψ_a (r_1,r_2,…,r_i,…,r_j,…,r_N )=ψ_a(r_1,r_2,…,r_i,…,r_j,…,r_N )$$ But for it to be symmetric does this have to be true for all $ij$ combinations or only one? (note that $P_{ij}$ is the exchanges the element $r_i$ for $r_j$)


title: Are isobaric, isochoric, and isothermal processes quasistatic by definition?

text: Let us start from definitions, as they are given in many sources. A _quasistatic process_ is an idealized processes that happen out so slowly that a system go through a sequence of states arbitrarily close to equilibrium. An _isobaric_ process is a process, in which the pressure remains constant. An _isochoric_ process is a process, in which the volume remains constant. An _isothermal_ process is a process, in which the temperature remains constant. Now, I could imagine how an isochoric process could not be quasistatic, but what about isobaric and isothermal processes? For instance, for the following specific example (from Count Iblis answer): > Suppose we put a mixture of hydrogen gas and oxygen gas in a conducting > cylinder that is kept at constant volume and in thermal contact with a heat > bath. There is also a device in the cylinder that will produce a spark, > igniting the gas. I indeed object (as Count Iblis expected) that the process of ignition is isothermal: when we ignite the mixture chemical reaction takes place, the huge amount of heat is liberated, and besides, a complicated movement of the gas mixture occurs. Thus, it does not make a lot of sense to speak about keeping the temperature of the system constant, since the temperature is different even in different regions of the system. This should not surprise us, since right after the ignition and for some time then the system is clearly not in equilibrium, and thus, the notion of the temperature of the system is not even defined. So, if we adopt this definition then, an isothermal process _imply_ a quasistatic one, since it does not make sense to talk about constant temperature in a process that is not quasistatic, since temperature is not even well-defined in intermediate non-equilibrium states of such process. In other words, the presence of the heat bath is not enough for the process described above to be isothermal: the process should also be performed slowly enough so that all heat liberated can be almost _instantaneously_ transferred to the surroundings. Only in such a case the temperature of the system is indeed _constant_ during a process. Clearly, the system should be in thermal equilibrium with its surrounding all the way, and thus, I think that for the process to be isothermal, it should be quasistatic. Am I right? P.S. Oh, how I feel like I see where I went wrong. An isothermal process indeed must happen at such a slow rate that the _thermal equilibrium_ is maintained, however, thermal equilibrium is not the whole story. As we know systems in thermodynamic equilibrium are always in thermal equilibrium, but the converse is not always true. So, since only thermal equilibrium is required for an isothermal process, it is not necessarily quasistatic. A quasistatic process is always arbitrarily close to _thermodynamic equilibrium_ , while isothermal is only in _thermal equilibrium_.


title: Observationally indistinguishable quantum states

text: What does it mean for 2 quantum states to be "observationally indistinguishable"? If I may venture a guess: Does that mean that the set of possible measured values are the same though the probability of measuring each eigenstate does not necessarily have to be equal? References will also be appreciated. Thanks.


title: Are these sites reliable?

text: First off, I am not a physicist, although I would have loved to become one. The simple fact is I lack the mathematical skills needed (and now I'm too old to acquire them to any sufficient level). Anyway, I do like to read about physics and maths as a hobby. There's lots of stuff around on the internet for lay people like myself. Browsing I stumbled upon these two sites. They look very interesting, but I am not well versed enough to either understand their deeper truths, or debunk them. So, I humbly ask the opinion of the people who frequent this website. What do you think of these theories: Vacuum mechanics and Autodynamics


title: How to transfer energy from a generator to a storeage battery

text: and thank you in advance for taking the time to read my question. To give an idea of my working level, I'm a 21 year old computer science student entering my senior year at college. It's been a few years since my Electricity/Magnetism course, and i'm a bit rusty on the Lorentz Force. I wanted to create a sort of "Human Powered Generator", ie. something as simple as a stationary bicycle turning a generator as I peddle. Now I know the "right hand rule", and can quite easily make a motor/generator with some wire and magnets. My question is about voltage/current. I was never clear on the effects of the strength of the magnet and its importance in the amount of potential power generated. In other words, 1.) If i'm turning a generator at a constant rate and it is lightning a bulb, if I magically replaced the magnets with ones twice as strong, what would happen? I'm assuming it gets twice as hard to turn, but outputs potentially twice as much power. In addition, I was never clear on the relationship between voltage and current in the lorentz force. 2.) While turning a generator at constant speed S with magnets of strength B and # of wire coils C, how much voltage/current is created? I know there are many variables involved here, perhaps such as the width of the rod the coils are wrapped around, thickness of coils, etc. Finally, 3.) If trying to charge a battery of V volts and A ampre-hours, what measures should I take to ensure safe delivery of energy to the battery? In other words, if I peddle the generator very rapidly, I expect lots of either current/voltage/both. I assume I need a voltage regulator of sorts, and I'm not sure if the current matters (I think its just how "much" energy there is, whereas voltage is the "pressure" or "strength" of the energy). I appologize if any assumptions I made are incorrect, i'm just going off of old knowledge. I tried wikipedia, but its all symbolic and I cant find a hard example (As in, I dont know how to find magnetic strength of a magnet, always denoted as B in the equation). Thanks to anyone who can answer any/all 3 of my questions! -John


title: Madelung transformation

text: What is the Madelung transformation and how is it used?


title: Why does breaking one bulb on a string of Christmas Lights cause all the rest to break in some lights?

text: If I break one bulb on a string of Christmas lights, or the filament goes out, then all of the rest seem to stop working. Why is this? However, I've also seen the opposite happen in some newer lights -- when one breaks, the others stay on. What makes this one different than the others?


title: Why do earphone pieces repel each other when music is on?

text: I know it has to do with electricity flowing and generating a magnetic field, but I would like a thorough explanation (with perhaps a picture). In particular:   * What is in the ear piece?    * Why do they repel as opposed to attract?    * How come my ear buds don't seem to do it?  Thanks in advance!


title: Hubble's law and conservation of energy

text: If all distances are constantly increasing, as Hubble's law say, then lots of potential energies of form ~$\frac{1}{r}$ changes, so how is the total energy of the Universe conserved with Hubble's expansion?


title: Is the world $C^\infty$?

text: While it is quite common to use piecewise constant functions to describe reality, e.g. the optical properties of a layered system, or the Fermi–Dirac statistics at (the impossible to reach exactly) $T=0$, I wonder if in a fundamental theory such as QFT some statement on the analyticity of the fields can be made/assumed/proven/refuted? Take for example the Klein-Gordon equation. Even if you start with the non- analytical Delta distribution, after infinitesimal time the field will smooth out to an analytical function. (Yeah I know, that is one of the problems of relativistic quantum _mechanics_ and why QFT is "truer", but intuitively I don't assume path integrals to behave otherwise but smoothing, too).


title: What causes an electron to return from a higher orbit to a lower orbit

text: Ultraviolet light can cause electrons from Hydrogen gas to jump from a lower energy orbit to a higher energy orbit. What causes the electrons to return from the higher energy orbit to the lower energy orbit by releasing photons? Would it be possible to maintain the electrons in the higher energy orbit? How? Would placing the hydrogen gas in an electric field generated between the parallel plates of a capacitor have this effect?


title: Simulating an orbit, primary is not at focus

text: I've been toying around with some -very- simple orbital simulators, mostly using preexisting physics libraries (I took a layman's stab at doing it with vectors too). The thing that is confusing me is that my orbits do not behave as in reality - the primary is always at the center of the ellipse rather than at one of the foci. I get the same result regardless of the engine or library I use. I've simply been putting in a primary and an orbiter, with the primary at the center of the layout. I plug in the formula $$F = G\frac{m_1m_2}{r_1^2}$$ with the force directed towards the primary. I've tried adjusting the time step, but I get the same result. I'm simply confused as to what would cause this. **Update** You're right, I was multiplying, I just didn't know it (something about the way the Construct2 interpreted my commands). I had it configured to apply the force which was essentially G(arbitrary) / distance * distance (didn't like it when I tried ^2). When I put distance squared in a variable and then divided by it things worked. Now I had tried this before in Panda3d with the same problem, so I'll have to go back and look at that.


title: About the work done by gravity to pull the object down to the earth surface

text: The following question is from an physics exercise and I know the answer and the way to solve the problem but just curious why my own way doesn't work. The question is asking "A 2500kg space vehicle, initially at rest, falls vertically from a height of 3500km above the Earth's surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth's surface." The way to solve that is to integrate the differential work from the force $F_g = GM_\text{earth}m/r^2$ such that $$W = \int_{6400 \times 10^3}^{(6400+3500)\times 10^3}F_g dr$$. It does solve the problem but my question is why can't I use the $mgh$ to estimate the work done by gravity?


title: Nomenclature: Yang-Mills theory vs Gauge theory

text: If you're writing about a theory with Yang-Mills/Gauge fields for an arbitrary reductive gauge group coupled to arbitrary matter fields in some representation, is it best to call it a Yang-Mills theory or a Gauge theory? I've heard that one is more likely to refer to a theory with no matter sector - but I can't remember which one! Or are the terms basically interchangeable in the context of quantum field theory?


title: Ask for help: the Reaction-Diffusion related problem

text: In container there are two kinds of molecules A and B which are distributed uniformly. Initially the quantities of A and B are $N_A$ and $N_B$ respectively A and B are distributed uniform, i.e., in any small space (of cause the space should be large enought that there are thousands of molecules in the space), the ratio of number A and B is equal to the ratio of total number of A and B. Assume the molecules are fixed all the time (i.e., we don't consider diffusion) and in every peroid, there is at most one molecule $B_j$ around molecule $A_i (i=1,2,...,N_A)$ be transformed to molecule A: if the molecule $B_j$ is on the boundary/neighbor of molecule $A_i$, the molecule $B_j$ would be transformed to A, otherwise (i.e., all molecules neighboring $A_i$ are molecule A) no molecule around $A_i$ be transformed. In addition, if there are several $A_k$ and $B_j$ on the neighbor of molecule $A_i$, the closest $B_j$ would be transformed to A. Now the problem is: Is there any analytical expression to show how many molecules (approximately) would be transformed to A in every peroid until all molecule B are transformed to molecule A? Thanks, Tang Laoya


title: Instantons and Borel Resummation

text: As explained in Weinberg's _The Quantum Theory of Fields, Volume 2, Chapter 20.7 Renormalons_ , instantons are a known source of poles in the Borel transform of the perturbative series. These poles are on the negative real axis, and the series remains Borel-summable as long as the coupling constant is not too large. However, instantons are objects in the _Euclidean_ version of QFT. What's the significance of the above Borel resummation in the Minkowski theory?


title: Bell polytopes with nontrivial symmetries

text: Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, v_i}$, possibly in a nondeterministic manner. We are interested in joint conditional probabilities of the form $p(r_1r_2\dots r_N|s_1s_2\dots s_N)$. Bell polytope is the polytope spanned by the probability distributions of the form $p(r_1r_2\dots r_N|s_1s_2\dots s_N) = \delta_{r_1, r_{1, s_1}}\dots\delta_{r_N, r_{N, s_N}}$ for all possible choices of numbers $r_{i,s_i}$ (in other words, each input $s_i$ produces a result $r_{i,s_i}$ either with probability 0 or 1, regardless of other players' inputs). Every Bell polytope has a certain amount of trivial symmetries, like permutation of parties or relabelling of inputs or outputs. Is it possible to give an explicit Bell polytope with nontrivial symmetries? (e.g. transformations of the polytope into itself that takes faces to faces and is not trivial in the above sense) In other words, I'm interested whether a specific Bell scenario can possess any "hidden" symmetries Bell polytopes in literature are usually characterized by their faces, given by sets of inequalities (Bell inequalities), which, however, usually do not have any manifest symmetry group.


title: Do all atoms in the universe gravitate each other?

text: I understand that matter will gravitate toward matter. (ex: Earth gravitates a satellite toward it, and the satellite toward Earth.) Does this always apply, regardless of distance? Take two atoms, the farthest apart in the universe. Do they gravitate toward each other, but at an infinitesimal rate? (And actual position displacement is likely in a different direction, due to stronger/closer forces)


title: Derivation of Eq. 7.12 in the review paper of Kraus

text: I'm reading "Lectures on black holes and the $AdS_3/CFT_2$ correspondence" by Kraus. http://arxiv.org/abs/hep-th/0609074 I don't know how one can obtain Eq.7.12. My stupid question is how to obtain this equation. After this equation, it is stated that "one has to take care to consider only variations consistent with the equations of motion and the assumed boundary conditions". What are the variations consistent with the equations of motion and the assumed boundary conditions? What Krasu says is as follows. To compute the bulk functional integral, we need to evaluate the bulk action for the solutions which contribute, including boundary counterterms if necessary. For an on-shell solution around the $AdS_{3}$ vacuum, one can evaluate the action at the $AdS_3$ vacuum by using the variation of the action with respect to the boundary metric $g^{(0)}$ and the gauge fields $A^{(0)}, \tilde A^{(0)}$ \begin{equation} \delta S= \int d^2x \sqrt{g}\,\left[ \frac12 T^{ij} \delta g_{ij}+\frac{i}{2\pi} J^{i} \delta A_{i} +\frac{i}{2\pi}\tilde J^{i} \delta \tilde A_{i} \right]~. \end{equation} where the superscript $(0)$ is omitted for brevity. Reexpressing this in complex coordinates of the boundary metric, we obtain: \begin{equation} \delta S=4\pi i \left(T_{ww}\delta \tau+T_{\bar w\bar w}\delta \bar\tau +\frac{\tau_2}{\pi} J_{w}\delta A_{\bar w}+\frac{\tau_2}{\pi} \tilde J_{\bar w}\delta \tilde A_{w}\right)~. \end{equation} One can integrate the above equation to get: \begin{eqnarray} \mathcal{S}(\tau) & = & \- 2 \pi i \tau \big(L_{0} - \frac{c}{24} \big) + 2 \pi i \bar\tau \big(\tilde L_{0} - \frac{\tilde c}{24} \big) \cr && \; - \frac{i\pi}{2} k \big( \tau A_{w}^{2} + \bar \tau A_{\bar w}^{2} + 2 \bar \tau A_{w} A_{\bar w} \big) \+ \frac{i\pi}{2} \tilde k \big( \tau \tilde A_{w}^{2} + \bar\tau \tilde A_{\bar w}^{2} + 2 \tau \tilde A_{w} \tilde A_{\bar w} \big) \, . \end{eqnarray} I would like to know the derivation of the last equation.


title: Magnetostatics of Current-Carrying wire

text: A question has been nagging at me about Faraday's Law as related to a wire with a constant current: If you have a circular loop of wire with some small resistivity, connected to a battery so that it has a constant DC current, the electric field inside the wire is going to be longitudinal and proportional to the current density. In particular, is going to be something like $\vec{E}=E_0\hat{\phi}$, choosing cylindrical coordinates with the origin at the center of the loop. The curl of this field is obviously nonzero, so applying Faraday's Law there should be a linearly growing magnetic field in the Z direction, at every point inside the wire. But this seems like nonsense; we're supposed to be in a steady-state, and it doesn't seem physically plausible to me that there is a field that can be growing arbitrarily large as we keep our battery plugged in, or that its growth should be affected by the shape of the wire loop. So I expect I am missing some contribution that would cancel this effect. But what it is? I feel like I must be missing something obvious, but can't for the life of me see what it is.


title: rotation of earth and changes in its diameter

text: could calculated the changes of the Earth's diameter cause the rotation?? I have seen 2 other posts about it but I couldn't understand their calculation and they were a little confusing and couldn't conclude anything special with calculation.


title: Spin matrix for various spacetime fields

text: Let $V^{\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu}$. We can write this like $V'^{\mu} = (e^{i\omega})^{\mu}_{\,\,\,\nu}V^{\nu}$ I think where $\omega$ denotes a rotation in some plane spanned by indices $\left\\{\rho \sigma\right\\}$, say. In 2D Euclidean space time, we can write the matrix representation of $\Lambda$ as $$\begin{pmatrix} \cos \omega & \sin \omega\\\\-\sin \omega&\cos \omega\end{pmatrix}$$ and in Minkowski space this changes to the 'hyperbolic' rotation. Linearising the above yields $$\begin{pmatrix}1&\omega\\\\-\omega&1\end{pmatrix} = \text{Id} + \begin{pmatrix} 0&\omega\\\\-\omega&0\end{pmatrix} = \text{Id} + \omega \begin{pmatrix} 0&1\\\\-1&0\end{pmatrix}$$ Now compare with the more general treatment: $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu} \approx (\delta^{\mu}_{\nu} + \omega^{\mu}_{\,\,\,\nu})V^{\nu}$, where $\omega^{\mu}_{\,\,\,\nu} \equiv (\omega^{\rho \sigma} S_{\rho \sigma})^{\mu}_{\,\,\,\nu}$ In 2D, the spin matrix $S$ when acting on vectors in 2D Euclidean space time is therefore the matrix multiplying $\omega$ above, which agrees with the single generator of the SO(2) group. If we continue with the general analysis, we obtain $V'^{\mu} - V^{\mu} = \omega^{\mu}_{\,\,\,\nu}V^{\nu} = \eta^{\mu \rho}\omega_{\rho \nu} V^{\nu} = \omega^{\rho \sigma} \delta^{\mu}_{\rho} \eta_{\sigma \nu} V^{\nu}$ Now use the antisymmetry of $\omega_{\rho \sigma}$ gives $$2(V'^{\mu} - V^{\mu}) = \omega^{\rho \sigma}(\delta^{\mu}_{\rho} \eta_{\sigma \nu} - \delta^{\mu}_{\sigma} \eta_{\rho \nu})$$ from which we can identify $S$ to be $\delta^{\mu}_{\rho} \eta_{\sigma \nu} - \delta^{\mu}_{\sigma} \eta_{\rho \nu}$. I am wondering how this agrees with the matrix I obtained above. Many thanks.


title: Why doesn't an electron feel an electric field, and thus accelerate whilst inside the drift tube of a linear accelerator?

text: Why is the resultant field strength inside the tubes of a linear particle accelerator zero (i.e. the particles move in constant velocity through the tube)? ![enter image description here](http://i.stack.imgur.com/t1aAA.png) As in the diagram, the particle accelerates **between** the space of first tube (l1) and second tube (l2), but no acceleration within the tube. The particle moves with constant speed in the tube. But, why? I read a few pages, some texts refer to electric screening. But I am confused about that.


title: Is the mathematical form of the acoustic diffusion equation present in other fields of physics?

text: We are working in the field of High Performance Computing and we have developed a very efficient parallel implementation for solving the Acoustic Diffusion Equation as described below: $$ \frac{\partial w(r,t)}{\partial t}-D\nabla^2w(r,t) + cmw(r, t) = P(t)\delta(r-r_s) $$ We now wonder whether similar equations are important in other fields of physics; then we would just make somo slight modifications of our parallel implementation, and we could apply it to other relevant fields in physics. Any ideas?


title: Why does a phononic gas require that to reach equilibrium phonon waves need to lose momentum when colliding?

text: I'm learning about phonons and I read that for a phononic gas to reach equilibrium the phonon waves need to lose momentum when they collide. However in an ideal gas the momentum is always preseved. Could anyone comment on this interesting requirement for phoninic gases?


title: Why can the entropy of an isolated system increase?

text: From the second law of thermodynamics: > The second law of thermodynamics states that the entropy of an isolated > system never decreases, because isolated systems always evolve toward > thermodynamic equilibrium, a state with maximum entropy. Now I understand why the entropy can't decrease, but I fail to understand why the entropy tends to increase as the system reach the thermodynamic equilibrium. Since an isolated system can't exchange work and heat with the external environment, and the entropy of a system is the difference of heat divided for the temperature, since the total heat of a system will always be the same for it doesn't receive heat from the external environment, it's natural for me to think that difference of entropy for an isolated system is always zero. Could someone explain me why I am wrong? PS: There are many questions with a similar title, but they're not asking the same thing.


title: Why must phonon-phonon interaction be anharmonic?

text: I'm learning about phonons and this is really interesting. Why can't I describe two phonons colliding as a harmonic oscillation? Why does it have to be anharmonic for the phonon model to work?


title: Edge states at high magnetic field (Quantum ballistic transport)

text: I am learning about edge states at high magnetic field (Quantum ballistic sample). What I understand so far from it is that at high magnetic field, Landau states arise. Then (I don't know exactly why?), edge states are formed. To enhance my understanding, can someone help me with these questions: *Why must the chemical potential cross a Landau level? *Why is the net current carried solely by these edge states? (I am only interested in intuition, no derivations)


title: Lorentz Invariance of Maxwell Equations

text: I am curious to see a simple demonstration of how special relativity leads to Lorentz Invariance of the Maxwell Equations. Differential form will suffice.


title: Imaginary number for extinction coefficient in complex refractive index

text: In complex refractive index on a material, $n=n'+ ik$, the imaginary part $k$ is physical meaning, as it shows absorption in the material but it is an imaginary. How we measure an imaginary values in physics with imaginary numbers?


title: Where does the energy for magnetizing a metal come from?

text: I saw a young man's science project. He asked this question in a different way "does magnetizing something reduce the magnetic field potential of the original magnet?" His answer experimentally was "no". This begs the question, was he right? Magnetizing an object clearly requires energy. It is now placed into an ordered state from which there are many other possibilities. Therefore, it has moved from a state of relative disorder to a state of order. The laws of thermodynamics say that this requires energy. Where does the energy come from?   1. Does the magnetic field strength of the original "permanent" magnet decrease?   2. Is there "work" performed (e.g. Force x distance) and therefore the energy comes from the work it takes to bring the non-magnetic material close and then tug it away from the "permanent" magnet?


title: Find the exact angle as a function of time for a rod that swings on a frictionless axle

text: This is a simple problem I thought of that I haven't been able to solve. _Given a rod of uniform mass attached to a fixed axle, find the angle it makes with the horizontal if it is dropped from rest along the horizontal._ $\theta(t) = ?$ ![a bad drawing](http://i58.tinypic.com/1zdm440.jpg) I had two approaches, both of them were verified by my professor. The first involves potential energies. Given that the axles are frictionless, we can say that the energy of the system at any point in time is constant. $K2 + U2 = K1 + U1$ Because the system starts at rest, K1 is 0. $K2 + U2 = U1$ The potential energies are entirely due to gravity, and if we focus on the center of gravity of the rod, the lowest point is at $\frac{L}{2}$ below the horizontal, so the potential energy, $U_{g}$ can be expressed as: $U_{g} = \frac{mgL(1 - \sin(\theta))}{2}$ Hence, $K2 = U1 - U2 = \frac{mgL(1 - \sin(0))}{2} - \frac{mgL(1 - \sin(\theta))}{2} = mgL\frac{1 - sin(0) - 1 + sin(\theta)}{2} = \frac{mgL\sin(\theta)}{2}$ Now, K2 can be expressed in terms of the angular velocity and rotational inertia of the system: $\frac{I\omega^{2}}{2} = \frac{mgL\sin(\theta)}{2}$ Since $I = \frac{mL^{2}}{3}$ $\frac{mL^{2}\omega^{2}}{3} = mgL\sin(\theta)$ $\frac{L\omega^{2}}{3} = g\sin(\theta)$ And since the angular velocity, $\omega$ is $\frac{d\theta}{dt}$, or $\theta^{'}$ $\frac{L\theta^{'2}}{3} = g\sin(\theta)$ $\theta^{'2} = \frac{3g\sin(\theta)}{L}$ Taking square roots and seperating variables leads to $\int \frac{d\theta}{\sqrt{\sin(\theta)}} = \int{\sqrt{\frac{3g}{L}} dt}$ **But the left side can't be integrated.Wolfram shows this** A simpler approach that was suggested to me uses torques. Since the sum of the torques is equal to the second derivative of the angle times the moment of inertia, we get $\sum{\tau} = I\alpha$ The only torque is gravity, which changes based off of the angle the rod is currently at. $mgL\cos{\theta} = \frac{mL^2}{3} \theta^{''}$ $g\cos{\theta} = \frac{L}{3} \theta^{''}$ $\theta^{''} - \frac{3g}{L}\cos(\theta) = 0$ Which is a second-order, nonlinear O.D.E. of the form $y'' = f(y)$ **I can't solve it,but here's something from Wolfram** I looked at the Pendulum article on Wikipedia, but I'd rather not have my answer involve an infinite series. Is there a closed form solution for this problem? EDIT: Maybe a combination of the two final equations could solve it? $k = \frac{3g}{L}$ $\theta^{'2} = k\sin(\theta)$ **and** $\theta^{''} = k\cos(\theta)$


title: Relation between impact parameter and distance of closest approach of a light ray in Schwarzschild Geodesics

text: The following wikipedia articles are incompatible :   * Two body problem / bending of light by gravity   * Schwarzschild geodesics / bending of light by gravity According to both articles, the equation describing the path of a photon in a Schwarzschild metric is : $$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$ with :   * $\varphi$ the deviation angle   * $r$ the distance between the photon and the mass   * $r_s$ the Schwarzschild radius and $b$ which is the distance of closest approach in the first article and the impact parameter in the second one. So I have two questions :   1. Which one is correct? (What is $b$?)   2. If $b$ is the impact parameter, what is the formula of the distance of closest approach $r_0$ as a function of $b$ (or inversely, the impact parameter $b$ as a function of $r_0$)?


title: In $\textbf{f} = -\boldsymbol{\nabla} u$, what is $u$?

text: I know that force is the negative gradient of the potential: $$\textbf{f} = -\boldsymbol{\nabla} u$$ where force $\textbf{f}$ is a vector and $u$ is a scalar. This is a relatively soft question, but what is $u$? I frequently hear it referred to as "the potential." But is it actually the potential _energy_? For example, suppose I have a system consisting of several classical particles that interact. Suppose that I can calculate the potential _energy_ at the position of each particle, because I know how they interact (e.g., by gravity, by Coulomb's Law, by the Lennard-Jones potential, and so on). Can I then determine the force $\textbf{f}_\boldsymbol{1}$ on particle 1 by simply calculating the negative gradient of the potential _energy_ $u_1$ at the position of particle 1?


title: Where does the electricity, generated by a solar panel, go if you don't use the electricity?

text: I'm sorry if this question is too trivial for this Q&A forum. I am a layman when it comes to physics (though I did cover the high-school physics courses). I was wondering what happens to the electricity, that is generated by your own solar panel, in case you don't use any electricity in your home. Does it turn from electrical energy into heat? Or does something else happen? Is the energy "lost" for practical purposes? A follow-up question: In case the energy is "lost" for practical purposes, wouldn't that be an enormous waste of energy? Is this what the "Smart Grid" network is intended for? To distribute the energy to other places where it _is_ needed, thereby minimizing the energy loss and CO_2 emissions?


title: Eddy currents and EMF?

text: Im confused, can both phenomenons happen at the same time?! A conductor the passes a powerful magnetic field at extremely high speeds, will induced EMF AND Eddy currents simultaneously?!


title: What is a "reggized gluon"?

text: I'm reading a paper in which the author used these words many times assuming that the reader knows what he is talking about. Can someone please explain what it is? What is the difference between a regular gluon and a reggized gluon?


title: Historical: Natural vs unnatural parity mesons

text: Quick question: In the old papers and text I occasionally see authors referring to mesonic states as having 'natural parity' or 'unnatural parity'. What was their motivation for classifying mesons this way? Thanks!


title: Could we survive if the sun were a black hole?

text: Ignoring the impossibility of the sun suddenly collapsing, and the energy release (which would kill us): If the sun suddenly turned into a black hole, could we survive if we had sufficient energy for heat, light, etc?


title: Wavefunction restrictions of odd potentials

text: So I was just reading back through Griffiths' "Introduction to Quantum Mechanics" and solving some of the problems for practice. There is a nice one (problem 2.1c for those playing at home) where you have to show that if the potential energy function $V(x)$ is an even function that $\psi(x)$ can always be taken to be even or odd. This is pretty straightforward, but it got me to thinking about the general restrictions on the wavefunction if $V(-x)=-V(x)$ (ie, it is odd). I decided just to use a one dimensional case. Starting with the time independent Schrodinger equation, and taking $x\rightarrow-x$ gives: $-\frac{\hbar^2}{2m}\frac{d^2\psi(-x)}{dx^2}+V(-x)\psi(-x)=E\psi(-x)$ Now use the fact that $V(x)$ is odd, and put the minus sign in an illuminating location: $-\frac{\hbar^2}{2m}\frac{d^2\psi(-x)}{dx^2}+V(x)(-\psi(-x))=E\psi(-x)$ Now we see that the Schrodinger equation is satisfied if $\psi(-x)=-\psi(-x)=0$. My suspicion: it is not consistent in the last line to have $V$ defined at $x$ and $\psi$ defined at $-x$. And this method, while it works nicely for the case of even potentials, is just meaningless for the case of odd potentials. My question: Does the fact that this is meaningless mean that we can't have odd potentials in Quantum mechanics (This seems silly, we can just go into the lab and make some, right?) or does it (more likely) just mean that this method tells us nothing, and that the wavefunctions can be even, odd or neither depending on the problem? Thanks :)


title: Observational Cosmology, light polarisation

text: Can anyone explain the purpose of measuring light polarisation, as carried out by the upcoming planck satellite?


title: Parity of a system composed of 2 particles

text: I have read that for a system of 2 particles, the total parity is given by: $P=P_1 P_2 (-1)^L$ where   * $ P_1, P_2$= insisec parity of particle 1, 2   * $L$ = relative angular moment what's the meaning of "relative angular moment"? Do I have to add the $l$ numbers of the two particles? And what if I have 3 particles?


title: Compute energy from displacement via time integration (using ODE)

text: How to compute energy from displacement given by solving an ODE via, e.g., ODE45. Say one has: $$ \ddot{z}+\zeta \dot{z}+z=\kappa\cos\left(\Omega t\right),\quad z=z\left(t\right) $$ and wishes to compute the relating energy in the frequency range $\Omega\in[0.8,1.2]$. One approach is to multiply with velocity (which yields the power) and integrate over 0 to $2\pi$ to obtain the energy: $$ E=\int_0^{2\pi}\ddot{z}\dot{z}dt+\zeta \int_0^{2\pi}\dot{z}^2dt+\int_0^{2\pi}z\dot{z}dt-\kappa\int_0^{2\pi}\cos\left(\Omega t\right)\dot{z}dt $$ I have solved the (first-mentioned) equation numerically with respect to displacement for each $\Omega\in[0.8,1.2]$ (with an arbitrary interval). Then, I can find the velocity to each of these displacement vectors by using, e.g., diff(...). Integration can probably be done with trapz(...). But what do I do with time $t$? Can I make some simplifications due to steady-state (I want to compare with analytical steady-state solution).


How do you get the equation of motion of a spherical pendulum WITHOUT using Lagrange or Hamilton equation? I have been searching for a proper explanation but all websites use Lagrangian and I don't know that. Can anyone give me a link or something?


title: Periodic Table and strong fields at high Z

text: There is a known "phenomenon" recalled by Greiner et al. in several of his books related to hypothetical elements with Z>172: in some point arount this Z, the nuclear field is strong enough to pop out an electron-positron pair from vacuum. What does it mean? Are elements beyond that Z "possible" or are they unstable? How important are quantum mechanics, special relativity and quantum electrodynamics there? In particular, does it mean that superheavy elements emit positrons at no cost from the continuum sea of negative energy?


title: water flow in a sink

text: When one turns on the tap in the kitchen, a circle is observable in the water flowing in the sink. The circle is the boundary between laminar and turbulent flow of the water (maybe this is the wrong terminology?). On the inside the height of the water is lower than on the outside. I'm sure that you have seen it many times, if not you can try it out yourself. I found out by experiment that the effect is most likely independent of the curvature of the sink, as it works in strongly curved bathroom sinks as well as in flat kitchen sinks. Why is there a circle, why not a gradual change and what is happening qualitatively and/or quantitatively in this system? As I have not found a satisfying answer to this problem yet, every time I turn on a tap, I am reminded of my own ignorance. It is really annoying.


title: Mathematical justification of Hartle-Hawking "no boundary" proposal

text: In Hartle-Hawking "no boundary" proposal it is proposed that Riemannian spacetimes rather than Lorentzian dominated the path integral near the big bang. Moments after the big bang however spacetimes with Lorentzian metrics started to dominate over the Riemannian. The dominance of the Riemannian spacetimes is characterized by positive definite metrics obtained by applying Wick rotation. Now to compute the path integral some approximations are made and the process of analytic continuation is applied. What bothers me is analytic continuation of an approximate holomorphic function is not guaranteed to remain analytic in another region. Why then this unreliable process applied--can one rely on this scheme which seems to be mathematically dubious?


title: Are the basic postulates of QM the only set of postulates that can give rise to a sensible semi-probabilistic physical theory?

text: Are the basic postulates of QM, such as complex Hilbert space, unitary evolution, Hermitian operator observables, projection hypothesis etc., the **unique and only** set of postulates that gives rise to a semi-deterministic and semi-probabilistic theory, in which the time evolution is non-degenerate? By non-degenerate, I mean different initial states never produce the same final state probabilities, which in QM is guaranteed by unitarity. Phrased in another way, is it possible to prove from some general principles, such as semi-determinism, semi-reversibility (not for collapse), causality, existence of non-compatible observables etc., that a physical theory must satisfy these postulates? In particular, is it possible to prove that complex numbers, or a mathematical equivalent, must be fundamental to the theory? I haven't studied anything about foundational issues of QM, so feel free to point out if I'm being a crackpot. I suppose this question may be similar to something like "can you prove that gravity must be a metric theory entirely from the equivalence principle?", whose answer is no, but I'll be glad if it turns out to be otherwise.


title: The correspondence between Grassmann number and 4-spinor

text: In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number. For two Grassmann number $\psi_1$, $\psi_2$, we can calculate $\psi_1\cdot\psi_2$ and it has the properity $\psi_1\cdot\psi_2=-\psi_2\cdot\psi_1$. While for two column vectors, we can't even define $\psi_1\cdot\psi_2$. So what's the correspondence between Grassmann number and column vector?


title: How does intensity differ from apparent luminosity in the context of photometry?

text: **Context** : photometry in astronomy. **Background** : The total luminosity $L$ of a star is the energy that radiates in all spatial directions in all wavelengths and is given by the following formula, where $F$ is the flux, $\lambda$ the wavelength and $R$ the radius of the star: \begin{align*} L = 4\pi R^2 \int_0^\infty F(\lambda) \mathbb{d} \lambda \end{align*} If we "divide" that total luminosity on a sphere with radius $d$ where $d$ is the distance between us, the observers, and the star, then we get the apparent luminosity, that is: \begin{align*} \ell = \frac{L}{4\pi d^2} \end{align*} and is measured in $W/m^2$. **Question** : How does the apparent luminosity, $\ell$, differ from the intensity $I$ ?


title: Basic of Capacitor

text: I've 2 capacitors; plate area, difference between plates and dielectric is same. Only thing is that the metal used in plates is different. Since the formula $\displaystyle C=\frac{\varepsilon A}{d}$ states, it won't affect but why?


title: Constant mass flux in fluid dynamics?

text: OK so I am having a conceptual crisis about Hydrodynamics. 1) Since mass should be conserved, is $\frac{dm}{dt}=0$ ? 2) But I know that the formula $\rho_1u_1A_1 = \rho_2u_2A_2$ exists... and doesn't that mean the mass flux is conserved? Where does this formula come from? 3) How can we derive an expression for $\frac{dm}{dt}=0$ from the continuity equation $\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{u})=0$


title: Lorentz force in rotating frame

text: This is the common problem of a charged particle moving in a static electric and magnetic field. Say $\textbf{E}=(E_x,0,0)$ and $\textbf{B}=(0,0,B_z)$. In the **inertial frame of reference** , the equation of motion is (1): \begin{equation} \frac{d \textbf{v} }{dt} = -\frac{q \textbf{B} }{m}\times \textbf{v} + \frac{q}{m}\textbf{E} \end{equation} We can find equations for $v_x$ an $v_y$ and see that the resulting motion is a circular orbit with a **constant** drift velocity $v_d=\frac{E_x}{B_z}$. Surely I should get the same answer if I solve the problem in a **rotating frame of reference**? I know that (2): $$ \frac{d \textbf{v} }{dt} \vert_{Inertial} = \frac{d \textbf{v} }{dt} \vert_{Rotational} + \boldsymbol{\omega}\times\textbf{v};$$ If I use Eq. (1) as the LHS of Eq. (2), and choose $ \boldsymbol{\omega}=-\frac{q \textbf{B} }{m}$, then I get (3): $$ \frac{d \textbf{v} }{dt} \vert_{Rotational} = \frac{q}{m}\textbf{E};$$ where $\mathbf{E}=(E_x\cos(\omega t), E_x\sin(\omega t),0)$ is the electric field in the rotating frame. Solving this:$ \textbf{v} \vert_{Rot} = \frac{E_x}{B_z} (\sin(\omega t)\textbf{i} - \cos(\omega t)\textbf{j})$ But I think I am missing the relation between $\mathbf{i}$ and $\mathbf{j}$ in the rotation and in the inertial frame... How do I get the drift velocity $v_d = \frac{E_x}{B_z}$?


title: Instabilities in the CDMT

text: Could anyone explain or refer to references on why the CDMT f(R) gravity model suffers from Instabilities any why the sign of ${\mu}^{4}$ matters.


title: Does Earth really have two high-tide bulges on opposite sides?

text: # The bit that makes sense – tidal forces My physics teacher explained that most tidal effect is caused by the moon rotating around the Earth, and some also by the Sun. They said that in the Earth and Moon system, the bodies are are in free fall about each other. But that points on the surface of Earth, not being at Earth's centre of gravity, experience a slightly different pulls towards the moon. The pull is a little greater if they are on the moon's side, and slightly lesser if they are on the side away from the moon. Once free fall is removed, on the moon side this feels like a pull towards the moon and on the the opposite side it feels like a repulsion from the moon. This makes sense to me, and is backed up by other questions and answers here, like this and also this Phys.SE question. # The bit that doesn't make sense – tidal bulges They also said that there are "tidal bulges" on opposite sides of the Earth caused by these forces. The bulges stay still relative to the moon and Earth rotating through the bulges explains why we get two tides a day. They drew a picture like this one… ![Tidal Bulges](http://need- media.smugmug.com/Graphics/Graphics/i-Cfkw5rH/1/XL/Tidal%20Bulge-XL.jpg) An image search for tidal bulges finds hundreds of similar examples. …But, if there is a tidal bulge on both sides of Earth, like a big wave with two peaks going round and around, how can an island, like Great Britain where I live, simultaneously have a high tide on one side and a low tide on the other? For example:   * Holyhead tide times on the West coast   * Whitby tide times on the East Two ports with tides 6 hours, or 180º apart. It's high tide at one while low tide at the other. But only about 240 miles by road. Great Britain is much smaller than Earth. It's probably not even as big as the letter "A" in the word "TIDAL" in that picture. * * * To prove this isn't just Britain being a crazy anomaly, here is another example from New Zealand:   * WESTPORT, New Zealand South Island   * Kaikoura Peninsula, New Zealand South Island Two ports that are 180º (6 hours) apart, but separated by just 200 delightful miles through a national park. New Zealand, unlike the UK, is in fairly open ocean.


title: Light cone and order of events?

text: If one event lies outside the light cone of another, can the events to some observers appear in a different order in one reference frame compared to the other, and is this the only time when this is the case? (I think this is right but am looking for verification)


title: Mandelstam variable equality derivation for $s$-channel process

text: I am supposed to show that the Mandelstam variable $s=(p_1^2 + p_2^2) = 4(P_{cm}^2 + m^2)$ for an electron/positron interaction, where $p_1$ is the incident 4-momentum of the electron and $p_2$ is the incident 4-momentum of the positron, $m$ is the electron/positron mass, and $P_{cm}$ is the magnitude of the 3-momenta of the incident and scattered particles in the center of mass frame. The derivation that my professor gave is as follows: $(p_1^2 + p_2^2) = (E_1 + E_2)^2 - (\vec p_1 + \vec p_2)^2$ In the center of mass frame, $\vec p_1 + \vec p_2 =0$ (since each are the regular 3-momenta of the particles), so $=(E_1 + E_2)^2 = (2E)^2$ since the energy of each particle is the same in the center of mass frame we just express each energy as $E$. But then the step I don't understand is the following: $E^2 = P_{cm}^2 + m^2$ which gets plugged in to $(2E)^2$ to get the final answer. I believe the equation in the line above comes from the relativistic energy-momentum relation but I don't see how we can change from the energy $E$ in the center of mass frame to $P_{cm}^2$, which generally is not in the center of mass frame. Is $E^2$ a Lorentz invariant? If so, I can understand how it works in to the derivation. Edit: as pointed out, $P_{cm}$ is not the momentum of the cneter of mass but the momenta int he center of mass frame. This means that the above paragraph is irrelevant. Thus, the energy-momentum relation holds because $P_{cm}$ is the momentum of each particle in the center of mass frame and so there is no jumping between frames as I thought previous.


title: Where are the time dilatational effects of orbital motion and gravitational acceleration equal?

text: Nearly four years ago, upon hearing of the observation of time dilation in two optical atomic clocks at an elevation one metre apart, due to acceleration towards earths centre of gravity by Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. I wondered where the point would be, at which time dilation caused by acceleration toward the earth gives way to time dilation caused by acceleration around the earth, via orbital velocity. (I mean the point at which the two effects are at equilibrium with each other.) I asked a friend who is a physicist, he got back to me almost immediately, fascinated by his back of an envelope calculations that this point is the Clarke orbit, or the geosynchronous orbit. If for instance the rotation velocity of earth were a slightly different speed, the result would be different, and no relationship apparent. I think there may be a very interesting relationship waiting to be revealed. Is there a physical or mathematical relationship? And if so, what is it? Not being a mathematician (I'm an artist) I'm not capable of doing the math, but this has fascinated me ever since.


title: Curious about this perpetuum concept

text: I know it's not possible to build a perpetual motion machine, but I still got this concept running in my mind.   If you take a (longer)copper coil, and insert an axis through it(right-angled) with a magnet fixed on it,   it would generate electricity when you spin the axis. The axis and magnet rotate, thus alteration of the flux, thus electricity. Of course normally there is a soft-iron core inserted in the coil to achieve better efficiency,   but it also brakes the movement. If you don't use a core and insert the axis+magnet into the coil,   you'd still generate a certain amount of electricity. The magnet is **only** braked by the friction of the axis and the air, but this friction is almost insignificant. So if you use good bearings to reduce this friction to a minimum, and you take a coil with as much as possible windings, and the strongest magnet you can find, you could _in theory_ build a perpetuum mobile right? I'm almost sure I'm not right but I'm asking anyway. Just curious.   Any comments or responses are welcome.


Do x-rays and gamma rays also contain photons like visible light does? If so, then what makes photons of visible light and other waves different? The rest mass of a photon is zero, but as it moves at the speed of light, its mass is infinite. By $E=mc^2$, its energy is infinite, but using formula $ E=h\nu$, we get finite energy. How is this happening?


title: Expectation values in QFT?

text: What is the meaning of different expectation values in QFT? For instance: $$\langle 0|{\cal O}(0)|q,s\rangle$$ or $$\langle 0|{\cal O}(0)|0\rangle$$ with ${\cal O}$ being some operator and $|q,s\rangle$ some single particle state with the written quantum numbers. For instance, in Srednicki's textbook I mostly see expectation values of the second kind, i.e. $\langle 0|{\cal O}(something)|0\rangle$. How come?