title: How to get time from equation of linear uniformly accelerated motion?

text: I have had a problem solving this equation for time (from Linear Uniformly Accelerated Motion (LUAM)): $$ s= v_0t + \tfrac{1}{2}at^2 $$ I'll appreciate if someone could provide me some step-by-step solution to this.


title: Could a photon travel faster than the speed of light in vacuum?

text: If the cosmic speed limit is the speed of light in vacuum, then what happens when a photon traveling through space meets gravity? Wouldn't gravity pull on the photon that's already traveling at max speed, thus making it ever-so- slightly faster?


title: Intuitive explanation of the waves superposition

text: When the two waves collide, why do they pass right through each other? Mathematically it's due to the principle of superposition: the sum of the two solutions of a wave equation is also a solution. But intuitively it's not clear why the waves would not, say, just cancel each other during the collision. What would be a convincing 'local' explanation - in terms of the individual particles in the medium (or segments of the medium), that move only due to the interactions with their neighbors? * * * * * * Here's a simple example - two colliding wave pulses in the opposite phase with equal amplitudes and wavelengths (animation). In this case explanation is more straightforward (below). How would one make a similar argument for a general case? In this special case the middle point acts as a fixed point, so that each wave pulse is 'reflected' from a hard boundary. In terms of individual segments: the one closest to the center is drawn to the equilibrium position by the force from the fixed point; it drags the next closest segment with it; etc. When they reach the equilibrium position they continue moving due to inertia and restart the wave in the opposite direction with the opposite phase. The reflected wave has the same amplitude and wavelength due to the symmetry around the equilibrium: the force on each segment at the opposite displacement is a mirror image of the original. * * * EDIT: elaborating on the question Consider one of the particles on the spring-connected string, like in the bead-spring model (picture). It only interacts with its two neighboring particles - left and right. Now we send two wave packets moving towards each other along the string. For concreteness, let's say we are looking at the particle one to the right from the middle particle (the position where the two wave packets arrive simultaneously and first meet each other). First our particle is moved by its right neighbor due to the wave from the right reaching it. It moves like any other particle in the wave path. But then the particle on its left exerts the force on it, and that one has already been influenced by the wave from the left. Again, all the particle 'knows' is the position of its left and right neighbor. The question is: by just looking at its neighbors, how does our particle 'figure out' to have the displacement that is always equal to the sum of the two constituent wave functions at that point?


title: Superradiance of electromagnetic waves

text: I have to do a calculation (problem 5 of chapter 12 in Wald) verifying the super-radiance of electromagnetic waves incident on Kerr black holes and have a few preliminary questions. As background: on pages 328-329 of Wald, there is a discussion of super- radiance achieved by Klein Gordon fields incident on Kerr black holes. Using the standard $(t,r,\theta,\phi)$ coordinates for the Kerr space-time, we consider a scalar wave of the form $\phi = \text{Re}[\phi_{0}(r,\theta)e^{-i\omega t}e^{im\phi}]$ with $0 < \omega < m\Omega_{H}$ where $m, \omega$ are positive constants and the constant $\Omega_{H}$ is related to the killing field tangent to the null geodesic generators of the event horizon by $\chi^{a} = (\partial_{t})^{a} + \Omega_{H}(\partial_{\phi})^{a}$; it physically represents the "angular velocity" of the event horizon. We then construct the conserved energy-current $J_{a} = -T_{ab}\xi^{b}$ (here $\xi^{a} = (\partial_t)^{a}$ is the time-like killing field) and calculate the time averaged flux for the scalar wave incident on the horizon i.e. we calculate $-\left \langle J_{a}\chi^{a} \right \rangle = \left \langle T_{ab}\chi^{a}\xi^{b} \right \rangle$. The energy-momentum tensor for the Klein Gordon field is given by $T_{ab} = \nabla_{a}\phi \nabla_{b}\phi - \frac{1}{2}g_{ab}(\nabla_{c}\phi \nabla^{c}\phi + \mathfrak{m}^{2}\phi)$ but $g_{ab}\xi^{a}\chi^{b} = 0$ on the horizon so we just end up with the expression $\left \langle (\xi^{a}\nabla_{a}\phi) (\chi^{b}\nabla_{b}\phi) \right \rangle$. Plugging in the above form for the scalar wave into this expression yields, after a very simple calculation, $\left \langle (\xi^{a}\nabla_{a}\phi) (\chi^{b}\nabla_{b}\phi) \right \rangle = \frac{1}{2}\omega(\omega - m\Omega_{H})\left | \phi_{0}(r,\theta) \right |^{2}$ so the energy flux through the horizon is negative i.e. the reflected wave carries back greater energy than the incident wave. This phenomena is called super-radiance. I must now show this also holds for electromagnetic waves. My main question is: what form of an electromagnetic wave do I consider? Would I consider a simple wave solution analogous to the scalar wave above i.e. $A_{a} = \text{Re}[(A_0)_{a}(r,\theta)e^{-i\omega t}e^{im \phi}]$? Secondly, would this wave fall under the regime of the geometrical optics approximation? That is, would the space-time derivatives of $(A_0)_{a}(r,\theta)$ be negligible? The reason I ask is, Wald never specifies the nature of the amplitude of the Klein Gordon wave above but in that case it isn't an issue since the derivatives of the amplitude never come into the expression for the time averaged flux ($\xi^{\mu}\nabla_{\mu}\phi = \partial_{t}\phi $ and $\chi^{\mu}\nabla_{\mu}\phi = \partial_{t}\phi + \Omega_{H}\partial_{\phi}\phi$). However, for an electromagnetic wave, the time averaged flux becomes $\left \langle T_{ab}\chi^{a}\xi^{b} \right \rangle = \left \langle F_{ac}F_{b}{}{}^{c}\chi^{a}\xi^{b} \right \rangle$ and since $F_{ab} = 2\nabla_{[a}A_{b]}$ this expression looks like it can potentially contain space-time derivatives of $(A_0)_{a}(r,\theta)$. As such, I would like to know if the geometrical optics approximation is valid since I can then drop any terms involving space-time derivatives of $(A_0)_{a}(r,\theta)$ that could potentially show up in the time averaged flux, saving me a lot of calculation. I'm still trying to figure out how to use Wald's hint (which says to first prove that $\mathcal{L}_{X}F_{ab} = -2\nabla_{[a}(F_{b]c}X^{c})$ for any vector field $X^{a}$, which is very easy to prove, and then says to use this to relate $F_{ab}\xi^{b}$ to $F_{ab}\chi^{b}$, which I still can't figure out) but I just want to have it clarified beforehand (so that it is less daunting from the start of course!) and [I]more importantly[/I] I want to make sure that the form $A_{a} = \text{Re}[(A_0)_{a}(r,\theta)e^{-i\omega t}e^{im \phi}]$ is actually the correct kind of wave solution to use for this problem as I am unsure of this as well. Thanks in advance!


title: In which direction does the current flow in a wire moving north to south along the Earth?

text: I stumbled upon this question and I can't seem to figure it out. I think you have to use the right hand rule, but it doesn't get me the right answer. The question is something along the lines of:                 In which direction does the current flow in a wire moving from the northern hemisphere        down towards the southern hemisphere. The wire is oriented east to west.      So, the magnetic field in the northern hemisphere points downward, and the wire is moving south, so also down. I don't see how I could apply the right hand rule to this, since my thumb and fingers would both point downward. Apparently the answer is east, and I could get that if the magnetic field moved down into the earth, and the wire moved south along the earth. Is this where my mistake it? I always thought that the magnetic field moved down from the north to south, not necessarily into the Earth.


title: Relationships between different measure of opacity

text: I'm reading some papers that compare different values for a materials opacity to a particular particle. The first is given as $\frac{dE}{dX}$, a single particles energy loss per unit column depth (X = $x\rho $) to a continuous process. Makes sense. So then the author goes on to compare his value to that of other authors, who have their opacities for the process expressed in $\kappa $ given in $cm^{2}/g $. They're both presented as measuring the same thing, but I usually think of $ \kappa $ as being an opacity for a large group of particles that are removed discretely from a beam, so that the total energy in the beam dies of exponentially (since energy loss per distance is proportional to the number of particles left in the beam). $\frac{dE}{dX}$ is due to a continuous process acting on all the particles, and so would have a roughly linear effect per distance. So my question is how does one compare the two values. What does $ \kappa $ mean in a context of continuous energy loss. Thanks


title: Pressure in Bernoulli's equation

text: ![Example Problem](http://i.stack.imgur.com/FzA1a.png)! I've been reading an introductory physics textbook. It has a chapter on fluids which I'm finding quite confusing. Specifically, I don't understand the meaning of the pressure terms $P_0$ and $P_1$ in the the equation: $P_0 + \frac{1}{2} \rho v_0^2 + \rho g h_0 = P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1$ I have been thinking about a particular problem which I have thought up (see image). The problem is as follows: Imaging a big lake 100m deep. At the bottom, there exists a small pipe which allows water to flow from the lake into a vacuum. I wish to find the velocity at which the water leaves through this small pipe. My idea to do this is to consider a fluid particle at the surface of the lake. This particle will experience atmospheric pressure $P_0 = P_{atm}$. Since the lake is large, $v_0 \approx 0$ and since the lake is 100m deep, $h_0 = 100$. Now consider a fluid particle in the pipe at the bottom of the lake. It would experience a pressure of $P_1 = 0$ since the pipe empties into a vacuum. We can set $h_1 = 0$ since I am free to choose the point from which I measure height. Now I can simply solve for $v_1$. Is this correct? The reason I am unsure is that I don't understand how to choose the values of $P_0$ and $P_1$. What I would like to understand the nature of the pressure terms in Bernoulli's equation shown above. For example, why is/isn't $P_1$ equal to $100 \rho g $, which would be the pressure exerted on the liquid in the tap since it is at depth 100m. Or is this equation for calculating pressure at a particular depth only valid in fluid statics? Finally, what is pressure (maybe this is too broad a question)? I believe it is a scalar since we can simply talk about the "Pressure" of a fluid at a particular depth, yet I use the equation $P = \frac{\mathbf{F}}{A}$ where $\mathbf{F}$ is a vector. Wouldn't this suggest that P is a vector? Any answers to any of my questions would be much appreciated.


title: Displacement of an object due to opposite and equal forces acting at different times?

text: This question is about mechanical waves in solids, the speed at which forces propagate through solids, and the displacement that might occur due to unequal starting times of equally sized forces. There is a metal box which contains two signal paths: a short one and a long one. Forces that can be generated (for example by a hammer) can travel through these paths. See the initial state in the image. Left is the long signal path, and on the right there's the short signal path. The waves are travelling some distance marked by the dotted yellow and light green lines on the blue box's edges. ![metal box with different time signal paths](http://i.stack.imgur.com/WzTsv.png) The red arrows represent forces. The plates next to it are the spots where the hammers hit. In situation a), the forces are equal in size and opposite to each other. Force 1 starts earlier than force 2. A few milliseconds later, force 2 starts. It's just a short hammer blast on both sides, just a smash and release. The left side of the box is now being pulled to the left, and the right side is pulled to the right. After some time, the mechanical waves meet in the middle. The forces travel at a finite speed; the speed of sound in the material. In situation b), the waves don't meet in the center but near the left edge. The times that the forces start are still unequal. Force 1 still starts before force 2, but now with even less time in between. My questions are:   * Even though the forces started at different times, is there any displacement of the metal box in any of the situations? Or is there any movement at all but is the net displacement zero?    * What happens when the waves (signals) meet? Do they annihilate and communicate to all particles that the velocity must become zero throughout the whole system?   * How can I visualize the movement of the particles of the box? For example in situation a), is the left side moving to the left, while the right side moves to the right and the center staying fixed? Is this like a stretching effect of the box?   * Does the left side of the box (with the longer signal path) affect the right side of the box by pulling harder to the left than the right side does because it is more massive, and thus creating a displacement to the left? Thanks for reading.


title: How/why can the cosmic background radiation measurements tell us anything about the curvature of the universe?

text: So I've read the Wikipedia articles on WMAP and CMB in an attempt to try to understand how scientists are able to deduce the curvature of the universe from the measurements of the CMB. The Wiki article on CMB explains merely the following: > The peaks contain interesting physical signatures. The angular scale of the > first peak [of the power spectrum of the cosmic microwave background > radiation temperature anisotropy in terms of the angular scale] determines > the curvature of the universe (but not the topology of the universe). I don't really see, or simply do not understand, why. I also searched Stackexchange for similar questions, like this one for example, but in that example, the marked answer simply says: > We can view the lumpiness of the Universe then by observing the cosmic > microwave background radiation at high resolution. ...which doesn't really explain anything. Can someone explain in non-mathematical or math-light terms why this data can tell us anything about the curvature of the universe?


title: Why some electrons in ESR aligned their spin towards external magnetic and some against

text: Why some electrons in ESR aligned their spin towards magnetic field and some against?


title: Why is density an intensive property?

text: I am still trying to understand what are intensive and extensive properties. Possibly someone can give a pointer to a decent text (preferably on the web), as I am not too happy (to say the least) with what I found so far on the web. I already asked here one question on this, which I finally answered myself. My new problem (among several others) is that _density_ seems to be one of the first properties taken as example of an intensive property. While it seems a good approximation of what I know about solids and liquids, it seems to me a lot more problematic with gas, as they tend to occupy all the available space you give them. But none of the documents I found seems to make any resriction regarding density of gas. It seems to me that my opinion (apparently contested) that velocity is an intensive property, may be easier to support than the intensiveness of density in the case of gas. Or to put it differently, I do not see why _pressure_ should be more intensive than _volume_ , while wikipedia lists pressure as intensive, but not volume. Ideal gas law states that $PV=nRT$, which apparently gives a pretty symmetrical role to $P$ and $V$. And density depends on pressure (actually using this same formula and molecular weight). If it were not for the fact that some principles seem to be based on the concept, such as the state postulate which I found on wikipedia, I would start wondering whether these are real concept in physics.


title: Quantization of a particle on a spherical surface

text: Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is $$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \theta \dot{\phi}^2 $$ The canonical momenta are $$P_\theta = \frac{\partial L }{\partial \dot{\theta }} = m R^2 \dot{\theta }$$ and $$P_\phi = \frac{\partial L }{\partial \dot{\phi }} = m R^2 \sin^2 \theta \dot{\phi }$$ The Hamiltonian is $$H = \frac{P_\theta^2}{2 m R^2} + \frac{P_\phi^2}{2 m R^2 \sin^2\theta }$$ Now start to quantize the system. We replace $P_\theta $ and $P_\phi $ as $-i\hbar \frac{\partial}{\partial \theta}$ and $-i\hbar \frac{\partial}{\partial \phi} $, respectiely, to obtain $$H = -\frac{ \hbar^2 \partial^2}{2 m R^2 \partial \theta^2} - \frac{\hbar^2 \partial^2 }{2 m R^2 \sin^2\theta \partial \phi^2 } $$ This is apparently wrong, it should be the total angular momentum! So what is the right procedure to quantize a system, especially a system in curvilinear coordinates?


title: Why is scale factor inversely proprotional to temperature?

text: In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature $$a \propto \frac{1}{T}$$ What is the reasoning behind this relation? People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation?


title: Explain internal energy and enthalpy

text: Internal energy and enthalpy. I am finding it hard to distinguish between the two. Confused. Can someone explain me the two terms and difference between them? I tried learning from wikipedia but it said for both the above terms that it is the total energy contained by a thermodynamic system, so Iam confused?


title: Nature of motion of a pendulum

text: Consider a pendulum suspended from the ceiling of a lift in free fall , if its displaced from its mean position , what will be its nature of motion? what i thought was that it would simply stick to the ceiling as the lift's in free fall; but on the contrary what my teacher said was that , as                 "g" = 0 it's  time period  will be "infinity".         Thus it implies it won't move at all. But is that even possible? please provide me with some reason or correct me if I'm wrong. any help appreciated. cheers.


title: Polarizing virtual photons to make fusion easier?

text: So, this is probably incredibly foolish, outlandish, etc but I figure I'd post it anyways. Basically I was thinking about how the Deuterium nuclei repel each other and how they do this by exchanging virtual photons. After doing some research I came to the conclusion that virtual photons can indeed be polarized (I think when doing the calculations, one sums over the polarizations of the exchanged photons to obtain the total amplitude) and therefore can be blocked by using a polarizer in the opposite "direction". By this logic would it be super easy to do fusion by figuring out the polarization states of exchanged photons between nuclei, and then using some sort of sunglasses style direction polarizer to negate the repulsive force? If this was possible then wouldn't a small amount of electric potential be required to fuse which would make such a theoretical fusor incredibly efficient? Also on the same thread, would it also be better to use magnets similar to a synchrotron to keep the particles of D2 on the right path when fusing and having the photon polarization state detected? I know this sounds pretty insane, and I'm probably missing something really simple, but just my 2 rupees. I know that most fusions also rely on the attractive force of virtual gluons in the strong interaction at less the half a femtometer, but it's attractive so would this come into play at all?


title: How long will a permanent magnet retain its properties when used in a votrexing magnetic field?

text: How long will a permanent magnet remain effective when used within a vortexing magnetic field? I am refering to both the magnets making the field, and the magnet in the center that spins within the field.


title: Is the center of mass in general relativity equal to the center of mass in newtonian gravity?

text: Consider 2 point masses $A,B$ a distance $d$ away from eachother without velocity or rotation spin. Is the center of mass in general relativity equal to the center of mass in newtonian gravity ? In newtonian gravity the center of mass is where the particles collide , if they collide. Is the center of mass in general relativity also the place where the particles collide , if they collide ? Or maybe the center of mass in general relativity is the place where the particles collide if they had no initial velocity or rotation spin ?


title: Free energy device - QEG. Is this for real?

text: I'm not a physicist but I ran across reports on the web of a so called over unity or free energy device that has been created that supposedly works. It's called the QEG. Does this seem legit or is it yet another failed attempt to prove that free energy exists? Of course this goes against the 1st law of thermodynamics. I suppose those who believe in such a device would say they aren't creating energy, they are pulling energy in from the ether?


title: Is it possible to calculate how fast something will roll down a hill?

text: If I have a wheel, I know it's mass and diameter and the slope of a hill. Can I calculate the time it will take to get to the bottom of the hill? I am doing a project for my science fair and I sent 5 wheels down a hill. I thought the biggest wheel would go fastest but it didn't. I don't know why. The middle wheel wheel went the fastest. It was right in the middle of the weight and diameter. The smallest and lightest wheel went the same speed and the heaviest and biggest wheel. I know the diameter and masses of the wheels. The length of the hill and the degrees of the slope. Is there a formula that can tell me how fast it should go? I did not push it, I let all the wheels roll just by letting go of them so there wasn't any force. If anyone could help me understand this my Mom will try to explain what you tell me.


title: Is it physically realistic to have an electric field and polarisation density but no displacement field?

text: Given a Lagrangian density that describes a classical dielectric in interaction with the EM field, I found the Euler-Lagrange equations, and in the case of the electric field, worked through to find that $\vec{P} = -\epsilon_0 \vec{E}$. For some reason alarm bells are going off in my head but I'm not sure why. This implies from the usual equation $\vec{D} = \epsilon_0 \vec{E} + \vec{P}$ that the displacement field is zero. I suppose this just means that inside this dielectric the polarisation density exactly compensates $\epsilon_0 \vec{E}$, but I'm feeling unsure. Could somebody reassure (/correct!) me?


title: Does the heat produced in an inductor affected by the total current in the resistance in the circuit?

text: The energy stored in an inductor depends on the inductance of the inductor and the current flowing through it. But, assume that the switch is opened. Then, does the heat generated in the inductor dependent on the total resistance in the circuit or just the resistance of the inducor? ![enter image description here](http://i.stack.imgur.com/4bCfa.png)


title: parity invariance of Einstein, Maxwell and Dirac Lagrangians

text: How can we show that Einstein, Maxwell and Dirac Lagrangians are parity invariant?


title: Electric potential due to a point charge in Gaussian/CGS units

text: I learned electrostatics in SI units. In SI, the electrostatic potential due to a point charge $q$ located at $\textbf{r}$ is given by $\Phi(\textbf{r}) = \frac{q}{4 \pi \epsilon_0 |\textbf{r}|}$. Now, the Griffiths electrodynamics textbook says, "Converting electrostatic equations from SI to Gaussian units is not difficult: just set $\epsilon_0 \rightarrow \frac{1}{4 \pi}$." So, in Gaussian/CGS units, apparently $\Phi(\textbf{r}) = \frac{q}{|\textbf{r}|}$. However, one textbook (Understanding Molecular Simulation, by Frenkel and Smit) says that the potential due to a point charge is $\Phi(\textbf{r}) = \frac{q}{4 \pi | \textbf{r} |}$. Did I make a mistake, or did Frenkel and Smit? Thank you.


title: Spectrum measurement

text: How can be spectrum of hydrogen measured (Lyman series, Balmer series, Paschen series and so on)? I mean schema of measurement circuit and the measuring technique (including all the steps needed). Is there difference between spectrum of atomic and molecular hydrogen?


title: Landau poles in dimension <4?

text: It is well-known that QED and $\Phi_4^4$ quantum field theory have a Landau pole and therefore are not asymptotically free. Is this specific to 4-dimensional QFT, or are there examples of theories with Landau poles in 2 and 3 dimensions?


title: How is the physical Lagrangian related to the constrained minimization Lagrangian?

text: If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is $$L = V(q) + \lambda C(q).$$ I have fairly solid intuition for this Lagrangian, namely that the energy gradient is normal to the constraint manifold iff we can annihilate the energy gradient using constraint derivative terms. In contrast, I have extremely poor intuition for the Lagrangian in the Lagrangian formulation of dynamics: $$L = T(q) - V(q),$$ where $T(q)$ is kinetic energy. How are the two Lagrangians related? In particular, can my intuition for the constrained minimization case be converted into intuition for the physical case?


title: Is abruptly phase change a dominant cause in observable interferece pattern?

text: In newton's ring, plane parallel film or in measuring thickness of a paper we see interference pattern confirming wave nature of light. Will interference be observable if we let there is no phase change in hard reflection? Is abruptly phase change a necessary condition for observing interference pattern?


title: Is the spin 1/2 rotation matrix taken to be counterclockwise?

text: The spin 1/2 rotation matrix around the z-axis I worked out to be $$ e^{i\theta S_z}=\begin{pmatrix} \exp\frac{i\theta}{2}&0\\\ 0&\exp\frac{-i\theta}{2}\\\ \end{pmatrix} $$ Is this taken to be anti-clockwise around the z-axis?


title: Light emission spectrum units

text: Do someone knows the units of the spectra provided here ? It seems obvious enough that it's said nowhere, but even Wikipedia and other sites are quite blurry on this point. So, is it power ($W$), radiance($\:\rm{W/m^2sr}$), or something else ? Thanks !


title: Electric dipole, error in calculation

text: Currently I am calculating the dipole moment of a metal sphere in a uniform electric field $E_0$ in z-direction. From here I know that the charge density look at page 15 is given by $ 3 \epsilon_0 E_0 cos(\theta)$ Now I wanted to calculate the resulting dipole moment by $ \int_0^R \int_0^{2\pi} \int_0^{\pi} r^3 sin(\theta) cos(\theta) 3 \epsilon_0 E_0 d\phi d\theta dr$ but in this case the integral over $\theta$ is zero and therefore this whole term will be zero which is somewhat strange, since there should be a dipole moment. So what am I doing wrong?


title: Why does my watch act like a mirror under water?

text: I have a digital watch, rated to go underwater to $100 \rm m$. When it is underwater it can be read normally, up until you reach a certain angle, then suddenly, it becomes almost like a mirror, reflecting all the light that hits it, almost perfectly (you cannot read the digits, the entire surface becomes reflective.) I've seen this happen with other waterproof watches too, so I don't think it's unique to mine or the specific model. I'm wondering what causes this? My physics teacher was stumped when I told him about this (we're doing lenses and imaging in physics right now.) I think it has something to do with internal refraction. I haven't been able to measure the angle it becomes reflective accurately, I estimate about 30-40 degrees. Near this critical point, it can be half and half reflective, where only half becomes a mirror, but it's always either reflective or not - never in between being reflective and non- reflective.


title: Why isn't ray optics applicable for marginal rays?

text: I have been studying light and going a little bit ahead I studied about ray optics and one thing each book had in notes is ray optics is applicable only for paraxial rays not marginals rays, I was wondering why not for marginal rays. I googled it and got answer but their language was very complexed and I simply could not understand them


title: Maximum-Entropy Method deconvolution implementation?

text: I am looking for an implementation of the Maximum Entropy Method for image deconvolution, ideally in MATLAB or Mathematica. I'm trying to reconstruct an image from fringe visibilities, as is often done in aperture-masking interferometric astronomy, but I have only found minimal pseudocode in the literature.


title: How to understand the Lagrangian of the standard model, effective or "fundamental"

text: I have a question about understanding the Lagrangian of standard model, should we view it as a "fundamental" or effective theory? The "fundamental" theory here means the theory with physical cutoff (maybe Planck scale) $\Lambda$; the effective theory here means the theory given by lowering the cutoff $\Lambda'<\Lambda$, (Wilsonian renormalization), and we pick up the leading terms as we saw in the action. **The reason to think the standard model Lagrangian is an effective action** We haven't tested the physics up to the physical cutoff (Planck scale?), this is an extra assumption that the standard model is valid up to physical cutoff. To the contrary, in the effective theory, the cutoff is much more reasonable (e.g. LHC scale). **The reason to think the standard model Lagrangian is a "fundamental" action** If we follow the renormalization flow, the coupling constant of QCD will grow in low energy scale, then the perturbation breaks down. We know from lattice calculations of hardron mass, that even in low energy, QCD works. The Wilsonian renormalization group transformation is formulated from perturbative expansion (may be this is just my limited knownledge, correct me if I am wrong). Since QCD agree with experiment beyond the regiom of perturbation, it has to be a "fundamental" action, than effective action. Similar attitude applies to the standard model. It seems the question is rather a mental setup. Since standard model works very well so far at LHC, it seems hard to distinguish these two opinions: whether viewing standard model action as "fundamental" or effective.


title: What the quantum spin refers to? How we calculate the angular momentum $\omega$ from the spin quantum number?

text: How we calculate the angular momentum $\omega$ from the spin quantum number, to know how fast a particle rotating on its own axis?


title: Equivalent definitions of vectors

text: Equivalent definitions of vectors. In maths a vector is an object that obeys some axioms of a vector space. But in physics a vector can be thought as an object which is invariant under rotations of the system's coordinates. Is there a way to prove this equivalence?


title: How much does the Earth's rotation affect a golf ball?

text: When launching rockets into space, they start straight but begin to lean, I assume due to the Earth's rotation. For something on a much smaller scale, such as a golf tee, is there a measurable trajectory alteration, or is it something so incredibly minuscule that there's virtually no effect?


title: Visualise the sound intensity

text: I'm studying Biophysics and my current subject is **sound**. One of the properties of sound is **intensity**. From my notes I can see the following definition: **Intensity Formula is:** $(I = w*m^{-2})$ or $(I = \frac{w}{m^2})$ where _w = amound of energy_ and _m = area_. **Definition** : Intensity is the amount of energy passing through an area of $1m^2$ perpendicular to the direction of sound wave propagation within 1 second. So I came with this picture: http://i.stack.imgur.com/3OZXH.jpg I know the picture is lame. What I care about is if the above definition is diplayed correctly here. I'm not a native English speaker and the word _perpendicular_ in this context confuses me. Thanks


title: Antimatter bomb

text: I stumbled upon this wikipedia article on antimatter weaponry. Being greatly appalled by the sad fact that large sums of money are being wasted on this, I could not stop myself from thinking for a moment about the physics behind it. If you somehow put together a gram of antimatter and a gram of matter all at the same instant and at high density (so the "annihilation efficiency" would be 100%), would there actually be any explosion? AFAIK, this would just produce large amounts of gamma photons, neutrino's etc., but there's be very little (anti)matter left to absorb the energy in the form of kinetic energy. In other words -- it would be a radiation bomb. There wouldn't even a flash of light to warn anyone. Would this indeed be the case? Or am I overlooking something here?


title: Physics, double Atwood machine

text: ![enter image description here](http://i.stack.imgur.com/pjaZh.jpg) According to this tutorial over here: http://commons.wikimedia.org/wiki/User:Phys1csf4n It is stated that $a_A + a_B + 2a_C = 0$ But I don't understand the logic used. Is there any way to prove this mathematically, instead of logically? It holds true for the two cases presented, but how do I know that it holds true for every single case? I can not think of the solution for some reason. I do know that $T_4 = T_3 = T_1 + T_2$ and $T_1 = T_2$, but there is no way to relate the acceleration because the masses are different.


title: Is the Graveyard Really so Serious?

text: Calculations in relation to black holes are solely in consideration of spacetime curvature and its effects. They are in total alienation with respect to the action of inertial agents[external forces].That world lines do not point outwards from a black hole is consistent with the fact that inertial agents are absent. Can the presence of inertial agents change the picture of the world lines? Let's consider the following situation: _An experimenter [source of "external forces" capable of producing non- geodesics] throws some mass into the black hole. Its surface area and volume increase.This is suggestive of outward world lines from the surface, causing the expansion----points on the surface should move outwards.The presence of "external forces" has modified the entire perspective of the situation._ The black hole picture takes into consideration only the blackhole. _Right from the outset the black hole is considered in isolation from the rest of the universe._ Any type of modification due to the presence of external agents--- artificial or natural ones do not figure in the blackhole calculations. Consideration of such factors can modify the picture of the black hole. Can technology prevent somebody near a black hole from falling into it? Can it remove the apprehensions of what we know to be a graveyard?


title: Effect of a wavefront deformation on the far-field diffraction pattern of a TEM00

text: By performing Matlab simulations on a TEM00 mode (approximated by a gaussian intensity profile with a flat wavefront), I got the impression that applying wavefront deformations (such as a single Zernike coefficient) on the input field had very minimal effects on the far-field diffraction pattern (computed with a Fourier-transform of the input field), even for deformations up to several tens of Lambda. The only exceptions seem to be the first+second order Zernikes (a linear phase gradient), which shift the beam as they should, and the fourth order Zernike (spherical deformation) which focuses the beam as expected. No significant effect was observed with the coefficients corresponding to astigmatism and coma, for instance. On the contrary, even small deformations (such as astigmatism and coma) applied to such a gaussian profile but with an added phase helix (phase given by 2*pi*Theta, where Theta is the angle in polar coordinates in the wave plane) have dramatic effects on the shape of the far-field pattern, which is normally a perfect donut (approximating an LG01 mode, the first non-trivial Laguerre-Gaussian mode). What puzzles me is that, from what I have seen in a simple experimental realization of the first case (laser beam coming out of a monomode fiber, wavefront deformation applied by a LC-SLM, focusing of the beam with a microscope objective, far-field diffraction pattern observed with a camera looking at the reflection of the beam on a microscope slide), even slight deformations have a very visible effect on the "TEM00" (ok, exactly how TEM00 the beam stays after going through all the optical elements is hard to say). Am I missing something ? (illustrations can be provided upon request)


title: Determining the spin of wavefunction

text: We all know that by uncertainty principle, location of a wave-particle is perfectly determined when uncertainty of momentum becomes infinite. (I also heard that in reality, it is almost impossible to determine the location of a wave-particle perfectly and the location has some uncertainty.) So, what happens to spin? Spin can be superpositioned, so I am wondering what happens to spin.


title: A good, concrete example of using "chaos theory" to solve an easily understood engineering problem?

text: Can anyone suggest a good, concrete example of using "chaos theory" to solve an easily understood engineering problem? I'm wondering if there is a an answer of the following sort: "We have a high level objective to design a system that does XYZ. To achieve this objective, we propose the following design D. If we look at the low level physics of our design, we see that the dynamics are nonlinear and exhibit "chaotic" behavior, but we can categorize the chaotic behavior as of being type ABC. Because we are able to understand that the chaotic dynamics are of the specific type ABC, although we can't make precise statements about every aspect of the system, we are still able to make the following "high-level" claims. And using those claims, we have designed a system in which,although certains parts are behaving "chaotically," the system still performs our desired objective XYZ very effectively." I'm not looking for answers of either the following sort: "Looking at the underlying physics we see that the dynamics are chaotic, but we can also see that if we introduce a mechanism EFG into our design, we can see that it will "dampen" the chaotic behavior and then leave us in a place where we can find a good solution." or "Looking at the underlying physics, we can see that certain aspects might be chaotic, but it turns out for the following reasons R1, R2, etc that underlying chaos has no bearing on our high level objective and we can create a solution in which we do not need to worry about the chaotic aspects." * * * An ideal answer might be something like "even though it's clear that this wing design is going to create a lot of turbulence, we can see that the turbulence will have a certain "structure", and thinking about this "structure" just a little bit, we can see that it's going actually work in our favor and greatly increase fuel efficiency." And course the problem is that it seems so implausible that turbulent flow ever works out this way -- rather it is something that is meant to be stamped away at all costs, but I imagine there must be some other example out there. I find most texts on nonlinear dynamics/chaos very reasonable in their mathematical development of why chaos occurs, and how it has certain structure, but while I find myself able to see ways of avoiding chaos or even ignoring it, I don't have a good idea of how I could use my knowledge of it's structure directly to my advantage. thanks!


title: What is the relation of the release time of fossil fuel emissions to climate change?

text: This question presupposes that   1. The amount of fossil fuels on earth is limited   2. Mankind will burn all of its fossil fuels eventually Based on these presuppositions, the total amount of CO2 released into the atmosphere from burning fossil fuels during time X (where X is the time it takes to use up all fossil fuels) is constant. Regulations of emissions would thus merely increase or decrease X while the total emissions remain constant. What beneficial and/or negative effects would a shorter X (no emission guidelines) thus have over a longer X (strict emission guidelines)?


title: Dehn twists and topological order

text: I am trying to understand notion of a "Dehn twist" and how it relates to topological order. In particular refering to http://arxiv.org/abs/1208.4834 it is stated that Xiao Gang Wen's paper on "Topological Order in Rigid States" (http://dao.mit.edu/~wen/pub/topo.pdf) is supposed to provide an introduction to "non-abelian adiabatic Berry phases associated with Dehn twists for a U(1) Chern Simons Theory". Skimming through the respective paper however I could not find the notion of a "Dehn twist" appearing at all? Maybe it appears under a different name or it is not given a name at all? I would be very happy for any support. Best.


title: Entropy: two explanations for the same quantity?

text: I studied thermodynamics and I saw the following definition for entropy: $$ \Delta S = \int_1^2 \frac{\text{d}Q}{T} $$ that we use to calculate $\Delta S$ for different types of transformations. In the last lecture we started to talk about entropy like measurement of disorder and information. The form of entropy becomes: $$ S = k \ln{W} $$ where $W$ is the number of microstates. At this point i felt lost and searching in internet I increased my confusion. I really can't see the relation of the two formulation of the same quantity. How they are related? Disorders means disorder on the microscopical structure of matter? What is the "information" carried by entropy?


title: Does a different opposing force affect work?

text: Suppose a man exerts $10~N$ as he lifts a $1~kg$ box a distance of $2~m$ against Earth's gravity. To determine work we can use the following equation: $$ W = F \cdot d \\\ W = (10~N) \cdot (2~m) = 20~J $$ The work in this case is $20~J$. Would work be the same if the man performed this task on the _moon_ rather than the _Earth_? Mathematically, the equation shows that if the man exerts the same force ($10~N$) over the same distance ($2~m$), then the work will remain the same ($20~J$) -- but I'm having trouble conceptualizing this. $$ MoonGravity < EarthGravity $$ The force opposing our movement when lifting the box on the moon would be _less_ than that on Earth since the gravity on the moon is far _smaller_. Conceptually, it seems that it'll be "easier" to lift the box on the moon, and thus should take _less work_. Where is the fault in my logic?


title: Energy Equation

text: I am reading a book on radiation and the author, while developing the equation of radiation transfer provides two forms of the equation, the second incidentally being more general and useful that the first:$$\rho C_v\frac{D T}{D t} =\nabla \cdot (k\nabla T) - p \nabla \cdot U - \nabla \cdot q_R $$ to $$ \rho C_p\frac{D T}{D t} =\nabla \cdot (k\nabla T) + T\beta\frac{D P}{D t} - \nabla \cdot q_R$$ (Note: derivatives are material derivatives) Question: (1) How are these two forms equivalent? (2) How is the second more useful?


title: Deriving infinitesimal time dilation for arbitrary motion from Lorentz transformations

text: I'm trying to derive the infinitesimal time dilation relation $dt = \gamma d\tau$, where $\tau$ is the proper time, $t$ the coordinate time, and $\gamma = (1-v(t)^2/c^2)^{-1/2}$ the time dependent Lorentz factor. The derivation is trivial if one starts by considering the invariant interval $ds^2$, but it should be possible to obtain the result considering only Lorentz transformations. So, in my approach I am using two different reference frames $(t,x)$ will denote an intertial laboratory frame while $(t',x')$ will be the set of all inertial frames momentarily coinciding with the observed particle, i.e. the rest frame of the particle. These frames are related by $$t' = \gamma \left(t-\frac{Vx}{c}\right),\quad x' = \gamma \left( x - V t\right),$$ where $V$ is some nonconstant (i.e. time dependent) parameter which is, hopefully, the velocity of the particle in the laboratory frame. Treating $x$, $t$ and $V$ as independent variables (for now) and taking the differential of the above relations, I obtain $$dt' = \gamma \left(dt-\frac{Vdx}{c}\right) - \frac{\gamma^3}{c^2}(x-Vt)dV,$$ and $$dx' = \gamma \left( dx - V dt\right) - \gamma^3 \left(t-\frac{Vx}{c}\right) dV.$$ Imposing either the definition of the rest frame $dx'=0$ or (what should be equivalent) $dx = Vdt$, the only way in which i obtain $dt = \gamma dt'$ is if $dV=0$. So, the derivation breaks badly at some point or I must be wrong in using some of the above equations. Which one is it?


title: potential energy change in conservative field

text: Is it true that potential energy of a particle always decreases in the direction of conservative force acting on it?


title: Newton's Law of Graviation: Why $G$ and not e.g. $\dfrac{1}{4\pi G_0}$?

text: I've been wondering, in Coulomb's Law, $k_e = \dfrac{1}{4\pi\epsilon_0}$. Therefore, why do we use $G$ in Newton's Law of Gravitation? What if the constant is more like Coulomb's Law, e.g. $G = \dfrac{1}{4\pi G_0}$ where $G_0$ is some constant. This would make Newton's Law of Gravitation look like the following: $$\bf{F}_{12} = -\dfrac{m_1 m_2}{4\pi G_0 |\bf{r}_{12}|^2} \bf{\hat{r}}_{12}$$ $GM = \mu$ is used for calculations but so could $\dfrac{M}{4\pi G_0} = \mu$. If this is not the case, what is the significance of this definition?


title: why there is no accuracy of the measured value of $G$?

text: With the advancement of Modern Technology still there is no accuracy of the measured value of $G$ Gravitational Constant, why!?


title: Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

text: Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using $$\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A),$$ where $PV$ denotes the Cauchy Principal Value, we get (taking only the imaginary part): $$\Im{F} = -\pi \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y\, \delta(f(x,y)) .$$ The trouble I got is that the zeros of $f(x,y)$ which I call $y^{\pm}$ seems to be outside integration range and hence the delta should yield zero. BUT here's what's funny: when I ignore all this and just perform the formal calculations (assuming I do it correctly) namely; replacing $\delta(f(x,y))$ with $$\frac{1}{\bigl\vert \partial f/\partial y\bigr\vert_{y=y^{\pm}}}\times(\delta(y-y^-)+\delta(y-y^+)),\ \ \ (1)$$ (where $|\partial f/\partial y|_\pm $ are equal) and assuming that $y^{\pm}\in[0,1-x]$ (which seems to be false) the two deltas just give $1+1 = 2$. Then the result seems to be correct, or at least it agrees with what I have calculated the same thing using a totally different method. Could this all just be a coincidence? I mean shouldn't the deltas produce zero if $y^{\pm}\notin[0,1-x]$, or I'm I using the wrong formula $(1)$?


title: Eigenfunctions in periodic potential

text: For Hamiltonian $\operatorname H$ and lattice translation operator $\operatorname T$, if $$\operatorname H\psi=E\psi, \qquad \operatorname T\psi=e^{ik\cdot R}\psi,$$ and $$\operatorname H\phi=E\phi, \qquad\operatorname T\phi=e^{ik\cdot R}\phi,$$ then $\psi=\phi$? Here $R$ is lattice vector, $\psi$ can be $\psi_k$ and $\phi$ can be $\psi_{k+K}$, and $K$ of reciprocal lattice.


title: Proof that flux through a surface is independent of the inner objects' arrangement

text: $$\Phi=\iint_{\partial V}\mathbf{g} \cdot d \mathbf{A}=-4 \pi G M$$ Essentially, why is $\Phi$ independent of the _distribution_ of mass inside the surface $\partial V$, and the shape of surface $\partial V$? That is, I'm looking for a mathematical justification for the characterization of flux as a kind of "flow" of gravitational field through a surface. The reason I worry about that is this. I have encountered the following proof of Gauss' Law for gravity: $$\iint_{\partial V}\frac{-MG}{r^2}\mathbf{e_r} \cdot \mathbf{e_r}d A=\frac{-MG}{r^2}\iint_{\partial V}d A$$ $$=-MG4 \pi$$ However, two assumptions are made in the above proof that I've not convinced myself hold true for the general case:   * The Gaussian surface is a sphere.   * It is assumed that all mass is concentrated at the centre of the sphere. Returning to my original question:   * Why is it true that when the Gaussian surface is deformed (not adding or removing any mass to inside the surface, of course), the amount of flux, $\Phi$, through the surface is equal to the flux in the spherical case?   * Why is it true that the positions of the masses inside the surface doesn't affect $\Phi$? **Edit:** removed reference to Poisson's equation, changed much of question to make it clearer. **Further edit:** I am happy that $$\iint_{\partial V}\mathbf{F}\cdot d \mathbf{A}=\iiint_{ V}( \nabla \cdot \mathbf{F})dV$$ However, my problem is with the following equation: $$\nabla \cdot \mathbf{g}=-\rho G 4 \pi $$ Which can be used to get $$\iint_{\partial V}\mathbf{g}\cdot d \mathbf{A}=\iiint_{ V}(-\rho G 4 \pi)dV=-M G 4 \pi$$


title: a wierd image of sun

text: Have you ever notice the sunset's image in the sea? It's like long light path to the end of the horizon! I've attached a sample of this: ![a long light path](http://i.stack.imgur.com/hB7L9.jpg) How can we explain this? I know that it can happen even in wave less sea or lake! Why is this path so continuous? I expected that sea should act like a mirror!


title: What is the Schechter luminosity function's domain of support?

text: I'm trying to fit a Schechter luminosity function to some data points, but it's not clear from this definition what the domain of support of the PDF should be. I'm familiar with the standard Pareto distribution $$p(x) = \frac{\alpha-1}{x_0}\left(\frac{x}{x_0}\right)^{-\alpha}$$ which is non-zero over $[x_0,\infty)$. The Schechter function has a similar form (using slightly different conventions from the Wikipedia article): $$p(x) \propto e^{-\frac{x}{x_0}}\left(\frac{x}{x_0}\right)^{-\alpha}$$ Can anyone confirm that this distribution has the same support as a standard Pareto distribution? I'm not sure how to normalise it, and the Wikipedia article is light on details.


title: Can one assign an equivalence principle of some kind to the EM field?

text: Introduction: Consider the EM field. There was a time when the field was defined in a similar manner to that of the gravitational field. This changed when the view on gravitation evolved to this new idea which suggested it was a consequence of space-time curvature.   Now consider the equivalence principle, it was due to this principle that the view on gravitation was changed from that of a classical field to one which suggested a more geometric nature. I cannot help but think that we should of owed the same respect to the EM field since it doesn't seem unreasonable to think that an some equivalence principle of some kind may be assigned EM field. If we may assign an equivalence principle to the EM field then perhaps we can define the electromagnetic field in some geometric way as well.   I understand and certainly agree that EM energy stored in a volume of a space will contribute to the mass-energy content in that volume and hence curvature but this is not what I am talking about, rather I am considering the possibility of a newer description of the EM field which may complement the geometric description of the gravitational field in a more explicit way. Consider gravity before it was described by Einstein, I claim that one could derive an gravitational tensor in the same way as one can derive an electromagnetic tensor $F^{\mu\nu}$. I say this because gravity possesses a gravito-magnetic property. This isn't surprising to me since magnetism is nothing more than a relativistic effect of static fields. What I am trying to say is that before Einstein, one could in principle derive a gravitational tensor which was analogous to the electromagnetic tensor yet after getting to this point _it would still remain a special-relativistic compatible classical description of a gravitational field (meaning it doesn't necessarily take gravitational time dilation into account)_ ; in this sense the electromagnetic tensor describes the EM field classically. Of course this is not important anymore since we DO have a more powerful set of machinery for describing the gravitational field. All this to me only suggests that the EM field could be better described at the macroscopic scales. Conclusions: 1) The principle of equivalence made it possible to assign a geometric description to the gravitational interaction. 2) If gravitational fields may be defined in a geometric way then perhaps the EM field can be as well (if a principle of equivalence may be defined for the EM field). Questions: 1) Is it impossible to assign an equivalence principle of some kind to the EM field? If not, why? 2) If it is possible to assign an equivalence principle to the EM field then what may be the first steps in constructing the EM field in a more geometric way? 3) Where is my logic / thought process flawed?


title: Why are objects opaque?

text: I have been searching the internet for answers to this question, but haven't found a convincing one. I would appreciate any response. I understand why objects are opaque/black. For example when ( _white_ ) light is incident on a closed book ( _let's say it's blue_ ), all wavelengths of visible light except blue are absorbed and hence we see it as blue. Now, from what I read, [0.] it is the electrons that absorb the ( _non-blue_ ) photons from the white light and jump to higher energy states. My questions are: 1 - Do the electrons stay in the same state? 2 - If they were to jump back to there original state, shouldn't they emit a photon that is exactly of the same wavelength as the one that was absorbed by it in the first place? 3 - If they are in the same state, why does the book continue to appear blue? It can't absorb additional energy unless it returns back to its original state, In doing so won't it emit a photon of the same wavelength it absorbed in the first place? 4 - And , finally, what part of the atom is reflecting the blue light? Is it a bouncing electron?


title: The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

text: Why does a particle with position $\textbf r(t)$ that satisfies $\textbf{v}(t) = \textbf{c}\times \textbf{r}(t)$ follow a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$?


title: Why does the sky suddenly look gray through this window?

text: I am looking to the windowed roof of my building, and I notice that the sky, which has few white clouds, sometimes looks completely gray, as if there was a huge cover of gray clouds, but there are not. By observing it and waiting for the change to happen, I notice it takes a matter of seconds : when the sun shines directly on the building (but not directly on me), the blue sky turns gray. I wonder if this is just an optical illusion because my vision of colour could be relative to its environment, or if there is serious glass/sun/refraction effect or something. I haven't found anything on that matter on the internet or on Stack Exchange, so I'm asking you what you think about it. Additional information : The top window is a very peculiar glass, it's basically a strong glass window (because it's almost horizontal, it has to resist hits) and looks like it is covered by a layer of opaque black, with big holes (surface of holes is greater than surface of black, from a distance you don't see the holes). Also, I'm taking the pictures from a windowed office, you see the sky through two glass windows, but the one closest to me looks basic and I didn't notice a difference when looking through one layer. You will also notice there is a blue light, some neon-like directed at the roof, but during the day its effect is negligible. Here are the pictures, first the sky through top window without sun hitting the window, then with the sun (The colours/ exposition differences don't seem to make a great difference compared to my naked eye observation. The pictures were taken only a few seconds apart, you can see the same cloud in both pictures. The gray sky isn't the most grayish I've observed, it gets even farther away from blue sometimes) : ![Sky through top window without sun hitting the window](http://i.stack.imgur.com/fbIyO.jpg) ![Sky through top window with sun hitting the window](http://i.stack.imgur.com/hBmW9.jpg)


title: What are the reasons why dark matter above the TeV scale appears to be ignored experimentally?

text: I understand that sub-TeV-scale dark matter would fit nicely with TeV-scale SUSY, but are there any other reasons for the large experimental blind spot between TeV-scale WIMPs and the much larger masses constrained by microlensing? Is it just an experimental difficulty, or is it some theoretical bias? I mean, from perusing arxiv it seems that it's not hard to come up with self-interactions that get around cosmological constraints, and it's not at all like self-interactions are somehow implausible.


title: Is the universe expanding faster than we could catch up with it?

text: Like the title says, could we ever catch up with the expanding universe?


title: Fermat's principle to prove the law of reflection

text: Fermat's principle tells that a light ray will follow a path from point $A$ to point $B$ so that the optical path length of this path is an extremum over neighboring paths. I wanted to use this principle to prove the law of reflection. They do this f.i. here: http://scipp.ucsc.edu/~haber/ph5B/fermat09.pdf (the pdf can also be found when googling "Fermat's principle and the laws of reflection and refraction"). They show that the path where $\theta_1 = \theta_2$ is indeed a minimum over neighboring paths in the same plane that also reflect on the mirror. But I don't really see how this prove is complete, because I can easily imagine a path from A to B, which deviates only infinitesimal from the path that reflects on the mirror, which has a shorter optical path length. For instance a path that reflects before the mirror. Because of this is doesn't seem to me that the path is really a minimum over all neighboring paths, we have only proven that it is over some neighboring paths. So I guess that I'm missing something here and I hope someone can explain me.


title: Angles on swing sets

text: I'm building a swing set for my children. All of the designs I've seen involve building two A-frames and connecting them at the top with a crossbar/beam from which hang the swings. The A-frames are placed perpendicular to the ground. In addition, it seems that the A-frames are commonly constructed at an angle of 12 degrees from the perpendicular to the ground. I have several questions:   1. Where does the 12 degree angle come from?   2. I've been on swingsets that move too much. I think this is due to a combination of unanchored posts and the angle between the legs of the frame being too little. How do I compute the angle that will provide optimum stability?   3. Shouldn't the frames be placed at an angle with respect to the ground (e.g. 70 degrees) rather than perpendicular, in order to prevent side-to-side motion?   4. How would I go about computing the above? Assumptions that I'm going on include:   * 2 swings   * Adults will swing (up to 200 lbs)   * A-frames will be anchored to concrete piers


title: Parallel-plate capacitor

text: I'm trying to grasp parallel-plate capacitors in a class of electromagnetic field theory. My book leaves out a lot and I couldn't figure it out from what I found on Google.   1. The book's procedure is to place a charge Q on the surface of the one conductor plate and -Q on the other. Now would this charge only be distributed over the surfaces of the plates that face each other, or on all four surfaces?   2. Is it possible to prove that any kind of source of potential difference connected to a capacitor can be represented as two opposite charges like that?   3. The book treats one of the charged surfaces as a conductor surface (or this is my understanding) and this yields, with $\Delta S$ being area, $E_n=Q/\epsilon\Delta S=\rho/\epsilon$ when using Gauss' law (if you place say a thin box around the charged surface because the electric field within the conductor is zero when in equilibrium). I find this way of doing it very odd because first you place Q on the one surface and -Q on the other, and then disregard that you've put -Q on the other surface. And I actually don't understand how come the -Q does not matter. How can this work?   4. A different I think more intuitive way of doing it that I've found online is to treat each conductor as a sheet of charge. Then you have twice the area to spread out the charge over, so you get $E_n=\rho/2\epsilon$ directed inwards from each surface but with different signs, the sum which is $\rho/\epsilon$ in between the plates, just what we want. I like this because it uses the setup with Q and -Q, plus it's easy to see how the fields cancel each other out. But why is it OK to consider a conductor to be a sheet of charge? Is it because the electric field will not be changed by passing through a conductor even though it will temporarily be zero within it? Thank you in advance.


title: How can mass affect spacetime?

text: In General Relativity Theory, mass can warp spacetime. However, in my view interaction only occurs between pieces of matter. Spacetime is not matter; how can it be affected by matter?


title: Can light exists in $2+1$ or $1+1$ spacetime dimensions?

text: Spacetime of special relativity is frequently illustrated with its spatial part reduced to one or two spatial dimension (with light sector or cone, respectively). Taken literally, is it possible for $2+1$ or $1+1$ (flat) spacetime dimensions to accommodate Maxwell equations and their particular solution - electromagnetic radiation (light)?


title: Schrodinger equation in term of Fokker-Planck equation

text: From Wikipedia on the Fokker-Planck equation: > $$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) > ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) > \frac{\partial }{\partial x}+D_2 \left( x,t\right) > \frac{\partial^2}{\partial x^2}\right] \delta\left( x^{\prime}-x\right) > \right) f\left( x,t\right).\qquad $$ > > _Integrate over a time interval $\varepsilon$,_ > > $$f\left( x^\prime ,t+\varepsilon \right) $$ $$~=~\int_{-\infty }^\infty \, > dx\left(\left( 1+\varepsilon \left[ D_{1}\left(x,t\right) \frac{\partial > }{\partial x}+D_{2}\left( x,t\right) \frac{\partial^{2}}{\partial > x^{2}}\right]\right) \delta \left( x^\prime - x\right) \right) f\left( > x,t\right)$$ $$\tag{2}+O\left( \varepsilon ^{2}\right).\qquad $$ OK, but Fokker-Planck equation for one dimension is usually $$\tag{0} \frac{\partial}{\partial t}f(x,t) = -\frac{\partial}{\partial x}\left[\mu(x,t)f(x,t)\right] + \frac{\partial^2}{\partial x^2}\left[ D(x,t)f(x,t)\right].$$ I was not able to understand how one gets from the original equation (0) to the above (1) and how does the first equation (1) lead to the second equation (2). Can anyone explain this?


title: Analytic form of the normalization constant for Laughlin wavefunction

text: Is there any analytic form of the normalization constant for Laughlin wavefunction $$\prod_{i < j} (z_i-z_j)^{1/\nu} e^{-\sum_i |z_i|^2/4}$$ where $\nu$ is the filling factor?


title: What physical processes may underly the collisional term in the Boltzmann equation, and how do they increase entropy?

text: Consider particles interacting only by long-range (inverse square law) forces, either attractive or repulsive. I am comfortable with the idea that their behavior may be described by the collsionless Boltzmann equation, and that in that case the entropy, defined by the phase space integral $-\int f \log f \, d^3x \, d^3v$, will not increase with time. All the information about the initial configuration of the particles is retained as the system evolves with time, even though it becomes increasingly harder for an observer to make measurements to probe that information (Landau damping). But after a long enough time most physical systems relax to a Maxwellian velocity distribution. The entropy of the system will increase for this relaxation to occur. Textbooks tend to explain this relaxation through a collisional term in the Boltzmann equation ('collisions increase the entropy'). A comment is made in passing that an assumption of 'molecular chaos' is being made, or sometimes 'one-sided molecular chaos.' My question is, how do the collisions that underlie the added term in the Boltzmann equation differ from any collision under an inverse square law, and why do these collisions increase entropy when it is clear that interactions with an inverse square law force do not generally increase entropy (at least on the time scale of Landau damping?) And finally, how valid is the so-called molecular chaos assumption? EDIT: I should clarify that, if entropy is to increase, then it is probably necessary to invoke additional short-range forces in addition to the long range inverse square law forces. I suppose I could rephrase my question as 'what sort of short-range forces are necessary to explain the collisional term in the Boltzmann equation, and how do they increase entropy when inverse- square law collisions do not?' If the question is too abstract as written, then feel free to pick a concrete physical system such as a plasma or a galaxy and answer the question in terms of what happens there.


title: Maximum speed higher than the speed of light

text: 1. I wanted to ask if higher speed than the speed of light will be discovered, will scientists be able to adjust the special relativity to the new situation?   2. I read that informations transmitted faster then the speed of light, but According to special relativity, $c$ is the maximum speed at which all matter and information in the universe can travel. Is there a contradiction here?   3. How the fact that the speed of light is the maximum speed for all matter, affects special relativity?


title: Gravitational Length Contraction

text: General Relativity predicts that a clock at rest in a gravitational field will run slower than a clock in free fall. Similarly, will a vertical ruler on the earth's surface be shorter than a ruler in free fall? Why or why not and by how much?


title: Why is it more convenient to consider space or time as a continuum?

text: I often find that phisicists and cosmologists make use of Planck's units. I have read propositions that sound like > " _...at the level of Planck's units many law of physics break down_ " > > " _...Planck time, the smallest observable unit of time...before which > science is unable do describe the universe_ " > > " _... it would become impossible to determine the difference between two > locations less than one Planck length apart_ " even in string theory: > "* Planck length is the order of magnitude of the oscillating strings that > form elementary particles, and shorter lengths do not make physical sense*". Notwithstanding this and the fact that in QM (almost) everything is quantizied (discrete) I read that in mainstream they are still considered not discrete, cf. Phys.SE link. I am not particularly aware of any pros, I see only cons; can someone tell me what are the compelling reasons to consider space and time not discrete? Is it a requisite of relativity? As a corollary, I suppose that they must be both discrete or non discrete, right?


title: How does liquid convert to gas on getting thermal heat energy?

text: Say for example, when we heat, water converts to steam gas. How does it happen? What happens underneath giving rise to breaking of bond between molecules in liquid state and spreading them in gas state..?


title: why egg cooks slowly in mountains?

text: A quick Google tells me "Because water boils at a lower temperature at high altitudes". But I am not being able to understand this answer and fill-in the gap. Like, how does an egg cook in the first place? What does it mean when we say that water boils at lower temperature? In fact, I would have thought otherwise that since water is able to boil at low temperature itself (which will be reached sooner), the egg will be able to cook sooner. What is wrong with this reasoning?


title: Readable books on advanced topics

text: I realise that there are already a few questions looking for general book recommendations, but the motivation and type of book I'm looking for here is a little different, so I hope you can indulge me. I enjoy reading quite a lot, and tend to prefer books that teach me something new, rather than straight up fiction. When I was younger I used to read a lot of popular science books, and at the time I thought these were teaching me something, but then I got older and went to university and studied theoretical physics, and that killed the genre for me. Now, I know that I can learn about a topic simply by seeking out review papers on it, or finding an appropriate textbook. That's what I do when I need to do something for work. However, most of the time this material does not make for light reading, and requires a significant amount of effort to work through, and is not necessarily the kind of thing I'd want to use as a way to relax. However, occasionally one comes across a very readable book on some aspect of physics. Nielsen and Chuang's Quantum Computation and Quantum Information strikes me as an example of this. There are probably better examples of this, but what are they? This brings me to my question: > Beyond the standard undergraduate topics, which areas of physics or > mathematics have books which both provide fairly comprehensive introductions > and which are actually enjoyable to read? To be clear, by "enjoyable" I mean something more than that they simply be accessible or well written. I mean that it should be something that I could read for relaxation rather than work. To some people this will probably sound like an insane thing to want to do. However modern physics is huge, and it bothers me that I don't know very much about, say, string theory beyond bosonic string theory, and if it is something I can learn more about in my spare time, then that would be great. * * * UPDATE: There appears to be some confusion in the answers over the type of book I am looking for. I am **not** looking for popularisations. Rather I am looking for something at the level of a graduate text, but only those which are particularly readable.


title: Whose reference frame to use for $d \theta$ near a black hole?

text: Using the Schwarzchild metric for a body circularly orbiting a nonspinning black hole (i.e. $dr=0$), the relation between $d\tau$, the time between two light pulses sent out infinitesimally close together, as measured by the object, and $dt$, the time between the pusles as measured by the observer far away from the black hole who recieves these pulses, is $$c^2 d\tau^2=\frac{c^2dt^2 }{1+\frac{r_s}{r}}-r^2d\theta^2$$ $$\left(\frac{d\tau}{dt}\right)^2=\frac{1 }{1+\frac{r_s}{r}}-\left(\frac{r \dot{\theta}}{c}\right)^2=\frac{1 }{1+\frac{r_s}{r}}-\left(\frac{v}{c}\right)^2$$ where $r$ is the reduced radius. However, which observer measures $d\theta$, and why? This will have measurable consequences for the value of $v$.


title: Finally photons got a mass, what now?

text: Finally photons have got a mass, and what we have got is a new state of matter. Most of you must have heard about this till now, for those who don't, visit this page. I am not a pro or something, but the circumstances seems interesting to me. Anything and everything we have learnt till now, about photon of course, is based on a speculation that the photos are mass less. What about the theories, are they still valid?


title: How fast is a true vacuum bubble going to expand?

text: If a true vacuum bubble will occur in our universe, how fast is it going to expand?


title: Can the same car with 100HP and 500HP produce the same acceleration on first gear?

text: My friend and I are debating about the acceleration of cars in first gear. He said that the same car (weight, tires, etc) produces the same acceleration with both 100HP and 500HP. I imagine that the moment when the tires catch the surface of the road both start with the same speed at moment 0, but the one with 500HP will have a higher acceleration in the next moment because it manages to spin the tires faster than the other one. If someone can explain me a little about this can be great.


title: Speed of sound at temperatures below 0 °C

text: How can the speed of sound be calculated for temperatures below 0 °C (down to -40 °C)? Does the calculation $v=331\ \frac{m}{s} + 0.6 \frac{m}{s°C} \times T$ still hold (where T's unit is °C)?


title: QCD in the non-perturbative regime

text: In the list of unsolved problems in physics.   * **Confinement:** the equations of QCD remain unsolved at energy scales relevant for describing atomic nuclei. How does QCD give rise to the physics of nuclei and nuclear constituents?    * **Quark matter:** the equations of QCD predict that a plasma (or soup) of quarks and gluons should be formed at high temperature and density. What are the properties of this phase of matter? what is this means that "non-perturbative QCD are the exploration of phases of quark matter, including the quark-gluon plasma."?


title: Raising Altitude by using Magnetic Torquer and Reaction Wheels

text: Suppose a satellite with a magnetic torquer and reaction wheels. I wonder if it is possible to use both in such a way that the overall effect would be to raise the satellite's altitude?


title: Having difficulties finding objects with Celestron Telescope

text: I have a Celestron Astromaster 114 EQ, which I saw was highly recommended as a great beginner scope. But I just can't seem to find anything in the sky with it. Yes, I've removed the lens cap, yes I've tried aligning with Polaris. Unfortunately, I think it is not properly balanced, and I can't seem to get it to balance, and I am totally unsure if my finderscope is aligned properly with the viewer. How can I check that, by the way? Everytime I blink the finder scope seems to have moved slightly (obviously it's my head, not the scope). How can I be sure to align it? Any tips?


title: What is a "Trojan Moon"?

text: Today I read an article about Saturn's moon Helene, and it was described as a "Trojan Moon" but no further explanation was given. It was difficult to even get any context about the term from the article. What is a Trojan moon and how did the term "Trojan" come about?


title: How to draw a star chart?

text: How does one draw a star chart like this? I have:   1. a star database with coordinates (right ascension - RA, declination - DEC),   2. observer coordinates (latitude, longitude) and   3. time of observation. I need a formula which considers these parameters (mapping star coordinates to a 2D image). e.g. starX = (90-abs(DEC)) * cos(RA) starY = (90-abs(DEC)) * sin(RA)


title: Is imagining 10 dimension video by Rob Bryanton has any invalid/wrong information?

text: I just watched this youtube video by Rob Bryanton and am wondering that is there anything invalid/wrong in term of real physics. I search for Rob Bryanton and it look like he has not-so-good reputation, for example here here and here His video look good, presentation is smooth, has citation to lots of famous physicists and I didn't have enough knowledge about this topic to be able to spot any error (if there it is). And it is hard to believe that **everything** he said is false. **Question.** Is there anything in his video wrong and how the correct information should be?


title: Wave propagation in an incompressible flow

text: Incompressible approximation of fluid flow is usually known to be lame in modeling the propagation of any disturbance in it, predicting a speed equal to infinity for the propagation of the disturbance. However, very recently I have entered the field of water waves and I just wondered when mentioned that they are easily obtaining finite speeds for the propagation of waves in water starting from the very incompressible equations! The speed of wave propagation is called the group speed which is defined by the relation $v_g=\tfrac{d\omega}{d\kappa}$ wherein, $\omega$ is the frequency and $\kappa$ is the wave number. For example consider the gravity waves, their governing equation is $\nabla^2\phi=0$ which compared to the equation of sound waves $\nabla^2\phi=\tfrac{1}{c^2}\tfrac{\partial^2\phi}{\partial t^2}$ should perhaps predict the speed $c\rightarrow\infty$ for the propagation of the waves in water but all the books that I saw use the relation already expressed for the group speed and no infinity appears anywhere in their calculations! How is it understandable? Maybe the question can be reduced to why the propagation speed is obtained from the relation given for $v_g$? Thanks in advance ;) * * * UPDATE 1. In the case it may help let me add that the answer usually used for $\phi$ in the gravity wave equation $\nabla^2\phi=0$ is a superposition of the harmonic modes of the form $$\phi(\vec{x},t)=Re\left[A\,e^{i (\vec{\kappa}\cdot\vec{x}-\omega t)}\right]$$ so that it contains $t$ although there is no time derivative present in the equation. * * * UPDATE 2. According to the comments partly from @VladimirKalitvianski a few points may be helpful:   * We have $∇⋅v⃗ =0$ which together with inviscid assumption in a small wave easily give the general equation $∇^2 ϕ=0$ having no time derivative in it. It's a kinematic equation like in potential flow modeling. It does not contain any steadyness assumption.   * In the first glance, in an incompressible liquid there may occur transversal waves (like surface waves), since they do not need the liquid to be compressed like it is the case for longitudinal waves (like sound waves), but instead due to liquid motion. But suppose a fluid particle is to move upward, the motion is not due to expansion thus immediately another particle should fill its previous position, and this will immediately affect its own neighborhoods, and this way particles one after another but immediately will sense the presence of the disturbance. Now look from a far distance to see again the wave has been propagated everywhere immediately with the infinity speed, even if the wave was transversal.


title: Query about Bernoulli's principle

text: We know that the lower atmosphere has high pressure and as we go up, the pressure decreases, if it's so then why doesn't all gases fly up into the upper atmosphere from the lower following Bernoulli's theorem? I do expect that gravitational effect on gases isn't that worth notable. Do correct me for my mistake if it exists!


title: Atomic structure and corresponding superpartner behavior

text: If all quantum particles have a superpartner, what happens, if this has been able to be speculated based on theory, to the superpartners when the corresponding partners start forming atoms?   * Is there an S-hydrogen, S-helium, etc.?   * Or are superpartners free particles that cannot form structure?    * Are there spatial interactions; folds/intersections in supersymmetric space that form S-matter? (edit-clarification) Alright, maybe the idea that when matter forms, s-matter forms is fundamentally incorrect. Perhaps I put that out there first, even though I figured it to be false, to be falsified first. However, that aside, these particles do exist (given assumption) and there must be some kind of interaction, lack of interaction, as in, these particles must be doing something. They can't just provide mass for these equations to work...even though that's why they were theorized in the first place, is that correct? There must be physical ramifications of these particles existing. Have we had any insights into the behavior of these particles? Do they always exist, do they decay, what is their state in their hidden dimension of space? Does supersymmetry breaking have behavioral ramifications other than just having unequal mass? Can we at all determine the composition of this hidden spatial dimension?


title: Do glass panes become thicker at the bottom over time?

text: Is it true that a glass window, that has been placed in a wall for about 10 years or more, is thicker on the bottom than on the top? I can vaguely remember my physics teacher saying that this was true. So if this is true, how is this possible?


title: Is compton scattering only effective by x rayphotons and gamma ray photons

text: Can Compton scattering work with any other forms types of photons, and why was the the Compton Scattering experiment only done with x ray photons?


title: Why must $\Psi (x,t)$ go to zero faster than $\frac{1}{\sqrt{|x|}}$?

text: Why must $\Psi (x,t)$ go to zero faster than $\frac{1}{\sqrt{|x|}}$ as $|x|$ goes to $\infty$? According to Griffiths' Introduction to Quantum Mechanics, it must. I don't understand why, and this is in his footnote (while talking about normalizability), so there's no explanation as to why this must be so.


title: is there an error in this paper ¿¿

text: http://vixra.org/pdf/0908.0079v1.pdf see equation (2.21) the reasoning to get this (2.21) goes from page 9 to page 14 should be the inverse of the potential denoted by $ x(V) $ proportional to the half-derivative ? i mean something $$ x(V)= C\frac{d^{1/2}}{dV^{1/2}}N(V) $$ instead of the author's claim $$ X(V)= A\frac{dN(V)}{dV} $$ here A and C are constants


title: Field operator in Schrödinger picture

text: In Schrödinger picture operators do not depend on time explicitly. Consider a free scalar field with Lagrangian density $${\cal L} ~=~ \frac{1}{2}\partial_{\mu} \phi\partial^{\mu}\phi-\frac{m^2}{2}\phi^2,$$ where $\phi=\phi(t,x,y,z)$. In quantum field theory we take it as an operator. Is this operator Schrödinger or Heisenberg one? How can we express free field Lagrangian in both pictures?


title: What does the Atomic Form Factor means?

text: I was reading about Nuclear Physics and the autor mentioned something about the Atomic form factor, something relationated with the Fourier Transform of the espacial distribution of the electric charge. but I don't know what is the meaning of that, ¿What's the physical interpretation of the atomic form factor?


title: Wave equation in the case of non-monochromatic light

text: monochromatic light refers to electromagnetic radiation of a single frequency, $\omega$ The wave equation in this case turns into the well known helmholtz equation.. How about for non-monochromatic light, how can we simplify the wave equation


title: Gamma Matrices in Dimensional Regularization

text: Prove that $tr\left(\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_5\right)=0$ when the spacetime dimension is not 4. What I have tried: We know that $\gamma_\alpha\gamma^\alpha=d\mathbb{1}$, so we can write: $tr\left(\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_5\right)=\frac{1}{d}tr\left(\gamma_\alpha\gamma^\alpha\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_5\right)$ Then I thought I could commute $\gamma^\alpha$ past two gammas because if $\alpha\notin\left\\{\mu,\,\nu,\,\rho,\,\sigma\right\\}$, then $\left\\{\gamma_\alpha,\,\gamma_\mu\right\\}=0$ and somehow show that I get minus of what I started with, using the cyclicality of the trace and that $\left\\{\gamma_5,\,\gamma_\mu\right\\}=0$. However, what I am not sure about is why can we always find such $\alpha$ so that $\alpha\notin\left\\{\mu,\,\nu,\,\rho,\,\sigma\right\\}$. I understand this is generally possible when $d\in\mathbb{R}\wedge d>4$, however, when $d\in\mathbb{C}$, this claim doesn't make any sense for me. Can anyone provide a rigorous proof of this claim which avoids the hurdle I mentioned above?


title: Product of exponential of operators

text: in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators $$e^{A}e^{B}=e^{A+B} $$ if and only if $$[A,B]=0$$ I remember my professor told use about looking for a differential equation but I don't remember the details, and I want to be able to prove it. Brute force the series doesn't seem to be a good idea. Any hint will be appreciated thanks.


title: What is the theoretical wattage output of a Tokamak fusion reactor?

text: By that I mean the complete radiative wattage of any type of energetic radioactivity or electromagnetic wave or even particle if that's what they output. My purpose is to compare this to the suns energy output into the Earth of 174 petawatts. How many Tokamak sized reactors would it take to simulate the sun. And after all their should be many smaller reactors in case one breaks.


title: Power to suspend a mass in air

text: How much power is needed to suspend a mass in the air? The four parts below deal with the same problem. I post them all here, should there be some subtle things I didn't see. ## The textbook problem A 9th edition college physics textbook has this as an example problem in its energy chapter (details cleaned-up): An elevator car ("a lift") and its passengers has a mass of 1800 kg and negligible friction force in all its moving part. How much power must a motor deliver to lift the elevator car and its passengers at a constant speed of 3.00 m/s? _Answer_ : Speed is constant, so $a=0$. Now let $F$ be the total force and $T$ be the force exerted by the motor. Then, $$ \sum F = T-Mg = 0 \qquad \Rightarrow \qquad T = Mg $$ with power = force x velocity, we then have $$ P = T\cdot v = Mg \cdot v $$ $$ P = 1800 \,kg\cdot9.8\,m/{s^2}\cdot 3.00\,m/s=5.29\times10^4\,{\textrm W} $$ And, what power must the motor deliver at the instant the speed of the elevator is $v$ if the motor is designed to provide the elevator car with an upward acceleration of 1.00 m/s per second? _Answer_ : Now $a=1.00 \,m/{s^2}$, and then $$ \sum F = T-Mg = Ma \qquad \Rightarrow \qquad T = M(a+g). $$ Proceeding as before, we have $$ P = T\cdot v = M(a+g) \cdot v $$ $$ P = 1800 \,kg\cdot (1.00+9.8)\,m/{s^2}\cdot 3.00\,m/s= 5.83 \times10^4\,{\textrm W}. $$ That's the end of the example. **My questions**. We might as well ask these:   1. How much power must the motor deliver to suspend the elevator and its passengers so it's kept at the constant height? That is, zero velocity? Certainly it can't be zero right? Because otherwise the elevator car will free-fall. But how much power is required?   2. What power must the motor deliver at the instant the speed of the elevator is $v=3.00$ (as before) if the motor is designed to provide the elevator car with a _downward_ acceleration of 1.00 m/s per second? I've been looking for an answer but to no avail. ## The helicopter problem This is _the_ real question that motivates this post. I've been asked by a friend who's going to build a model-helicopter. "Ok, I have a 2000 W engine for my 50 kg model-helicopter. Assuming 100% efficiency, will it able to float 1 m above ground for at least 1 minute?" I can't answer that. "Ok then, what is minimum power required for the engine if I want it to float 1 m above ground, assuming 100% efficiency?" I'll ask physics.se ## The pulley problem This is a simpler version of the above two problems. Using a rope and a pulley, what is the minimum power required for a man to suspend a 1 kg mass in the air 1 m above ground for 1 second? ## The table or string problem There's not much problem here. Suspending a 1 kg mass 1 m above the ground? Simple. Put it on a table, or hang it with a string. No movement. No power required. What's happening here?


title: Comparing unknown temperature scales

text: Source: Principles of Physics by Resnick, Halliday, Walker. $9^{th}$ edition. Chapter 18. Problem 6. On a linear $X$ temperature scale, water freezes at $-125.0^0X$ and boils at $360.0^0X$. On a linear $Y$ scale, water freezes at $-70.00^0Y$ and boils at $-30.00^0Y$. A temperature of $50.00^0Y$ corresponds to what temperature on the $X$ scale? **My approach** : I have a formula for such problems- $$\frac{X-(freezing \;point\; of\; water\;in\;X)}{(boiling\;point\;of\;water\;in\;X)-(freezing\;point\;of\;water\;in\;X)}=\frac{Y-(freezing \;point\; of\; water\;in\;Y)}{(boiling\;point\;of\;water\;in\;Y)-(freezing\;point\;of\;water\;in\;Y)}$$ So, $$\frac{X+125}{360+125}=\frac{Y+70}{-30+70}$$ Putting $Y=50.00^0$ in this equation, I get $X=1330.00^0$. **Approach of the solution manual** : Since the temperature scales are linear, we get $$Y=mX+c$$ From the given data, we can form two equtions and solving them simultaneously, we get $$m=0.08,\;\;c=-60$$ So, $$Y=0.08X-60$$ Putting $Y=50.00^0$, we get $X=1375.00^0$ **Is there any error in my method?**


title: Immediately after the Big Bang, was the universe in a state of extremely low or extremely high entropy?

text: Phase space theory suggests that the largest course-graining region, $p$, in a phase space, $P$, is the point in the phase space with the highest entropy. As such, it is in thermal equilibrium with the course-graining regions surrounding it. Now, when an intensity vs frequency graph is obtained for the CMB (Cosmic Microwave Background), a curve almost identical to Planck's curve representing black-body radiation is seen. The temperature of the CMB is observed to be almost uniform across the universe, and the curve of black-body radiation represents a point of thermal equilibrium for a system. Since the CMB has the same curve as black-body radiation, this would suggest that at the birth of the cosmic microwave background, the universe was in a state of thermal equilibrium. Referring back to the phase space spoken of in the first paragraph, since the universe was in a state of thermal equilibrium, it was at a point of highest entropy. However, surely due to the Second Law of Thermodynamics, the entropy of the universe would need to decrease - so it wouldn't be possible for the entropy of the universe to be in such a high state so soon after its birth - so it should surely begin in a low state of entropy. So what's the deal here? Did the universe begin in a really low state of entropy, or a really high one? Are all my explanations here logical or quite inaccurate?


title: What is the mechanism for graphene to conduct so well?

text: If metals have always been the best conductors, what is it about graphene that makes it such a good conductor in the plane? Specifically, in the metals silver is better than copper. I always assumed that the further away the valence electron was, the easier mobility would be for the same structure, but gold is worse than silver. Now, it appears that graphene is better than silver. What is it about graphene that makes it so conductive, and can a wire theoretically be fabricated out of graphene? Given that thickness seems to change its properties, would there have to be dielectric separating each layer in such a wire? Finally, why isn't Silicon by analogy a better conductor than Carbon? And please don't give an answer stating that the band gaps are lower for graphene, I want an explanation of why a particular element has higher or lower bandgaps.


title: Ice cube left in water at 0 °C for a thousand years

text: Let's say we take an ideal calorimeter with a liter of water at +5 °C and throw a 5×5×5 cm cube of ice having temperature of –70 °C. I choose these initial parameters so that the stationary temperature would come to zero after melting about 10 grams of ice. The calorimeter is then closed an isolated. How would the contents of this calorimeter evolve with time? How would it look like after a day? a year? a thousand years? Let's suppose that ice never touches walls and floats in the middle. ![illustration of the system](http://i.stack.imgur.com/F9LS5.png) This may sound as a "homework" problem, but it's just a question I came up with and I found it to be quite a complicated one — so I formulate it this way just to be specific. I suppose that the ice cube would eventually recrystallize to take the least- energy shape of some hexagonal kind. What would this shape be like? I have never heard of a single hexagonal ice crystal. I have seen large single crystals of quartz, but not of ice. How can I estimate the rate of recrystallization? Can you point me to some relevant papers on this? Also, as surface tension of ice is about two times less than that of water, the least-energy state of the system would have all it's water-air surface frozen. Am I right? How can I estimate the depth of the surface ice? Another factor is that top of the ice is always above water. Eventually it would recrystallize through sublimation and desublimation. But the rate would be different from that in water. How can this rate be estimated?


title: How quickly does air return to a vacuum chamber?

text: I know there are a lot of variables that affect how quickly this would occur. Any information on the topic would be highly appreciated. For sake of argument, lets say 1 square foot chamber, lets say medium vacuum, and a 1/16 square inch opening. If you could provide me with any formulas or thoughts on the topic that would be very helpful. Also, how would the speed change with a different gas, say hydrogen? Or even, a liquid such as water? Thanks


title: modern physics :emission spectrum

text: If in a hydrogen atom all possible transactions take place. The ratio of maximum frequency to minimum frequency is 135/7.what is the principal quantum number of excited state. The max frequency case will be from n to 1 where n is the principal quantum number of excited state,but what will be the min frequency case how to prove that it will be from 2 to 1??


title: What is the weekly average of coronal mass ejections the sun sends out?

text: What is the average number of coronal mass ejections that the sun sends out each week, including those that are not directed towards Earth?


title: How many measurements should be done?

text: I am measuring time of a computer operation. The operation should run roughly same time each time I measure it. How many times should I measure it to get good average and standard deviation?


title: Why can Hiroshima be inhabited when Chernobyl cannot?

text: There was an atomic bomb dropped in Hiroshima, but today there are residents in Hiroshima. However, in Chernobyl, where there was a nuclear reactor meltdown, there are no residents living today (or very few). What made the difference?


title: Has everything we see happened in the past?

text: Whatever we see is basically based on the light that hits our eyes, right? When we look at the Moon we are looking at the Moon as it was couple of seconds ago, as the light takes some seconds (~2 seconds I believe) to reach us. In a way aren't we ALWAYS seeing the past as, no matter how close we are to something, we can never be at distance 0. Am I wrong in thinking about it like that?


title: A problem says : a car starts from rest and undergoes a constant acceleration. It travels 5 m in the....?

text: A problem says : > _A car starts from rest and undergoes a constant acceleration. It travels 5 > m in the t interval 0 s to 1 s, find the displacement of the car during the > time interval from 1 s to 2 s?_ the answer is 15m and I wonder why it is! Because the acceleration in the interval from 0 to 1 will be 5 m/s^2 ..and from 1 to 2 will be 10 m/s^2 Therefore the displacement will be 10m not 15. What do you think?


title: Apparent Lamb Shift paradox

text: If the lamb shift causes the $^2$S$_{\frac{1}{2}}$ state to be higher in energy than $^2$P$_{\frac{1}{2}}$, why does the $^2$S$_{\frac{1}{2}}$? state not decay to $^2$P$_{\frac{1}{2}}$ Is there a selection rule that forbids this or is there a more subtle effect at work?


title: What is the best evidence/theory about the first generation of stars

text: In particular, during what range of years after the Big Bang did the stars form? What is the expected range of masses of these stars and what is the expected lifetime before they supernova? I assume these stars resulted in the re-ionization of ISM (interstellar medium), so what is the evidence and estimates of the age of the re-ionization era?


title: The force on an infinite line of charge perpendicular to a on a rod of charge

text: Essentially the question is to calculate the force exerted by a charged rod perpendicular to an infinite line of charge: $\vec{F}_{\text{rod}\rightarrow \text{line}} =\dots$ My question is it okay to just find $\vec{F}_{\text{line}\rightarrow \text{rod}} =\dots$, basically calculating the force 'the opposite way', since this seems easier to do?


title: Lack of scale in Schrödinger equation for square-inverse potential

text: I see that if we set our potential in schrodinger equation to be a inverse- square dependence we don't have a typical unit of length as we have for hydrogen atom. $$-{\hbar^2\over 2m}\nabla^2\psi + {\alpha\over r^2}\psi=E\psi$$ Is clear that ${2m\over \hbar^2}\alpha$ is dimensionless. When I treat this system as a simple system, I found some divergences and continuous spectra for bound states. I found a promiscuous relationship with the squared angular momentum eigenvalues (in units of $\hbar^2$) and the strength $\alpha$ of the potential to. This relationship is so promiscuous that if I forget the spectra of angular momentum and derive this by behaviour of homogeneous polynomials of $(\cos(\phi)\sin(\theta);\sin(\phi)\sin(\theta);\cos(\theta))$ near to $r=0$ I have a freedom to set the eigenvalues for embrace the strength coupling or part of it. $$L^2\psi\rightarrow [l(l+1)\hbar^2-2m\beta]\psi$$ when in this case, $\psi\rightarrow r^lY_l(\theta,\phi)$,with $Y_l(\theta,\phi)$ is a homogeneous polynomial of oder $l$, this clear means that $l$ is an integer. And $\beta$ is a arbitrarily real number. If I put $\beta=\alpha$ then the force is erased by the centrifugal force.   * This mean that we have to consider the Renormalization Group ideas?    * What is the relationship with the angular momentum and the renormalization techniques?