If kinetic energy is a function of the square of velocity, how can gravitiational acceleration be constant?

Kinetic energy is defined as: \begin{equation} E_k = \frac{mv^2}{2} \end{equation} Because of the squared velocity term in there, the faster you're going, the more energy it takes to further accelerate. (This is why it takes longer to get your car from 30 to 60 than from 0 to 30, for example.) However, unlike normal methods of acceleration that add (more or less) constant energy over time to an object, the force of gravity is defined as imparting a constant _acceleration_ over time to an object. How does that work? Does that imply that the faster an object is already moving (in the direction of the mass exerting gravity on the object), the stronger gravity pulls at it?


How is the energy of a lightning calculated?

I was wondering how is the energy of a lightning measured? Is it based on the luminous intensity and/or the generated sound? What about the generated heat and energy spent in the production of ozone? Is it taken into account? I looked here and here, but I didn't find an answer that satisfies me.


Where does Computer Science background students fit in Theoretical Physics

I am basically an Electronics student - background in computer science (that's where I want to work). I applied for an internship in USA in a research institute where the group is focused in Theoretical Condensed Matter Physics, Chemical Physics, Physical Chemistry, Materials Science. I mentioned my areas of interests as: Computational Science, Machine Learning, Web development My skills as: Python, C, Django, Java, etc.. I got selected. Now, I would like to know where could possibly a CS background guy would actually work on? I am looking for a detailed answer


Relation between a Quasistatic and a reversible process

Why is it that if a process is reversible, it is quasi-static? Does it mean that then the process is also non-dissipative if it is quasistatic?


Is there any role of steering in turning?

![enter image description here](http://i.stack.imgur.com/jqmeF.png) I am aware of the practical requirement of turning the steering wheel to turn a car while going on a straight and even curved road. But in the proofs of turning of car on leveled or banked curves, we see the tyres **straight**. Assuming a road where a car enters tangentially to a circle with some velocity satisfying the conditions for turning (no skidding), is there no reason in an ideal case to steer the car ? What all reasons in practical life make it necessary to steer ?


Simple examples for exchange and correlation

Is there an easy, in the best case intuitive, explanation of the difference of exchange and correlation? Is there a simple way to distinguish whether a certain contribution is due to exchange or correlation?


Ideally speaking, will a rolling disk with no slipping come to a stop because of friction from the ground?

Consider a rotating disk on a horizontal plane with static friction. The contact point of the disk with the plane has null instantaneous velocity. Assuming the center of the disk has fixed $v_0$ velocity at a time $t_0$, is it right to conclude that the disk will stop in a time $t_1 > t_0$ by effect of the momentum of friction force? (By the Euler second law we get a negative angular acceleration)


What is the boundary condition of graphene flake with zigzag edges?

It is a question about free carrier behavior in graphene flakes. (or may be called charge confinement) Say if we have a perfect hexagonal free standing graphene flake terminated with zigzag edges. Since there are two subatoms (let's call them A,B) in every unit cell. Edges will be terminated, let say, ABABAB (as shown in the figure). ![figure of graphene flake with zigzag edge](http://i.stack.imgur.com/gt1tJ.png) My question is how to calculate (understand) electron wavefunction in this system (only considering relative low energy case, charge carrier can be seen as massless fermions). Another way to ask it what type of **boundary condition** one shall use for those two types of edges? According to this paper(pdf), scattering at the zigzag boundary dosen't couple the two valleys. I think we can only consider one valley picture(is this true?), which means the most simple Dirac equation $-i\hbar \nu_F\nabla \Psi_i=E\Psi_i$, where $i=A,B$ (or Klein-Gordon eq). For simplicity, spin effect here can also be ignored(no magnetic field). I am not only asking about the edge states, also for higher energy states. What confusing me is how to distinguish this from the particle in a box case (relativistic particle, here I mean). Intuitively, I think electron will behave as billiard in side of the flake. But there could be two cases, either electrons bouncing between A,B edges, or they do it only in one type of edge, such as only bouncing at edges terminated with subatom A. Can it be understood this way? If yes, which of the two picture is the real one? Hope it is clear. Thanks.


What do we consider "Perpetual Motion"

I know this is a bad question to most serious Physics but I have a question about what is considered “Perpetual motion.” The Foucault pendulum in the UN consists of sphere that passes directly over a raised metal ring at the centre that contains an electromagnet, which induces a current in the copper inside the ball. This supplies the necessary energy to overcome friction and air resistance and keeps it swinging uniformly. Now the swing of the pendulum is induced but the 36h 45m clockwise shift generated bay the earths rotation is perpetual or as long as the earth rotates. Is this assumption correct? Does a generator that works on tidal movements not fall in to the same assumption?


Doubts about the definition of mass

I'm having some problems understanding what are the possible definitons of mass and how they are related to each other. In Classical Mechanics, we can distinguish between inertial and gravitational mass (although they are supposed to be the same $m_i=m_g$): $$F=m_ia$$ $$F=G\frac{Mm_g}{r^2}$$ On the other hand, in Special Relativity, mass is: $$p^\mu p_\mu=m^2$$ But in Quantum Mechanics, I think none of these definitions should be valid (Newton's second law doesn't hold, gravity can't be described in QM and QM isn't relativistic). So, what mass should we use in Schrödinger's equation, when we state that: $T=\frac{p^2}{2m}$? In QFT, the correct mass is $p^\mu p_\mu=m^2$, isn't it? Finally, are there more possible definitions of mass, besides the description in String Theory (maybe in General Relativity, I've read that there are problems defining energy in a localised point, that we can only evaluate the total energy)? Are some definitons more fundamental than the rest? EDIT: What about this definition I've found in QM $m=\hbar (\frac{d^2E}{dk^2})-1$?


Nanorobots. What stops us from producing them yet?

If we can already predicts accuratelly motion on molecular levels, what stops us from developing small robots to, for instance, navigate through our blood vessels looking for cancerous cells and destroying them? What are the predictions for when we will able to do this? If someone is very interested on this goal and wants to participate on research, what should exactly he study? Programming, chemistry, physics, quantum mechanics? What could help at all? Books/resources welcome.


Kinetic energy of Electron

I had to find the kinetic energy of electron with wavelength $2$ pm. I used the formula $$ KE = \frac{p^2}{2m} = \frac{h^2}{\lambda^2 2m}$$ which gave me result, $KE = 376.9$ KeV. But the answer given in the book is $292$ KeV. Am I making a mistake in calculating it?


Comparing Static Frictions

In this figure, which of the static frictional forces will be more? ![enter image description here](http://i.stack.imgur.com/L9tHZ.png) _My aim isn't to solve this particular problem_ but to learn how is static friction distributed . Since each of the rough-surfaces are perfectly capable of providing the $-1N$ horizontal frictional force but why don't they ? This is kind of ambiguity that who will provide a bigger share in total static friction. And as the surface have different $\mu$, so we can't even invoke symmetry.


Non-SHM oscillatory motion

![enter image description here](http://i.stack.imgur.com/muwVu.png) How to solve these kind of questions , where $|F| \propto x^2$? How to find time period and velocity type related things to the oscillatory motion? $$m\dfrac{d^2x}{dt^2}=F=-\dfrac{dU}{dx}=-3kx|x|.$$ But after this $$m\dfrac{d^2x}{dt^2}=-3kx|x|.$$ What is general solution of this ODE? I think it would give the $x$ in terms of $t$ and I would be able to get time period from it.


Vibrational noise for scientific apparatus on a satellite

Say we're running an experiment on a satellite in _some_ earth orbit. For example, maybe we're imaging something with transmission electron microscope or we're catching atoms in ion traps --- really anything. In practice, and otherwise keeping things very simple on the satellite to avoid mechanical / etc. noises, what kind of vibrational noise profile will we have relative to what's possible on earth? What would be the primary cause of the main noise source (e.g. on earth this might be the building's ventilation system followed by geological sources, etc.)?


What happens when a ferromagnetic object encounters a field too strong for it?

Just thinking aloud ... It is possible for a star to grow so large it collapses under it's own gravity. Along a parallel path (so to speak) when a conductor carries a current too large for it to sustain, it burns out. Can a ferromagnetic object be imparted a field too strong for it? What happens then? I would begin to guess the dipoles within the magnet can only be magnetized so far; a stronger field wouldn't have an effect. But surely the same argument could have applied to a wire conductor?


What makes thrust reversers different from fans on a sailboat?

Why do thrust reversers on jet planes work? By Newton's third law, it seems like the force exerted by pushing air through the nozzle should cancel the force from redirecting the air forwards. The problem seems identical to the case of a fan on a sailboat, with "fan" replaced by "jet engine" and "sail" replaced by "thrust reverser."


Calculate the latitude / longitude coordinates of the location where the Sun is at the zenith

There are plenty of resources showing how to calculate zenith and azimuth of the Sun when the time and the location are given. However, I need to calculate the location where the sun is at the zenith for a given time and date, not the other way around. Is there a more or less easy way to do this calculation?


Is it possible to calculate specific velocities from acceleration?

I'm trying to understand a sample problem from a current (published 2011) high school physics textbook, but I really think the question is flawed. The task is to convert an acceleration-time graph (AT-graph) into a velocity-time graph (VT-graph). The AC-graph is given as: ![enter image description here](http://i.stack.imgur.com/tTSbF.jpg) Then the book gives the following instructions for finding velocities from accelerations: ![enter image description here](http://i.stack.imgur.com/kztRH.jpg) Here's my first problem. Where did they get the formula $v=(\Delta a)(\Delta t)$? Isn't $\Delta a$ a more advanced concept called "jerk"? (a concept that this textbook never introduces, by the way). Shouldn't it be $\Delta v = a\Delta t$? And if so, we're not finding $v$ (which, correct me if I'm wrong, is impossible from the data given), but merely $\Delta v$. Then they show the following calculations: ![enter image description here](http://i.stack.imgur.com/ZzrCh.jpg) This chart is confusing to me. Is the time column an instant of time or a time interval? (I think it's a time interval). It seems to me that they're calculating $\Delta v$, not $v$, in which case it's impossible to draw a VT graph. But that's what they do next anyway: ![enter image description here](http://i.stack.imgur.com/TMsNF.jpg) Any feedback is appreciated.


What type of substances allows the use of the Ideal Gas Law?

I know that I can use the ideal gas law with pure gases or pure liquids. But can I also use the ideal gas law at saturated gases and saturated liquids as long as they aren't two phase substances?


Special conformal transformations and locality

In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal transformations, which are not linear. I feel that this non - linearity could cause a conflict with locality, in the following sense:   * Let us consider two wave packets 1 and 2, separated by a space-like interval, corresponding to impulsions P1 and P2. If we make a linear transformation, for example a Lorentz transformation, we obtain impulsions P'1 and P'2. Now let us consider the total system 1 + 2; it has an impulsion P, which is equal to P1 + P2 because both systems are not causally related. By this same Lorentz transformation, the total impulsion P becomes P', but because the Lorentz transformation is linear, we have P' = P'1 + P'2, and P' is linear in P'1 and P'2, and is linear in P1 and P2. This is coherent with the fact that there is still no interaction between the systems 1 and 2.   * On the other hand, imagine we had taken, for example, a pseudo-Lorentz transformation , which would be, for example, quadratic, then, in the expression of P', we would have had cross-terms of type P1.P2, what would mean an interaction between systems not causally connected, which is incoherent. So, I need a detailed argumentation, to infirm or confirm my feeling : non- linear special conformal transformations could cause a conflict with locality.


Any interesting physics theroies/equations to be used in a video game?

I am designing a little game based on Newtons Graviational Law. Are there any theories that I can use to create games? thanks


Speed of a dynamic hydraulic system

We all have noticed that changing the temperature of the water in the shower is not instantaneous, rather the result is felt when the water that was in the tap works its way up to the showerhead. However, changing the pressure does feel instantaneous. I wonder at what speed the change in pressure propagates in the direction of flow in a flowing fluid. My guess would be at the speed of sound in the medium, based on the fact that it is faster than the speed of the flow of the water (like the temperature is), yet obviously at or below C. I am obviously missing the correct keywords to google for as I cannot find any references to this phenomenon. What is the answer to the query, or better yet, how could I have found this information (short of empirically)? Thanks.


Resultant wave in the Michelson-Morley experiment

I was wondering when the two waves in the experiment, don't they lead to constructive interference? Consider the first case in the image. The two combining waves should lead to constructive interference. ![Wikipedia](http://i.stack.imgur.com/u6zXO.gif) Source: Wikipedia. **Edit** Sorry for the confusion. What I wanted to ask is what John replied. But John how will they have half the intensity. The wavelength will be the same and so the frequency even if the wave splits. But after recombination the frequency should increase. Shouldn't it?


Tubelights+power lines pictures?

I've come across many pictures like these, sometimes in chain emails reporting the dangers of power lines. Another claim is that they run on "wasted" energy. The explanations given are that the wires set up a field, and thus a potential difference. This p.d. gives current in the wires. Now, I can't help but feeling skeptical. I can't see how the tubes would stay lit for the time it takes to set up the whole grid. Here's my logic: If current flows through the tubelight, there will be a net charge distribution on the caps. At one point, the charge distribution will be such that it creates its own potential, halting the flow of light. This works even if one end is grounded. An easier way to analyse this is a capacitor (GP: G for ground, P for power lines), with a bulb inside it. Each terminal of the bulb is connected to a plate (A and B). There's an optional grounding wire W. In this situation, it's obvious that eventually the plates AB will get enough charge (same charge density as PG incidentally) to create zero p.d. Current should be there for a brief moment in time. ![diagram](http://i.stack.imgur.com/eJGEf.png) I understand that this is a DC situation, but for power lines, I think that something similar should happen. Or maybe not. I'm clear that the potential difference across the live and neutral switches polarities, but I'm not sure how the potential field is outside the wire. In fact, I feel that there shouldn't be a V/E field, only a magnetic field. If it's true that the tubelights stay lit, then how is the energy "wasted energy"? I feel that even if the tubelight can stay lit, the energy is being drawn from the power lines. This looks just like how one can naively say that there is energy wasted in an open socket; when in reality plugging something in draws more energy from the system (and can dim other appliances). **Update** : So my questions are thus:   1. I'd like a clearer explanation of the potential and E field outside an AC power line.   2. Is the energy "wasted energy"?


Could we run an electric car on electric eels?

Could we run an electric car on a tank full of electric eels? I've heard electric eels are around 400 to 500 volts and can keep shocking for up to an hour. They also have a 10 volt system to sense with, which might be good for the headlights and cd player.


Sum of acceleration vectors

If a point mass has some accelerations $\mathbf{a_1} $ and $\mathbf{a_2} $, why is mathematically true that the "total" acceleration is $\mathbf{a}= \mathbf {a_1}+\mathbf {a_2}$?


Inhomogenous schrodinger equation

Please help me out in solving this inhomogeneous Schrodinger equation in Cylindrical co-ordinates [You may suggest if I have to go for mathematics]: $$ \ddot R + \frac1r\dot R+\left(b^2-d^2-\frac{a^2}{r^2}\right) R - \frac{c^2}r \delta(r-r_0) R ~=~ F e^{ikr}. $$ This is only the radial part.


Terminology question about energy

I'm looking for the appropriate term to use for what gets "used up" as potential energy is converted to heat and work. For example, some of the the energy in solar radiation is converted by photosynthesis to chemical energy; much of the rest is lost as waste heat. The bio-chemical energy is used to power various biological functions until it has all been lost to waste heat. At the various steps along this path what term would you use for what there is less and less of? It can't be "energy" since that's conserved. It's potential energy, but is that really a standard term for what's lost when potential energy is converted to waste heat?


Higgs stability in Standard Model

I am a little unclear on what ramifications a negative quartic at high energies has on our world at low energies. (1) First of all, is it that there is a second, isolated minimum that appears at higher energies, or is it just that the original minimum at $246$ GeV was modified under RG? (2) Assuming its just a single minimum that was modified under RG: Now say the SM is correct at low energies with a 125 GeV higgs. Then we use RG to evolve up to some higher energies where the quartic is negative but the higgs potential is stabilized at some other (larger) field value by high-dimensional operators. Doesn't this new potential minimum just evolve back into the minimum at $v = 246$ GeV under RG evolution down to lower energies? I don't see why this is such a big deal, the vev shouldn't be RG invariant to begin with no? Measured Higgs mass and vacuum stability and http://motls.blogspot.com/2012/07/why-125-gev-higgs-boson-isnt-quite.html?m=1 are related, but I don't think it addresses these particular questions (or at least in terms I understand). EDIT: Just to clarify my confusion. According to Terning's 'Modern Supersymmetry' pgs 78-79 in the MSSM the higgs gets a negative mass squared, and hence a vev, radiatively using the initial condition that all the masses are positive at high energies. This means at high energies the higgs potential is stable, and at low energies its unstable. So if you run the MSSM at low energies back up to high energies it will look like you are expanding around the wrong vacuum as well? Isn't this the same thing as what is happening in the Standard Model?


Analysis of the impulse of 2 colliding carts under the effect of magnetic repulsion

![enter image description here](http://i.stack.imgur.com/IdJCw.jpg) Hi there! I have a question about an experiment that was conducted. It is related to momentum. 2 carts were put on a track on opposite sides. They were then propelled towards one another at approximately the same speed.Please note that on each end of each cart, there was placed a negative magnet, so as soon as they came close to each other, repulsion commenced. Two sensors were set on a frequency of 20 Hz to record their motion. After collecting the data, the data for the difference in momentum was graphed through the equation m2v2 - m1v1. The group did this to calculate the approximate average frictional force of the system by producing a linear fit. The slope of this line proved to be the average friction of the system. This was an extremely simple experiment and did not prove at all difficult. However, there is one question I have been asking myself that is driving me crazy. I can't seem to find the answer (Please note that our instructor did not ask us for this). Why are there spikes in the graph of Δp (change in momentum) very near the moment of collision (actually there is no "collision" because of the magnetic repulsion). Is it because the frequency was set only to 20Hz and thus since the instance of collision was so tiny, the sensors couldnt pick up data correctly? Is it because for the brief moment that the carts cease to move friction with the rail is static instead of kinetic and thus requires more force? Some useful information: mass of cart1: 255.9g mass of cart2: 251.1g Please note the equation: Δp= Δt*Favg and p = mv An answer would be greatly appreciated as this question has been driving me crazy.


What is the upper-limit on intrinsic heating due to dark matter?

Cold dark matter is thought to fill our galactic neighborhood with a density $\rho$ of about 0.3 GeV/cm${}^3$ and with a velocity $v$ of roughly 200 to 300 km/s. (The velocity dispersion is much debated.) For a given dark matter mass $m$ and nucleon scattering cross section $\sigma$, this will lead to a constant collision rate of roughly $r \sim \rho v \sigma / m$ for every nucleon in normal matter. The kinetic energy transferred to the nucleon (which is essentially at rest) will be roughly $\Delta E \sim 2 v^2 \frac{M m^2}{(m+M)^2}$, where $M \approx 1$ amu $\approx 1$ GeV/c${}^2$ is the mass of a nucleon. The limits for light ($m \ll M$) and heavy ($m \gg M$) dark matter are $\Delta E_\mathrm{light} \sim 2 v^2 \frac{m^2}{M}$ and $\Delta E_\mathrm{heavy} \sim 2 v^2 M$. This leads to an apparent intrinsic heat production in normal matter $\tilde{P} \sim r \Delta E / M$, which is measured in W/kg. The limits are $\tilde{P}_\mathrm{light} \sim 2 \rho v^3 \sigma m / M^2$ and $\tilde{P}_\mathrm{heavy} \sim 2 \rho v^3 \sigma / m$. **What existing experiment or observation sets the upper limit on $\tilde{P}$?** (Note that $\tilde{P}$ is only sensibly defined on samples large enough to hold onto the recoiling nucleon. For tiny numbers of atoms--e.g. laser trap experiments--the chance of any of the atoms colliding with dark matter is very small, and those that do will simply leave the experiment.) * * * The best direct limit I could find looking around the literature comes from dilution refrigerators. The NAUTILUS collaboration (resonant-mass gravitational wave antenna) cooled a 2350 kg aluminum bar down to 0.1 K and estimated that the bar provided a load of no more than 10 $\mu$W to the refrigerator. Likewise, the (state-of-the-art?) Triton dilution refrigerators from Oxford Instruments can cool a volume of (240 mm)${}^3$ (which presumably could be filled with lead for a mass of about 150 kg) down to ~8mK. Extrapolating the cooling power curve just a bit, I estimated it handled about $10^{-7}$ W at that temperature. **In both cases, it looked like the direct limit on intrinsic heating is roughly $\tilde{P} < 10^{-9}$W/kg.** However, it looks like it's also possible to use the Earth's heat budget to set a better limit. Apparently, the Earth produces about 44 TW of power, of which about 20 TW is unexplained. Dividing this by the mass of the Earth, $6 \times 10^{24}$ kg, **limits the intrinsic heating to $\tilde{P} < 3 \times 10^{-12}$W/kg.** Is this Earth-heat budget argument correct? Is there a better limit elsewhere? * * * To give an example, the CDMS collaboration searches for (heavy) dark matter in the range 1 to 10${}^3$ GeV/c${}^2$ with sensitivities to cross sections greater than 10${}^{-43}$ to 10${}^{-40}$ cm${}^2$ (depending on mass). A 100 GeV dark matter candidate with a cross-section of 10${}^{-43}$ cm${}^2$ would be expected to generate $\tilde{P} \sim 10^{-27}$ W/kg, which is much too small to be observed. On the other hand, a 100 MeV dark matter particle with a cross-section of $10^{-27}$ cm${}^2$ (which, although not nearly as theoretically motivated as heavier WIMPs, is not excluded by direct detection experiments) would be expected to generate $\tilde{P} \sim 10^{-10}$ W/kg. This would have shown up in measurements of the Earth's heat production. * * * EDIT: So it looks like I completely neglected the effects of coherent scattering, which has the potential to change some of these numbers by 1 to 2 orders of magnitude. Once I learn more about this, I will update the question.


What is the domain of validity of the Omega equation?

The Omega eqation is used in meteorology for estimate the value of the vertical velocity. Considering that the horizontal velocity field and density is given by observation. Under which condition this equation can be used for recover the vertical velocity and what is order of error between the real vertical velocity field and the one recovered by the Omega equation?


How to make mirrors give more authentic image of myself

My friend bought two normal mirrors from some shop probably IKEA, looking like this one: ![enter image description here](http://i.stack.imgur.com/xmJ6w.jpg) My images are different from the two. One is a little fatter and shorter, and the other a little thinner and taller. I wonder what can possibly be the reasons?   1. They basically look the same, smooth and flat surfaces, almost the same size (one has a label saying 37x134cm), except one having a little thicker decoration border than the other. The one with thinker edge is the one giving me fatter and shorter image.   2. I don't have other things to place them, so I just make both of them stand on the floor, with a small angle lying against the wall. Although the two mirrors are placed in the same way, will different placement of the mirrors change my image?   3. What kinds of mirrors (size, ...) and how to place them will give me more realistic image of myself?


Measuring concentration

I am trying to measure the concentration of a Volatile Organic Compound (VOC) inside a liquid using Gas Chromatography (GC). The liquid contains particles that cannot be sent to a GC, so I am using the head-space method (allowing the organic compound to evaporate into the gas above the liquid--- see this link:http://www.chem.agilent.com/Library/applications/5990-3285EN.pdf)-- I let the contents of the liquid equilibrate with the air in the vial at 50 C-- and then I use the air as feed for the GC. However, the concentration in the air is rather small ~ 10-20 ppm. Is there a technique that can be used to increase the concentration of the VOC's? I have wanted to use a smaller vial to collect my liquid, but the liquid splashes about and contaminates the cap of the vial, which makes me scared to inject a needle into it as I may wreck the needle. **Update** I tried to repeat the experiment with the vial at a higher temperature ~70 C. This lead to the needle clogging. I am unsure of the cause of the needle clogging but I suspect it may have something to do with water evaporating and taking some of the polymer particles with it. Once in the needle, the particles could get dry and solidify.


Number of microstates of discretized paths

> _Let us consider a square grid, which has been rotated by 45deg. On this > grid we deﬁne a path, the directed polymer, which starts at the origin ($t = > 0$) and extends in the positive $t$-direction (at each grid point the path > goes either left or right; and steps in the negative $t$-direction are not > permitted). All paths of length $N$ end at the same $t$-position. The > distance of the endpoint to the $t$-axis is characterized by a number $m$. > Each path represents a microstate and the endpoint (i.e. $m$) the > macrostate._ How can I find the number of microstates, say, $W(m)$? I've tried that $W(m) = 0.5 N! m$, but I'm not sure. I do understand the problem, but I'm not sure how to solve it.


Can a web community write papers?

the internet has changed science drastically. Not only in terms of distributing knowledge e.g. via online encyclopedias as wikipedia and freely available sources of publications as arXiv but also as a tool for communication through email and lively discussions in forums, newsgroups and communities like StackExchange. Scientific success nowadays greatly depends on the ability to build up teams with different scientific background and experience. To my mind, the ideal researcher has to seek for people helping him to tackle parts of his problems in which he is not an expert instead of spending months on something, someone else could have been done in days. So, can't **we** be this ideal researcher, or: Can a community on a webpage do research by openly presenting ideas and working them out? Or is this not possible due to "stealing" of ideas, lacking institutional structures and the like? What do you think? Robert


Could temperature have been defined as $-\partial S/\partial U$?

When coming up with a definition of temperature, it's typical to start with an empirical definition that a system with a hotter temperature tends to lose heat to a system with a colder temperature. Combined with the second law of thermodynamics, that leads to the condition that for two systems in thermodynamic equilibrium, $$\frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2}$$ (consider it implicit throughout this question that volumes are constant). So temperature has to be some function of $\partial S/\partial U$. Furthermore, we'd like the object that loses the heat to be the one with the hotter temperature, so temperature has to be inversely related to $\partial S/\partial U$. Of course the conventional choice is $\frac{1}{T} = \frac{\partial S}{\partial U}$, but if we were rederiving thermodynamics and statistical mechanics from scratch, is there any reason we couldn't have chosen $$B = -\frac{\partial S}{\partial U}$$ instead? Sure, it would lead to many common temperatures being negative, but suppose we're willing to accept that. Is there any other useful property that the standard definition of temperature has which wouldn't be shared by $B$? Would it _necessarily_ lead to nonlinear thermometers, for example? As Michael Brown pointed out in a comment, this alternative definition is just $B = -k\beta$, so I guess another roughly equivalent way of saying what I'm asking would be, is there a practical reason not to use $\beta$ or $B$ as temperature instead of $T$? I've already looked at this question and this one and some others, but any close-to-relevant answers there seem to be saying merely that it's conventional to define temperature the way it is defined, not that we couldn't have done it differently.


Relationship between height and velocity in conservation of mechnical energy

I'm a high school physics student, and we recently did a lab on the conservation of energy where we measured the speed of a marble at varying heights on a rollercoaster track. We were supposed to graph the height and speed, and I ended up with something resembling a square root graph ![Original graph](http://i.stack.imgur.com/Tp8ZP.png) We were required to linearize the data (so that we could have a single slope), so I squared the heights and got a slope of about $-0.11\,cm^{-1}\,s^{-1}$: ![Linearized graph](http://i.stack.imgur.com/MJl1A.png) I don't understand what the slope is supposed to signify. I tried to take the derivative of velocity with respect to height in the equation $E = mgh + \frac{1}{2}mv^2$, where E is constant, and got a cryptic answer of $\frac{dv}{dh} = \frac{-g}{v}$. Can someone help me understand what's going on here?


Change of variables in path integrals

I need to evaluate a path integral which involves a set of fields $X=\left\\{ \psi_i \right\\}$: $$ I = \int \prod_i \mathcal{D} \psi_i e^{-S \left[ \left\\{ \psi_i \right\\} \right] } $$ In order to simplify the treatment I'd want to make a change of variables, and express the integrand as a function of a new set of fields, let's call it $Y=\left\\{ \phi_i \right\\}$. My problem is: how do I implement the change of variables? Can I extend the multi-dimensional case to the continuum and include the determinant of the Jacobian of the transformation in the integral, i.e. $$ J_{ij} = \frac{\delta X_i}{\delta Y_j} $$ Moreover I'd want to defined one or more of the $Y_i$ variables as containing some derivatives of the original $X_i$ ones. My intuition leads me to write down the Jacobian in momentum space, where each derivative is then replaced by a $- \mathrm{i} k_\mu$ then evaluate it, also in momentum space. I would also say that as long as the Jacobian does not depends on the fields, it can be taken outside of the integral sign, i.e. it is a multiplicative constant. Is my intuition right? Any suggestion, or any reference to a textbook is welcomed. In particular I couldn't find in any textbook I own the cases in which one of the redefined fields depends upon the derivative of one the original fields. I know that this a mathematical question more than a physics question; however I decided to post it here because (usually) physicist are more used to manipulating path integrals.


Nature of frictional force

I was thinking about a situation where a person in standing on the ground with some friction. The frictional force is directly proportional to the normal reaction acting on him by the ground. Assume that he leaned forward i.e his center of mass is not vertically up the point of contact of him with the ground.Then, does the frictional force change? My thoughts:Although the person has leaned forward the force $mg$ acting on him will be vertically downward. Since normal reaction is always normal to the surface of contact its magnitude will remain same and thus frictional force will remain same.But, I have seen athletes starting the running race with a leaning forward position which would be mostly for increasing the friction between their shoes and the track. So, I'm in a dilemma. Please help.


The double-trace deformation effect in AdS/CFT

Let me use this paper as the reference for this. I want to understand better the argument at the bottom of page 6. If the bulk $AdS$ metric is written as $\frac{1}{r^2}(dr^2 + A(r)ds_{boundary}(x)^2)$ then the free massive scalars fields on it are of the form, $\phi = r^{\Delta_{+}}(\alpha(x)) + r^{\Delta_{-}}(\beta(x))$,near $r=0$. (...and the numbers $\Delta_{\pm}$ depend on the dimensionality of the AdS and the mass of the field..)   * As per their notation their "regular" boundary condition in the bulk corresponds to setting $\beta(x)$ (the coefficient of $r^{\Delta_{-}}$) to zero on the boundary and "regular" boundary condition is setting $\alpha(x)$ (the coefficient of $r^{\Delta_{+}}$) to zero. How is this compatible with the fact that on using the regular boundary condition the dual CFT necessarily needs to have a term of the form $\int d^d x \beta(x)O(x)$ where $O$ has dimension $\Delta_{+}$ and $\alpha = \langle O \rangle$? I thought that they themselves said that in the regular scenario the $\beta$ is being set to $0$ at the boundary - then what is this $\beta(x)$ that is featuring in the boundary action?   * Below equation 3.19 they argue that the when the double-trace deformation of the boundary CFT is switched off it is seeing the "irregular" scenario and when it is turned to infinity it is seeing the regular scenario.  But how does one argue that the regular scenario is the the IR fixed point and the irregular scenario is the UV fixed point for the boundary CFT? Where is that argument?   * Rethinking the section 4 of this paper in a different way - Suppose one wants to calculate the determinant of the bulk theory $det(-\nabla^2 + m^2)$ by taking product of eigenvalues. One wants to say calculate the determinant when the bulk has been quantized with the $\Delta_{+}$ boundary condition. (...one can presumably ask the same question with $\Delta_{-}$ as well..) If the bulk is $AdS_{d+1}$ then one can see that the small-r asymptotics ($r=0$ is the AdS boundary in the Poincare patch) of the harmonics are of the form, $# r^{a} + # r^{b}$ for some values $a$ and $b$ (which depend on the eigenvalue and $d$). Now knowing the above small-r asymptotics of the harmonics how does one pick out which of these will contribute to the above determinant with say the $\Delta_{+}$ boundary condition? Can someone schematically sketch how the calculation looks like?


Coordinate Systems in Loop Integrals

Let us consider a two-point two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2}, p)$, where $k _ {1}$, $k _ {2}$ and $p$ are four-dimensional vectors in Euclidean space and $p$ and $-p$ are the external momenta. There are three angles present in $f$: the angle $x$ between $k _ {1}$ and $k _ {2}$, the angle $y$ between $p$ and $k _ {1}$ and the angle $z$ between $p$ and $k _ {2}$. Which coordinate system is most suitable for consistently performing the integrals? For instance, for the one-loop integral $\int \mathrm{d} ^ 4 k \, f(k, p)$, I take $p$ to be the z-direction. Then $p = k \cos \omega$, where $\mathrm{d} V = |k| ^ {3} \, \sin ^ {2} \omega \, \sin \theta \, \mathrm{d} |k| \, \mathrm{d} \omega \, \mathrm{d} \theta \, \mathrm{d} \phi$ is the four- dimensional spherical volume element and $|k|$ is the Euclidean norm of $k$. Is there a two-loop analog coordinate system which takes into account the three aforementioned angles?


Idea of Covering Group

1. $SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?    2. I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$?


Why are we forced to choose a specific value for $\pi$ field in Nambu-Goldstone phenomenon?

In the sigma-model of spontaneous symmetry breaking, we have degenerate vacuum states. But if we don't pick up a particular value of VEV, we won't have any symmetry breaking. As I read from a book, in field theory at finite volume, there would be no problem; but at infinite volume, we are forced to choose a value of the $\pi$ field. But why?


Derivation of the Gauss-Codazzi equation

I'm interested in the derivation of the Gauss equation (Gauss-Codazzi). Usually we consider the definition of the Riemann tensor on the hypersurface. $$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c$$ where $D$ is the connexion associated to the induced metric ($h$) on the hypersurface. We have \begin{align} D_aD_bw_c=D_a\Bigl[h_b^{~d} h_c^{~e}\nabla_d w_e\Bigr]=h_a^{~f}h_b^{~g}h_c^{~e}\nabla_f\Bigl[h_g^{~d}h_k^{~e}\nabla_dw_e\Bigr] \end{align} where $\nabla$ is the connexion of the metric ($g$) in the full space. Hence after some algebra we can arrive at the desired result. What I don't understand is why can we not do: $$ D_aD_bw_c=D_a\Bigl[h_b^{~d} h_c^{~e}\nabla_d w_e\Bigr]=h_b^{~d} h_c^{~e}D_a\Bigl[\nabla_d w_e\Bigr] $$ because $D_a h^b_c=0$. But in that case I would have $$ D_aD_bw_c=h_b^{~d} h_c^{~e}D_a\Bigl[\nabla_d w_e\Bigr]=h_b^{~d} h_c^{~e}h_a^{~\mu}h_d^{~\nu}h_e^{~\sigma}\nabla_\mu\Bigl[\nabla_\nu w_\sigma\Bigr]=h_a^{~\mu}h_b^{~\nu}h_c^{~\sigma} \nabla_\mu\nabla_\nu w_\sigma $$ Therefore we would have $$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c=h_a^{~\mu}h_b^{~\nu}h_c^{~\sigma} ~^{(n)}R_{\mu\nu\sigma}^{~~~~~~~~d}w_d$$ which is not the desired result. So I understand the standard derivation, but I don't understand why in the way that I wrote, it doesn't work ?


Time-ordering in QFT

In Srednicki _QFT_ page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition amplitude. How can this be justified mathematically? And if I understand this correctly, if time ordering is not used, then a term like $$\langle 0\mid a_{1}\left ( -\infty \right )a_{2}\left ( -\infty \right )a_{1^{\tilde{}}}^{\dagger}\left ( \infty \right )a_{2^{\tilde{}}}^{\dagger}\left ( \infty \right )\mid 0\rangle $$ this mean that there's a contribution that depend on the amplitude of transition from the final momenta to the initial momenta but quantum mechanically the final momenta are not known in advance so such a term can't contribute to the process. Does this explain the use of time-ordering physically?


Scattering in quantum wires

I'm working on a project involving scattering of particles in networks of quantum wires off of point scatterers. The problem that I am having is that even in fairly simple configurations of quantum wires there is nontrivial scattering even in the absence of any scatterer. For example, if you consider a T-junction or a single wire with a bend in it there must be an S-matrix just from the particle scattering off of the walls of the wire itself. How can I calculate the transition amplitudes for this kind of scattering?


Collision problem: Finding the final speed of the collider

> A particle of mass $M$ moving in a straight line with speed $v$ collides > with a stationary particle of the same mass. In the center of mass > coordinate system, the first particle is deflected by 90 degrees. Find the > speed of the second particle after collision in the laboratory system. ## My effort: Considering the lab frame, we have, by conservation of linear momentum $$Mv=Mv_{1f}+Mv_{2f}\tag1$$ which implies that $$v=v_{1f}+v_{2f}\tag2$$ We want to know $v_{2f}$. Therefore, we have to find $v_{1f}$. From my textbook, I get $$v_{1f}=v+v_{1f}'\tag3$$ where $v_{1f}'$ is the post-scattering velocity of the incident particle in the center of mass system and $v$ is the velocity of the center of mass in the laboratory system. Also from my text, I get $$V=\frac{v_{1i}'}{2}=\frac{v}{2}\tag4$$ Therefore, it looks like I have everything I need to solve the problem except $v_{1f}'$. All I am given is that the post-scattering angle of the incident particle in the center of mass frame is 90 degrees. How do I find the velocity of the same particle in terms of the incident velocity in the lab system, $v$?


Canonical transformations in Hamiltonian mechanics

How to prove that in the new Hamiltonian, which is formed by any of the generator function will not contain $Q$ (transformed from $q$)? I.e. new Hamiltonian will only be a function of $P$ (transformed from $p$).


Perturbation of a Schwarzschild Black Hole

If we have a perfect Schwarzschild black hole (uncharged and stationary), and we "perturb" the black hole by dropping in a some small object. For simplicity "dropping" means sending the object on straight inward trajectory near the speed of light. Clearly the falling object will cause some small (time dependent) curvature of space due to its mass and trajectory, and in particular, once it passes the even horizon, the object will cause some perturbation to the null surface (horizon) surrounding the singularity (intuitively I would think they would resemble waves or ripples). Analogously to how a pebble dropped in a pond causes ripples along the surface. Is there any way to calculate (i.e. approximate numerically) the effect of such a perturbation of the metric surrounding the black hole?, and specifically to calculate the "wobbling" of the null surface as a result of the perturbation,maybe something analogous to quantum perturbation theory? Or more broadly, does anyone know of any papers or relevant articles about a problem such as this?


Reconciling Units in Jeans' Criterion Formula

In "Physics and Chemistry of the Solar System" Jeans' Criterion is given as: $\frac{GmM}{R_c} = \frac{3mkT}{2}$ ... To me this suggests that on the left we have Joules, and on the right we have kg$\cdot$joules. I then went to some old lecture notes and found that the professor derived the Jean's Radius from: $\frac{GM}{R_c} = \frac{3kT}{2}$ which also has (as far as I can tell) an issue with units. Can anyone help my understand what I'm missing with this equation?


Oil drop experiment--How was the result so accurate?

I remember while learning about Millikan's oil drop experiment and being pretty skeptical about the setup. I know that there is a lot of controversy regarding manipulation of data, but the fact is; he still got a pretty close answer. I mean, we can get the correct answer by manipulation of data if we know the answer already. If we don't all we can do is 'reduce' the error box (which Millikan purportedly did). But I'm surprised that he still got an answer close to 1% of the actual value. I personally would have expected an error of maybe even an order of magnitude for these reasons:   * Viscosity of air: The viscosity is dependent upon humidity and other factors. I know he _did_ use the wrong value here.   * $g$: How was this known to the accuracy required for this experiment? It varies across the Earth, and besides, could it be measured to whatever accuracy using the equipment available at that time?   * Velocity of the oil drop: I don't know how fast the oil drops fell, but to me, terminal velocity looks pretty fast. How did he accurately measure the velocity? A micrometer eyepiece takes time to focus, and more time to take a reading. And anyways micrometer scales aren't that precise, are they?   * Potential difference: How did he accurately know this?   * "Selecting one drop": How was this done? If all drops had a charge as a small integral multiple of q, there should be more drops with the same charge which behave the same. How can you get rid of those?   * Densities of air/oil: Air density varies. Oil density may not be known accurately.   * Electric field: Between plates, E is not exactly constant or equal to $V/d$ There are fringing effects which may alter the value of E. IMO since the distance between the plates is comparable to the area (it look like it), fringing effects are pretty significant.   * Stokes law is only applicable for perfectly spherical bodies. Is a falling oil drop perfectly spherical? Basically I'm skeptical of how one can get such a tiny number with 1% accuracy with all those uncertainities. Could someone convince me how he _did_ get it so accurately? Were there any measures he took to minimise these errors (I can't find any)?


Please recommend a good book about physics for young child (elementary school aged)

I'm looking for a book that would be appropriate for an advanced elementary school aged kids (say, 6-11 YO) **describing the basics of physics (or sciences in general) in entertaining way**.   * The structure of the book should preferably (though not necessarily) be "Curious question" followed by entertaining-but-advanced for their-age physics answer. As a good example of a chapter/story I had in mind: "Why can the soap wash off dirt/grease?" "Why can the water be poured into containers?" "How does a plane fly?" "How does a rocket fly?"   * The topics that must be covered are atoms/molecules, how they go together to make various matter etc...   * Doesn't have to be 100% limited to physics - could include chemistry, biology, other natural sciences.  P.S. For those fluent in Russian, i'm looking for an English equivalent of the soviet popular kid's physics/science book "почему вода мокрая").


Working of dry cell battery!

please help me out in this query!! actually i need to know why when we use a battery or dry cell, As we know it consists of :- 1)electrolyte 2)cathode and anode. but i need to know as electrolyte is consisting of a salt (we take it as NaCl salt for it ) and the ions is formed from the salt due to potential we make on both ends of the battery, but why it cant happen as if the ions can directly form without connecting the battery into a wire , because the two terminals consists of positive and negative ions which can directly put force on NaCl compound and can break it ,but why their is a need to create a potential by attaching the battery with a wire externally? In other words:- Why molten salt as an electrolyte cant ionise normally in a cell without connecting the cell with a wire outside? regards :- Harjot singh


Graph Theory and Feynman Integrals

In Vladimir A. Smirnov's book _Analytic Tools for Feynman Integrals_ , Section 2.3, the alpha representation of general Feynman integral takes the form $$ F_{\Gamma}(q_1,\ldots,q_n;d) = \frac{i^{-a-h}\pi^{2h}}{\prod_l\Gamma(a_l)} \int_0^{\infty}\mathrm{d}\alpha_1 \ldots \int_0^{\infty}\mathrm{d}\alpha_L \prod_l\alpha_l^{a_l-1} \mathcal{U}^{-2} Z e^{i\mathcal{V}/\mathcal{U} - i\sum m_l^2\alpha_l} $$ where $\mathcal{U}$ and $\mathcal{V}$ are defined as sums running over trees and 2-trees of the given Feynman graph. I know that $\mathcal{U}$ is equivalent to $\det{A}$ in the $4h$-dimensional Gauss integrals, but I can't figure out how it can be expressed in the language of graph theory. Could anyone provide some help? References on the topic of graph theory and Feynman integrals are also desired. Thanks a lot!


How exactly does gravity work?

The electromagnetic force and strong and weak forces require particles like photons and gluons. But in case of gravity there is no such particle found. Every _mass_ bearing object creates a gravitational field around it, and whenever another mass bearing object enters its field the gravitational force comes into operation. If all other forces of nature have some particles associated with them why should gravity be an exception? And if there is no such particle, what exactly is the gravitational field and how does it spread over an infinite distance and cause the gravitational force to operate? * * * **Note:** I am a high school student and have not studied quantum mechanics.


Choosing proper centripedal force in Newtonian gravitational forces

I am trying to do an orbit simulation, I took Newton's law of Newtonian gravitational forces. It seems to work correct, but I can't really determine the velocity of an object needed to stay in free fall. I didn't want to use the masses in our planet systems, as the values are to high (in kg) So when my earth's mass is 1, what should be the masses be of the orbiting satellites and what should the initial velocity be to stay in freefall, when I want to altitude be 50 units (meters). It would be nice, if you could give me some hints!


Is there a graphical representation of the Lorentz transformation equations?

I always loved theoretical physics as a kid and when I came upon this site while seeking computer advice via superuser I had to stick my silly little head into an oasis of intelligence. I have often wondered what it would look like to view an object moving via the Lorentz transformation equations from a nominal speed of less than $c$ towards $c$ at an acceleration that a human could survive, let's say $3g$. And the space-time distance could be 100 light years. Would anyone be willing to produce a graph of this 1 dimension? I have tried to visualize it since I read "A Sophisticate's Primer of Relativity" 2nd Ed PW Bridgman when I was 19. As a novice I always wanted an answer from a higher power and now I think I can get a response.


Partition Function - TOTAL internal energy vs Average Energy

Given a partition function Z, the books sometimes uses the terms 'total internal energy' and 'average energy' exchangeably. It confuses me to no end. On one hand they say that Internal energy is average energy: $$\bar U = -\frac{\partial ln Z}{\partial \beta}$$ Then they use this to calculate specific heat: $C_v = (\frac{\partial U}{\partial t})_V$. On the other hand they say internal energy is $$U = \sum \epsilon_p n_p = \int \epsilon \space n_p g_{(\epsilon)} d\epsilon$$ where $n_p$ is mean number of particles with energy p, $g_{(E)}$ is density of states. Then they use this to calculate heat capacity. Which is which?!


Problem in progresive wave

In this progressive wave  we need to find out the phase difference. If we define it by $\phi$ in the x distance, we get . $$\phi = \frac{2 \pi}{\lambda}x$$ But when we write the prograsive wave equation we , write as, $$y= A \sin (\omega t - \phi)$$ A author says that this phase difference is less than the phase difference in the left side. My problem is, we can insert also + sign. But what is the physics for inserting the plus or minus sign. Thanks


Examples to illustrate temperature dependant radiation by examples of temperature to color relation

I'm looking for a set of examples to **illustrate** the relation of temperature and color of "glowing" bodies.   It should allow to build an intuitive understanding of this relation, so it's not about spectra, but just color in terms of visual appearance. The examples should cover infrared to somewhere above visual light if possible (or reasons why it's not possible). * * * Some good temparature/color-relation-examples froom comments/answers: An example for about **2500K** could be a **tungsten wire** of an incandescent lamp. It is not a good example because the wire in gas is not a good model for a black-body, as C. Towne Springer explains in his answer.   He provides an an **electric kiln** as an example for about **1400K** giving very good black-body approximation. As said, it's about the illustration, not about perfectly matching the spectrum of a black body - "similar enough to look the same" would qualify. From Robin's comment, the **sun** could be good as example for about **5800K**.


How long does it take for the speed of the train to decrease to $1~\text{m/s}$?

Here's the full problem: > _The speed of a train coasting on a level track satisfies the differential > equation $\mathrm{d}V/\mathrm{d}t = kv^2$. The initial speed of the train is > $v(0)=10~\text{m/s}$ and the speed is decreasing at the rate of > $-1~\text{m/s}^2$ when $v=5~\text{m/s}$. How long does it take for the speed > of the train to decrease to $1~\text{m/s}$? Under this model, when does the > train stop?_ I'm very bad with word problems and have been struggling quite a bit. I'd go to my school's tutoring center but currently we are on labor day vacation. -_- EDIT 1: I've attempted it a few times and here's my answer: $vknot=10~\text{m/s}, v=5~\text{m/s}, a=-1~\text{m/s}^2$ I found $k = -1/25$ and the proceeded to integrate the $\mathrm{d}V/\mathrm{d}t = kv^2$. The equation I then got was $v=kv^2t + vknot$ and plugged all my conditions in. My answer was $t=5~\text{s}$. I have no idea if I am correct as word problems always throw me off. EDIT 2: I found k=-1/25 by plugging into the dV/dt equation. dV/dt is the same as acceleration and it's stated that our acceleration is -1m/s^2, leading to this: $-1 = k(5)^2$ ===> $-1 = 25k$ ===> $-1/25 = k$. I don't know if this was valid to do, but I wasn't sure how to approach so made the assumption that it was valid. EDIT 3: After realizing the error with my integration, after integrating I got the formula $v=v(vknot)kt + vknot$. Time was then t=2.5 after plugging in my information. Now I just don't know how to tell when the train stops...


Why are black hole singularities stable?

The Friedmann equations says that huge matter densities lead to huge expansion rates. In Newtonian gravity, two massive point particles separated by an infinitesimal distance will experience an enormous force. So what is different about a black hole?


Uniform Acceleration

> _A steel ball is allowed to roll from rest down a 20.0 m long slop and over > a 25 m long level surface. It takes 4.0 seconds to roll down the slope. > Calculate the speed at the bottom of the slope._ I found the average speed on the slope to be 50 m/s. But, I'm not sure what formula to use to find the speed at the bottom of the slope.


Minimizing the energy of a Slater determinant: why are the Lagrange multiplier elements of a Hermitian matrix?

If I want to minimize the energy of a Slater determinant subject to the constraint that the spin orbitals are orthonormal (as in the Hartree-Fock approximation), I can use Lagrange's method of undetermined multiplier, i.e. $$L[\\{\chi_{a}\\}] = E_{0}[\\{\chi_{a}\\}]-\sum_{a=1}^{N}\sum_{b=1}^{N}\varepsilon_{ba}([a|b]-\delta_{ab})$$ where $\\{\chi_{a}\\}$ are the spin orbitals, $E_{0}$ is the ground state energy, $[a|b]$ is the overlap integral between spin-orbitals $\chi_{a}$ and $\chi_{b}$ and $\varepsilon_{ba}$ is a Langrange multiplier. By setting the first variation of $L$ to zero I can proceed to minimize the energy of a single determinant $|\Psi_0\rangle=|\chi_{1}\chi_{2}\cdots\chi_{a}\chi_{b}\cdots\chi_{N}\rangle$. A problem in Szabo and Ostlund's "Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory" asks me to prove that the Lagrange multipliers are elements of a Hermitian matrix, i.e. $\varepsilon_{ba} = \varepsilon_{ab}^{*}$, given that $L$ is real and $[a|b] = [b|a]^{*}$. How would one go about proving this? Where I am starting at the moment is that since $L$ is real, then $L = L^{*}$. If I find $L^{*}$ then making this equal to $L$ will show that it is necessary for the Lagrange multipliers to be elements of a Hermitian matrix. However, following through with this doesn't seem to get me to a point where I can say that $\varepsilon_{ba} = \varepsilon_{ab}^{*}$.


No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\\ \nu = E/h .$$ equate a specific frequency and wavelength to a particle, yet we know that a wave packet is a linear combination of an infinite range of frequencies and wavelengths. How is it that we (or nature) choose one frequency and wavelength out of the range? Does this have to do with the collapse of the wave packet when measured? And if so, is the resulting measured frequency a random outcome? Similarly, when an electron jumps from one energy level to another in an atom, it emits a photon of frequency $$ \ \nu = \Delta E/h .$$ Since the photon is not a pure sinusoidal wave, how can a single frequency be ascribed to the photon?


Lorentz transformations of the polarization vector

![enter image description here](http://i.stack.imgur.com/jt63k.png) Let $\bf{n}'$ be a unit vector in the direction of a wavevector in the plasma rest frame and $\bf{B'}$ be a unit vector along the magnetic field in the plasma rest frame. The electric field of a linearly polarized electromagnetic wave is directed along the unit vector $\hat{e}=\bf{n'} \times \hat{B'}$, and the magnetic field of the wave is along the unit vector $\hat{b'}=\bf{b}\times \hat{e'}$, such that the Poynting flux along is $\hat{e'}\times \hat{b'}$ directed along $\bf{n'}$. We give a Lorentz boost to the explosion frame to find the electric field $e$ there, normalize it to unity, and project e$$ on some given direction (e.g. , along the projection of the flow axis on the plane of the sky). Fields in the wave expressed in terms of the direction of a photon in the explosion frame $\bf{n}$ is \begin{equation} \hat{e'} ~=~\frac{\bf{b} \times \hat{B'}}{\Gamma (1-\bf{n}\cdot\bf{v})}+\frac{1+\Gamma (1-\bf{n}\cdot \bf{v})}{(1+\Gamma(1-\bf{n}\cdot \bf{v}))} \end{equation} where $\Gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the bulk Lorentz factor. Would some one help to deduce the above equation?


Higgs Boson mass in Electron volts?

Im no physics genius here, I was just interested in the Higgs Boson so I was reading this article : How the Discovery of the Higgs Boson Could Break Physics I came across this > Furthermore, all indications are that scientists will find that the Higgs > **weighs 125 gigaelectronvolts (GeV)** – or about 125 times more than a > proton – which means that it sits exactly where the Standard Model expected > it to be. How can weight be measured in electron-volts?


Question on dielectric sphere

The question is: Suppose the field inside a large piece of dielectric is $E(0)$ so that $D(0) =$$\epsilon(0)E(0)$ + $P$. Now a small spherical cavity is hollowed out of the material. Find the displacement($D$) in the center of the cavity in terms of $D(0)$ and $P$. Now, the question had a hint: carving out a cavity is the same as superimposing an object of the same shape but opposite polarization. Using that hint the field at center of sphere = $E(0)$ - field due to uniform polarization of sphere. So we get, $E=E(0)+P/(3\epsilon(0))$ To get $D$ it should be $\epsilon(0)E$ and not $\epsilon(0)E$ + $P$. Why is it so? We have a field $E(0)$ which gives rise to polarization $P$ of the dielectric material. So we get $D(0)$. So now while calculating $D$ at center why don’t we consider $P$? Is it because that, if I do consider $P$, then I would somehow consider its contribution twice?


Gravity force between two objects with different mass

We know thanks to Newton that: $$F=G\frac{m_1\cdot m_2}{r^2}$$ Where $G$ is the Gravitational Constant that is about $6.673\cdot10^{-11}$ $m_1$ and $m_2$ are the masses of two different objects $r$ is the distance between them. We also know that $F=ma$ so can we write this? : $$ma=G\frac{m_1\cdot m_2}{r^2}$$ If so what is $m$ in the first term of the equation? $m_1$ or $m_2$? Thanks a lot for help!!!


What happens when molten metal cools within a strong magnetic field? I don't know what more say, don't remember anything relevant to this in my uni text books.


Calculating an energy of an electron with known De Broglie wavelength (why can't we calculate it similar than we do it for a photon)

Lets say we have an electron with known De Broglie wavelength $\lambda$. Can anyone justify or explain why we calculate its energy $E$ using **1st** the De Broglie relation $\lambda = h/p$ to get momentum $p$ and **2nd** using the invariant interval to calculate $E$: \begin{align} p^2c^2 &= E^2 - {E_0}^2\\\ E &= \sqrt{p^2c^2 + {E_0}^2} \end{align} Why we are not alowed to do it like we do it for a photon: \begin{align} E=h\nu = h\frac{c}{\lambda} \end{align} These equations return different results.


Uniqueness of the number of degrees of freedom

As per my knowledge, degrees of freedom of any physical system are the number of independent quantities(coordinates) which need to be specified in order to specify the state of a system uniquely. However, my question is, why is there a unique number of degrees of freedom for any system? Or, to put it differently, if a set of $n$ independent coordinates describes the state of the system completely, why is it true that any other set of $n$ independent coordinates also describes the state of the system completely? And why can there not be a set of $m$ independent coordinated which also describe the state of the system completely, where $m>n$ ?


How to solve bound states of 2D finite rectangular square well?

I want to solve bound states (in fact only base state is needed) of time- independent Schrodinger equation with a 2D **finite** rectangular square well \begin{equation}V(x,y)=\cases{0,&$ |x|\le a \text{ and } |y|\le b$ \\\ V_0,&\text{otherwise}}.\tag{1}\end{equation} $$\Big[-\frac{\hbar^2}{2m}(\partial_x^2+\partial_y^2)+V(x,y)\Big]\psi(x,y)=E\psi(x,y)$$ At first glance, this problem is simple. It seems that the solution is variable-separable and can be written as $\psi(x,y)=f(x)g(y)$. Then $$ \frac{f''(x)}{f(x)}+\frac{g''(y)}{g(y)}+\frac{2m}{\hbar^2}(E-V)=0.$$ Let $E=E_x+E_y$ and $V=V_x+V_y$, then the problem is reduced to two 1D problems $$\cases{f''(x)+\frac{2m}{\hbar^2}(E_x- V_x)f(x)=0\\\g''(y)+\frac{2m}{\hbar^2}(E_y-V_y)g(y)=0}.$$ However, how to determine $V_x$ and $V_y$ in the 2D space? A definitely wrong method is making $$ V_x=\cases{0,&$|x|\le a$\\\V_1,&$|x|>a$}\text{ and }V_y=\cases{0,&$|y|\le b$\\\V_2,&$|y|>b$}\tag{2}.$$ In fact, the potential Eq. (2) is equivalent to two independent "1D finite square well" problems in $x$ and $y$ direction respectively. However, a careful reader will note that the potential Eq(2) is DIFFERENT from Eq(1), which means that the potential Eq(2) is NOT what we want. It's not a rectangular well, but as following ![Potential of Eq\(2\), but NOT a 2D square well.](http://i.stack.imgur.com/IO5Xu.png). Then, I find that a variable-separable bound state for finite 2D square well does not exist. Although analytical solutions exist in each region with a constant potential, problems occur when matching boundary conditions to keep the continuity of $\psi(x,y)$. Unlike matching boundary condition at descrete points in 1D, in 2D we have to match boundary conditions along lines, e.g., $$ f_1(a)g_1(y)=f_2(a)g_2(y)$$ in the boundary between $x<a$(region 1) and $x>a$ (region 2). This leads to $$ g_1(y)/g_2(y)=f_2(a)/f_1(a)=constant.$$ Matching all boundaries this way will lead to that $\psi(x,y)$ have to be 0 outside the well. But this cronsponds to the case of INFINITE well. It's not the solution of finite well. Then I think no solutions exist under the separating-variable method. Then, the question is, beyond separating-variable method, how to solve this problem? BTW: Does anyone know that what kind (shape) of 2D well is solvable for bound states and how? (Potential with circular symmetry is excluded, because I know how to solve it. I want to find another shape of 2D well which is solvable.)


Total angular momentum - single electron

I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero. My book says that total atomic angular momentum is $\mathbf{J}=\mathbf{L+S}$ and its magnitude $J=\sqrt{j(j+1)}\hbar$ where quantum number $j$ can have half-integer values on an interval $|\ell-s|\leq j \leq|\ell+s|$. There is allso a quantum number $m_j$ which can have half-integer values on an interval $-j \leq m_j \leq j $. So lets say I have a single electron in an orbital $\scriptsize\boxed{\ell=1}$ _(p orbital)_. In this case $j=\tfrac{3}{2},\frac{1}{2}$ while $m_j=\tfrac{3}{2},\frac{1}{2},-\tfrac{3}{2},-\frac{1}{2}$ and I can calculate two magnitudes for $J$: \begin{align} J&=\tfrac{\sqrt{15}}{2}\hbar\\\ J&=\tfrac{\sqrt{3}}{2} \hbar \end{align} Now there is this weird vector sum image that I don't understand completely: ![enter image description here](http://i.stack.imgur.com/vvecP.jpg) On the left image we have $J=\tfrac{\sqrt{15}}{2}\hbar$ while on the right we have $J=\tfrac{\sqrt{3}}{2}\hbar$. It seems that vector $\mathbf{L}$ together with its magnitude $L=\sqrt{\ell(\ell+1)}\hbar = \sqrt{2}\hbar$ is the same in both cases - this makes sense as quantum number $\ell$ doesn't change. But how do we explain the change in $\mathbf{S}$?


Is the cosmological redshift caused by the Planck mass increasing?

The standard explanation for the cosmological redshift is that photons emitted from far away galaxies have their wavelengths lengthened as they travel through the expanding Universe. But perhaps the photons do not lose energy as they travel but rather the atoms in our detectors are more energetic in comparision with the atoms that emitted those photons a long time in the past leading to an apparent redshift effect? **Addition** (having had a comment exchange with @rob, see below) : My hypothesis is that the Planck mass $M_{pl} \propto a(t)$ where $a(t)$ is the Universal scale factor. **Addition 2** Of course if the Planck mass $M_{pl}$ is changing then $G=1/M^2_{pl}$ is changing so that we no longer have standard GR! I've asked this question before, see Cosmological redshift interpretation, but this time I'm including a little bit of theory to back up my hypothesis. For simplicity let us assume a flat radial FRW metric: $$ds^2=-dt^2 + a^2(t)\ dr^2$$ Consider the null geodesic path of a light beam with $ds=0$ so that we have: $$dt = a(t)\ dr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ Now at the present time $t_0$ we define the scale factor $a(t_0)=1$ so that we have: $$dt_0 = dr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ Substituting equation (2) into equation (1) we have: $$dt = a(t)\ dt_0$$ In order for the interval of time $dt$ to stay constant as the scale factor $a(t)$ increases we must have the corresponding interval of present time $dt_0$ varying inversely with the scale factor: $$dt_0 \propto \frac{1}{a(t)}$$ Thus as cosmological time $t$ increases, and the Universe expands, equal intervals of cosmological time $dt$ correspond to smaller and smaller intervals of present time $dt_0$. Now the energy of a system is proportional to the frequency of its oscillation which in turn is inversely proportional to its oscillation period: $$E(t) \propto \frac{1}{dt}$$ The corresponding energy of the system in terms of the present epoch $t_0$ is given by $$E(t_0) \propto \frac{1}{dt_0}$$ $$E(t_0) \propto a(t)$$ Thus an atom at time $t$ is a factor $a(t)$ times more energetic than the same atom at time $t_0$. As the energy scale is ultimately set by the Planck mass then the Planck mass must be increasing as the Universe expands: $M_{pl} \propto a(t)$. This effect alone would account for the gravitational redshift of distant galaxies without the assumption that photons travelling from those galaxies lose energy due to wavelength expansion. Addition: I believe this hypothesis leads to a linear cosmological expansion $a(t)\propto t$ (see comments below).


Is there any usage of $x^x$ in physics?

I am so excited about the function: $x^x$. I would like to know if there is any usage in the physics world of this function. Is there any formula that uses $x^x$?


Flow velocity out of a pipe

For a compressible homoentropic gas is the flow velocity out of the pipe the same as the velocity into the pipe?


Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we assume that a particle feels a force $F = -Kx$ like the classical version, and we define $K$ as the curvature of $U(x)$ at the point of stable equilibrium ($x = 0$). That is, $$K = \left.\frac{d^2 U}{dx^2}\right|_a$$ where $x = a$ is the general point of stable equilibrium. It then uses the formula $\omega = \sqrt{\frac{K}{m}}$, though this is a classical equation. I don't understand why this is technically valid. I understand that that formula comes from the classical description $-Kx = ma = m \frac{d^2x}{dt^2}$ where $x = A\sin(\omega t + \phi_0)$. However, with quantum mechanics, why can we say that it oscillates by that description? We're not even dealing with a particle; it has wave and particle properties described by the wavefunction, so why can we immediately say that $U(x) = \frac{1}{2}Kx^2 = \frac{1}{2}m\omega^2x^2$? When we solve the Schrodinger equation, we find that the particle isn't limited to the interval $[-A, A]$, so why is it valid to use a formula that is derived from assuming that? Any help is much appreciated!


Explicit solutions to 2-d Dirac Equation

The 2-d Dirac equation without any constants is represented usually as $$i*dt (\phi) = D (\phi)$$ where $D = m\sigma_2-i\sigma_1dx-i\sigma_3dy$. Where can I find explicit closed form solutions to this equation? If there are none, are there perturbation-like non-exact solutions? Thanks alot.


Anomalies for not-on-site discrete gauge symmetries

If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in general there are discrete symmetries that cannot be gauged. For example, Freed and Vafa in http://link.springer.com/article/10.1007%2FBF01212418 discuss how in 1+1d one needs the pullback of a certain class in $H^3(G,U(1))$ to have some trivial periods. So, is the converse true--that if $G$ can be gauged then there is a formulation of the theory where $G$ acts ultralocally? In other words, is a symmetry with no ultralocal action necessarily anomalous? And if so, can we see this anomaly as an explicit class in $H^3(G,U(1))$ for 1+1d theories, for example? It seems to me like the answer is yes. If we have no anomaly and go ahead and gauge $G$, then we can put the result on a lattice where the $G$ gauge field will live on edges. These edge variables will have the flatness condition that the start and end vertex variables differ by the action of the edge variable (an element of $G$). It seems like $G$ shouldn't act anywhere else, since in some sense gauging $G$ is a type of "fat quotient" of the theory by $G$. Thus, if we take this lattice formulation and forget the gauge field, we end up again with the original theory, but now with an ultralocal action of $G$. What remains is how to quantify this anomaly in the cohomology group.


Twisted Nematic liquid crystal: extent of degradation due to DC drive?

It is commonly accepted wisdom that a Twisted Nematic LCD must always be driven using AC instead of DC, because DC will supposedly denature the LC. I have two questions:   1. Does this effect only happen at typical drive voltages of 8V ~ 9V? If the LCD is driven with a lower voltage, around 3V, is the degradation not as bad?   2. At 3V, how many hours does it take to degrade the LC? I've run a test for 500 hours with no bad effects. Thanks!


What is meant by "net charge"?

Lets consider a system of two opposite charges separated by a certain distance (dipole), if we ask what is the net charge for this system? the answer would be zero. The net charge (what I have come to know until now) is said to be addition of sign of charges irrespective of the distance between the charges. Lets consider the following case:   According to Gauss's law the total electric flux through a closed surface enclosing a charge is equal to $\frac{1}{\epsilon_0}$ times the magnitude of charge enclosed. In the below figure we have a $+q$ charge enclosed in the guassian surface, so there is a flux linked with the guassian surface, which will be equal to $\frac{1}{\epsilon_0}$ times the $+q$ charge. ![enter image description here](http://i.stack.imgur.com/eiu87.gif) Now, replace the $+q$ charge with the dipole in which charges are separated by a certain distance. Magnitude of electric field ($E$) due to an electric dipole at a distance $r$ from its centre in a direction making an angle $\theta$ with the dipole is given by the equation,$$E=\frac{1}{4\pi\epsilon}.\frac{p\sqrt{3\cos^2\theta+1}}{r^3}$$   where, $p=2aq$ ($2a$ is the distance of separation of the charges $q$). Electric field around the dipole will be zero if and only if distance of separation of the dipole charges is zero. Although we can't expect field lines to be perpendicular to the surface, we can expect some contribution due to the electric field components (perpendicular to the surface) of the enclosed dipole (with charges separated by a distance). From the dipole equation it follows that there exists electric field around dipole when charges are separated by distance, so there is flux through the surface which implies that there is charge enclosed in the surface (from gauss law). But net charge of dipole is said to be zero. By gauss law, net charge is not zero. Then what is net charge? From gauss law we can directly find the $q$. Can we say this as net charge? Then, is it that net charge can't be said directly just by adding the signs and is it a function of electric field? If I have misunderstood, pardon me and explain. Sometimes the problem might not be understood well due to communication error, if any correction required, comment on the part.


What object is quantized in quantum gravity?

In theories of quantum gravity, which object is it that is quantized? Working on field theories, I expect the quantization to mean the promotion of a classical field to an operator valued field that works on some kind of Fock space. I heard in QG the metric gets quantized. This would imply, by my above reasoning, that the metric and thereby spacetime itself is an operator working on a Fock space of spacetime excitations. While I can accomodate this picture well with my idea of graviational waves, how does it rely to spacetime in general? Especially, how do the other quantum fields relate to this operator-spacetime, since they have an intrinsic dependence on spacetime?


Force of exploding pumpkin

Okay so in physics class we are going to explode a pumpkin by placing some kind of bomb inside of the pumpkin. Well us students have to come up with an apparatus to measure the force of the explosion, i am in desperate need of some help or some ideas of some solutions and what i need to measure to find the force of the explosion. I know that i need to find the mass of the chunks of pumpkin, the position of the pumpkin chunks from start to finish, and the time it took for the chunks to reach a certain point. What else do i need to know and/or find? PLEASE HELP


Straight line on 8-axis

How would a straight line appear in a 8-axis co-ordinate system. Will it continue to look like a straight line or will it appear like zig-zag line. We can see that straight line appears like a dot in one-dimension, straight line in two-dimension graphs.


Modeling an Adiabatic Expansion (EES)

How would one model an isentropic (reversible) adiabatic expansion using thermodynamic software in order to determine isentropic work? What I am looking for is an actual pressure-volume curve for the isentropic expansion of a gas. Approach using EES: 1) Start with a known gas state/scale (Mass, Volume, Pressure, Temperature, Enthalpy, Entropy, etc.)   2) Update gas volume by a small increase in volume dV   3) Work done = Pressure * dV   4) Keep entropy constant   5) Recalculate new gas state based on the new Specific Volume (Volume/Mass) and Entropy   6) Repeat until final state has been reached. Summing the work done in this method results in significantly less work (~85%) than is possible through isentropic expansion. The max isentropic work is calculated through   Wmax = Mass * (h_initial - h_final) Even with very small step sizes (thousands of steps), this approach doesn't work. Can anyone shed some light as to why these work values don't match up? Code:               Procedure Adiabatic Expansion(P, T, P_final : Work_max, Work)       Vol=1       v=Volume(CarbonDioxide,P=P,T=T)       s=Entropy(CarbonDioxide,P=P,T=T)       h_initial=Enthalpy(CarbonDioxide,P=P,T=T)       Mass=Vol/v       Vol_step=1/10000       Work=0            repeat       Vol:=Vol + Vol_step       v:=Vol/mass       Work_step:=(P)*Vol_step       Work:=Work + Work_step       P:=Pressure(CarbonDioxide,v=v,s=s)       until (P <= P_final)            h_final=Enthalpy(CarbonDioxide,P=P,s=s)       Work_max=mass * (h_initial - h_final)       End


Does this statement make any sense?

I am asking this question completely out of curiosity. The other day, my roommate, by mistake, used 'Light year' as a unit of time instead of distance. When I corrected him (pedantic, much), he said the following: "Units are relative. And according to Fourier Transforms, units can be changed so Light year is a unit of time." That got me thinking and I read up Fourier Transforms on wikipedia but couldn't find anything about using a unit in one domain as a unit for another measurement. I do agree that units (particularly, base units are relative. eg: the meter), but does his statement make any sense? EDIT Thank you everyone for all the answers. It isn't so much to in it in or prove a point as it is to understand the concept better. Anyways this is his response after I showed him this thread. Any comments would be appreciated. His response: Nevermind, for the first time I accept I was wrong. BUT using lightyears to measure time is possible. My example didn't make sense bacause I was wrong when I meantioned that I'm still measuring dist. If you have a signal in time domain and ...take the FT, I get a signal which DOES NOT HAVE to be in frequency domain. Clarify this to the guy who posted last. Now the new signal is in a domain defined by me and so is its units. This signal although not equal to the original signal, still represents that if ya take an inverse FT. So, the idea of time will still be there. Now coming back to our case: lightyears here is not the lightyears you are used to read when dealing with distance. It represents time.


How to find exponential values

I have a power equation for flow between two circular disks on a common axis as stator and rotor, which follows, $$P = \frac{n^a}{G}\tag{1}$$ Where:   * $P$ = power,   * $n$ = revolution (1200 rev/min),   * $G$ = gap between two plates (0-2.5 mm) I have experimental data showing Power vs Gap curve at a constant revolution of 1200 rmp. How do I get the value of $a$ and how do I plot Eqn (1)?


What causes a ring-like image around light of the moon?

I just encountered an interesting image in sky. As you can see in following images there was a ring-like image around light of the moon. I don't know if it was clouds but it was looking like it is far above the clouds. I got curious and only place I know I could get some answers is here. It is much more clear in first photo, other ones are a bit blocked by clouds in between Here are some photos I took: ![Photo 1](http://i.stack.imgur.com/s4cBL.jpg) ![Photo 2](http://i.stack.imgur.com/zWijW.jpg) ![enter image description here](http://i.stack.imgur.com/6c4tf.jpg)


If acceleration causes relative time dilation does the eventual deceleration reverse it? For example: traveling to Alpha Centauri Based on me reading this site: http://www.convertalot.com/relativistic_star_ship_calculator.html -


Solving a statically indeterminate truss

I'm trying to solve the statically indeterminate truss shown below and I'm having a little trouble. ![truss](http://i.stack.imgur.com/7tLWa.png) $H$, $P$ and $\beta$ are given. The material is aluminum (density is 2700 ${kg/m^3}$) and has a cross section of 0.002 $m^2$. We are instructed to take the weight of each member into account by distributing $\frac12$ of each member's weight to its end joints. I can easily solve for $f_1$, $f_3$, $f_4$, $f_5$ and the support reactions using statics (method of joints), but I can't figure out how to go about solving for the remaining members. The main point of my question is, how do you solve statically indeterminate trusses? I assume I'll have to use some other property such as deformation. I've only posted the homework question to provide background as to why I am asking. Thanks for any help.


On characteristic X-ray spectra and relative intensity

I am looking at the following image (from http://www.schoolphysics.co.uk/age16-19/Atomic%20physics/X%20rays/text/X_ray_spectra/index.html): ![enter image description here](http://i.stack.imgur.com/Z270f.gif) and I want to know, of the two curves, which one represents the element with the higher atomic number. That is, I understand that X-rays are scattered and that the peaks are characteristic of a material. What I am less clear on i why one curve should be above another relative to their atomic numbers (I am going to assume the cutoff voltage on the left is identical even though in this picture it isn't). The reason I am confused is that I am told that that $\frac{1}{\sqrt{\lambda}} \propto Z$ which is all very well, but on this graph it would seem to indicate that the upper line is a heavier (higher Z) material. Is hat the case, or is intensity actually proportional to $\frac{1}{\sqrt{\lambda}}$ as well? I would have thought the lower curve would be a higher Z since it usually take less energy to knock electrons off of elements further up the periodic table. Anyhow, what clarification might be offered is appreciated.


What is escape velocity? In reality, how can something no longer be under the gravitational influence of something else?

Isn't G a continuous function and although you leave the immediate vicinity of the earth with an escape velocity won't it always exert a force, however small it may be. Won't that force eventually pull the object back to the earth (assuming the absence of other objects)


Can horizontally and vertically polarized light combine to become circularly/elliptically polarized light?

Well, we know that circularly/elliptically polarized light is made up from orthogonal components. So is it possible then to create circularly/elliptically polarized light by combining horizontally and vertically polarized light? It seems to make perfect sense to me. EDIT: Clarification: I mean by creating circularly/elliptically polarized light in the laboratory (though beam-splitters ?, I don't know) by combining different light beams of horizontally/vertically polarized light.


A Cosmological horizon at the Hubble radius?

I have calculated that if one extends a rigid ruler into space by a fixed proper distance $D$ then a clock at the end of the ruler, running on proper time $\tau$, will run more slowly than time $t$ at the origin by a time dilation factor: $$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - H^2 D^2 / c^2}}$$ where $H$ is the Hubble parameter. If one substitutes in Hubble's law, $v = H D$ (the theoretical law which is exact), one finds the following satisfying result that $$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - v^2 / c^2}}.$$ Although this looks like a result from special relativity I derived it by combining the FRW metric line element and an equation for the path of the end of the ruler, $\chi = D / R(t)$, where $\chi$ is the radial comoving coordinate of the end of the ruler, $D$ is a fixed proper distance and $R(t)$ is the scalefactor. Does this prove that there is a cosmological horizon at the Hubble radius $D=c/H$ where the proper time $\tau$ slows down to a stop compared to our time $t$? This seems to be a general result which is true regardless of cosmological model. Details of Calculation The general FRW metric is given by: $$ds^2 = -c^2 dt^2 + R(t)^2\left[d\chi^2+S^2_k(\chi)d\psi^2 \right]$$ where $d\psi^2 = d\theta^2 + \sin^2 \theta d\phi^2$ and $S_k(\chi)=\sin \chi$,$\chi$ or $\sinh \chi$ for closed ($k=+1$), flat ($k=0$) or open ($k=-1$) universes respectively. The scale factor $R(t)$ has units of length. Consider a ruler of fixed proper length $D$ extending out radially from our position at the origin. The path of the far end of the ruler in comoving coordinates is $$\chi = \frac{D}{R(t)}$$ Differentiating this equation by proper time $\tau$ gives us: $$\frac{d\chi}{d\tau} = - \frac{D}{R^2} \frac{dR}{dt} \frac{dt}{d\tau}.$$ Using the FRW metric we can find a differential equation for the path of the far end of the ruler. We substitute in $ds^2=-c^2d\tau^2$ (end of ruler has a time-like path), $d\psi=0$ (the ruler is radial) and divide through by $d\tau^2$ to obtain: $$c^2\left(\frac{dt}{d\tau}\right)^2 - R(t)^2 \left(\frac{d\chi}{d\tau}\right)^2 = c^2.$$ Substituting the expression for $d\chi/d\tau$ into the above equation we find: $$c^2\left(\frac{dt}{d\tau}\right)^2 - D^2 \left(\frac{\dot R}{R}\right)^2 \left(\frac{dt}{d\tau}\right)^2 = c^2.$$ Using the definition of the Hubble parameter $H=\dot{R}/R$ we finally obtain: $$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - H^2 D^2 / c^2}}.$$


Are these data enough to solve this simulataneity problem

observers: **S** and **S** '. S stands **in midway** between two events (say, 2 flashes) located **20 metres apart**. He sees two flashes occurring **simultaneously**. **S'** notes a **20ns difference** in them. **S'** is moving along x axis with uniform velocity What is this velocity in S frame? delta t equation leads to a 4th degree polynomial equation of v, so I must be wrong. Please help.


Educational physics stories with a funny punchline

As a physicist, when I'm giving lectures or during colloquiums, I usually find it necessary to have some appropriate and related jokes in hand. All of the best teachers have some, and they use them when they have to; so I think that it is a necessity of good teaching. On the math side we have this gem(although being closed and reopened several times); however on the physics side I couldn't find much. Hopefully this topic will help me and many others with this regard.


RC Parallel Circuits

I am trying to solve the following exercise: * * * A capacitor 27 and $\mu$F is connected to a battery of 21 V through a resistance of 14 M$\Omega$. 1) What 's the charge on his parallel-plate? 2) To reduce such a charge of 16%, I connect a wire in parallel with the condenser. What will be now the ddp across the parallel-plate capacitor? 3) What resistance the new wire must present to ensure this outcome? Solutions: q = 576$\mu$C; $V_c$ = 17.64 V; $R_2$ = 2.666667K$\Omega$ * * * I succeeded to find the solution for the question 1) and 2) but not the 3). Here is what I tried: the picture should be like this: ![enter image description here](http://i.stack.imgur.com/YVOob.jpg) $R_1$ = 14 M$\Omega$ $V_f$ = 21 V c = 27 $\mu$F * * * q = c * $V_f$ = 576 $\mu$ C $V_c$ = $\frac{q}{c}$* (1 - 16/100) = 17.64 V $R_2$ resistance in parallel to the capacitor (1) $R_2$ * I = $V_c$ (2) $V_f$ = ( $R_2$ + $R_1$ ) * I I tried to solve (1) and (2) to find the $R_2$ but the result I found is different from the solution. Any hints? Thanks.


complex numbers in optics

I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers? I suppose this is quite fundamental, but I do not find the answer in my course ... Thanks, Isaac


Angular Displacement

If something is rotating about a point and it covers a complete circle, should we take its angular displacement as 360 degree or 0? Please give link to some established material on this subject which defines your answer, whether it is that it should be taken as 0 or 360 degrees. Question that led to this problem : The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. 1. What is its angular acceleration, assuming the acceleration to be uniform, 2. How many revolutions does the wheel make during this time. In the solution to above question the author(s) for solving part 2 have used equation of motion and mentioned that they have obtained angular displacement for it, it basically implied that the equations of motion used for rotation provide angular displacement which does not become zero on returning to original point which is not followed when we consider it analogous to displacement of linear motion. Addendum : Even here ( Angular displacement and the displacement vector ) the selected answer says that on completing the circle, angular displacement is 360 degrees, is there some established text to support this ? Similarly here ( http://www.ask.com/question/calculating-angular-displacement ) something else is said in the answer, is then angular displacement ambigous and hence has no correct definition, if there is please guide me to some established text.


Splitting light into colors, mathematical expression (fourier transforms)

I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function $G(x)$ of photons hitting a certain area to include just photons of a certain wavelength, say red light. I feel like this could be accomplished using a Fourier transform and de Broglie's law, but I'm not sure. Can someone please help, just for a general Gaussian function $G(x)$? More information: Basically, given a function that gives the number of photons hitting a certain area, I want a mathematical way to determine how many of those photons are of a specific frequency (such as red light). G(x) is defined as the integral in a gaussian slit experiment (that is, a double slit experiment with the path integral of a gaussian probability) such as in Feynmann's Path Integrals and Quantum Mechanics or in arxiv.org/pdf/1110.2346.


picture and isomorphisms

I found a paper by A. Belopolsky : "Picture changing operators in supergeometry and superstring theory" where he says there exists a possible physical state in one picture that doesn't exist in another picture. What A.B. does is essentially defining the picture formalism on the BRST complex and the de-Rham complex saying that the main idea would be that there is an even and an odd ghost number, picture changing involving the change in the odd ghost number. What strikes me here is the splitting of the complex on the odd number. I am not sure I get the full idea about what this means? Is it a sort of split-exact complex? And then why are the isomorphisms on the cohomology such a big deal? Shouldn't that be natural?


Surface potential resulting from the charge transfer between insulator and conductor

Common non-contact electrostatic voltmeters are used to measure the surface potential. Imagine we place a charged insulator on top of the conducting plate (insulated from the surrounding) and measure the resulting potential from the bottom of the plate. How is the measured potential related to the charge transferred from the insulator? To my understanding, the charge will be transferred from the insulator to conductor only at their interface. Therefore, the potential detected by the voltmeter at conductor surface should be only due to this portion of charge transferred. Is that correct? Does the conductivity affect the surface potential detected? If yes, what is the relationship? I do not have much experience in electrostatic events, I am a chemist trying to understand the above situation.


Why is electric current dangerous to humans?

How does a strong electric current harm our body? A strong electric current will posses a great charge. But how does that charge injure us?


Calculating the components of angular momentum of a rigid body

You have a rigid body with 1 fixed point in space (the origin). It's self-explanatory how to get the following equation for the angular momentum: > $\vec L = \sum_n m_n\vec r_n\times\vec v_n$ Where you take the sum of all "$n$" indicates all "$n$" points of mass.   This can be transformed into: > $\vec L = \sum_n m_n(\vec\omega(r_n^2)-(\vec\omega.\vec r_n)\vec r_n)$ Now, out of this, I have to get the following: > For component "$i$" of this equation, you get:   >  $L_i = \sum_n m_n(r_n^2\omega_i-x_{ni}\sum_j\omega_jx_{nj})$ I understand that for component "$i$", the scalars $m_n$ and $r_n^2$ are the same as for the other components, I also understand that for component "$i$" I need to take the $\omega_i$ component.   But what is meant with everything that follows after that? What does the "$x$" indicate for example? Does the "$j$" indicate the other components?


How was the universe created?

I do not know much beyond high school Physics. Thus, I am asking this question from almost layman's perspective: > _What, as per the best of our existing knowledge and widely accepted among > the scientific community, are the models of creation of the universe, and > what is their core argument?_


Scattering Amplitudes in Centre of Mass Frame

I'm reviewing page 59 of the QFT notes here and am a little confused by a reference frame argument. You can compute the second order probability amplitude term for nucleon-nucleon scattering to be $$-ig^2\left[\frac{1}{(p_1-p_1')^2-m^2+i\epsilon}+\frac{1}{(p_1-p_2')^2-m^2+i\epsilon}\right](2\pi)^4\delta(p_1+p_2-p_1'-p_2')$$ in a scalar field theory approximation. Now the author argues that we may remove the $i\epsilon$ terms by moving to the centre of mass frame. Here he says that $p_1=-p_2$ in this frame and that $|\vec{p_1}|=|\vec{p_1'}|$ by conservation of momentum. He continues to claim that the four-momentum of the meson is hence $k=(0,\vec{p}-\vec{p'})$ so $k^2<0$. Quite what $p,p'$ are I don't know exactly. I don't understand this argument at all. Surely in the centre of mass frame, the sum of all momenta $p_i$ and $p_i'$ is zero (.)? Also where has the second constraint come from? I don't see how morally you could get more than my claim (.). Could someone explain this argument to me? Very many thanks in advance.


Thermal equilibrium and non correlations

I read in a book on quantum fluctuations and quantum noise that, at thermal equilibrium the classical canonical variables are uncorrelated, ie: $$\langle xp\rangle=\langle x\rangle\langle p\rangle$$ But I am not sure to understand the sense of _at thermal equilibrium_ , for me it just means $$\langle x^2\rangle=\langle p^2\rangle=\frac{T}{2}$$ in the correct units with $T$ the temperature. On the other hand what I can derive easily in the canonical ensemble is: $$\langle xp \rangle = Z^{-1}\int {\cal{DxDp}}\;\text{e}^{-\beta H(x,p)}xp$$ and $$\langle x\rangle\langle p \rangle = Z^{-2}\int {\cal{DxDpDx'Dp'}}\;\text{e}^{-\beta H(x,p)}\text{e}^{-\beta H(x',p')}xp'$$ which are equal if we have the following equality $$H(x,p) = H_1(x) + H_2(p)$$ separability of $x$ and $p$ in the Hamiltonian, which seems unrelated to the previous hypothesis of thermal equilibrium (used here by taking the mean over the canonical distribution of course).


How far can a voice carry through the desert at night?

Let's use these conditions: temp = 7C, relative humidity = 10%, air pressure = 85 kPa, elevation = 1500m, shouting at 100dB (not sure if this is a reasonable volume). Given these conditions, what distance could a human voice travel and still be mostly understandable? How far to be heard at all? I haven't done any physics for a long time, and I really don't know how to go about this. Thanks.