question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>In the pilot's introductory book "Stick and Rudder" it claims that a nose-up glide is possible. It doesn't state how, why or when. It implies it's possible to do and maintain a constant forward velocity.</p>
<p>Is this possible? I really don't see how, unless the aircraft has what I assume would be an extremely unusual design, where the wings would have to have a reverse angle of incidence of the common designs. </p>
<p>Glancing through clancy's Aerodynamics, it seems that the force of lift acts upward, and slighly behind the normal of the chord. Given a glide has no thrust, I can't see how the net forces could balance with the drag to maintain forward velocity if the nose is up.</p>
<p>Thanks</p> | g12495 | [
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<p>The party line of the anthropic camp goes something like this. There are at least $10^{500}$ flux compactifications breaking SUSY out there with all sorts of values for the cosmological constant. Life takes a lot of time to evolve, and this is incompatible with a universe which dilutes away into de Sitter space too soon. The cosmological constant has to be fine-tuned to the order of $10^{123}$. Without SUSY, the zero point energy contribution from bosons and fermions would not cancel naturally.</p>
<p>However, superstring theory also admits N=2 SUGRA compactifications which have to have an exactly zero cosmological constant. Surely some of them can support life? I know there are a lot more flux compactifications out there compared to hyperKahler compactifications, but does the ratio exceed $10^{123}$? What probability measure should we use over compactifications anyway? Trying to compute from eternal inflation leads to the measure problem.</p> | g12496 | [
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<p>Ok, I've stumbled onto what I think is a bit of a paradox. </p>
<p>First off, say you had some computer in a very fast(near light speed) centrifuge. You provide power to this computer via a metal plate on the "wall" of the centrifuge's container, so it works similar to how subways and streetcars are powered.</p>
<p>If the computer normally would consume 200 watts, how much power would it consume at say 1/2 of light speed? Would it consume 400 watts from our still viewpoint?</p>
<p>Also, what if you were to be capable of communicating with this computer? Would the centrifuge-computer receive AND transmit messages faster from our still viewpoint? I'm a bit lost in even thinking about it.</p> | g12497 | [
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<p>Okay, I'm asking a question similar to this one here: <a href="http://physics.stackexchange.com/q/4404/2330">Time Dilation - what happens when you bring the observers back together?</a>. Specifically, I am curious about a specific angle on the second part of his question, regarding when two moving frames of reference (FoR) are brought back together, and how "it" knows which one should be still young.</p>
<p>The accepted answer on that question says that it is whichever FoR experienced the forces of acceleration/deceleration. But, isn't that the whole point of relativity, is that it's all ... well, relative; that it is not possible to say with a certainty that it was the traveler in the spaceship who was accelerating/decelerating? </p>
<p>Isn't it the case that it is just as legitimate to say that the universe and people on the planet accelerated/decelerated and the traveler in the spaceship was stationary? This would therefore then lead to that the universe and planet-side people should remain young and the spaceship occupant should be old, no? </p>
<p>Does dilation (temporal-spatial) just generally apply to the smaller of the two FoRs, or is there some other system or rule which "decides" which FoR gets dilated?</p> | g968 | [
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<p>How to do this?</p>
<p>Show that if the tension $F$ in a string is changed by a small amount $\mathrm dF$, the fractional change in frequency of a standing wave, $\frac{\mathrm df}{f}$ is given by:</p>
<p>$$\frac{\mathrm df}{f}~=~0.5\frac{\mathrm dF}{F}.$$</p> | g12498 | [
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<p>This question is raised because of the recent incident in a London subway station involving a stroller falling onto the tracks. The stroller was probably 10 feet (or 3 m) from the edge of the platform but was moved by the wind as the train departed and went into the tunnel.</p>
<p>Assume the weight will likely be 10 kg in total.</p>
<p>Also,
- what if the wheels of the strollers are locked?
- What if they are unlocked?</p> | g12499 | [
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<p>So I have the potential energy defined as:
$$U(x)=U_0[2(\frac xa)^2-(\frac xa)^4]$$
And I want to find the angular frequency of oscillations about the point of stable equilibrium. I found the point of stable equilibrium to be 0, but now I do not know how to find the angular frequency. Normally I would look to use
$$E=K+U=\frac 12m \dot x^2+U(x)$$
But I do not know where an $\omega$ would end up popping out. </p>
<p>I a not looking for a solution, just a hint.</p>
<p>EDIT:: My attempt:</p>
<p>Using conservation of energy, I get that
$$\frac 12 mv^2=U(a)=U_0$$
And then
$$\omega=kv=\frac {2\pi}{\lambda}\sqrt{\frac{2U_0}{m}}$$
Where $\lambda$=2a</p> | g12500 | [
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<p>Look at any Kruskal–Szekeres coordinate plot of the Schwarzschild solution. It shows the same mass everywhere. Yet the two sides cannot talk to each other, in that no information, particles, etc can cross the wormhole throat. So how do the sides 'know' to be the same mass?</p>
<p>So is there a way to draw a Kruskal–Szekeres plot with the masses unequal on each side? In other words, would the geometry of space play nice and smooth at the interface between regions II and III, where different mass solutions are right next to each other?</p>
<p>Another way of putting this is that if you overlap two K-S diagrams with different mass, M1 in region I && II, and M2 in region III and IV, and then do an embedding diagram, will you see something different than the single mass version. </p>
<p>Another way of putting it.
The Schwarzschild solution is static, and unique. So can you sew two of them with dissimilar masses together coherently? If not, then it would seem that another - non static - solution is in order, which would be surprising, since there is only one parameter (M) to be non static. </p>
<p>Look at say <a href="http://www.csun.edu/~vcmth00m/embedding.pdf">http://www.csun.edu/~vcmth00m/embedding.pdf</a> or similar for diagrams.</p> | g12501 | [
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<p>Of course it should have dimension $2n$.</p>
<p>But any more conditions?</p>
<p>For example, can a genus-2 surface be the phase space of a Hamiltonian system? </p> | g12502 | [
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<p>How dense does a gas (Argon in particular ) have to be to in order to ionize it using electron bombardment and weak magnetic fields. Is there a correlation with the density of a gas and the easiness to ionize the gas ?</p> | g12503 | [
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<p>I was studying for my exam and looking at the chapter which talks about Potential-energy graphs.</p>
<p>Let's take this as an example:<br>
<img src="http://i.stack.imgur.com/F0iVj.jpg" alt="Potential-Energy Graph"></p>
<p>My book states that: "If the object is in $B$ and has a total energy of $0$ then it can only vibrate between the points $A$ and $C$."</p>
<p>Which makes sense because if it went beyond e.g. $C$, that would mean $U > E$, which is 'impossible' because then $K<0$.</p>
<p>But having read about imaginary time and quantum tunneling (I don't really understand the concepts though.) I had the following thought: If $K<0$ that would mean that I have an imaginary value for my speed $v$ and since imaginary time exists that could mean $t$ has an imaginary component. It made sense in my head because I somewhat understand that quantum tunneling means a particle can get to certain positions which their initial position and energy wouldn't allow in classical mechanics.</p>
<p>Are the two at all related or is this too farfetched and totally unrelated?<br>
I checked Wikipedia and didn't find much.</p> | g12504 | [
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<p>I am trying to derive Feynman rules from a given Lagrangian and I got stuck on some vertex factors. What for example is the vertex factor that corresponds to the four-scalar interaction that is decribed by the following Lagrangian?</p>
<p>\begin{equation}
L = -\frac{1}{4} g_3^2 \phi^\dagger \lambda^a \phi \chi^\dagger \lambda^a \chi + \frac{2}{9} g_1^2 \phi^\dagger \phi \chi^\dagger \chi \,,
\end{equation}</p>
<p>where $\phi,\chi$ are complex scalar (color triplet) fields, $\lambda^a$ are the Gell-Mann matrices, and $g_1,g_3$ are the coupling constants corresponding to $\text{U}(1)$ and $\text{SU}(3)$ respectively. </p>
<p>If we would have only had the second term here, say, then the vertex factor would simply be found by "dropping" the fields and multiplying by $i$. But now there are two terms contributing, and in the first term the Gell-Mann matrices even mix the color components of the scalar triplets. So how do I proceed in this case? </p>
<p>And could anyone give me some general strategies on how to derive vertex factors for "complicated" interactions? For example, I also find it tricky to get the sign right if there is a derivative in an interaction. </p>
<p>(If you are interested in the context of this Lagrangian, for $\phi = \tilde{u}_R$ and $\chi = \tilde{d}_R$ this Lagrangian describes the interaction between two up squarks and two down squarks in a supersymmetric theory.)</p> | g12505 | [
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<p>I recently asked a <a href="http://physics.stackexchange.com/questions/128471/how-to-simulate-rotational-instability">question</a> about modeling instability in a rotating rigid body. I now realize that I was mentally confounding two different effects:</p>
<ol>
<li><p>The "Dzhanibekov effect" in which a rigid object with three different moments of inertia appears to tumble when spun around the intermediate axis. It ends up oscillating in a rather complex-looking pattern.</p></li>
<li><p>The tendency of an object (e.g. a fluid-filled cylinder) to change its spin axis to that with the greatest moment of inertia.</p></li>
</ol>
<p>I've successfully <a href="http://highfrontierblog.com/2014/07/30/rotational-dynamics/" rel="nofollow">reproduced</a> effect 1, which in the end is a relatively simple (if somewhat surprising) result of conservation of angular momentum.</p>
<p>Effect 2, however, is <em>not</em> something that occurs with ideal rigid bodies. It happens only when there is some mechanism for energy loss -- say, whipping antennas (as in the famous <a href="http://en.wikipedia.org/wiki/Explorer_1#Results" rel="nofollow">Explorer 1</a> satellite), or movement of an internal fluid (as in <a href="http://youtu.be/BPMjcN-sBJ4?t=27s" rel="nofollow">this video</a>).</p>
<p>I've found explanations saying that in these cases, rotational energy (aka angular kinetic energy) is lost, though angular momentum remains (in some way) unchanged. I'd like to model this effect. I imagine it is a matter of transferring some momentum from one axis to another, but in what way?</p> | g12506 | [
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<p>I need to plot the time evolution of the total angular momentum in an accretion disk. This confuses me because I thought this should be constant, since angular momentum has to be conserved?</p>
<p>I'm given the angular velocity $\Omega=(GM/R^3)^\frac{1}{2}$ where $M$ is the mass of the central object, and that the disk is made up of annuli of matter lying between $R$ and $R+\Delta R$ with mass $2\pi R\Delta R\Sigma$, where $\Sigma(R,t)$ is the surface density at time $t$ (I calculated the surface density numerically at different times in the previous question, so I assume this has to be used in my answer).</p>
<p>So</p>
<p>a) Why does total angular momentum change?</p>
<p>and</p>
<p>b) How do I know what function of $R$, $\Omega$, and $\Sigma$ represents total angular momentum?</p> | g12507 | [
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<p>The missile plume can be considered a cold plasma. If a radio signal passes across the plume it's attenuated. Obviously the attenuation depends on the frequency of the signal. Where can I find some information about the behaviour of the RF across the cold plasma? Thanks</p> | g12508 | [
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<p><a href="http://en.wikipedia.org/wiki/Work_%28physics%29" rel="nofollow">Work done</a> is defined as the dot product of force and displacement. </p>
<p>However, intuitively, should it not be the product of force and the time for which the body is acted upon by the force (force * time) because while time is independent of force applied, displacement is not.</p>
<p>Were these formulae (for work and energy) actually derived based on some physical understanding or are they just constructs to understand forces better?</p> | g12509 | [
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<p>Let's consider a spring which is subjected to forced vibrations:
$$
F = F_0 \cos(\omega t)
$$
Is the resonance frequancy $\omega_0$ of the spring dependent on the amplitude $F_0$? </p>
<p>I ask this because I am currently conducting tests with a plate which is forced to vibrate in the Z-direction orthogonally to its plan, thanks to a shaker, and it turns out that the resonance frequency of the plate is different for different values of the shaker amplitude (a higher amplitude gives a higher resonance frequency)</p>
<p>Thank you.</p> | g12510 | [
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<p>Redhead claims in his paper "More ado about nothing" (<a href="http://link.springer.com/article/10.1007%2FBF02054660" rel="nofollow">http://link.springer.com/article/10.1007%2FBF02054660</a>) that number operators associated with different space points (at fixed time) fail to commute, and hence are not physically meaningful.</p>
<p>However, Halvorson, in his paper "Reeh-Schlieder defeats Newton-Wigner" (<a href="http://arxiv.org/abs/quant-ph/0007060" rel="nofollow">http://arxiv.org/abs/quant-ph/0007060</a>), section 3.1, claims that operators $N(x)=a^\dagger(x)a(x)$ are not even mathematically well-defined. However I can't understand in what sense his argument using phase invariance proves that such operators are not well defined: we are simply taking the product of two unbounded operators. This product might indeed not have a clear physical sense (more precisely no "nice" localisation properties), but this was more or less Redhead's claim.</p>
<p>So basically I'm trying to understand if $N(x)$ is not associated to any local algebra and hence is not physically meaningful or really mathematical ill-defined, and if so what would be a clear argument to prove it.</p> | g12511 | [
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<p>When a system is expressed in terms of creation and annihilation operators for bosonic/fermionic modes, what exactly is the physical meaning of the order in which the operators act?</p>
<p>For example, for a fermionic system with states $i$ and $j$, $c_i c_j^\dagger$ is different from $c_j^\dagger c_i$ by a sign change, due to anticommutativity. I understand the mathematics of this, but what does it mean intuitively?</p>
<p>The former would be described as destroying a particle in state $j$ "before" creating one in state $i$, but what does "before" actually mean in this context, since there's no notion of time?</p>
<p>As another (bosonic) example, $a_i^\dagger a_i$ is clearly different from $a_ia_i^\dagger$, since acting the former on a vacuum state $|0\rangle$ gives zero while the for the latter, $|0\rangle$ is an eigenstate, but again, what is the physical interpretation?</p>
<p>My normal interpretation of commutativity as a statement regarding the effect of a measurement on a state fails here since creation/annihilation are obviously not observables.</p>
<p>I hope the question makes sense and isn't too abstract!</p> | g12512 | [
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<p>I face some trouble solving Maxwell's equations inside a cylinder with perfect conductor boundaries (in 3D) ?
We work with cylindrical coordinates $(r, \phi, z)$ and we make the assumption that fields have a sinusoidal "$e^{i\omega t}$" time dependence. Note that we have a $\phi$ symmetry.
First, and in any coordinates system, by taking the rotational and injecting one equation in the other we reduce Maxwell's equations to the following,
$$
\nabla\times\nabla\times E = -\partial_t^2 E = \omega^2 E
$$
In vacuum, from the $curl curl$ identity, it leads,
$$
\nabla\times\nabla\times E = \nabla(\nabla . E) - \nabla^2 E = - \nabla^2 E
$$
Where $- \nabla^2 E$ is the laplacian operator applied to each coordinate.</p>
<p>Now, in cylindrical coordinates, we can only compute the $z-$coordinate since, in this case we get the wave equation,
$$
\nabla^2 E_z = \omega^2 E_z
$$
For the other coordinates, the change of coordinates introduce other terms such that (for the $\phi-$ coordin.
ate)$\frac{E_r}{r^2} - \frac{2}{r^2}\frac{\partial E_\phi}{\partial\phi}$.</p>
<p>Then, a fastidious step consists in performing a separation of variable which leads us quite easily to the solution for every separated variable and also to the Bessel differential equation which brings its solution, the Bessel function. </p>
<p>Together with boundary conditions we can get the solution according to $z$ but what about the other coordinates ?</p> | g12513 | [
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-0... |
<p>Book: Classical mechanics (textbook) by Douglas Gregory (cambridge publications)</p>
<p>Law of mutual interaction states that when two particle (let it be P1 and P2) interacts, the particle (P1) induces an instantaneous acceleration (a21) on particle P2 and the particle P2 induces an instantaneous acceleration (a12) on particle (P1).</p>
<p>If the (inertial)masses of the particles are same, then the magnitude of acceleration be the same, and the ratios of acceleration will be constant ( for this case it is 1)(consistency relation) That is what Newton's third law says.</p>
<blockquote>
<p>My question is, for different (inertial)masses the ratio will be constant ( but not unity) ( it does not satisfy consistency relation) Am i right?</p>
<p>If yes My question is consistency relation is important in classical mechanics?</p>
</blockquote> | g12514 | [
0.0675794705748558,
-0.001229644869454205,
0.005158524960279465,
0.005831622984260321,
0.05209057033061981,
0.07097648829221725,
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0.01086646318435669,
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0.04759427532553673,
0.005188833922147751,
-0.026649195700883865,
-0.07486586272716522,
-0.00912... |
<p>I saw somewhere about being able to measure the velocity, period and radius of a binary star orbit by looking at red shift and blue shift. </p>
<p>I understand it but can someone give me an example of calculations etc done to calculate the velocity, period and radius of a binary stars in orbit?</p> | g246 | [
-0.023010151460766792,
0.007856679148972034,
0.012737930752336979,
0.009978879243135452,
0.005942576099187136,
-0.01759609766304493,
-0.03522719442844391,
0.01158791221678257,
-0.015053747221827507,
-0.019092006608843803,
0.007516808807849884,
0.061751190572977066,
0.03037337027490139,
0.0... |
<p>I have read about <a href="http://en.wikipedia.org/wiki/Zeno%27s_paradoxes" rel="nofollow">Zeno's arrow paradox</a> that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we cannot calculate velocity at any instant in the real world (of course it can be done by using calculus) but how can this be possible? What is the intuition behind this concept?</p> | g12515 | [
0.04356680065393448,
0.02083163894712925,
-0.010965758934617043,
0.06978541612625122,
0.05299847945570946,
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0.07283281534910202,
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-0.016032474115490913,
-0.01693084090948105,
-0.0011106666643172503,
0.05723525956273079,
-0.0... |
<p>We know that after the big bang the temperature was about 10^32 K.<br>
But now the average temperature of the universe is about 4 K. What happened to the temperature at that time? Where did it go?</p> | g12516 | [
0.05913180857896805,
-0.028593774884939194,
0.00892097968608141,
0.027783676981925964,
-0.06619198620319366,
0.008787940256297588,
0.01666242443025112,
0.06314603984355927,
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-0.0353303998708725,
-0.012355516664683819,
0.00917599257081747,
0.013689404353499413,
0.019015... |
<p>A teacher has recommended me to read this book in order to prepare for a project I am doing. Anyway, I feel that I should need a book in order to prepare for this one. Any suggestions?</p> | g89 | [
0.029028620570898056,
0.054572444409132004,
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-0.03417528048157692,
-0.0068410406820476055,
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0.09399796277284622,
0.031094443053007126,
0.004803389310836792,
0.04382934421300888,
-0.053... |
<p>The wave equation can be solved using Fourier transform, by assuming a solution of the form of
$$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$
and then reducing the equation to the <a href="http://en.wikipedia.org/wiki/Helmholtz_equation" rel="nofollow">Helmholtz equation</a>.</p>
<ul>
<li><p>What are the presumed <em>restrictions</em> on the solution, when solving the equation this way? (e.g., on <em>time</em> boundary condition) I mean can this method give the most general solution (given some boundary conditions)? What <em>features</em> does the solution obtained this way have?</p></li>
<li><p>Does it have any difference with solutions obtained using Laplace transform? (The very same above questions, for Laplace transform.)</p></li>
</ul> | g12517 | [
0.006727401167154312,
0.009652498178184032,
0.007910706102848053,
-0.02721480093896389,
0.06306302547454834,
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0.048222415149211884,
0.01184749137610197,
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-0.03208531066775322,
-0.031961482018232346,
0.03740670904517174,
0.016601962968707085,
0.... |
<p>Assume that you are holding a cage containing a bird. Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage?<br>
Does it make a difference whether the cage is completely closed or it has rods to let air pass? </p> | g792 | [
-0.01676429808139801,
0.03428295999765396,
0.01750972867012024,
0.03002852015197277,
0.00030174307175911963,
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-0.02143903821706772,
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0.010308722034096718,
-0.07221177965402603,
0.025... |
<p>I simulate scenario of two contacting elastic spheres, where their radii and the distance of their centers is known, and repulsive force should be computed. Material parameters are also given. Most contact mechanics references (also the Wikipedia page on <a href="http://en.wikipedia.org/wiki/Contact_mechanics" rel="nofollow">Contact mechanics</a> give equations for determining $a$ (contact area radius) when the force $F$ is already known.</p>
<p>Simpler contact models such as Hertz are able to express force as function of overlap of undeformed spheres ($F=\frac{4}{3}E^*\sqrt{R^*}\delta^{\frac{3}{2}}$), but the more complicated ones always only give an equation for $a$ supposing $F$ is already known; it does not seem that $F$ can be analytically expressed from those equations - e.g. in the Carpick-Ogletree-Salmeron model with $a=a_0(\beta) \left(\cfrac{\beta + \sqrt{1 - F/F_c(\beta)}}{1 + \beta}\right)^{2/3}$.</p>
<p>So, in the case of more complicated models such as the COS one:</p>
<ol>
<li>do I need to find $F$ iteratively so that the equation for $a$ is satisfied, or am I overlooking something obvious?</li>
<li>What is the relationship between $a$ and the overlap $\delta$?</li>
</ol>
<p>I will be also grateful for any good reference.</p> | g12518 | [
0.026972696185112,
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0.01199202798306942,
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0.03014443814754486,
0.0014541537966579199,
-0.043318651616573334,
0.03127780929207802,
-0.035... |
<p>I'm having trouble understanding just what the hyperbolas on a space-time diagram actually signify. From what I understand, these hyperbolas trace out the equation $S^2=x^2-(ct)^2$, which is the equation for a hyperbola. Fine so far. Furthermore, the plot is supposed to represent the value of $S$, which is the invariant interval. </p>
<p>When I think of a spacetime event, though, I think of a line that begins at the origin and that moves out the upper right of the diagram, in between the $ct$-axis and the 45% Null line of the speed of light. I can see how different frames of reference could represent different coordinate shifts of the Spacetime diagram which would yield different values of $x$ and $ct$, keeping the value of $S$ invariant. What I am struggling with is understanding what the hyperbolic shape of the plot of $S$ on the spacetime means. What is it saying? How do I read it? Assuming we are only dealing with time-like events, what do different values of $S$ signify? Is there only that single hyperbola that the value of $S$ falls along for a given spacetime event, or is that curve part of a larger family of curves? </p> | g12519 | [
0.010066160932183266,
0.039054084569215775,
-0.037573400884866714,
0.006478507071733475,
0.012789248488843441,
0.02051856927573681,
0.037923429161310196,
0.011663475073873997,
-0.02329910546541214,
-0.02837447263300419,
0.00203274586237967,
0.02391861379146576,
0.08033868670463562,
-0.0230... |
<p>Suppose you had 80% up quarks, and only 20% down quarks. How would this affect stellar formation?</p> | g12520 | [
-0.02005974017083645,
-0.005898645147681236,
-0.008371393196284771,
-0.05777422711253166,
0.024287378415465355,
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-0.056354962289333344,
0.03666508570313454,
0.027763530611991882,
0.024295179173350334,
-0... |
<p>Using Special relativity theory, of course. Can Lorentz transformations to "tell" something about it?
"Wikipedia's" article:
<a href="http://en.wikipedia.org/wiki/One-way_speed_of_light">http://en.wikipedia.org/wiki/One-way_speed_of_light</a> .</p> | g12521 | [
-0.020788069814443588,
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0.047144003212451935,
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-0.016907071694731712,
0.023528676480054855,
0.02457275427877903,
0.02535998448729515,
0.02... |
<p>In Vladimir I. Arnold's <em>Lectures on Partial Differential Equations</em>, Chapter 3 <em>Huygens' Principle in the Theory of Wave Propagation</em>, which is devoted to the proof of Huygens principle (original one by Huygens, maybe not by Fresnel), it's said that</p>
<blockquote>
<p>the contact elements of space-time that are tangent to the large front belong to the retardation hypersurface.</p>
</blockquote>
<p>By <em>contact elements of space-time tangent to the large front</em>, he means roughly the tangent hyperplanes of the large front, if the space-time is just simply $\mathbb R^3\times\mathbb R$ (It's generally a manifold $M^{n+1}=B^n\times\mathbb R$, where $B^n$ is the manifold of physical space). By <em>the large front</em>, he means the geometric object in the space-time that describes the poropagation of fronts of perturbations by means of a single hypersurface in space-time with a more general retardation hypersurface in the manifold of contact elements of space-time, or simply just the graph of all fronts in the space-time. By <em>retardation hypersurface</em>, maybe he means <em>Fresnel hypersurface</em>, i.e. a hyperplane in the manifold of co-oriented contact elements of space-time defined by the field of cones of possible velocities. He says that</p>
<blockquote>
<p>In geometric optics, <em>a cone of possible velocities of motion</em> is prescribed at each point of the space-time manifold $M$.</p>
</blockquote>
<p>For example, in the flat space of vacuum, the light cone is prescribed at each point of the space time.</p>
<p>And the points of <em>Fresnel hypersurface</em> are just the tangent planes to the cones of possible velocities of motion in the manifold of contact elements, i.e, the bundle of $PT^*M\to M$ or co-oriented $ST^*M\to M$.</p>
<p>In order to understand these concepts, a solid physical maturity seems needed, which I haven't, so I want to fix the gaps in my knowledge of physics. First,</p>
<blockquote>
<p>What's the local law of propagation of disturbances in the geometric optics?</p>
</blockquote>
<p>Second,</p>
<blockquote>
<p>What's the physical meaning of the <em>Fresnel hypersurface</em>? Is it well-known or related to other well-known terms?</p>
</blockquote>
<p>I need some clues. Thanks!</p> | g12522 | [
0.028445305302739143,
0.02776646427810192,
-0.008479313924908638,
-0.013093989342451096,
0.04137609153985977,
0.007113107014447451,
0.10623477399349213,
0.002300581429153681,
0.04022888094186783,
0.004770607687532902,
0.045641668140888214,
0.033292971551418304,
0.026560619473457336,
0.0053... |
<p>Can you give a basic explanation of what is crystal field anisotropy ?</p>
<p>What is the reason to arise ?</p>
<p>In spin ice it forces the dipoles to point in the local 111 direction.</p>
<p>For partially filled rare earth atoms hund rule requires S and L max.
This leaves (2s +1)(2l+1) degeneracy which is partially lifted from the LS coupling.
When inserted in anisotropic field inside crystal the
expectation value , or mean value of L is 0 = 0 and the L is quenched leaving only S so is forced to point in some local direction, but this is if the field removes the deneracy ?</p>
<p>Is it this or it is much more complicated ?</p> | g12523 | [
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0.09834621101617813,
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0.001017666538245976,
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0.017286095768213272,
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... |
<p>I'm going to be brief, I just saw a Discovery Channel show that showed a lot of interesting phenomena around lightning (like <a href="http://en.wikipedia.org/wiki/Lightning#Elves" rel="nofollow">elves</a>, how cool is that(!)), and got me wondering.</p>
<p>1) Thinking of lightning as a purely mechanical phenomenon, I would think the elves are the "other side of the momentum balance". What I mean is this: somewhere in some cloud formation, an event happens that triggers a lightning flash. This means a ton of electrons (and other, associated particles) that start a <em>very</em> fast journey to the surface of the Earth. Their momentum must be balanced by particles going in the opposite direction, hence <a href="http://en.wikipedia.org/wiki/Lightning#Elves" rel="nofollow">elves</a>. Am I right?</p>
<p>2) Taking this further, is it possible that the "trigger" for the lightning flash could be atomic/molecular fusion forced by extremely high electrical fields, that may only exist for a nanosecond, causing a ton of energy to be transferred to the electrons around the atoms/molecules, and we have lightning. I would then think of the energy necessary to force the fusion to occur as a quantum fluctuation as in $$\Delta E \Delta t \leq \hbar$$</p>
<p>3) Is every arc caused by a particle collision, somewhat like in a particle accelerator? It seems logical to me: there are more than enough particles in the air to collide with, and all the light could well be some form of Brehm or Cherenkov radiation.</p>
<p>The question I seem to be asking is where to find good scientific theory/information about lightning. Some say we know a lot about it, but I haven't found any good papers explaining it. I have a pretty good background in physics (1st year Master student) and am not afraid of serious literature. Thanks!</p> | g12524 | [
0.016410106793045998,
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0.07264405488967896,
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-0.014249797910451889,
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0.0281970351934433,
0.041512638330459595,
0.004... |
<p>I got an integral in solving Schrodinger equation with delta function potential. It looks like </p>
<p>$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$</p>
<p>I'm trying to solve this by splitting it into two integrals</p>
<p>$$\int_{-\infty}^{x_0 - \epsilon} \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x} + \int_{x_0 + \epsilon}^{\infty} \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$</p>
<p>and then do the limit $\epsilon\to 0$. Could you tell me how to solve this integral please? I used Mathematica, it gave out a weird result.</p> | g12525 | [
-0.003287888364866376,
0.04483413323760033,
0.006638206075876951,
-0.03076954372227192,
0.052499618381261826,
0.013549655675888062,
-0.014144733548164368,
0.07070819288492203,
-0.014032470062375069,
-0.006584770977497101,
-0.03110501542687416,
0.028508836403489113,
0.0019342333544045687,
0... |
<p>Yang's theorem states that a massive spin-1 particle cannot decay into a pair of identical massless spin-1 particles. The proof starts by going to the rest frame of the decaying particle, and relies on process of elimination of possible amplitude structures.</p>
<p>Let $\vec\epsilon_V$ be the spin vector of the decaying particle in its rest frame, and let $\vec\epsilon_1$ and $\vec\epsilon_2$ be the polarization 3-vector of the massless particles with 3-momenta $\vec{k}$ and $-\vec{k}$ respectively.</p>
<p>In the literature, I've seen arguments saying that</p>
<p>$\mathcal{M_1}\sim(\vec\epsilon_1\times\vec\epsilon_2).\vec\epsilon_V$, and $\mathcal{M_2}\sim(\vec\epsilon_1.\vec\epsilon_2)(\vec\epsilon_V.\vec{k})$ don't work because they don't respect Bose symmetry of the final state spin-1 particles.</p>
<p>But, why is $\mathcal{M_3}\sim(\vec\epsilon_V\times\vec\epsilon_1).\epsilon_2+(\vec\epsilon_V\times\vec\epsilon_2).\epsilon_1$ excluded? Sure, it's parity violating (if parent particle is parity even), but that's not usually a problem</p>
<p>Thanks</p> | g12526 | [
0.008321776986122131,
-0.030131932348012924,
0.01289419550448656,
-0.007589345332235098,
0.01980217546224594,
0.03040781244635582,
0.06930000334978104,
0.059078991413116455,
0.006452830508351326,
0.028046391904354095,
-0.004751535132527351,
0.04290369898080826,
-0.08838016539812088,
-0.039... |
<p>I am interested to know how does <a href="http://en.wikipedia.org/wiki/Center_of_mass" rel="nofollow">CG</a> of vehicle plays role in the Fuel economy and the vehicle performance. Does CG of vehicle has anything to do while accelerating of your vehicle. I am a student, henceforth I would feel glad to know about these issues. Can any one give me the link that containing these topics.</p> | g12527 | [
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0.06653058528900146,
-0.019496245309710503,
0.02032294310629368,
0.05226164683699608,
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0.04920246824622154,
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-0.005628868006169796,
0.037059515714645386,
-0.04196571186184883,
0.048816125839948654,
0.01... |
<p>How do you calculate the Poynting vector for a laser given it's power? I know for a sphere you can just take the power, and divide it by 4$\pi R^2$, but I don't know what I would do for a laser. Would I have to take into account the aperture size?</p> | g12528 | [
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-0.0037247336003929377,
0.009351642802357674,
-0.0326872281730175,
-0.04601810872554779,
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0.021637437865138054,
-0.009255370125174522,
0.018125122413039207,
-0.0036599510349333286,
... |
<p>I have a fridge compartment, the lowest temperature can be set to to about 0c. Is it possible I could bring it down to -2c, -3, -4 c etc by simply adding water bottles which were previously frozen in a freezer compartment. Would the temp be similar throughout the entire fridge or would it fluctuate significantly?</p> | g12529 | [
-0.0002094585361192003,
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0.021280985325574875,
0.06113887578248978,
0.016831884160637856,
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-0.002771236700937152,
-0.027701837942004204,
0.0074424403719604015,
0.0018443374428898096,
0... |
<p>My sister got this homework that she don't understand, me neither. Can anyone help out?
Here's the problem:</p>
<p>Give the x-component and y component of the following vectors:</p>
<p>a. A = 7.0 cm, E</p>
<p>b. B = 5.7 cm, S</p>
<p>c. C = 5.5 cm, 30 degrees E of N </p>
<p>d. D = 5.5 cm, 60 degrees S of E </p>
<p>Get the resultant of the four vectors, R = A + B + C + D.</p>
<p>Thanks in advance :) </p> | g12530 | [
0.060967642813920975,
-0.03395257145166397,
-0.01577572152018547,
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0.032549601048231125,
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0.045071084052324295,
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0.020591668784618378,
0.008365713059902191,
-0.013106978498399258,
0.025231685489416122,
-0.006243621930480003,
... |
<p>recently I have had <a href="http://physics.stackexchange.com/questions/4372/does-entropy-apply-to-newtons-first-law-or-does-acted-upon-always-require-an-e/4380#4380">some exchanges with @Marek</a> regarding entropy of a single classical particle. </p>
<p>I always believed that to define entropy one must have some distribution. In Quantum theory, a single particle can have entropy and I can easily understand that. But I never knew that entropy of a single rigid classical particle is a well defined concept as Marek claimed. I still fail to understand that. One can say that in the classical limit, the entropy of a particle in QT can be defined and that corresponds the entropy of a single classical particle. But I have difficulty accepting that that gives entropy of a single Newtonian particle. In my understanding, If a system has entropy then it also should have some temperature. I don't understand how one would assign any temperature to a single classical particle. I came across <a href="http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2751v3.pdf">a paper where there is a notion of "microscopic entropy"</a>. By no means, in my limited understanding, it corresponded to the normal concept of entropy. I am curious to know, what is the right answer. </p>
<p>So, my question is, is it possible to define entropy of a single classical particle?</p> | g12531 | [
-0.004577907733619213,
-0.023265251889824867,
0.00006345914880512282,
-0.029880577698349953,
0.03877266123890877,
-0.004149602726101875,
-0.02573736570775509,
0.05782955512404442,
-0.0420074537396431,
-0.004366131965070963,
-0.011941843666136265,
0.026392484083771706,
-0.029847659170627594,
... |
<p>This question is inspired by the following comment:</p>
<blockquote>
<p>the strings in string theory are relativistic and on a large enough piece of world sheet, the internal SO(1,1) Lorentz symmetry is preserved. That's why a string carries not only an energy density ρ but also a negative pressure p=−ρ in the direction along the string.</p>
</blockquote>
<p>by Lubos at the end of his answer to the question <a href="http://physics.stackexchange.com/questions/3343/what-is-tension-in-string-theory">"What is tension in String Theory?"</a>.</p>
<p>I don't see how having a $SO(1,1)$ symmetry for the worldsheet leads to a negative pressure. I have the following questions:</p>
<ol>
<li><p>Why does a string carry negative pressure?</p></li>
<li><p>Can a gas of strings then be treated as a substance which has a negative equation of state:</p>
<p>$$ w = \frac{p}{\rho} = -1 $$</p></li>
<li><p>If this is possible then the next natural question is: what are the implications for the cosmological constant problem?</p></li>
</ol>
<p><strong>Background material to give some context for questions 2 and 3:</strong></p>
<p>One part of the cosmological constant problem is an explanation of what form of matter could seed cosmological expansion. A cosmological constant term $\Lambda$ in the action for GR is equivalent to having a medium which satisfies the negative equation of state (relation between density and pressure). One simple example of matter with a negative equation of state is a homogenous, isotropic scalar field [for details see any book on cosmology with a chapter on inflation]. If strings carry negative pressure and if a string gas can be treated as matter with $w\sim-1$ then one would have a far more natural alternative to a scalar field.</p> | g12532 | [
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0.0006326169823296368,
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0.01588955894112587,
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-... |
<p>I went (on vacation) to the beach, The sea was very calm (just like solar system) There was one person in a fishing boat, Suddenly a huge wave came to shore...</p>
<p>Is it possible that a gravitational wave of space - time hit the solar system in a same way?</p> | g12533 | [
0.010440383106470108,
0.07265155017375946,
0.016703851521015167,
0.016218455508351326,
0.03656408190727234,
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-0.05747440084815025,
0.03924316540360451,
0.034076400101184845,
0.042808178812265396,
0.001... |
<p>1) What is the difference between these two momentum operators:
$\frac{\hbar}{i}\frac{\partial}{\partial x}$ and
$-i\hbar\frac{\partial}{\partial x}$?
How are these two operators the same?</p>
<p>My textbook says that $\frac{\hbar}{i}\frac{\partial}{\partial x}$ is the mathematical operator acting on $\Psi$ that produces the $x$ component of the momentum. </p>
<p>2) What is an operator? By operator, do they mean like addition, subtration, differentiation? Things of that nature? So if I take the partial with respect to $x$ of $\Psi$ and then multiply that whole thing by $\hbar / i$ then I should get the $x$ component of the momentum? </p> | g12534 | [
0.03635410591959953,
-0.014776683412492275,
-0.00278042396530509,
0.03379018232226372,
0.05713392049074173,
0.014065304771065712,
0.034282464534044266,
0.003476748475804925,
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0.0014636488631367683,
-0.01634746417403221,
-0.03781241923570633,
-0.035233065485954285,
-0.... |
<p>I'm going through an introduction to many-body theory and I am getting tripped up on the formalism. I understand quantities such as $\hat {N} = \sum_{i}\hat{n}_{i}=\sum_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}=\int d^{3}x\psi^{\dagger}(x)\psi(x)$ but struggling with things interpreting things like the kinetic energy of the system</p>
<p>Specifically, how is it that one goes from the creation/annihilation formalism to the field operators? If you have a general many-body Hamiltonian,</p>
<p>$$
H=\sum_{i,j}(t_{ij}+U_{ij})\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{1}{2}\sum_{i,j,k,m}\left\langle i,j \right | f^{(2)} \left| k,m \right\rangle\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{m}\hat{a}_{k}
$$</p>
<p>Where we have for the total kinetic energy and the two-particle interaction term</p>
<p>$$
T=\sum_{i,j}t_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}
\\
t_{ij}=\left\langle i \right | t \left| j \right\rangle
\\
F= \frac{1}{2}\sum_{\alpha \neq \beta}f^{(2)}(\mathbf{x}_{\alpha},\mathbf{x}_{\beta})
$$</p>
<p>In terms of field operators, the Hamiltonian is given as</p>
<p>$$
H=\int d^{3}x\left(\frac{\hbar^{2}}{2m}\nabla\psi^{\dagger}(\mathbf{x})\nabla\psi(\mathbf{x})+U(\mathbf{x})\psi^{\dagger}(\mathbf{x})\psi(\mathbf{x})\right)+\frac{1}{2}\int d^{3}xd^{3}x'\psi^{\dagger}(\mathbf{x})\psi^{\dagger}(\mathbf{x}')V(\mathbf{x},\mathbf{x}')\psi(\mathbf{x}')\psi(\mathbf{x})
$$</p>
<p>I can't really see the transformation (nor do I have a good intuition for it) between the creation and annihilation operators and the field operators formalism for the Hamiltonian. Why do we have the two-particle interaction "sandwiched" between the field operators, but the annihilation/creation operators do not follow the same pattern? </p>
<p>I am aware of basic quantum mechanics, commutation rules, as well as the Fourier transform. I need help developing an intuition for writing down a field operator Hamiltonian. When I read the field operator Hamiltonian, the story that I get is: There are some field operators that create and annihilate, and integrating over them with a energy density yields a total energy term. </p>
<p>But I get lost in the details. For instance, although it has been simplified by IBP above, the kinetic energy term acts on the annihilation operator before the creation operator acts on it. What is the meaning of the motif $H=\int d^{3}x\psi^{\dagger}(\mathbf{x})\hat{h}\psi(\mathbf{x})$?</p> | g12535 | [
0.06088646128773689,
0.06136980280280113,
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0.026891957968473434,
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0.027748674154281616,
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0.00839502364397049,
0.04557179659605026,
-0.0161... |
<p>First we have Discretization of the Transport Equation
$$
\frac{\partial \rho \phi}{\partial t} + \nabla(\rho U \phi) - \nabla (\rho \Gamma_\phi \nabla \phi) = S_\phi (\phi)
$$</p>
<p>In Finite Volume Method it looks like:
$$
\int_t^{t+\Delta t} \left[ \frac{\partial}{\partial t} \int_{V_p} \rho \phi dV + \int_{V_p} \nabla \cdot (\rho U\phi)dV - \int_{V_p} \nabla \cdot (\rho \Gamma_\phi \nabla \phi)dV \right]dt = \int_t^{t+\Delta t} \left(\int_{V_p}S_\phi(\phi)dV \right)dt
$$</p>
<p>and after transformations:</p>
<p>$$
\int_t^{t+\Delta t} \left[ \left( \frac{\partial \rho \phi}{\partial t} \right)_p V_p + \sum_f F\phi_f - \sum_f(\rho \Gamma_\phi)_f S.(\nabla\phi)_f \right]dt=\int_t^{t+\Delta t}(SuV_p+S_pV_p\phi_p)dt
$$</p>
<p>and time discretization:
$$
\frac{\rho_p\phi_p^n-\rho_p\phi_p^o}{\Delta t}V_p+ \frac{1}{2}\sum_f F\phi_f^n-\frac{1}{2}\sum_f(\rho\Gamma_\phi)_f S.(\nabla\phi)^n_f+ \frac{1}{2}\sum_f F\phi_f^o-\frac{1}{2}\sum_f(\rho\Gamma_\phi)_f S.(\nabla\phi)^o_f = SuV_p + \frac{1}{2}S_pV_p\phi_p^n+\frac{1}{2}S_pV_p\phi_p^o
$$</p>
<p>For every cell we can make equation:
$$
a_p\phi^n_p+\sum_N a_N \phi_N^n = R_p
$$</p>
<p>But how we can get elements of matrix $A$ if we don't know $\phi$?
$$
[A][\phi]=[R]
$$</p>
<p>Here Hrvoje Jasak wrote that every coefficient $ a_p $ includes the contribution from temporal derivative, convection and diffusion terms. But what formula of $a_p = ...$?
<a href="http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/docs/HrvojeJasakPhD.pdf" rel="nofollow">http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/docs/HrvojeJasakPhD.pdf</a></p> | g12536 | [
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0.010407296940684319,
0.05577019229531288,
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0.0033263368532061577,
-0.06305503845214844,
0.049759555608034134,
0.025625020265579224,
0.05... |
<p>I'm having trouble grokking the relationship between a winch's pull/torque and a spring's potential energy.</p>
<p><strong>I would like to compress a spring using an electric winch and figure out how far it will be able to launch an object.</strong></p>
<ol>
<li><p>If the spring's maximum load is higher than the winch's force, do the other details of the spring even matter? Specifically, if the electric winch has 2000-lbs of pull, is the maximum potential energy it can store in the spring 2000-lbs (8896-N) of force, regardless of the spring's length or constant?</p></li>
<li><p>What is the correct formula for finding the height which the spring will launch an object (of let's say 1-pound) straight into the air (after it has been compressed up to the winch's max pull of 2000-lbs)?</p></li>
</ol> | g12537 | [
0.016091695055365562,
0.0023882114328444004,
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0.03513109311461449,
0.019267747178673744,
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0.01990104466676712,
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-0.022427303716540337,
-0.05363214761018753,
-0.021855121478438377,
0.050756845623254776,
-0... |
<p>This theorem is number 44 of chapter 14 in Barret O'Neil's book "Semi-Riemannian Geometry (with applications to relativity)". The proof given, in particular the use of another theorem to justify the last passage, does not convince me. Here is it. How do I prove it?</p>
<p><strong>Lemma</strong></p>
<p>Let $S$ be a closed achronal spatial hypersurface in spacetime $M$. Let $q\in D^+(S)$, there is a geodesic $\gamma$ from $S$ to $q$ which has lenght $\tau(S,q)$, is normal to $S$ and does not have focal ponts of $S$ before $q$. ($\gamma$ is time-like aside for the trivial case $q\in S$).</p>
<p><strong>Notations</strong></p>
<p>$D(S)$ is the Cauchy development of $S$
$D(S)=D^+(S)\cup D^-(S)$, where, for example, $D^+(S)=\{q\in M:$ every past pointing inextensible causal curve from $q$ intersects $S\}$</p>
<p>$\tau(S,q)=\sup\{lenght(\alpha):\,\alpha$ is a causal curve from a point in $S$ to $q\}$</p>
<p>Now I know/I can prove that</p>
<p>-D(S) is open and globally hyperbolic</p>
<p>-There is a geodesic of the desired lenght normal to $S$</p>
<p>My question is this: how do I show that there are no focal points before $q$ (just because $q\in D(S)$)?</p> | g12538 | [
0.05816261097788811,
0.007686247117817402,
-0.008240660652518272,
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0.09198839217424393,
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-0.06180581450462341,
0.030508169904351234,
0.043130822479724884,
0.049264341592788696,
-0.0090... |
<p>Goldstein's Classical Mechanics has a puzzling few sentences in his discussion of orbits. </p>
<p>Referring to the case of orbit where the energy is low enough for the orbit to be bounded, he says :"This does not necessarily mean that the orbits are closed. All that can be said is that they are bounded, contained between two circles of radii $r_1$ and $r_2$ with turning points always lying on the circles."</p>
<p>Doesn't "bounded" automatically mean "closed"? The object cannot escape from the attractive force and hence returns over and over. At least, that is my understanding of the terms. Wikipedia says "The orbit can be open (so the object never returns) or closed (returning), depending on the total energy (kinetic + potential energy) of the system." But it also says "Orbiting bodies in closed orbits repeat their paths after a constant period of time." So the only way out I see is if a closed orbit is a special case of non-precessing bounded orbit.</p> | g12539 | [
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-0.04588928446173668,
0.006727974861860275,
0.006927901413291693,
0.01928049698472023,
-0.031... |
<p>In my physics textbook, it says that for any pulse, if $\Delta x$ becomes smaller, $\Delta k$ becomes larger where $k$ refers to $2\pi/\lambda$ and $x$ is x-axis displacement, as described by $\Delta x \Delta k \approx 1$. Why is it like this?</p> | g12540 | [
0.07746991515159607,
0.04003412649035454,
-0.0031115144956856966,
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0.03353473171591759,
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-0.0635092481970787,
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0.020539605990052223,
0.04882388934493065,
-0.038... |
<p>I have read about anomalies in different contexts and ways. I would like to read an explanation that unified all these statements or point-views:</p>
<ol>
<li><p>Anomalies are due to the fact that quantum field theories (and maybe quantum mechanical theories with singular potential) must be <em>regularized</em> and it can be possible that none regularization procedure may respect all the symmetries of the classical theory. For instance, in <em>Fujikawa</em> point-view one cannot find a regularized functional <em>measure in the path-integral</em> which preserves the symmetries of the classical action.</p></li>
<li><p>Anomalies are due to the fact that quantum theory requires an <em>ordering prescription</em> for operators and it may happen that none ordering prescription respects the symmetries of the classical theory. Here in principle one could have anomalies in quantum mechanics with non-singular potentials.</p></li>
<li><p>Anomalies are due to the fact that <em>generators of the symmetry do not leave invariant the domain of definition of the Hamiltonian</em> and thus, although the formal commutator of those generators with the Hamiltonian vanishes, the charges are not conserved due to the extra surface term that appears in the exact Heisenberg equation. It is not clear to me if in this case the anomaly is already present in the classical theory.</p></li>
<li><p>Anomalies are due to the emergence of <em>central charges</em> in the algebra of conserved quantities. In this case one can have classical anomalies if it is in the algebra of Poisson brackets, or quantum anomalies if it happens at the level of quantum commutators.</p></li>
</ol>
<p>I would like you to share examples and relations between the previous perspectives, and discussions about their equivalence. The only I see is the following relation:</p>
<p>Ordering of operators ----> different definition of path integral measure.</p>
<p>Ordering of operators ----> delta functions ---> regularization.</p>
<p>Ordering of operators ----> different algebra of conserved charges.</p> | g12541 | [
0.07134667783975601,
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0.02182725816965103,
0.06220627576112747,
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0.04033242538571358,
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-0.08958111703395844,
-... |
<p>I'm reading <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.44.7880&rep=rep1&type=pdf" rel="nofollow">"Symplectic geometry and geometric quantization"</a> by Matthias Blau and he introduces a complete set of observables for the classical case:</p>
<blockquote>
<p>The functions $q^k$ and $p_l$ form <em>a complete set</em> of observables in the sense that any function which Poisson commutes (has vanishing Poisson brackets) with all of them is a constant.</p>
</blockquote>
<p>I wonder why is it so? That is why do we call it a complete set of observables? As I understand it means the functions satisfying the condition above form coordinates on a symplectic manifold, but I don't see how.</p> | g12542 | [
0.07647114247083664,
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0.0025585805997252464,
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0.05002560839056969,
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0.02866210974752903,
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-0.002217019209638238,
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-0.0009120891336351633,
-0.018465692177414894,
0... |
<p>Reading <a href="http://www.dnr.state.oh.us/Portals/11/bainbridge/DMRM%202%20Chapter%202%20-%20Hydraulic%20Fracturing%20Analysis.pdf" rel="nofollow">this</a>, I can intuitively understand that fractures propagate along the path of least resistance, creating "width in a direction that requires the least force". However, it is less intuitive (to me, at least) that "a fracture will propagate parallel to the greatest principal stress and perpendicular to the plane of the least principle stress". In my mind, I visualize a rock between the two plates of a press. I can see the forces balancing until a critical force is applied. At this point, how do I visualize the stresses? Do I have to analyze the stresses at a molecular level or is macroscopic description sufficient?</p> | g12543 | [
0.01664593443274498,
0.06868218630552292,
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0.06466159224510193,
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0.024182690307497978,
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0.03036154806613922,
-0.04155810549855232,
0.04539938643574715,
-0.045... |
<p>I am a second year undergraduate and studying quantum mechanics from sakurai's 'Modern Quantum Mechanics'. Is it a good idea to solve problems from sakurai, which are mostly mathematical in nature? I need a textbook that has physically relevant problems, maybe going even into condensed matter, or field theory in its exercises. This would probably help me to appreciate and understand qm better. Sorry if this question is too localised but I just had to post it.</p> | g12 | [
0.019829977303743362,
0.021699965000152588,
0.008540346287190914,
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0.015523924492299557,
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-0.0044962335377931595,
0.021591830998659134,
0.04473871365189552,
-0.047871120274066925,
0.0072413417510688305,... |
<p>I know of events that are happening about 45 KM away from me which are said to be 210 or 213 dB at 75 meters distance from <a href="http://listverse.com/2007/11/30/top-10-loudest-noises/" rel="nofollow">multiple</a> <a href="http://www.makeitlouder.com/Decibel%20Level%20Chart.txt" rel="nofollow">sources</a>. I think that I can hear them, so I did the obvious:</p>
<p>$$ \frac{I_2}{I_1} = \left( \frac{d_1}{d_2} \right)^2$$</p>
<p>$$ I_2 = I_1 * \left( \frac{d_1}{d_2} \right)^2\text{ dB}$$</p>
<p>For my values:</p>
<p>$$ I_2 = 213 \left( \frac{75}{45000} \right)^2\text{ dB}$$</p>
<p>$$ I_2 = 213 \left( \frac{1}{600} \right)^2\text{ dB}$$</p>
<p>$$ I_2 = \frac{213}{360000}\text{ dB}$$</p>
<p>$$ I_2 \approx 0.6 \cdot 10^{-3}\text{ dB}$$</p>
<p>Well, <strong>0.6 e-3 dB</strong> I should not be able to hear!</p>
<p>However, going by the rule of thumb <a href="http://www.sengpielaudio.com/calculator-levelchange.htm" rel="nofollow">"twice the distance, minus three decibels"</a>:</p>
<p>$$
75 * 2^x = 45000
$$</p>
<p>$$
2^x = 600
$$</p>
<p>$$
x = \log_2(600)
$$</p>
<p>$$
x \approx 9.2
$$</p>
<p>$$
213 - 3 \cdot 9.2 = 213 - 27.6 = 185.4
$$</p>
<p>The value of <strong>184 dB</strong> seems too loud, and additionally does not agree with the previous method.</p>
<p>Lastly I tried tool for <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/isprob2.html" rel="nofollow">estimating Sound Levels With the Inverse Square Law</a> from the usually-terrific HyperPhysics website. This tool gave a value of
<strong>157 decibels</strong>! <a href="http://www.sengpielaudio.com/calculator-distance.htm" rel="nofollow">Another on-line tool</a> gives the same result.</p>
<p><strong>Which result should I trust?</strong></p> | g12544 | [
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0.... |
<p>Consider a column of fluid of length $L$, with initial density $\rho_0$ and initial velocity ($u_0 =0$) everywhere. Now at time $t=0$ gravity is switched on. No-slip boundary conditions are assumed at both end of the fluid column.</p>
<p>We know that after a while column will attain a steady state with fluid everywhere at rest and density as exponential function of distance from either end.</p>
<p>Continuity equation is
\begin{eqnarray}
\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} = 0
\end{eqnarray}</p>
<p>Navier-stokes equation for fluid in one dimension is</p>
<p>\begin{eqnarray}
\rho\left[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} \right] &=& -\frac{\partial P}{\partial x} + f_{external} \nonumber \\
&=& -\frac{\partial\rho}{\partial x}c^2_s - \rho g
\end{eqnarray}
Here I assume shear forces are zero since the system is one dimensional. </p>
<p>In the steady state $u=0$ and $\frac{d\rho}{dt} = 0$ so we get,
\begin{eqnarray}
\frac{d\rho}{dx}c^2_s &=& - \rho g \\
\frac{d\rho}{\rho} &=& - dx \frac{g}{c^2_s} \\
\rho &=& \rho'\exp\left(-\frac{g}{c^2_s}x \right)
\end{eqnarray}</p>
<p>where $\rho'$ is evaluated by mass conservation equation.
\begin{eqnarray}
\rho_{0} L = \int^{L}_0\rho'\exp\left(-\frac{g}{c^2_s}x \right)dx
\end{eqnarray}</p>
<p>Where I assume hydrostatic pressure($P$) is proportional to density ($\rho$).
Is it possible to solve these equations(assuming they are correct) as a function of time?
To start with, I tried to get velocity ($u$) profile for the time very close
to initial time. When the time is really small $t<<1$, For the Navier-Stokes equation we assume spatial variation in density($\rho$) and velocity($u$) is yet to develop, so that we get
\begin{eqnarray}
\frac{du}{dt} &=& -g \\
u &=& -gt \hspace{0.5cm} t <<1
\end{eqnarray}</p>
<p>I am not sure if it is allowed to assume initial spatial variation small compared to time variations in the system. Even if allowed, I am not able to go any further.</p>
<p>Also I feel the solution for density and velocity depend upon viscosity of the fluid but viscosity appears nowhere in the formulation. Do I need to include shear forces? </p> | g12545 | [
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0.011673737317323685,
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... |
<p>I'm studying the book "Classical Mechanics" by Goldstein together with a coursebook my professor provided.</p>
<p>I'm having trouble grasping how to intuitively determine what the rate of change of a vector is affected by.</p>
<p>Earlier on in the book, I saw a body with 3 frames of reference:<br>
$Oxyz$ which is just an arbitrary coordinate system.<br>
$O'x'y'z'$ which has a fixed origin in a point of the body and fixed axes that rotate corresponding to the body.<strong>(Body fixed?)</strong><br>
$O'xyz$ which has a fixed origin in a point of the body and fixed axes that are parallel with the original coordinate system.<strong>(Space fixed?)</strong></p>
<p>Now, in my professors book it says the following:</p>
<blockquote>
<p>It is clear that there are 2 sources of time-dependence in the Carthesian components of a general vector in the (body-fixed) coordinate system:<br>
On one hand: The intrinsic time-dependence of the vector.<br>
On the other hand: The time dependence of the basisvectors $\vec n'_i$ of the moving coordinate system, with respect to which the Carthesian components (orthogonal projection of the vector onto $\vec n'_i$) will be determined.</p>
</blockquote>
<p>I'm confused as to what exactly is meant by this paragraph.<br>
What exactly is the intrinsic time-dependence of the vector? To my knowledge, in the body-fixed coordinate system every vector stays constant, so there is no rate of change.<br>
I understand that the $\vec n'_i$ vectors move through space, but during a translation they wouldn't change since they are their orientation stays the same. During a rotation however I understand they would change.</p>
<p>What am I getting wrong here? I have a feeling I'm mixing these frames of reference up since it's not so clear in my book. I think that I'm misunderstanding how these frames of reference behave as the body moves.</p> | g12546 | [
0.03210686519742012,
0.005774299148470163,
0.0007701566792093217,
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0.04257713258266449,
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0.0461437813937664,
0.029416069388389587,
0.01470... |
<p><em>Note: I struggled to decide the appropriate site for this question, between <a href="http://physics.stackexchange.com/">http://physics.stackexchange.com/</a>, <a href="http://avp.stackexchange.com/">http://avp.stackexchange.com/</a>, <a href="http://math.stackexchange.com/">http://math.stackexchange.com/</a>, and even <a href="http://linguistics.stackexchange.com/">http://linguistics.stackexchange.com/</a>. However, I felt that the Audio-Video Production and Linguistics communities might not provide the mathematically inspired answer I'm looking for, and the Mathematics community might dismiss the question as being ill-defined—I thought the Physics community might come up with important points and counterpoints to practical applications, in addition to having mathematical background.</em></p>
<hr>
<p><strong>Is it possible to estimate the number of people in a room from a limited number of simultaneously recorded audio samples?</strong></p>
<p>I'm searching on Google Scholar by combinations of keywords like <a href="/questions/tagged/recording" class="post-tag" title="show questions tagged 'recording'" rel="tag">recording</a>, <a href="/questions/tagged/audio" class="post-tag" title="show questions tagged 'audio'" rel="tag">audio</a>, <a href="/questions/tagged/extrapolate" class="post-tag" title="show questions tagged 'extrapolate'" rel="tag">extrapolate</a>, <a href="/questions/tagged/crowd" class="post-tag" title="show questions tagged 'crowd'" rel="tag">crowd</a>, <a href="/questions/tagged/number" class="post-tag" title="show questions tagged 'number'" rel="tag">number</a>, <a href="/questions/tagged/voices" class="post-tag" title="show questions tagged 'voices'" rel="tag">voices</a>, etc. and not coming up with any papers related to my question. It's made me wonder whether there's some mathematical or physical impasse that makes the task <em>necessarily</em> impossible. If so, what are the principles that exclude this possibility? If not obviously impossible, could someone perhaps point me to some research regarding this particular problem?</p>
<p>To be more specific, though I'm sure it's naive, my idea was to take two stereo recordings from different locations in a crowded room, and to analyze certain linguistic features like the rate of hisses (or other utterances and features of a language that are easy to isolate), to derive a statistically useful range estimating the number of persons in the crowd. (Of course, such a system would have to be calibrated to a uniform language or dialect.)</p>
<p>I'm aware of various <em>difficulties</em>, both technical (such as how to synchronize the recordings, how to achieve high sensitivity with low noise, etc.) and fundamental (such as how to distinguish between one person speaking profusely and others barely speaking), but I'm more trying to find out if one or more principles of mathematics or physics regarding an audio signal itself, in its "flattened" form, might preemptively shut down this idea. I'm afraid I don't know enough physics myself to explore the problem :-(</p>
<p>Thank you in advance for any insights or references.</p> | g12547 | [
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0.044167403131723404,
... |
<p>So, I was reading about the Casimir effect. Two mirrors facing each other attract to each other in a vacuum. The reason is due to pressure exerted on those mirrors from the multitude of EM waves (like light) outside of them, while in between those mirrors there are less waves present since long waves can't fit. The closer you bring those mirrors, the less waves are present and the stronger is the external pressure on them.</p>
<p>I always imagined waves as being made out of segments, like in the classic Nokia game Snake. I thought that the first segment of a wave to hit a reflective surface would be reflected first, while the last would be reflected last, and thus in between two mirrors the entire wave would simply fold. The first part hitting the first mirror would be reflected to the second mirror and hit it while the 5th part is only about to reach the first mirror.</p>
<p>Now I see I was wrong. Why?</p> | g12548 | [
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0.006015360821038485,
0.00977... |
<p><em>Note</em>: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a recommendation, I'd love to hear it.</p>
<p>Anyway, here goes:</p>
<p>I'm reading a paper (<a href="http://arxiv.org/abs/1407.2945" rel="nofollow">Arxiv</a>) in which a critical part is the derivation of a limiting form of a distribution---the relevant equations are (7) and (8). Specifically, given the negative binomial distribution </p>
<p>$$P(s,t|s_0) = \binom{s-1}{s_0-1}(e^{-\kappa t})^{s_0}(1-e^{-\kappa t})^{(s-s_0)},$$</p>
<p>with $s,s_0,\kappa,t>0$, the authors claim that in the limit as $t\to\infty$,</p>
<p>$$P(s,t\to\infty|s_0) = \frac{1}{\Gamma(s_0)s_0^{-s_0}}s^{s_0-1}\exp(-s_0s),$$</p>
<p>where $\Gamma(x)$ is the usual Gamma function. This is precisely a $\mathrm{Gamma}(s_0,1/s_0)$ density in the location-shape parameterization.</p>
<p>I am flabbergasted how they arrived at that. If I try putting the exact distribution $P(s,t|s_0)$ into Mathematica for a given value of $s_0$ and let $t\to\infty$, I decidedly do not observe that asymptotic result; rather, the mean and width of the distribution grow without limit as $t$ does. The result I obtain from taking a limit on the first expression recapitulates what I see in Mathematica, but it looks like</p>
<p>$$P(s,t\to\infty|s_0) = \frac{1}{\Gamma(s_0)(e^{-\kappa t})^{-s_0}}s^{s_0-1}\exp({-se^{-\kappa t}})$$</p>
<p>i.e., a $\mathrm{Gamma}(s_0,e^{\kappa t})$ density. If I continue to take the limit to $t\to\infty$, I won't get a distribution at all.</p>
<p>This is a pretty central result of their paper, the paper's now been published in <em>PRL</em>, and it would be a pretty severe typo to make, so my first instinct is that I'm misunderstanding what they mean by "asymptotic limit." </p>
<blockquote>
<p>Can you provide any clarification as to why the authors claim the second equation is a limit of the first equation? In what sense is this true? Or what am I missing?</p>
</blockquote>
<p>If you agree with me, then I'll feel more confident going to the paper authors and asking for clarification.</p> | g12549 | [
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0.0... |
<p>If we positioned a mirror 1 light year away from earth and shot a particle of light at the mirror so that it would reflect and come back to earth, how long would it take for us to receive that particle of light back to earth and if I jumped on this particle of light for the trip how would it effect how much time has gone by for me and the people waiting for me back on earth</p> | g12550 | [
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<p>So I have an equilibrium phase diagram of steel and I am asked to 'Calculate the proportion of pearlite in the microstructure of 0.4 wt% C steel just below the eutectoid temperature (727 °C).'</p>
<p>I have been using the forumlas</p>
<p>$Wa=\frac{Cb-Co}{Cb-Ca}$%</p>
<p>$Wb=100-Wa$%</p>
<p>$Ca=weight$% of (Carbon in this example) at the first intersection point on the eutectoid line.</p>
<p>$Cb=weight$% of (Carbon in this example) at the second intersection point on the eutectoid line or the bottom of the 'V'. I don't know how to explain any better sorry.</p>
<p>So looking at my diagram, I see that $Ca=0.022wt$% Carbon and $Cb=0.76wt$% Carbon, and $Co$ is just 0.4%.</p>
<p>So $Wa=\frac{0.76-0.4}{0.76-0.022}$%</p>
<p>$=0.487805%$%</p>
<p>This is where I am confused. What exactly is Wa? Is that how much ferrite/austenite/cementite that is in the pearlite then? Or how much pearlite is at that point compared to ferrite? And in any case, what does that mean Wb is?</p>
<p>Any help would be much appreciated. Thanks </p> | g12551 | [
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0.02023557759821415,
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<p><strong>Question:</strong><br>
3 boys push a small car 25 m up a hill inclined at 19° to the horizontal. The car has a weight of 860 N and they push it at a steady speed of 0.5 m s<sup>-1</sup> against an opposing force of friction of 70 N acting down the hill.
<img src="http://i.stack.imgur.com/2Hp6Z.png" alt="enter image description here"></p>
<ol>
<li>By considering all the forces acting on the car, calculate the net force.</li>
<li>Determine the size of the boys' combined push force.</li>
<li>Determine the combined average power output of the boys.!</li>
</ol> | g12552 | [
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0.02926647663116455,
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<p>There are theoretical predictions for SM Higgs decay branching ratios. What about the experimental results on BRs?</p>
<p>All that I found was measurements of signal-strengths $\mu = \frac{(\sigma \cdot BR)_{\text{meas}}}{(\sigma \cdot BR)_{\text{SM}}} $ and coupling strengths $\kappa$. </p>
<p>Why are there no results on branching fractions up to now? And how do you actually measure the signal strength $\mu$ without measuring the branching ratio $BR$?</p>
<p>I would like to have a little detailed information about the actual differences between measuring $BR$, signal strength $\mu$ and coupling strength $\kappa$.</p> | g12553 | [
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<p>In another Physics stack exchange thread here, <a href="http://physics.stackexchange.com/questions/119381/spin-matrix-for-various-spacetime-fields[/url]">Spin matrix for various spacetime fields</a> I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. I then tried to relate this to the spin-1/2 electron system, but it appears vectors representing states for that system transform under the Pauli matrices instead. I believe this is because those sets of matrices, when scaled properly, yield the correct eigenvalues for the operators. </p>
<p>However, I also noticed that $$D(\omega) = \text{Id} + \omega \begin{pmatrix} 0&1\\-1&0 \end{pmatrix} = \text{Id} + i \omega \begin{pmatrix} 0&-i \\i&0 \end{pmatrix},$$ so I seemed to have made contact with one of the Pauli matrices, (i.e the scaled form of which the spin states do transform). It looks like the above is the infinitesimal version of $\exp(i\omega \hat n \cdot \sigma)$ with $$\hat n \cdot \sigma = \sigma_2 = \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}$$ which seems to mean $\hat n = (0,1,0)$. In particular, c.f Sakurai P.159, eqn (3.2.3), he says that the physical spin 1/2 electron system does transform under the operator $D(\phi) = \exp(-iS_z\phi/\hbar)$ which is analogous to what I have above with the replacements $\phi \rightarrow \omega$ and $ S_z/\hbar \rightarrow \hat n \cdot \sigma$.</p>
<p>So is there really a connection to the matrix I derived and the spin 1/2 electron system? I do actually think not, since the matrix I derived in that other thread came from analyzing the rotation of vectors in space-time but the spin states live in another space. On the other hand, the analysis above makes me think otherwise at the same time.</p>
<p>Many thanks for clarification!</p> | g12554 | [
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-... |
<p>Now I'm trying to give an answer of the question. Satellite rotates about earth by the gravitational force of earth. Now if we place the satellite in such an orbit so that it rotate in opposite to earth rotation and having a greater angular rotational speed. </p>
<p>Now there is an existence of earth magnetic field. Though it is very poor but strength of magnetic field is not the major factor of energy conversion. So as in electrical machine rotor rotates and stator gives magnetic field and electricity generates is it not possible to extract gravitational energy by using satellite as rotor and geomagnetic field as stator? how much it is effective? </p>
<p>Edited: OK eventually this question becomes duplicate. Now I am asking how can we store this energy. Can we store this energy? or not.</p> | g82 | [
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<p>I don't want to open a debate about whether cell phones can cause cancer,
I read the thread:<br> <a href="http://physics.stackexchange.com/questions/12069/could-cell-phone-radiation-cause-cancer">Could cell-phone radiation cause cancer?</a></p>
<p>For the sake of this question let's assume there's a chance for cell-phones
to increase chances of cancer</p>
<p>Now, visible light is higher frequency and higher energy radiation than mobile phones,<br>
You can take a look at the spectrum of elctromagnetic radiation:<br> <a href="http://en.wikipedia.org/wiki/File:EM_spectrum.svg" rel="nofollow">http://en.wikipedia.org/wiki/File:EM_spectrum.svg</a></p>
<p>I'm thinking of going to photodynamic therapy (it's a form of treatment that uses intense visible light for skin conditions - including acne)</p>
<p>Now, if visible light is higher frequency & higher energy than radio
waves, is there a reason that this kind of treatment would not raise
risk of cancer if cell-phones might?</p>
<p>these are some specifics about the light used (it is blue light):<br>
Blue Intensity: 26273.0 uW/cm2 <br>
Blue Peak Wavelength: 415.20 nm</p>
<p>Thanx a lot to anyone giving thought to this question.</p> | g12555 | [
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<p>I heard there were powerful convective movements in clouds which were responsible for increasing the size of water droplets or ice crystals. My question is: do the same movements appear outside of clouds? Or in other words: what's so special about a cloud that makes it host these convective movements? Couldn't warm, dry air also create those movements?</p>
<p>If my question is not clear, don't hesitate in asking me for details.</p>
<p><strong>Update:</strong> Another way to phrase the question would be: Are convective movements in clouds different than those not in clouds?</p> | g12556 | [
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<p>I have been studying solitons theory to make a note on </p>
<blockquote>
<p>dimension analysis for solitons</p>
</blockquote>
<p>At first I have derived one space dimensional kink solution for soliton theory.
I want to go to higher dimension for solitons.</p>
<p>What theories would be fundamental for analyzing multi-dimension of soliton theory?</p> | g12557 | [
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-0.013362538069486618,
0.05717312544584274,
0... |
<p>It seems X-ray absorption spectroscopy is usually ascribed to the interation between photons and inner electrons. Does it mean inner electrons are much preferred by X-ray photons to outer electrons? If so, why? Thank you very much!</p> | g12558 | [
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<p>Materials with large coefficients of static friction would be cool and useful. Rubber on rough surfaces typically has $\mu_s\sim1-2$. When people talk about examples with very high friction, often they're actually talking about surfaces that are sticky (so that a force is needed in order to separate them) or wet (like glue or the tacky heated rubber used on dragster tires and tracks). In this type of example, the usual textbook model of <a href="http://en.wikipedia.org/wiki/Friction#Dry_friction">Coulombic friction</a> with $\mu_s$ and $\mu_k$ (named after Mr. Coulomb, the same guy the unit was named for) doesn't apply.</p>
<p>I was looking around to try to find the largest $\mu_s$ for any known combination of surfaces, limiting myself to Coulombic friction. This <a href="http://robotics.eecs.berkeley.edu/~ronf/Gecko/prl-friction.html">group at Berkeley</a> has made some <a href="http://www.eecs.berkeley.edu/IPRO/VIF/Papers/Fearing1.pdf">amazing high-friction surfaces</a> inspired by the feet of geckos. The paper describes their surface as having $\mu\sim5$. What's confusing to me is to what extent these surfaces exhibit Coulombic friction. The WP Gecko article has pictures of Geckos walking on vertical glass aquarium walls, and it also appears to imply a force proportional to the macroscopic surface area. These two things are both incompatible with the Coulomb model. But the Berkeley group's web page shows a coin lying on a piece of glass that is is nearly vertical, but not quite, and they do quote a $\mu$ value. This <a href="http://www.pnas.org/content/99/19/12252.full">paper</a> says gecko-foot friction involves van der Waals adhesion, but I think that refers to microscopic adhesion, not macroscopic adhesion; macroscopic adhesion would rule out the Coulomb model. (The WP Gecko article has more references.)</p>
<p>So my question is: what is the highest coefficient of <em>Coulombic</em> static friction ever observed, and does the Berkeley group's substance qualify?</p> | g12559 | [
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0.010478541254997253,
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0.042275913059711456,
-0.... |
<p>What makes it a good idea to use RMS rather than peak values of current and voltage when we talk about or compute with AC signals.</p> | g12560 | [
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<p>Thinking about the equivalence principle, is there a nice, simple way to show that a local, freely falling frame in Schwarzschild spacetime is described by the Minkowski metric</p>
<p>$$ds{}^{2}=c^{2}dt^{2}-dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta\left(d\phi\right)^{2}.$$</p>
<p>I thought if I could describe a test body (moving radially inwards with an acceleration due to gravity) using the Schwarzschild metric</p>
<p>$$ds^{2}=\left(1-\frac{2GM}{c^{2}r}\right)c^{2}dt^{2}-\frac{dr^{2}}{1-\frac{2GM}{c^{2}r}}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2},$$</p>
<p>the Minkowski metric would somehow pop out (with $d\theta = d\phi = 0$). I substituted $ar=-GM/r$
into the metric ($a$
is the acceleration due to gravity) to get</p>
<p>$$ds^{2}=\left(1+\frac{2ar}{c^{2}}\right)c^{2}dt^{2}-\frac{dr^{2}}{1+\frac{2ar}{c^{2}}},$$</p>
<p>which, on Earth for example with $2ar\ll c^{2}$, is pretty close to the Minkowski metric. Is this valid/correct?</p> | g12561 | [
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0.0035649691708385944,
-0.07878527045249939,
0.006336858030408621,
0.014362323097884655,
0.013470935635268688,
0.014013468287885189,
0.... |
<p>I am trying to calculate lattice parameter of ZnO (wurtzite). I have found out the formula for calculating lattice parameter of this system from a research article.</p>
<p>$\frac{1}{^{d^{2}}}=\frac{H ^{2}+ K^{2}}{a^{2}}+\frac{L^{2}}{c^{2}}$</p>
<p>I have this jcpds data of ZnO for carrying out calculations</p>
<pre><code>a (Å): 3.2489
b (Å): 3.2489
c (Å): 5.2062
No. h k l d [A] 2Theta[deg] I [%]
1 1 0 0 2.81000 31.820 80.0
2 0 0 2 2.60000 34.467 70.0
3 1 0 1 2.48000 36.191 100.0
4 1 0 2 1.91000 47.569 70.0
5 1 1 0 1.62000 56.783 80.0
6 1 0 3 1.48000 62.728 80.0
7 2 0 0 1.41000 66.229 50.0
8 1 1 2 1.38000 67.861 80.0
</code></pre>
<p>what i have done so far</p>
<p>for calculating parameter <strong>c</strong> i used this formula</p>
<p>$c=l*d$</p>
<p>for 002 plane of above data i got a <strong>c</strong> of <strong>5.198</strong> which is in accordence with jcpds <strong>c=5.2062</strong></p>
<p>But while calculating parameter <strong>a</strong> i am facing some difficulty. i used below formula to calculate <strong>a</strong> lattice parameter (hk0)</p>
<p>$a=\sqrt{H^{2}+K^{2}}*d$</p>
<p>for 110 of the above data i got a lattice parameter <strong>a=2.29</strong> but in jcpds data <strong>a=3.2489</strong></p>
<p>Have i done some thing wrong in my calculations?!!</p> | g12562 | [
0.011370147578418255,
0.002763741184026003,
0.0007920458447188139,
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-0.026982951909303665,
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0.025895211845636368,
-0.004659147933125496,
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0.00601931381970644,
-0.01639789342880249,
0.058215245604515076,
0.037487152963876724,
-... |
<p>Suppose we perform a double-slit experiment with a detector placed at a position of minimum intensity (maximum destructive interference), off-center where the path lengths differ by half a wavelength. The light source is alternately turned on and off (or blocked and unblocked near the source) and the intensity over time is recorded. I interpret the uncertainty principle to mean that there will be a peak in intensity at the times when the switch is flipped (whether on-to-off or off-to-on). i.e., it will look something like this (in ASCII art):</p>
<pre><code> __________'-----'__________'-----'__________
</code></pre>
<p>Is this correct? I have had trouble convincingly explaining my reasons for thinking so. What will be the measured intensity over time and why?</p> | g12563 | [
0.017921775579452515,
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-0... |
<p>Between 1947 and 1962 the US conducted 105 tests of nuclear weapons in the <a href="http://en.wikipedia.org/wiki/Pacific_Proving_Grounds" rel="nofollow">"Pacific Proving Grounds"</a>. I'm wondering how much radiation exposure resulted on the west coast of the US. These were part of the <a href="http://en.wikipedia.org/wiki/Nuclear_weapons_and_the_United_States#Nuclear_testing" rel="nofollow">1056 nuclear bombs that the US has ignited</a> over the years (most underground, but two notably in Japanese cities).</p>
<p>So how much radiation exposure in the US was caused by the 105 nuke tests in the Pacific? Should the inhabitants of the West coast have taken iodine pills?</p>
<p>I'd also like to know how much fallout these bombs produced, as compared to the reactor steam releases in Japan, 2011.</p> | g12564 | [
-0.003566332161426544,
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-0.02348247729241848,
-0.018670642748475075,
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0.... |
<p>A recent experimental paper measures a difference between the top quark and anti-top quark masses:</p>
<p>Fermilab-Pub-11-062-E, CDF Collaboration, <em>Measurement of the mass difference between $t$ and $\bar{t}$ quarks</em></p>
<blockquote>
<p>We present a direct measurement of the
mass difference between $t$ and
$\bar{t}$ quarks using $t\bar{t}$
candidate events in the lepton+jets
channel, collected with the CDF II
detector at Fermilab's 1.96 TeV
Tevatron $p\bar{p}$ Collider. We make an
event by event estimate of the mass
difference to construct templates for
top quark pair signal events and
background events. The resulting mass
difference distribution of data is
compared to templates of signals and
background using a maximum likelihood
fit. From a sample corresponding to an
integrated luminosity of 1/5.6 fb, we
measure a mass difference,
$\mathrm{M}_{t} -
> \mathrm{M}_{\bar{t}}$ $= -3.3 \pm
> 1.4(\textrm{stat}) \pm 1.0(\textrm{syst})$, approximately two standard deviations away from the CPT
hypothesis of zero mass difference.
This is the most precise measurement
of a mass difference between $t$ and
its $\bar{t}$ partner to date.</p>
</blockquote>
<p><a href="http://arxiv.org/abs/1103.2782" rel="nofollow">http://arxiv.org/abs/1103.2782</a></p>
<p>This seems to pile on to the recent evidence showing differences between the masses of the neutrinos and anti-neutrinos. But unlike neutrinos, quarks can't be Majorana spinors. So what theoretical explanations for this are possible?</p> | g12565 | [
-0.006980782374739647,
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-0.02824283018708229,
0... |
<p>I can't seem to find any info on connected rigid bodies by a joint. Can someone explain the basics to me? I'm trying to do a little research to find out how feasible it would be to implement 3d ragdoll physics for my first person shooter game. </p> | g12566 | [
0.034736454486846924,
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0.0070922463200986385,
0.02629232220351696,
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-0.04379160702228546,
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<p>I'm trying to understand what would really happen when large quantities (e.g., 10g) of anti-matter collide with matter. The normal response is that they'd annihilate each other and generate an expanding sphere of gamma ray photons. </p>
<p>However, thinking about it in more detail, what I see is that the anti-electrons annihilate first against the electrons. Let's assume the energy release in that case is not sufficient to noticeably change the momentum of the projectile. Then the nuclei penetrate the electron-annihilation plasma, and since antiprotons attract protons, their trajectory is changed. However, the nuclei are so small and so widely separated that presumably they just orbit each other as the electron clouds annihilate, and eventually enough energy is generated that the ionic plasma of nuclei and anti-nuclei just expands, with a small fraction of them actually ever combining.</p>
<p>In other words, we don't actually see total conversion happening -- only a small fraction of the total mass in an anti-matter/matter collision is turned into gamma rays.</p>
<p>Is that what actually would happen? (The ideal answer would be a video!)</p> | g487 | [
0.03514964506030083,
0.04771079495549202,
0.02618543431162834,
0.0024634022265672684,
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0.016291599720716476,
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0.039215631783008575,
-0.03980... |
<p>So I have learned in class that light can get red-shifted as it travels through space. As I understand it, space itself expands and stretches out the wavelength of the light. This results in the light having a lower frequency which equates to lowering its energy.</p>
<p>My question is, where does the energy of the light go? Energy must go somewhere! </p>
<p>Does the energy the light had before go into the mechanism that's expanding the space? I'm imagining that light is being stretched out when its being red-shifted. So would this mean that the energy is still there and that it is just spread out over more space?</p> | g12567 | [
0.03476540744304657,
0.06926033645868301,
0.01221853494644165,
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0.... |
<p>From what we know now about Quantum Mechanics and Elements, could we simulate <strike>life</strike> <em>the Universe</em> at a Quantum to Element level? </p>
<p>If we can't assume enough to create a sim, what fundamentals are we missing? (Gravity?Knowing the limit of Elements? Knowing the limit of Molecules? String Theory?)</p> | g12568 | [
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0.0751153826713562,
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-0.053557202219963074,
0.009704949334263802,
0.02011835388839245,
0.029104724526405334,
-0.0456... |
<p>the Lienard-Wiechert green functions have future and past null cones of radiation. Maxwell equations allow for a continuous range of mixtures between the retarded and advanced components, but we have observed so far only the retarded emission components</p>
<p>or so it goes the story, but is that really accurate? It looks to me the advanced component is not radiating at all but actually absorbing; if a reverting wavefront is arranged to converge where a electron is going to be, then it will be left afterward with <em>more</em> energy, not less, and the mixture will be temporarily reversed by this artificial arrangement of incoming radiation, with an advanced absorbing component and a retarded radiative component which will be zero or very small</p>
<p>Does it make sense an advanced component that is radiating, i.e: the electron is left will less energy? by symmetry under time reflection, the existence of radiating advanced wavefronts would imply the existence of <em>absorbing</em> retarded wavefronts (i.e: retarded wavefronts of negative electromagnetic energy) which we don't see either</p> | g12569 | [
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0.030495459213852882,
0.01170065626502037,
-0.03147561103105545,
0.025676758959889412,
0.04450244456529617,
0.016255341470241547,
-0.03... |
<p>Okay, I've got a little bit of a layman's question here.<br>
We're doing a bit of spring cleaning in our office and we've found a cabinet with boxes upon boxes of stored wires, so naturally, this discussion arose...</p>
<p>Picture a normal, bog-standard wire, with a plastic outer coating. Now, quite often when these wires are stored, they will wrapped up and twisted, to effectively make a coil.</p>
<p>I was just wondering what the effects of this type of storage would have.</p>
<p>What if you had a 15m wire and only used the each end to cover about a single meter (leaving 14m still twisted and wrapped in the middle), what the effect of the electrical current running through this have?</p>
<p>Thanks for helping us settle a mild dispute!</p> | g12570 | [
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0.05538720265030861,
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0... |
<p>I think this question has its place here because I am sure some of you are "self-taught experts" and can guide me a little through this process.</p>
<p>Considering that :</p>
<ul>
<li>I don't have any physics scholar background at all.</li>
<li>I have a little math background but nothing too complicated like calculus</li>
<li>I am a fast learner and am willing to put many efforts into learning physics</li>
<li>I am a computer programmer and analyst with a passion for physics laws, theories, studies and everything that helps me understand how things work or making me change the way I see things around me.</li>
</ul>
<p>Where do I start ? I mean.. Is there even a starting point ? Do I absolutly have to choose a type of physic ? Is it possible to learn physics on your own ?<br></p>
<p>I know it's a general question but i'm sure you understand that i'm a little bit in the dark here.</p> | g40 | [
0.03424053266644478,
0.03392859920859337,
0.024760644882917404,
0.04026305675506592,
0.05009077116847038,
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0.02095358446240425,
0.030843043699860573,
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0.007592407055199146,
0.011771243065595627,
0.014788053929805756,
-0.01903... |
<p>The task is to draw the two 4th-order Feynman diagrams of:</p>
<p>$$e^- + \mu^+ \to \nu_e + \bar{\nu}_\mu $$</p>
<p>I drew the first one as (time left->right):</p>
<pre><code> mu^+ \bar{\nu}_\mu \nu_\mu
--<---------<----------<-----
/ /
/W^- /Z^0
/ /
/ /
--->-------<--------<-----
e^- \nu_e \nu_e
</code></pre>
<p>which is correct. I drew the second one with a $W^+$ instead of a $W^-$, with the $W^+$ going the other direction (from the $mu^+$ to the $e^-$). However, the solutions gives the second diagram differently: rather, the order (in time) of the $W^-$ and the $Z^0$ is interchanged. I agree that this should be a valid feynman diagram as well for the process, but I don't understand why my alternative with the $W^+$ would be illegal. As far as I can see, it obeys all conservation laws.</p>
<p>Anyone?</p> | g12571 | [
-0.000383447710191831,
-0.021751631051301956,
0.02246369607746601,
-0.00019677523232530802,
0.06981537491083145,
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0.0018010273342952132,
0.0006700478261336684,
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0.06350130587816238,
0.033068038523197174,
0.01788996532559395,
-0.020046507939696312,
... |
<p>Consider the hamiltonian $H=\frac{p_x^2}{2m}$ in 1-D. It is invariant under $p_x \rightarrow -p_x$.
Again, this hamiltonian also has translational symmetry. Which one of these two is responsible for doubly degenerate energy eigenfunctions for a given energy $E$? I think it is the first one (Should I call it parity symmetry?). Am I right? </p>
<p>We know any symmetry appears as a degeneracy in quantum mechanics. Right? Then what is the manifestation of translational symmetry?</p> | g12572 | [
0.029946165159344673,
0.025614125654101372,
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0.04829481989145279,
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0.039816778153181076,
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-0.016264766454696655,
-0.02796308323740959,
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0.008564355783164501,
-0... |
<p>What are the units for <a href="http://en.wikipedia.org/wiki/Thermal_conductivity" rel="nofollow">thermal conductivity</a> and why?</p> | g12573 | [
-0.0008661191677674651,
0.024613142013549805,
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0.04759654030203819,
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0.02116730995476246,
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0.02061791718006134,
-0.08783739060163498,
0.05566917732357979,
0.02041427232325077,
0.024211... |
<p>As the title states, can a <a href="http://en.wikipedia.org/wiki/Miller_index" rel="nofollow">Miller index</a> for a cubic structure have 4 digits? If I have a structure with intercepts (2,8,3) on the x-y-z axes respectively, the following Miller index would be (12,3,8), which is not 3 digits.</p> | g12574 | [
0.023699481040239334,
0.01502877939492464,
-0.03140324726700783,
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0.005659632850438356,
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0.02282729186117649,
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-0.013899478130042553,
-0.009985395707190037,
0.02853020839393139,
-0.0001646179734962061,
-0.... |
<p>If centripetal acceleration is towards the center, then why - when you spin a bucket of water (a classic demonstration) - does the water not get pushed out but rather stays in the bucket without spilling?</p> | g12575 | [
0.07337995618581772,
0.03381974250078201,
0.014280397444963455,
0.013676696456968784,
0.005692527163773775,
0.03669101744890213,
0.10403243452310562,
0.013115782290697098,
-0.02083062008023262,
0.002709794556722045,
0.023778969421982765,
-0.017364704981446266,
0.0009170124540105462,
0.0175... |
<p>Carbon detonation is a characteristic event of Type 1a Supernova (EDIT: where an accreting white dwarf near the Chandrashankar limit of 1.4 solar masses explodes), an extremely important standard candle for cosmology. An area of active research is designing computer simulations to model supernova spectra and light curves and fit these to ones obtained observationally to better understand the effect of trace elements and characteristics of the explosion (asymmetry, companion star properties, etc) in order to provide better distance estimates to get more accurate constraints on cosmological parameters (Hubble constant, Dark Energy equation of state, etc). </p>
<p>But it would be very interesting (and extremely cool) if there was a way of generating carbon detonations in a laboratory situation in order to study these effects. What sort of temperature/pressure range is necessary to generate a carbon detonation? Would it be in the range of experimental apparati? Or, on the extreme sides of things, a large thermonuclear device?</p> | g12576 | [
0.04270084574818611,
0.015139221213757992,
0.016615230590105057,
0.01015480887144804,
0.04605923220515251,
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0.03835628926753998,
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-0.027488572522997856,
-0.006016010884195566,
-0.005861592013388872,
0.03083173744380474,
-0.022... |
<p>The Earth turns with a very high velocity, around its own axis and around the Sun.
So why can't we feel that it's turning, but we can still feel earthquake.</p> | g12577 | [
0.03716972842812538,
0.058487340807914734,
-0.008188874460756779,
0.05270996689796448,
0.026623714715242386,
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0.0506075844168663,
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-0.09755155444145203,
0.012269378639757633,
0.044272322207689285,
0.05038391053676605,
0.00294... |
<p>Cosmological Inflation was proposed by Alan Guth to explain the flatness problem, the horizon problem and the magnetic monopole problem. I think I pretty much understand the first two, however I don't quite understand how a period of exponential expansion fully explains monopole problem. </p>
<p>From Weinburg's Cosmology, the issue is essentially that various grand unified theories predict that the standard models $SU(3)\times SU(2)\times U(1)$ arose from the breaking of an original simple symmetry group. For many of these theories, a crazy particle known as a "magnetic monopole" is created at a certain energy (sometimes quoted at around $M = 10^{16} GeV$). So my question is why does a period of rapid expansion somehow or other result in a low density of magnetic monopoles (assuming they exist/existed at all)? </p>
<p>I would think, like in nucleosynthesis, that the primary factor in monopole creation is energy density, and since inflation is still a "smooth" process, at some point the universe would hit the proper energy density to create magnetic monopoles. How does the rate of expansion at the time they were created effect overall present density?</p> | g12578 | [
0.0475543849170208,
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0.026286618784070015,
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0.025638800114393234,
-0.017805276438593864,
-0.019556237384676933,
0.057219650596380234,
0.0065... |
<p>I know what an Hilbert Space is, but I'm not sure what exactly is the single particle hilbert space; I understand it as the space of all possible states of the particle; does it matter if you're talking about an electron, a neutron or a quark, or is it a particle in the 'abstract'? How does one construct it?</p> | g12579 | [
-0.024982169270515442,
0.035577017813920975,
-0.0028975738678127527,
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0.008116427809000015,
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0.0060906135477125645,
0.02139049768447876,... |
<p>I live at roughly 52.4,-2.1. On a sunny evenings I've often looked at the moon and the sun and noticed that the light part of the moon does not appear to line up with the sun. For example, at about 17:00 GMT on 13 Mar 2011, I noticed the half moon was facing toward a point roughly 10-20 degrees above where the sun appeared to be. Why?</p> | g12580 | [
0.0010539935901761055,
-0.049823589622974396,
0.014193491078913212,
0.043673623353242874,
0.01018716674298048,
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0.027861151844263077,
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0.016618086025118828,
-0.024851853027939796,
0.061906710267066956,
-0.006923815235495567,
-0.0045153554528951645,
... |
<p>We don't normally consider the possibility that massless particles could undergo radioactive decay. There are elementary arguments that make it sound implausible. (A bunch of the following is summarized from Fiore 1996. Most of the rest, except as noted, is my ideas, many of which are probably wrong.)</p>
<ul>
<li><p>1) Normally we state the lifetime of a particle in its rest frame, but a massless particle doesn't have a rest frame. However, it is possible for the lifetime $\tau$ to be proportional to energy $E$ while preserving Lorentz invariance (basically because time and mass-energy are both timelike components of four-vectors).</p></li>
<li><p>2) The constant of proportionality between $\tau$ and $E$ has units of mass<sup>-2</sup>. It's strange to have such a dimensionful constant popping up out of nowhere, but it's not impossible.</p></li>
<li><p>3) We would typically like the observables of a theory to be continuous functions of its input parameters. If $X$ is a particle of mass $m$, then a decay like $X\rightarrow 3X$ is forbidden by conservation of mass-energy for m>0, but not for m=0. This discontinuity is ugly, but QFT has other cases where such a discontinuity occurs. E.g., historically, massive bosons were not trivial to incorporate into QFT.</p></li>
<li><p>4) In a decay like $X\rightarrow3X$, the products all have to be collinear. This is a little odd, since it doesn't allow the clear distinction one normally assumes in a Feynman diagram between interior and exterior lines. It also means that subsequent "un-decay" can occur. Strange but not impossible.</p></li>
</ul>
<p>So what about less elementary arguments? My background in QFT is pretty weak (the standard graduate course, over 20 years ago, barely remembered).</p>
<ul>
<li><p>5) The collinearity of the decay products makes the phase-space volume vanish, but amplitudes can diverge to make up for this.</p></li>
<li><p>6) If $X$ is coupled to some fermion $Y$, then one would expect that decay would correspond to a Feynman diagram with a box made out of $Y$'s and four legs made out of $X$'s. If $Y$ is a massive particle like an electron, $P$. Allen on physicsforums argues that when the energy of the initial $X$ approaches zero, the $X$ shouldn't be able to "see" the high-energy field $Y$, so the probability of decay should go to zero, and the lifetime $\tau$ should go to infinity, which contradicts the requirement of $\tau\propto E$ from Lorentz invariance. This seems to rule out the case where $Y$ is massive, but not the case where it's massless.</p></li>
<li><p>7) If $X$ is a photon, then decay is forbidden by arguments that to me seem technical. But this doesn't forbid decays when $X$ is any massless particle whatsoever.</p></li>
<li><p>8) There are some strange thermodynamic things going on. Consider a one-dimensional particle in a box of length $L$. If one $X$ is initially introduced into the box with energy $E=nE_o$, where $E_o$ is the ground-state energy, then it undergoes decays and "undecays," and if I've got my back-of-the-envelope estimate with Stirling's formula right, I think it ends up maximizing its entropy by decaying into about $\sqrt{n}$ daughters at a temperature $\sim \sqrt{hE/L}$. If you then let it out of the box so that it undergoes free expansion, it acts differently from a normal gas. Its temperature approaches zero rather than staying constant, and its entropy approaches infinity. I may be missing something technical about thermo, but this seems to violate the third law.</p></li>
</ul>
<p>So my question is this: <strong>Is there any fundamental (and preferably simple) argument that makes decay of massless particles implausible?</strong> I don't think it can be proved completely impossible, because Fiore offers field theories that are counterexamples, such as quantum gravity with a positive cosmological constant.</p>
<p>References:</p>
<ol>
<li>Fiore and Modanese, "General properties of the decay amplitudes for massless particles," 1996, <a href="http://arxiv.org/abs/hep-th/9508018">http://arxiv.org/abs/hep-th/9508018</a>.</li>
</ol> | g12581 | [
0.03477208688855171,
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0.012041171081364155,
-0.05316467210650444,
0.016444437205791473,
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0.03188520669937134,
0.04064764454960823,
0.013... |
<p>I had the intuition that, in classical mechanics, when the trajectory
of a body is known, then analysis of its motion can be done in the
linear space of that trajectory, if all forces are projected on the
tangent of the trajectory.</p>
<p>My original idea was to bypass the consideration of angular momentum,
and work only with linear momentum. I even wrote a <a href="http://physics.stackexchange.com/questions/75641">question for a
formalization of this idea</a>, with no success, which surprised me.</p>
<p>Luckily, I found a hint on how to get started on it, and I managed to
write that formalization and <a href="http://physics.stackexchange.com/questions/75641#75830">answer my own question</a>, with some very
elementary differential geometry. And I realized that I get a coherent
problem of mechanics, but in a 1-dimensional world, which may even
include several interacting masses (sharing the same trajectory), and
has momentum conservation.</p>
<p>But my mathematical knowledge and ability stops there.</p>
<p>So my question is the following. If I am analysing the motion of
masses, with the knowledge that they never leave a known surface, can
I play the same trick, projecting all forces on tangents to that surface, so that I can analyse my
problem in a 2-dimensional space, and benefit of some simplifications.
For example an angular momentum is a scalar in 2D. A problem is
obviously that coordinates are harder to define on a non-developable
surface.</p>
<p>Can there be a concept of inertial frame in this projection space.</p> | g12582 | [
0.0667916014790535,
-0.01297506783157587,
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0.0070949094370007515,
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-0.06616881489753723,
0.022790025919675827,
-0.057211726903915405,
0.004319700412452221,
-0.007... |
<p>A flop transition changes the second homotopy group of a Calabi-Yau compactifation, but not the fundamental group or the number of connected components. Can the number of connected spatial components change in string theory? Can a part of space pinch off never to interact with the rest of space?</p> | g12583 | [
-0.012493148446083069,
0.019141970202326775,
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0.021700365468859673,
... |
<p>In mathematics, we talk about tangent vectors and cotangent vectors on a manifold at each point, and vector fields and cotangent vector fields (also known as differential one-forms). When we talk about tensor fields, we mean differentiable sections of some tensor power of the tangent or cotangent bundle (or a combination).</p>
<p>There are various natural differentiation operations, such as the exterior derivative of anti-symmetric covariant tensor fields, or the Lie derivative of two vector fields. These have nice coordinate-free definitions.</p>
<p>In physics, there is talk of "covariant derivatives" of tensor fields, whose resulting objects are different kind of tensor fields.</p>
<p>I was wondering, what is the abstract interpretation of the general notion of a covariant derivative in terms of (tensor products of) tangent vectors and vector fields.</p> | g12584 | [
0.026882601901888847,
-0.020310115069150925,
-0.03223694860935211,
-0.022477535530924797,
0.04204949736595154,
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-0.018700482323765755,
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-... |
<p>If we apply the Friedmann Lemaître equation to the Universe, we find a critical density $\rho_c$ : if the actual density $\rho$ is under it, the Universe will continue to expand, if it is higher than it, it will end in a Big-crunch.</p>
<p>For what reason (mathematically or physically), can't we apply Friedmann-Lemaître to the Universe's volume associated to the Earth and conclude that this portion of space-time is shrinking because the density is higher than $\rho_c$ ?</p> | g12585 | [
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-0.06880665570497513,
-0.006710696499794722,
-0.05030277743935585,
0.08838824927806854,
0.... |
<p>What is the parametric equation guiding the geometry of a <a href="http://en.wikipedia.org/wiki/File%3aFerrofluid_Magnet_under_glass_edit.jpg" rel="nofollow">ferrofluid</a> under a magnetic field? See also <a href="http://en.wikipedia.org/wiki/Ferrofluid" rel="nofollow">this</a> Wikipedia page.</p>
<p>From previous research, <a href="http://en.wikipedia.org/wiki/Maxwell%27s_equations" rel="nofollow">Maxwell's Equations</a> and <a href="http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" rel="nofollow">Navier-Stokes Equations</a> were previously used but I am not sure how they are being combined to create this stunning geometry.</p>
<p>If the 3d model is too complex, is there a 2D pointed parabola with a curved crest equation which we might use?</p> | g12586 | [
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-0.014794638380408287,
-0.01557242963463068,
0.006032786797732115,
0.07647040486335754,
-0.0354... |
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