question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context:</p>
<blockquote>
<p>An arbitrary rotation of the coordinate frame will transform the
tensor $T$ into a tensor $T'$, whose components $T_{ij}'$ are, in the most
general case, linear combinations of all the components $T_{ij
}$.
However, it is always possible to find certain subgroups of the
components $T_{ij}$ (formed by linear combinations thereof) <strong>that transform
among themselves</strong> under a rotation. These components are called the
irreducible tensor components.</p>
</blockquote>
<p>Can you please explain that scientifically and linguistically?</p> | g13692 | [
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<p>The reasons for choosing length, mass, time, temperature, and amount as base quantities look (at least to me) obvious. What I'm puzzling about is why current (as opposed to resistance, electromotive force, etc.) and luminous intensity (as opposed to illuminance, emittance, etc.) were chosen to be base quantities. Does it have something to do with them being the easiest to measure?</p> | g13693 | [
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<p>Consider the following system. The given friction coefficients are for static. Let $g=10$ (meter per square second).</p>
<p><img src="http://i.stack.imgur.com/8zp7O.png" alt="enter image description here"></p>
<p>If the $F(t)=10t$, for example, determine the friction forces $f_{AB}(t)$ (between the boxes $A$ and $B$) and $f_{BF}(t)$ (between the box $B$ and the floor).</p>
<p>I am confused how the friction forces grow reacting the external force $F(t)$. Could you explain the concept to solve this kind of problem? It is not a homework for sure.</p> | g13694 | [
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<ul>
<li>I placed a box in front of a plane mirror</li>
<li>I looked at the image and smudged a bit of Vaseline on the mirror as a mark to indicate where I thought where the virtual image was. </li>
<li>After that, I repositioned myself at different places of my room. One thing I noticed was that I the image was changing position to.</li>
</ul>
<p>I read on this website (<a href="http://www.physicsclassroom.com/class/refln/Lesson-1/The-Line-of-Sight" rel="nofollow">http://www.physicsclassroom.com/class/refln/Lesson-1/The-Line-of-Sight</a>)</p>
<p><em>"The precise image location of the object is the location where all lines of sight intersect regardless of where the eye is located"</em></p>
<p>This was quoted from the website. So, apparently, the position of the image does not change regardless of the position of my eye.</p>
<p>My QUESTION:</p>
<p>When I was trying this out, why did it seem to me that the image was also changing position as I changed my position?</p> | g13695 | [
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<p>a friend of mine thinks that by buying a product called <a href="http://www.humanfirewall.com/orderrayguard.htm" rel="nofollow">rayguard</a> he can effectively shield against electrosmog.</p>
<p>On the website, there are multiple evidences that this might work. However the product seems to be just created of one plastic chasi with two spools inside. The functionality of the product is described as following:</p>
<blockquote>
<p>With the RayGuards, the solution therefore is not based upon
electronics but on physics. In a radius depending on the model of
device used, all frequencies are absorbed.</p>
<p>This spectrum of frequencies is being put through a dividing and
analysing system. There the compen- sation of the harmful waveparts is
made.</p>
<p>To explain it better, we will use the example of a camera. Compare the
effect of the RayGuard to the polarising filter. Disturbing effects of
reflection are being cleared by the filter and the foto looks clear
and colourful again.</p>
</blockquote>
<p>I do not claim that I am a physicist , however after some research my personal opinion on this product is that it is not so simple to shield against radiation. My questions to you are:</p>
<ul>
<li>Is it possible with the rayguard to shield effectively against radiation? If it is a scam? Why "not"</li>
<li>From a physical standpoint, how to shield properly against radiation? What device would be needed to shield effectively against for example mobile phone radiation</li>
<li>Is the everyday radiation from a physical standpoint "unhealthy"?</li>
</ul>
<p>I really appreciate for your reply!</p>
<p>PS.: I know that this question is not the typical physician question which is asked here, however it would be great if you could give me some hard facts/theories to prevent my friend from wasting a lot of money!</p> | g13696 | [
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<p>I am curious as to how much voltage should be applied to create a specific charge. Is there a formula to calculate it, and what are the parameters that can affect the relation between voltage and charge created in that object?</p>
<p>P.S: I haven't taken course on this subject, so I don't know the details of this subject.</p> | g13697 | [
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<p>I seem to be confused about the nature of forces as vectors, in the basic Newtonian mechanics framework. </p>
<p>I know what a vector is as a mathematical object, an element of $R^3$. I understand that if a vector is drawn in a usual physical way as an arrow in space, it can be seen as a mathematical vector by translating it to begin at 0 and seeing where the arrow tip ends up. Generally it seems the word "vector" is used in such a way that a vector remains the same vector if it's translated arbitrarily in space (always corresponding to the same mathematical vector).</p>
<p>But now let's say I have a solid object, maybe a metal cube, with some forces acting on it: I push at it with a stick in the center of one facet, it's held by a rope in a different corner, etc. To specify each force that is acting on the cube it doesn't seem enough to specify the vector: I also need to specify the place of application. The cube behaves differently if I push it in the center as opposed in the corner etc.</p>
<p>I'm reading through J.P.Den Hartog's Mechanics that teaches me how to find the resultant force on the cube. I need to sum forces one by one using the parallelogram law, but I should always be careful to slide each force along its line of application, until two forces meet. I <em>could</em> just translate them all to start at the same point and add, but then I won't find the right resultant force, only its direction and magnitude; I will still need to find its line of application (maybe using moments etc.)</p>
<p>So let's say I'm calculating the resultant force "the right way": by sliding arrows along their lines until tails meet, adding, repeating. What am I doing mathematically? (it's not vector addition, that would correspond to just translating them all to 0 and adding) What mathematical objects am I working with? They seem to be specified with 4 free parameters: 3 for direction/magnitude of the vector and 1 more to displace it to the correct line of application; the location at the line of application seems irrelevant according to the laws of statics.</p> | g13698 | [
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<p>Here's an explanation from a book that I read:</p>
<p><img src="http://i.stack.imgur.com/9q7I7.png" alt="enter image description here"></p>
<p>Now, I have a few problems with this explanation as well as with the "cone method". I guess that all my problems boil down to the concept of infinitesimal.</p>
<ol>
<li><p>The book says "a normal to $dA$ makes an angle $\phi$ with a <strong>radial</strong> line from $q$". Now, the problem is that I can take another radial line intersecting $dA$ which will make a different angle with the area.</p></li>
<li><p>The strength of the electric field passing through the projected area ($dA \cos \phi$) is different from $\vec{E}$ because the projected area is closer to the charge, yet the book seems to ignore that. The same goes for the "cone method":</p></li>
</ol>
<p><img src="http://i.stack.imgur.com/mXMH1.png" alt="enter image description here"></p>
<p>Books calculate proportion between the areas and the radii using the same radius for $dA \cos \phi$ and $dA$ although the distance of the projected area from the charge is different than from the original area to the charge.</p>
<p>So how can I solve these contradictions? Is there a strict mathematical proof that an electric flux through any area inside a cone (regardless of its orientation) is the same?</p> | g13699 | [
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... |
<blockquote>
<p>A ball of mass 0.37 kg is thrown upward along the vertical with a
initial speed of 14 m / s, and reaches a maximum height of 8.4 m.
a) What is the work done by air resistance on the ball? b) Assuming that the
air resistance performs roughly the same job during the descent,
Calculate the speed of the ball when it returns to the starting point.</p>
</blockquote>
<p>How can I calculate the work of the air resistance?</p> | g13700 | [
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<p>With <a href="http://en.wikipedia.org/wiki/Jones_calculus" rel="nofollow">Jones vectors and matrices</a> one can describe the change in polarization of a EM wave. What is the convention of the reference coordinate system; Is it fixed or does it change whenever the direction of the wave changes?</p>
<p>In other words: I have a +45° linear polarized wave traveling in $+z$ direction towards a metallic mirror where it becomes reflected back in $-z$ direction.</p>
<p>One can now either choose a fixed coordinate system so that the polarization remains +45° also for the back-travelling wave (but with an additional phase of $\pi$). Or one can choose a coordinate system "attached" to the wave so that here the reflected wave again travels in positive z direction but now with a -45° polarization (and an additional phase of $\pi$) in this attached coordinate system.</p>
<p>What is the conventional choice?</p> | g13701 | [
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<p>Conformal gravity is an "alternative" theory of gravity, where instead of using the Einstein-Hilbert action composed of the Ricci scalar, the square of the conformal Weyl tensor is used. It was originally designed to arrive at the inflationary cosmological models without the use of dark energy. </p>
<p>However, it was later noticed that the galactic rotation curves of a certain matter distribution commonly seen in galaxies can also be accurately predicted using the conformal gravity and without the use of dark matter, but where in addition to total luminosity of the galaxy and its mass used in the matter distribution, two new constants appear. However, both of the constants turn out to be universal and are found to be equal for all galaxies (within the errors permitted by the deviation from the assumed baryonic matter distribution).</p>
<p>A while ago it was widely reported that the Bullet Cluster lensing effects rule out the alternative theories of gravity and provide evidence for dark matter. Does conformal gravity sufficiently explain the lensing effects observed in Bullet Cluster or is it similarly ruled out?</p> | g13702 | [
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<p>I don't understand the solutions to a problem about blackbody radiation and was wondering if anybody could help me out.</p>
<p>Here is the question:</p>
<blockquote>
<p>The sun can be considered as a blackbody radiation source at temperature
T = 5778 K. Radiation from the sun which is incident on the earth is reflected by the
atmosphere such that the intensity hitting the earth's surface is reduced by a factor R.
Some of the radiation emitted from the earth's surface is reflected by the atmosphere such
that only a fraction A leaves the atmosphere. If A = R = 0:1, what temperature would
the earth be? </p>
</blockquote>
<p>Then in the solutions they state:
This is obtained by first trying to find the power from the sun which passes through unit area at the earth's radial distance. This is given by:</p>
<p>$\frac{4 \pi r_s^2}{4 \pi d_e^2}\sigma T_s^4$</p>
<p>where $r_s$ is the radius of the sun, $d_e$ is the distance between the earth and sun and $T_s^4 = 5778K$ is the temperature of the sun.</p>
<p>I know that $\sigma T_s^4$ is from the Stefan-Boltzmann law, and that $4 \pi r_s^2$ is the surface of the sun. What I don't understand is why the distance to the earth is important.
Thanks in advance!</p> | g13703 | [
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<p>I think this is analogues for the <a href="http://en.wikipedia.org/wiki/Woodward_effect" rel="nofollow">Woodward effect</a>, but macroscopic:</p>
<p>We assume a spacecraft consisting of a broomstick, a donut and lots of gear for storing and transfering mechanical energy.
Take the donut, spin it till the outer edge moves at relativistic speed (We assume a tough glazing), and push it along the broomstick, brake it's spin and then the donut, move donut forward along broomstick, repeat.
Since the donut is heavier when it's being accelerated backwards then when it's braked/accelerated forward, there should be a net momentum. Since nothing leaves the broomstick, this has to be wrong. But where?</p> | g13704 | [
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<p>I am reading Smolin's book <a href="http://rads.stackoverflow.com/amzn/click/0195126645" rel="nofollow">Life of the Cosmos</a>. One part which I can hardly understand is the argument for cosmological natural selection as a scientific principle.</p>
<p>Smolin keeps on saying that cosmological natural selection is a falsifiable scientific theory because the assumption that our universe is the most black hole friendly universe and so on.... I have great difficulty in following his line of argument, can anyone summarize it here?</p> | g13705 | [
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<p>Why is it that creation and annihilation operators ($a_+$ and $a_-$) can only be defined for the problem of quantum harmonic oscillator and nothing else? Can any other bound state problem be solves using $a_+$ and $a_-$?</p> | g13706 | [
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<p>For the Klein Gordon nonlinear equation,</p>
<p>$$ u_{tt}- \Delta u +f(u)=0,$$</p>
<p>how could I use Noether's theorem to prove that there is a conserved quantity? I.e.,</p>
<p>$$ (\Pi _{k} )_{t} - \rm div(j_{k})=0 $$
for $k=0,1,2,3$, where</p>
<p>$$ \Pi _{k} = \int _{R^{3}}p(x,y,z,t)dv $$
is the density of four-momentum.</p> | g13707 | [
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<p>Firstly I think I am right in saying that TV broadcast are sent via electromagnetic waves which means they are sent via photons, how is that even possible?</p>
<p>And then the main questions, how can you broadcast such large amounts of data over the airwaves and produce photorealistic TV and a PC cannot.</p>
<p>I understand a bit about offline and real-time rendering, and I guess it's that for a PC game you have to interact which using a lot more processing power. So how does TV do it, will PC games ever get to photorealistic levels?</p> | g13708 | [
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<p>As we know that induction cooker works on the principal of induction of current in a conducting plate. So I just wanted to know what will happen if we place salt water in a plastic container on the induction cooker will it get hot as it is also a conductor or nothing will happen?</p> | g13709 | [
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<p>I'm a third year mathematics undergrad, and have just started the module General Relativity and spacetime geometry, I also have a keen interest in black holes.</p>
<p>However I would like to know why and how the mass of the Kerr black hole is proportional to it's angular momentum, and also inversely proportional to it's Schwarzchild radius?</p>
<p>Thank you in advance for any help!</p> | g13710 | [
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<p>I have a circuit that has 4 wires and 2 following each other <a href="http://en.wikipedia.org/wiki/Toffoli_gate" rel="nofollow">Toffoli gates</a>.</p>
<p>I have permutation matrix for each Toffoli gate (A and B).</p>
<p>Do I have to multiply that 2 matrices to get the entire permutation matrix of that 2 Toffoli gates?</p>
<p>And if I do that is $A\times B$ or $B\times A$?</p> | g13711 | [
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<p>On <a href="http://en.wikipedia.org/wiki/Einstein_notation" rel="nofollow">Einstein notation</a> with multiple indices: For example, consider the expression:</p>
<p>$$a^{ij} b_{ij}.$$</p>
<p>Does the notation signify,</p>
<p>$$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$</p>
<p>i.e. you sum over every combination of the indices? Or do you sum over the indices at the same time, i.e. they take on the same values:</p>
<p>$$ a^{00} b_{00} + a^{11} b_{11} +... ?$$</p> | g13712 | [
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<p>in my atomic physics lecture notes it is said that depending on the values of the exchange operator $K_{ij}$ there can be different kinds of magnetism, say, ferromagnetism and antiferromagnetism, how can I see that?</p> | g13713 | [
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-0.022784722968935966,
-0.018495401367545128,
0.043533653020858765,
0.0015000627608969808,
-0.025165965780615807,
0.026552889496088028,
0.03324352204799652,
0.004098272416740656,
-0.02603737823665142,
0.0363151840865612,
0.0290579404681921,
0.018... |
<p>Examining the following graph, I am trying to understand the power plot. The power appears to take on a negative value when the current changes direction or the voltage changes polarity. Negative power does not make sense to me. In a proper representation of the power curve shouldn't the power remain above the dotted line?
Additionally, since the power is a product of voltage and current, I would think that the power plot should be a maximum value during where the plots of the voltage and current intersect, and should be greater in magnitude the the plots of the voltage and current.</p>
<p><img src="http://i.stack.imgur.com/CalKa.png" alt="enter image description here"></p> | g13714 | [
0.003739429172128439,
-0.007673340383917093,
-0.03624695912003517,
0.00322278356179595,
0.015628311783075333,
0.01382497139275074,
-0.023020025342702866,
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-0.0030220807529985905,
0.05286116153001785,
0.03115609847009182,
0.033434923738241196,
0.0... |
<p>I'm not sure if this is the best place to ask this question, but as it's related to the terminology of nuclear physics I thought it would probably be a logical place to start.</p>
<p>I'm currently writing my thesis, and in the section on radioactive decay I talk about the mass number and activity of radionuclides. The problem is that the notation for both these terms is <em>A</em>, which is likely to become confusing later on. Can anybody recommend a commonly accepted alternative symbol/notation for either term, or suggest a way that I differentiate between the two in the text and nomenclature/glossary?</p>
<p>Many thanks!</p> | g13715 | [
0.03306128457188606,
-0.03111141547560692,
0.009986130520701408,
-0.035599805414676666,
0.048545923084020615,
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0.0324857123196125,
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-0.027986427769064903,
0.031363073736429214,
0.024656984955072403,
0.09129729866981506,
0.0052... |
<p>We know that the molecule of hot water($H_2O$) has more energy than that of cold water (temperature = energy)
and according to Einstein relation $E=mc^2$ ,this extra energy of the hot molecule has a mass.
Does that make the hot molecule heavier?</p> | g13716 | [
0.04415161535143852,
-0.01260374579578638,
0.015306389890611172,
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0.018129082396626472,
0.0... |
<p>If I use a telescope to observe a plane in the sky, how can I find the altitude of the plane(altitude of the plane with respect to the ground)?</p> | g13717 | [
-0.0024651845451444387,
0.0005514414515346289,
0.00020485233108047396,
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0.01124588493257761,
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0.0023736953735351562,
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-0.00205772113986313... |
<p>Supersymmetry should cancel the radiative corrections to the Higgs mass and solve the hierarchy problem. If Supersymmetry would be unbroken, would the higgs mass be 0?</p> | g13718 | [
0.08471617102622986,
0.02112804353237152,
0.006472461391240358,
0.02625264786183834,
0.008739674463868141,
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0.009662386029958725,
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0.0775... |
<p>Suppose we have
$$[Q^a,Q^b]=if^c_{ab}Q^c$$
where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have
$$[P^a,P^b]=0$$
where P's are generators of a Lie algebra associated to an Abelian group. We have the following relation between these generators
$$[Q^a,P^b]=if^c_{ab}P^c$$</p>
<p>I would like to know what we can say about the following trace. Is it equal to zero?
$$tr([Q^a,P^b]Q^c P^d)$$</p>
<p>Cheers!</p> | g13719 | [
0.011654906906187534,
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0.03131299838423729,
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0.0024464584421366453,
-0.037848733365535736,
0.018186267465353012,
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<p>Consider a coupled harmonic oscillator with their position given by $x_1$ and $x_2$. Say the normal coordinates $x_{\pm}={1\over\sqrt{2}} (x_1\pm x_2)$, in which the harmonic oscillators decouple, exist.</p>
<p>When you quantize this theory, the corresponding operators are $\hat x_1,~
\hat x_2,~ \hat x_{\pm}$. The operators $\hat x_1,~\hat x_2$ commutes and
$$\hat x_{\pm}\equiv{1\over\sqrt{2}} (\hat x_1 \otimes \hat 1_2 \pm \hat 1_1\otimes \hat x_2),$$
where $\hat 1_1,~\hat 1_2$ are identity operators.
So this operator $\hat x_{\pm}$ acts in a direct product basis given by
$$|x_1,x_2\rangle=|x_1\rangle \otimes |x_2\rangle$$
where $|x_1\rangle$ and $|x_2\rangle$ are eigen states of $\hat x_1$ and $\hat x_2$ respectively. </p>
<p><strong>But the state $|x_1,x_2\rangle$ is eigenstate of both $\hat x_{\pm}$.</strong> </p>
<blockquote>
<p><strong>How can we construct unique eigenstates of the operator $\hat x_{\pm}$ in the basis $|x_1,x_2\rangle$? Or is it that it is not possible?</strong></p>
</blockquote> | g13720 | [
-0.011186769232153893,
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-0.0... |
<p>If someone has short or long sight, is it possible to tune image on a computer monitor in such way, that a person could see it sharp as if they were wearing glasses? If not, will 3d monitor make it possible?</p> | g486 | [
-0.005836624186486006,
0.02421623095870018,
0.007774458732455969,
0.009157627820968628,
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0.02084922231733799,
-0.013191056437790394,
0.04842570051550865,
0.06396448612213135,
0.037183091044425964,
... |
<p>Reynolds-averaged Navier–Stokes equations (RANS) is one of the approaches to turbulence description. Physical quantities, like for example velocity $u_i$, are represented as a sum of a mean and a fluctuating part:</p>
<p>$$
u_i = \overline{u_i} + u'_i
$$</p>
<p>where the Reynolds averaging operator $\overline{\cdot}$ satisfies, among the others, relations:
$$
\overline{\overline{u_i}} = \overline{u_i}, \qquad \overline{u'_i} = 0
$$
which distinguish it from other types of averaging. In fluid dynamics Reynolds operator is usually interpreted as time averaging:
$$
\overline{u_i} = \lim_{T \to \infty} \frac{1}{T}\int_t^{t+T} u_i \,dt
$$</p>
<p>The above construction seems to be universal for me and is likely to be used in other areas of physics. Where else does one encounter Reynolds averaging?</p> | g13721 | [
-0.03294799104332924,
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0.0440763495862484,
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0.0084538534283638,
0.020865... |
<p>I was told that if a plane takes less time to travel from the China to US as opposed to the other direction due to rotation of the earth. I suspect this is incorrect however. </p>
<p>From this scenario I came up with the following thought experiment. Imagine two people sitting on a train. One person is sitting with his back (A) facing the front of the train and the other is facing toward the front of the train (B). They are throwing a ball to one another. They pass by an observer who is standing outside as the train passes (C). How does each person view the ball?</p>
<p>My intuition tells me that (A) and (B) should perceive the speed of the ball the same while (C) should see the ball move faster towards (B) than (A). However some people have told me that time taken for the ball to reach (B) when thrown by (A) should be less than the time taken for the ball to reach (A) when thrown by (B) due to the relative velocity of the speed of the ball and the train. Is this true?</p> | g13722 | [
0.0362178310751915,
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0.03493158146739006,
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0.04250907897949219,
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0.04847070574760437,
-0.00398... |
<p>I have to fit some data to a power law</p>
<p>$$ F=\alpha q^{\beta}$$</p>
<p>being $q$ and $F$ the experimental data points. What I usually do is taking logs so that</p>
<p>$$ \ln(F) = \beta \ln(q)+\ln(\alpha)$$</p>
<p>and apply least squares with uncertainties in both variables. How may I approach this in the case that some of the values of $q$ (and eventually $F$) are negative?. If it helps $\beta$ should be compatible with a value of $1$, but I want a general approach.</p>
<p>Edit:</p>
<p>The law to be fitted is Coulomb's law, $F$ is the force between to charged spheres. One of them has fixed charge, the other is a variable we call $q$. The proportionality constant $\alpha$ is related to $\varepsilon_0$ with a known expression in powers of $\beta=\frac{R}{d}$ where $d$ is the distance between the centers of the spheres and $R$ is their radius. So, I want to determine, from experiment, both $\alpha$ and $\beta$ and compare them with $\varepsilon_0$ and $1$.</p>
<p>The actual values are </p>
<p>$$
\begin{array}{|c|c|}
\hline
q / \mathrm{nC} & F / \mathrm{mN} \\ \hline
-26,8 \pm 0.8 & -1.5 \\ \hline
-18.8 \pm 0.5 & -1.1 \\ \hline
-9.9 \pm 0.2 & -0.6 \\ \hline
9.9 \pm 0.2 & 0.5 \\ \hline
18.8 \pm 0.5 & 1.0 \\ \hline
26.8 \pm 0.8 & 1.5 \\ \hline
35 \pm 2 & 2.1 \\ \hline
\end{array}
$$</p>
<p>with the uncertainty in the last decimal place for the forces.</p> | g13723 | [
0.08496087789535522,
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0.07928714156150818,
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0.003739491803571582,
0.02233249507844448,
-0.0021951082162559032,
-0.018897397443652153,
... |
<p>A recent <a href="http://what-if.xkcd.com/7/" rel="nofollow">XKCD What-if</a> article mentions the situation where each additional kilogram of cargo to LEO requires an additional 1. 3 kilograms of fuel, which in turn requires fuel to carry (simplification). I wonder then, if an additional half-kilo object, <a href="http://www.blogcdn.com/www.tuaw.com/media/2008/03/ipodcloseup.jpg" rel="nofollow">such as an iPod</a> will have a large impact on the amount of fuel that engineers will load into the ET. Is the ET, and other rockets, always 'topped off' regardless of payload, or is the amount of fuel loaded carefully measured? What tolerances are acceptable? <strong>How much excess fuel is typically in the rocket / tank when the payload is separated?</strong></p> | g13724 | [
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0.024786055088043213,
0.006251493003219366,
-0.09568246454000473,
0.002413678215816617,
-0... |
<p>Suppose there is a particle in space. When we measure the position of that particle, we get a particular value with a probability that can be calculated from the wave function. But, according to the <a href="https://en.wikipedia.org/wiki/Copenhagen_interpretation" rel="nofollow">Copenhagen interpretation</a>, where was the particle before the measurement? Was it in a superposition of all possible positions?</p> | g13725 | [
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0.024139421060681343,
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0.03322922810912132,
-0.005... |
<p>I do not understand how any state in Hilbert Space $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$ of dimension $\text{dim}(\mathcal{H}_A)\times\text{dim}(\mathcal{H}_B)$ can be decomposed in the Schmidt basis when the Schmidt basis has size $\text{min}\{\text{dim}(\mathcal{H}_A),\text{dim}(\mathcal{H}_A)$}.</p>
<p>Thank you in anticipation of your help.</p> | g13726 | [
0.011296250857412815,
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-0.03973862901329994,
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-0.006678338162600994,
... |
<p>I had never seen this system until today, and I'm really confused. I've read the wikipedia article about it but I still don't know how to change between this and the international system. For example, studying emision of radiation of an electric dipole: $\wp$, one gets to an equation for the magnetic potential given, in Gauss' system, by:</p>
<p>$$A=\frac{1}{cr}\dot\wp$$</p>
<p>How could I change that? I don't know how to deal with all the $4\pi$ that appear over there, and how are $\mu$ and $\epsilon$ constants omitted.</p>
<p>Thanks</p>
<p>EDIT: From the table at that <a href="http://en.wikipedia.org/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units#Various_extensions_of_the_CGS_system_to_electromagnetism" rel="nofollow">wiki article</a> I can just change the $c^{-1}$ by $\mu_0/4\pi$, but given that gaussian system just changes a lot of constant by 1, I have doubts about that.</p> | g13727 | [
-0.03440086171030998,
0.03794465214014053,
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-0.0466042086482048,
0.03750678151845932,
0.026207974180579185,
0.016266394406557083,
0.01955713890492916,
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0.028755132108926773,
-0.013721124269068241,
0.08136247098445892,
0.05379100516438484,
-0.0105... |
<p>When I go to, for example, a museum I try to take some pictures.</p>
<p>Sometimes the museum staffs forbid me to use a flash. Do you know the reason? I don't think it is related to photo-electric effect, right?</p> | g13728 | [
-0.03718378022313118,
0.07782673090696335,
-0.007924909703433514,
0.03604112192988396,
0.03708955645561218,
0.0009193560108542442,
0.0715036541223526,
0.06231960654258728,
0.00735310697928071,
-0.026363223791122437,
-0.0024257434997707605,
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-0.0051326751708984375,
0.0... |
<p>I'm reading this book "Introduction to Quantum Fields in Classical Backgrounds" by Mukhanov & Winitzki, and there in the chapter 8 "The Unruh Effect" they introduce 3 reference frames.</p>
<p><strong>Laboratory Frame</strong>: "is the usual inertial reference frame with the coordinates $(t,x,y,z)$".</p>
<p><strong>Proper Frame</strong>: "is the accelerated system of reference that moves together with the observer, we shall also call it the accelerated frame".</p>
<p><strong>Comoving Frame</strong>: "defined at a time $t_0$ is the <em>inertial</em> frame in which the accelerated observer is instantaneously at rest at $t=t_0$. (Thus the term 'comoving frame' actually refers to a different frame for each $t_0$)".</p>
<p>Now, I don't understand why they say that the observer's proper acceleration at time $t=t_0$ is the 3-acceleration measured in the comoving frame at time $t_0$. Could you explain why? Also I don't understand completely the definition of comoving frame.</p> | g13729 | [
0.04538590833544731,
0.02123563177883625,
0.00858728401362896,
-0.014264213852584362,
0.07042364776134491,
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0.049547914415597916,
0.060672540217638016,
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0.014850911684334278,
0.0032297573052346706,
0.0022619247902184725,
-0.015230998396873474,
0.007... |
<p>Newton's Third Law states that <em>Whenever any force is exerted by a body#1 on any other body#2, another force which is equal in magnitude and opposite in direction is exerted on body#1 by body#2.</em> In theory, suppose a boy is pushing a rock heavier than him on ice. Due to the relative lack of friction, the boy would probably be pushed back more than the rock was pushed forward. How is this possible? How can an inanimate object create a force?</p> | g13730 | [
0.025428378954529762,
0.07772503048181534,
0.022844675928354263,
-0.033118050545454025,
0.08752370625734329,
0.07737643271684647,
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0.021319998428225517,
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0.00022287220053840429,
-0.06510494649410248,
-0.035625360906124115,
-0.007239006459712982,
-... |
<p>Suppose one has a circuit consisting of an inductor $L$ and resistor $R$ in series where $L$ and $R$ are known, passes an alternating voltage of frequency $\omega$ through it and that one wishes to calculate the mean power dissipated in the resistor.</p>
<p>Let the RMS voltage across the series combination be $V_0$. Then the RMS current through the components will be $I=\frac{V_0}{i\omega L + R}$ and the mean power dissipated in $R_2$ will be $\overline{P}=I^2R$. However, at this point, $I$ involves a complex quantity. How do you calculate the mean power? Do you calculate the magnitude of the complex current?</p>
<p>With very many thanks,</p>
<p>Froskoy.</p> | g13731 | [
0.002464671153575182,
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0.0015239209169521928,
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0.0021897493861615658,
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0.024022838100790977,
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0.02046389691531658,
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0.07170843333005905,
0.021891644224524498,
0.... |
<p>Found this at the gas station yesteday - got me thinking...</p>
<p><img src="http://i.stack.imgur.com/51Kz3.jpg" alt="enter image description here"></p> | g13732 | [
-0.028355011716485023,
-0.014844224788248539,
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-0.01774662733078003,
0.02932014688849449,
-0.053155601024627686,
0.0018728218274191022,
-... |
<p>While studying about band theory of semiconductors, I observed that when the electrons were excited from the valence band to the conduction band, they left behind holes in the valence band. From my existing knowledge, I believe that the valence electrons alone occupy the valence band which tells me the valence band is negative. For the holes to exist, the valence band has to be neutral. So, why is the valence band neutral? </p>
<p>If my reason for the valence band to be negative was wrong, I would like to know the reason. </p> | g13733 | [
0.02115766704082489,
0.01978110522031784,
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0.03620617091655731,
0.07931942492723465,
0.018685689195990562,
0.02354886382818222,
0.07141561806201935,
-0.007394794840365648,
0.02560693398118019,
0.06069489195942879,
-0.0068781147710978985,
0.00957... |
<p>I'm not very knowledgeable about physics generally, but know that nothing can escape a black hole's gravitational pull, not even light (making them nearly invisible?). </p>
<p>My question is: <strong>What has been obtained from direct observation of a black hole to prove their existence?</strong> Please note that I'm not questioning the existence of black holes, but am just curious as to how we have come to realize it.</p> | g16 | [
0.011284894309937954,
0.026313282549381256,
0.024534771218895912,
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0.05347450077533722,
0.030932648107409477,
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0.0392390675842762,
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-0.029807841405272484,
0.008526922203600407,
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0.049814604222774506,
0... |
<p>In <a href="http://davidsenouf.com/wp-content/uploads/2012/01/The-quaternion-group-and-modern-physics-Girard.pdf">The Quaternion Group and Modern Physics by P.R. Girard</a>, the quaternion form of the general relativistic equation of motion is derived from</p>
<p>$du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / d s ) + a (du/ds ) {a_c}^*$</p>
<p>as</p>
<p>$du/ds = - [ ( a_c \frac{da}{ds})u - u ( a_c \frac{da}{ds})^*]$. </p>
<p>where $u$ is a "minquat" of the form $(ct,ix,iy,iz)$ and a is an arbitrary function satisfying $a{a_c}=1$</p>
<p>This is said to "correspond" to the equation of motion of GR. It is an incredibly elegant formulation of the equation.</p>
<p>I've tried applying this formula to $a=\cos(s)-i\sin(s) \hat{i}$ and, unless I've made some errors-and I probably did, this results in $x(1+\cos(2s))+ ict(1+\sin(2s))$ for the first two terms.</p>
<p>1.What is the physical significance of these terms?</p>
<p>2.Does this really represent the GR equations of motion?</p>
<p>*3. Does the set of arbitrary functions satisfying $a{a_c}=1$ represent a general relativity symmetry group?</p>
<p>4.What other functions satisfying the condition $a{a_c}=1$ should I use? (I've also used $e^{i\theta}$ and this switched $x$ and $ct$ which is interesting, but-again-what does it actually represent?) </p>
<p>5.Can I use this equation to derive any standard results? I would like to see the centrifugal and other fictitious forces come out of it maybe.</p>
<p>I've found that the Runge Lenz vector is used to calculate orbits in Newtonian Gravity and GR, but I'm baffled by the introduction of this vector in this case.</p>
<p>Can it be translated into a standard formulation of the equations of motion?</p>
<p>Any information or advice for tackling this would be appreciated.</p> | g13734 | [
-0.038254786282777786,
0.03090083599090576,
-0.0062530203722417355,
-0.0032147683668881655,
0.05561678111553192,
0.02430236153304577,
0.07042521238327026,
-0.04639240354299545,
-0.031504690647125244,
-0.029991107061505318,
0.005239980295300484,
-0.02931654080748558,
0.03497406095266342,
-0... |
<p><strong>Introduction</strong></p>
<p>I have the following data for a galaxy: apparent magnitude in $U,B$ and $V$ Johnson photometric system:</p>
<p>$$m_{U}=17.51 \\m_{B}=17.08 \\m_{V}=16.74 $$</p>
<p>The galaxy is such that the galactic hydrogen column has $N(H)=6 \cdot10^{22} \mbox{ cm}^{-2}$. I have to take into account galactic extinction, $A_{J}$ with $J=U,B,V$ and this is the work so far:</p>
<p>$$m^{0}_{J}=m_{J}-A_{J}$$</p>
<p>where $m^{0}_{J}$ is the magnitude without dust. I have the following formula for the colour excess</p>
<p>$$E(B-V)=-0.055+1.986\cdot10^{-22}N(H)[\mbox{ cm}^{-2}]=0.0642 $$</p>
<p>Also, it can be written in term of the galactic extinctions</p>
<p>$$E(B-V)=A_{B}-A_{V} $$</p>
<p>and the following hint</p>
<p>$$ \boxed{\displaystyle\frac{A_{V}}{E(B-V)} \approx 3.1}$$</p>
<p>which implies</p>
<p>$$ A_{V}=0.199, A_{B}=0.2632$$</p>
<p>Now another given hint is that:</p>
<p>$$ \boxed{\displaystyle\frac{E(U-V)}{E(B-V)} \approx 1.6}$$</p>
<p>therefore </p>
<p>$$A_{U}=0.3017$$</p>
<p>and I have $m^{0}_{J}$. Now the flux $f_{J}$ is related to the apparent magnitude via</p>
<p>$$ m^{0}_{J}=2.5 \log (f_{J} (m=0))-2.5 \log (f_{J})$$</p>
<p><strong>Question</strong></p>
<p>How can I calculate $f_{j}$ such that $m=0$, or in general $f_{J}$?</p>
<p>The numerical answer is</p>
<p>$$ f_{U_{0}}=2.32 \cdot 10^{-4} \mbox{ Jy} \\ f_{B_{0}}=7.48 \cdot 10^{-4} \mbox{ Jy} \\ f_{V_{0}}=8.72 \cdot 10^{-4} \mbox{ Jy}$$</p>
<p>but I can't get that result.</p> | g13735 | [
0.01115375105291605,
-0.047468941658735275,
-0.010753818787634373,
-0.04661636799573898,
0.010688086040318012,
0.002645617350935936,
0.012306608259677887,
-0.004295384045690298,
-0.02112676203250885,
0.015486915595829487,
-0.017164869233965874,
0.06870248913764954,
0.041954994201660156,
0.... |
<p>I'm reading about the second quantization formalism. I can see the advantages of using number states to represent multiparticle states. Here's my question:</p>
<p>Let's say we're given a single-particle basis of plane waves with spins (so each particle can be described by some wavevector $k$ and some spin $\sigma$). Extending this formalism to $N$ such particles and writing this in occupation number representation, one could write states like $|n_1, n_2, \ldots, n_i,\ldots\rangle$. In doing so, what does each number $n_i$ mean? Does it mean, for example, that there are $n_i$ particles in some state $\{k_i, \sigma_i\}$? If that's the case, then supposedly we have enumerated as many $k$ and $\sigma$ combinations as could possibly exist, and only put non-zero values for those that describe any of the $N$ particles. Or is there another interpretation that I'm missing?</p>
<p>Any references that clarify this material are also welcome. Most books quickly skip from single-particle wavefunctions to those of many particles by saying something along the lines of "each $n_i$ means there are $n_i$ particles in state $i$," but I'm not sure what exactly this state describes.</p> | g13736 | [
0.008755505084991455,
0.04864059016108513,
-0.011372855864465237,
-0.03885367140173912,
0.05153634399175644,
-0.018165186047554016,
0.008339515887200832,
0.03709839656949043,
0.01928398758172989,
-0.0076920706778764725,
-0.05223527178168297,
-0.01110424567013979,
0.01978437975049019,
0.013... |
<p>In a nucleus, the strong nuclear force causes interactions between protons and protons, between neutrons and neutrons, and between protons and neutrons. What do we know about this interaction?</p>
<p>Related:</p>
<ul>
<li><a href="http://physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit">Why don't electrons crash into the nuclei they "orbit"?</a> </li>
<li><a href="http://mathoverflow.net/questions/119495/mathematical-proof-of-the-stability-of-atoms/119528">Mathematical proof of the stability of atoms</a> </li>
<li><a href="http://physics.stackexchange.com/questions/78078/why-do-nucleons-feel-a-repulsive-force-when-less-than-1-fm">Why do nucleons feel a repulsive force when less than 1 fm?</a> </li>
<li><a href="http://physics.stackexchange.com/questions/75403/what-is-the-the-ehrenfest-oppenheimer-rule-on-the-statistics-of-composite-system">What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?</a></li>
</ul> | g13737 | [
0.029040303081274033,
0.08566984534263611,
-0.024378066882491112,
-0.02110004611313343,
0.09126269072294235,
0.021967537701129913,
-0.03126314654946327,
0.025236789137125015,
-0.006232847925275564,
-0.07023277133703232,
-0.01842663623392582,
-0.00386825785972178,
-0.04771639406681061,
0.02... |
<ol>
<li><p>In spin ice systems magnetic monopole-like excitations are sources or sinks of $H$, not the $B$ field, why is that? Is it because the strings carries magnetic moment $M$ and not solenoidal $B$ filed like the Dirac string.</p></li>
<li><p>How is it different from Dirac construction of magnetic monopole?</p></li>
<li><p>In the Dirac case we still have incoming flux from the solenoid and out going from the monopole that will result in 0 for any sphere around the end. How he goes around this (why is it justified to not count the singularity) and why is not possible for the spin ice case?</p></li>
</ol> | g13738 | [
0.016214504837989807,
0.059960465878248215,
-0.013463149778544903,
-0.029036087915301323,
0.08266809582710266,
0.03184511885046959,
0.057369790971279144,
0.051585473120212555,
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-0.006715833675116301,
-0.08590057492256165,
0.00895311962813139,
0.0004146837454754859,
0.01... |
<p>My question is specifically about how to use sources? For an interacting theory with one field, one puts a $J(x)\phi(x)$ term in the exponential in the path integral for $W[J]$. I now have two different fields ($\phi_1(x)$ and $\phi_2(x)$) and a number of interactions in the Lagrangian involving a combination of fields up to fourth order. To calculate the appropriate Green's functions and Feynman rules, should I use two different sources so that I have terms $J_1(x)\phi_1(x) + J_2(x)\phi_2(x)$?</p>
<p>Hope someone can help!</p> | g13739 | [
0.04578644037246704,
0.022429609671235085,
0.0005699293105863035,
-0.02686030976474285,
0.0346810519695282,
-0.014599681831896305,
0.011142815463244915,
0.008423597551882267,
-0.03259877488017082,
0.02411261945962906,
-0.01706209033727646,
0.019055200740695,
0.013553695753216743,
0.0259237... |
<blockquote>
<p><em>Suppose we have two masses, $m_1$ and $m_2$, where $m_2$ is at rest, and $m_1$ is headed directly towards $m_2$. I would like to write the ratio of the masses as a function of the angle.</em> </p>
</blockquote>
<p>Using Conservation of Energy, Conservation of Momentum, and applying the Law of Cosines to the the triangle that the initial momentum vector and final momenta vectors form, I obtain these three equations:</p>
<p>(1) $ m_1^2 v_1^2 = m_1^2 v_1^2~' + m_2^2 v_2^2~' - (m_1 v_1)(m_2 v_2) \cos \theta$ (Law of Cosines)</p>
<p>(2) $m_1 v_1^2 = m_1 v_1^2~' + m_2 v_2^2~'$ (Cons. of Energy)</p>
<p>(3) $m_1 \mathbf{v}_1 = m_1 \mathbf{v}_1~' + m_2 \mathbf{v}_2~' $ (Cons. of Mom.)</p>
<p>Taking the magnitude of the Conservation of Momentum equation, and squaring the result, yields</p>
<p>(4) $m_1^2 v_1^2 = m_1^2 v_1^2~' + m^2 v_2^2~' + (m_1 m_2)(v_1~' v_2~') \cos \theta$ </p>
<p>I would like to try to eliminate the variables $v_1~'$ and $v_2~'$, seeing as in a experimental setting they might be a little more difficult to measure; however, I would like to keep the variable $v_1$, because it is a little more easy to determine--suppose the moving particle is an electron, then $m_1$ is known, and $v_1$ can be easily determine by knowing the accelerating voltage.</p>
<p>I was hoping that someone might guide me towards the correct path, I certainly would appreciate it.</p> | g429 | [
0.036896221339702606,
-0.05477968230843544,
0.0040188925340771675,
-0.004969235975295305,
0.05194932967424393,
-0.013320728205144405,
0.014560936018824577,
0.005363804753869772,
-0.03260987624526024,
0.07205887883901596,
0.05268722027540207,
0.02168716862797737,
0.020614685490727425,
0.010... |
<p>In the hierarchy of theories, first comes hamiltonian theory, from which one deduces kinetics theory, and at last thermodynamics and fluid theories. From a kinetics point of view, entropy and temperature are macroscopic parameters with no equivalent at the microscopic level. However, thermodynamical entropy and information entropy being isomorph, there has been a general trend to generalize thermodynamical properties to non Hamiltonian systems and use entropy as a fundamental property.
Prigogine had tried at the end of its life to give entropy a fundamental status and to redefine hamiltonian mechanics, starting from entropy. Does anyone know if there has been any successful attempts in this direction ?</p> | g13740 | [
0.016861334443092346,
0.009360001422464848,
-0.015814129263162613,
-0.02561364509165287,
0.008420178666710854,
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0.016115253791213036,
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0.047874536365270615,
0.024741098284721375,
0.02353096939623356,
0.017206314951181412,
0.0... |
<p>Two very intriguing papers recently appeared on the arXiv, claiming that one can use "superqubits" -- a supersymmetric generalization of qubits -- to violate the Bell inequality by more than standard quantum mechanics would allow. (That is, they claim one can violate the Tsirelson bound, which says that the CHSH game can be won quantum-mechanically with probability at most cos<sup>2</sup>(π/8) ~ 0.85.) The first paper is by <a href="http://arxiv.org/abs/1206.6934">Borsten, Bradler, and Duff</a> and the second is by <a href="http://arxiv.org/abs/1208.2978">Bradler</a>.</p>
<p>Alas, I remain deeply confused about the physical meaning of these results, if any. As the authors define them, "superqubits" seem to involve amplitudes that can be Grassmann numbers rather than just complex numbers. While I know next to nothing about the topic, that <i>seems</i> like a fundamental departure from "supersymmetry" in the sense that high-energy physicists use the term! I take it that supersymmetry is "just" a proposed new symmetry of nature, alongside the many other symmetries we know, and doesn't involve tampering with the basic rules of quantum mechanics (or with spatial locality). In particular, in supersymmetric theories one still has unit vectors in a complex Hilbert space, unitary transformations, etc.</p>
<p>If that's correct, though, then what on earth could superqubits have to do with supersymmetry in physics---besides perhaps just repurposing some of the same mathematical structures in a totally different context? Is there <i>any</i> possibility that, if nature were supersymmetric in such-and-such a way, then one could do an actual experiment that would violate Tsirelson's bound?</p> | g13741 | [
0.02947697974741459,
0.09123172610998154,
0.017008332535624504,
0.03085051104426384,
0.015399216674268246,
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0.00017457817739341408,
0.015827476978302002,
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-0.024889519438147545,
-0.03308229148387909,
-0.03149021789431572,
-0.023581279441714287,
0... |
<p>Suppose you have an object with zero for the value of all the derivatives of position. In order to get the object moving you would need to increase the value of the velocity from zero to some finite value. A change is velocity is acceleration, so the value of the acceleration would have to increase from zero to some value. You would also need to increase the jerk from zero to some value to have a change in acceleration. Is there an infinite series of higher derivatives of position for this to work? Or am I missing something?</p> | g13742 | [
-0.0031573737505823374,
0.07017513364553452,
0.00356824672780931,
-0.03180958703160286,
0.03331374749541283,
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0.03607305884361267,
0.05632852017879486,
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-0.04567703232169151,
-0.08996494859457016,
-0.0037023562472313643,
0.06081153079867363,
0.008... |
<p>I am attempting to estimate the elasto-optic coefficients ($p_{11}$ and $p_{12}$) of $\mathrm{TiO}_2$ and $\mathrm{ZrO}_2$, where $p_{11}$ and $p_{12}$ refer to the elements of a strain-optic tensor for a homogeneous material as given in Hocker (Fiber-optic sensing of pressure and temperature, 1979).</p>
<p>I have <a href="http://www.usna.edu/Users/physics/mungan/Publications/SBSreview.pdf">found a document</a> which specifies that the longitudinal elasto-optic coefficient ($p_{12}$) can be estimated using the Lorentz-Lorenz relation that it gives as</p>
<p>$$p_{12} = \frac{(n^2 - 1)(n^2 + 2)}{3n^4}$$</p>
<p>however no reference is given, and other sources give the Lorentz-Lorenz relation as something rather different. For example <a href="http://en.wikipedia.org/wiki/Lorentz%E2%80%93Lorenz_equation">Wikipedia</a> says that the equation relates the refractive index of a substance to its polarizability and gives it as</p>
<p>$$\frac{n^2 - 1}{n^2 + 2} = \frac{4\pi}{3}N\alpha$$</p>
<p>which bares only a vague relation to the earlier equation. </p>
<p>Does anyone know of any other ways in which to estimate the elasto-optic coefficients of a material?</p> | g13743 | [
-0.03174138441681862,
-0.045585259795188904,
-0.0011882686521857977,
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0.042895253747701645,
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0.01590328849852085,
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0.02065405808389187,
0.0880470871925354,
-0.004196864552795887,
0.02258492261171341,
0.024291228502988815,
-0... |
<p>What is the sign of the <a href="http://en.wikipedia.org/wiki/Work_%28thermodynamics%29#Formal_definition" rel="nofollow">work</a> done on the system and by the system?</p>
<p>My chemistry course book says, when work is done on the systems, it is taken positive. When work is done by the system, it is taken as negative.</p>
<p>My physics course book says quite the opposite. It says that when work is done on the system, it is taken negative. When work is done by the system, it is taken as positive.</p>
<p>Why do they differ? Are they taking it in different sense?</p> | g13744 | [
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0.041776612401008606,
0.01... |
<p>I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the <a href="http://iopscience.iop.org/1742-5468/2004/06/P06002/fulltext/" rel="nofollow">entanglement entropy of 1+1D quantum field theory</a> (<a href="http://arxiv.org/abs/hep-th/0405152" rel="nofollow">arXiv</a>). The identity is given below:</p>
<p>$$
\int^\infty_0 x I_{\alpha}(x)K_{\alpha}(x) dx \,\, ``=" \frac{\alpha}{2}.
$$</p>
<p>Here $I_\alpha(x)$ and $K_{\alpha}(x)$ are the modified Bessel functions of the fist and second kind. The above identity is implicitly used in Eq.4.19 and Eq.4.22 of the said paper. However, the above identity looks false to me as the left hand side is apparently divergent. Recall that we have the following asymptotic expansion:</p>
<p>$$
I_\alpha (x) \sim \frac{e^{x}}{\sqrt{2\pi x}};\\
K_\alpha (x) \sim \sqrt{\frac{\pi}{2 x}}e^{-x},
$$</p>
<p>from which we obtain</p>
<p>$$
xI_\alpha(x)K_\alpha(x) \sim \frac{1}{2}.
$$</p>
<p>I believe that some sort of regularization has been done on the integral or I simply miss something. I hope someone here could help me understanding this identity.</p>
<p>EDIT: I think the following might be a useful hint.</p>
<p>The above integral rises from the expression of $\int^\infty_0 rG(r,r)dr$, where $r$ is the radial coordinate and $G$ is the usual Green's function (see Eq.4.18). Now let us consider $\int^\infty_0 G((1+\epsilon)r,r)rdr$, motiviated by the short distance expansion. The corresponding integral is:</p>
<p>$$
\int^\infty_0 x I_{\alpha}((1+\epsilon)x)K_{\alpha}(x) dx = \frac{(1+\epsilon)^{-\alpha}} {(1+\epsilon)^2-1} = \frac{1}{2\epsilon}-\frac{1}{2}(\alpha+\frac{1}{2})+O(\epsilon).
$$</p>
<p>The above identity can be found in Gradshteyn & Ryzhik's table, 7th Edition, entries (6.521.2), (6.521.8), and (6.521.9).</p>
<p>Now we take the limit $\epsilon\to 0$. The first term of the above is divergent, which could be interpreted as the usual UV divergence part. The second term, however, is finite. If somehow we can argue that it is meaningful to drop the first term and take the $\alpha$-dependent as the "physical" answer of the integral, then we are done.</p> | g13745 | [
0.01228413451462984,
0.004744038917124271,
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-0.052313514053821564,
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0.00800047256052494,
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-0.0036216480657458305,
-0.0798838883638382,
-0.009560124948620796,
-0.02967030555009842,
0.... |
<p>We have this formula for centripetal acceleration -
$a = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r}$</p>
<p>but in case of <em>usual</em> acceleration i know that speed in $t_1 = v_0 + a \cdot (t_1 - t_0)$</p>
<p>but in circular case i don't understande the nature of acceleration, speed is always the same if motion is uniform, but acceleration is not zero.</p>
<p>EDIT:
Suppose T is 1. $t_0 = 0, t_1 = .25$, so $v$ was rotated for $\frac{\pi}{2}$. So $\frac{\bar{a}}{4} = \bar{v_1} - \bar{v_0}$ and $|a| = 4\sqrt{v^2 + v^2} = 4\sqrt{2} v$, but with previous formula magnitude of a is $\frac{v^2}{4r}$
.</p>
<p>ANSWER:
physical meaning of $a$ is the length of an arc swept out by velocity vector!</p> | g13746 | [
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0.011006995104253292,
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0.02107321284711361,
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0.012793024070560932,
0.007627048995345831,
0.03307740390300751,
-0.0002499626134522259,
-0.02... |
<p>Let´s consider a curved universe that is very small, say 20 square meters and not expanding. </p>
<p>If you stood at the middle of this tiny universe and looked forward, you wouldn´t see any walls, since it is curved. But you would see your back.
If you looked left you would see your right side of your body, and when looking right you would see the left side of you body. </p>
<p>Is that how it would be?</p>
<p>Or would the photons that your body emits be all over the space in the room, so you would consider yourself to be disseminated all over the room so it would be impossible to see the shape of your own body?</p> | g13747 | [
0.003202544990926981,
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0.01388526801019907,
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-0.08079694956541061,
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0.04797809198498726,
0.03568830341100693,
0.01598667... |
<p>Apparently, in Los Alamos scientists handled sub critical masses of plutonium (for example the <a href="http://en.wikipedia.org/wiki/Demon_core" rel="nofollow">demon core</a>) with little or no protection. Richard Feynman and others mentioned that plutonium spheres were warm to the touch, which seems to imply that they actually touched them. Slotin (one of the victims of the demon core) performed experiments wearing blue jeans and cowboy boots.</p>
<p>Are certain plutonium isotopes safe to handle (as long as they don't go critical) or is it just that people back then didn't know any better?</p> | g13748 | [
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0.04002669081091881,
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0.008873521350324154,
0.0323803685605526,
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0.01710481569170952,
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-0.01124912966042757,
0.0023116343654692173,
0.0030628154054284096,
-0.007205502595752478,
-0.0... |
<p>Here's my problem and the work I've done. The time is already past for me to submit the answer, but I want to know where I went wrong and why I was wrong.</p>
<p>The 2-kg box slides down a vertical wall while you push on it at a 45 degree angle from below. Both the box and the wall are wood. What magnitude of force should you apply to cause the box to slide down at a constant speed? </p>
<p>The coefficient of kinetic friction for wood-wood is 0.2.</p>
<p>The vertical forces acting on the box are:</p>
<p>$F_{box}$(sin 45) - $F_{friction} - mg$ = 0</p>
<p>where</p>
<p>$F_{box}$ = force acting on the box</p>
<p>$F_{friction}$ = frictional force opposing the motion</p>
<p>$m$ = mass of the box</p>
<p>$g$ = acceleration due to gravity</p>
<p>Hence</p>
<p>$F_{friction} = F_{box}$(sin 45) - $mg$ --- call this Equation 1</p>
<p>The normal force acting on the box is as follows:</p>
<p>$F_{box}$(cos 45) = $F_n$</p>
<p>and since</p>
<p>$F_{friction}$ = µ$F_n$, then</p>
<p>$F_{friction}$ = µ($F_{box}$)cos 45 --- call this Equation 2</p>
<p>Setting Equation 1 = Equation 2,</p>
<p>$F_{box}$(cos 45) - $mg$ = µ($F_{box}$)cos 45</p>
<p>Simplifying the above for "$F_{box}$"</p>
<p>$F_{box}$(cos 45) - $F_{box}$(µ*cos 45) = $mg$</p>
<p>$F_{box}$(sin 45 - µcos 45) = $mg$</p>
<p>and solving for "$F_{box}$"</p>
<p>$F_{box}$ = $\frac{mg}{cos 45 - µcos 45}$</p>
<p>Substituting appropriate values and calculating for "$F_{box}$"</p>
<p>$F_{box}$ = 34.65N </p>
<p>The system says that the solution is 23N. How did they get that and where is my mistake?</p> | g13749 | [
0.0887921079993248,
0.06663324683904648,
-0.008145016618072987,
0.011833187192678452,
0.049397315829992294,
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0.07869260758161545,
0.011985443532466888,
-0.09185260534286499,
-0.044560614973306656,
-0.03344603255391121,
0.016794485971331596,
-0.02346644550561905,
0.008... |
<p>Almost all optical phenomenon can be explained considering a fluctuating electric field. Is there any optical phenomenon which can't be explained without considering two fluctuating fields, electric and magnetic?</p> | g13750 | [
0.032523564994335175,
0.029171478003263474,
0.0004377492587082088,
0.04205378144979477,
0.05539673939347267,
0.017784245312213898,
0.0222998708486557,
0.03203576058149338,
0.0585218146443367,
-0.014732442796230316,
0.009752757847309113,
0.0131591921672225,
0.030152950435876846,
0.033417563... |
<p><img src="http://i.stack.imgur.com/3pHzS.png" alt="enter image description here"></p>
<p>I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does "correspond" mean? What is the exact definition of $|X_{a}\rangle$? Thanks in advance for any help.</p> | g13751 | [
-0.0006703998660668731,
-0.007108771707862616,
-0.010097290389239788,
-0.06350241601467133,
0.025390852242708206,
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0.030144285410642624,
0.07493548095226288,
-0.018594015389680862,
-0.014250378124415874,
-0.0014763911021873355,
0.009552115574479103,
-0.00289760273881256... |
<p>In the minimal-GMSB model, the messenger fields transform under the MSSM gauge group and connect a so-called hidden sector to the visible sector. These meesenger fields (left-handed chiral supermultiplets in this case) break SUSY and introduce the soft terms in the MSSM. I do not understand how to integrate out these fields in the superspace formalism. How do the gauginos and scalars in the MSSM gain masses? The superpotential was,
$W= W_{breaking} + W_{mess} + W_{MSSM}$ ,
$W_{mess} = y_{l}Sl\bar{l} + y_{q}Sq\bar{q}.$
I do not see a coupling between the MSSM fields and the messenger fields. It happens by 1-loop and 2-loop contribution is what every book says but I am unable to see this. Can someone point out a reference if not a direct explanation of how to integrate out the messenger fields?</p> | g13752 | [
-0.010680494830012321,
0.005335134919732809,
-0.012263156473636627,
-0.020853862166404724,
-0.006123825442045927,
0.07514440268278122,
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0.028041014447808266,
0.05697016417980194,
-0.038739971816539764,
-0.05405566468834877,
-0.015453958883881569,
0.0015137047739699483,... |
<p>The question might sound elementary. However, the thing is this. Changing Magnetic field produces a changing electric field. That changing electric field would produce a changing magnetic field. So, is it apt to say that changing magnetic field produces an electric field and a magnetic field? Or should we just stop at saying that it produces an electric field?</p> | g13753 | [
0.048858512192964554,
0.027624323964118958,
0.01920928619801998,
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0.05106740817427635,
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0.004489122424274683,
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0.03858323395252228,
-0.0... |
<p>E-L condition:</p>
<p>$$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$</p>
<p>Where $p=\frac{\partial L}{\partial \dot{q}}$</p>
<p>Are the following steps valid:</p>
<p>$$\frac{\partial q}{dt} dp=\partial L$$</p>
<p>$$\dot{q} \: dp = \partial L$$</p>
<p>$$ \int \dot{q} \: dp = L+C $$</p>
<p>By integration by parts the LHS becomes:</p>
<p>$$ \int \dot{q} \: dp = \dot{q} p-\int p \: d \dot{q} = \dot{q} p- \int \frac{\partial L}{\partial \dot{q}} \:d \dot{q}=\dot{q} p-L+C_1$$</p>
<p>Substituting this back into the LHS:</p>
<p>$$\dot{q} p-L=L+C_2$$</p>
<p>If steps are valid, then this indicates that the Legendre transformation of the Lagrangian is just the Lagrangian plus some constant $C_2$, and that the Hamiltonian is thus $H=L+C_2$. Seems pretty fishy. If it is not fishy (if above steps are valid), then this question: Since it is derived from the E-L condition, does this result imply that the action of all functions that are their own Legendre transformation (plus a constant) is stationary?</p> | g13754 | [
0.054632239043712616,
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0.04574231430888176,
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0.0813915953040123,
0.031643010675907135,
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0.021067503839731216,
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0.048400282859802246,
0.01763913594186306,
0.0... |
<blockquote>
<p>A uniform rod$(m,l)$ is standing vertically on a horizontal frictionless surface. Gravity is downwards and uniform. I give its upper end a little push and off it goes. I want to find the Normal Force(and all other variables) as a function of time. </p>
</blockquote>
<p>Here's what I do :</p>
<p>$mg-N=ma$</p>
<p>Where a is downward acceleration(and the only) of centre of mass wrt ground.</p>
<p>Next, I think of conservation of energy. I assume it rotates $\theta$ from vertical</p>
<p>$mg\frac{l}{2}(1-\cos\theta)=K.E.$</p>
<p>I have problem writing expression of its Kinetic Energy. I have studied about instantaneous centre of rotation.</p>
<p>So, I can write its speed and rotational kinetic energy about it as $K.E.$? Also How do I find relation between $\omega$ (angular velocity about that instantaneous centre) and velocity of rod? </p> | g13755 | [
0.07135328650474548,
-0.022581787779927254,
-0.02836596593260765,
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0.03693770617246628,
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0.10170052945613861,
0.049363162368535995,
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0.016402002424001694,
0.01368158869445324,
0.022067023441195488,
-0.01949966512620449,
-0.003... |
<p>Holevo's theorem says that no more than n bits can be stored (and retrieved) in n qubits. Indeed, allowing error can't improve this either -- the probability of retrieving the correct information is no better than that which could be transmitted in the same number of bits and guessing at the rest.</p>
<p>Superdense coding is one way around this bound: if the receiver shares n maximally entangled qubits with the sender, the sender can manipulate them such that when she gives the receiver her n qubits the receiver can obtain 2n bits of information. Perhaps this is not surprising, though, since he has to measure 2n qubits to get the data.</p>
<p>Is this the limit of quantum information capacity? That is, say sender and receiver share a large number N of entangled qubits and (after judicious manipulation and selection) the sender gives n of them to the receiver. Can more than 2n bits be transmitted in this way?</p>
<p>It would seem that the answer is "no", but I'd like a double-check. I'm very much a beginner, just working through Michael Nielsen's tutorials and Scott Aaronson's book. This question is similar to <a href="http://physics.stackexchange.com/q/38922/2818">another question here</a> but my question is different and not answered there.</p> | g13756 | [
-0.022190913558006287,
0.09988834708929062,
0.00016234954819083214,
-0.016668343916535378,
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-0.006454525515437126,
-0.009931061416864395,
-0.05944434925913811,
-0.0032333408016711473,
-0.01146183814853... |
<p>I still can't understand that the phase can be a constant until now.</p>
<p>If the phase is constant, from the </p>
<p>$$y(x,t) = a \times \sin(phase)$$</p>
<p>the shape of wave will be a line parallel to x-axis.But I know that I am wrong,how to explain it?</p> | g13757 | [
0.03610766679048538,
0.03647714853286743,
0.00718541257083416,
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0.05515880510210991,
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<p>This is in reference to slide 19 of this <a href="http://cosmology.lbl.gov/talks/Pajer_13.pdf" rel="nofollow">talk</a></p>
<p>"<em>As always in Effective Field Theory, the theory becomes predictive when there are more observables than parameters</em>" </p>
<ul>
<li><p>Can one explain what this exactly means? May be if you can refer me to examples or literature where this is explained?</p></li>
<li><p>In this case it seems that this translates into having 2 fields ($\delta$ and $v$) and hence 3 2-point functions to regularize but one seems to have only two counterterms. Then how does this make sense? </p></li>
</ul>
<p>Isn't the quoted line basically mean that there is a meaningful interpretation of the scenario with having more correlations to regularize than counter-terms? What is the meaning? </p> | g13758 | [
0.06492859870195389,
0.0490194596350193,
-0.02163139544427395,
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0.0821380466222763,
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0.04751461744308472,
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0.003413229947909713,
-0.0483657605946064,
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0.02375130169093609,
-0.0158782... |
<p>Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories for the particles that travel through it depending on their spin.</p>
<p>My Question: At what point is the spin determined when it is in superposition? When the particle starts to feel the magnetic field? Or only when the trajectory is measured in the detector?</p>
<p><a href="http://physics.stackexchange.com/questions/86117/what-is-the-spin-state-of-a-spin-1-2-particle-when-it-comes-out-of-a-stern-gerla">This</a> is a similar question, however it does not answer my question.</p> | g13759 | [
0.03562214970588684,
0.008692081086337566,
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0.0038309399969875813,
0.025030914694070816,
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0.07864972203969955,
0.05099206790328026,
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-0.02045007422566414,
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0.01738259568810463,
0.04039843752980232,
-0.014... |
<p>When drawing ray diagrams for concave mirrors, I was advised to :</p>
<ol>
<li><p>draw a ray that is parallel to the principle axis, the reflected ray will pass through the focus</p></li>
<li><p>Draw a ray through the center (C), the reflected ray will be same as the incident ray but opposite in direction.</p></li>
</ol>
<p>I understand the reasoning behind point 2. as the incident angle will equal to zero. However, I don't understand why every single incoming parallel ray will pass through this certain point called the focus. Why not choose any other point. Is there any proof as to why this happen or was this just experimentally found?</p>
<p><strong>Main question:</strong> why do parallel rays pass through the focus?</p> | g13760 | [
0.016471151262521744,
0.015275395475327969,
0.012340482324361801,
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0.03790893778204918,
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0.054277386516332626,
0.013916335068643093,
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-0.010012554004788399,
0.023304635658860207,
0.05437023937702179,
0.054037805646657944,
-0.... |
<p><a href="http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=35&category=253&page=show_problem&problem=3491" rel="nofollow">Source</a></p>
<p>Given the coordinates of $n$ 3D joints ($1kg$ each) connected by $m$ rods. Assume rods have zero mass and joints with $z=0$ are fixed to the ground while others are free to move, will the shape be move or not? If not, will it be stable?</p>
<p>The totals force at each joint has to be zero for the shape to be stable. the force at each node is</p>
<ol>
<li>weight of the ball which is $1 \times 9.8067$ </li>
<li>tension force in the rods connecting that node.</li>
</ol>
<p>How can the tension force be computed?</p> | g13761 | [
-0.006316169630736113,
-0.00015266015543602407,
-0.004992357455193996,
-0.02677287720143795,
0.04419197514653206,
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0.015177159570157528,
0.04402117431163788,
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-0.014577602967619896,
-0.02048533409833908,
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-0.006884186994284391,
... |
<p>I am asked to calculate the variance of the kinetic energy in the ground state of the harmonic oscillator. That requires $\langle T^2\rangle$. This is the same as $\langle p^4\rangle$. My question is do I have to go about and calculate
$$\langle 0|(a-a^\dagger)^4|0\rangle$$
Or is there a shortcut.</p> | g13762 | [
0.04735949635505676,
0.01668240688741207,
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0.02555651217699051,
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0.04920044168829918,
0.056363366544246674,
-0.06680861860513687,
0.0073003522120416164,
-0.005487049464136362,
0.021424785256385803,
-0.031115613877773285,
0.0... |
<p>It is given here that the maximum safe speed of a vehicle on a banked road having coefficient of friction $0$, is $V =\sqrt{R g \tan \theta}$, where $\theta$ is the angle of banking. I can't understand how circular motion is possible without friction. </p> | g13763 | [
0.04273097589612007,
0.04098823666572571,
0.01164067629724741,
0.062051787972450256,
0.09972589462995529,
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0.08386239409446716,
-0.025187397375702858,
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-0.015203360468149185,
-0.04788406193256378,
-0.038206830620765686,
-0.03164464980363846,
-0.0... |
<p>Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the conservation of this vector?</p> | g13764 | [
0.015805352479219437,
-0.01570698246359825,
-0.0026977078523486853,
-0.0316070057451725,
-0.004307036753743887,
0.01734509877860546,
0.0595969595015049,
0.04576355963945389,
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0.01981709524989128,
-0.059452179819345474,
0.004303588066250086,
-0.06866078823804855,
0.030... |
<p>Why don’t photons interact with the Higgs field and hence remain massless?</p> | g13765 | [
0.06225265562534332,
0.021768862381577492,
0.030162865296006203,
0.02108936384320259,
0.018116099759936333,
0.06448980420827866,
0.008371842093765736,
0.05413734167814255,
-0.004791505169123411,
-0.02880348451435566,
-0.02856886014342308,
-0.01584889180958271,
-0.015340057201683521,
0.0548... |
<p>Do you know cases where chaos theory is actually applied to successfully predict essential results? Maybe some live identification of chaotic regimes, which causes new treatment of situations.</p>
<p>I'd like to consider this from the engineers point of view. By this I mean results that are really essential and not just "interesting". For example one might say weather calculation effects are explained with chaos theory, but an "engineer" might say: "So what? I could have told you without a fancy theory and it provides no added value since there is nothing I can change now."</p> | g13766 | [
0.04836084321141243,
0.01770724169909954,
0.02402466908097267,
0.026784071698784828,
0.023970115929841995,
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0.015096701681613922,
0.029087482020258904,
0.059863585978746414,
-0.010609169490635395,
0.05214402824640274,
-0.03414261341094971,
0.05126241594552994,
0.035999... |
<p>What can be a <strong>scientific</strong> solution to the Q-measurement problem (other than many worlds idea)? Can it be somehow verified through experiment?</p> | g13767 | [
0.0009562621708028018,
0.048564471304416656,
-0.003500083228573203,
-0.015136019326746464,
0.020408637821674347,
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-0.031218377873301506,
-0.014361215755343437,
0.013956457376480103,
-0.027738209813833237,
0.0031539436895400286,
0.0067329322919249535,
0.01160893682390451... |
<p>Consider this <a href="http://www.youtube.com/watch?v=MYNJITf9brg" rel="nofollow">youtube video of a guy lifting a car.</a> (skip to 2:15).</p>
<p>I checked, and these cars (PT Cruiser) weigh $1417\:\rm{kg}$. <strong>What would the equation be to estimate how much weight he is actually deadlifting?</strong> The lever would obviously be assisting as he is translating a smaller movement to a larger one. Is the $1417\:\rm{kg}$ is also not being completely lifted as the car is tilting on the axis of it's front wheels? What would the equation look like if the inputs were the </p>
<p>1) length of the lever, </p>
<p>2) the weight of the car, and I guess</p>
<p>3) length of the car.</p>
<p>I'm guessing the longer the car the more weight he would be lifting as he is getting less assistance as that unstable centre of gravity helps him pull up the car. I also checked and the length of the car is $4.2\rm{m}$.</p> | g13768 | [
0.004002382978796959,
0.048636242747306824,
-0.01639961265027523,
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0.00011507219460327178,
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0.06058160215616226,
0.013423091731965542,
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0.03420336917042732,
-0.04193362593650818,
-0.06195588409900665,
0.012297072447836399,
-0... |
<p>I am facing the following question. It is well known that power laws arise in many situations in nature. They arise even in thats physical systems that are completely deterministic (e.g. sand piles). But power law is a probability distribution function and can be thought as generated from a stochastic variable. What is the relation between a power law distributed variable and a deterministic dynamical system? In other words why I can see statistical distribution in a deterministic system? </p> | g13769 | [
0.08085393905639648,
0.017160095274448395,
0.0012929923832416534,
-0.0028023049235343933,
0.05198957771062851,
0.05150866508483887,
-0.03838707506656647,
-0.007900255732238293,
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-0.03680901974439621,
-0.019752822816371918,
-0.02556547522544861,
0.04686247184872627,
0.0... |
<p>Why is oil a poor conductor of heat? Does it help in insulation?</p> | g13770 | [
0.019350050017237663,
0.023706693202257156,
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-0.02624444290995598,
0.00251202704384923,
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0.035517387092113495,
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-0.019443577155470848,
-0.041495952755212784,
-0.02927260659635067,
-0.019871335476636887,
... |
<p>I was reading this answer by madame anna v: </p>
<blockquote>
<p>You are right, the planetary model of the atom does not make sense when one considers the electromagnetic forces involved. The electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus.</p>
<p>One of the reasons for "inventing" quantum mechanics was exactly this conundrum.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Bohr_model" rel="nofollow">Bohr model</a> was proposed to solve this, by stipulating that the orbits were closed and quantized and no energy could be lost while the electron was in orbit, thus creating the stability of the atom necessary to form solids and liquids. It also explained the lines observed in the spectra from excited atoms as transitions between orbits. </p>
</blockquote>
<p>But I don't understand one assumption: Why would an electron in an orbit be accelerating continuously and would thus radiate away its energy and fall into the nucleus in a classical model?</p>
<p>Also in a planetary model planets would be also accelerating and thus fall into the sun? </p>
<p>I mean for example a planet would accelerate towards the sun, but because it will then have bigger velocity, it will escape a little bit the sun, and then it will accelerate towards the sun... and the hole process repeats again. But why wouldn't that be true for an electron and a proton?</p>
<p>Thank you.</p> | g430 | [
0.023960039019584656,
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0.017048304900527,
0.09779619425535202,
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-0.032356441020965576,
0.054398998618125916,
0.04110505431890488,
0.036889996379613876,
-0.019043... |
<p>I understand that the inter-galactic space is expanding but galaxies themselves are not. What is happening to the space within a galaxy? Is it fixed by the gravity of the galaxy or is it expanding and moving out around it?</p>
<p>Using the coins on a balloon analogy, is the balloon surface under the coins (galaxies) stuck to the coin and constrained or is it free to expand out, slipping underneath the coins leaving them in the same place (and the same size)?</p> | g56 | [
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-0.011738304048776627,
-0.0... |
<p>I am reading <a href="http://rads.stackoverflow.com/amzn/click/3540736174" rel="nofollow"><em>Free Energy Calculations: Theory and Applications in Chemistry and Biology</em> by Chipot and Pohorille</a>. At the beginning of the text (page 19, for example), the authors define the Helmholtz free energy $A$ and changes $\Delta A$ in it for two states (states 0 and 1):</p>
<p>$$\begin{align}
A&=-\beta^{-1} \ln Q(N, V, T) \;\;\;\textbf{(1.14)} \\
\Delta A&=-\beta^{-1} \ln \frac{Q_1}{Q_0} \;\;\;\textbf{(1.15)} \\
\Delta A&=-\beta^{-1} \ln \frac{Z_1}{Z_0} \;\;\;\textbf{(1.16)}
\end{align}$$</p>
<p>where $Q_i$ is a partition function:</p>
<p>$$Q_i = \frac{1}{N! h^{3N}} \int_{\Gamma_i} \exp[-\beta \mathcal{H}(d\vec{x}, d\vec{p})] d\vec{x} d\vec{p}_x$$</p>
<p>and $Z_i$ is a configurational integral:</p>
<p>$$Z_i = \int_{\Gamma_i} \exp[-\beta U(\vec{x})] d\vec{x}$$</p>
<p>where $U$ is the potential energy.</p>
<p>Next (page 20), the authors write a derivation for $\Delta A$, in the framework of <strong>free energy perturbation (FEP)</strong>:</p>
<blockquote>
<p>Equation (1.15) indicates that our ultimate focus in calculating
$\Delta A$ is on determining the ratio $Q_1/Q_0$, or equivilently
$Z_1/Z_0$, rather than on individual partition functions. On the
basis of computer simulations, this can be done in several ways. One
approach consists of transforming (1.16) as follows:</p>
<p>$$\begin{align} \Delta A&=-\beta^{-1} \ln \frac{\int \exp[-\beta U_1(\vec{x})] d\vec{x}}{\int \exp[-\beta U_0(\vec{x})] d\vec{x}} \\ &=-\beta^{-1} \ln \exp\left\{-\beta [U_1(\vec{x}) - U_0(\vec{x})]\right\} P_0(\vec{x}) \\ &=-\beta^{-1} \ln \langle \exp\left\{-\beta [U_1(\vec{x}) - U_0(\vec{x})]\right\} \rangle_{0} \end{align}$$</p>
<p>Here, the systems 0 and 1 are described by the potential energy
functions, $U_0(\vec{x})$ and $U_1(\vec{x})$, respectively.
Generalization to conditions in which systems 0 and 1 are at two
different temperatures is straight forward. $\beta_0$ and $\beta_1$
are equal to $(k_B T_0)^{-1}$ and $(k_B T_1)^{-1}$, respectively.
$P_0(\vec{x})$ is the probability density function of finding system 0
in the microstate defined by positions $\vec{x}$ of the particles:</p>
<p>$$P_0(\vec{x}) = \frac{\exp[-\beta_0 U_0(\vec{x})]}{Z_0} \;\;\;\textbf{(1.19)}$$</p>
</blockquote>
<p>My question is, what happened to the integrals in going from (1.18) to the next step? $P_0(\vec{x})$ only contains an integral in its denominator (i.e., in $Z_0$), and not also in the numerator. Could you please perhaps provide one more step between equation (1.18) and the equation that immediately follows it? Thanks for your time.</p> | g13771 | [
0.04731791466474533,
-0.0071830786764621735,
-0.02472708187997341,
-0.016652323305606842,
0.010744191706180573,
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-0.027335455641150475,
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0.02571575529873371,
0.043750736862421036,
... |
<p>A string with a fixed frequency vibrator at one end forms a standing wave with 4 antinodes when under tension T1. When the tension is slowly increased, the standing wave disappears until tension T2 is reached with no resonances occurring between the two tensions. How many antinodes are there in this new standing wave?</p>
<p>There would be 8 antinodes, right? Because as you jump up to the next frequency, the number of antinodes doubles.</p>
<p>Actually, no, it would be 5 antinodes, right?</p> | g13772 | [
0.05809013172984123,
0.0097210593521595,
0.00138916727155447,
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0.06710430979728699,
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-0.023898549377918243,
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-0.... |
<p><em>I am wondering can someone help to solve second part which extends first part;</em></p>
<p>The power radiated by the sun is ${3.9*10^{26}}_{watt}$. The earth orbits the sun in a nearly circular orbit of radius $1.5 *10^{11}m$. The earth’s axis of rotation is tilted by $27^o$ relative to the plane of the orbit , so sunlight does not strike the equator perpendicularly. What power strikes a $0.75$ $m^2$patch of flat land at the <strong><em>equator</em></strong>?</p>
<p>$P_0 = I*4*π*r^2$</p>
<p>$P = I*A*cos(\alpha)$</p>
<p>then</p>
<p>$P=\frac{P_0 * A *cos(\alpha)}{ 4*π *r^2}$</p>
<p>then</p>
<p>$P=\frac{(3.9*10^{26})(0.75)(cos(27))}{(4*π*(1.5*10^{11})^2)}=920_{watt}$</p>
<p>~ How much power strikes a patch of the same area located at latitude (at the same time of year) of 43° S (like Christchurch, New Zealand) or 43° N?</p> | g13773 | [
0.02965894341468811,
0.017856812104582787,
-0.04440128058195114,
-0.033562298864126205,
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0.007911888882517815,
0.04187377169728279,
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0.016609860584139824,
0.03702981770038605,
0.0692320466041565,
-0.03565426915884018,
0.006922... |
<p>I have not been able to find any information about this on the Internet. I am a middle-schooler, 14, who self-studies physics, and I know up to and including ODEs, and some of the calculus of variations, enough so that I can take a variation of the action. However, I have absolutely zero budget, and no ability to get textbooks. So, I was wondering how a physicist would quantize a classical field theory, such as Gauss's law for gravity, which I'm currently trying to create a quantum field theory for, as part of my plan to create a possible theory of quantum gravity, by adding in the relativistic corrections later.</p> | g13774 | [
-0.030545061454176903,
-0.003172624623402953,
-0.020346317440271378,
-0.0011500394903123379,
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-0.007438203319907188,
-0.05752259120345116,
-0.013018769212067127,
0.021396595984697342,... |
<p>I was sitting down yesterday and saw my phone vibrate on a side, and it moved about a centimetre per vibration.
I wondered why it moves, and thought perhaps that the side it was on had a slight slope, because it tended to move in the same direction every time.
So I then though, what if the side was perfectly flat? Well, then I considered the fact that the phone's centre of mass would make it move in a biased direction (is this correct?).</p>
<p>So I then thought, what if the phone's centre of mass was in the exact centre, and the slop was perfectly flat?</p>
<p>Would the phone still move?</p>
<p>And if so, what factor would make it move in a particular direction?</p> | g13775 | [
0.04739686846733093,
-0.007134116720408201,
0.0006782291457056999,
0.0004709010536316782,
0.06994981318712234,
0.040893346071243286,
0.018868837505578995,
-0.0052663967944681644,
-0.004669311922043562,
-0.03385530784726143,
-0.037298981100320816,
0.007626939099282026,
0.04136047884821892,
... |
<p>There are many things with different textures that appear white – salt, beer foam, paper, shaving cream, snow, talcum powder, white paint, etc. The most common answer is all of the frequencies must be reflected in order for us to perceive white, which I already know. There must be more underlining details why such a huge variety of things are white. Furthermore, if you wet paper, snow or talcum powder, they appear off-white or gray. Why is this?</p> | g13776 | [
0.04744984954595566,
0.07399316877126694,
-0.0007060414645820856,
-0.03636877238750458,
0.08748769760131836,
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0.03207550570368767,
0.026140568777918816,
0.00315676792524755,
-0.02545822039246559,
0.03878772631287575,
0.018561171367764473,
0.05821321904659271,
0.0566504... |
<p>I know how an air gap behaves: for lower voltages the resistance is extremely high, until spark voltage is achieved, at which point resistance rapidly drops, creating electric arc. Now, how does that behave if we remove air?</p>
<p>Let's assume a part of electronic circuit that is a vacuum pipe with two electrodes at its ends. The electrodes are otherwise separated by an extremely (infinitely) highly resistive material. What would the I:U graph of voltage across the void gap versus amperage of current flow look like for element like this? What would it depend on, besides voltage (and possibly distance between the electrodes)?</p>
<p>As I understand some charge flows through vacuum between oppositely charged electrodes but it's nowhere near as straightforward as in case of normal conductors. How does it look like? What are the caveats? Is there a plain equivalent in common circuitry that behaves just the same as a void gap for common electrical characteristics (voltage-resistance curve etc) ?</p> | g13777 | [
0.035456523299217224,
0.0027859818655997515,
-0.03321578726172447,
0.03430047631263733,
0.01123217586427927,
0.034597430378198624,
0.00782062578946352,
0.03359875828027725,
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-0.011824198067188263,
-0.0026374030858278275,
0.05734049901366234,
-0.013859758153557777,
-0... |
<p>...speaking of Perseid meteor showers...How close would a Perseid meteorite have to be in order to be heard by the human ear? Is it even possible to hear a Perseid meteorite? If they come close enough, would we hear "ka-boom!", or would it sizzle, or do they <em>always</em> burn up before we hear them?</p> | g13778 | [
0.028188854455947876,
0.0037033422850072384,
0.012554554268717766,
-0.06988178938627243,
-0.04473663493990898,
0.0529777929186821,
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0.03477725759148598,
0.015610985457897186,
-0.007795247714966536,
-0.007679739501327276,
0.036132026463747025,
0.029040217399597168,
-0.0... |
<p>The problem:</p>
<blockquote>
<p>In a car crash, a passenger with a mass of 82kg is not wearing a
seatbelt. The car is travelling at 45km per hour. What impulse must
the car's airbag provide in order to stop the passenger's motion?</p>
<p>Explain why hitting an airbag will cause less injury than if a
passenger hits the dashboard.</p>
</blockquote>
<p>Here is what I did:</p>
<p>I thought impulse meaning force, so to find force I can use $F = m\Delta v/\Delta t$. But I don't know what $t$ is. How can I find $t$?</p> | g13779 | [
0.01915649138391018,
0.044620782136917114,
0.008844523690640926,
0.016191542148590088,
0.044253237545490265,
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0.003490216564387083,
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-0.07677829265594482,
0.02835393324494362,
-0.03573911637067795,
0.02616126276552677,
-0.01647... |
<p>I'm currently in the process of writing a 2.5D application that should display the perceived size of an object. For example, When I have a ball that has a diameter of 1 meter, how big would it appear if the ball would be 5 meters away from me? I know that there is angular diameter, but how can I translate the angle that I get to a size or scale? </p> | g13780 | [
-0.0011377099435776472,
-0.047870490700006485,
-0.008882596157491207,
-0.10780028253793716,
-0.042926300317049026,
-0.01696062460541725,
0.024058479815721512,
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-0.0368744395673275,
-0.004519515670835972,
-0.01893165335059166,
0.008661533705890179,
0.12023798376321793,
... |
<p>Why the color motion picture film, or a color photographic slide, appears as black?
The film appears black but when the light from the projector lamp hit it then you can see all the colors.</p> | g13781 | [
0.07826147973537445,
0.014406617730855942,
0.018006134778261185,
0.009989961050450802,
0.04051290825009346,
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0.08061660826206207,
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0.02372581511735916,
0.044229812920093536,
0.0540568046271801,
0.04865635186433792,
0.036911... |
<p>In a transformer assumed to be transformer, power in the primary is equal to power in the secondary. So in a sense, the power in the secondary is 'fixed'. Output voltage in the secondary is also fixed by the turns ratio and the voltage in the primary. Then, The current tapped from the secondary coil should also be fixed $I=\frac{P}{V}$.</p>
<p>Doesn't the resistance in the circuit connected to the secondary coil also affect the current tapped from the secondary coil? So if a load resistance in the secondary coil circuit is varied, would the current still be fixed the same $I=\frac{P}{V}$ or would it change accordingly with $I=\frac{V}{R}$</p> | g13782 | [
0.012655920349061489,
-0.0714842826128006,
0.0013841360341757536,
-0.0571245513856411,
0.03387182578444481,
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0.032268647104501724,
0.015165457502007484,
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0.012771748937666416,
-0.06278135627508163,
0.030716679990291595,
-0.03954307734966278,
-0.00... |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/27641/entanglement-in-time">Entanglement in time</a> </p>
</blockquote>
<p>I heard that there is an experiment that uses quantum entanglement to try to send messages back to the past. I am having a hard time understanding how such experiments would work theoretically. </p>
<p>Can anyone offer me some insights toward these types of experiment? </p> | g431 | [
-0.03120935522019863,
0.034388840198516846,
0.01509490329772234,
-0.040335919708013535,
0.05240194872021675,
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0.019243933260440826,
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0.032222162932157516,
-0.0216397512704134,
0.04599791765213013,
0.003765504341572523,
0.009... |
<p>First of all sorry for my English - it is not my native language.</p>
<p>During my engineering studies at the university the thermodynamics professor told us that the "second law of thermodynamics is not true for a low number of molecules".</p>
<p>At that time scientists were already talking about micro technologies where single molecules do some work. (Today such technologies already exist).</p>
<p>Now I'm wondering if a <a href="http://en.wikipedia.org/wiki/Perpetual_motion">perpetual motion machine of the second kind</a> (a machine gaining energy by cooling down the environment) would be possible in nanotechnology. As a result such microchips would lower the entropy of the universe.</p>
<p>I was thinking about the following theoretical experiment:</p>
<p>Brown's molecular movement was discovered by observing objects in fluids in amber stones. Due to Brown's molecular movement these objects are moving. The energy comes from the environmental heat. If the object was a small, very strong magnet and metallic parts are placed near the amber then the moving magnet should induce eddy currents in the metallic parts. This would mean that there is a heat energy flow from the amber to the metallic parts even if the metallic parts are warmer then the amber.</p>
<p>Now the question: Would such a device be possible in nanosystem technology or even microsystem technology?</p>
<p>(I already asked a physics professor who only said: "maybe".)</p> | g13783 | [
0.05905582010746002,
0.03548956289887428,
0.021554997190833092,
0.007194632664322853,
0.034308791160583496,
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0.01892230659723282,
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-0.023087235167622566,
-0.006150334142148495,
0.03519768267869949,
-0.014275801368057728,
-0.... |
<p>Today's liquid crystal shutter glasses, when in the "transparent" state, exhibit only 40% light transmission.</p>
<p>They work using two polarizer layers, one which is liquid crystal and goes {vertical <-> horizontal} under electronic control, and the other which is static horizontal. Thus, even when the polarizers are aligned, the total system is still a horizontal polarizer, which necessarily discards half the light, setting a theoretical upper bound of 50% transmission for unpolarized scenes.</p>
<p>However, a friend of a friend working at a commercial lab has offhandedly remarked that a liquid crystal shutter with > 90% transmission in the transparent state is now possible. Unfortunately, he is under NDA and cannot explain the technique.</p>
<p>Given that > 90% transmission is possible, how can this be achieved? What would be the layers in the stack-up, and which kinds of liquid crystals and/or polymers would be used? Try to stick to materials and techniques that today's manufacturers already use.</p> | g13784 | [
-0.0022438617888838053,
0.030843596905469894,
0.02241038717329502,
0.04521169140934944,
-0.016600504517555237,
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0.004492846317589283,
0.018832804635167122,
0.031132444739341736,
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0.0070... |
<p>I have studied for a couple of months now a mathematical model of the energy of a configuration of immiscible fluids situated in a fixed container such that the fluids fill the container. In other terms, I considered a partition of the container into $2,3,...n$ sets. I will present the $3$ fluid case here.</p>
<p>So, in my hypothesis the container $\Omega$ is partitioned in $S_1,S_2,S_3$, the three fluids, with prescribed volumes $v_i$ and prescribed densities $\rho_i$. I took into account in the formulation of the energy the interfacial tensions, the gravity, and the contact of the fluids with the container $\Omega$ (these are not my ideas; they are taken from other similar mathematical articles). I will denote $P_\Omega(S)$ the perimeter of $S$ situated in the interior of $\Omega$ and $P_{\partial \Omega}(S)$ the perimeter of $S$ which is situated on the boundary of $\Omega$. I will not be very rigorous in what I'm about to write: I will write, for example $P_\Omega(S_i\cap S_j)$ the perimeter of the intersection $S_i\cap S_j$ even if as set theory intersection, this is void. Still, I think that the idea will be clear.</p>
<p>So, the formula for the energy of the configuration, which I found in other articles too is:</p>
<blockquote>
<p>$$\mathcal{F}(S_1,S_2,S_3)= \sum_{1\leq i<j\leq 3}\sigma_{ij}P_\Omega(S_i \cap S_j)+\sum_{i=1}^3 \beta_i P_{\partial \Omega}(S_i)+\sum_{i=1}^3 g \rho_i\int_{S_i} z dV$$
where $\sigma_{ij}$ is the interfacial tension between $S_i$ and $S_j$ and $\beta_i$ is something I do not understand entirely, but it should take into account the effect of the walls of $\Omega$ (which in some cases, like thin tubes are not negligible). The last term is the potential gravitational energy (which was explained <a href="http://physics.stackexchange.com/questions/20410/potential-energy-in-a-gravitational-field">here</a>).</p>
</blockquote>
<p>My goal is that using this mathematical model of the phenomenon to prove the existence result of the minimal energy configuration, and to be able to obtain some numeric results, which will show how the final configuration looks like.</p>
<p>In the mathematical model, the inequalities $\sigma_{ij}+\sigma_{jk} > \sigma_{ik}$ are assumed. These assumptions seem physically reasonable, after reading the following <a href="http://jcp.aip.org/resource/1/jcpsa6/v66/i8/p3667_s1?isAuthorized=no" rel="nofollow">article of J. Cahn</a></p>
<p>My confusion is about what the $\beta_i $ mean. In my intuition, $\beta_i$ should express that fluid $S_i$ 'likes' to be in contact with the wall or 'doesn't' like to be in contact with the wall. There are two things I have in mind for $\beta_i$.</p>
<ol>
<li>Can we consider $\beta_i$ the interfacial tension between the wall and the fluid $S_i$? In this case $\beta_i$ would be positive and very large, verifying the same triangle inequality conditions $|\beta_i-\beta_j| \leq \sigma_{ij}$. Given these inequalities, a proof can be given for the existence in the case of $3$ fluids (and probably for $n$ fluids; work in progress).
Can we consider $\beta_i$ the interfacial tension between the wall and the fluid $S_i$? In this case $\beta_i$ would be positive and very large, verifying the same triangle inequality conditions $|\beta_i-\beta_j| \leq \sigma_{ij}$. Given these inequalities, a proof can be given for the existence in the case of $3$ fluids (and probably for $n$ fluids; work in progress). It seems reasonable that surface tensions between the wall and the fluids determine somehow the way the fluids interact with the wall: If the interfacial tension is very large, then the wall tries to reject the fluid and the equilibrium configuration will try to minimize the wall contact with such fluids, and maximize contact with fluids which have less interfacial tension. My confusion comes also from the fact that $\beta_i$ is not considered to be positive in the mathematical model; I'm not sure why is this. Maybe it is just a way of saying that "the mathematical model works even if $\beta_i$ are negative".</li>
<li>$\beta_i$ could be the wetting coefficient, $\beta_i=\cos \theta$ (ref: <a href="http://books.google.fr/books?id=_ydSF_XUVeEC&printsec=frontcover&hl=fr#v=onepage&q&f=false" rel="nofollow">Molecular theory of Capillarity</a>, pag 9) where $\theta$ is the <a href="http://en.wikipedia.org/wiki/Contact_angle" rel="nofollow">equilibrium contact angle</a>.</li>
</ol>
<p>My questions are:</p>
<blockquote>
<ol>
<li>Is this mathematical model valid?</li>
<li>Is the energy formula physically reasonable? If not, what changes should be made?</li>
<li>What is the correct interpretation for $\beta_i$?</li>
<li>Can we expect that the equilibrium configuration (which exists in experiments) must be a minimizer for the energy of the system?</li>
</ol>
</blockquote> | g13785 | [
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0.0013947978150099516,
0.008875121362507343,
-0.029381662607192993,
-0.04191888868808746,
... |
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