question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>I have performed a calculation the tensor of elastic constants (or stiffness tensor) for a given crystalline material. From there, I calculated various elastic properties, such as Young’s modulus, shear modulus, etc. I have, however, received advice to analyze it by</p>
<blockquote>
<p>further decomposing the elastic constant into contributions from the
bare terms and phonon relaxation terms</p>
</blockquote>
<p>I don't have any more context than that, and I have not been able to identify what these two terms (“bare terms” and “phonon relaxation”) refer to. Does anyone here know?</p>
<hr>
<p>PS: I don't know if it helps, but the elastic tensor was calculated using quantum chemistry calculations, by measuring the energy of the solid as a function of strain.</p> | g14448 | [
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<p>The s orbital have higher probability to be closer to the core and feels larger attraction than the p orbital and on average is further away and in addition p has repulsive potentilal l(l+1)h^2/2mr^2. But is there name for this effect and is it this the only way to explain ? </p>
<p>What is the usual energy difference between these levels and how is measured, the transitions are highly forbidden ? </p>
<p>Does quantum electrodynamics separate s from p ? In Lamb shift I think I does not but how about the multielectron atoms ?</p>
<p>In real life if hydrogen was not a molecule but atom is s lower that p, and is it opposite in all multielectron atoms ? There is no hund rule for this ?</p> | g14449 | [
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<h1>Background</h1>
<p>This issue has been bothering me for a few days now. It's actually part of my homework, but I've already finished them and in a sense they're not part of the question here. What I would like to know is more of a theoretical question about this formulation.</p>
<p>Now my professor's book and many online resources (<a href="http://www.continuummechanics.org/cm/tractionvector.html" rel="nofollow">sample</a>) maintain the same mathematical relation, hence my dilemma.</p>
<h1>Problem</h1>
<p>Consider the following:</p>
<p><img src="http://i.imgur.com/bTaKei0.png" alt="Problem"></p>
<p>Under axial loading, the stresses can now be computed for any section. The question posed is what if we make an imaginary section at an <em>inclined</em>, its normal defined by an angle $\theta$. I will not write every detail here, but its clear that we can use $A$ as a placeholder for the area and we can define $\sigma_x$ to be $\sigma_x = \frac{F}{A}$ and therefore in the diagram below we will use only stress.</p>
<p><img src="http://i.imgur.com/u4PU86j.png?1" alt="Problem"></p>
<p>Now, I will refrain working with vector quantities, I will only use scalar relations. If you take a look at this triangle, its easy to see that the area is projected to the section and we can get the relationship:</p>
<p>$$ A' = \frac{A}{\cos(\theta)} $$</p>
<p>Now the rest of the equations go by the books:</p>
<p>$$ T = \frac{F}{A'} = \frac{F}{A}\,\cos(\theta) = \sigma_x\,\cos(\theta) $$
$$ T_n = T\,\cos(\theta) $$
$$ T_s = T\,\sin(\theta) $$</p>
<p>My dilemma starts here. First of all <em>notice</em> how in each of the above expressions the value for the length of the object is not taken into consideration. Also, we're working with areas and forces hence the length itself is not part of the equation at all even as a <em>contained</em> information. Now that would be all fine, if it were not for the following section(to simplify, I've taken the origin to be at the center of the object).</p>
<p><img src="http://i.imgur.com/CZYfjZA.png" alt="Section C"></p>
<p>Now as you can see this section of the object gets very interesting. Clearly the formula for $A'$ does not hold anymore because the area itself is shrinking on the left side. To make things even worse, notice how the area on the right appears. The two areas are clearly linked, in this case, the sum of the two should be equal to the original area - from the fact that the height appearing on the right is equal to the height subtracting from the left. Calculating the angle at which the breaking point is made, that is when the section plane cuts the corner is easily calculated using:</p>
<p>$$ \tan(\theta_k) = \frac{l}{h} $$</p>
<p>Where $ \theta_k $ stands for the critical angle, just so we can refer to it. Now the question that is on my mind is what is the distribution of the stresses on this section. Clearly there is stress on the left, there's stress on the right. Now I am confused as to the distribution of stress at this angle.</p>
<h1>My Work</h1>
<p>Now, I've attempted a solution here. First, I am in a tight spot between assumptions. Clearly the original force $F$ must still be present here. But the force distribution is confusing me at this point. The best idea I could come up with, is the following. I would be grateful for your review.</p>
<h2>FBD</h2>
<p><img src="http://i.imgur.com/0DIhQMU.png" alt="Problem 3"></p>
<p>I've only written here $F_1$ and $F_2$ for visualization purposes, because personally I don't think that we will need the $F_2$. Knowing that $F_1 = F$ it's obvious that they will cancel each other out, and they will also cancel the moments they create against the newly formed centroid on the trapezoid. Hence I think the only remaining task is to compute the areas of each individual piece and compute stresses.</p>
<h2>Areas</h2>
<p>Computing the individual areas for the cuts, since we won't need the area at where supposedly $F_2$ was located, we will ignore that for the time being. Though it will come naturally as we try to find the left area.</p>
<p>The inclined plane area is the easiest to find, clearly, if we project the length $l$ to the top of the smaller rectangle we can see that using trigonometric relations we can find:</p>
<p>$$l\,'=\frac{l}{\sin(\theta)}$$</p>
<p>Multiply that by the base and you've got the area $A'=b l\,'$. To find the areas of the left and right sections one must notice that the amount we deviate from the original height we add to the right side. So if we're to project the length like we did last time we get this:</p>
<p><img src="http://i.imgur.com/9m1MbQ0.png" alt="Area2"></p>
<p>From the figure above it's clear that the following relation holds true:</p>
<p>$$ h = h' + 2\,h_2 $$</p>
<p>From trig we know $h'=l\cot(\theta)$. From this, $h_2$ is defined by:</p>
<p>$$ h_2 = \frac{1}{2}(h - l\cot(\theta)) $$</p>
<p>Now $ h_1 = h - h_2 $ and after some mathmagic we should get the following:</p>
<p>$$ h_{1/2} = \frac{1}{2}(h\pm l\,\cot(\theta)) $$</p>
<p>Now let's multiply this by the base and we've got ourselves $A_1 = h_1\,b$.</p>
<h2>Stress</h2>
<p>Now we have two stresses, namely $\sigma_{x_1}$ and $T$. Let's analyze the relations. The forces clearly have an eccentric deviation from the centerline. Now, is this going to be a problem? I seem to think that it won't be, and I base my logic on the fact that they will cancel the bending moments as we said above. The formulas are:</p>
<p>$$\sigma_{x_1} = \frac{F}{A_1}$$
$$T = \frac{F}{A'} $$</p>
<p>Now obviously these are general equations, one could always dig deeper and I believe we could even express these stresses as a function of elementary stresses, namely $\sigma_x$ and some form of this other <em>tangential</em> stress $F/(l\,b)$.</p>
<h2>Appendix</h2>
<p>To be completely accurate, the formula for $\theta_k$ seems to be a bit more complex, since there are four corners on the cut (in our case). Knowing that paired corners are symmetric to the x-axis we can use this equation:</p>
<p>$$\theta_k(n) = |\arctan(\frac{l}{h}) + \pi\,n|$$</p>
<p>where $n\in(-2,-1,0,1)$. Is this ugly or is it just me?</p>
<h1>Closing</h1>
<p>I hope to get some review on the ideas here, I'd really love to know how would this get solved correctly. And please someone, help me fix the appendix formula.</p> | g14450 | [
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<p>Suppose that a matrix</p>
<p>$$A ~=~ x_1 B + x_2 C$$</p>
<p>is a linear combination of two self-adjoint matrices $B$ and $C$.</p>
<p>I'm interested in when $A$ represents a physical quantity. </p>
<p>When the linear combination is a complex combination, then $B$ and $C$ have to be commutable for $A$ to represent any physical quantity, cf. <a href="http://physics.stackexchange.com/q/38255/2451">this</a> Phys.SE post. </p>
<p>Now suppose that $x_1$ and $x_2$ are real. What happens in this case? If $B$ and $C$ are noncommutable, does $A$ still represent physical quantity?</p> | g14451 | [
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... |
<p>The galaxy moves of the celestial sphere. It is given that proper speed is transverse to the observer and it must to find this speed in the moment of light emission. The motion is in the FLRW universe. Does it relate to the comoving distance?</p> | g14452 | [
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<p>I know that this is a simple question! But I would like to know the details.
How we can show that the term</p>
<p>$$A_\mu(x)\dot{x}^\mu$$</p>
<p>is global and local Poincare invariant? Where $A_\mu(x)$ is supposed to be a four-covector and
This term is a part of reparametrization-invariant square-root Lagrangian
$\mathcal{S}=\int d\tau[-mc\sqrt{-(\dot x^\mu)^2}+\frac{e}{c}A_\mu\dot x^\mu]$
where $x^\mu$ are parametric equations of the physical trajectory $x_i(t)$. We assume $\frac{dt}{d\tau}>0$ and $A^\mu$ is electromagnetic four-vector potential. </p> | g14453 | [
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<p>Recently, I've been watching "The Big Bang Theory" again and as some of you might know, it's a series with a lot of scientific jokes in it - mostly about Physics or Mathematics. I understand most of the things mentioned in the series and whenever I don't understand a joke, I just look up the knowledge I'm missing on wikipedia - e.g. I learnt about Schrödinger's cat in this way.</p>
<p>However, at one point, I did not know how to proceed, which is why I'm asking this question. At a Physics quiz, the participants are asked to "solve" the following equation:</p>
<p><img src="http://i.stack.imgur.com/GXfQe.jpg" alt=""Solve the equation!""></p>
<p>The solution turns out to be $-8 \pi \alpha$. My questions are: What is the meaning of this equation? How does one obtain it? And of course, how does one solve it and is the solution given by the university janitor (in the series) correct?</p>
<p>Also, I am sorry if this is an equation found in Physics, I didn't know.</p> | g14454 | [
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<p>I'm studying from Goldstein's <em>Classical Mechanics</em>. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 \dots m$. Then it also holds that $\sum \lambda_\alpha f_\alpha = 0$. Using Hamilton's principle (i.e. that the action must be stationary), we get that</p>
<p>$$\delta \int_1^2 L\ dt =
\int_1^2 dt\ \sum_{k=1}^n \left(\frac{\partial L}{\partial q_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q_k}}\right)\delta q_k = 0.
$$</p>
<p>But we can't get Lagrange's equations from this because the $\delta q_k$ aren't independent. However, if we add this with $\sum \lambda_\alpha f_\alpha = 0$, it follows that</p>
<p>$$\delta \int_{t_1}^{t_2} \left(L +\sum_{\alpha=1}^m \lambda_\alpha f_\alpha\right)\ dt = 0.$$</p>
<p>And then Goldstein says that</p>
<blockquote>
<p>The variation can now be performed with the $n\, \delta q_i$ and $m\, \lambda_\alpha$ for $m+n$ independent variables.</p>
</blockquote>
<p>Why have the variables suddenly become independent? First we had $n$ dependent variables, why do we now have $m+n$ independent ones?</p> | g14455 | [
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<p>Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$.
For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint?
The fundamental representation is defined as identifying $T^a$ the original generator. The adjoint representation is defined as identifying $T^a$ as structure constant. For $SO(3)$ case, generators seem to be same as structure contant. </p>
<p>Reference:</p>
<ol>
<li>Srednicki QFT</li>
</ol> | g14456 | [
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<p>I'm only in high school (junior) so goes easy on me. But, how do physicist refute these paradoxes? Considering there are a number of theories regarding the origin of the universe and some postulate that the universe is infinite, I assume they've come across this. Just looking for a direct answer to what should be a relatively simple question. Here are the two paradoxes (for those unaware):</p>
<p>Zeno's paradox </p>
<p>"A similar paradox arises if the past is infinite. If there exists an infinite past, we would never have the present day. If there was an infinite set of past events and each event requires the previous event to occur, would we ever have the present? Of course not. This is because if today is dependent upon the fact that yesterday happened, and there is an infinite set of these dependencies (i.e. forever) - today will have not occurred. This is similar to the library book example mentioned earlier."</p>
<p>Hilbert's hotel</p>
<p>"If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning.</p>
<p>Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems.</p>
<p>The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupied by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accommodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible."</p> | g14457 | [
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<p>Say I have a metric representation $g_{\mu\nu}$ in a coordinate system $x$ and I want to find the representation of the metric in a new set of coordinates $y = y(x)$. I know how to do this if you are given $x(y)$, as in <a href="http://physics.stackexchange.com/questions/16681/metric-tensor-of-coordinate-transformation">this post</a>. </p>
<p>$g_{\mu' \nu'} = \frac{\partial x^{\mu}}{\partial y^{\mu'}} \frac{\partial x^{\nu}}{\partial y^{\nu'}} g_{\mu \nu}$ -------- $(1)$</p>
<p>But what if I'm only given $y=y(x)$, and it's tricky to figure out $x=x(y)$? Is there a method that uses partial derivatives $\frac{\partial y^{\mu'}}{\partial x^{\mu}}$ instead? Or is $(1)$ the only way?</p> | g14458 | [
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<p>Can you explain to me what causes the <a href="http://en.wikipedia.org/wiki/Buoyancy" rel="nofollow">buoyant force</a>? Is this a result of a density gradient, or is it like a normal force with solid objects?</p> | g14459 | [
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<p>I wanted to know that if FIRE is Matter or Energy. I know that both are inter-convertible. but that doesn't mean there is interconversion taking place just like nuclear fusion or fission. If it is matter, then its has to occupy space and have mass. I really doubt that fire has mass. If it is energy, then i don't get it. Please enlighten me from my ignorance.</p> | g546 | [
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<p>In the Physics Today article by Avron et.al. "<em>A Topological Look at the Quantum Hall Effect</em>" Physics Today (2003) it is suggested that to observe ordinary Hall effect, planar geometry is preferred to bar geometry as below. I am trying to understanding the reason behind this. Can someone help me? Any reference would be helpful too.</p>
<blockquote>
<p>hasty experiment . . . though without success.”Hall made a fresh start and designed a different experiment, aimed at measuring, instead, the magneto-
resistance—that is, the change of the electrical resistance due to the magnetic field. As we now know, that is a much harder experiment, and it too failed. Maxwell appeared to be safe. Hall then decided to repeat Rowland’s experiment.Following his mentor’s suggestion, Hall replaced the original metal conducting bar with a thin gold leaf, to compensate for the weakness of the available magnetic field.</p>
</blockquote> | g14460 | [
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<p>Galileo is reputed to have demonstrated that unequal masses fall at an equal rate by dropping them off the tower of Pisa and observing that they hit the ground simultaneously. Has anyone tried to push this experiment to the limit at small scale or is the result observable from some other experiment? </p> | g14461 | [
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<p>Earlier I studied about the three states of matter-gaseous,liquid ans solid. Then, I came to know about Plasma and Bose-Einstein Condensate. Now, scientists are trying to explain superconductivity as a state of matter. But, I would like to know on what factors does scientists rely to decide whether a particular state is a state of matter? </p> | g14462 | [
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<p>I think I understand how pressure works with gases. More molecules bouncing around -> more random impacts -> stronger force. </p>
<p>But I realized to my embarrassment that I don't understand how solid things press on each other, on a molecular level. Say I put a block of iron on my head. If I put another one on top of it, I feel twice the weight. The two blocks together can tear through thin paper where one block can't. But the contact between my head (or paper) and the blocks is just a very thin layer of atoms of the lower block's structure. If the lower block doesn't move when I put the upper one on it, what causes this thin layer to "press" on my head (or paper) more? When the two blocks together tear through thin paper, where does the force come from that acts on the paper molecules - it can't be gravity from the upper block, right? And how come that whatever this source of pressure is only depends on the weight of the upper block, and not on what it's made out of, iron or wood?</p> | g14463 | [
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-0.0498967319726944,
0.015951547771692276,
-0.03986499458551407,
-0.021111104637384415,
-0.004208298400044441,
-0.05647377669811249,
0.0017672926187515259,
-... |
<p>For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four times, is the identity transformation. Wikipedia has a good description of this.</p>
<p>What do we get when we apply the transform multiple times to a distribution?
For example applying it twice would look like this:</p>
<p>$ \mathscr F \mathscr F T(f) = T(\mathscr F \mathscr F f) = T( f_-) = ? $</p>
<p>Where $f_- = f(-x)$. </p>
<p>Here $\mathscr F$ is the fourier transform $T$ is a distribution, and $f$ is a function. </p> | g14464 | [
0.035286493599414825,
0.015119753777980804,
0.005354760680347681,
-0.08854453265666962,
0.06200810521841049,
-0.057554904371500015,
-0.04046579822897911,
0.06324829161167145,
-0.027972904965281487,
-0.029341109097003937,
-0.10369057953357697,
0.005131756421178579,
0.011014988645911217,
-0.... |
<h2>Introduction</h2>
<p>I'm trying to make some strawberry liqueur. The recipe I got says I should put alcohol, strawberries and sugar in a bowl and leave it soaking for 4-5 weeks. I read that making liqueur is not a chemical reaction, but only a solvent (mostly alcohol) acting on some fruit, herb or nut. So, intuitively, I would add the sugar only after the soaking time. I wonder if that makes a difference.</p>
<h2>Question</h2>
<p>If I dissolve material A in alcohol until it reaches saturation, how much of material B can I still dissolve in the solution? </p>
<ul>
<li>Nothing</li>
<li>As much as with pure alcohol</li>
<li>Something in between</li>
</ul>
<p>None of the materials are gaseous and both are soluble in alcohol.</p> | g14465 | [
0.0373748280107975,
0.03624497726559639,
-0.008814742788672447,
0.00868041068315506,
0.018510576337575912,
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<p>We have
$$X^\textrm{t}gX = 0 \iff X^\textrm{t}L^\textrm{t}gLX = 0,$$
where $X$ is a column vector of length four, $L$ is a non-singular $4 \times 4$ matrix, 't' denotes matrix transpose, and
$$g = \left(\begin{matrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{matrix}\right)\,.$$
doesn't it immediately follow that $g = L^\textrm{t}gL,$ since $$X^\textrm{t}gX = 0 \iff X^\textrm{t}\left(L^\textrm{t}gL\right)X = 0?$$
Why or why not? I ask because the proof in my book takes up an entire page, so I have a feeling that this argument is not sound.</p>
<p>It shouldn't matter for the question, but $X$ and $L$ come from the following equations:
$$X = \left(\begin{matrix}
ct_2 \\ x_2 \\ y_2 \\ z_2
\end{matrix}\right) - \left(\begin{matrix}
ct_1 \\ x_1 \\ y_z \\ z_1
\end{matrix}\right)\,,$$ where $t_1$, $x_1$, $y_1$, $z_1$ and $t_2$, $x_2$, $y_2$, $z_2$ are the inertial coordinates of two events, and
$$\left(\begin{matrix}
ct \\ x \\ y \\ z
\end{matrix}\right) = L \,\left(\begin{matrix}
ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime
\end{matrix}\right) + C,$$ which gives the Lorentz transformation from the primed to the unprimed inertial coordinate system.</p> | g14466 | [
0.025372112169861794,
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0.03434259071946144,
-0.0300989281386137,
0.01990526169538498,
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0.08779694139957428,
-0.006345036439597607,
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-0.007285936735570431,
-0.04582557827234268,
0.02066596783697605,
-0.04849562793970108,
-0.0... |
<p>I'm not a physicist or cosmologist, so I hope I am asking the right question with the right words here. </p>
<p>My question regards the word "energy" as it pertains to quantum vacuum states, and to concepts such as Lawrence Krause's "something from nothing" theory. </p>
<p>When I hear the word "nothing" I envision a "void" that is an empty set. But Krause, Eva Silverstein and J. Richard Gott (2 hr YouTube video of debate on "nothing" hosted by Neil deGrasse Tyson - last url segment = watch?v=sNh-pY3hJnY) seem to be saying, if I understand correctly, that this void has a non-zero energy state, and that quantum fluctuations create "something" out of nothing because of this.</p>
<p>I have to say that this sounds a bit to me like hand waving and weasel words, but perhaps my ignorance is committing an injustice.</p>
<p>But if this is what is happening, is this "nothing" actually a void? Energy is something, and that would make "nothing" contain something. And I fail to see how this could be so.</p>
<p>The question: What, exactly, IS "energy." In usage in this context it sounds a lot like phlebotinum.</p> | g14467 | [
0.02455281838774681,
0.029313866049051285,
0.015246523544192314,
-0.032669033855199814,
0.01893039233982563,
0.04430842027068138,
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0.024986600503325462,
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-0.05396401509642601,
0.020212765783071518,
0.0006576970918104053,
-0.010786101222038269,
0.01... |
<p><a href="http://en.wikipedia.org/wiki/Jenga" rel="nofollow">Jeng</a>a is a game place with wooden blocks stacked on top of one another in an alternating pattern. Players take turns removing blocks from any layer and placing them on top. </p>
<p><img src="http://i.stack.imgur.com/NxBEl.jpg" width="200"></p>
<p>As the game progresses the tower gets higher and higher until it collapses.</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/3/34/Jenga.gif"></p>
<p>For a given configuration of blocks is there a way to calculate wither the tower is going to collapse? </p>
<p>I also want to know why some pieces are easier to remove than others... </p> | g14468 | [
0.020696137100458145,
0.05386791378259659,
-0.0009644666570238769,
-0.008760666474699974,
-0.020094579085707664,
-0.08596864342689514,
0.07436404377222061,
-0.007594535127282143,
-0.03520342707633972,
0.046515967696905136,
-0.04593788832426071,
-0.0553864985704422,
0.032028138637542725,
0.... |
<p>A mobile phone move aside when it vibrates. How is that happening ? and most importantly is it possible to make any changes to the vibration motor to stop moving when vibrating or any other methods to stop that ?
I have already read following links
<a href="http://physics.stackexchange.com/questions/61436/would-a-phone-move-upon-vibration-in-a-completely-uniform-situation">Would a phone move upon vibration in a completely uniform situation?</a>
<a href="http://physics.stackexchange.com/questions/17169/how-does-a-mobile-phone-vibrate-without-any-external-force">How does a mobile phone vibrate without any external force?</a></p> | g14469 | [
0.03875400125980377,
-0.011765718460083008,
0.006342057604342699,
0.036614008247852325,
0.006969204172492027,
0.017931537702679634,
0.0068661305122077465,
0.04641105607151985,
-0.0054062483832240105,
-0.017083503305912018,
-0.027235738933086395,
-0.035171546041965485,
0.008871239610016346,
... |
<p>The nuclear fusion taking place inside the stars opposes its gravitational self collapsing force. But, how does physicists calculate it? I just know the classical gravitational theory and not a bit of general relativity. So, I would like to take the situation in a classical way.
For example, in terms of Lyttleton-Bondi Model for the Expansion of the Universe the expansion of the universe is explained if matter has a net charge. Imagine a spherical volume containing un-ionized atomic hydrogen gas of uniform density. Assume the proton charge is $ke$ (where $e$ is the charge of an electron (this situation is taken for net charge of matter)). How to calculate the gravitational self collapsing force of this so that I could calculate the value of $k$ for which the electrostatic repulsion opposes it and the volume doesn't collapse on itself? I know how to calculate the gravitational force between masses or simple shapes(found using integration). But,I have no idea on how to calculate this. </p>
<p>(Even though Lyttleton-Bondi Model for the Expansion of the Universe model is discarded I would like to proceed with this as I don't know any high level math or other theories except classical mechanics taught in high school).</p> | g14470 | [
-0.009323800913989544,
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0.018230296671390533,
-0.03912196308374405,
-0.013473025523126125,
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0.038519907742738724,
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-0.0022861119359731674,
0.019916189834475517,
-0.011230829171836376,
0.01615193858742714,
0.... |
<p>According to Einstein's Mass-energy equivalence,</p>
<p>$ E = mc^2$ OR
$ m = \frac E{c^2}$..... (1)</p>
<p>and According to Newton's Second Law of motion,</p>
<p>$ F = ma$ OR $m = \frac Fa$ ..... (2)</p>
<p>If we compare eq. (1) and eq. (2), we obtain; </p>
<p>$\frac E{c^2} = \frac Fa$..... (3)</p>
<p>If we multiply both the sides of eq. (3) with $c^2$, we get;</p>
<p>$E = \frac Fac^2$ ..... (4)</p>
<p>Is the above relation valid? </p> | g14471 | [
0.026003748178482056,
0.005777283105999231,
0.0036638593301177025,
0.013059468939900398,
0.02540828287601471,
0.005274440161883831,
0.013442891649901867,
0.006913942284882069,
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-0.015316197648644447,
-0.032095931470394135,
-0.03411396965384483,
0.06592065840959549,
0.0... |
<p>People wait for decades to catch the chance of a solar eclipse to observe the sun. </p>
<p>Why cannot they do it every noon? </p> | g14472 | [
-0.058216605335474014,
0.07607322186231613,
0.0037602109368890524,
0.038662977516651154,
0.000231295358389616,
0.033123377710580826,
0.018227988854050636,
0.04304521158337593,
-0.019259655848145485,
-0.029562674462795258,
0.054336536675691605,
0.007367781363427639,
0.03499792143702507,
0.0... |
<p>The hydroelectricity plants extract the potential energy of highly deployed massive object (water) as it falls down. Without turbine, all that energy would be converted into speed (kinetic energy) at the bottom of the waterfall and further into heat. The turbine produces energy by slowing water down. </p>
<p>The efficiency of turbine, how much energy is extracted by turbine, can be charactarized by the exhaust speed: the faster is the output stream, the less efficient our turbine is since not all speed/energy is extracted. So, slower the turbine spins, the higher is its the efficiency. The extraction is 100% when turbine does not spin and no electricity is produced at all. So, there must be a trade-off between the efficiency and amount of the output, the trade-off determined by the turbine spinning speed (exhaust speed). How is it decided? </p>
<p>I read that large <a href="https://en.wikipedia.org/wiki/Water_turbine#Efficiency" rel="nofollow">modern water turbines operate at mechanical efficiencies greater than 90%</a>. Since couple of percent losses are inevitable whatever you do, it seems that they say that theoretical efficiency is 100%. Identical efficiency is provided by <a href="https://en.wikipedia.org/wiki/Switched-mode_power_supply#Explanation" rel="nofollow">switching power supply converters, which are 100% efficient in theory</a>. I understand the secret exploited by SMPS. My question is how similar, 100% energy extraction, is achieved through the turbines, which seem to operate linearly (spinning at the same pace) rather than switching mode pumping. What is the water release speed when 100% energy extraction is achieved?</p>
<p>This question is actually is not limited to water wheels. Today wind turbines are becoming more popular and I am curious how do you extract all power from the wind flow. If turbine spins quckly, the air is realased at high speed, which means that you do not slow down the flow, which means that it makes no work. On the other hand, if if you stop the flow completely, your turbine stops and no power is extracted either. What is the optimal turbine speed?</p>
<p>tags: engeneering, efficiency</p> | g14473 | [
0.03398468345403671,
0.017279300838708878,
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0.0603010319173336,
0.010479109361767769,
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0.03470809385180473,
-0.04451196640729904,
-0.005696864798665047,
0.02634907141327858,
0.029840... |
<p>My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576.
I have understand the equation (92.1) but don't understand that how they guessed the potential in equation (92.2).<img src="http://i.stack.imgur.com/YIlt3.jpg" alt="enter image description here"> </p>
<p>EDIT:
Contrast the above potential the author used this <a href="http://arxiv.org/abs/0802.3525" rel="nofollow">potential in equation (2)</a> $$U(\phi)= \frac{1}{8} \phi^2 (\phi -2)^2.$$
My query is , are we using different potential just for our convenient using? or I need to redefine the $\phi^4$ theory to get the potential ?</p> | g14474 | [
0.00958536472171545,
0.020932823419570923,
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0.010645843110978603,
0.032824236899614334,
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-0.01513578463345766,
-0.012265714816749096,
0.05461134761571884,
0.030380461364984512,
0.038... |
<p>Recently, I read an article on time reversed laser. I don't know why they call it a time reversed. I have a doubt that why they use two laser in the device. And what is an anti-laser?
The device absorbs the particular frequency of light which means some energy is lost. What happens to the lost energy?</p> | g14475 | [
0.06912481784820557,
0.020702356472611427,
0.01743575558066368,
0.04604939743876457,
0.024969980120658875,
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-0.014876356348395348,
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0.04932887852191925,
0.0182931087911129,
-0.01539... |
<p>I have heard that when the speed of the object increase, the mass of the object also increase. (<a href="http://physics.stackexchange.com/q/45320/">Why does an object with higher speed gain more (relativistic) mass?</a>)</p>
<p>So inertia which is related to mass, increase with speed?</p>
<p>So, if I accelerate on a bus, my mass will increase and my inertia will increase for a while on the bus, until the bus stops?</p> | g14476 | [
0.018579619005322456,
0.030502399429678917,
0.022734293714165688,
0.05118749663233757,
0.09542594850063324,
0.04141973704099655,
0.01645246520638466,
0.0701526403427124,
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-0.017678992822766304,
-0.03698737919330597,
-0.015684006735682487,
-0.013275772333145142,
0.03340... |
<p>Right now I'm working on a hot air balloon in Physics formulas. At the moment it's flying up like it should, but once it is at a certain height we want it to go back down.</p>
<p>After some research about hot air balloons we found out that they drop down by opening a valve at the top. So what we want to do is once the balloon gets at a certain height it will open the valve and drop down again. We can get the size of the valve and that it opens at a certain height. </p>
<p>How can we calculate the speed of cold air going into the valve? We know the speed of the balloon, the size, temperature of the air in and outside of the balloon, air pressure etc. Is there some kind of formula we can use to calculate how fast the cold air will flow in and then cool of the air in the balloon making it drop down.</p> | g14477 | [
0.044031716883182526,
0.009620020166039467,
0.020231831818819046,
0.021742315962910652,
0.016514470800757408,
0.000544615148101002,
0.039220403879880905,
0.005770158488303423,
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-0.016235707327723503,
-0.03235409036278725,
0.09234973043203354,
0.035712726414203644,
0.01... |
<p>In $φ^4$ theory we often write the Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$</p>
<p>If I want to write from the Relativistic Lagrangian then it takes $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V \tag{2}$$
but how will I convert this equation to equation like (1) ?</p>
<p>EDIT: I just want to get Equation (1) from equation (2)</p>
<p>EDIT by joshphysics: What motivates choosing to study the $\phi^4$ potential as opposed to other potentials?</p> | g14478 | [
0.0476810559630394,
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0.0026062747929245234,
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0.039007533341646194,
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0.004228974226862192,
0.0060483189299702644,
0.04268614575266838,
0.018914083018898964,
0.054... |
<p>For a region devoid of charge, maxwell's equation yields $\nabla \cdot \mathbf{E} = 0$ which still allows a constant field. So why is in electrostatics for the vacuum always $\mathbf{E} = 0$ assumed?</p> | g14479 | [
0.0713508129119873,
0.006458658259361982,
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0.005490539595484734,
0.006780378054827452,
-0.003785... |
<p>I've been learning about the theta parameter of QCD and I'm confused about the fact that it's supposed to be very small but at the same time some sources say that the Yang-Mills theory should be invariant to 2$\pi$ shifts in $\theta$</p>
<p>Sources that says that $\theta$ is small:</p>
<p>(1)<a href="http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDoQFjAB&url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-8964.pdf&ei=a1WTUdz_EaWb0QXktoHABg&usg=AFQjCNEdna5RTSrzhXErQyw74zQXwgGxKg&sig2=0-NnhPOC6D5gh29txNkd_g&bvm=bv.46471029,d.d2k">The CP Puzzle in the Strong Interactions</a> (See page 6)</p>
<p>(2) <a href="http://arxiv.org/pdf/hep-ph/0011376v2">TASI Lectures on The Strong CP Problem</a> (See page 19)</p>
<p>Sources that say that Yang-Mills theory is invariant under $2\pi$:</p>
<p>(3) <a href="http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CEQQFjAC&url=http://www-fp.usc.es/~edels/SUSY/aSuSy.pdf&ei=3FuTUZm3I4zy0gWc-4H4DQ&usg=AFQjCNFZz0xfuFOPFfT8Qw2brECRRNflVQ&sig2=oihdamaQIOw5jdq1sU-3wQ&bvm=bv.46471029,d.d2k">Notes on Supersymmetry</a> (See page 6)</p>
<p>(4) <a href="http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDEQFjAA&url=http://www.physics.uc.edu/~argyres/661/fgilec.pdf&ei=ulyTUZ7oFIzB0gXSv4GoBg&usg=AFQjCNF7s2q88iISSJDg3Vas0xe0dF7ckg&sig2=GDmHvVT4GTWFR4xd3D3aFg&bvm=bv.46471029,d.d2k">Non-Perturbative Dynamics Of Four-Dimensional Supersymmetric Field Theories</a> (See page 4)</p>
<p>Now source (1) (amongst others) tells us that there exists a term of the form: </p>
<p>$\theta n = \theta (1/32 \pi^2) \int d^4 x \ \epsilon^{\mu\nu\rho\sigma} \ F^a_{\mu\nu} F^a_{\rho\sigma}$</p>
<p>in the Lagrangian of our QCD, where the $\int d^4 x \ \epsilon^{\mu\nu\rho\sigma} \ F^a_{\mu\nu} F^a_{\rho\sigma}$ part gives CP violation. Thus we require $\theta = 0$ if want no CP violation. However to me this implies that the theory is not invariant under a shift in $2\pi$ of $\theta$ as $\theta = 2\pi$ would not make the above term disappear (you'd end up with $2\pi n$ rather than $0 \times n = 0$).</p>
<p>Similarly how can we say that $\theta$ has to be small if $\theta$ can be shifted by any multiple of $2\pi$ to give the same theory? Is it the case that when we say '$\theta$ is small' we actually mean '$\theta$ modulo $2\pi$ is small'. Also is it the case that the above equation some how disappears for all values of $\theta$ that are multiples of $2\pi$, and not only $\theta = 0$? </p>
<p><a href="http://arxiv.org/pdf/1105.0413v2">Theta vacuum effects on QCD phase diagram</a> - page 2 seems to imply that the CP violating term disappears for all value of $\theta$ that are multiples of $\pi$.</p> | g14480 | [
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<p>If I have two operators A and B living in the Composite Hilbert Space $H_I \bigotimes H_{II} $ and I want to take the partial trace of $C=AB$ over the subspace $H_I$, i.e., $Tr_I[AB]$, is there any identity that can help me do this in terms of $Tr_I[A]$ and $Tr_I[B]$. Actually what I am interested in is the partial trace of the commutator $[A,B]$.</p> | g14481 | [
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<p>For a pure dipole situated at the origin, I don't understand why the dipole moment vector is
$$
\vec{p} = p\cos \theta \hat{r} - p\sin \theta \hat{\phi}
$$
Since the vector does not have any $r$ in it, the value of the vector is same any distance away from the origin if the angle is constant, which doesn't make sense to me. It would be very helpful if someone could derive it because Griffiths uses this value without explanation. </p> | g14482 | [
0.0005225026980042458,
-0.009138834662735462,
-0.014927765354514122,
-0.024320539087057114,
0.03537886589765549,
-0.01782241277396679,
0.05278141051530838,
0.029281698167324066,
-0.03620089218020439,
-0.024662967771291733,
-0.06766583770513535,
0.01298004761338234,
-0.019641095772385597,
-... |
<p>Why is the earth's iron core stationary, while the liquid matal circles around it creating the magnetic shield. Don't understand how can the entire planet rotate where as the planet's center is stationary. Anyone could explain this?</p> | g14483 | [
0.012340282090008259,
0.05736863240599632,
0.005384783260524273,
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0.01216206606477499,
0.01611... |
<p>It is known that there are real numbers that can't be calculated (non-computable numbers). Quite probably that some physical phenomena (it is possible still undetected) depend on this numbers. Whether means it, what we can't construct the theory of this phenomenon and-or (if probably to construct) to do predictions (i.e. to check up this theory in practice)?</p> | g14484 | [
-0.02112279273569584,
0.048288892954587936,
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0.03... |
<p>Regarding a question on <a href="http://bricks.stackexchange.com/a/466/132">another Stack Exchange site</a> which identifies a weighted Lego brick to be some kind of metal, I wanted to know how to identify a metal, perhaps limited to household objects like scales or a bowl of water to measure volume. </p> | g14485 | [
0.08020709455013275,
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0.006916368380188942,
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0.0497385673224926,
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0.03201... |
<p>I know this may actually be a chemistry question, but I don't know where else to ask.</p>
<p>When I have a soda plastic bottle, I turn it upside down so the soda fills the empty space, and bubbles start growing on the walls of the plastic.</p>
<p>So, why does this happen?
I can think of something like a film forming around the empty space, and it reacts with the liquid, if so, what is it? why does it react that way?</p> | g14486 | [
0.07787352055311203,
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0.03698491305112839,
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<p>For a physics major, what are the best books/references on Greens functions for self-studying?</p>
<p>My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas.</p> | g14487 | [
0.02652624249458313,
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<p>Each spring enormous amounts of water rise up in trees and other vegetation. What causes this stream upwards? </p>
<p>Edit: I was under the impression that capillary action is a key factor: the original question therefore was: what are the fundamental forces involved in capillary action?</p>
<p>Can perhaps anybody make an estimation of the amount of gravity energy that is involved in this process earth-wide each season?</p> | g14488 | [
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<p>I'm following Zettili's QM book and on p. 39 the following manipulation is done,</p>
<p>Given a localized wave function (called a wave packet), it can be expressed as $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi}} \int_{-\infty}^{\infty} \phi(k) e^{i(kx-\omega t)} dk$$ Now use the de broglie relations: $p = \hbar k$ and $E = \hbar \omega$ and define $\tilde{\phi}(p) = \phi(\frac{k}{\hbar})$.</p>
<p>This should yield $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi \hbar}}\int_{-\infty}^{\infty} \tilde{\phi}(p) e^{i(px-E t)/ \hbar} dp$$ but I get $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi} \hbar}\int_{-\infty}^{\infty} \phi\biggl(\frac{p}{h}\biggr) e^{i(px-E t)/ \hbar} dp$$ when I make the change-of-variable $k=\frac{p}{\hbar}$. What am I missing?</p> | g14489 | [
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<p>I would like to derive the following two well-known formulas that work for crystal lattice [1]:
$$
F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G})
= {1\over\Omega_\mathrm{cell}} \int_{\Omega_\mathrm{cell}} f(\mathbf{x})
e^{-i\mathbf{G} \cdot \mathbf{x}}\,d^3 x
$$
$$
F^{-1}[\tilde f(\mathbf{G})] = f(\mathbf{x}) = \sum_{\mathbf{G}}
\tilde f(\mathbf{G}) e^{+i\mathbf{G} \cdot \mathbf{x}}
$$
Specifically, I want to derive them from the general 3D Fourier transform (the other way is to simply plug the second formula into the first, one obtains a delta function and obtains an identity --- see this <a href="http://physics.stackexchange.com/questions/88169/">question</a> where I have worked this out in details, but here I don't want to use this approach).
How do I derive the second formula?</p>
<p>Following [1], here is how to derive the first formula from the basic definition of a 3D Fourier transform, divided by the volume of the crystal $\Omega_\mathrm{crystal}$ (to make it finite):</p>
<p>$$
F[f(\mathbf{x})] \equiv \tilde f(\boldsymbol\omega)
= {1\over\Omega_\mathrm{crystal}}\int_{\Omega_\mathrm{crystal}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot
\mathbf{x}}\,d^3 x =
$$
$$
= {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} \int_{\Omega_\mathrm{cell}}
f(\mathbf{x}+\mathbf{T}(\mathbf{n}))
e^{-i\boldsymbol\omega \cdot (\mathbf{x}+\mathbf{T}(\mathbf{n}))}\,d^3 x =
$$
$$
= {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} \int_{\Omega_\mathrm{cell}} f(\mathbf{x})
e^{-i\boldsymbol\omega \cdot (\mathbf{x}+\mathbf{T}(\mathbf{n}))}\,d^3 x =
$$
$$
= {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} e^{-i\boldsymbol\omega \cdot \mathbf{T}(\mathbf{n})} \int_{\Omega_\mathrm{cell}} f(\mathbf{x})
e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x =
$$
$$
= {1\over\Omega_\mathrm{crystal}} N_\mathrm{cell} \int_{\Omega_\mathrm{cell}} f(\mathbf{x})
e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x =
$$
$$
= {1\over\Omega_\mathrm{cell}} \int_{\Omega_\mathrm{cell}} f(\mathbf{x})
e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x
$$
In here, the function $f(\mathbf{x})$ is periodic: $f(\mathbf{x}+\mathbf{T}(n_1, n_2, n_3)) = f(\mathbf{x})$ and the sum $\sum_\mathbf{n} e^{-i\boldsymbol\omega \cdot \mathbf{T}(\mathbf{n})} =
\sum_\mathbf{n} 1 = N_\mathrm{cell}$ for $\boldsymbol\omega=\mathbf{G}$, where $\mathbf{G}$ are reciprocal space vectors (defined by $e^{i\mathbf{G} \cdot \mathbf{T}(\mathbf{n})} = 1$). For $\boldsymbol\omega\neq\mathbf{G}$, the sum is bounded, and so in the limit $\Omega_\mathrm{crystal}\to\infty$ the factor before the integral sign above goes to zero.</p>
<p>For the second formula, there is no hint in [1] how to proceed. Here is my best effort so far:
$$
F^{-1}[\tilde f(\boldsymbol\omega)] = f(\mathbf{x})
= {\Omega_\mathrm{crystal}\over(2\pi)^3}\int_{-\infty}^{\infty}
\tilde f(\boldsymbol\omega) e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega
=
$$
$$
= {\Omega_\mathrm{cell}N_\mathrm{cell}\over(2\pi)^3}\int_{-\infty}^{\infty}
\tilde f(\boldsymbol\omega) e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega
=
$$
$$
= {N_\mathrm{cell}\over\Omega_\mathrm{BZ}}
\sum_{\mathbf{G}}
\int_{\Omega_\mathrm{BZ}}
\tilde f(\mathbf{G}+\boldsymbol\omega)
e^{+i(\mathbf{G}+\boldsymbol\omega) \cdot \mathbf{x}}\,d^3 \omega
=
$$
$$
= {N_\mathrm{cell}\over\Omega_\mathrm{BZ}}
\sum_{\mathbf{G}} e^{+i\mathbf{G} \cdot \mathbf{x}}
\int_{\Omega_\mathrm{BZ}}
\tilde f(\mathbf{G}+\boldsymbol\omega)
e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega
= \cdots
$$
Here $\Omega_\mathrm{BZ} = {(2\pi)^3 \over \Omega_\mathrm{cell}}$ is the volume of the Brillouin zone. I have moved the integration over $\boldsymbol\omega$ to the Brillouin zone. As you can see, it's quite close, but I can't figure out how to finish it. Any ideas?</p>
<p>[1] Martin, R. M. (2004). Electronic Structure -- Basic Theory and Practical Methods (p. 642). Cambridge University Press.</p> | g14490 | [
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0.0440874807536602,
-0.02... |
<p>In the <a href="http://en.wikipedia.org/wiki/Drude_model" rel="nofollow">Drude model</a> they derive a formule for the conductivity of a conductor.
I wonder though how the main free time $\tau$ is defined in this formula.
Wikipedia says that it is "the average time between subsequent collisions".
But I have two possible interpretations of this: </p>
<ol>
<li>the average time an electron travels before colliding (which it seems to imply).</li>
<li>The average amount of time electrons have been travelling at a given time $t$ ($\tau$ will be substantially smaller in this definition).</li>
</ol>
<p>The first definition seems to me how they describe it, while the second definition seems to be implied by the formulas. I wonder the same for "mean free path", which seems to be analogous.</p> | g14491 | [
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0.019010677933692932,
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0.045101966708898544,
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0.017620917409658432,
0.04079968109726906,
... |
<p>So, in my previous question <a href="http://physics.stackexchange.com/questions/88001/where-does-this-formula-for-prediction-of-a-multiple-wave-come-from">Where does this formula for prediction of a multiple wave come from?</a>, I get that using this picture: </p>
<p><img src="http://i.stack.imgur.com/NPiCC.png" alt="enter image description here"></p>
<p>we have so far written the time it takes for a multiple to travel in the form:</p>
<p>$$T_2 = T_1 + \left( \frac{1}{v} \frac{2h}{\cos (\theta)} \right)$$</p>
<p>where</p>
<p>$$\cos(\theta) = \frac{h}{\sqrt{\frac{x^2}{4} + h^2}}.$$</p>
<p>AFter substituting in for $\cos(\theta)$, this can then be re-written as</p>
<p>$$T_2 = T_1 + \sqrt{\tau^2 + \frac{x^2}{v^2}}$$</p>
<p>where $\tau = \frac{4h^2}{v^2}$. From here though, it then says that we can represent this in the frequency domain as:</p>
<p>$$R h(\omega) e^{2 \pi i f (T_1 + \sqrt{\tau^2 + \frac{x^2}{v^2}}}$$</p>
<p>where $R$ is some constant including decay and $h(\omega)$ is the source wavelet.</p>
<p>I don't understand how they have managed to transform the previous equation into this one. Where has the exponential come from?</p> | g14492 | [
0.007301573641598225,
0.0029040018562227488,
-0.02170894481241703,
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0.05787212401628494,
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0.05607251450419426,
0.04763367399573326,
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-0.03569129481911659,
-0.05859631672501564,
0.018331104889512062,
0.04308247193694115,
0.011... |
<p>I was always taught that any accelerating charge produces radiation, but I don't think this condition is sufficient condition. For instance, any free charge on Earth is accelerated due to Earth orbiting the Sun but it doesn't produce radiation. </p> | g14493 | [
0.05785328149795532,
0.08493455499410629,
0.01736992411315441,
0.018118901178240776,
0.025380311533808708,
0.0661168023943901,
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-0.03542632982134819,
0.03678091615438461,
0.037276070564985275,
-0.000033293676096946,
0.0120... |
<p>I came accross a question about verifying the transverse of electric field in Peskin and Schroeder's QFT p179.</p>
<p>Given</p>
<blockquote>
<p>$$ \mathcal{A}^{\mu}(\mathbf{k}) = \frac{ -e}{| \mathbf{k} | } \left( \frac{p'^{\mu}}{k \cdot p'} - \frac{p^{\mu}}{k \cdot p} \right) \tag{6.7} $$</p>
</blockquote>
<p>and </p>
<blockquote>
<p>$$ \mathcal{E} = -i \mathbf{k} \mathcal{A}^0 (\mathbf{k}) + i k^0 \mathcal{A}(\mathbf{k}) \tag{6.9} $$</p>
</blockquote>
<p>it is said</p>
<blockquote>
<p>you can easily check that the electric field is transverse: $\mathbf{k} \cdot \mathcal{E}(\mathbf{k}) =0 $</p>
</blockquote>
<p>I tried to vertify this almost trivial relation and I didn't get it...
Here is my attempt
$$ \mathbf{k} \cdot \mathcal{E}(\mathbf{k}) = \left( \frac{ -e}{| \mathbf{k} | } \right) \left[ - i \mathbf{k} \frac{ \mathbf{k} p'^0 - k^0 \mathbf{p}'}{ k \cdot p'} + i \mathbf{k} \frac{ \mathbf{k} p^0 - k^0 \mathbf{p}}{ k \cdot p} \right] \tag{1}$$</p>
<p>I may use $p^{\mu}=(p^0, \mathbf{0})$ (since the particle is initially at rest) and $k^0=|\mathbf{k}|$. It seems that Eq. (1) is still not necessarily zero. Would you provide, at least, any hint how to show Eq. (1) is zero? </p> | g14494 | [
0.06773684918880463,
0.010001679882407188,
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0.0870584174990654,
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0.008087549358606339,
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0.010894249193370342,
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0.027677156031131744,
0.0034114629961550236,
0.00... |
<p>If an astronaut leaves planet $A$ for planet $B$ at speed $v$, will the time (measured by the astronaut's clock) that it takes for the astronaut to reach planet $B$ be less than the distance between the planets divided by the speed because of length contraction?</p> | g14495 | [
0.010366859845817089,
0.07922133058309555,
0.0013381440658122301,
-0.009032395668327808,
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0.039892468601465225,
0.027291400358080864,
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0.0038734592963010073,
-0.024950528517365456,
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-0.027786055579781532,... |
<p>I was reading about WiTricity (<a href="http://en.wikipedia.org/wiki/WiTricity" rel="nofollow">http://en.wikipedia.org/wiki/WiTricity</a>) a technology developed by MIT to wirelessly transmit electricity through resonance, and I have this question:</p>
<p>Given the phenomenon of resonant inductive coupling which wikipedia defines as: </p>
<blockquote>
<p>the near field wireless transmission of electrical energy between two
coils that are tuned to resonate at the same frequency. <a href="http://en.wikipedia.org/wiki/Resonant_inductive_coupling" rel="nofollow">http://en.wikipedia.org/wiki/Resonant_inductive_coupling</a></p>
</blockquote>
<p>And the Schumann resonances of the earth ( ~7.83Hz, see wikipedia), would it be theoretically possible to create a coil that resonates at the same frequency or one of it's harmonics (7.83, 14.3, 20.8, 27.3 and 33.8 Hz) to generate electricity?</p>
<p>I have a feeling that these wavelengths may be too big to capture via resonance (they are as large as the circumference of the earth if I understand it correctly), so alternatively would it be possible to create a coil that resonates with one of the EM waves that the sun sends our way?</p> | g14496 | [
-0.004816057160496712,
-0.0008691713446751237,
0.004769127815961838,
-0.017266394570469856,
0.002331033581867814,
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0.02788684144616127,
-0.011300977319478989,
0.01621430553495884,
0.02174164541065693,
-0.005796285346150398,
... |
<p>Do you think it is possible to earn a PHD degree in Theoretical Physics on your own? I have a M. Sc. in Applied Computer Science but I really want to go in the direction of Physics. Is there a way one could perform a significant work in this field without a team of other PHD-students and/or lab experiments? Is it even officially possible to acquire the PHD degree in Physics if one has B. Sc. in Applied Mathematics and Ms. Sc. in Applied CS only?</p> | g14497 | [
-0.008733521215617657,
0.05058293417096138,
0.011287519708275795,
0.013874308206140995,
0.006385683082044125,
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0.02335427515208721,
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0.01489215437322855,
0.009375154040753841,
0.02265201322734356,
0.01240199338644743,
0.008... |
<p>Question: In order to protect himself from the rain, a person is standing holding an umbrella at right angle to the horizontal surface. The rain is falling at 10m/s when the velocity of the wind is zero. Suddenly, wind begins to blow at a speed of 20m/s, towards 30 degrees south of west. Now by what angle does the person have to turn his umbrella in order to protect himself from the rain? </p>
<p>What I did:
Split the speed of wind into components:
$$v(x) = 20\cos30$$
$$v(y) = 20\sin30 + 10$$</p>
<p>Thus, I found the angle that the resulting rain makes with the vertical:
$$\tan\theta = \frac{10\sqrt{3}}{20} = \frac{\sqrt{3}}{2}$$
$$\theta = \arctan\frac{\sqrt{3}}{2}$$</p>
<p>However, the answer is given to be $\arctan2$, where have I made a mistake?</p> | g14498 | [
0.05906971916556358,
-0.0341203473508358,
0.007415079511702061,
-0.05630914866924286,
0.07079971581697464,
0.015422086231410503,
0.10460110008716583,
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-0.011782640591263771,
0.004615101031959057,
0.02724311500787735,
-0.020794929936528206,
-0.04... |
<p>When taking a picture with old fashioned film what sets the resolution of the picture? Is it the wavelength, or the chemical makeup of the film?</p> | g14499 | [
-0.02416829764842987,
-0.005992575082927942,
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0.0054888175800442696,
0.02386835403740406,
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0.013602238148450851,
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-0.04224611818790436,
-0.011917575262486935,
0.025916507467627525,
0.010577964596450329,
... |
<p>Can anyone give examples of mechanics problems which can be solved by <a href="http://en.wikipedia.org/wiki/Lagrangian_mechanics" rel="nofollow">Lagrange equations</a> of the <a href="http://www.encyclopediaofmath.org/index.php?title=Lagrange_equations_%28in_mechanics%29&oldid=17218" rel="nofollow">first kind</a>, but not the <a href="http://www.encyclopediaofmath.org/index.php?title=Lagrange_equations_%28in_mechanics%29&oldid=17218" rel="nofollow">second kind</a>?</p> | g14500 | [
0.08526469022035599,
0.006495836656540632,
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0.05848349630832672,
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0.03155342862010002,
0.0032170661725103855,
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0.03500397503376007,
0.0... |
<p>I understand what are virtual photons and the difference between the real and virtual photons.</p>
<p>However, I am not able to clearly distinguish the difference between the '<a href="http://en.wikipedia.org/wiki/Phonon" rel="nofollow">phonons</a> and virtual photons'. Both are involved in non radiative energy transfer. </p> | g14501 | [
0.014846784062683582,
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0.008203980512917042,
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0.03580032289028168,
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0.04076722264289856,
-0.014063038863241673,
-0.001... |
<blockquote>
<p>A disk with a rotational inertia of 5.0 kg·m2 and a radius of 0.25 m
rotates on a fixed axis perpendicular to the disk and through its
center. A force of 2.0 N is applied tangentially to the rim. As the
disk turns through half a revolution, the work done by the force is:</p>
</blockquote>
<p>This was another question on my test that I missed. I didn't know where to get started. I know the work done is the integral of the torque. However, I don't know how to get a function that I can integrate from this example. Any ideas?</p> | g14502 | [
0.054583825170993805,
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-... |
<p>Do surfaces that reflect visible light efficiently also reflect UV light? If not, are there surfaces that do? </p>
<p>(I have a large array of UV LEDs that I need to make larger and more diffuse, so I'm considering reflecting it off nearby surfaces and into some diffuse material.)</p> | g14503 | [
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0.... |
<p>Apologies if this question is a bit too chemistry-flavoured.</p>
<p>In electron paramagnetic resonance spectroscopy, there's a practically ubiquitous convention of plotting the first derivative of the absorption with respect to field, rather than the simple absorption. Why is this?</p> | g14504 | [
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0.01526... |
<p>So it has been asserted that a Stirling engine with proper regeneration can be made reversible. It will consist of two isothermal quasi-static processes connected by two constant-volume processes. The working gas is supposed to be heated/cooled through infinitesimal temperature differences (thus preventing the creation of entropy).</p>
<p>I'd like to know an explanation on the working gas properties during each step of the cycle. What are the state conditions for the gas at each of the points in the cycle (including inside the regenerator)? I've been looking for a the past week and found nothing.</p> | g14505 | [
-0.008219130337238312,
-0.03787141665816307,
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<p>As described in the wikipedia article on <a href="http://en.wikipedia.org/wiki/Wave_function_collapse#History_and_context" rel="nofollow">wave function collapse</a>, the mathematical formulation of quantum mechanics postulates that wave functions change according to two processes:</p>
<ol>
<li>When not being observed, a wave function will evolve (deterministically) according to some relevant differential equation, e.g., Schrödinger's equation. </li>
<li>Upon measurement, a wave function will collapse (probabilistically) to one of its component eigenfunctions. </li>
</ol>
<blockquote>
<p>What I'd like to understand is how quickly after a measurement a particle's wave function will wander away, according to (1), from whatever eigenfunction it collapsed to.</p>
</blockquote>
<p>As a concrete example (I hope!), consider <a href="http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment#Sequential_experiments" rel="nofollow">two Stern-Gerlach experiments set up in sequence</a>, both arranged to measure the z component of spin, in which we feed to the second experiment only the spin up half of the beam from the first experiment. When one learns about this in an introductory QM course, it is said that the observed output from the second experiment will still be all spin up. But is this still true if the experiments are spaced really far apart? How will the time evolution in (1) affect the outcome then? (Or is there something I have missed that prevents time evolution from affecting the intermediate beam?) If the observed outcome from the second experiment does depend on the distance between the experiments, then it seems this would give a way to measure (at least in principle) how rapidly a pure eigenstate can degenerate (by measuring, for example, the distance between the two experiments needed to get an output beam from the second experiment which is 75% up, 25% down). </p> | g14506 | [
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0.... |
<p>A cannon is placed at the bottom of a cliff 85 m high. If the cannon is fired straight upward, the cannonball just the reaches the top of the cliff. </p>
<p>a) Calculate the initial speed of the cannonball.
b) Suppose a second cannon is placed at the top of the cliff and fired
horizontally with the same initial speed as part (a). Prove numerically that the range of this cannon is the same as the maximum range of the cannon from the base of the cliff.</p>
<p>My work for part A
<img src="http://i.imgur.com/TQMSiZY.jpg" alt="work">
How do I know what the velocity of the ball is at the top of the cliff? </p> | g14507 | [
0.08117172122001648,
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0.01650480180978775,
0.008003... |
<p>I would like some help in solving this problem. </p>
<p><img src="http://i.stack.imgur.com/fjVLX.png" alt="Height is 15 cm , base is 88 cm"></p>
<p>Height is 15 cm , base is 88 cm</p>
<p>My attempt:
$$(119.3 + m)(\sin(\theta)) = (88*9.8)$$
$$ 15^2 + 88^2 = 89.269^2 $$</p>
<p>$$ \sin^{-1} = 15/89.269 $$</p>
<p>$$ (119.3 + m)(9.6733) = (88*9.8) $$</p>
<p>$m = -30.14$ ....Am I missing something? I feel like it has something to do with the force of friction...</p> | g14508 | [
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0.04467744007706642,
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-0.0280271... |
<p>Can retrocausality resolve the paradoxes of quantum mechanics? The Copenhagen interpretation presumes something has no property until it is measured, but retrocausal interpretations explain that away by claiming that thing has a definite property before measurement, but the choice of which property is affected by the configuration of the measuring apparatus in its future. The philosopher Huw Price has written a lot about this. Nonlocal entanglements are also explained away because entangled pairs are connected by going to the past, then back up to the future. Or going to the future, then back to the past without travelling faster than light at any time.</p> | g14509 | [
0.022111831232905388,
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0.01... |
<p>Engineering textbooks constantly use the concept of 'phase space' (see e.g. <a href="http://www.cs.cmu.edu/~baraff/sigcourse/notesc.pdf" rel="nofollow">http://www.cs.cmu.edu/~baraff/sigcourse/notesc.pdf</a>). That is, they think of the state of a mechanical system as a high-dimensional geometric space. For example, for a system of particles, the phase space has $6n$ dimensions, where $n$ is the number of particles.</p>
<p>My understanding is that the notion of phase space initially developed among theoretical physicists (possible in conjunction with quantum mechanics? or possibly before that point?). Therefore, I'm wondering when and how engineers began to adopt this terminology and concept from physicists.</p>
<p>It might have something to do with differential equations, since the concept of representing the state of a mechanical system as a point in a space becomes obvious when you view ODE's in terms of flows on euclidean spaces.</p>
<p>NOTE: The phase space concept was invented by theoretical physicist and mathematician Willard Gibbs.</p> | g14510 | [
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0.049344077706336975,
-... |
<p>This is prompted by the strong claims made in <a href="http://www.sciencemag.org/content/332/6034/1170.full.html">Science 332, 1170 (2011)</a> to have observed trajectories of photons, "something all of our textbooks and professors had always told us was impossible". I'm suspicious of this claim because they work entirely within quantum theory, and it seems to me that in making weak measurements at two points on what they say is a trajectory, they do nothing else than make unsharp measurements of incompatible observables, a concept that was first introduced, to my knowledge, by Busch and Lahti in <a href="http://prd.aps.org/abstract/PRD/v29/i8/p1634_1">Phys. Rev. D 29, 1634–1646 (1984)</a>, and that is now much used in quantum information. Certainly observables that are at time-like separation are in general incompatible, so an unsharp measurement approach would seem applicable. </p>
<p>A brief account of the Science paper mentioned above can be found <a href="http://www.physorg.com/news/2011-06-quantum-physics-photons-two-slit-interferometer.html">here</a>, amongst other places, and a few quotes can be found <a href="http://physicsandphysicists.blogspot.com/2011/06/observing-average-trajectories-of.html">here</a>, but, this being Science, I couldn't find a preprint. I was first alerted to this paper by <a href="http://www.bbc.co.uk/news/science-environment-13626587">a page at the BBC</a>. The claim highlighted by ZapperZ, "Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm–de Broglie interpretation of quantum mechanics (8), although the reconstruction is in no way dependent on a choice of interpretation", seems to me particularly egregious, insofar as both their choice of what experiment to do and of what to do with their raw data seem partly driven by the form they want to present their results in. It seems entirely possible to present the data in terms of unsharp measurements and POVMs, which would give relatively less support to the de Broglie-Bohm interpretation.</p> | g14511 | [
-0.00658615306019783,
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0.03542786464095116,
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0.0094... |
<p>Hi I read the following puzzle from an old text book long time ago. However it doesn't provide the answer. So what is the solution?</p>
<p>Let's suppose a car is going to park to a garage and the garage is exactly the same size of the the static size of the car. Now the car is running in a speed that in its point of view the garage is just 1/10 of its static size. So it will consider the garage is too small to fit. On the other hand, the garage manager consider the car is only 1/10 of the garage size so it definitely fit.</p>
<p>The question is to explain what will ACTUALLY happen if the car try to park in the garage, and what the garage manager and the driver will see. </p> | g14512 | [
0.038800738751888275,
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0.021710988134145737,
... |
<p>There's ample direct evidence for the existence of galactic and stellar mass black holes. However, there is no such direct evidence of primordial black holes, those formed after the Big Bang. A recent paper [http://arxiv.org/abs/1106.0011] describes oscillations the Sun might undergo if it encountered a primordial black hole. The theory that primordial black holes exist hasn't had experimental backing. Why should one believe they actually exist? Is this science or "guesstimating?"</p> | g14513 | [
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0.0434313... |
<p><a href="http://physics.stackexchange.com/questions/290/what-really-allows-airplanes-to-fly">A previous question by David Zaslavsky</a> was a request for a broad, "how things work" type of explanation of the lift of an airfoil. The answers given there are enlightening, but don't address a more specific question I have.</p>
<p>First a summary of my current understanding. There is no logical reason to assume that parcels of air take equal time to travel above and below the wing. If we did make such an assumption, Bernoulli's principle would make all kinds of incorrect predictions. "How things work" explanations (e.g., <a href="http://www.allstar.fiu.edu/aero/Flightrevisited.pdf" rel="nofollow">this one</a>) often speak as though Bernoulli's principle is one effect, and then there are some other effects, such as conservation of momentum; it's implied that these effects are additive, and it's often stated that the Bernoulli effect is negligible. But in fact Bernoulli's principle is, under certain hypotheses, equivalent to conservation of energy. Conservation of energy and conservation of momentum are not effects to be added. They're two physical laws, both of which need to be obeyed.</p>
<p>It's only under certain hypotheses that Bernoulli's principle is equivalent to conservation of energy. It requires nonviscous flow, and the version given in freshman physics textbooks also assumes incompressible flow, although there is also a version for compressible flow. The WP article on Bernoulli's principle states that it's usually valid for Mach numbers lower than about 0.3, which I guess might be valid for birds and some small planes (cruise speed for a Cessna 172 is Mach 0.25), but not passenger jets.</p>
<p>Based on this, I can envision two logical possibilities:</p>
<p>(1) The hypotheses of Bernoulli's principle hold for birds and some small planes. The equation gives at least an approximately correct result for the net force on the wing, but there is no obvious, simple way to know what velocities to assume.</p>
<p>(2) The hypotheses of Bernoulli's principle fail in these cases.</p>
<p>Which of these is correct? If #2, which hypothesis is it that fails? Is the incompressible-flow version inapplicable but the compressible-flow version OK?</p> | g14514 | [
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0.04... |
<blockquote>
<p>A man walking in rain at a speed of 3kmph find rain to be falling
vertically. When he increases his speed to 6 kmph, he finds rain
meeting him at an angle of 45 deg to the vertical. What is the speed
of the rain??</p>
</blockquote>
<p>My approach:</p>
<p>$ \overline{V_{\operatorname{rain}}} - \overline{V_{\operatorname{man}}} = \overline{V_{\operatorname{rain relative to man}}}$ </p>
<p>$\Longrightarrow \overline{V_{\operatorname{rain}}} = \overline{V_{\operatorname{rain relative to man}}} + \overline{V_{\operatorname{man}}} $ </p>
<p>Equating velocity of rain wrt ground in both cases, we get</p>
<p>$9+ x^2 = 36+ x^2 - 12x \frac{1}{\sqrt{2}}$
where x is velocity of rain wrt to man.</p>
<p>so x becomes $9\frac{\sqrt{2}}{4}$
but that means velocity of rain wrt ground is (according to the first case, where velocity of man is 3 kmph) :</p>
<p>$\sqrt{9 + {(9 \frac{\sqrt{2}}{4}})^2}$</p>
<p>which is the wrong answer. Is it a calculation mistake?? Or is my basic idea incorrect? Please help!</p> | g14515 | [
0.06840521097183228,
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0.028013629838824272,
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0.029183460399508476,
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... |
<p>I have a question, has the <strong>Universe</strong> been found to come in Discrete Quantum, like Quantum Physics or is it Continuous in Nature?</p>
<p>I was wondering if time was like a <strong>Continuum</strong>, like the fluid in a soft drink for example, and if it "comes into existence" in <strong>Discrete Quantum</strong>, as if someone was taking a "sip" from that drink?</p>
<p>Is there any merit to this kind of <strong>analogy</strong>, or is it all unknown?</p>
<p>I've heard there comes a point when space/time becomes what appears to be <strong>indeterminate</strong> and almost radical departure from the norm of common experience.</p> | g446 | [
0.0012992927804589272,
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0.007101933006197214,
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<p><img src="http://i.stack.imgur.com/1bLlk.png" alt="elevator pulley system"></p>
<p>This is the scenario where my mass is $60 kg$, the mass of the elevator is $30kg$, and due to a malfunction, I have to hold myself and the elevator at rest. The question is, if there is a weighing scale under me, (of negligible mass), then what force will it measure (in newtons)?</p>
<p>My approach:
The total mass of the system is $90kg$ which is approximately $900 N$ for equilibrium, tension in the rope is $900N$ because I am giving a downward force of $900N$, so according to Newton's 3rd Law, upward force on me by the rope must be $900N$. This means I should go up, and the weighing scale must not show any reading. But that is not among the options to choose from. Guidance rather than a direct answer would be appreciated.</p> | g14516 | [
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0.020518096163868904,
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<p>How can any matter contract to its <a href="https://en.wikipedia.org/wiki/Schwarzschild_radius" rel="nofollow">Schwarzschild radius</a> if gravitational time dilation clearly states that all clocks stop at that point. So any contraction any movement would stop. If that is so why all this talk about objects which can never form in the first place?</p> | g447 | [
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0.02... |
<p>Say you have a magnetic field $\vec{B}=(0,0,B_0)$. Then the Schrodinger Equation Hamiltonian for a spin-2 particle of charge $e$ moving in this field is:</p>
<p>$$H = \frac{1}{2m}[\vec{p}-e\vec{A}]^2-\vec{\mu}\cdot\vec{B},$$</p>
<p>where $\vec{A}=(-\tfrac{1}{2}y,\tfrac{1}{2}x,0)$ is the magnetic vector potential.</p>
<p>You can find the speed by finding this commutator: $\frac{d\vec{r}}{dt}=i[H,\vec{r}]$.</p>
<p>When I did out this calculation, I ended up with $\frac{\vec{p}-e\vec{A}}{m}$. However, I'm told that I should have gotten $\frac{\vec{p}}{m}+\vec{\omega}_c\times\vec{r}$, where $\vec{\omega}_c$ is the cyclotron frequency. This has the value $\vec{\omega}_c=-\frac{e\vec{B}}{m}$, but if you plug that in, you get $\vec{\omega}_c\times\vec{r}=-\frac{2e\vec{A}}{m}$. Why is this off by a factor of 2?</p> | g14517 | [
0.006333140190690756,
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0.03232048451900482,
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<p>Let the Einstein-Hilbert action be rewritten as a functional of the tetrad $e$ (units shall be set to $1$) such that
$S_{EH}(e)=\int \frac{1}{2}\epsilon_{IJKL}~e^I\wedge e^J\wedge F^{KL}(\omega(e))$, where $\epsilon$ is the Levi-Civta-symbol as usual, $F$ is the curvature of the spin connection $\omega$ and $I,J,K,L$ denote internal indices, indicating that the object carries a rep. of the Lorentz group.
How do I get back to the usual action $S_{EH}(g)=\int~d^4x\sqrt{-g}R$? I am trying to figure it out using identities but can't finish till the end. Could anybody give a little more detailed account, please? Thank you.</p> | g14518 | [
-0.04811345785856247,
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<p>There are countless of hilarious youtube videos like <a href="http://www.youtube.com/watch?v=pegHuqSHCRg" rel="nofollow">this</a> one here, in which you can see that the carousel causes people to be pulled outwards. </p>
<ul>
<li><p>What force is actually working on them, and why does it pull them outwards (instead of for example inwards)?</p></li>
<li><p>Can you give me some other (notable) example of this force in everyday life?</p></li>
</ul> | g14519 | [
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-0... |
<p>When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
i\hbar\frac{d}{dt}\hat{U}=H\hat{U}.
$$
And in this case Hamiltonian in Heisenberg picture ($H_{H}$) is just the same that Hamiltonian in Schrödinger picture ($H_{S}$), i.e. it commutes with $\hat{U}$.
Now i have Hamiltonian that depens explicitly on time. Specifically,
$$H_{S}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega \hat{q}^2-F_0sin(\omega_0t)\hat{q}$$.</p>
<p>And in my problem I need to calculate $H_H$ (Hamiltonian in Heisenberg picture).</p>
<p>I've found that differential equation for $\hat{U}$ (I've mentioned about it above.) has generally solution in the form (with $U(0)=1$)
$$U(t)=1+\xi\int\limits_0^t H(t')\,dt'+ \xi^2\int\limits_0^t H(t')\,dt'\int\limits_0^t' H(t'')\,dt''+\xi^3\int\limits_0^t H(t')\,dt'\int\limits_0^t' H(t'')\,dt''\int\limits_0^t'' H(t''')\,dt'''+...$$</p>
<p>So my questions are:</p>
<ul>
<li>Is there other ways to calculate $\hat{U}$, could give a link or tell me about them?</li>
<li>If you know form of the solution for my case, please tell me.</li>
<li>If you know any articles or papers articles on this topice, please link them to me, too.</li>
</ul> | g14520 | [
-0.05201373249292374,
0.05454521253705025,
-0.01634844020009041,
-0.0016344727482646704,
0.010879972018301487,
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0.0808926448225975,
0.03164622187614441,
-0.0025072554126381874,
0.023699477314949036,
-0.05897318944334984,
0.010390393435955048,
-0.005696300882846117,
0.... |
<p>After watching the recent "space jump" a question arose. Why can a balloon not float into space? Can one be made/designed to do this?</p>
<p>Next, everything in orbit is falling back to earth. It only maintains its altitude with speed, about 26,000mph give or take. This number means what, 20,000+ mph relative to what? </p>
<p>If the shuttle or station was to stop or approach zero mph it would fall to Earth, right? The balloon was at 128,000 feet and was only traveling at the Earth's rotational velocity of 1600mph. As viewed from the ground zero mph since it came down almost in the same stop. So what happens between say 130,000 feet and space that you need to increase your speed by 24,000mph?</p>
<p>Last if we were at a high enough orbit and "stopped", where we would not fall back to Earth, would we be able to watch the earth leave us. Could we wait there for a year for the Earth to return? How far out would this be?</p> | g14521 | [
0.03955185040831566,
0.06055484339594841,
0.01889888569712639,
0.018827103078365326,
-0.0028425499331206083,
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0.02502780221402645,
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-0.08774899691343307,
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0.047146547585725784,
0.020662600174546242,
-0.013... |
<p>This question came up as an exercise in a first year undergraduate course I was a TA for. It turned out to be a lot more difficult (impossible?) than anticipated...</p>
<blockquote>
<p>Two platforms of mass $M_1$ and $M_2$ ($M_1\neq M_2$) are connected by
a spring of constant $k$, and are initially at rest with the spring
unstretched, sitting on a frictionless surface. A man of mass $m$
stands on one platform and begins to run, always with a constant speed
$v$ measured relative to the platform he is running on. What is the
maximum speed reached by the other platform, relative to the ground?</p>
</blockquote>
<p>My intuition is telling me that I need to know something about how the man gets from rest to his constant speed (instantaneously? very slowly? with some smooth acceleration?) to solve this, but I haven't been able to prove to myself that this is required. If this is a requirement, I think the most reasonable assumption would be that he reaches his full speed instantaneously.</p>
<p>What I'm most interested in is whether this problem can be tackled with a typical freshman's toolbox - simple arguments around conservation of energy/momentum, no/very limited differential equations, basic calculus. I can see an easy way to get an upper bound on the maximum speed from energy/momentum considerations, but I don't see a way to check if this speed is ever reached.</p> | g14522 | [
0.0758446529507637,
0.009993054904043674,
0.021761579439044,
0.02468433417379856,
0.03744436055421829,
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0.014885392040014267,
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-0.04684605076909065,
0.0287358835... |
<p>Wikipedia <a href="http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking#Chiral_symmetry" rel="nofollow">states</a> that the spontaneous breaking of chiral symmetry <em>"is responsible for the bulk of the mass (over 99%) of the nucleons"</em>.</p>
<p>How do the nucleons gain mass from the spontaneous breaking of chiral symmetry? Why don't leptons gains mass from it? What is the role of the Higgs field in this all?</p> | g14523 | [
0.030617188662290573,
0.04271697998046875,
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0.009954564273357391,
0.07612001150846481,
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0.0031777899712324142,
0.0406... |
<p>The DMV manual says that </p>
<blockquote>
<p><em>The faster you go, the less time you
have to avoid a hazard or collision.
The force of a 60 mph crash is not
just twice as great as a 30 mph crash;
it’s four times as great!</em></p>
</blockquote>
<p>My physics is quite rusty, so I could not figure it out. I guess the above statement is correct, but how do we prove it?</p>
<p><strong>Edit</strong></p>
<p>I figured this out myself, but alternative methods or new ways of understanding are still welcome.</p> | g436 | [
0.04625893011689186,
0.06294197589159012,
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0.03617733344435692,
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0.0025466633960604668,
-0.023872245103120804,
0.01146798487752676,
-0.0342206209897995,
0.0334809273481369,
-0.031605903059... |
<p>I am trying to understand the Feynman path integral by reading the book from Leon Takhtajan.</p>
<p>In one of the examples, there is a full explanation of the calculation of the propagator </p>
<p>$$K(\mathbf{q'},t';\mathbf{q},t) = \frac{1}{(2\pi\hbar)^n} \int_{\mathbb{R}^n} e^{\frac{i}{\hbar}(\mathbf{p}(\mathbf{q'}-\mathbf{q})-\frac{\mathbf{p}^2}{2m}T)} d^n\mathbf{p},\quad T=t'-t.$$</p>
<p>in the case of a free quantum particle with Hamiltonian operator</p>
<p>$$H_0 = \frac{\mathbf{P}^2}{2m},$$</p>
<p>and the solution is given by</p>
<p>$$K(\mathbf{q'},t';\mathbf{q},t) = \left(\frac{m}{2\pi i \hbar T}\right)^{\frac{n}{2}} e^{\frac{im}{2\hbar T}(\mathbf{q}-\mathbf{q'})^2}.$$</p>
<p>Could you please help me to understand how to perform the calculation in the case where the Hamiltonian is given by</p>
<p>$$H_1 = \frac{\mathbf{P}^2}{2m} + V(\mathbf{Q})$$</p>
<p>where $V(\mathbf{Q})$ is the potential defined by</p>
<p>$$
V(\mathbf{Q})=\left\{
\begin{array}{cc}
\infty, & \mathbf{Q} \leq b \\
0, & \mathbf{Q}>b. \\
\end{array}
\right.
$$</p>
<p>Update :</p>
<p>I've read the article provided by Trimok, and another one found in the references, but I am still annoyed with the way the propagator is computed. I may be mistaken, but it seems that in that kind of articles, they always start the computation from the scratch, without using what they already know about path integral.</p>
<p>I am actually trying to write something about the use of path integrals in option pricing. From Takhtajan's book, I know that for a general Hamiltonian $H=H_0 + V(q)$ where $H_0 = \frac{P^2}{2m}$, the path integral in the configuration space (or more precisely the propagator) is given by</p>
<p>\begin{equation}
\begin{array}{c}
\displaystyle K(q',t';q,t) = \lim_{n\to\infty}\left(\frac{m}{2\pi\hbar i \Delta t}\right)^{\frac{n}{2}} \\
\displaystyle \times \underset{\mathbb{R}^{n-1}}{\int \cdots\int} \exp\left\{\frac{i}{\hbar}\sum_{k=0}^{n-1}\left(\frac{m}{2}\left(\frac{q_{k+1} - q_k}{\Delta t}\right)^2 - V(q_k)\right)\Delta t\right\} \prod_{k=1}^{n-1} dq_k.\\
\end{array}
\end{equation}
I would like to start my computation from this result, and avoid repeating once again the time slicing procedure. So du to the particular form of the potential, I think I can rewrite the previous equation as
\begin{equation}
\begin{array}{c}
\displaystyle K(q',t';q,t) = \lim_{n\to\infty}\left(\frac{m}{2\pi\hbar i \Delta t}\right)^{\frac{n}{2}} \\
\displaystyle \times \int_0^{+\infty} \cdots\int_0^{+\infty} \exp\left\{\frac{i}{\hbar}\frac{m}{2}\sum_{k=0}^{n-1}\frac{(q_{k+1} - q_k)^2}{\Delta t}\right\} \prod_{k=1}^{n-1} dq_k.\\
\end{array}
\end{equation}
Then I need a trick to go back to full integrals over $\mathbb{R}$ and use what I already know on the free particle propagator. However, since the integrals are coupled, I don't find the right way to end the calculation and find the result provided by Trimok.</p>
<p>Could you please tell me if I am right or wrong ? Thanks.</p> | g14524 | [
0.05098186433315277,
0.034719664603471756,
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0.05108... |
<p>I'm very new to quantum mechanics. I'm thinking of writing a quantum computer simulator, would the following work?</p>
<ul>
<li>Each qubit is stored as a single bit,</li>
<li>For each operation, the qubits involved are transformed into a complex vector of amplitudes of size 2^N. This will result in a vector containing 2^N-1 0s and one 1.</li>
<li>This vector is multiplied by the unitary matrix representing the operation.</li>
<li>The resulting vector is squared element-wise.</li>
<li>An outcome is picked using the elements of the result as probabilities and the qubits are set according to this outcome.</li>
</ul>
<p>I'm not concerned with the running time or the memory cost of the algorithm. What I am concerned about is whether this would result in a physically accurate simulation.</p> | g14525 | [
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0.03452983498573303,
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0.0041574835777282715,
... |
<p>I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying.<br>
However, I've run into a small problem while studying "Free vibrations of a linear triatomic molecule". (We're assuming all 3 atoms are in 1 straight line and the forces they exert on each other are represented by springs with constant values for $k$).</p>
<p>My potential energy is described as: </p>
<p>$V=\frac{k}{2}(\eta_1^2+2\eta_2^2+\eta_3^2-2\eta_1\eta_2-2\eta_2\eta_3)$<br>
Where $\eta$ is a coordinate relative to the equilibrium position.</p>
<p>And 'hence' the tensor has the form: </p>
<p>$\vec V=\begin{bmatrix} k & -k & 0\\ -k & 2k & -k\\ 0 & -k & k \end{bmatrix}$</p>
<p>How do I go from 1 to the other and vice versa?<br>
I've tried googling terms like "Equation to tensor", "Potential energy tensor", "Absolute value of tensor" etc. but they didn't yield anything usable for me.</p> | g14526 | [
0.02847907319664955,
0.05145018920302391,
0.006616330239921808,
-0.023362936452031136,
0.046957556158304214,
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0.0459463894367218,
-0.020979614928364754,
-0.026616236194968224,
0.061122406274080276,
-0.0... |
<p>Here's the problem I am trying to find a solution to:</p>
<blockquote>
<p>Consider a heavy point particle of mass $m_2$ fixed at a point (preferably the origin) and another light point particle of mass $m_1$ at an initial separation of $r_i$ from the fixed particle. There are no other bodies that can interfere with their interaction. Given any time $>t$, what will be the separation $r$ of the two particles at that time?</p>
</blockquote>
<p>I tried to solve this myself with the following procedure:</p>
<p>$$E = K_E + P_E$$ where $E$ is the constant total energy, $K_E$ and $P_E$ the kinetic and potential energies of the light particle respectively. Elaborately,
$$\frac{-Gm_2m_1}{r_i}=\frac{1}{2}m_1v^2-\frac{Gm_2m_1}{r}$$ Solving for $v$ gives me
$$v=\sqrt{2Gm_2(\frac{r-r_i}{rr_i})}$$ after that
I get to a halt when this appears
$$t=\frac{1}{\sqrt{2Gm_2}}\int{\sqrt{\frac{rr_i}{r_i-r}}dr}$$
which I cannot solve further, So I post this question in Mathematics Stack Exchange, and a satisfying answer to the integral they give is
$$t=r_i\sqrt{r_i}\ln\left(\sqrt{r}+\sqrt{r-r_i}\right)+\sqrt{r(r+r_i)}-\frac{r_i\sqrt{r_i}\ln\left(\sqrt{r_i}\right)}{\sqrt{2Gm_2}}-(r-r_i)^2$$
which doesn't seem correct by the way. Differentiating this again must get me back to where i started but I don't get there. Is there any other way to solve such a question or is this the best that can be achieved?</p> | g14527 | [
0.05115624517202377,
0.0007135880878195167,
0.025294719263911247,
-0.0009541631443426013,
0.013667166233062744,
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0.03978712484240532,
0.01321030966937542,
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0.053595706820487976,
-0.013234487734735012,
0.07701709121465683,
-0.050910331308841705,
-... |
<p>Is it practical to attempt to build a 3D hologram generator that is full color and big enough to recreate a watermelon full size? If so, is real-time control feasible?</p> | g14528 | [
0.005305808503180742,
0.08422563225030899,
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-0.0053316145204007626,
0.007989364676177502,
-0.04097258299589157,
0.013080473057925701,
-0.0... |
<p>UT1, UTC, TAI, TDB, or what?</p>
<p>I need to determine the time difference between a given observation and the epoch from which certain constants apply. I typically work with the J2000.0 epoch. This is to determine how much time has passed and how far motions have propagated.</p>
<p>I recently learned that when making planetary observations, I should convert the observation time into TDB time for improved accuracy (Jean Meeus). But I also found out that we use Earth-based time for sidereal time computations, so the rule doesnt apply for sidereal time... but unfortunately which Earth-based time scale to use was not specified in my research.</p>
<p>There are a half dozen or so. UT1, UTC, TAI to name just a few. They differ from as much as 0.00017 seconds to as much as 1 minute 6 seconds. Now, when I compute sidereal time, I am consistently in error by anywhere from 20 to 35 seconds, depending on the resource I compare to. And I cannot seem to resolve this inconsistency, no matter how complicated or detailed my math gets; I find I am getting more and more out of synch with the quoted, expected value with each new variable I factor in (exactly opposite Id expect).</p>
<p>I LOVE exactness. This is an exercise in conceptual understanding of the variables at play and in the technicality of it all; practicality is not an issue here, Im just trying to figure it all out with as much precision as possible.</p>
<p>Any help at finding as precise a value as possible for sidereal time would be appreciated.</p>
<p>I have been using my local clock time and date (local civil time). Factoring out daylight savings if applicable, converting to GMT (time zone offset). I assume that my result is the time in UT or UTC (I may be wrong). While my resource (Astronomical Algorithms, Jean Meeus) assumes that UT and UTC are identical to one another, so I dont concern myself with any more details. Then I use the equations in chapter 12 and (?)22, on sidereal time and nutation.</p>
<p>In doing this I get results that differ by as much as 35 seconds, max, from that which Im quoted on sites like WolframAlpha, other EDU and GOV (time keeping) sites, as well as iPod Touch apps that purport to display sidereal time.</p> | g14529 | [
-0.013158340007066727,
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0.015233265236020088,
0.009019236080348492,
-0.03547213226556778,
0.048262350261211395,
... |
<p>The following does not include all scientific details and parameters, only a common summary of "thoughts". What is scientifically wrong with this summary? </p>
<p>When you take your beer and tap the top of a friends beer bottle, the beer shoots out the top as the CO2 is taken out of solution and rebounds off the bottom of the bottle. If the liquid in the bottle does not have anything in solution, the tap on the top of the bottle results in the explosion of the bottom of the bottle. This is from the cavitation that was created at the bottom of the bottle, and the pressure wave that propogates through the liquid to collapse the cavitation. When the cavitation collapses, the momentum of the water that has now been shifted down hits the bottom and blows out the bottom of the bottle. It does not seem that the energy imparted with the relatively light tap on the top is enough to violently blow out the bottom of the bottle. Is there more energy in the collapse of a cavitation than the energy required to create the cavitation in the first place? Or is it just the momentum of the liquid that blows out the bottom of the bottle? Pits and dings in metal pump blades are caused by cavitation collapse. What is and where does that power come from?</p> | g14530 | [
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-0.0008568926714360714,
0.04291832074522972,
-0.026150217279791832,
0.007906671613454819,
0.049... |
<ol>
<li>See the scan attached below. Brown, in his QFT book, argues a certain way to do an integral. I understand that 1.8.13 or equivalently 1.8.14 can be performed once analytic continuation is done. I also understand that under this transformation the LHS of 1.8.12 would not change when viewed as an integral over $C^2$. But, I do not understand why the value of the integral should not alter when viewed as an integral over $R^2$ as Brown claims and then uses to finish the evaluation of the integral.</li>
<li>A similar kind of argument is used while doing complex scalar field theory where we treat $\phi$ and $\phi^*$ as independent fields. For a while, I have understood it as first complexifying them such that the the two fields combined now live in $C^2$ and then solving the variational problem in the complexified space. And then in the end we identify the two fields as conjugate of each other, that is, we project them over to $C$ again. The fact that extremas in the variation in complexified space maps bijectively to extrema in the variation in the original problem looks kind of a mathematical accident to me. But, as often happens in mathematics, there is a deep reason behind such coincidences. So, I wonder if somebody can shed the light on some deep reason for the fact that we can use this kind of trick of complexification and projection. It isn't obviously clear to me why this trick can be used unless this above argument is produced which seems like some magic is happening behind the curtains.<img src="http://i.stack.imgur.com/Fxl8O.jpg" alt="enter image description here"></li>
</ol> | g14531 | [
0.04427690804004669,
-0.008083250373601913,
-0.00646198308095336,
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0.0177248977124691,
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0.03526010736823082,
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-0.014539637602865696,
-0.048616793006658554,
-0.0005617258721031249,
-0.017594698816537857,
... |
<p>The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_3 ~=~ (\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}),$$ </p>
<p>are mathematically symmetric in the sense that (like $i$ and $-i$) they can be universally exchanged with each other in several ways without altering any mathematical result. However, the visual forms for these three matrices are unexpectedly diverse (e.g., only $\sigma_2$ uses $i$ and $-i$). My understanding of physics history is that Pauli (and also Dirac) developed his matrices by trial and error, rather than by applying any specific theory.</p>
<p>Does a deeper theoretical explanation exist for why these very different visual representations of spin are nonetheless interchangeable in multiple ways?</p> | g14532 | [
0.0015811914345249534,
0.010234791785478592,
-0.025469180196523666,
-0.026806918904185295,
0.039094191044569016,
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0.07416051626205444,
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0.04064951464533806,
-0.03744500130414963,
0.029466208070516586,
0.0584370493888855,
0.00... |
<p>I understand that if the wire is not aligned with the magnetic field, it won't rotate, but I'm still confused on how so.
Also, which direction is the current flowing? </p> | g14533 | [
-0.01500350795686245,
0.009249803610146046,
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0.024798989295959473,
0.0738440677523613,
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0.04830746352672577,
-0.032080963253974915,
-0.012... |
<p>I've been trying to solve this using the method the prof. taught us, and I happen to know the answer but I can't reach it no matter how many times I've tried. The circuit in question is below:</p>
<p><img src="http://i.stack.imgur.com/X6wgG.jpg" alt="image"></p>
<p>I am asked to use Maxwell's circulating current theorem to find out the current at the $20 \Omega$ resistor. My method was to write out all the loop equations:</p>
<p>$20 = 15I_1 + 10I_2$</p>
<p>$10 = 15I_1 + 25I_2 + 15I_3$</p>
<p>$10 = 15I_2 + 35I_3$</p>
<p>then solve by method of elimination. My answers are as follows:</p>
<p>$I_1 = 2.44A$</p>
<p>$I_2 = -1.66A$</p>
<p>$I_3 = -0.43A$</p>
<p>The answer for $I_3$ is $-0.57$.</p>
<p>Am I on the right path? If not, can someone point out where I am going wrong and why? Thanks </p> | g14534 | [
0.03263173624873161,
0.006320952903479338,
0.0012252859305590391,
-0.04165632650256157,
0.046550244092941284,
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0.10595116019248962,
-0.01061759702861309,
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0.061628468334674835,
-0.03978032246232033,
0.06655599921941757,
-0.0627712830901146,
0.04256... |
<p>In many aerial war face movies they show that aircraft is locked by missile ,particularly in movie <strong><em>behind enemy line</em></strong> " F/A-18 Hornet" is locked by 2 <strong><em>SAM missile</em></strong>,pliot try to break the lock but aircraft get shot down. <br>
i find wikipedia have some <a href="http://en.wikipedia.org/wiki/Missile_lock-on" rel="nofollow"> information</a> about this.<br>
i dont understand this concept ,how this locking work and how pilot break the lock?</p> | g14535 | [
-0.04515143856406212,
0.007105579134076834,
0.014815472066402435,
0.06570026278495789,
0.023752737790346146,
0.05177519470453262,
-0.034628644585609436,
0.007445129565894604,
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0.005505877546966076,
0.0... |
<p><img src="http://i.stack.imgur.com/fzIUA.png" alt="enter image description here"></p>
<p>This is the question, and the answer is given as 2).</p>
<p>Now my basic doubt is, are they asking us the direction of the total contact force on the object by the floor? Or, are they asking the direction of the static friction on the block by the floor? What is it? Because, the normal reaction between any two surfaces in contact is ALWAYS perpendicular to the two surfaces. </p>
<p>Please help me, and please answer this before closing it as being too specific.</p> | g14536 | [
0.08049479126930237,
0.015012442134320736,
0.013515434227883816,
0.021464765071868896,
0.044107794761657715,
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0.03261187672615051,
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-0.05339683219790459,
-0.006990641355514526,
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0.018916109576821327,
0.008... |
<p>The phenomenon of observing one entangled particle and noticing the other take on corresponding values... Does this take a finite speed at all or is it instantaneous?</p> | g14537 | [
0.024512439966201782,
0.03953506425023079,
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0.08210837841033936,
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0.019620954990386963,
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0.0034940331242978573,
0... |
<p>For a given Hamiltonian, is the space of histories of a classical system the same as the symplectic manifold? </p>
<p>Do I have to take care of gauge equivalences and if so, is this only an issue for fields (not for trajectories)?</p> | g14538 | [
0.028186507523059845,
0.008912080898880959,
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0.01109382789582014,
0.012695051729679108,
0.0398603230714798,
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0.038822... |
<p>When thinking about how the <a href="http://physics.nist.gov/cgi-bin/cuu/Value?asil" rel="nofollow">lattice constant of silicon</a> can be given up to eight decimal places without a remark for the temperature I realized that, it seems</p>
<p><em>most insulators and semiconductors seem to expand less than metals when exposed to heat.</em></p>
<p>Is there an intuitive connection between the band structure and the thermal expansion?</p> | g14539 | [
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<p>There is <a href="http://astarte.csustan.edu/~tom/SFI-CSSS/info-theory/info-lec.pdf" rel="nofollow">a book from Tom Carter</a> on entropy. In the Economics I application (page 111), he ingeniously computes that the distribution of fixed amount of M money over N individual tends to </p>
<p>$$p_i = {1 \over T}\, e^{-i/T}$$</p>
<p>The temperature, $T = M/N$, is the average amount of money per individual. </p>
<p>The author calls this a <em>Boltzmann-Gibbs distribution</em>. I wanted to find a plot and compare it with Pareto distribution and <a href="http://en.wikipedia.org/wiki/Black-body_radiation#Planck.27s_law_of_black-body_radiation" rel="nofollow">Planck black body radiation</a>. The problem is however, and this is my question, Wikipedia does not have it! Actually, there is an article on <a href="https://en.wikipedia.org/wiki/Boltzmann_distribution" rel="nofollow">Boltzmann distribution</a>, which is also called Gibbs distribution, but I see nothing similar to $p_i = 1/T\, e^{-i/T}$ there. At least I cannot see how ${N_i \over N} = {g_i e^{-E_i/(k_BT)} \over Z(T)}$ with $N=\sum_i N_i,$ and partition function, $Z(T)=\sum_i g_i e^{-E_i/(k_BT)}$ can be reduced to it. I do not see a plot $p_i(T)$ either. Are they related?</p> | g14540 | [
0.028868714347481728,
0.04195242375135422,
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<blockquote>
<p><em>A car is on a banked curve, following a path which is part of a circle with radius $R$. The curve is banked at angle $\theta$ with the horizontal, and is a frictionless surface. What is the speed the car must go to accomplish this?</em></p>
</blockquote>
<p>What I don't understand about this problem is why we assume there is only the normal force and the gravitational force on the vehicle. From that point onwards, I have no trouble following the solution. </p>
<p>The way I see it, if we're considering only that there exists a normal force and a gravitational force, something the car is doing (accelerating?) must be "adding" to the normal force. Or is it possible that the car could just be coasting, and it could keep a constant height along the banked curve?</p> | g14541 | [
0.07184944301843643,
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0.025481758639216423,
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0.018990879... |
<p>If you look at the Feynmann Diagram of an electron capture:
<img src="http://i.stack.imgur.com/3mS4E.jpg" alt="enter image description here"></p>
<p>Whe W+ boson turns the electron into a neutrino. How is this possible? I thought the the boson carries the positive charge and converts the electrons into a positron. Why is this possible?</p> | g14542 | [
0.010125030763447285,
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<p>In the perturbation theory for non-degenerate levels, the energy $E_n(\lambda)$ of an eigenstate $|\psi_n(\lambda)\rangle$ of the hamiltonian $\mathcal{H}=\mathcal{H}_0+\lambda \mathcal{H}_1$ (where $\mathcal{H}_0$ is the unperturbed hamiltonian with eigenstates $E_n^0$ and $\lambda\mathcal{H}_1$ is the perturbation) is described by the equation</p>
<p>$$E_n(\lambda)=E_n^0+\dfrac{\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle+\lambda^2\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle+\lambda^3\langle\phi_n|\mathcal{H}_1|\psi_n^2\rangle+\dots}{1+\lambda a_n^{(1)}+\lambda^2 a_n^{(2)}+\dots}.$$</p>
<p>Here, $|\phi_n\rangle$ is the $n$'th eigenstate of the unperturbed hamiltonian, and $a_n^{(p)}=\langle\phi_n|\psi_n^p\rangle$, with $|\psi_n^p\rangle$ being the $p$'th correction to the $n$'th eigenstate. The zeroth correction of course equals the eigenstate of the unperturbed hamiltonian: $|\psi_n^0\rangle=|\phi_n\rangle$.</p>
<p><strong>My question is the following</strong>: how does one go from the above equation to this?
$$\begin{array}{r l}
E_n(\lambda)&=E_n^0+\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle\\
&+\lambda^2\left[\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle-\langle\phi_n|\mathcal{H}_1|\phi_n\rangle a_n^{(1)}\right]\\
&+\lambda^3\left[\langle\phi_n|\mathcal{H}_1|\psi_n^2\rangle-\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle a_n^{(1)}-\langle\phi_n|\mathcal{H}_1|\phi_n\rangle a_n^{(2)}\right]+\dots
\end{array}$$</p>
<p>The book I'm working with tells me this is done by <em>"Expanding this expression in powers of $\lambda$"</em>, but I can't recognize the procedure. Of course the first term $E_n^0$ is of order $\lambda^0$, and the second term $\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle$ is of order $\lambda^1$, and these can be taken from the original equation right away, but I can't find out how the subtractions in the third and fourth term got there.</p> | g14543 | [
0.011099688708782196,
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0.... |
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