question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>Assuming 2.3% per year exponential growth of human energy consumption (so, roughly speaking, that corresponds to multiplying consumption tenfold every century), it's argued that human annual energy consumption would be equal to annual solar energy output in 1400 years time. Is that correct?</p>
<p>For reference, human energy consumption is currently about $2 \times 10^{13} W$</p>
<p>(source: <a href="http://physics.ucsd.edu/do-the-math/2011/07/galactic-scale-energy/" rel="nofollow">http://physics.ucsd.edu/do-the-math/2011/07/galactic-scale-energy/</a> )</p> | g12771 | [
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<p>Let a,b be two charged particles.
$$\vec{r}_a(0)=\vec{0}$$ $$\vec{r}_b(0)=r\hat{j}$$ $$\vec{v}_a(t)=v_a \hat{i}$$ $$\vec{v}_b(t)=v_b\hat{j}$$</p>
<p>In which both $v_a$ and $v_b$ $<<c$.</p>
<p>Then </p>
<p>$$\vec{E}_{ab}(0)=\frac{q_a}{4\pi \epsilon r^2}\hat{j}$$</p>
<p>$$\vec{B}_{ab}(0)=\frac{\mu q_av_a}{4\pi r^2} \hat{k}$$</p>
<p>$$\vec{E}_{ba}(0)=-\frac{q_b}{4\pi \epsilon r^2}\hat{j}$$</p>
<p>$$\vec{B}_{ba}(0)=\vec{0}$$</p>
<p>Note that $v_a$ and $v_b$ $<<c$ thus a and b almost obey Coulomb's law. Moreover, $\vec{j_i}(\vec{r})=q_i\delta(\vec{r}-\vec{r}_i)\vec{v}_i$ hence BS law can be applied.</p>
<p>Hence </p>
<p>$$\vec{F}_{ab}(0)=q_b(\vec{E}_{ab}+\vec{v}_b \times \vec{B}_{ab})$$
$$=\frac{q_a q_b}{4\pi \epsilon r^2}\hat{j}-\frac{\mu q_av_a v_b}{4\pi r_b^2} \hat{i}$$</p>
<p>But</p>
<p>$$\vec{F}_{ba}(0)=-\frac{q_aq_b}{4\pi \epsilon r^2}\hat{j}$$</p>
<p>Consequently
$$\vec{F}_{ab} \ne -\vec{F}_{ba}$$</p>
<p>This result contradict to Newton's 3rd law!! But I cannot find any error... It troubled me.</p> | g12772 | [
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<p>Can anyone explain to me Hamiltonian mechanics relation to conservation of energy? I'm not very good at mathematics, and I know it's important into understanding Hamiltonian mechanics. However, can it be explained in a simple way?</p> | g12773 | [
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<p>I'm having a hard time to grasp one simple idea.
Let us say we have a moving point charge, q, with velocity v, in the x direction.
Now calculating both the Magnetic and the Electric fields is easy: We know that B=0 in the charge's frame, hence we obtain B in the lab's frame.</p>
<p>Now, let us say we would want to know the same fields, but with a stationary charge, positioned above the moving charge (perpendicular). I could not just surmise that there is no Magnetic field in the moving charge's frame since, in turn, the other one is now moving in the other one's frame, so in both frames there is Magnetic field.</p>
<p>I would be more than grateful if someone could settle this, thanks.</p> | g12774 | [
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<p>It is easy to drink cold drinks, soft drinks with straw but when I tried to drink tea with straw then that was very unpleasant and even for time I lost the sense from my tongue.</p>
<p>My question is why it is difficult to drink hot liquids with straw?(Drinking cold liquids is very pleasant!!!) </p> | g12775 | [
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<p>When you bring a large iron plate and a magnet, the magnet attracts the iron plate and it tends to slide itself to the center. When I place it on the edge, it always aligns at the center why is that?</p> | g12776 | [
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<p>What's the significance of large-scale anomalies in CMB that are confirmed by Planck? I've read somewhere that the cold spots can provide support for string theory or it may be due to a parallel universe . To what extent is that true ? Are there models that can explain these anomalies?</p> | g12777 | [
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<p>Is it possible to calculate the force exerted on a bound, infected red blood cell under various shear stresses? The strength of an adhesive interaction is normally measured by SPR or AFM, however using microfluidics, you can increase the shear stress until the cell dislodges, therefore, can you use this data to calculate force?</p> | g12778 | [
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<p>In a previous post [ <a href="http://physics.stackexchange.com/questions/57901/">Noether theorem, gauge symmetry and conservation of charge</a> ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied to a global $U(1)$ gauge symmetry, or via the covariant (or minimal, or Weyl, or ...) substitution with a $U(1)$ local gauge symmetry and antisymmetry of the electro-magnetic field applied to the equations of motion. I nevertheless still get confused about something: what is a charge ? More precisely, how to charge a field ?</p>
<p>Indeed, if one believes that only the global gauge construction is able to demonstrate the conservation of the charge, one has to admit that the charge is not defined that way ! The same demonstration applies for the conservation of the particle number. Indeed, I used in [ <a href="http://physics.stackexchange.com/questions/57901/">Noether theorem, gauge symmetry and conservation of charge</a> ] a Lagrangien for <em>uncharged</em> particles to show the conservation of a particular current through Noether theorem. </p>
<p><strong>So the question becomes: how to charge this particle ?</strong> In particular: how to charge this particle and still conserve the global gauge ?</p>
<p>NB: In the local gauge construction, it seems to be easier to give a charge to the system, but then only the second Noether theorem applies, and people get annoyed by that. </p>
<p><strong>To conclude, a</strong> (possibly important) <strong>remark about my point of view:</strong> I would like to understand the problem of "charging the field" in a condensed matter perspective, when I believe we have no a priori idea of the charge of the quasi-particles appearing in our effective theories. Can we overcome this problem ? Of course, any point of view (especially the opinion of QFT physicists who might have already tackled this problem) is warmly welcome :-) And the question "how to charge a field ?" is just a warm-up for this more complicated one "how to define the charge of an effective, emergent field ?" which is entirely subsidiary question for the moment.</p>
<p><strong>IMPORTANT EDIT</strong> I just became aware of this question [ <a href="http://physics.stackexchange.com/questions/21103/">How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?</a> ] which is strongly related, and well answered. I nevertheless think one can continue to discuss here about the condensed matter problem. Indeed, there are some people who believe that a superconductor (among other emergent phases of matter) have charge neutral excitations [see e.g. <a href="http://dx.doi.org/10.1103/PhysRevB.41.11693" rel="nofollow">http://dx.doi.org/10.1103/PhysRevB.41.11693</a> which is too old to be on arXiv, but this one: <a href="http://arxiv.org/abs/cond-mat/0404327" rel="nofollow">http://arxiv.org/abs/cond-mat/0404327</a> rephrases it in a different way.]. So the question follows: how can these excitations at low temperatures become again charged at higher temperature ? How can we measure charge current from an uncharged excitation ? How can we "charge" a field after all ?</p> | g12779 | [
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<p>I am currently working out the energy required to create a particle anti-particle pair from a collision of a proton travelling along the x-direction with an anti-proton which is at rest. The particle has a mass $m_q$.</p>
<p>Conservation of 4-momentum in the rest frame of the target anti-proton ($c=1$):</p>
<p>$p_1=(m_\text{p},0,0,0)$</p>
<p>The moving proton has this minimum energy, the quantity we are trying to find:</p>
<p>$p_2=\left(E, \sqrt{E^2-m_p^2 },0,0\right)$</p>
<p>Then what I find confusing is the terminology, it says that $p_{q\bar{q}}=\left(\sqrt{p^2+4 m_q^2 },p,0,0\right)$</p>
<p>Where has this $p$ come from and what is it? Is it the momentum of the center of mass of the $q\bar{q}$ system? Also, we don't seem to have taken much care about reference frames here, or rather I havn't thought about them really which is worrying.</p>
<p>Finally, it says that the minimum energy, E, the $q\bar{q}$-pair will be at rest in the CMF frame. What is this frame referring to? The center of mass of the particle/anti-particle pair? I understand that the threshold energy to produce these will be a combination of the energy of the proton and their rest masses but I don't know how we have the equation (above)</p> | g12780 | [
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<p>I am trying to calculate the field of an infinite flat sheet of charge
(a plain with uniform charge density $\sigma$) using the superposition
principle.</p>
<p>I know that the field of an infinite line charged with uniform charge
density $\lambda$ is $E(r)=\frac{2\lambda}{r}$.</p>
<p>I want to consider the plain as infinite number of lines from $-\infty$
to $\infty$.</p>
<p>I wrote that
$$
dE=\frac{2\lambda}{r}
$$</p>
<p>and since $\lambda=\sigma dl$ we get
$$
dE=\frac{2\sigma dl}{r}
$$</p>
<p>$$
E(r)=\int_{-\infty}^{\infty}\frac{2\sigma dl}{r}
$$</p>
<p>Which is clearly wrong: First, I still have $r$, secondly I get that
the above equals to
$$
\frac{2\sigma}{r}\int_{-\infty}^{\infty}dl
$$</p>
<p>which does not converge.</p>
<p>What are my mistakes, and how do I get the correct result: $2\pi\sigma$
?</p> | g12781 | [
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<p>Light travels with a speed of $3\times10^8{m\over s}$. My question is that was the light initially accelerating or it archived the speed in an instance? If it was accelerating then why it did not accelerate beyond that? </p> | g159 | [
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<p>I was talking with a friend of mine, he is a student of theoretical particle physics, and he told me that lots of his topics have their foundations in statistical mechanics. However I thought that the modern methods of statistical mechanics, for example the renormalization group or the Parisi-Sourlas theorem, come from the methods of quantum field theory or many-body techniques (Feynman diagrams and so on). I notice that books also regarding modern concepts, such as spin glasses, don't require any other knowledge then basic calculus.</p>
<p>Can someone explain which is the relation between these subjects?</p>
<p>What topics should I study of field theory or similar to have a deep understanding in statistical mechanics?</p> | g12782 | [
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<p>A charged disk or sphere will create currents around its turning axis if a rotation is added. The total current can then be calculated by adding all concentric currents together. Every current (I take the example of a thin charged disk, as it is the most easiest casus) is as big as:</p>
<p>$$
dI = dQ / dt = f*\sigma*2 \pi *rdr
$$ </p>
<p>So the total charge would be something like:</p>
<p>$$
I = \omega\int_0^R \sigma r dr
$$</p>
<p>With $R$ the radius of the disk,$f$ the frequency, $\omega$ the angular speed (so we can leave the $2\pi$ out) and $\sigma$ the charge density.</p>
<p>In my physics book they consider $\sigma$ as a constant. But I was wondering: as electrons (who determine the charge) have mass, they also have a centrifugal force, which would imply that the charge density is NOT the same at every distance from the centre, so we would have to define a $\sigma$ in function of $r$ (so $\sigma$ is <strong>not</strong> just total charge on total surface).</p>
<p>Would this centrifugal force give considerable different results and if so, could anyone show me a calculation of the difference in total current? Also if the difference is not considerable for a small disk, would it be for a very large disk, because the centrifugal force is bigger at a bigger radius (for example a disk with radius 1000km)?</p> | g12783 | [
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<p>For an electronics experiment, I began wondering about the <strong><a href="http://en.wikipedia.org/wiki/Hot_plate" rel="nofollow">electric hot-plates</a></strong> (specifically the temperature dependence over time). If I were to measure the given temperature over time, I assume that the temperature would <em>not</em> increase linearly. In fact, I suspect the initial heating rate will only decrease.</p>
<p>Do anyone have some intuition regarding a model for this? An answer giving an ODE is fine too...</p> | g12784 | [
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<p>Basic facts: The <a href="http://en.wikipedia.org/wiki/TauTona_Mine">world's deepest mine</a> is 2.4 miles deep. Railguns <a href="http://en.wikipedia.org/wiki/Railgun">can acheive a muzzle velocity</a> of a projectile on the order of 7.5 km/s. The Earth's <a href="http://en.wikipedia.org/wiki/Escape_velocity">escape velocity</a> is 11.2 km/s.</p>
<p>It seems to me that a railgun style launch device built into a deep shaft such as an abandoned mine could reasonably launch a vehicle into space. I have not run the calculations and I wouldn't doubt that there might be issues with high G's that limit the potential for astronauts on such a vehicle, but even still it seems like it would be cheaper to build such a launch device and place a powerplant nearby to run it than it is to build and fuel single-use rockets.</p>
<p>So, what is the possibility of a railgun assisted orbital launch? </p>
<p>What am I missing here? Why hasn't this concept received more attention?</p> | g464 | [
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<p>I've read a few explanations about why stress concentration occurs at sharp corners but I don't find the explanations intuitive.</p>
<p>Can anyone explain it perhaps using an analogy such as atoms "holding hands" with neighbor atoms or a similar easy to understand analogy?</p>
<p>Most explanations I've shown just show the result of a finite-element-analysis based stress analysis. This shows the emergent property but doesn't explain the underlying mechanism that causes the stress to concentrate at corners.</p>
<p>Better understanding the cause of the crashes of the De Havilland Comet jet aircraft due to metal fatigue at the corners of their rectangular windows is the context for this question. See:</p>
<p><a href="http://www.cracked.com/article_19623_6-small-math-errors-that-caused-huge-disasters.html" rel="nofollow">http://www.cracked.com/article_19623_6-small-math-errors-that-caused-huge-disasters.html</a></p>
<p>The answer I'm looking for should not contain any math.</p> | g12785 | [
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<p>Is there any book you recommend for graduation exam/higher studies? I have an exam on physics this week so I need some books to brush up my knowledge. Is there any book you recommend for that purpose? Other than Nelsons Parker Physics.
What I ment here is a book which has all in one <strong><em>like</em></strong> <strong>"Advanced Level Physics by Michael Nelkon (Author), Philip Parker (Author)"</strong>
<a href="http://rads.stackoverflow.com/amzn/click/043592303X" rel="nofollow">http://www.amazon.com/Advanced-Level-Physics-Michael-Nelkon/dp/043592303X</a>
A book with all exercises alog with the lesson. For Advanced Level Examination.</p> | g98 | [
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<p>I'm making several assumptions, not sure if any are correct:</p>
<ul>
<li>there is a black hole at the center of a galaxy</li>
<li>the black hole is eating the galaxy</li>
</ul>
<p>Eventually the galaxy will be gone, right?</p>
<p>Has this been observed? Do we know what happens afterwards?</p>
<p>Posting here since astronomy got merged into physics</p> | g405 | [
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<p>Considering that on the atomic level objects consists of densely spaced positively and negatively charged particles, does not the acceleration of those objects lead to Bremsstrahlung of those particles? And although the monopole field is zero, couldn't higher order multipole radiation escape and cause the inertia? I would think at least a small effect like this must happen, even if it doesn't explain all of the inertia. </p> | g12786 | [
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-0.... |
<p>I need to estimate a drag race quarter mile time given the car's weight, <a href="http://en.wikipedia.org/wiki/Brake_horsepower#Brake_horsepower" rel="nofollow">bhp</a> and preferably the drive (FWD, RWD, <a href="http://en.wikipedia.org/wiki/Four-wheel_drive" rel="nofollow">4WD</a>). I know $v(t) = ds/dt$ and $a(t) = dv/dt = d^2s/dt^2$, but how can I get the function $s(t)$ to calculate $v(t)$ and $a(t)$? I thought maybe <a href="http://www.ajdesigner.com/phphorsepower/horsepower_equation_power_to_weight_ratio.php" rel="nofollow">Power to Weight ratio</a> would help for solving the acceleration but I have no idea what to do with it and maybe the drive would just be a constant which is then subtracted from the calculated time. Can you help me?</p> | g12787 | [
0.02943132072687149,
0.008688715286552906,
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0.000006768001640011789,
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0.059389740228652954,
-... |
<p>My question is about two equations regarding uniform spheres that I've run into:</p>
<p>$V=\frac{GM}{r}$</p>
<p>... and ...</p>
<p>$U = \frac{3}{5}\frac{GM^2}{r^2}$</p>
<p>$V$ is unknown to me, and is described (in Solved Problems in Geophysics) as "the gravitational potential of a sphere of mass M." I also found it online called "the potential due to a uniform sphere."</p>
<p>$U$ is what I've seen before and I know it by the descriptions "sphere gravitational potential energy" or "gravitational binding energy."</p>
<p>My understanding is that $U$ is the amount of energy required to build the sphere piece by piece from infinity. I also recognize $GMm/r$ as the gravitational potential between two masses.</p>
<p>Can someone explain the difference between these concepts? How can $GM/r$ be the "gravitational potential of a sphere"? Isn't that what $U$ is?</p>
<p>Thank you very much.</p> | g12788 | [
0.031575046479701996,
0.0050319647416472435,
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0.052483174949884415,
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0.04531882703304291,
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<p>I am trying to figure out why vibrations (say, from an engine) loosen screws. It seems to me that there is evident symmetry between loosening and tightening a screw. I am wondering what breaks this symmetry.</p> | g12789 | [
0.07312291860580444,
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0.03... |
<p>Why uniform constant magnetic fields can not exert net force on a piece of iron whatever strong it might get?</p> | g12790 | [
0.030585438013076782,
0.0497148223221302,
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0.0622074119746685,
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<p>Apparently, light is just a certain wavelength, or "the visible spectrum" of electromagnetic waves. If I recall correctly, my physics teacher explained to me that electromagnetic waves are basically consisted of two, interchanging parts. The "electric" and the "magnetic" parts of the wave are somehow manifesting each other in synchronized intervals, or they are (more probably) causing each other and canceling themselves out in the process. I checked wikipedia and the transition seems to be somehow interpolated (but then again, many things in nature are).</p>
<p>So, which part of the mirror actually reflects the wave? Which of these two parts? Both? How come the wave doesn't get heavily distorted in the process?</p>
<p>I guess the actual electrons of atoms of silver play a role, but why isn't every material reflective, then? Because is isn't "perfectly" flat? If I lined up atoms of a non-metal element in a perfect plane (maybe several rows, actually), would it reflect light just as mirrors do?</p> | g92 | [
0.004362943582236767,
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<p><strong>Just wondering:</strong></p>
<p>We know that, in its current form of the $SU(2)_L\times U(1)$, the electroweak theroy rides a wave of huge success. However, is it not possible that the correct <strong>simple</strong> group unification could reveal physics that the current model doesn’t? </p>
<p>For example, it could reveal the existence of a unique gauge boson, like the photon for EM ('EM-photon') produced by $U(1)$ invariance, which might be called the 'EW-photon.’ This EW-photon, with the right group structure, could then produce the $W^+,W^-,Z_0$ and the EM-photon and the known physics at the E-W scale?</p>
<p>Does anybody know whether there is any activity on this, other than Technicolour? Links will be greately appreciated.</p> | g12791 | [
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<p>I'm here asking about real or though experiments (i.e., physical effects) where, at least in principle, one can see some consequence of a non-Lorentz-invariant vacuum state in an otherwise Poincare invariant theory.</p>
<p>Let me develop the question. Assume a theory in which the Hamiltonian (that closes Poincare algebra with the rest of generators) $H$ acts non-trivially on the vacuum state $|0>$, where 'non-trivially' simply means:</p>
<p>$$H|0> = E \, |0>$$</p>
<p>with $E$ a positive constant. Since the Hamiltonian transforms as the temporal component of a 4-vector, the vacuum state is not Lorentz invariant. Therefore, the theory is not Lorentz invariant (it is usually claimed that this is an additional condition besides the Poincare algebra). However, I'm not able to see any consequence of this fact. I think that this does not affect any cross-section or decay rate.</p>
<p>I think that one can redefine the Hamiltonian so that $H'=H-E$ and $H'|0>=0 \,$ (this does not affect the Poincare algebra if one also redefines the boost generators properly). I know this seems obvious (this redefinition is usually done in canonical quantization when one doesn't adopt normal ordering), but I've just read this in this forum:</p>
<p><a href="http://physics.stackexchange.com/a/8360/10522">http://physics.stackexchange.com/a/8360/10522</a></p>
<blockquote>
<p>However, in special relativity, energy is the time component of a
4-vector and it matters a great deal whether it is zero or nonzero. In
particular, the energy of the empty Minkowski space has to be exactly
zero because if it were nonzero, the state wouldn't be
Lorentz-invariant: Lorentz transformations would transform the nonzero
energy (time component of a vector) to a nonzero momentum (spatial
components).</p>
</blockquote>
<p>One has a family of vacuum states related by Poincare transformations which are unitary, I don't think this is a problem... what do you think?</p>
<p>Added:
1) I'm not thinking about a Poincare invariant Lagrangian with a potential of the form $(A^2(x)-v)^2$, where $A_{\mu}(x)$ is a vectorial field that acquires a vacuum expectation value $v$. Assume that every field has zero vev.</p>
<p>2) I'm looking for Lorentz violating effects instead of vacuum energy effects unless you argue that these vacuum energy effects (Lamb shift, spontaneous emission, etc.) break Lorentz symmetry.</p> | g12792 | [
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<p>I've seen electromagnetics formulated in the potentials ($\bf A$, $\varphi$) (and their magnetic counterparts) in the Lorenz gauge for a long time. Justifications for using these include that they are less singular than the fields ($\bf E$, $\bf H$) and reduce the redundancy of Maxwell's equations. All the fields derive from two vector and two scalar potentials (total of 8 potentials). In older engineering works (1950s-1969), I've found they don't use the ($\bf A$, $\varphi$) potentials at all, and instead use a set of two vector potentials called the Hertz vectors by the authors, usually noted by ${\bf\Pi}$, from which all the fields derive. They seem to have an additional level of regularity to them because the ($\bf A$, $\varphi$) potentials derive from them. There are also a total of only 6 scalar quantities between the two, from which all the fields derive.</p>
<p>Modern authors seem to prefer the dyadic Green's functions. I haven't studied these in any depth, but it appears to me that in this formulation, all the fields derive from 5 scalar quantities? Not sure about this one so feel free to correct me. But it got me thinking about my questions:</p>
<p>Is 3D classical EM, how many fundamental constraints are there in Maxwell's equations? What is the minimum number of quantities that can be defined, from which all the fields can be said to derive? I suspect the differential forms / tensor formulation will be necessary to get to the bottom of this, but if there is an interpretation in terms of 3D vector calculus, that'd be great.</p> | g12793 | [
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0.001... |
<p>How can we determine during a lunar eclipse whether the earth moves faster or the moon using minimum instruments?</p> | g12794 | [
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<p>I didn't know where to begin with this problem. I eventually found a solution online, which is why I'm reposting this question with an answer. I was wondering if anyone can explain the one question I have.</p>
<p>A self-contained machine only inputs two equal steady streams of hot and cold water at temperatures $T_1$ and $T_2$. Its only output is a single high-speed jet of water. The heat capacity per unit mass of water, $C$, may be assumed to be independent of temperature. The machine is in a steady state, and the energy in the incoming jets is negligible.</p>
<p>$(a)$ What is the speed of the outgoing jet in terms of $T_1$, $T_2$ and $T$, where $T$ is the temperature of the outgoing jet?</p>
<p>$(b)$ What is the maximum possible speed of the jet?</p>
<p>$(a)$ The heat intake per unit mass is $\Delta Q = (C(T_1 - T) - C(T - T_2))/2$ </p>
<p>Since the system is in a steady state, $\frac{v^2}{2}=\Delta Q$, therefore $v=\sqrt{C(T_1 + T_2 - 2T)}$</p>
<p><strong><em>why is that the case? Why does all the heat go towards its kinetic energy per unit mass? That is my one question</em></strong></p>
<p>$(b)$ $\Delta S = \frac{C}{2}[ln(\frac{T}{T_1})+ln(\frac{T}{T_2})] \ge 0$</p>
<p>So $T\ge \sqrt{T_1 T_2}$</p>
<p>and $v \le v_{max} = \sqrt{C(T_1 + T_2 - 2\sqrt{T_1 T_2})}$</p>
<p>What an insidious problem, no?</p> | g12795 | [
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<p>I have currently been working with a sample that "appears to" decrease its resistance when I cover it and protect it from light.</p>
<p>Basically it presents the opposite behaviour of a photoresistor.</p>
<p>What kind of phenomenon may cause this? Or it has to be a measurement error?</p>
<p>EDIT: I am talking about typical room light from fluorescent lamps on the ceiling.</p>
<p>Experimental Setup:
Sample: A dense, sintered, ceramic pellet (with a disc shape, 1 cm in diamater and ~3mm thick). The faces of the disc were carefully painted with conductive silver ink to make the electrical contacts. An Agilent 4294A impedance analyzer was used to measure the electrical properties with a 0.5 V AC signal. This was done at air, in a room with the ceiling lights turned on. Approaching my hands to the sample, blocking the light, the resistance increases.</p> | g12796 | [
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<p>When I studied physics some time ago my teacher explained that if we consider the gravitational atraction not instantaneous, such as the General Relativity says, the planets would be attracted towards the position were other planets were before and not the current one, getting spiral orbits instead of ellipses.</p>
<p>What was the solution to this problem? Because we do observe elliptic orbits.
I guess when we use speed "c" we can't continue using newton laws and we need the realtivity.</p>
<p>Many articles, even in the Wikipedia, say that the speed of the gravitons must be much higher than "c".<br>
Is it a complet nonsense or some academics believe it?</p>
<p>regards</p> | g12797 | [
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<p>In General Relativity, if the system accelerates, the inside of the system and the outside of the system will have different speed of time.</p>
<p>Where is the boundary of the system? </p>
<p>If a human accelerates closer to the speed of light, does that mean the human exist inside of their specific system? </p> | g12798 | [
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<p>The concept of relativistic energy comes from it's conservation in relativistic mechanics for an elastic collision.</p>
<p>It seems to me that another possible derivation could equate the energy of a single particle before, with the kinetic energy after of N particles after the energy is distributed amongst them in the form of the Boltzman distribution, as well as individually being of the form $mv^2$.</p>
<p>Are there any problems in doing this?</p> | g12799 | [
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<p>Why is it that the distance between two light rays changes as they pass from one material to another? </p>
<p>It must have something to do with the change in refractive index, but to me it seems the ''ends'' of the beam are refracted by equal amounts. </p>
<p>Can you explain this? See the picture below.</p>
<p><img src="http://i.stack.imgur.com/DxMtq.png" alt="enter image description here"></p> | g12800 | [
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<p>I am having a hard time in finding out what exact light media laser shows use. I am trying to build a laser show myself. I know that the laser light is reflected off these particles in such a way that that it makes the laser line "viewable" in <em>all directions</em></p>
<p>Can somebody explain to me how exactly do the collection of particles make it viewable in all directions and what exact conditions are necessary? Does the angle of the incoming laser light matter? Does the size of the particles matter? Does the uniformity of how the particles are dispersed matter? Would water vapor work?</p>
<p>I have tried using a fog machine, but the red laser that I am using only reflects off of the fog particles in a way that makes it viewable <em>only from a certain perspective</em>. This would not be a good show to the people standing in one side of the room vs. another.</p> | g12801 | [
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<p>D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero.</p>
<p>This is obviously true for the constraint of a rigid body where all the particles maintain a constant distance from one another. It's also true for constraining force where the virtual displacement is normal to it.</p>
<p>Can anyone think of a case where the virtual displacements are in harmony with the constraints of a mechanical system, yet the total work done by the internal forces is non-zero, making D'Alembert's principle false?</p> | g12802 | [
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<p>The equation describing the force due to gravity is $$F = G \frac{m_1 m_2}{r^2}.$$ Similarly the force due to the electrostatic force is $$F = k \frac{q_1 q_2}{r^2}.$$ </p>
<ol>
<li><p>Is there a similar equation that describes the force due to the strong nuclear force? </p></li>
<li><p>What are the equivalent of masses/charges if there is? </p></li>
<li><p>Is it still inverse square or something more complicated?</p></li>
</ol> | g12803 | [
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<p>In "The Elegant Universe," two individuals, George and Gracie, are noted to be moving relative to one another in space and wearing clocks. They sync their clocks upon passing each other. Each observes the other's clock to be ticking more slowly. Gracie communicates with George via cell.</p>
<blockquote>
<p>Gracie communicates her time to George as he recedes into the distance. Initially, Gracie thinks that George will hear her reported time before his clock reaches the same time. But then, she takes into account travel time and realizes the travel time will more than compensate, and George will actually hear her time after his clock has passed that time.</p>
</blockquote>
<p>Then, the author notes that</p>
<blockquote>
<p>Gracie realizes that even if George takes the travel time into account, he will conclude from Gracie's communication that her clock is running slower than his.</p>
</blockquote>
<p>By this last statement, does it mean that the time difference will be greater than what can be accounted for by mere communication travel time? I would think that some of the time difference is due to Gracie (from George's perspective) transmitting the time after that time has already elapsed on his clock. After all, Gracie's clock is moving more slowly (due to time dilation) relative to his clock and that accounts for the extra time difference (beyond the communication travel time). Do I have that correct?</p> | g12804 | [
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<p>It is written everywhere that gravity is curvature of spacetime caused by the mass of the objects or something to the same effect. This raises a question with me: why isn't spacetime curved due to other forces or aspects of bodies?</p>
<p>Why isn't it that there are curvatures related to the charge of a body or the spin of particles or any other characteristics? </p> | g366 | [
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<p>I was talking with a friend recently about concave mirrors, which frequently invert the reflected image - I think we were playing with a spoon. I raised the question, if you had a mirror whose shape could be easily controlled so you could move it from convex through being flat to being concave, at what point does the image flip from being the right way up to being inverted - is there some sort of discontinuity? If not how would this transition occur?</p> | g12805 | [
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0.... |
<p>Imagine a manifold $\mathbb{R}^{1,3}\times{}B$ where $B$ is a compact group-manifold with isometry group $U(1)\times{}SU(2)\times{}SU(3)$.</p>
<p>Let's consider the Dirac equation for a massless Spinor field.</p>
<p>$iD_{1+n}\Psi=0$</p>
<p>and separating the compact space part with the 4-space part</p>
<p>$i(D_4+D_{compact})\Psi=0$</p>
<p>we see that the compact dirac operator might be regarded as a mass operator for the 4 dimensional spinor. </p>
<p>Imagine we are interested in solutions whose mass is zero, that is, solutions whose eigenvalues of the internal Dirac operator is zero.</p>
<p>I am trying to understand the impossibility of arranging these zero mode fermions on complex representations of the isometry group, but for that first I need to understand why the fermions obtained by dimensional reduction must form a representation (be it real or complex) of my isometry group.</p>
<p>This is asserted in <a href="http://www.sciencedirect.com/science/article/pii/0550321381900213">this</a> paper (Witten 1981). On the third paragraph of the ninth page it is said</p>
<blockquote>
<p>If the ground state is a product of four dimensional Minkowski space with one of the $M^{pqr}$ (for our purposes this just means a compact group manidold with the desired isometry group), then the zero modes of the Dirac operator in the internal sace will automatucally form multiplets of $U(1)\times{}SU(2)\times{}SU(3)$ since this is the symmetry of the internal space. </p>
</blockquote>
<p>Could someone please be more explicit of why this is so?</p> | g12806 | [
-0.02637936733663082,
-0.020273694768548012,
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0.0... |
<p>The nerve impulse transmission is specifically a biophysical process. Under a <em>resting</em> stage, the membrane is already polarised (presence of charge on either side leading to a potential difference across it, due to its finite capacitance). Changes in this polarisations mediated by several agents, cause some part of this membrane to be depolarised (inversion of the charges in that localised area). This changes the potential difference across that region and starts ion flow from the depolarised region of the membrane to the adjacent polarised region, inducing the same depolarisation there. Myelinating a neuron is similar to lowering the capacitance of the neuron. Myelinating the neuron causes subsequent action potentials to be spatially separated, and the conduction of voltage between them occurs primarily through ion flow along the neuron. <a href="http://en.wikipedia.org/wiki/Action_potential" rel="nofollow">Here</a> are some nerve conduction basics.</p>
<blockquote>
<p>Now my actual question. <strong>Modelling a neuron as a simple one dimensional
<em>membrane/cable</em>, how can a lowered capacitance lead to a faster conduction of the voltage perturbation i.e a change in voltage
(depolarisation), which is initially limited to a small localised
region? This conduction of the voltage change can occur through the
ion flow along the inner side of the membrane and also due to long
distance changes in potential due to the altered distribution of
charges along the membrane</strong>. A very good physical modelling of the
neuron is given <a href="http://www.scholarpedia.org/article/Electrical_properties_of_cell_membranes" rel="nofollow">here</a> and <a href="http://www.scholarpedia.org/article/Cable_theory" rel="nofollow">here</a>, but because of the quite
complicated mathematical nature of the modified telegrapher's equation
which appears as the final answer, I am unable to understand how
lowered capacitance increases the speed of voltage conduction? </p>
</blockquote>
<p>Tell me if this question is off-topic or too biological to be answered here.</p> | g12807 | [
0.0068192570470273495,
-0.012386521324515343,
-0.01129316259175539,
0.021253440529108047,
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0.0005693738930858672,
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0.042411256581544876,
0.... |
<p>I'm trying to better understand the modal content and dynamics of laser cavities. I'm specifically interested in mode-locking, although this question is pretty general and applies to CW lasers too.</p>
<p>I understand that the modal spacing, Δf, in a cavity is set by cavity geometry and the gain bandwidth, ΔF, is set by the gain medium. Therefore, the maximum number of modes that can oscillate is ΔF/Δf. </p>
<p>However, what determines how many modes will actually be excited, and when some are (i.e. standard lasing operation), which modes dominate?</p>
<p>To quantify the modal content of a laser, I appreciate you can using a scanning Fabry perot interferometer connected to an oscilloscope and visualise each mode, providing the resolution and free spectral range are selected carefully.</p>
<p>However, I wonder if there are any easier ways of interfering this, say electrically? For example, I wonder if shining the laser onto a photodetector (with bandwidth ~ 1 GHz say) will cause interference between modes, creating difference frequencies, which can then be visualised on an electrical spectrum analzyer. </p>
<p>So, if I had 3 excited modes in my cavity, spaced by 10 MHz, then would I see 2 peaks on the electrical signal analyzer: one at 10 MHz (for the difference 30-20 MHz and 20-10 MHz) and a lower amplitude peak at 20 MHz (for the difference 30-10 MHz)?
And similarly, if my laser were single-frequency, could we deduce that there is no interference on the photodetector as there is only one incident mode, so the only signal generated is in the THz regime (due to the wavelength of visile light, say), which can't be detected by the photodetector. Hence, we'd see no signal on the electrical spectrum analyzer?</p>
<p>Any guidance / references much appreciated. Thanks</p> | g12808 | [
-0.04102461785078049,
0.03336078301072121,
-0.002590550109744072,
0.028186524286866188,
0.024654841050505638,
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0.08950572460889816,
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0.008538980036973953,
-0.03541219234466553,
-0.021117743104696274,
-0.013932515867054462,
0.06592713296413422,
0.... |
<p>Let $\Omega$ be an observable and let $\Psi = \sum a_i\phi_i$ be a decomposition of a state of a system (satisfying the Schrödinger equation) in eigenfunctions of $\Omega$ (assume for simplicity it is a finite linear combination and there is no degeneracy). Measuring the observable $\Omega$ will give the value $\lambda_i$, the eigenvalue of state $\phi_i$, with probability $|a_i|^2$, in which case the wavefunction will collapse to $\phi_i$.</p>
<p>However, there is no reason why $\phi_i$ should be a state of the system. How is that possible? A few ways out occur to me:</p>
<ol>
<li>the state instantanously "evolves" to a new, valid state</li>
<li>such a measurement is somehow not possible</li>
</ol>
<p>Concretely, consider a particle in a 1-D box, and let $\Omega$ be the momentum operator. Then the state can be written as an "linear combination" (in this case an integral combination) but no state of definite momentum is a solution.</p>
<p>If we imagine the walls of the box, the infinite potential barriers, to be finitely wide, then if 1) is the case, it looks like the measurement of the momentum could cause the particle to tunnel through an infinite barrier (it would collapse to a state with uniform position probability, and evolve to a state that is nonzero on both sides of the barrier).</p>
<p>What is actually going on? Or did I just totally misunderstand something?</p>
<p><strong>EDIT:</strong> Strictly speaking such a $\phi_i$ always is a state of the system (only the time-evolution is governed by the Schrödinger equation). To be more precise would be to remark that it seems that measurement could make it go into a state of (extremely) much higher potential without any other effort than making that measurement.</p> | g12809 | [
-0.013136114925146103,
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0.0... |
<p>I'm reading Peter Atkins' book, <em>Galileo's Finger</em>, and in the chapter on energy, he makes the points that the conservation of momentum stems from the shape of space (that it's smooth and not lumpy) and that the conservation of energy stems from the shape of time (that it's smooth and not lumpy). I'm not totally clear on how the shape of spacetime leads to the conservation laws. Could someone elucidate the relationship, in layman's terms?</p> | g12810 | [
0.07290864735841751,
0.02019435353577137,
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0.04660945013165474,
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0.029879797250032425,
0.028250141069293022,
-0.05658518895506859,
0.07081174850463867,
0.012694... |
<p>I got this question from playing games like Halo and Borderlands (I know kinda dumb but raised a good question) in which the primary protection is an electric shield. Now I'm wondering if it would be possible to use energy to create a barrier in which physical objects either get disintegrated or deflected. Id imagine magnetism would come into play somewhere.</p>
<p>This kind of technology would be arguably the most useful invention because it would enable us to protect ourselves, not in war, but in situations that normal protection couldn't. Such as on spacecraft and rockets, common collisions and asteroids wouldn't be an issue. I would imagine the amount of protection, size, and duration would depend on the amount of energy available but if you had a system in which it's self propelling and generating more than it's using(such as a car batter and multiple alternator) then that wouldn't be an issue. Also I'd imagine the device would be small because the shield would appear and disappear with the addition and removal of the energy source.</p>
<p>So basically, is it theoretically possible to make a shield or barrier out of energy and if so, are we close or very far from this kind of breakthrough? thanks in advance</p> | g12811 | [
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0.03825780376791954,
0.010402671061456203,
-0.003852891968563199,
-0.03... |
<p>This is an approach to anomalies which seems unfamiliar to me..</p>
<ul>
<li><p>Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu \nu} \rangle $. Is there any standard such $W$? </p></li>
<li><p>Now one says that this $W$ must satisfy the following R.G equation, </p></li>
</ul>
<p>$(\mu \partial_\mu + 2\int d^dx g^{\mu\nu}\frac{ \delta }{\delta g^{\mu \nu} } )W = 0$</p>
<p>Where does this come from?</p>
<ul>
<li>So substituting the first equation into the second and defining $A_{anomaly} = \langle T^\mu_\mu \rangle$ we get, $\mu \partial_\mu W = -2\int d^dx A_{anomaly}$. And one one wants to take two derivatives of the L.H.S w.r.t the metric to get the LHS to look like, $\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle $. </li>
</ul>
<p>But I don't know what is this theorem which is now being used to say that this derivative of the 2-point function of the stress-tensor will necessarily have the following form, </p>
<p>$\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle = \frac { C_T} { 4(d-2)^2 (d-1)}\Delta^T_{abcd} \mu \partial_\mu (1/x^{2d-4} ) $</p>
<p>where $C_T$ is some number and $\Delta^T_{abcd} = \frac{1}{2}[S_{ac}S_{bd} + S_{ad}S_{bc}] - \frac{S_{ab}S_{cd} }{d-1} $ , where $S_{ab} = \partial_a \partial_b - \delta_{ab} \partial^2 $</p>
<p>Can someone kindly help as to what is the motivation/proof of the above structure? Is something similar known for derivatives of higher point correlations of the stress-tensor? </p> | g12812 | [
0.05817759782075882,
-0.04796190559864044,
-0.047057244926691055,
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0.050087835639715195,
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0.07049392908811569,
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-0.03860722854733467,
-0.03908354789018631,
0.05310603603720665,
0.0378759503364563,
-0.069... |
<p>Initially at time $t=0$, a solid sphere slides with velocity $v$ along a horizontal surface. The coefficient of friction is u. Find the required time for the sphere to stop sliding (the sphere will then be rolling). </p>
<p>Ive tried using Newtons equations of motion but I'm not sure what the final velocity will be?
At first I thought it may be zero because it will be rolling but obviously it will still have some translational velocity. I've also found the work done by the friction on the sphere and taking this figure away from the initial kinetic energy leaving (what I thought) would be the rotational kinetic energy that the ball has. However I can't get to the answer. I'm almost certain I'm just missing a small trick but I have no idea where to go with it.</p> | g12813 | [
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0.006998639088124037,
0.022809095680713654,
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-0.002152773318812251,
-0.039544232189655304,
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0.04605865478515625,
-0.01... |
<p>Let's assume we have a 1 kg cube on a flat uniform surface. The coefficients of friction between them are $\mu_k = 0.25$ and $\mu_s = 0.50$. This cube is moving at 1 m/s in the y direction, with whatever force is needed to overcome friction being applied. There is no force or motion in the x direction. Assume $g = 10\, {\rm m / \rm s^2}$ in the z direction. </p>
<p>Now, if a force is applied in the x direction which type of friction applies, static or kinetic? Edit: To clarify, if the cube is moving forward, and gravity is acting down, the new force is being applied to the right.</p>
<p>My rudimentary model of friction is that it acts like two pieces of sandpaper where the bumps can interlock when there is not motion, but when they are moving over top of each other they don't have time to settle in. In that model, I would think kinetic friction would apply, as the direction of movement shouldn't matter.</p>
<p>However, I realize that in reality friction is a result of molecular attractive forces, and probably a lot of other more complicated things I'm unaware of. Because of this, I suspect the answer may lie somewhere between static and kinetic friction.</p> | g12814 | [
0.06790412962436676,
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0.0192... |
<p>If a particle is placed on a rotating turntable then the particle has a tendency to slip tangentially with respect to the underneath surface... So the friction should act tangentially to the particle... Why then does it act towards the center... Please explain with respect to inertial frame</p>
<p>Friction opposes relative slipping between surfaces... When the particle is initially placed on the rotating turntable... the surface under it has a tangential velocity. So, in order to oppose this slipping friction must act tangentially... How then does it act radially inward?</p> | g64 | [
0.04412547126412392,
-0.031602438539266586,
0.010654747486114502,
-0.03292613476514816,
0.057587865740060806,
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-0.04585014656186104,
0.... |
<p><img src="http://i.imgur.com/EAVEwED.png" alt="HW Problem"></p>
<p><strong>1.) What is the acceleration of the hanging mass? (Indicate the direction with the sign of your answer.)</strong></p>
<p><strong>2.) Determine the tension in the cord. (Indicate the direction with the sign of your answer.)</strong></p>
<p>Seems like the simplest problem but I cant seem to get the right answer. Here's my approach: </p>
<hr>
<p>$n $ = normal force on $m_1 $</p>
<p>$f_s $ = friction force on $m_1 $</p>
<p>$T $ = tension in the cord</p>
<p>$\mu_k $ = coefficient of kinetic friction</p>
<hr>
<p><strong>Problem 1.)</strong></p>
<p>$(1)\quad n = m_1 g = (40kg)(9.8m/s^2)$</p>
<p>$(2)\quad n = 392 N$</p>
<p>$(3)\quad f_s = \mu_k n = (0.22)(392N)$</p>
<p>$(4)\quad f_s = 86.24N$</p>
<p>$(5)\quad T = m_2 g - f_s$</p>
<p>$(6)\quad T = (13.4kg)(9.8m/s^2) - 86.24N$</p>
<p>$(7)\quad T = 131.32N - 86.24N = 45.08N$</p>
<p>$(8)\quad m_2 a_{m_2} = m_2 g - T$</p>
<p>$(9)\quad (13.4kg) a_{m_2} = 131.32N - 45.08N$</p>
<p>$(10)\quad (13.4kg) a_{m_2} = 86.24N$</p>
<p>$(11)\quad a_{m_2} = 86.24N/13.4kg = 6.44m/s^2$</p>
<p>$6.44m/s^2$ is wrong, though. Could someone point me in the right direction? Any feedback is welcome, and thanks in advance</p>
<p><strong>SOLUTION:</strong></p>
<p>Equation (5) should be $T-f_s = m_1 a$ </p>
<p>$T-f_s = m_1 a$ </p>
<p>becomes (after rearranging) </p>
<p>$T-(40kg)a = 86.24N$ </p>
<p>and</p>
<p>$m_2 a = m_2 g - T$ </p>
<p>becomes (after rearranging) </p>
<p>$T + (13.4kg)a = 131.32N$. </p>
<p>Then we throw those two equations into a matrix: </p>
<p>\begin{bmatrix}
1 & -40 & 86.24 \\
1 & 13.4 & 131.32
\end{bmatrix}</p>
<p>Solve, and our answers will be:</p>
<p>$a = 0.84m/s^2$</p>
<p>$T = 120.01N$</p> | g12815 | [
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-0.011176020838320255,
-0.029009239748120308,
0.031422507017850876,
-0.045824456959962845,
... |
<p>I'm having trouble understanding how Carroll (Spacetime and Geometry p.296) explains the effect of a passing gravitational wave on test particles.</p>
<p>If we have two geodesics with tangents $\vec{U}$, $\vec{U'}$ that begin parallel and near each other, and $\vec{S}$ is a vector connecting one geodesic to another at equal affine parameter values, then the equation of <a href="http://en.wikipedia.org/wiki/Geodesic_deviation" rel="nofollow">geodesic deviation</a> is:
\begin{align*}
\frac{D^2}{d\tau^2}S^\mu = R_{\ \ \nu\rho\sigma}^\mu U^\nu U^\rho S^\sigma
\end{align*}
We work in the weak-field limit and the transverse-traceless gauge. If we assume our particles on the geodesics are moving slowly, then $\vec{U} \approx (1,0,0,0)$, so:
\begin{align*}
\frac{D^2}{d\tau^2}S^\mu = R_{\ \ 00\sigma}^\mu S^\sigma
\end{align*}
Now the bit I don't understand is how Carroll is able to turn the double covariant derivative on the left into a simple double-derivative with respect to $t$:
\begin{align*}
\frac{\partial^2}{\partial t^2}S^\mu = R_{\ \ 00\sigma}^\mu S^\sigma
\end{align*}
Carroll's reasoning is that <em>"for our slowly moving particles we have $\tau = x^0 = t$ to lowest order"</em>, but I don't know what he means. I just don't understand why the Christoffel symbols vanish in the covariant derivatives. I have read several books about this. Some say the Christoffel symbols vanish because we work in a local inertial frame... but then why doesn't the Riemann tensor on the RHS also vanish?</p> | g12816 | [
0.05839410796761513,
0.031310535967350006,
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0.049850232899188995,
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-0.04081117734313011,
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0.021810539066791534,
0.050418343394994736,
0.01... |
<p>For two non-interacting particles, with eigenfunctions $\phi_{n1}(x1)$ and $\phi_{n2}(x2)$ in a one-dimensional potential well $V_{(x)}$ with n = 1,2,....</p>
<p>Consider two spinless non-identical particles:</p>
<p>What is the degeneracy of the ground state and first excited state?</p>
<p>I'm thinking it should be 0 for both.</p>
<p>Now Consider two spinless identical particles,</p>
<p>I'm thinking degeneracy for ground state is 0, and first excited state is 2. (since ground state both n = 0, and first excited state it can either be 1,0 or 0,1)</p> | g12817 | [
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-0.008759614080190659,
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0.002202539937570691,
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0.060... |
<p>In "Mechanics" book by Landau and Lifshitz, $\S18$ "Scattering", problem 7 on deducing the form of scattering field given the effective cross-section as a function of scattering angle an equation is obtained:</p>
<p>$$-\pi\text{d}\log w=\text{d}\left(\frac s w\right)\int_0^{s^2/w^2}\frac{\chi'(x)\text{d}x}{\sqrt{\frac{s^2}{w^2}-x}}.$$</p>
<p>Before this, several variables have been introduced: $s=1/r, x=1/\rho^2, w=\sqrt{1-\frac U E}$, where $U>0, U(\infty)=0$.</p>
<p>They then say:</p>
<blockquote>
<p>This equation can be integrated immediately if the order of integration on the right-hand side is inverted. Since for $s=0$ (i.e. $r\to\infty$) we must have $w=1$ (i.e. $U=0$), we have, on returning to the original variables $r$ and $\rho$, the following two equivalent forms of the final result:
$$w=\exp\left\{\frac1\pi\int_{rw}^\infty\cosh^{-1}\left(\frac{\rho}{rw}\right)\frac{\text{d}\chi}{\text{d}\rho}\text{d}\rho\right\}=\exp\left\{-\frac1\pi\int_{rw}^\infty\frac{\chi(\rho)\text{d}\rho}{\sqrt{\rho^2-r^2w^2}}\right\}.$$</p>
</blockquote>
<p>As I understand, they integrate taking $\frac s w$ as variable of integration. This seems OK, and I tried to take integral with the seemingly sensible limits from $0$ to $\frac s w$ (i.e. variable upper limit), but the result for RHS appears quite strange: denoting $\frac s w=\beta$, here's what I get after switching order of integration:</p>
<p>$$\int_0^\beta \text{d}x\chi'(x)\int_0^\beta\frac{\text{d}\beta}{\sqrt{\beta^2-x}}=\int_0^\beta\text{d}x\chi'(x)\left(\log\left(\beta+\sqrt{\beta^2-x}\right)-\log\sqrt{-x}\right).$$</p>
<p>Given that $x=1/\rho^2$, the RHS seems to become complex, which makes me think I do something wrongly. I suspect that I'm taking wrong limits of integration, but L&L don't specify which ones are to be taken! Supposedly the readers <em>could</em> determine this themselves, but I'm not as smart.</p>
<p>So, what are the right limits of integration here?</p> | g12818 | [
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-0.00... |
<p>I was wondering if a photon is divisible. </p>
<p>If you look at a photon as a particle, then you may be able to split it (in theory).</p>
<p>Is it possible and how do you split it?</p> | g602 | [
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0.01942460611462593,
-0.0270... |
<p>I wanted to know the importance and applications of the study of the diatomic molecule in physics</p>
<p>I know that the question may not make much sense to you, but while exposing the solution of relative motion of the diatomic molecule, some students asked me about their applications and could not answer. On the internet, I have not found anything useful, so I turn to you. Thanks for your attention. </p> | g12819 | [
-0.0012531098909676075,
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0.042450714856386185,
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0.0850495994091034,
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-0... |
<p>This is related to my previous Phys.SE <a href="http://physics.stackexchange.com/q/52894/2451">question</a> on the derivation of the phase function - upon sifting and scanning through 600 pages of John Strutt's collected work, there is <strong>absolutely no mention</strong> of Rayleigh's phase function (related to <a href="http://en.wikipedia.org/wiki/Rayleigh_scattering" rel="nofollow">Rayleigh scattering</a>) quoted by a million other sources with no derivation or justification. </p>
<p><a href="http://archive.org/details/scientificpapers04rayliala" rel="nofollow">Rayleigh's scientific papers (1899.) @ archive.org</a></p>
<p>I know it is some sort of godforsaken approximation for which nobody on the Web has a justification or derivation for and it's really bothering me. I have no problem with empirical data reconstruction, but I want to see this "empirical data" which everyone just throws around.</p>
<p>Why do people say something is "defined". Nothing in physics is defined, it's not a recipe for an exotic top-of-the-head meal, throw three cosines, five g and you're there. It has to be constrained by nature in some way.</p>
<p>Please, assist!</p> | g12820 | [
0.01146937720477581,
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0.014007466845214367,
-0.06815642863512039,
0.039877623319625854,
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-0.04828336089849472,
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0.01927071437239647,
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<p>Suppose I have two plates with a viscous fluid in between. I slide them in the same direction (a direction in their own plane), one at $5 \,\text{m/s}$ and the other at $6 \,\text{m/s}$. Due to the relative velocity, there will be a viscous damping force tending to equalize the velocities of the plates.</p>
<p>Now suppose I have two concentric cylindrical shells, the second with a radius $6/5$ that of the first. I put viscous fluid between them and rotate the entire system such that the cylinders have tangential velocities $5 \,\text{m/s}$ and $6 \, \text{m/s}$. Because the system is executing rigid rotation, there will be no viscous damping.</p>
<p>Suppose I take a small cube of the viscous fluid in either scenario. In both cases, there's a similar velocity gradient across the cube, but only in one case does the cube dissipate energy. Why?</p>
<p>I can work this out by writing the Navier-Stokes equation and transforming to cylindrical coordinates, but I am still having a hard time visualizing the kinematics such that this becomes clear to me.</p> | g12821 | [
0.07387932389974594,
0.013669426552951336,
0.013713337481021881,
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0.02... |
<p>If the earth is rotating at some <a href="https://en.wikipedia.org/wiki/Earth%27s_rotation#Angular_speed" rel="nofollow">$465~\text{m}/\text{s}$</a> at the equator and that's really fast.</p>
<ol>
<li>Shouldn't we in that case be in orbit with the earth just not fast enough? </li>
<li>How fast do we need to move?</li>
<li>If the earth stopped rotating (gradually), shouldn't gravity increase significantly?</li>
<li>Do satellites need extra speed when they are released or does it just depend on their inertia from the earth's rotation?</li>
<li>If the satellite decelerated against the earth direction of rotation will it fall? </li>
<li>If the earth rotated extremely fast, shouldn't the earth's crust along with everything else get ejected away by the centrifugal force?</li>
</ol>
<p>I'm sorry if the questions cover many matters but I preferred to ask them all at once instead of asking each separately.</p> | g12822 | [
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<blockquote>
<p><em>Two straight ladders, AC and BC, have equal weights $W$. They stand in equilibrium with their feet, A and B, on rough horizontal ground and their tops freely hinged at C. Angle CAB = 60$ ^o$, angle CBA = 30$^o$ and AB = $l$ (see diagram). Find the vertical reactions at A and B.</em></p>
</blockquote>
<p><img src="http://i.stack.imgur.com/dTUt5.png" alt="illustration of mechanical problem"></p>
<p>Resolving moments about A:</p>
<p>$$\frac12l\sin30^o\cdot\sin30^o\cdot W + (l-\frac12l\sin60^o\cdot\sin60^o)W = lN$$</p>
<p>$$\implies N = \frac{3W}4$$</p>
<p>Resolving moments about B:</p>
<p>$$\frac12l\sin60^o\cdot\sin60^o\cdot W + (l-\frac12l\sin30^o\cdot\sin30^o)W = lR$$</p>
<p>$$\implies R = \frac{5W}4$$</p>
<blockquote>
<p>Find the magnitude and direction of the force exerted on BC by AC.</p>
</blockquote>
<p><strong>I couldn't work out how to solve this?</strong></p>
<p>The required answer is $\frac12W$, 30$^o$ above horizontal.</p> | g12823 | [
0.06268052011728287,
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0.028384745121002197,
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0.03108479455113411,
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<p>If I have a system consisting of 2 Einstein solids (<code>A</code> and <code>B</code>) is it equivalent to say that maximizing the multiplicity of the system is the same as setting the temperature to $\infty$? I have two reasons to support this Idea.</p>
<p>1) The total multiplicity of the system can be expressed as:
$$\Omega_{tot} = \Omega_A(q_A,N_A) \Omega_B(q_B,N_B)$$
Which is the product of the individual multiplicities, and $q = q_A + q_B = $ Total quanta of energy shared between the system, and $N = N_A + N_B =$ total number of quantum harmonic oscillators.</p>
<p>I wish to maximize the multiplicity with respect to oscillator <code>A</code>'s total internal energy ($U_A)$
$$\left( \frac{ \partial \Omega_{tot} }{ \partial U_A } \right) = 0 \implies \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = \frac{ \Omega_{tot}'}{ \Omega_{tot}} = 0$$</p>
<p>This is true because Ln(x) is a monotonic function. But it should be noted that the definition of temperature is such that:</p>
<p>$$\frac{1}{kT} = \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = 0 \implies T= \infty$$</p>
<p><code>k</code> here is the boltzman constant. </p>
<p>2) This makes some intuitive sense. On the Gaussian distribution as a function of Internal energy, the maximum multiplicity lies on the peak. The change in temperature gives a quantification of the direction of heat flow. If the maximum multiplicity is attained, then heat cannot flow any higher, it can only flow downward. This corresponds to infinite temperature. </p>
<p>I hope my intuition here is correct.</p> | g12824 | [
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0.026492653414607048,
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0.007019486743956804,
0.006126977503299713,
0.00... |
<p>I have a very naive question about the notion of time-reversal symmetry applied to topological insulators that are studied in experiments. If I understand correctly, the exsistance of time-reversal symmetry is crucial for the occurence of topologically protected surface states in topological insulators. At least this is what the theory of topological insulators tells us. But I have a problem with applying this theory to real materials. In real world there is always dissipation and therefore there is an "arrow of time" and time-reversal symmetry does not hold (right?). So how can we talk about time-reversal symmetry in real materials that we call topological insulators? Please help me find where I went wrong in my argument. I realize that there must be something wrong but can't figure out what exactly. </p> | g12825 | [
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-... |
<p>I read this example in a popular math book I was browsing, and the author does not give any physical explanation. </p>
<p>The example: Consider a rock falling freely under gravity. There are no viscous or dissipative effects. We will find the average speed at the end of 5 seconds.</p>
<p><strong>Time-domain.</strong> $v = 32 t \mbox{feet/sec}$
Average over $5$ seconds $$\langle v\rangle_\text{time} = \frac{1}{5} \int_0^5 32 t \mathrm{d}t = 80\text{ feet/sec}$$</p>
<p><strong>Space Domain.</strong> At $t =5 $ sec, $y = 400$ feet. $$\langle v\rangle_\text{space} = \frac{1}{400}\int_0^{400} 8\sqrt{y}\; \mathrm{d}y \approx 107\text{ feet/sec}$$</p>
<p>Sorry about the use of imperial units. But I just dont understand the physics of this. The source uses this only as a demonstration of Cauchy Schwarz inequality. Is there something elementary I am missing out here?</p>
<p>Essentially, he derives $$\frac{1}{T}\int_0^T v(t) \mathrm{d}t \leq \frac{1}{L}\int_0^L v(y)\mathrm{d}y$$</p>
<p>The math is ok to follow, but my physical intuition just cant get it somehow.</p> | g12826 | [
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0.059... |
<p>Suppose I look at a parallel plate capacitor in its rest frame and calculate the electrostatic energy, $E$.</p>
<p>Next, I look at the same capacitor in a primed frame boosted in the direction perpendicular to the plane of the plates. In this frame, the $E$-field is the same strength, there is no magnetic field, and the volume over which the $E$-field extends is less by a factor $1/\gamma$. This suggests $E' = \frac{1}{\gamma} E$, but relativity states that energy transforms as $E' = \gamma E$. </p>
<p>Where is the missing energy?</p> | g12827 | [
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0.012195510789752007,
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-0.017746... |
<p>Let's say I turn on a <a href="http://en.wikipedia.org/wiki/Van_de_Graaff_generator" rel="nofollow">Van de Graaff</a> which creates a large positive charge. Now let's say I have an object with a positive charge in my hand and I start walking toward the Van de Graaff from $x$ meters away. If I walk an arbitrary distance towards the Van de Graaff, I am doing work and the charge in my hand is gaining potential energy. Now somebody standing next to the Van de Graaff suddenly turns it off. What happens to that potential energy of the charge? By the conservation of energy it should be conserved somehow, but it appears not to be. Any help is appreciated.</p> | g12828 | [
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0.004427121952176094,
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0.05179384723305702,
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0.008313496597111225,
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<p>Suppose we have some tensor with components $T^{ij}$. Then suppose that we also have $T_{ij}$. </p>
<p>When would $T^{ij}T_{ij} = (T^{ij})^2 = (T_{ij})^2$?</p> | g12829 | [
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0.024168822914361954,
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0.032190997153520584,
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<p>A month or two ago I purchased a “Skywatcher Explorer 130P Newtonian Reflector” telescope - which so far is brilliant when looking at the Moon, esp. using a neutral density filter. However, when looking at Venus or Mars - which are very easy to view and beautiful to see through the eyepieces - they seem slightly blurred (almost impossible to get a sharp focus) but more frustratingly they seem to have a lot of flare coming out of the planet like an ‘X’ where the planet is at the centre. </p>
<p>For some reason Jupiter did not have any issues and I was able to see it sharply and its moons with no flare.</p>
<p>This issue is there with both eyepieces : 10 mm & 25 mm that were provided with the scope.</p>
<p>Can someone please assist me and/or advise me as to how I can go about remediating the flare and sharpness issue? Is it to do with the fact that I do not have any planetary filters? If so, what will I need please? Any links, esp. online stores in the UK, will be very helpful. Is it to do with the apparent brightness of the planets?</p> | g499 | [
-0.020691314712166786,
-0.020947398617863655,
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0.048166558146476746,
0.06863806396722794,
0.0104... |
<p>I am trying to find a formula that will enable me to calculate the illumination of the moon down to one thousandth of a percent, given that the Gregorian year, month, day, and hour is known. Can anyone help me with this formula?</p> | g12830 | [
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0.025747742503881454,
0.028706436976790428,
0.0020761368796229362,
0.052739690989255905,
... |
<p>We know that each gas giant planet was warmest when it was young. This warmth came from <a href="http://en.wikipedia.org/wiki/Internal_heating" rel="nofollow">internal heating</a> from both radioactive decay and from gravitational potential energy. This warmth, in turn, means that the planet emits blackbody flux.</p>
<p>So this raises some interesting questions. Among them - <a href="http://www.quora.com/Are-there-certain-alignments-in-the-solar-systems-orbits-that-create-additional-interesting-effects-on-our-planets-moons-or-sun/answer/Alex-K-Chen" rel="nofollow">how much of this early daytime flux affected the day-to-day temperatures of their moons</a>? </p> | g12831 | [
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0.011441175825893879,
0.028436873108148575,
0.03106272965669632,
-... |
<p>I'm busy studying for my Physics exam this evening and I've come across a vector problem that I cant quite solve. Any help would be appreciated!</p>
<p>We are given the following vectors:</p>
<pre><code>A = 5i - 6.5j
B = -3.5i + 7j
</code></pre>
<p>We are also told that a vector C lies in the xy-plane and is perpendicular to A. The scalar product of B and C is equal to 15. With this information we have to calculate the components of the vector C.</p>
<p>I have tried calculating the magnitude of B and then trying to solve the equation:</p>
<pre><code>|B||C|cos{theta} = 15
</code></pre>
<p>But I didn't get very far. I have a feeling I'm just not seeing something and making a careless error.</p>
<p>Thanks in advance!</p> | g12832 | [
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0.049807846546173096,
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0.0226274561136961,
0.03571547567844391,
0.033069126307964325,
-0.012599245645105839,
-0.02... |
<p>I'm tutoring high school students. I've always taught them that:</p>
<blockquote>
<p>A charged particle moving without acceleration produces an electric <em>as well as a magnetic field</em>.</p>
</blockquote>
<p>It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?</p> | g346 | [
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-0.0193... |
<p>I know I've seen zodiacal light, especially at my old club's best dark-site. However, I'm pretty sure I've yet to see the <a href="http://en.wikipedia.org/wiki/Gegenschein" rel="nofollow">gegenschein</a>. How good does the seeing have to be to be able to see that?</p> | g12833 | [
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-0.015... |
<p>Here is a typical question which I have been asked many times while giving public lectures in various places. While I know one of the paths like Diploma, Masters and PhD but sometimes this is not so obvious for that eager young person who thinks to go on that path. So what that would be?</p> | g12834 | [
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<p>I remember a method to make a reasonable estimate of the magnitude of a given star X by using two other stars of known magnitude as references. The method used evaluation phrases like "the star X and the reference star appear to have the same brightness, and this sensation remains even after prolonged observation" or "X and the other star appear to be the same, but after some time, X appears slightly brighter" or "X and A are completely different in brightness". Each of these phrases had a value (IIRC from 1 to 5) and then with a formula you could estimate the magnitude of X from the magnitude of the reference stars and the values obtained from the comparison phrases.</p>
<p>Do you know the name of the method, and the exact protocol ? </p> | g12835 | [
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0.01... |
<p>Lets consider it case by case:</p>
<p>Case 1: Charge particle is at rest. It has electric field around it. No problem. That is its property.</p>
<p>Case 2: Charge particle started moving (its accelerating). We were told that it starts radiating EM radiation? Why? What happened to it? What made it do this? </p>
<p>Follow up question:
Suppose a charge particle is placed in uniform electric field, it accelerates because of electric force it experiences. Then work done by the electric field should not be equal to change in its kinetic energy right? It should be equal to change in K.E + energy it has radiated in the form of EM Waves. right? But then, why don't we take energy radiated into consideration why solving problem (I'm tutoring grade 12 students. I never encountered a problem in which energy radiated is considered.)</p>
<p><a href="http://physics.stackexchange.com/questions/65335/how-does-moving-charges-produce-magnetic-field">How does moving charges produce magnetic field?</a></p> | g437 | [
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0.05320208519697189,
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-0.014... |
<p>I am looking at peltier heaters/coolers, and all of them are thin wafers like the one pictured below.</p>
<p><img src="http://i.stack.imgur.com/zZBXg.jpg" alt="Peltier device"></p>
<p>This works fine for some things (ie. a microchip or a beer coaster) but not for others (ie. anything elongated). Assume, for the sake of argument, that I wish to heat a screwdriver to melt the ice off of frozen screws while I unscrew them. I want to put a peltier device inside of the handle of the screwdriver (as this is the most practical place to conceal it). The problem is that the only point of contact will be with the roughly 0.5cm diameter of the 'bit'. So my question now becomes whether I should somehow use a thermal conductor to connect the peltier device to the screw bit (if so, how inefficient is this), or whether to use a multi level peltier device (in an attempt to elongate the design). Alternatively, is there another solution I have not thought of? </p>
<p>I have included a crude drawing to further clarify my question. Also, ignore for the moment that there is no heat sink in this design.</p>
<p><strong>Option 1: Using a thermal conductor</strong>
<img src="http://i.stack.imgur.com/tDCMy.jpg" alt="enter image description here"></p>
<p><strong>Option 2: Using a multi-level peltier device</strong>
<img src="http://i.stack.imgur.com/979q8.jpg" alt="enter image description here"></p> | g12836 | [
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... |
<p>Here goes: </p>
<ul>
<li>imagine a spacecraft with engines shut off in a free fall towards some celestial body x.</li>
<li>let's say the distance to surface is $100 000$ $\text{meters}$. </li>
<li>initial vertical velocity is $500$ $\text{m/s}$. </li>
<li>gravitational pull experienced by the body is obviously given by $F=G \frac {m_1 m_2}{r^2}$.</li>
<li>acceleration of said craft is calculated as $a = \frac{F}{m}$.</li>
</ul>
<p>Given constant $F$ and therefore $a$, impact velocity and time are easy to calculate, but how exactly do I solve this, considering that gravitational pull and acceleration are constantly changing as we approach the surface?</p>
<p>Key question in <a href="http://physics.stackexchange.com/questions/60492/kinematics-with-non-constant-acceleration">Kinematics with non constant acceleration</a> is "To what value does the velocity v of the particle converge to <strong>as x approaches infinity</strong> if the particle starts at some point x0?"</p>
<p>The answer by David H involves the following formula $v_f = \sqrt{\frac{2k}{x_0} + v_0^2}$</p>
<p>In my case x goes from 100000 to 0 m (surface), in other words, there is a clearly defined endpoint which is not equal to infinity. Therefore I do not see how David's formula is applicable in my situation. </p> | g406 | [
0.035194091498851776,
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0... |
<p>We normally think of the "default" or "root" state of things as being on the positive side of the spectrum. For example, we don't normally use a <code>+</code> symbol to indicate the sign of positive numbers, whereas negative numbers are invariably preceded with <code>-</code> to indicate their negative sign. Similarly, positive temperatures (in °C or in F) are those that we tend to encounter more typically.</p>
<p>Why, then, do we define the basic unit of <em>electricity</em> as something with a negative charge?</p> | g258 | [
0.025644565001130104,
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0.05677955597639084,
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0.... |
<p>When I boil water in the kettle, it makes a nice cup of tea. Sometimes I need to use a microwave because a kettle isn't available. I boil the water in the mug and it looks pretty normal, but when I drop in the teabag the water froths up and looks foamy. I don't see what the chemical difference is here, so I assume it must be some physical difference. I have noticed this with multiple types of tea and multiple microwaves, the results being consistent so it's not just a weird microwave or something like that. </p>
<p>What is the reaction here and how/why does it occur?</p>
<p>Here is a photo of the 'fizzy' looking tea just after dunking in the teabag. </p>
<p><img src="http://i.stack.imgur.com/DxXfZ.jpg" alt="enter image description here"></p> | g12837 | [
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0.13664... |
<p>I am a visual person, so it's hard to imagine the information I keep getting, but shouldn't time be a constant? </p>
<p>If you were traveling at the speed of light and your able to cover $299{,}792{,}458$ meters in one second, aren't you just going faster than anything else? Why does time slow down as you approach high speeds? </p>
<p>Think about it on a $x-y$ <code>2D</code> grid. I start off from point $(0,0)$. Then I travel at $299{,}792{,}458\ \mathrm{m/s}$ horizontally, so my $x$ after traveling one second should be at $299{,}792{,}458$. If each $x$ point is one meter apart, wouldn't this mean you are just traveling so fast that your eyes are all blurred up until the point where you slow down to a speed where your eyes could distinctively see whats going on around you, yet nothing is slowed down, it's just you were able to physically reach that speed within a second. So time doesn't stop its always constant in the sense that it never stops or increases in speed, it's just ticks a constant pace.</p>
<p>If I throw a ball at a speed and it will travel at that same speed, but if I threw the ball at the speed of light, I can pretty much make the ball reach the location faster than the time it should of reached mathematical wise, which that doesn't even make any sense.</p> | g12838 | [
0.07224346697330475,
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0.006887208670377731,
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0.04837395250797272,
... |
<p>We know that the sun loses an amount of it's mass equivalent to the amount of energy it produces, according to the E=mc² equation.
so the sun is losing mass every second. Does this affect the space-time curvature it creates. Or does this affect the distance between the sun and the Earth. Does losing mass affects the gravity of sun or other planets?</p> | g12839 | [
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-0.02702154964208603,
0.02113831788301468,
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0.06459575146436691,
0.0551... |
<p>Suppose we meet a person wearing glasses.<em>Can we determine whether the person is short-sighted or long-sighted</em>?</p>
<p>However,due to courtesy,we are not allowed to ask him to try the glasses and in general we can't even make any mention of them.</p>
<p><strong>Attempt at the solution:</strong>
I tried using the image of the eyes but since practically for normal glasses we are at infinity.So the images appear almost the same.</p> | g12840 | [
-0.002050754614174366,
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0.05022... |
<p>I have a very basic question about photons in double-slit experiment. I am not good at math, and have some Quantum mechanics knowledge. Math free explanation would be very good to me.</p>
<p>When we do double slit experiment with single photon at a time, it will be detected at some location of screen. Over the time period the accumulation of detected locations show interference pattern. The reason for the interference pattern is that photon's location in not well defined, the presence of photon in particular place can be determined by prbability and this probability is presented as a wave.
that probability wave is splitted in to two while passing the two slits, collide with each other and causes interference pattern.</p>
<p>Question 1: Is this reason for single photon interference correct?</p>
<p>Question 2: How is the probability wave behaving while two-photons passing double-slit at a time? Are 4 probability waves (2 for each) interfering?</p> | g12841 | [
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0.0330... |
<p>Water inside bucket is rotated (by spoon or something) to flow in circular motion. An object kept in the bucket tends to be at the center of the bucket. Why is that?</p> | g196 | [
0.023744937032461166,
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<p>From my limited understanding of quantum entanglement, it seems like qubits act the same way as pseudo-random-number-generators (except as far as we can tell, these ones really are random). When you entangle two qubits, you give them both the same seed.</p>
<p>Then a measurement is like running an (otherwise deterministic) program. As long as Alice and Bob both run the same program on their two (identically seeded) <a href="http://en.wikipedia.org/wiki/Pseudorandom_number_generator" rel="nofollow">PRNG</a>'s, they'll both get the same results. There's no teleportation here, they just both have access to the same information.</p>
<p>Why is this referred to with the fantastic name "<a href="http://en.wikipedia.org/wiki/Quantum_teleportation" rel="nofollow">Quantum Teleportation</a>"? Since Alice and Bob can't pass information to each other through their PRNG's (they're just copies after all, they're not really linked).</p> | g12842 | [
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0.00024395853688474745,
... |
<p>Coherent states of light, defined as</p>
<p>$$|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{n!}|n\rangle$$</p>
<p>for a given complex number $\alpha$ and where $|n\rangle$ is a Fock state with $n$ photons, are usually referred to as the most classical states of light. On the other hand, many quantum protocols with no classical analog such as <a href="http://www.nature.com/nature/journal/v421/n6920/abs/nature01289.html" rel="nofollow">quantum key distribution</a> and <a href="http://arxiv.org/pdf/quant-ph/0008057v1.pdf" rel="nofollow">quantum computing</a> can be implemented with coherent states.</p>
<p>In what sense or in what regime should we think of coherent states as being 'classical' or 'quantum'?</p> | g12843 | [
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0.0074987150728702545,
0.034... |
<p>Suppose that I escaped the gravitational field of earth.</p>
<p>Then: am I going to be pulled by Sun's gravity?</p> | g12844 | [
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<p>In QFT one may refer to a particle as a representation of the Lorentz group (LG). More accurately - every particle is a quantum of some field $\phi(x)$ that belongs to some representation of the LG. <em>I wonder if that is really neccesary?</em> Arguments are usually like following:</p>
<p>Let $L$ be some Lorentz tranformation and $\Lambda(L)$ is the corresponding transformation of the field $\phi(x)$ , i.e. under the transformation $x\to x'=Lx$ field $\phi(x)$ goes to $\phi'(x')=\Lambda(L)\phi(x)$. Then under two succsessive transformations $L_2,L_1$ field $\phi(x)$ goes to $\Lambda(L_2)\Lambda(L_1)\phi(x)$. But one can also find himself in the same reference frame via transformation $L_2L_1$ and therefore observe the field $\Lambda(L_2L_1)\phi(x)$. Now demanding $\phi'(x')$ to be the same thing in both cases (since we arrived in the same reference frame) one concludes that
$$\Lambda(L_2)\Lambda(L_1)=\Lambda(L_2L_1)\,\,\,\,\, (1)$$. In mathematical language one says that tranformations of the field $\phi(x)$ constitute the representation of the LG.<br>
Well i do not see this to be neccesary. At least not within the "proof" that was been given. I've intentionally used the word "observe" above. Usually in physics not all the information that is encoded in the field value is virtually observable. For instance let $\phi(x)$ be a complex scalar field. Then an overall phase of the $\phi(x)$ is not observable. So i can assume that two ways of going to new reference frame may differ by the "gauge" tranformartion since it does not affect observables. In the case of the complex scalar field one can try to weaken (1) to something like
$$\Lambda(L_2)\Lambda(L_1)=\Lambda(L_2L_1)e^{i\alpha(L_1,L_2)}$$
where $\alpha(L_1,L_2)$ is the real number depending on Lorentz tranformations $L_1,L_2$.
My questions are: </p>
<ol>
<li><p>is it really possible for the physical fields to behave as was described?</p></li>
<li><p>if so, does their behavior differ (at classical or quantum level) from the "ususal" fields behavior?</p></li>
</ol> | g12845 | [
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0.0... |
<p>What's the scattering matrix for a PBS (polarization beam splitter) ?
Is it just unitary ?
If one polarization never couples into another polarization (then there's a lot of zeroes in that 4x4 matrix) - is that impossible somehow ?</p> | g12846 | [
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<p>In <a href="http://physics.stackexchange.com/questions/6209/paradoxical-interaction-between-a-massive-charged-sphere-and-a-point-charge">PSE here</a> electrons are added to a sphere and gravitational modifications are expected. </p>
<p>My question is:<br>
Is there any <strong>experiment</strong> that show that a negatively charged object is source of a stronger gravitational field than the same uncharged object? </p>
<p>In other words:<br>
The active gravitational mass of an electron is equal to his passive gravitational mass <strong>by experiment</strong>?</p>
<p>added, for clarification:<br>
from here <a href="http://physics.gmu.edu/~joe/PHYS428/Topic7.pdf" rel="nofollow">activ/passive grav mass</a>:<br>
- active gravitational mass: establishes the field<br>
- passive gravitational mass: responds to the field<br>
- by experiment: a carefully designed setup that evidences how electrons interact in the presence of a gravitational field. Does it exists already? If no, is it doable? etc.</p> | g12847 | [
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-0.00... |
<p><a href="http://en.wikipedia.org/wiki/Tolman_surface_brightness_test" rel="nofollow">http://en.wikipedia.org/wiki/Tolman_surface_brightness_test</a></p>
<p>It says:</p>
<blockquote>
<p>In a simple (static and flat) universe, the light received from an object drops inversely with the square of its distance, but the apparent area of the object also drops inversely with the square of the distance, so the surface brightness would be independent of the distance. In an expanding universe, however, there are two effects that reduce the power detected coming from distant objects. First, the rate at which photons are received is reduced because each photon has to travel a little farther than the one before. Second, the energy of each photon observed is reduced by the redshift. At the same time, distant objects appear larger than they really are because the photons observed were emitted at a time when the object was closer. Adding these effects together, the surface brightness in a simple expanding universe (flat geometry and uniform expansion over the range of redshifts observed) should decrease with the fourth power of (1+z).
To date, the best investigation of the relationship between surface brightness and redshift was carried out using the 10m Keck telescope to measure nearly a thousand galaxies' redshifts and the 2.4m Hubble Space Telescope to measure those galaxies' surface brightness.[1] The exponent found is not 4 as expected in the simplest expanding model, but 2.6 or 3.4, depending on the frequency band.</p>
</blockquote>
<p>I'm trying to understand where $(1 + z)^4$ comes in.</p>
<p>Specifically, it says:</p>
<blockquote>
<p>there are two effects that reduce the power detected coming from distant objects.</p>
</blockquote>
<p><strong>Question</strong></p>
<p>Do the two effects each contribute by $(1 + z)^2$, which are then multiplied together?</p>
<p>It also says </p>
<blockquote>
<p>First, the rate at which photons are received is reduced because each photon has to travel a little farther than the one before.</p>
</blockquote>
<p>Let's say just this factor was removed. Would the predicted brightness then be reduced by $(1 + z)^2$?</p> | g12848 | [
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-0.09140152484178543,
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0.042240552604198456,
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0.029... |
<p>In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:</p>
<ol>
<li><p>The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$</p></li>
<li><p>The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$</p></li>
<li><p>A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass)</p></li>
</ol>
<p>My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately.</p>
<p>PS. The question is a little related to a more general one I posted yesterday: <a href="http://physics.stackexchange.com/questions/116628/explicit-form-of-the-entropy-production-in-hydrodynamics">Explicit form of the entropy production in hydrodynamics</a> </p> | g12849 | [
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... |
<p>How is the image in a mirror created without a lens or pinhole?</p>
<p>From every point in the world there are infinitely many rays going out.
How come the image on a mirror is ok, when there is no pinhole to capture
exactly one ray from each world point, or a lens to focus all rays coming
from one world point onto one image point? I mean, in computer vision we
use these models to create images, so the mirror example here confuses me a lot.</p> | g12850 | [
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-0.02009519375860691,
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0.014186028391122818,
0.06288444250822067,
0.0271... |
<p>How is the image in a mirror created? Are there infinitely many light rays?</p>
<p>My motivation for the question is from image processing.
We work with images as discrete 2D functions, as matrices.
Spatial sampling is done, and also quantization.
Then I thought, well that is far from perfect, I wonder
how does God create images? And then I thought of a real example,
well, the mirror! The 3D world is projected on a 2D plane, the mirror.
So how many rays arrive on the mirror? Infinitely many?
Does God do spatial sampling or is the image in the mirror continuous?</p> | g12851 | [
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0.0... |
<p>Is there a simple way to explain the difference between a classical and truly quantum correlation to a non-quantum person who has basic understanding classical correlation? </p>
<p>I mean without invoking quantum mechanics, a simple CHSH type Bell inequity can be explained without it--perhaps with a little more math than I'm looking for, but more importantly non-classical correlation does not necessarily imply Bell violation. </p>
<p>Has anyone found a successful approach to explaining quantum correlation to a non-expert audience? Even partially successful approaches would be helpful.</p> | g12852 | [
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0.0444... |
<p>I hope you can clear up my following confusions.</p>
<p>In Girardello's and Grisaru's paper (Nuclear Physics B, 194, 65 (1982)) where they analysed the most general soft explicit supersymmetry breaking terms, they explicitly mentioned that $\mu \psi \psi$ is not soft, where $\psi$ is the fermionic component of a scalar superfield. This means that such a term will give rise to quadratic divergences.</p>
<p>However, in Stephen Martin's SUSY primer, on page 49, he explictly says the following:</p>
<blockquote>
<p><em>One might wonder why we have not included possible soft mass terms for the chiral supermultiplet fermions, like $L = −\frac{1}{2} m_{ij}\psi^i\psi^j + c.c.$. Including such terms would be redundant; they can always be absorbed into a redefinition of the superpotential and the terms ...</em></p>
</blockquote>
<p>This would seem to suggest that adding explicit chiral fermion mass terms is not problematic, since it is equivalent to a redefinition of the superpotential and the other soft breaking terms (e.g. scalar masses), both of which don't give rise to quadratic divergences.</p>
<p>I seem to see a contradiction here, but I am sure it is due to some subtlety I am too blind to notice.</p> | g12853 | [
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<p>On earth, using <a href="http://en.wikipedia.org/wiki/Earth_sheltering">earth-sheltering</a> techniques can significantly reduce the temperature fluctuations on a structure. Would the same statement be true as well on the Moon? Does the Moon's core still contain significant heat?</p> | g12854 | [
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<p>Assuming we originated from a single infinitely dense point in space time in the big bang, how can there be parts of the universe that we can't see as the light has not reached us yet, if nothing can travel faster than the speed of light relative to our frame of reference? Is that because the universe can expand faster than the speed of light? Or am I missing something?</p> | g10 | [
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<p>I can find various speeds and estimated durations listed at numerous places but none specifically describe the frame of reference. </p>
<p>Possible options as example of kind of answer I expect. </p>
<ul>
<li>Local Galactic cluster </li>
<li>Distance quasars </li>
<li>The cosmic background radiation?</li>
</ul>
<p><strong>--------- UPDATE ---------</strong></p>
<p>Thanks AIB and voithos. Lot of reading for me.</p>
<p>Though technically, I still don't have an answer that meets the following criteria. </p>
<ol>
<li>rotational velocity(average preferably) of sol around best estimate of center of Milky-way galaxy. </li>
<li>publicly available reference(I don't have immediate access to some of the books given) </li>
<li>frame of reference external to Milky-way galaxy.</li>
</ol>
<p>As I note below the only reference (wmap5basic_reprint.pdf) I can read that uses an external frame of reference doesn't specifically state the vector is rotational (despite wikipedia article assuming such). The topic is barely touched on in that paper.</p>
<p>I realised the speeds are variable. What I had not realised is that the whole idea of a (relatively) clearly defined x orbiting y system doesn't really scale up well from the local solar system to the galactic scale. The galaxy is more like a whirlpool or tornado compared to the "clockwork" appearance of the solar system. Although both the solar system and galaxy are constantly(very slowly) changing "fluid" rotational systems, the galaxy is obviously far more fluid than the solar system. Also we have not yet been able to observe anything about it's center.</p>
<p>Or in other words, we are not "orbiting" the galaxy, we are part of the galaxy.</p>
<p>I suspect the topic is more in the realm of "fluid dynamics" than "orbital mechanics"</p>
<p>I've accepted AIB's answer as the most enlightening to me personally.
Also, it would appear I have wiki-sidebar blindness. Apologies for that.</p>
<p>FYI<br>
The paper referencing the speed relative to the CMB, as mentioned in wikipedia article can be found here <a href="http://cmbdata.gsfc.nasa.gov/product/map/dr3/pub_papers/fiveyear/basic_results/wmap5basic_reprint.pdf" rel="nofollow">http://cmbdata.gsfc.nasa.gov/product/map/dr3/pub_papers/fiveyear/basic_results/wmap5basic_reprint.pdf</a>
The relevant section appears to be 7.3.1.
"... implies a Solar System peculiar velocity of 369.0 ± 0.9kms-1 with respect to the CMB rest frame."
Although it's not obvious to me what vector that velocity is along.</p>
<p>Though <a href="http://arxiv.org/abs/astro-ph/9312056" rel="nofollow">Dipole Anisotropy in the COBE DMR First-Year Sky Maps</a> gives a specific velocity(including vector) for the local galactic group in relation to the CMB rest frame
"implied velocity of the Local Group with respect to the CMB rest frame is 627 +/- 22 km/s toward (l,b) = (276 +/- 3 deg, 30 +/- 3 deg)."</p>
<p>FYI<br>
Other reference frames that are external to the local galaxy are
"The Supergalactic coordinate system"</p> | g12855 | [
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