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<p>Basically, given a set of noisy observations for the apparent magnitude of a Cepheid variable, how is this fit to a curve which allows the period, and therefore distance, to be found? Cepheids' luminosity isn't sinusoidal. My first though was to use a Fourier approximation, i.e. fit using least sqaures error to $$L(t) = \sum_{k=0}^na_i\sin(k\omega t)+b_i\cos(k\omega t)$$</p>
<p>For some small $n$, to find the most accurate value of $\omega$, but this model would have serious problems with overfitting to the noisy data. So how is this accomplished? Two possible ideas: first would be to have some sort of addition to the error function which penalizes large coefficients as is usually done with things like logistic regression. The other would be to fit a Fourier approximation to a much less noisy set of observations and then fit just a scalar multiple and new $\omega$ to this curve. What is usually done? I think it's a pretty imporant question to get these values accurately, as we base our determination of Hubble's constant from the periods of Cepheids. </p> | 6,471 | [
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<p>I couldn't find the definition of a Nicolai map. </p>
<p>What is it and what is a simple example which helps understanding it?</p> | 6,472 | [
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<p>Are the boundaries of a black hole, the event horizon, similar to the boundary at the end of our observable universe?</p> | 6,473 | [
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<p>I have the usual equations of electrostatics in macroscopic media expressed in Gaussian units:</p>
<p>$$\nabla\cdot\vec{D}=4\pi\rho$$
$$\nabla\times\vec{E}=0$$
$$\vec{D}=\vec{E}+4\pi\vec{P} \tag1$$</p>
<p>My teacher wants me to convert these Equations to the SI system using the following transformation rules:</p>
<blockquote>
<p>$$\vec{E}\rightarrow\frac{\vec{E}}{k_1}$$
$$\vec{D}\rightarrow\frac{\vec{D}}{k_1k_4}$$</p>
<p>Where $k_1=\frac{1}{4\pi\epsilon_0}$ and $k_4=\epsilon_0$,</p>
</blockquote>
<p>which gives:</p>
<p>$$\vec{E}\rightarrow 4\pi\epsilon_0\vec{E} \tag2$$
$$\vec{D}\rightarrow 4\pi\vec{D}. \tag3$$</p>
<p>Applying this to my electrostatics equations I get:</p>
<p>$$\nabla\cdot\vec{D}=\rho$$
$$\nabla\times\vec{E}=0$$
$$\vec{D}=\epsilon_0\vec{E}+\vec{P},$$</p>
<p>which is consistent with what I find in books. If now I suppose that $\vec{P}=\chi\vec{E}$, doing the appropriate manipulations to the equation (1) I obtain:</p>
<p>$$\vec{D}=\epsilon\vec{E} \quad , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+4\pi\chi$$</p>
<p>If now I apply the transformation rules (2) and (3) to this in order to express it in SI units I get:</p>
<p>$$\vec{D}=\epsilon\epsilon_0\vec{E} \,\,\ , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+4\pi\chi$$</p>
<p>But this is not consistent with what I find in books, which is:</p>
<p>$$\vec{D}=\epsilon\epsilon_0\vec{E} \,\,\ , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+\chi$$</p>
<p>The only way to obtain the solution that appears in the books is to assume that $\vec{P}=\epsilon_0\chi\vec{E}$ (This is the accepted equation in SI units) and do the same I did before (Replace this in (1) and use the transformation rules).</p>
<ol>
<li><p>My question is: is there any way of obtaining the equations in SI units without using the relation $\vec{P}=\epsilon_0\chi\vec{E}$? (using only the transformation rules)?</p></li>
<li><p>And if this is not possible, how do I convert between $\vec{P}=\epsilon_0\chi\vec{E}$ (SI units) and this $\vec{P}=\chi\vec{E}$ (Gaussian units)?</p></li>
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<p>This is a question I have heard quite some contrary opinions, so I want to ask it here, as it deals with physics in principle:)</p>
<p>The question is basically that, if having a unheated intermediate (in between) will reduce the insulation as compared to a direct outside wall?</p>
<p>This might be a little abstract so I will give the real case situation here:</p>
<p>Situation:
I have a flat with:</p>
<ul>
<li>a) 1x a living room (is heated)</li>
<li>b) 1x a kitchen (is not heated)</li>
<li>c) 1x a small hallway-room in between a) b) (also not heated)</li>
<li>b and c) are to be the "intermediate room" which the question refers to.</li>
</ul>
<p>Further explained there are doors:</p>
<ul>
<li>One door between a) and b) </li>
<li>Another between b) and c).</li>
<li>the doors are closed.</li>
</ul>
<p>As far as I understand given the basic situation above I assume that not heating the kitchen (nor the hallway) will not reduce the insulation. This is the insulation that the heated living room would have with regards to the outside world.
In my opinion the temperature of the kitchen is not a matter for the insulation but only the characteristics of the outside walls.</p>
<p>This sketch shows the setup:
<img src="http://i.stack.imgur.com/DtXi8.png" alt="sketch showing the setup"></p>
<p>But differenly: Would start heating my kitchen help the insulation anything?<br>
And to that extend: Would start heating my kitchen help me conserve heating cost (so if I heat room a "living" 100% in one case and in another I heat both rooms 50% and 50%?</p>
<p>I have done some thinking already and I am convinced that the question can be addressed physically. If nonetheless the question can be improved, please tell me how via comments. Else feel free and motivated to give the inside in an answer.</p> | 6,475 | [
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<p>I've read something about this and I conclude that it happens because of the <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow">uncertainty principle</a>. But I don't understand very well the meaning of that.<p></p>
<p>I mean, it's very abstract that the speed, $\vec{v}(t)$, and position, $\vec{r}(t)$, of a particle can't be known at the same time. I don't understand that statement.</p> | 6,476 | [
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<p>I am interested in knowing how much is one eV of energy. Everywhere I found are the technical definitions. Can anybody please tell me how much is this much energy. I need something which I can feel. I mean how much work I can do with 1 eV? Can I drive a 1000cc car for 1hour? Any of example in context of real life usage would be interesting.</p> | 6,477 | [
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<p>A friend of mine claims to have been able to surf the Internet without fuss on a Wi-Fi connection while performing NMR on samples he was analyzing. I would have thought the strong magnets needed for this would have washed out Wi-Fi signals due to the radio waves emitted.</p>
<p>How plausible is his claim?</p>
<p>Or, to be more general, what kind of (electro?)magnet would it take to interfere with mobile devices?</p> | 6,478 | [
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<p>Let's imagine the following situation: </p>
<p>At an initial moment $t=0$, a large water drop with diameter for example $D=10\ \text{cm}$ is placed in deep space (Say an astronaut is experimenting). Let's the initial temperature of the drop be moderate $T_0=283 K$ and the drop itself is at rest at $t=0$. What will happen with the drop? Maybe it will decompose into smaller droplets while boiling? Or maybe it is going to flash-freeze and the ice shell is forming? Or maybe something else?</p> | 6,479 | [
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] |
<p>How can astronomers say, we know there are black holes at the centre of each galaxy?</p>
<p>What methods of indirect detection are there to know where and how big a black hole is?</p> | 744 | [
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<p>If one draws a circle on a sphere and measures the ratio of the diameter to the circumference, that value varies depending on the diameter of the circle compared to the diameter of the sphere it is drawn on (for a circle much smaller than the sphere it's drawn on, the ratio will converge to $\pi$, whereas for a circle the same size as the sphere, the ratio will be 2).</p>
<p>Does the same or similar hold for curved 4-dimensional space? As I approach a massive body, will the ratio of the circumference of a circle to its diameter change?</p>
<p>I found a number of sometimes conflicting statements online, including</p>
<p><a href="http://www.physicsforums.com/archive/index.php/t-9869.html" rel="nofollow">http://www.physicsforums.com/archive/index.php/t-9869.html</a></p>
<p><a href="http://mathforum.org/library/drmath/view/55198.html" rel="nofollow">http://mathforum.org/library/drmath/view/55198.html</a></p>
<p><a href="http://www.last-word.com/content_handling/show_tree/tree_id/2339.html" rel="nofollow">http://www.last-word.com/content_handling/show_tree/tree_id/2339.html</a></p> | 6,480 | [
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<p>Let's imagine I'm very far from any massive objects, so my local space-time is <a href="http://en.wikipedia.org/wiki/Minkowski_space" rel="nofollow">Minkowskian</a>. Off in the distance is a black hole, far enough away that it doesn't noticeably curve space-time near me, but close enough for me to see it. Its entropy is proportional to the area of its event horizon. Another observer is moving past me at a high relative velocity. Looking at the same distant black hole as me, does she see it as having the same surface area (and hence the same entropy) as I do?</p>
<p>Naïvely, the event horizon should Lorenz-transform into an oblate spheroid, contracted in the direction of motion but unchanged in perpendicular directions, so it should have a smaller surface area and a smaller entropy. Is this correct (which would suggest that entropy is not Lorenz-invariant after all), or does the event horizon transform in a different way that preserves its surface area?</p> | 6,481 | [
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<p>What is the difference between <a href="http://en.wikipedia.org/wiki/Single-mode_optical_fiber" rel="nofollow">single mode</a> and <a href="http://en.wikipedia.org/wiki/Multi-mode_optical_fiber" rel="nofollow">multi mode</a> optical fibres? First off, I guess that by <em>modes</em> we mean the spatial modes of the electric (or magnetic?) field right?</p>
<p>Now: what makes a fibre able to support more than a single mode? I mean, what aspect of its structure corresponds to which mode(s) can be transmitted?</p> | 6,482 | [
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<p>Are there any methods of direct detection for black holes?</p>
<p>I'm not referring to gravitational lensing, or measuring the orbits of a star in a binary pair.</p>
<p>Is there any way of directly 'seeing' them? </p> | 6,483 | [
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<p>Occasionally when I make coffee in my french press I experience something odd. It happens pretty infrequently but certainly enough to be curious about. I have the grounds ready in the carafe. The water just heated in an electric kettle. I pour the water over the grounds and place the plunger on top. Then, maybe 1/4 inch into plunging, the water "explodes" out the top like a busted sprinkler head.</p>
<p>I'm curious what's going on here. Is the slight pressure increase (it can't be much) - or simply the plunging action - adding enough energy to the system to induce a flash boil? I've ruled out that the additional volume that the plunger itself adds is not enough to make the coffee overflow (this is a normal setup in every other respect where plunging would simply deliver a refreshing hot beverage).</p> | 6,484 | [
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<p>Recently I have read one book where there was some incomprehensible proof of the Pauli's spin-statistics theorem. I want to ask about a few details of the proof. </p>
<p><strong>First</strong>, the author derives commutation (anticommutation) relations like $[\hat {\psi} (x), \hat {\psi} (x')]_{\pm}$ for arbitrary moments of time for scalar, E.M. and Dirac theory cases. He notices that all of them depend on the function
$$
D_{0} = \int e^{i(\mathbf p \cdot \mathbf (\mathbf x - \mathbf x'))}\frac{\sin(\epsilon_{\mathbf p}(t - t'))}{\epsilon_{\mathbf p}}\frac{d^{3}\mathbf p}{(2 \pi)^{3}}, \quad \epsilon^{2}_{\mathbf p} = \mathbf p^{2} + m^{2},
$$
which (as it can be showed) is Lorentz-invariant. For example, it is not hard to show that for fermionic field
$$
[\Psi (x), \Psi^{\dagger } (x')]_{+} = \left( i\gamma^{\mu}\partial_{\mu} + m\right)D_{0}(x - x').
$$</p>
<p><strong>Second</strong>, he assumes that for the case of arbitrary integer spin $2n$ there exists a function $\Psi(x)$, for which
$$
[\hat {\Psi}_{a} (x), \hat {\Psi}^{\dagger}_{b}(x')]_{\pm} = F^{\ 2n}_{ab}\left(\frac{\partial}{\partial x}\right)D_{0}(x - x'),
$$
and for the case $s = 2n + 1$ there exists a function $\Psi(x)$, for which
$$
[\hat {\Psi}_{a} (x), \hat {\Psi}^{\dagger}_{b}(x')]_{\pm} = F^{\ 2n + 1}_{ab}\left(\frac{\partial}{\partial x}\right)D_{0}(x - x'),
$$
where $F_{ab}^{\\ k}$ refers to the $\frac{\partial}{\partial x}$ polynomial of rank $k$ and the author (at this stage of the proof) doesn't clarify the sign of commutator. </p>
<p>How can one argue such a generalization from spin $0, \frac{1}{2}$ and $1$ cases on the arbitrary cases of spin value? It is a very strong assumption, because formally it almost proves Pauli's theorem.</p> | 6,485 | [
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<p>I was thinking, if the space is infinite, what if there are infinite number of spaces, inside our universe? I mean, everyone knows that black holes exist, but nobody knows what happens when you get sucked in.</p>
<p>Therefore, I believe that inside the black holes there must be some kind of molecule compression which bursts out inside the black hole after molecules get overcrowded and can't compress anymore. That could be actually how Big Bang happened. Furthermore, after the bursting, something has to happen. Probably after billions of years new black holes inside the former black hole start forming, while the former one expires.</p>
<p>The Big Bang should happen right after the black hole itself collapses, actually expires, because black hole finds itself in position where there is nothing so close that it can suck in, therefore it cannot gain mass anymore. In other words, if nothing were to enter the black hole, Hawking Radiation would allow it to expire.</p>
<p>The best of all is that every universe has multiple black holes, which means even more universes to come alive. The other universe, the one that you're not in, is now another dimension.</p>
<p>Based on my beliefs and some research, I believe that we are the outcome of some another universe. Now, my question is, is my belief plausible, based on on facts that I may not know.</p>
<p>On the other hand, if black holes aren't guilty for the Big Bang,may it be that our scales are to small? That the cosmic egg was actually so big that at a specific moment it collapsed and made itself over again? What if Big Bang is yet to happen and that we are part of it?</p> | 6,486 | [
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<p>I was reading the <a href="http://www.google.com/search?as_q=formalism&as_epq=in+in" rel="nofollow">"in-in" formalism</a> (or "closed time path formalism" used in condensed matter physics) in cosmology created by Schwinger in 1961, and there is a saying： "they care about correlation functions instead of S-matrix scattering amplitudes".
When I learn QFT, these two things are almost the same thing and are related by <a href="http://en.wikipedia.org/wiki/LSZ_reduction_formula" rel="nofollow">LSZ formula</a>.
Why they use in-in instead of in-out? what's the difference between correlation functions and S-matrix?</p> | 6,487 | [
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<p>Now in disordered organics, the band picture is thrown out the window, from what I can tell (due to lack of symmetry). But don't HOMO/LUMO levels basically take the place of conduction/valence bands in molecules? In a organic system (a lot of molecules with no order), then, I am correct to believe that the HOMO and LUMO levels broaden into a Guassian density of states? However, the LUMO 'band' does not act like a conduction band in that the states are still localized. From my reading, it appears that the Fermi Level is in the LUMO 'band'. My question is, how can the LUMO 'band' have filled states since a single molecule of course has no electrons in the LUMO. What am I missing?</p> | 6,488 | [
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<p>I just did a back-of-the-envelope calculation, which surprised me. I calculated the difference in acceleration (due to repelling like-charges) experienced by two sides of an electron the size of the classical electron radius, when placed one angstrom from another electron. I used purely classical formulas:</p>
<p>$$F = m_{e}\Delta a ~=~ \frac{k_e q_e^2}{r^2} - \frac{k_e q_e^2}{(r+r_e)^2},$$ </p>
<p>Where $m_e$, $q_e$ are the mass and charge of the electron, $r_e$ is the classical electron radius, and $\Delta a$ is the difference in acceleration (the tidal effect) between the two sides of the electron. </p>
<p>Using $r = One\,\, Angstrom = 10^{-10}m$, I get:
$$\Delta a ~=~ 1.5 \times 10^{18} m/s^2.$$ </p>
<p>In other words, assuming I didn't make a mistake, the electromagnetic tidal effect is enormous!</p>
<p>This brings up some questions: </p>
<ol>
<li><p>Would this effect be measurable if the electron were not point-like (or far smaller than $r_e$)? </p></li>
<li><p>Can we prove a particle is nearly point-like by considering electromagnetic tidal effects like the above? </p></li>
<li><p>Are these sorts of effects studied or considered at all in QFT?</p></li>
</ol> | 6,489 | [
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<p>I have great trouble in understanding simultaneity in special relativity. Let me illustrate it with a concrete example.</p>
<p>Assuming there is a train, its two end points are $A$ and $B$, the length of the train is $x$. The train moves at speed $v$. Assuming the train is moving in the direction from $A$ to $B$.</p>
<p>For a ground observer observing the train movement, he notices that two lightnings strike simultaneously at $A$ and $B$ when the middle part of the train $O'$ passes through right in front of him. In other words, the ground observer is located at the middle part of the train($O'$) when the lightnings strike simultaneously at $A$ and $B$.</p>
<p>Now there is another moving observer sitting inside the train and he sits right in the middle of the train ($O$, equidistant from $A$ and $B$). Does this moving observer think that the lighting happens at the same time? If no, how much time has passed before he notices lightning at $B$, after he had observed lightning at $A$?</p> | 6,490 | [
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] |
<p>I'm curious about how the material absorb the light and reflect the light back as colors in a sense of Quantum Mechanics (Quantum Electro Dynamics)<br>
Does Hadron related to the absorbs of photon ? or are there any other factors ?</p> | 6,491 | [
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<p>The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function.</p>
<p>I am wondering if someone can give me an interesting, or useful, example of a Hamiltonian system for which $X$ is not the cotangent bundle of a manifold. </p> | 6,492 | [
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<p>Is there any classical/ quantum mechanical proof for the non-existence of magnetic monopoles?
Or is it just lack of experimental evidence that has led us to the conclusion that monopoles do not exist, when in fact certain other theories predict their existence/ non-existence?</p> | 6,493 | [
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<p>Okay, some textbooks I came across, and a homework assignment I had to do several years ago, suggested that the reason we can skate on ice is the peculiar $p(T)$-curve of the ice-water boundary. The reasoning is that due to the high pressure the skates put on the ice, it will melt at temperaturs below $273 K$ and thus provide a thin film of liquid on which we can skate. It was then mentioned as fun fact that you could ice-skate on a planet with lakes of frozen dioxide because that gas has the $p(T)$-curve the other way round.</p>
<p>My calculations at that time told me that this was, pardon my french, bollocks. The pressure wasn't nearly high enough to lower the melting point to even something like $-0.5$ degrees Celsius.</p>
<p>I suppose it is some other mechanism, probably related to the crystal structure of ice, but I'd really appreciate if someone more knowledgeable could tell something about it. </p> | 22 | [
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<p>Does anyone know how the acuity of your vision translates to a difference in limiting magnitude?</p>
<p>e.g., the kind of answer I'm looking for would be
"For each factor of 2 improvement in your vision (20/80 to 20/40, 20/40 to 20/20, 20/20 to 20/10, etc., your personal limiting magnitude for point sources increases by 1, independent of seeing or sky brightness."</p>
<p>(except I just made those numbers and conditions up out of thin air)</p>
<p>I also implicitly introduced the ansatz of a smooth logarithmic dependence. Yes, no, approximately so?</p> | 6,494 | [
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<p>How do scientists calculate that density? What data do they have to calculate that?</p> | 6,495 | [
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<p>The nebulae we see in the night sky are forming new stars.</p>
<p>The stars are eating up the nebulae and there is no obvious process in which those nebulae are being created to compensate for that.</p>
<p>Shouldn't the universe have run out of those nebulae a long time ago?</p> | 6,496 | [
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<p>I was reading the article <em><a href="http://www.bbc.co.uk/news/science-environment-14372708">Oxygen finally spotted in space</a></em> today in which it stated </p>
<blockquote>
<p>Oxygen is the third most abundant element in the cosmos, after
hydrogen and helium.</p>
</blockquote>
<p>Why would <a href="http://en.wikipedia.org/wiki/Oxygen">oxygen</a> take the third spot when it is so heavy (relative to the five elements ignored for it)? It would seem logical to me that the third most abundant element would be <a href="http://en.wikipedia.org/wiki/Lithium">lithium</a> or <a href="http://en.wikipedia.org/wiki/Beryllium">beryllium</a> as hydrogen and helium smash into each other.</p> | 6,497 | [
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<p>Neutrons have a measureable magnetic dipole momentum from their intrinsic spin. Is it possible to slow down and catch the neutron by imposing a force by an inhomogeneous magnetic field. I think the force the neutron experiences in a magnetic field is
\begin{align}
\bar{F} = \bar{\nabla}(\bar{m}\cdot\bar{B}).
\end{align}</p> | 6,498 | [
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<p>I am having trouble in understanding why in $Z_2$ topological insulators odd number of Kramers' pairs on one edge are protected by time reversal symmetry against elastic backscattering while even number of such Kramers' pairs are not. I am aware of the argument saying time reversal invariant perturbation can only couple edge states pairwise, which can be used to prove the claim above, but I do not understand why this argument is true. To me, it seems as long as on one edge there are more than one pairs of edge states, there can be backscattering between those pairs.</p>
<p>An explicit example is, suppose on one edge there are three degenerate Kramers' pairs, where two of them have spin-up left-going and spin-down right-going states and the other one has spin-down left-going and spin-up right going states, as shown in the figure.</p>
<p><img src="http://i.stack.imgur.com/0OZuz.png" alt="The three degenerate Kramers' pairs on one edge, where pairs #1 and pairs #2 have spin-up (red) left-going and spin-down (blue) right-going states while pair #3 has spin-down left-going and spin-up right going states"></p>
<p>Now consider the following perturbation:</p>
<p>\begin{equation}
V=
\left(
\begin{array}{cccccc}
0&0&0&0&0&a\\
0&0&0&0&a&0\\
0&0&0&0&0&a\\
0&0&0&0&a&0\\
0&a&0&a&0&0\\
a&0&a&0&0&0
\end{array}
\right)
\end{equation}
where the first three rows represent $|1,1\rangle$, $|2,1\rangle$ and $|3,1\rangle$ and the last three rows represent $|1,2\rangle$, $|2,2\rangle$ and $|3,2\rangle$, respectively. </p>
<p>This perturbation is time reversal invariant and it couples $|1,1\rangle$ and $|2,1\rangle$ to $|3,2\rangle$, and $|1,2\rangle$ and $|2,2\rangle$ to $|3,1\rangle$, so it causes backscattering. This perturbation can be realized simply by a spin-independent potential, as long as the above configuration of the edge states can be realized.</p>
<p>However, it has odd number of Kramers' pairs on one edge, which leads to a contradiction. What did I miss?</p>
<p>I also appreciate if someone can point out the relation between numbers of pairs of Kramers' pairs on one edge and figure 3 of this paper:</p>
<blockquote>
<p>M. Zahid Hasan, and Charles L. Kane. “<a href="http://dx.doi.org/10.1103/RevModPhys.82.3045" rel="nofollow">Colloquium: Topological insulators</a>.” <em>Reviews of Modern Physics</em> <strong>82</strong>, no. 4 (2010): 3045. (<a href="http://arxiv.org/abs/1002.3895v2" rel="nofollow">arXiv</a>)</p>
</blockquote> | 6,499 | [
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<p>What's the entropy of the universe today? How does one go about calculating this? I've heard the statement that black holes account for the bulk of the entropy in the universe today, but don't know why this would be true or the relationship between black holes and entropy.</p> | 568 | [
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<p>A few years ago in an astronomy course, we calculated some (transverse?) velocity of a moving object and got super luminal results. The answer was apparent and not physical velocity of the object. Hence no problem. But at the moment, I don't recall the solution to this apparent issue. Anyone? </p> | 6,500 | [
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<p>If time in systems moving with different speed goes differently, does speed of entropy change differ in these systems?
(is "speed of entropy change" a valid term? can we compare them?)</p> | 6,501 | [
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<p>What is the parity eigenvalue of the $W^{\pm}$ boson, or is it even an eigenstate? I have not found any source that discusses this. I have seen some lists of particles with their parity eigenvalues, but the $W^{\pm}$ and $Z^{0}$ bosons are always left out. </p> | 6,502 | [
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<p>I want to know what mathematical theories I should be aware of for a deep understanding of the standard particles model.</p> | 223 | [
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<p>Is there a particular reason, why the <a href="http://icecube.wisc.edu/" rel="nofollow">Icecube experiment</a> has been installed at the South pole and not at the North pole?</p> | 6,503 | [
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] |
<p>If I have a parallel circuit with two resistors ($R_1=48 \Omega$ and $R_2=16 \Omega$) - and $R_1$ has a $0.1 A$ current running through, is it possible to calculate the current through $R_2$ without having the voltage given, an if so, how?</p> | 6,504 | [
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<p>I understand normal force to be the perpendicular force to a surface of contact. However, I have come across a problem which has caused me to rethink this. </p>
<p>My initial understanding of force is demonstrated by the following diagram of an object sliding down an inclined plane.</p>
<p><img src="http://i.stack.imgur.com/RKqcg.gif" alt="enter image description here"></p>
<p>Here you can see that $F_N = F_g * cos(\theta)$</p>
<p>Now look at this banked curve problem.</p>
<p><img src="http://i.stack.imgur.com/kdsmP.png" alt="enter image description here"></p>
<p>This is taken directly from Wikipedia <a href="http://en.wikipedia.org/wiki/Banked_turn" rel="nofollow">here</a>. It says that $F_N = F_g/cos(\theta)$ which is bigger than in the first scenario.</p>
<p><strong>Why is the <em>normal force</em> different here?</strong></p>
<p>The normal force comes from Newton's third law, as a reactive force. Before there is any centripetal accel, $F_{N}$ was $F_{g}∗cosθ$, but after there is centripetal accel, $F_{N}$ changes to $F_{g}/cosθ$. This makes $F_{N}$ <strong>larger in the second scenario</strong> even though there is no more force acting in the direction of the ground. What causes the increase in $F_{N}$? <em>Something must push against the ground in order</em> for $F_{N}$ to push back.</p>
<p>My thoughts and attempts at solving this:</p>
<p>There has to be a force acting in the direction of the road in order to increase the value of the normal force. I think this force comes from the fact that the <em>car's velocity is causing a force on the road</em>, pushing in to the road because the car's velocity wants to go straight, but the road is shaped in a curved bank so the car cannot go straight; the normal force of the road pushes back, which allows for the centripetal acceleration. </p>
<p>I've tried accounting for the difference in $F_{N}$ and I've found that $F_{N}$ in the second scenario is $Fg*cos\theta + Fg*tan\theta*sin\theta$ the difference being the second term. I haven't been able to describe the difference in $F_{N}$ in any other way. I believe the difference in $F_{N}$ should should somehow be related to the centripetal acceleration. </p>
<p>I think the normal force is a result of the car's (the ball in this case) velocity against the circular road causing the road to push back. I lack the mathematical ability to describe it. </p> | 6,505 | [
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<p>Ok so I am an A2 physics student, and for one of my pieces of coursework I conducted a practical investigation, my topic being the factors affecting the period and swing of a bifilar pendulum.</p>
<p>The only useful information I was able to find on the topic was here: <a href="http://voyager.egglescliffe.org.uk/physics/gravitation/bifilar/bif.html" rel="nofollow">http://voyager.egglescliffe.org.uk/physics/gravitation/bifilar/bif.html</a></p>
<p><b>What I need to do is explain simply why increasing d</b> in the diagram below, the distance between the threads <b>on support A, while keeping d on bar B the same, will increase the period.</b></p>
<p><img src="http://physics.dorpstraat21.nl/images/expts/bifilar%20pendulum1.png" alt=""></p>
<p>In the link I posted above, Faysal Riaz seems to explain this in this section:</p>
<pre><code>"Therefore the restoring Couple, CR, which acts towards the equilibrium position so negative, is given by:
CR = (-Mgθd2)/4y
Applying Newton’s Second Law for the rotational motion of the rod, which is of constant mass:
(Id2θ)/dt2= (-Mgθd2)/4y
∴(d2θ)/dt2= (-Mgθd2)/4Iy"
</code></pre>
<p>The problem is, I have no clue what this means.
<b>Can anyone simplify this so I could understand it better, or explain why decreasing d in the way I described above, increases the period in a few lines of basic mechanics if that's possible?</b></p> | 6,506 | [
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<p>This capacitance contain 4 dielectric as shown in the figure dielectric 1 in half sphere and 2,3 in for 1/4 of the sphere and the fourth one in the last 1/4 of the sphere as shown and I want to find total capacitance.
i think that 2nd dielectric parallel with potion of 1st and portion of 4th then this group series with (3rd dielectric and other portion of 1st and 4th one which are also parallel) is the correct answer as the voltage between a,b is the same at any dielectric as the electric field is the same from boundary condition ET1'=ET2=ET4' AND ET1''=ET3=ET4''
<img src="http://i.stack.imgur.com/4tJ6n.png" alt="enter image description here"> is it correct</p> | 6,507 | [
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<blockquote>
<p>A linearly polarized plane wave at 100 MHz is propagating in the $z$ direction. The electric field vector makes an angle of 30° with the $x$-axis. Its peaks amplitude is measured to be $2.0\:\mathrm{ V m}^{-1}$. Write down equations for the electric field and magnetic fields components of the wave as a function of distance, $z$, and time $t$, measured in meters and seconds respectively. Assume the phase term is zero.</p>
</blockquote>
<p>Since the phase term is zero, I got that $E(z,t)=2\cos(kz-ωt)$. I think I should use $ω=2πf$ and $k=2πf/c$, but how can I split the electric field into $x$ and $y$ components? Also, I think $B(z,t)=E(z,t)/c$, so is the $x$ component of $B(z,t)$ equal to the $x$ component of $E(z,t)/c$? The $x$ component of the electric field at any time is $|E|\cos(30°)$ and the $y$ component of the electric field at any time is $|E|\sin(30°)$.</p> | 6,508 | [
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<blockquote>
<p>An object falls from a height $h$ above water through air with negligible drag. In the water, the upward buoyancy exactly balances the downward gravitation force. The only remaining force on the body in the water is a drag force with magnitude $cv^2$ per unit mass, $c$ is a constant, and $v$ is the velocity. Show that at depth $d \geq 0$ the velocity is $\sqrt{2gh} e^{-cd}$</p>
</blockquote>
<p>I'm taking everything downwards to be positive, so above the water I have the equation:</p>
<p>$$v\dfrac{dv}{dy} = g$$</p>
<p>and thus $\int_0^{v_0} vdv = \int_h^0gdy $ where $v_0$ is the velocity when it hits the water. From here I already go wrong, because solving this I get $v_0^2 = -2gh$ which implies the particle goes back up which is absurd. I looked at the solutions and then took everything upwards to be positive instead, and ended up with the integral $\int_0^{v_0} v dv = -\int_h^0 gdy$ which goes on to give the correct answer. Could someone explain how I modelled the particle above the water incorrectly?</p> | 6,509 | [
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<p>The situation looks as follows:
<img src="https://dl.dropboxusercontent.com/u/76821907/incline.png" alt="situation"></p>
<p>The incline is moving rightwards with a constant acceleration<br>
<img src="https://dl.dropboxusercontent.com/u/76821907/latex1.png" alt="a=g"></p>
<p>The forces on the picture are:
mg - weight, N - normal force, F - inertial force (already shown), since it'll be easy to analise the problem in a noninertial frame.</p>
<p>The question is: what's the trajectory of the block, i.e. how the block will be moving?</p>
<p>I will show my hypothesis; please tell me if it's right or not:</p>
<p>I've calculated the vector sum of the forces acting on the block when it's on the incline (bold letters symbolize vectors, bold <strong>i</strong> and <strong>j</strong> are unit vectors):</p>
<p><img src="https://dl.dropboxusercontent.com/u/76821907/CodeCogsEqn.png" alt="net force"></p>
<p>Let's say theta = 30 degrees, and the incline is infinitely long. In this scenario, the net force looks like this:<br>
<img src="https://dl.dropboxusercontent.com/u/76821907/incline2.png" alt="situation"></p>
<p>The net force is such, that the block will actually loose contact with the incline. But then the normal force will disappear and the net force will become:<br>
<img src="https://dl.dropboxusercontent.com/u/76821907/incline3.png" alt="situation"></p>
<p>This force brings the block back down. So my hypothesis is, that for a theta equal to 30 degrees the block will be "jumping" down.
Is it right?</p> | 6,510 | [
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<p>Forgive me if this is a newb question but I am not a trained scientist, much less a physicist. I'm just curious and would like to know if there's a unit to measure impact force.</p>
<p>I know the newton is the amount of force required to accelerate a mass of one kg at a rate of 1m/s^2 but what about the impact itself?.</p> | 6,511 | [
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<p>In Perkin's book Particle Astrophysics (page 144): I do not understand how one comes to the following expression (the second equality with $r$) for the Helium mass fraction due to the Big Bang Nucleosynthesis:</p>
<p>$$Y= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}= \frac{2r}{1+r} $$
where $r=N_\text{n}/N_\text{p}$.</p>
<p>The first equality follows from the fact that He is (approximately) 4 times heavier than H:
$$Y = \frac{m_\text{He}}{m_\text{He}+m_\text{H}}= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}.$$</p>
<p>However I can't derive the second equality relating $Y$ to $r$:</p>
<p>$N_\text{He}= 2N_\text{p} + 2N_\text{n}$ and $N_\text{H}= N_\text{p} + N_\text{n}$</p>
<p>$$Y= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}= \frac{8(N_\text{n}+N_\text{p})}{9(N_\text{n}+N_\text{p})}\quad???$$</p> | 6,512 | [
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<p>As discussed <a href="http://www.math.poly.edu/courses/projective_geometry/chapter_three/node1.html" rel="nofollow">here</a> the complex projective space $\mathbb{C}P^n$ is the set of all lines on $\mathbb{C}^n$ passing through the origin. It would seem natural to assume that any $\mathbb{C}P^n$ can be viewed as a tangent space to a point in an equivalently parametrized space. So for a point $p$ on some manifold $M$ the set of all tangent vectors at $p$ is:</p>
<p>$$T_pM$$ </p>
<p>As discussed <a href="http://www.math.binghamton.edu/paul/601-S06/AS3.pdf" rel="nofollow">here</a>, the union of all points on a manifold is written as:</p>
<p>$$TM = \bigcup_{p\in M} T_pM$$
and is known as the tangent bundle.</p>
<p>Also, $\pi$ is used to represent the map of from the tangent bundle to the manifold such that $\pi: TM \rightarrow M$ and for each $p \in M$ :</p>
<p>$$\pi^{-1}(p)=T_pM$$ </p>
<p>Where $\pi^{-1}(p)$ is a vector space of dimension $n = \dim M$ (confirming as stated above that the tangent space will have same dimension, or equivalent number of parameters as the underlying manifold). </p>
<p>Ordinary spacetime is often described as $\mathbb{R}^{3,1}$ and it is known that euclidean plane $\mathbb{R}^{2}$ can be described by a complex number in $\mathbb{C}^{1}$, and $\mathbb{R}^{1,1}$ can be described with <a href="http://en.wikipedia.org/wiki/Split-complex_number" rel="nofollow">split-complex numbers</a> in $\mathbb{R}^1\oplus \mathbb{R}^1$. When one considers the <a href="http://en.wikipedia.org/wiki/Lorentz_boost#boost" rel="nofollow">Lorentz boost</a>, it is tempting to think of ordinary space as being $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$ and described by two variables $\left( z , \mathring{z} \right)$ where $z$ is a complex number and $\mathring{z}$ is a split complex number.</p>
<p>Since there are exactly three $2$-dimensional unital algebras, complex numbers, split-complex numbers and <a href="http://en.wikipedia.org/wiki/Dual_numbers" rel="nofollow">dual numbers</a>, and the algebra of dual numbers is isomorphic to the exterior algebra of $\mathbb{R}^1$, and there are two separate $\mathbb{R}^1$ in $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$, it is tempting to add an additional set of dual numbers so that we can parameterize space as $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ where $ \mathring{v}$ is the dual number for the first $\mathbb{R}^1$ and $ \mathring{w} $ is the dual number for the second $\mathbb{R}^1$. Dual numbers are noteworthy as having "fermionic" directions of "bosonic" directions.</p>
<p>If I wanted to keep my new set of coordinates, $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ how best would I describe the tangent space of this "ad hoc" manifold? Would I be able to describe this in $\mathbb{C}P^n$, if not, why not? If only because of compactness, why can't tangent spaces be compact? Could we describe it as pseudo-tangent?</p> | 6,513 | [
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<p>I have a little problem with the potential energy of a spring... I hope you can help me!</p>
<p>I have two coupled pendula, given by two masses $m$ fixed to two rigid bars (that haven't any mass) and with length $L$ and $2L$. The two bars are at the distance $d$. The masses are connected each other with a spring of elastic costant $k$ and initial length $d$. The bar keeps the horizontal position in every istant of the motion. I consider as coordinates $\theta_1$ (that is the angle formed by the first bar with the $z$ axes) and $\theta_2$ (that is the angle that the second bar forms with $z$ axes). </p>
<p>Why the potential energy of the spring is given by $$U=\frac {1}{2}kL^2 (\sin \theta_1-\cos \theta_1 \tan \theta_2)$$ and not by $$U=\frac{1}{2}kL^2(2 \sin\theta_2-\sin \theta_1)^2$$? </p>
<p>I have used the formula
$$U=\frac{1}{2}k (\Delta x^2)$$ where $\Delta x$ is the variation of length of the spring, but I can't understand why in U there is $\tan \theta_2$... where am I wrong?</p> | 6,514 | [
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<p>I am reading the paper "Bosonization in a Two-Dimensional Riemann-Cartan Geometry", Il Nuovo Cimento B Series 11
11 Marzo 1987, Volume 98, Issue 1, pp 25-36, <a href="http://link.springer.com/article/10.1007%2FBF02721455" rel="nofollow">http://link.springer.com/article/10.1007%2FBF02721455</a></p>
<p>Equation 4.8' on p. 34 suggests a particular transformation law for the measure under the Weyl scaling 4.8.
I am concerned with the fact that this law is somehow dependent on the Dirac operator in given metric. Certainly this general form is expectable (it involves a product of $\exp(\sigma)$ over all points), but I want to understand where the particular factors come from. Zeta-function of Dirac operator suggests that it is obtained via some zeta-regularization of something.</p>
<p>Does anybody know what is this all about (may be a sketch of derivation), or have a reference?</p> | 6,515 | [
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<p>For resonance to occur, is it true that the force lags behind the motion by $\pi/2$? I saw some notes written that the motion lags behind the force by $\pi/2$ which makes no sense to me.
As I watched many videos and I worked out the motion, it always happens before the force pushes.
E.g. if the force is $F=\cos\omega t$ and $x = A\cos(\omega t -\pi/2)$, is it true that force lags behind the motion?</p>
<p>If it's $\pi/2$ , does there happen anything special?</p>
<p>Also, I realised that if there is no damping, we can only get 180 or 0 phase difference, which is quite counterintuitive to me. Can anyone give any example to help me feel better?</p> | 224 | [
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<p>As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger equation can make general relativity and quantum mechanics closer</p>
<p>$$ \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. $$</p>
<p>My question is can we derive Schrödinger equation from this one and what are solutions of this equation representing actually in physical relativistic sense? Which part in equation is associated to relativistic mechanics? </p> | 6,516 | [
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<p>Just a curious question, do astrophysicist use the SI units, for example in this equation, </p>
<p><img src="http://i.stack.imgur.com/Mj11w.png" alt="enter image description here"> </p>
<p>$r = 5pc$, will this be converted to meters?
And what does this $\nu$ stand for? </p> | 6,517 | [
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<p>Obviusly a controlled-not gate is possible, is a not-controlled gate possible?</p>
<p>I need a gate to flip the first qbit and leave the second unchanged, but in literature I have never seen such a gate.</p>
<p>Would it look like $$
\left(\begin{matrix}
1 & 0 & 0 &0 \\
0 &1& 0 & 0 \\
0 &0&0&1 \\
0&0&1&0
\end{matrix}\right) $$</p>
<p>Also would a 4x4 phase changing gate look like:
$$
\left(\begin{matrix}
1 & 0 & 0 &0 \\
0 &1& 0 & 0 \\
0 &0&0&e^{i\theta} \\
0&0&e^{i\theta}&0
\end{matrix}\right) $$</p> | 6,518 | [
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<p>In QM, the wave function (in the Copenhagen interpretation) is not an actual physical wave but a device to derive probabilities about the outcomes of experiments. The wave function encodes all the information about the system we want to derive predictions for. Predictions are about future measurements. Once the measurement has been performed and the result is known, we adjust accordingly our expectation: the so-called collapse of the wavefunction just took place (let me add, in our minds). This subjective knowledge about the predictions of QM is crucial to avoid problem with causality in relativity when studying entangled systems. Fine. </p>
<p>What I am a bit confused about is what QM says about the past, rather than the future. What is the analog picture that QM gives about the state of a system in the past? What does QM say about the conditional probabilities of events? What does QM tell about, say, cosmology and the far past of the universe when e.g. string theory becomes relevant? I hope it is not a trivial, naive, question.</p> | 6,519 | [
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<p>Bacteriochlorophyll (BChl) are pigments that occur in the photosynthetic mechanisms of bacteria. I am studying some papers on the excitonic properties of BChl's, and the term $Q_y$ transition comes up a lot, but I haven not found any explanation of its meaning. </p>
<p>Can someone explain to me the meaning of this term?</p> | 6,520 | [
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<p>On <a href="http://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html" rel="nofollow">this page</a> right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space: </p>
<p>$$
\begin{split}
f(x) &= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} A(k) e^{ikx} dk\\
A(k) &= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f(x) e^{ikx} dx
\end{split}
$$</p>
<p>while the second set is connection between $x$ (position) and $p$ (momentum):</p>
<p>$$
\begin{split}
\psi(x) &= \frac{1}{\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty} \phi(p) e^{i\frac{p}{\hbar}x} dp\\
\phi(p) &= \frac{1}{\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty} \psi(x) e^{-i\frac{p}{\hbar}x} dx\\
\end{split}
$$</p>
<hr>
<p><strong>Q1:</strong> How do i derive the second set out of first one?</p>
<p>I know De Broglie relation $p = k \hbar$. Hence from $\exp[\pm ikx]$ in the first set we get $\exp \left[\pm i \frac{p}{\hbar} x\right]$ in the second set of equations. This is clear to me. What i dont know is how do we get from $1/\sqrt{2\pi}$ in the first set to $1/\sqrt{2 \pi \hbar}$ in the second set. Where does a $\hbar$ come from?</p> | 6,521 | [
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<p>I was trying to naively draw a parallel between special relativity and the Heisenberg uncertainty principle. I try to understand uncertainty principle as a consequence of 4-position and 4-momentum being conjugate variables in phase space, and this arises from a Lagrangian that looks like this: $p_\mu dx^\mu$. If in special relativity, the Lagrangian looks like this: $mc^2 d\tau$, then could I say $\Delta m\Delta\tau\ge\frac{\hbar}{2c^2}$?</p>
<p>I know that ultimately I would need a formalism in terms of mass and proper time operators to get a formalism of quantum mechanics this way, but I know that you can derive the uncertainty principle just from Fourier analysis and assuming that position (or time) and momentum (or energy) are conjugate variables. If mass and proper time are conjugate variables in special relativity, could one write an uncertainty relation between $mc^2$ and proper time? </p>
<p>If this is correct, would it imply a particle's mass and proper time cannot be known simultaneously? What would be some consequences of this?</p> | 6,522 | [
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<p>I am trying to understand <strong>the Bernoulli's theorem:</strong></p>
<ul>
<li><strong>$\frac{p}{\rho}+\frac{1}{2}u^2+\phi$ is a constant along a streamline</strong></li>
</ul>
<p>I got that:</p>
<p>$\frac{\partial u}{\partial t}$ + ($\nabla \times u)\times u$ = $-\nabla(\frac{p}{\rho}+\frac{1}{2}u^2+\phi)$</p>
<p>For a steady flow: $\frac{\partial}{\partial t}$ = 0 and then:</p>
<p>($\nabla \times u)\times u$ = -$\nabla H$ with the scalar: $H$ = $\frac{p}{\rho}+\frac{1}{2}u^2+\phi$</p>
<p><strong>now, I didn't understand the next steps:</strong></p>
<p>Taking the “dot product” of ($\nabla \times u)\times u$ = -$\nabla H$ the left hand side vanishes, as ($\nabla \times u)\times u$ is perpendicular to $u$ and we get:</p>
<p>$(u \cdot \nabla)H = 0$</p>
<p>This implies that $H$ is constant along a streamline</p>
<p><strong>Can someone explain me this thing in other words please?</strong></p> | 6,523 | [
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<p><a href="http://en.wikipedia.org/wiki/Dislocation" rel="nofollow">Dislocation</a> (like <a href="http://en.wikipedia.org/wiki/Dislocation#Screw_dislocations" rel="nofollow">screw</a> or <a href="http://en.wikipedia.org/wiki/Dislocation#Edge_dislocations" rel="nofollow">edge</a> dislocation) is not a 'real' thing, while Newton's laws only apply to a real object (no matter macroscopic, like stars, or microscopic, like atoms). </p>
<p>In the derivation of <a href="http://solidmechanics.org/text/Chapter5_9/Chapter5_9.htm#Sect5_9_5" rel="nofollow">Peach-Koehler force</a> (stress acting on a dislocation), I understand that the force is actually acting on those atoms around the dislocation, which is equivalent to acting on the dislocation in the mathematical sense. However, based on the above point (treating dislocation as a 'real' thing), some books directly use force balance and other mechanical analysis on dislocation. </p>
<p>Can we treat dislocation as a 'real' thing in all mechanical cases just as the void in a solid when analysing electrical properties? Is there some way to think about it easily other than a complicated mathematical argument?</p> | 6,524 | [
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<p>I know thought that a crystal, if annealed for long enough at high enough temperatures, will "shed" its higher index (higher energy) planes, leaving only its most stable form. This, I wrongfully assumed, was the shape of its primitive cell. Some crystals however, notably pyrite, do tend to take the shape of their primitive cell. So my question is this:</p>
<p>What can you tell about a crystals lattice structure merely by looking at its shape? And when does a crystal take the shape of its Primitive Cell?</p> | 6,525 | [
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<p>I have looked at some of <a href="http://www.staff.science.uu.nl/~hooft101/" rel="nofollow">'t Hooft</a>'s recent <a href="http://www.staff.science.uu.nl/~hooft101/gthpub.html" rel="nofollow">papers</a> and, unfortunately, they are well beyond my current level of comprehension. The same holds for the discussions that took place on this website. (See, for example, <a href="http://physics.stackexchange.com/questions/34217/why-do-people-categorically-dismiss-some-simple-quantum-models">here</a>.) I therefore tried to imagine what these papers might be about in my own terms. The following is one such imagination.</p>
<p><strong>Is, possibly, the essence of his recent papers that 't Hooft forces <a href="https://en.wikipedia.org/wiki/Probability_amplitude" rel="nofollow">probability amplitudes</a> to be of the form ${\Bbb {Q}}e^{2\pi i{\Bbb Q}}$,* which, I presume, is <a href="http://en.wikipedia.org/wiki/Dense_set" rel="nofollow">dense</a> in $\Bbb C$? That is, does 't Hooft provide an unfamiliar, and possibly cumbersome, <a href="https://en.wikipedia.org/wiki/Interpretation_of_probability" rel="nofollow">interpretation of <em>probability</em></a>, which nonetheless might be considered appealing and/or insightful, especially since the set of allowed probability amplitudes is <a href="https://en.wikipedia.org/wiki/Countable_set" rel="nofollow">countable</a> in such an interpretation?</strong></p>
<p>Is this the gist of his models? Or am I a long way off?</p>
<p>*I'm using ${\Bbb {Q}}e^{2\pi i{\Bbb Q}}$ as a notational shortcut for $\left.\left\{re^{2\pi i \theta}\,\right|\, r,\theta\in\Bbb Q\right\}$.</p> | 6,526 | [
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<p>Voltage is the work done per unit charge. Given by:</p>
<p>V = W/q</p>
<p>Electron volt is the maximum kinetic energy gained by the electron in falling through a potential difference of 1 volt. Given by:</p>
<p>K.E (max) = eV</p>
<p>When there is a potential difference (voltage) of 10V between two points, it means that we are doing 10 joules of work per unit charge (electron). </p>
<p>My question is that if we are doing 10 joules of work on an electron than why isn't the kinetic energy of electron equal to 10 joules? Why do we multiply charge of electron with Voltage to get the kinetic energy gained by the electron?</p> | 6,527 | [
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<p>I have the following problem for an astrophysics course:</p>
<blockquote>
<p><em>A star is seen through a rather dusty region of space has its
brightness dimmed by +1 magnitude/kpc, which makes it seem further
away than it actually is. If the observed apparent magnitude of the
star is $m_v = +4.0$ and absolute magnitude $M_v = -4.5$, determine
the distance using the extinction coefficient ("A" in the distance
modulus equation).</em></p>
</blockquote>
<p>This seems to be a relatively simple problem using the following equation which was given to us in class:</p>
<p>$$
m - M = 5\log{\frac{d}{10}} + A(\lambda)
$$</p>
<p>The issue comes in that our professor did not say anything about the $A(\lambda)$ term. I figure that I could go through deriving it in some way, but that is something that this course should not require. How do I find this term?</p>
<p>Also, our professor is not the greatest, so does anyone know of any good resources for astrophysics that they have used? More complex resources are actually preferred since I am a physics major as well.</p> | 6,528 | [
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] |
<p>How can one prove the Bianchi identity of a non-Abelian gauge theory? i.e.
$$
\epsilon_{\mu \nu \lambda \sigma}(D_{\nu}F_{\lambda \sigma})^a=0
$$</p> | 6,529 | [
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<p><a href="http://en.wikipedia.org/wiki/Holeum" rel="nofollow">Here</a> is the wikipedia article. The basic idea is that the primordial density of microscopic black holes was high enough that many were able to form stable bound states before decaying through Hawking radiation (similarly to how unstable neutrons were able to form stable bound states with protons during BBN). These "holeums" would be dark matter candidates that would be much smaller than what is ruled out through gravitational lensing experiments (which rule out ordinary primordial black holes that must be large enough not to have evaporated already).</p>
<p>The <a href="http://arxiv.org/abs/gr-qc/0308054" rel="nofollow">papers</a> by Chavda that apparently introduced the idea are not well written or formatted, so I can understand why they are perhaps ignored. On the other hand the basic idea seems sounds to me. Is there a simple reason why it is not?</p> | 6,530 | [
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<p>You often hear intergalactic space is an example for a very good vacuum. But how vacuos is space between galaxy clusters and inside a huge void structure? Are there papers quoting a measurement/approximation method (building the difference of a very near known and far away similar spectral source)? Are the rest particles mainly Hydrogen, He,...? Is it important at all to know the average density of intergalatic space in cosmological research?</p> | 262 | [
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<p><img src="http://i.stack.imgur.com/0nuvB.jpg" alt="enter image description here"></p>
<p>In order to workout the method for establishing the formula of thin lens, my teacher says that the optical path is:
$PA + AQ = PS_1 + nS_1S_2 + S_2Q$ ($n$ is the refractive index of the lens)<br>
Why is she multiplying $S_1S_2$ by $n$? </p>
<p>Also, she says that for spherical surface $S_1$, with centre of curvature $C_1$, we can write from geometry,<br>
$h^2 = 2(S_1C_1 - OS_1) OS_1 $
How is she deriving this?</p> | 6,531 | [
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<p>Why do we have to divide uncertainty of the measurement by $\sqrt{3}$?</p>
<p>For example we have the uncertainty of the measurement with a ruler being the smallest scale of 0.01cm, and it is not our uncertainty but we first divide it and so it makes our final uncertainty value.</p>
<p>Why is that this way?</p> | 6,532 | [
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<p>In recent exoplanet meeting <a href="http://seagerexoplanets.mit.edu/next40years.htm" rel="nofollow">The Next 40 Years of Exoplanets</a>, it was mentioned a few times that the field/topic is becoming saturated.</p>
<p>In what ways is it becoming saturated, and can you see the effect of this in the quality of papers being published?</p> | 6,533 | [
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<p>From the Apollo missions we know that the moon is covered with dust. Where does it come from? Is it from the erosion of the moon rock? By what? Or by accretion of dust from space? Which comes from where?</p> | 6,534 | [
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<p>Suppose a police car is standing by a wall. The siren light is rotating and it will hit the wall and reflect back to the car. Does the reflected light show a Doppler effect? </p> | 6,535 | [
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<p>What is the most precise data for neutron-antineutron production by one photon (hitting a target in the laboratory system)?</p>
<p>and/or</p>
<p>What is the most precise data for neutron-antineutron
annihilation to two photons?
Is this data available online?</p> | 6,536 | [
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<p>One of the key datasets of the <a href="http://arxiv.org/abs/1403.3985" rel="nofollow">recent BICEP2 results</a> is the B mode power spectrum shown below. The existence of these B modes implies the existence of gravitational waves prior to inflation. </p>
<p>My question is: what is the relationship between the multipole number of the B mode and the wavelength of the gravitational waves which generated it? My naive expectation is that it is one to one with some sort of scaling factor which includes information about the expansion of the universe and the conversion between spherical and linear coordinates.</p>
<p><img src="http://i.stack.imgur.com/ilsKc.png" alt="B mode power spectrum from ref. 1"></p> | 6,537 | [
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<p>Could it be said (as my question implies) that elementary particles (can) exist in three quite different states: wave, point-particle and superposition?</p>
<p>So, a wave or a point-particle could be 2 manifestations of an elementary particle in "our-macrocosmos dimension" and a superposition would be a third state of a particle in "quantum-microcosmos dimension"?</p>
<p>Does this makes any sense? </p> | 6,538 | [
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<p>I am working on the following physics problem and have run into some trouble</p>
<p><img src="http://i.stack.imgur.com/Qcwhu.png" alt="enter image description here"></p>
<blockquote>
<p>The figure above shows particles $1$ and $2$, each of mass $m$, attached to the ends of a rigid massless rod of length $L_1 + L_2$, with $L_1 = 20cm$ and $L_2 = 80cm$. The rod is held horizontally on the fulcrum and then released. What are the magnitudes of the initial accelerations of (a) particle $1$ and (b) particle $2 \space ?$</p>
</blockquote>
<p>My Approach: </p>
<p>So I first considered the net torque when the system is at rest so that I could get to the angular acceleration using the equation $\tau_{net} = I \alpha$. Given that particle $2$ would induce clockwise motion, it is given a negative sign while particle $1$ is given a positive sign because it induces counter-clockwise motion so $\tau_{net} = F_{t2}r_2 – F_{t1}r_1 = I \alpha$. (Where $F_{ti}$ represents the tangent force acting on particle $i$) </p>
<p>Solving for alpha we have that $\alpha = \frac{F_{t2}r_2 – F_{t1}r_1}{I} = \frac{mgL_2 - mgL_1}{I}$. The next step is then to find the rotational inertia. Now after consulting with my solutions manual I see that this can be found by simply treating the fulcrum as the axis of rotation, but I didn't see this approach when solving the problem. Instead I used the parallel axis theorem $I = I_{com} + Mh^2$. Now even though this approach is a waste I'm trying to figure out why I didn't arrive at the same answer anyway, so I've included my work computing the rotational inertia in this way. </p>
<p>Computing Rotational Inertia Using Parallel Axis Theorem</p>
<p>First I computed the center of mass of the rod as follows: $x_{com} = \frac{m_w * 0 + m_w* 0.80 m}{m_w + m_w}= 0.4m$ (Note: I use $m_w$ to denote mass while I use $m$ to denote distance). Next I computed $I_{com}$ as follows: $I_{com} = \Sigma \space m_{wi} \cdot \space r_i^2 = 2\space m_w \cdot (0.4 m)^2$. </p>
<p>And by the parallel axis theorem $I = I_{com} + Mh^2 = 2\space m_w \cdot (0.4 m)^2 + 2\space m_w (0.2m)^2$. After inputting this value into the original equation for angular acceleration I arrive at an invalid value. What have I done wrong? Also, how do we know definitively when to use the parallel axis theorem? </p>
<p>Any help understanding my problem would be appreciated greatly </p>
<p>Note: When either rotational inertia value is inputted into the angular acceleration formula the masses will cancel so using the correct approach and canceling the mass $\frac{I}{m_w} = (0.2m)^2 + (0.8m)^2 = 0.68 m^2$ while using my original approach I have $\frac{I}{m_w} = 2((0.4 m)^2 + (0.2m)^2) = 0.4m^2$</p> | 6,539 | [
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<p>I want to combine three spin half particles and this is what I have so far.</p>
<p>I used the lowering operator $J_{-}$ on the top states and found the following states fine:</p>
<p>$$|\frac{3}{2},\frac{3}{2}\rangle , |\frac{3}{2},\frac{1}{2}\rangle , |\frac{3}{2},0\rangle , |\frac{3}{2},-\frac{1}{2}\rangle , |\frac{3}{2},-\frac{3}{2}\rangle $$</p>
<p>So that is the combination of three spin $\frac{1}{2}$ particles is equivalent to a spin $\frac{3}{2}$ particle, right?</p>
<p>In the case where I did it for combining two spin $\frac{1}{2}$ particles I found this was equivalent to a spin 1 particle and an additional spin zero particle.</p>
<p>So my question is, is there a singlet or anymore that accompany what I found for the case where I combined three spin $\frac{1}{2}$?</p> | 6,540 | [
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<p>I'm watching a lot of basketball <a href="http://www.ncaa.com/march-madness" rel="nofollow">this month</a>. A common event is the ball going part way into the hoop and then coming out again. Announcers sometimes claim that the ball was "halfway through" when it rims out. Thinking about it, with enough rotation and friction I wouldn't be surprised if the ball <em>could</em> fall that far and not go through. How far can a basketball fall without being certain to fall all the way through?</p>
<hr>
<p>Assuming we are talking about <a href="http://en.wikipedia.org/wiki/Basketball_%28ball%29#Modern-day_specifications" rel="nofollow">NCAA men's basketball</a> the <strong>relevant data:</strong></p>
<pre><code>Circumference of the ball: 29.5-30 inches (749–762 mm)
Weight of the ball: 20-22 ounces (567–624 g)
Bounce of the ball: 49-54 inches (1245–1372 mm) when dropped 6 feet (1829 mm)
Diameter of the rim: 18 inches (457 mm)
Coefficient of friction*: 1.2
</code></pre>
<p>The coefficient of friction between the rim and the ball was estimated in <a href="http://books.google.com/books?id=optom1F4uLEC&pg=PA707&lpg=PA707&dq=basketball+coefficient+of+friction&source=bl&ots=6wKffLkVnr&sig=s17DXd37E2oGTiP2-L7St3rMIK0&hl=en&sa=X&ei=ho0sU7TwBtLEoASRtoCIAw&ved=0CEMQ6AEwAw#v=onepage&q=basketball%20coefficient%20of%20friction&f=false" rel="nofollow">The Engineering of Sport 7: Vol. 1
By Margaret Estivalet, Pierre Brisson</a>. I assume the synthetic cover is used. For leather, the coefficient of friction was estimated at 0.5.</p>
<p>For the purpose of this question, let's assume any rotational speed is possible. (It should be possible to estimate what humans can achieve, but obviously there is a limit.) Also, assume that gravity (and air pressure, if it matters) is at sea level.</p> | 6,541 | [
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<p>As from the title, I'm not too sure how they are related. Definition is that <a href="http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines" rel="nofollow">streamlines</a> are instantaneously tangential to the velocity vector of the field. Why would a streamline that shows direction be a tangent to the velocity? </p> | 6,542 | [
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<p>There are two identical LC oscillators with electronic counters attached indicating how many times they have oscillated (from the time they are turned on).
They are turned on simultaneously and one is kept on earth and other one is hurled to outer space at very large speeds on a rocket and brought back to earth after traveling several million miles. Does relativity say they will show two different counts ? (One traveled will show less than the one on earth) </p> | 6,543 | [
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<p>While discussing this question (<a href="http://physics.stackexchange.com/q/82296/">Does light have an unending journey?</a>) I stumbled on the fact that light's speed is constant only in inertial frame.</p>
<p>What I happened to do was add up the expansion of universe to the theory. It goes as follows: </p>
<p>Suppose from a source we emit a photon, at the same time, we (source) are moving away from the point of launch and so is the point that too with acceleration, now lets consider the moment when we are so far from the point of fire that the distance between the 2 points (source and firing point) is so much that the space with their respect to each other is moving with speed more than speed of light, certainly since the points are also constantly being separated they have a relative velocity of separation which just happens to be more than speed of light, now by the time the fired photon must have travelled even further and its speed now with respect to the source must be greater than speed of light. </p>
<p>Can this be taken as a prove for the variance of speed of light from inertial frames?</p> | 6,544 | [
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<p>I have somewhat of an understanding for other physical quantities, but as far as <a href="http://en.wikipedia.org/wiki/Entropy" rel="nofollow">entropy</a> goes I only know it to be "disorder". Why is the change in entropy formula an appropriate/useful definition, moreover, why is the equation for entropy an appropriate statement for entropy. With things like volume and pressure, at least, I have a natural inkling as to what they are. What <em>is</em> entropy, more than disorder. Why are there no units? Why are these definitions correct?</p> | 6,545 | [
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<p>The Earth rotates about it's own axis. Do the geomagnetic field lines rotate due to this rotation or not? </p> | 6,546 | [
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<p>How did <a href="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion" rel="nofollow">Newton's third law</a> came into being? Was it his original finding like the second law? Or was it more of a restatement of someone else like the 1st one coming from Galileo? In either case what initiated the thought of what is now known as Newton's 3rd law?</p> | 6,547 | [
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<p>According to the 2nd law of thermodynamics, no system can be a 100% efficient. Looking at the universe as it's own system, is it an exception to the rule (a system that doesn't lose any energy)? If not, are there any theories that suggest where the energy go?</p> | 6,548 | [
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<p>Is there an unit of color charge? I haven't found it, so I suppose that it doesn't exist, if this is right, why? Isn't it supposed that every measurable quantity can be expressed in terms of base quantities? What about flavour charge?</p> | 6,549 | [
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] |
<p>Euteictic freeze crystallization is a method where an electrolytic solution is cooled and separated into a stream of (relativly) clean, pure ice and a salty brine. I know anectdotally of wine concentrates that where made similiar. Now I wonder wether a suspension of solids can be separated in a similiar way, and if so under what circumstances, and if not, why.</p>
<p>Some half baked thoughts of mine: </p>
<ul>
<li><p>Anecdotally, if I put soup into the the freezer, no easily visible separation occurs. </p></li>
<li><p>With comparativly large suspended particles, a growing ice crystal may not be able to 'push' particles away and encapsulate them instead. </p></li>
<li><p>With salty solutions, the brine is denser than the water or ice and separates by gravity - while in my soup, er suspension, some solids will settle at the top with the ice and become frozen in it.</p></li>
<li><p>The whole thing my simply be a matter of time, with slower freezing allowing for cleaner crystals </p></li>
</ul> | 6,550 | [
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<p>I was solving a question about inclined planes, and it got me thinking:</p>
<p>Imagine a scenario where, an object is on an inclined plane and the inclined plane is on a frictionless surface, and is itself frictionless. </p>
<p>Now, if the mass of the object is comparative, or maybe, even more than the mass of the plane, do you think the plane will slide backward and the object will slide forward? If yes, please explain with a free body diagram which force is responsible for that, and if no, then why does our intuition tell us that this will happen?</p>
<p><img src="http://i.stack.imgur.com/jwT1G.jpg" alt="enter image description here"></p>
<p>This is all I know....</p> | 6,551 | [
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<p>I apologize if this question is trivial, but I am new to physics and am struggling with some of the basic concepts.</p>
<p>Working in $\mathbb{R}^2$ with standard coordinates $(x,y)$, suppose we have a particle of mass $m$ moving on a curve $(x(t),y(t))\in\mathbb{R}^2$. It's tangent vector (velocity vector) is $$x^\prime(t)\frac{\partial}{\partial x}+y^\prime(t)\frac{\partial}{\partial y} \ \ \ \ \ \ \ \ \ \ (1)$$This particle's kinetic energy is $\frac{1}{2}m\left((x^\prime(t))^2+(y^\prime(t))^2\right)$. Also, suppose we have some conservative force $F$ so that $F=\left(\frac{\partial U}{\partial x},\frac{\partial U}{\partial y}\right)$ where $U$ is some smooth potential $U:\mathbb{R}^2\to\mathbb{R}$.</p>
<p>Anything I've read says the kinetic energy in polar coordinates is $$\frac{1}{2}m\left((\dot r)^2+(r\dot\theta)^2\right)$$ and the force in the $r$ and $\theta$ directions are $$F_r=-\frac{\partial U}{\partial r} \ \ \ \ \text{ and } \ \ \ \ F_\theta=\frac{1}{r}\frac{\partial U}{\partial \theta}$$</p>
<p>For the second point, I don't understand what it means to say force in the $r$ or $\theta$-direction. It's clear the force in the $x$-direction is just the first component of $F$, but is the force in the $r$-direction just the first component of $F$ in polar coordinates? I don't see how that really makes sense. Also, computing $\frac{\partial U}{\partial x}=\frac{\partial U}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial U}{\partial \theta}\frac{\partial \theta}{\partial x}$ (and similarily $\frac{\partial U}{\partial y}$) I can see where these terms pop up, but don't get how to put the concepts together.</p>
<p>For the first point, I don't understand how they are getting these equations, and especially how they get them so fast! If you use the change of variables formula (i.e. $\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial x}\frac{\partial}{\partial \theta}$ and so on) on equation $(1)$, compute $x^\prime , y^\prime$, and collect like terms you get that the velocity vector above is $\dot r\frac{\partial}{\partial r}+r\dot\theta\frac{\partial}{\partial\theta}$. This takes some work but in this form it makes sense, to me, to say that the kinetic energy in polar coordinates is $\frac{1}{2}m\left((\dot r)^2+(r\dot\theta)^2\right)$. But any book I've read just computes this extremely quick by saying $$x^\prime(t)=\dot r\cos\theta +r\dot\theta\sin\theta \ \ \ \ \text{ and } \ \ \ \ y^\prime(t)=\dot r\sin\theta-r\dot\theta\cos\theta$$I see sometimes $\hat r=(\cos\theta,\sin\theta)$ and $\hat\theta=(-\sin\theta,\cos\theta)$ but how can you have a "basis" that changes at every point?</p>
<p>Any help would be greatly appreciated!!</p> | 6,552 | [
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<p>I hope this isn't a dumb question, but...</p>
<p>If we have two fixed sine waves, both of which have a frequency range of +1 to -1, with a ratio between the waves of wave(1):3 to wave(2): 1, what does the Schrodinger equation tell us (if anything) about the relationship between the waves?</p>
<p>Equally, if the waves are no longer fixed (ie, time dependent) and wave(1) has complete nine oscillations, what does Schrodinger predict about the relationship between the waves for oscillation number 10?</p> | 6,553 | [
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<p>SI system uses all (that I know) measurement basic units as 1 (single) instance: meter, second, ampere, etc, <strong>except the KILOgram</strong>. It already defined with 1000 multiplier (kilo).</p>
<p>It prevents from using usual multiplier prefixes: <em>mega, giga, tera</em>, ... Though we sometimes use miligrams or micrograms. All points to the "gram" as a basic unit. But for some reason it isn't the case.</p>
<p>Does anybody have explanation of this fact? Why kilogram and not gram was decided to be a basic unit?</p> | 6,554 | [
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<p>Is temperature solely a function of a kinetic energy? If a solid and a gas are at thermal equilibrium at a temperature of 20 degrees Celsius, the solid has much less kinetic energy than the gas. How can the temperatures of both be the same? What is keeping the solid at the same temperature as the gas?</p> | 6,555 | [
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<p>I am doing a project with some small miniature incandescent light bulbs (like a <a href="http://vcclite.com/_pdf/T-1%203_4%20Midget%20Flange%20Base.pdf" rel="nofollow">CM7333</a>).</p>
<p>Sorry for not providing more links, the system will only allow 2.</p>
<p>The power source is a <a href="http://www.datasheets.pl/batteries/E11A.pdf" rel="nofollow">E11A</a> battery.</p>
<p>The issue I am having is the bulb is not bright enough. We need to basically double or triple its current light out put or lumens. Theses bulbs are rated in MSCP. something Candle Power I presume. </p>
<p>These little bulbs are available with different amp, volt, and MSCP ratings, as precribed in this [chart][3].</p>
<p>Ok, so it would seem a simple matter of getting a bulb with an increased amp rating or filament design (I want to use the same battery in my design, which is 6v) so the total watts is higher and hence the bulb will burn brighter. From the [chart][4] </p>
<p>I could say grab a CM3150 which indicates a MSCP rating 3 times higher than the current bulb I am working with, for the same Volts and amps. I assume it brighter because the filament is a lighter duty design, which burns brighter. At least that's my way of wrapping my mind around it.</p>
<p>This were I run into my question or were I need some education. These little batteries seem to have some kind of current limiting capability or attribute. I have reviewed the technical data but its not clear to me how many amps the battery can supply.</p>
<p>I don't know how to properly determine how many amps my little battery will provide. I know it says "38 mAh to 3.0 volts", but I dont know how to properly apply that. The data sheet also states the drain as ".5 mA continuous", if I ma reading it correctly. Does that mean the battery can provide .5 mAs. or .0005 of an amp? Is it saying HALF a miliamp? or half a amp? half an amp sounds like a lot and half a milliamp sounds tiny.</p>
<p>So, in closing I hope I asked my question in a way that can be understood. Basically, I need to understand the maximum out put of that battery in even divisions of the amp. Like my bench top power supply. .01 .357, etc. Do these little guys have a current limit over time. I don't think they can discharge all their energy in a second... I think its the current limiting that is preventing my bulb from burning brighter... I dont know</p>
<p>Thanks,
Robert</p> | 6,556 | [
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<p>I do not understand this because angular momentum is $L=I\omega$ ($I$ is moment of inertia;$\omega$ is angular velocity) but it I have also seen equations where $L= rmv\sin(x)$. I do not understand how these are related, could someone please explain the connection?</p> | 6,557 | [
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<p>If I were to get a conductor e.g. a piece of copper wire or aluminium and connect it to one pole of a battery (let's take the positive pole for example), will electrons be removed from the conductor forming Cu or Al ions? Or does the flow only start when the circuit is completed?</p>
<p>I think as soon as a pole is connected, some physical change takes place. Let's say you connect both poles, there needs to be a potential difference, and I would think that difference would need to be "felt" by the other side for the electrons to flow. But if the second pole was only connected later, there needs to be a potential difference, so the metal would be charged (or I believe, ionised). Am I right in saying this?</p>
<p>Also, if electrons are flowing through a wire, can you say the wire is ionised, or am I getting this wrong?</p> | 6,558 | [
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<p>Is putting a balloon that is charged up against a wall and having it stick polarization AND charging by temporary induction, or just polarization?</p> | 6,559 | [
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] |
<p>Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics.</p> | 6,560 | [
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<p>The sphere has radius $R$ and is missing its "pole" - meaning that in the area $\theta\leq\alpha$ there is nothing. The object has a homogenous charge density $\sigma=\frac{Q}{\pi R^2}$</p>
<p>I'm trying to derive what the field inside and outside is. It should be a Legendre polynomials excercise. I know this is an axially symetrical problem, so the general solution of the Laplace equation should be:
$$\phi(r,\theta)=\sum_{l=0}^{+\infty}\left(A_l r^l+B_l r^{-(l+1)}\right)P_l(\cos\theta)$$
Where $P_l$ are the Legendre polynomials. In my case, I shoud separate the results in two areas:
$$\phi(r<R,\theta)=\sum_{l=0}^{+\infty}A_l r^l P_l(\cos\theta)$$
$$\phi(r>R,\theta)=\sum_{l=0}^{+\infty}B_l r^{-(l+1)}P_l(\cos\theta)$$
in order for the potential not to diverge in $r=0$ or $r\rightarrow+\infty$
There's a hint in the book to use the fact, that potential should be continuous and that the difference of derivatives (i.e. the difference in electrical intensity: $[\vec{E}]$) in the direction of the normal is the charge density.</p>
<p>The potential continuity is clear. It gives the following condition:
$$\sum_{l=0}^{+\infty}\left(A_l R^l-B_l R^{-(l+1)}\right)P_l(\cos\theta)=0\ \ \ \ \forall\theta\in[0,\pi]$$
which implies
$$\frac{A_l}{B_l}=\frac{1}{R^{2l+1}}$$
I'm not sure how to use the electrical intensity condition, because
$$\frac{\partial \phi(r\rightarrow R_-,\theta\leq\alpha)}{\partial r}-\frac{\partial \phi(r\rightarrow R_+,\theta\leq\alpha)}{\partial r}=0$$
$$\frac{\partial \phi(r\rightarrow R_-,\theta>\alpha)}{\partial r}-\frac{\partial \phi(r\rightarrow R_+,\theta>\alpha)}{\partial r}=\sigma$$
I'm not sure how to deal with the fact, that the condition is differenct for $\theta\leq, >\alpha$.</p>
<p>The result should be:
$$\phi(r<R,\theta)=\frac{Q}{8\pi\epsilon_0}\sum_{l=0}^{+\infty}\frac{1}{2l+1}\left[P_{l+1}(\cos\alpha)-P_{l-1}(\cos\alpha)\right]\frac{r^l}{R^{l+1}}P_l(\cos\theta)$$
$$\phi(r>R,\theta)=\frac{Q}{8\pi\epsilon_0}\sum_{l=0}^{+\infty}\frac{1}{2l+1}\left[P_{l+1}(\cos\alpha)-P_{l-1}(\cos\alpha)\right]\frac{R^l}{r^{l+1}}P_l(\cos\theta)$$
Can you tell me how to deal with the other condition?</p>
<p>Thanks</p> | 6,561 | [
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] |
<p>The dimensions of circulation $\int_C \vec{v}\cdot d\vec{r}$ seem strange, but if you include
(even a constant) density $\rho$, then $\int_C \rho\vec{v}\cdot d\vec{r}$ has dimensions
the same as action/volume. Is there any significance to that? Is there any heuristic way to
think about circulation which helps understand the dimensions?</p> | 6,562 | [
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<p>I used to live in Boston. Near my complex, there was an apartment complex with lots of our friends. Anyways, that place had faulty heating most of the time; mainly in the corridors. They were pretty stuffy.</p>
<p>One thing I'd noticed was that in our complex, static shocks were common but not too strong. Touching the elevator panel never did anything; and getting a doorknob shock was common but the shock was mild. One didn't need to avoid these.</p>
<p>In this other building, on the other hand, the elevator panels (IIRC nearly the same) gave shocks that hurt, and so did the doorknobs. I used to use my sleeve to touch things; and I'd be very wary of them.</p>
<p>The two complexes were nearly identical except for shape and the fact that one of them had faulty heating.</p>
<p>My question is, is there a correlation between faulty heating and buildup of static? A heat difference between the corridors and the inside room could generate a thermo emf in the doorknob (as well as in the elevator panel as a cooler shaft is on the other side). But, a thermo emf won't really build up static electricity, would it?</p> | 6,563 | [
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<p>I am just beginning to learn magnetism and my book used two ways to define the force caused by the magnetic field, brushing over the latter.
The first:</p>
<p>$$F = q v B \sin (\theta).$$</p>
<p>And:</p>
<p>$$\vec{F} = q(\vec{v} \times \vec{B})$$</p>
<p>where the $ \times $ is the cross product. After looking up the <a href="http://en.wikipedia.org/wiki/Cross_product" rel="nofollow">cross product</a>, I found that it was defined as:</p>
<p>$ \vec{a} \times \vec{b} = |\vec{a}||\vec{b}| \sin (\theta) \vec{n}$
where $\vec{n}$ is the unit vector found via the "right hand rule." </p>
<p>While I have no doubt the right hand rule is a useful tool, I wonder if there is a more "on paper" way to find the direction of the force a magnetic field applies to a charged particle. Basically, shouldn't there be a way to avoid the right-hand rule and to do it all via equations?</p> | 6,564 | [
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