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<p>Recently I was taking a look at a <a href="http://www.youtube.com/watch?v=eGguwYPC32I" rel="nofollow">video</a> explaining the existence of fourth spatial dimension and thereupon that infinitely many spatial dimensions are possible. Also it showed that what Einstein told about time being a dimension itself was also incorrect as time is always present, even in one dimension. So how's it all possible, is the statement by Einstein not correct?</p> | 6,657 | [
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<p>I am just wondering say if there is an expedition where some astronauts are sent to the moon, how do they choose the trajectory for the spaceshuttle (or whatnot)? I mean there are many possible trajectories depending on the initial velocity at which the vessel is launched. There must be some sort of optimal trajectory they choose right? What are the factors they take into account when choosing the trajectory? Things like costs, stability and so on? What does stability mean in this context? </p> | 6,658 | [
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<p>I'm having a little trouble understanding Shor's algorithm - namely, why do we throw away the result f(x) that we get after applying the F gate? Isn't that the answer we need?</p>
<p>My notation:
$\newcommand{\ket}[1]{\left|#1\right>}$</p>
<p>$F(\ket x \otimes \ket0) = \ket x \otimes \ket{f(x)}$ , </p>
<p>$f(x) = a^x \mod r$</p> | 6,659 | [
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<p>Let $| 0 \rangle$ and $| 1 \rangle $ be the states of qubit. Let $\hat{\sigma_x}$, $\hat{\sigma_y}$, $\hat{\sigma_z}$ be Pauli matrices:
$$
\hat{\sigma}_{x} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \;\;\;
\hat{\sigma}_{y} = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right), \;\;\;
\hat{\sigma}_{z} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).
$$
Using the representation $\hat{I} = | 0 \rangle \langle 0 | + |1\rangle \langle 1| $ I tried to receive the similar representation for Pauli matrices:
$$
\hat{\sigma}_x = | 1 \rangle \langle 0 | + |0\rangle \langle 1|, \\
\hat{\sigma}_y = -i| 1 \rangle \langle 0 | + i|0\rangle \langle 1|, \\
\hat{\sigma}_z = -| 0 \rangle \langle 0 | + |1\rangle \langle 1|, \\
$$
Is it correct? I'm not sure if it is possible but I assumed that $|0\rangle = (0,1)^{T}$ and $|1\rangle = (1,0)^{T}$ and I looked at $|0\rangle \langle 1|$ as on outer product $(0,1)^{T}(1,0)$. I'm afraid that I use matrices instead of operators etc. Help me please to understand the topic.</p> | 6,660 | [
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<p>I was wondering if I can say to a layman that <strong>"upon throwing the ball on a wall an enormously large number of times, there is a small probability that the ball will go through the wall"</strong>, while explaining quantum tunneling (alpha decay example is abstract and artificial for a layman).</p>
<p>My doubt if whether the wall region can be modeled as a finite potential barrier (infinite potential barrier - which is not of Dirac delta form - will not allow tunneling). </p>
<p>Also, the wall seems to have all the other characteristics of the artificial barrier potential we set up in quantum mechanics, am I missing anything?</p> | 6,661 | [
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<p>I am calculating the propagator of the free particle on a sphere : $K(\theta_f \phi_f t_f; \theta_i \phi_i t_i)$. The wavefunctions in this case are the spherical harmonics $Y_{lm}(\theta, \phi)=\frac{e^{im \phi}}{\sqrt{2 \pi}}{\Omega(\theta)}$. So the kernel is
$$K=\sum e^{\frac{-i E_n (t_f-t_i)}{\hbar}} \frac{e^{im (\phi_f - \phi_i)}}{{2 \pi}}{\Omega(\theta_f)}\Omega(\theta_i)$$</p>
<p>The sum is over $m=-l, -l+1, ....., l$, and $l=0,1,2.....$</p>
<p>How do I calculate this sum? </p> | 6,662 | [
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<p>I understand that gravity is viewed as flowing as a river pushing objects down on the body of a planet. If that is the case and earth is a sphere, where does the gravity go when it hits the center of the planet? Does it get compressed? Is it destroyed? </p> | 6,663 | [
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<p>I am trying to calculate the heat loss of an aquifer. An aquifer is basically a hot-cold storage $80\: \mathrm{meters}$ under the ground. It is an open system and utilizes groundwater. By pumping for example warm water ($70\: \mathrm{C}$) under the ground this can be stored for half a year and be used in the winter to heat a building.</p>
<p>The problem is that I'm not quite sure how to calculate the amount of heat lost during those 6 months.
I was thinking of using $\frac{dQ}{dt} = \mathop{UA}(T_2-T_1)$</p>
<p>Where $T_2-T_1$ is the temperature difference of the ground soil and water.</p>
<p>However I am working with an undefined 3D shape.</p>
<p>So if someone could point me in the right direction as to how to find out how much heat is lost during 6 months of storage, that would be great!</p> | 6,664 | [
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<ol>
<li><p>In a previous question, I was given an answer:
"A quick Google suggests that the triple point of Hydrogen is 13.8K and the triple point of Neon is 24.6K, so neither can exist as liquids at temperatures low enough to form BECs." CVB Note. It was inferred later in the answer, that a gas can never exist as a liquid, below its triple point.</p></li>
<li><p>From a separate source, I have a table, which shows that at 0.1atm press, the vap/liq point of Helium is 2.5066K. Being from, webbook.nist.gov (Note the ".gov" - See also comments below), I would assume that this is correct.
The Vap/liq point of Helium at atmospheric pressure is 4.22K, so Helium's Vap/liq point has changed with a pressure change. I know that Helium is different in many ways, from other gases. One of these differences, is that it has no triple point.</p></li>
<li><p>I seem to remember, from somewhere, that although water's boiling and freezing points can change with pressure change, this only applies to water and not other elements.
This appears to be untrue, in that Helium also seems to have the same property.</p></li>
</ol>
<p>My question is:<br>
Which of these three are correct, and does any other element, other than Helium, have the property, of its Vap/liq point, changing, when it's pressure changes.?</p> | 6,665 | [
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<p>$erg$ is an Energy unit, which means that it is equal to $\frac{mass{\cdot}distance^2}{time^2}$. If I want to calculate the kinetic energy of a body in erg units, in what units should the distance and mass be?</p>
<p>By "what units" I mean $kg$ or $gram$ and $cm$, $m$ or $km$ not $inch$ vs. $m$.</p> | 6,666 | [
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<p>How to show that $\hat {S}$-operator must be lorentz-invariant operator?</p>
<p>$$
|\Psi (t)\rangle = \hat {S} | \Psi (0) \rangle , \quad \hat {S} = \hat {T}e^{-i\int \hat {H}_{I}d^{4}x}.
$$
I have read that this result follows from the unitarity of the Poincare group operator $U_{0}(\Lambda , a)$ and the covariance of S-matrix $S_{out, in} = \langle out| \hat {S}|in \rangle$, but I don't understand how do we conclude that from this follows that $U_{0}(\Lambda , a)\hat {S}U^{-1}_{0}(\Lambda , a) = \hat {S}$.</p>
<p>I'm not interesting in the derivation of lorentz-invariance of S-operator from causality principle at this moment.</p> | 6,667 | [
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<p>Why light travels in straight paths here on earth? What's the real cause that makes light photons to go in a straight line? and what are the factors that could change the path of light externally? (I'm excluding reflection, refraction and deviation here)</p> | 6,668 | [
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<p>In school-level tasks, when (almost) all substances are linear, homogeneous and isotropic, we have $D=\epsilon E$, $H=B/\mu$ and thus Maxwell "in material" equations (1) say how $E$ and $B$ depend on time given known dependence of $\rho$ (free charge density) and $j$ (free current density). Here they are in CGS unit system:
$$\left\{\begin{aligned}
\text{div} D=4\pi\rho\\
\text{div} B=0\\
\text{rot} E = -\frac{1}{c}\frac{\partial B}{\partial t}\\
\text{rot} H = \frac{4\pi}{c}j+\frac{1}{c}\frac{\partial D}{\partial t}
\end{aligned}\right.$$
Also we know continuity equation $\partial \rho/\partial t + \text{div} j=0$. But this is not enough to determine, how j will change over time or in statical case, how $j$ is distributed in the conductor. What are other equations for $j$? Are there any for some "ideal case"?</p>
<p>For example, I don't know actually, is the following task correct or under determined:</p>
<blockquote>
<p>Electric current I flows along infinite cylindrical conductor. Inner radius is $r$, outer is $R$, magnetic constants of all substances are given ($\mu_1,\mu_2, \mu_3$ from inside out). Find magnetic field ($B$ and $H$) and current distribution in a conductor.</p>
</blockquote>
<p><strong>The question:</strong> Is there any "standard" equations for $j$? Particularly, is the task above well-determined?</p> | 6,669 | [
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<p>I have a question about cosmology. At popular level people explain the time of decoupling of matter and radiation as the moment when temperature falls enough for nuclei and electrons to recombine into atoms. People say that the Universe became “transparent”. The photon cross section by an electro-neutral system is smaller than Thomson's one, i.e., technically it means that the mean free path became very large or even infinite.</p>
<p>However, there is another mechanism for the mean free path to become infinite. It is because during its expansion the universe becomes less dense. Let me explain it by example. Imagine that you are in the forest where the diameter of tree is $a$, and mean distance between trees is $b$. What is the mean diameter of the observed area? It is proportional to $b^{2}/a$, I suppose. Now let's imagine that our forest is in the expanding universe, i.e., $b$ grows (linearly, for example) with time while $a$ remains constant. Then at some moment “the diameter of the observed area” starts to grow faster than the speed of light, i.e., becomes infinite. </p>
<p>It implies that the recombination is not necessary for the decoupling. Did the recombination start earlier than the moment I described above? Or both mechanisms (recombination and density fall) are equally important for the decoupling?</p> | 6,670 | [
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<p><a href="http://en.wikipedia.org/wiki/Potts_model">Wikipedia</a> writes:</p>
<blockquote>
<p>In statistical mechanics, the Potts
model, a generalization of the Ising
model, is a model of interacting spins
on a crystalline lattice.</p>
</blockquote>
<p>From combinatorics conferences and seminars, I know that the Potts model has something to do with the <a href="http://en.wikipedia.org/wiki/Chromatic_polynomial">chromatic polynomial</a>, but it's not clear to me where it arises (and for which graph it's the chromatic polynomial of).</p>
<blockquote>
<p>Question: What does the chromatic polynomial have to do with the Potts model?</p>
</blockquote> | 6,671 | [
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<p>Two bodies with similar/different mass orbiting around a common barycenter.</p>
<ul>
<li>What is force between them, where $F_{12}$ is the force on mass 1 due to its interactions with mass 2 and $F_{21}$ is the force on mass 2 due to its interactions with mass 1?</li>
<li>What is relation between $F_{12}$ and $F_{21}$?</li>
<li>What is total force between them?</li>
</ul> | 6,672 | [
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<p>I was just wondering what happens in a circuit in terms of different types of energy transformations. If you apply a voltage to a circuit then electrons start moving (very slowly). Since the electrons want to flow to the positive terminal they will have electrical potential. But the electrons are also now moving once the voltage is applied as there is a current so would they have kinetic energy as well? </p>
<p>Some information I have found online says that electrons collide with atoms in a bulb and this is why the filament heats up. But this would imply that kinetic energy is being transformed into heat and this can't be correct because surely any change in kinetic energy would alter the current flowing? Also, in a series circuit, if you measure the potential difference between any 2 points after all of the loads then you always get zero. I was just wondering why it is zero because don't the electrons keep moving even after passing through all the load in order to get back to the power source so surely they cannot do this without some form of energy?</p> | 6,673 | [
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<p>Under <a href="http://en.wikipedia.org/wiki/Active_and_passive_transformation" rel="nofollow">active transformation</a>, the particle moves. On the other hand, for a passive one, the coordinate is just relabel. </p>
<p>I've read that the passive one will not affect the potential energy and the active one will change it. This is the case for two massively different size objects, like earth and a ball. But basically there will be no difference between passive and active transformation if the two objects making the system are comparable in size.</p>
<p>My question is how to prove that and which mathematical framework one should use?</p> | 6,674 | [
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<p>I want to make a home experiment where I roughly explain the phenomenon of water remaining in a straw if you close one end of the straw.</p>
<p>So I'm thinking that the weight of the water is pulling it down, but the pressure underneath the straw is keeping it up. If there was no finger on the top, the pressure would also be pushing it down so it would fall.</p>
<p>Would this be a valid way to show this?</p>
<p>Mass of water: volume*density: $\pi r^2*H*1000$</p>
<p>Weight of water: $\pi r^2*H*1000*9.8$</p>
<p>Pressure*area = force</p>
<p>force= $101325*\pi r^2$ (weight of water)</p>
<p>And I somehow want to combine this with the upward force of pressure in the room.</p>
<p>Am I on the right track?</p> | 6,675 | [
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<p>I've just read <a href="http://image.gsfc.nasa.gov/poetry/ask/a11024.html">here</a> that:</p>
<blockquote>
<p>Equatorial radius = 6378.16 kilometers. Polar radius = 6356.78
kilometers, so the difference in circumference is 71.1 kilometers. It
is not a perfect sphere, but kind of pear-shaped.</p>
</blockquote>
<p>How correct is that information and what exactly are Equatorial radius and Polar radius with diagram if possible?</p> | 6,676 | [
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<p>I'm a bit confused about the difference between these two concepts. According to Wikipedia the <a href="http://en.wikipedia.org/wiki/Fermi_energy">Fermi energy</a> and <a href="http://en.wikipedia.org/wiki/Fermi_level">Fermi level</a> are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level?</p> | 916 | [
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<p>I'm a bit confused about how solar cells work. </p>
<p>My understanding is that there is a p-n junction. A photon is absorbed which creates an electron-hole pair, and the idea is to separate the electron and hole before they recombine in order to produce current.</p>
<p>n-type materials are doped with extra electrons and p-type materials are doped with holes. So why is that in all the diagrams I see, when charge separation occurs, electrons are conducted through the n-type material and holes are conducted through the p-type material? Shouldn't it be the other way round?</p> | 6,677 | [
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<p>In <em>Gravitation and Cosmology</em>, S.Weinberg states the following:</p>
<p>$$\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} \propto \epsilon^{\alpha \beta \gamma \delta} $$</p>
<p>and the argument is that the left-hand side must be odd under a single permutation of the indices. I don't see why this is true. Say I interchange $\alpha\leftrightarrow \beta$:</p>
<p>$$\Lambda_{\epsilon}^{\beta}\Lambda_{\zeta}^{\alpha}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} $$</p>
<p>I don't see why the above expression must satisfy</p>
<p>$$\Lambda_{\epsilon}^{\beta}\Lambda_{\zeta}^{\alpha}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda}=-\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} $$</p>
<p>Any hint will be appreciated.</p> | 6,678 | [
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<p>When 'constructing' the usual de Sitter space in $\mathcal{M^5}$ by invoking the contraint $-X^{2}_{0} +X^{2}_{1} +X^{2}_{2} +X^{2}_{3} + X^{2}_{4} = \alpha^2$ we quickly see that we end up with a hyperboloid suspended in Minkowski space.</p>
<p>Most literature continues by invoking different coordinates on this embedding, varying from flat/open/closed slices corresponding to different solutions of the Friedmann equations of a universe with solely $\Lambda$ and curvature.</p>
<p>I am not quite sure how to interpret these different coordinate systems physically; I can see what effect invoking them might have, but I am a tad confused due to the fact that we first map $X^{2}_{0}$ along the (arbitrary) $z$-axis and then continue to map a new variable $\tau$ for time on another axis. What allows us to create an embedding and invoke arbitrary coordinates on it within it's 'larger' Minkowski space?</p> | 6,679 | [
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<p>Consider a nanoscopic wire (with radius $R$) of superconducting material. The wire lies along the $z$-axis and a magnetic field $\mathbf{H}_a = H\mathbf{e}_z$ is applied. The magnetic field is too weak to destroy superconductivity or to induce vortices and we assume that the superconducting parameter $\psi$ can be put equal to the constant value $\psi_{\infty}$ in the material (and of course, zero outside).</p>
<p>Now, I want to use the second Ginzburg-Landau equation to find the vector potential $\mathbf{A}$ (and the magnetic induction field $\mathbf{B} = \nabla\times\mathbf{A}$ from this potential). With the assumptions mentioned above the second Ginzburg-Landau equation simplifies to<br>
$\nabla\times\left(\nabla\times\mathbf{A}\right) = \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2\mathbf{A}$<br>
where<br>
$\psi(\mathbf{r}) = \left\{\begin{array}{lr}
\psi_{\infty} & \mathrm{in\,nanowire} \\
0 & \mathrm{outside}
\end{array} \right.$<br>
This expression is simplified even further by using the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$, leading to the following vector laplace equation in the case outside the wire:<br>
$\nabla^2\mathbf{A} = 0$<br>
Since the system has cylindrical symmetry, I'm working in cylindrical coordinates. The scalar equations associated with the general vector equation are then:<br>
$\left\{\begin{array}{rcl}
\dfrac{\partial^2A_{\rho}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\rho}}{\partial\phi^2} + \dfrac{\partial^2A_{\rho}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\rho}}{\partial\rho} - \dfrac{2}{\rho^2}\dfrac{\partial A_{\phi}}{\partial\phi} - \dfrac{A_{\rho}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\rho} \\
\dfrac{\partial^2A_{\phi}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\phi}}{\partial\phi^2} + \dfrac{\partial^2A_{\phi}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\phi}}{\partial\rho} + \dfrac{2}{\rho^2}\dfrac{\partial A_{\rho}}{\partial\phi} - \dfrac{A_{\phi}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\phi} \\
\dfrac{\partial^2A_z}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_z}{\partial\phi^2} + \dfrac{\partial^2A_z}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_z}{\partial\rho} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_z
\end{array} \right.$<br>
while for the case outside the wire the right hand sides are zero. These equations are coupled and I'm not sure how to efficiently solve them. So my question is: could anyone point me in the right direction as to how I should try to solve this vector laplace equation in cylindrical coordinates? Or are there perhaps any possible further assumptions I failed to make? I'm not looking for a complete answer by the way, just a push down the right path, a method.</p>
<p>Note: I've asked for a method to solve the vector laplace equation in cylindrical coordinates with less context on math.se yesterday (<a href="http://math.stackexchange.com/questions/253263/solving-the-vector-laplace-equation-in-cylindrical-coordinates">Solving the vector Laplace equation in cylindrical coordinates</a>) but I thought it might be better to put this here, so I can describe the full physical problem. Perhaps the question on math.se can/should be deleted then?</p> | 6,680 | [
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<p>polarization could be easily imagined in classical model: direction of E vector. is there any simple image for polarization of single photon?</p> | 6,681 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/33916/what-is-the-escape-velocity-of-a-black-hole">What is the escape velocity of a Black Hole?</a> </p>
</blockquote>
<p>This is gravitational speed for earth: $$v=\sqrt {g_{e}r}.$$
What is gravitational speed for a <a href="http://en.wikipedia.org/wiki/Black_hole" rel="nofollow">black hole</a>? I want an approximate speed.</p> | 281 | [
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<p>I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or integral of other functions? Whereas the position and momentum wavefunctions are essentially different but related. They must preserve expectation values like the relationship of classical mechanics, $<p>=m~\frac{d<q>}{dt}$ (where $<p>$ and $<q>$ are now expectation values).</p>
<p>For example, a momentum wave packet that has a positive expectation value constant over time implies a position wave packet that moves over time in some direction. Simply saying there is Fourier transform seems to obscure this important relation.</p> | 6,682 | [
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<p>An object of a given mass falls from an unknown height. If the force exerted by the object on contact with the ground is known, how would you ascertain the height from which the object fell?</p> | 678 | [
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<p>Previous posts such as <a href="http://physics.stackexchange.com/questions/907/type-of-stationary-point-in-hamiltons-principle">this</a> ask about types of stationary point in Hamilton's Principle. There is, however, another aspect to discuss: the question as to whether the extremal path is unique.</p>
<p>One geometric way to envisage this is to assume that multiple paths are simultaneously extremal. I believe that this is an explanation for lenses, but I have not seen lenses explained as multiple classical solutions to Hamilton's Principle. (The multiple paths being the 360 degrees of rays between source and focus, etc also demonstrable through Fermat's principle.)</p>
<p>One can generalise lenses, but also consider a simpler case. Let the surface of a sphere be the action (phase space) surface which is minimized in classical paths. Thus (ignore antipodals here) between two points A and B the geodesic is the unique classical path. In quantum form the WKB approximation would no doubt have constructive maxima on this path.</p>
<p>However if the sphere has a disk (containing that geodesic) cut out, the shortest path now has exactly two choices: around one or the other rim from A to B. Presumably WKB would maximize the quantum paths on these two (although I havent proved this). If so then classically we have a quantum-like phenomenon: a particle has a choice in going from A to B. Experimentalists might see this and wonder whether the particle went from A to B via the LHS, the RHS or both....</p> | 6,683 | [
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<p>I've read material claiming it comes from the Higgs boson fails while others claim it is from the tensions of quarks in the gluon field... I am only a 15 year old kid in high school so please "dumb it down" </p> | 229 | [
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<p>After watching the first episode of wonders of the solar system, one question came up which is not explained.</p>
<p>Bryan Cox says that ultimately the universe will be devoid of matter, so not even a single particle will actually exist. This however, seems to be a state where entropy is zero. There is no entropy since there's no disorder. </p>
<p>On the other hand the 2nd law of thermodynamics says that the disorder increases.</p>
<p>So, I am pretty sure that I am thinking wrong about this in someway.</p>
<p>I can imagine that the answer might be that an empty universe has an infinite amount of entropy but it feels pretty counter-intuitive for me.</p> | 6,684 | [
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] |
<p>what is physics behind Water drops during falling from a tap.
<a href="http://upload.wikimedia.org/wikipedia/commons/b/bd/Water_drop_animation_enhanced_small.gif" rel="nofollow">water drop animation</a></p>
<p>A drop or droplet is a small column of liquid, bounded completely or almost completely by free surfaces.</p>
<ul>
<li>Why Water drops falling into spheres (during falling from a tap in vacuum )?</li>
</ul>
<p>please note that a drop of water made of $H_2O$ atoms.
<a href="http://upload.wikimedia.org/wikipedia/commons/5/51/2006-01-21_Detaching_drop.jpg" rel="nofollow">Drop of water</a></p> | 6,685 | [
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<p>In our de Sitter phase, the cosmological constant is tiny. $10^{-123}M_P^4$. Suppose there is another phase with a lower vacuum energy. Is de Sitter phase still stable? The tunneling bubble radius has to exceed the de Sitter radius. Suppose a metastable decay to such a bubble happened. Take that final state, and evolve back in time. It's unlikely to tunnel back because of exponential suppression factors. Light cones are dragged outward in expanding de Sitter at such radii, so, by causality, the bubble radius has to keep shrinking back in time until at least the de Sitter radius. This contradicts our earlier assumption.</p>
<p>What about engineering a phase transition? Form a small bubble and stuff it with enough matter in the new phase with sufficient interior pressure to keep the bubble from shrinking. It collapses to a black hole if the radius R is much greater than $M_P^2/T$, which is much less than the de Sitter phase. The black hole then evaporates.</p>
<p>Even if the cosmological constant in the new phase is large and negative, the tunneling radius still has to be larger than the de Sitter radius because of the hyperbolic geometry of AdS means the volume of the new phase is only proportional to the domain wall area?</p>
<p>If we assume there is a large matter density in the de Sitter phase, it has to be very very large to make the tunneling radius smaller than the de Sitter radius.</p>
<p>Is this deSitter phase stable and not metastable?</p> | 6,686 | [
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<p>What is the difference between <a href="http://en.wikipedia.org/wiki/Parallel_universe" rel="nofollow">parallel universe</a> and <a href="http://en.wikipedia.org/wiki/Multiverse" rel="nofollow">multiverse</a>? Is it parallel universe or universes?</p> | 6,687 | [
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<p>I'm beginning learn quantum mechanics. As I understand, <em>state</em> is a map $\phi$ from $L^2(\mathbb R)$ such that $|\phi|^2$ describes probability density of a particle's position. By integrating $|\phi|^2$ we can get the probability of a particle's position in a subset of $\mathbb R$.</p>
<p><strong>My question is</strong>. Why we need $\phi$? Why just not speaking about <em>probabilities</em> itself, i.e. probability measures $\mathsf \Phi$ on $\mathbb R$ such that $\Phi(A)$ is the probability of particle's position in $A\subset \mathbb R$.</p>
<p>Moreover, why we speaking about <em>observables</em> as <em>operators</em>. Why not just speaking about <em>random variables</em> on $\mathbb R$. (I know that observable isn't a arbitrary operator; but may be the restrictions can be expressed in terms of random variables?)</p>
<p>Many expressions in quantum mechanics via integrals with observables and states, as I understand, just express expectation, variation and standard deviation of such random variables w.r.t. probability that define the states. For example, Heisenberg's inequality just inequality for product of standard deviations of position and momentum.</p> | 6,688 | [
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<p>Looking at the classical Doppler effect there is one generalized equation, and they have a velocity of source, and observer.
In the relativistic version, there is only one velocity taken into account, and, in my book, there are two separate equations for receding and approaching sources.
These two formulas are leading me to be confused about the sign of the velocity, among other things. Is there a version with only one, generalized formula?
And why is only one velocity taken into account, as opposed to the two from the classical version?</p> | 6,689 | [
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<p>I'm reading <em>Music, Physics and Engineering</em> (H. Olson 1967), and contains models of musical instruments as electric circuits, i.e. uses the analogies between electrical circuits and mechanical (mass-spring-dashpot) systems. (The idea is that the reader will be more familiar with the electrical circuit representation.)</p>
<p>In some of these systems he uses (without defining) a "quadripole" circuit element. A specific instance is to represent the sound board of a guitar (or other string instrument).</p>
<p>I've seen some vague references that it performs some sort of amplification, but I would like to known a reasonable approach for mathematically representing it's effect within an electrical circuit.</p>
<p>Another possibility is that the term "quadripole component" has fallen out of favor, and there is some other more modern name for this type of thing.</p> | 6,690 | [
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<p>Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations).</p>
<p>It is clear that</p>
<ul>
<li>in biological (population) models usually pregnancy introduces a delay term, or</li>
<li>in disease transition in networks the latent period introduces a delay, or</li>
<li>in engineering in feedback problems signal processing introduces the time delay.</li>
</ul>
<p>I would like to see a list of answers where each answer contains one reference/example.</p> | 6,691 | [
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<p>It is clear that if you push on some object, there is reaction of the same force. But is it the same energy? Thanks a lot.</p> | 6,692 | [
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<p>My friend puzzled me with this: Say I have a container that contains a vacuum and another container inside. And this inner container contains air. Also the outer container material is just barely strong enough to not implode due to the vacuum. If I remove the air from the inner container to outside of the outer container (for example through a tube that can be sealed), would the outer container implode?</p>
<p>My friend says it will implode because the total volume of the whole thing is decreased which leads to decreased inside air pressure. Mathematically, it sounds right. But I thought it was weird because there was really no interaction between the air in the inner container and the outer container in the first place. So my answer is that it will not implode.</p>
<p>But I'm not sure myself. What would actually happen and why? </p>
<p>EDIT:
I actually think it will implode if the inner container was a balloon or something due to having less area for the remaining air molecules in the "vacuum" to bounce around. My question is for an inner container that doesn't change shape like a balloon. </p> | 6,693 | [
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<p>In Higgs mechanism, we take the combination of LH $SU(2)$ doublet and RH singlet along with Higgs doublet so that the overall weak hypercharge and weak isospin is zero to be $SU(2) \times U(1)$ invariant. I am pretty confident about my understanding on weak hypercharge calculation. But it seems weak isospin calculation is not quite clear. Specially when the terms comes from Weinberg's dimension five operator for leptons such as $\overline{L_{iL}^\mathcal{C}}$, Higgs $SU(2)$triplet or fermion $SU(2)$ triplet. To arrange them in $SU(2) \times U(1)$ invariant way how should I calculate weak isospin? </p> | 6,694 | [
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<p>I'm confused about the Second Law of Thermodynamics. The Second Law of Thermodynamics prohibits a decrease in the entropy of a closed system and states that the entropy is unchanged during a reversible process.</p>
<p>Then why do we say that $\Delta S = \int_a^b{\frac{dQ}{T}}$ for a reversible process? Doesn't the second law simply state that $\Delta S = 0$?</p>
<p>(I'm a high school student teaching himself the principles of thermodynamics, but I am struggling with more challenging material due to my poor understanding of these basics)</p> | 6,695 | [
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<p>I have a mutipartite system; the subsysyems having interacted. Now, to find reduced density matrix for any single subsystem, is the order of partial traces(over remaining subsystems) important?</p> | 6,696 | [
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<p>I'm getting some weird results from a calculation I'm doing and quite honestly, I'm pretty sure it's my fault. I do have an apparatus involved for the experimental process for my lab but I don't think it's that. I've come to the conclusion that my notes do not contain the right equations for these calculations. </p>
<p>For the theoretical masses I know the total mass of the point masses and the distance from the axis to the masses. I'm pretty sure the equation of this is either 1/2mr^2 or just mr^2.
The 1/2Mr^2 is from my notes but I think the correct answer would be mr^2. However the hanging mass lies on an apparatus that is a cylinder so 1/2Mr^2 could be correct.</p>
<p>The experimental part of the lab involved an apparatus that looked like a wheel which lowered a hanging mass on a string by turning the wheel (also by gravity). For the point mass and apparatus combined, I know the hanging mass, slope, and radius. I also know this data for the apparatus.
I started out by finding the Force (which I was told is also equal to the torque) using the equation F=m(g-a). I know a is acceleration, but I'm not exactly sure how that factors out. From there I'd probably use τ = I α but I know the angular acceleration nor how to calculate it. I'm pretty sure it involves integrals. Please help me with this, it's been three days and I still can't figure this out. </p>
<p>The next part of my lab involves calculating the experimental rotational inertia of the ring and disk, and I know the hanging mass, slop and radius of each one. Would that be where 1/2Mr^2 comes in?</p> | 6,697 | [
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<p>What is relation of fraction of binary stars with spectral class (mass)? For example, how many binary stars are among O,B,A,F,G,K,M stars separately?</p> | 6,698 | [
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<p>Why do the electrons revolve around the nucleus in circular motion not like the earth which revolves around the sun in an eliptical motion?</p> | 6,699 | [
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<p>I'm not expecting any rigor in the following and the answers...since we're dealing with Dirac deltas in the context of QFT. </p>
<p>Consider the integral </p>
<p>$$
\int d^4q\ \Theta(q_0)\Theta(p_{3,0}+q_0)\ \delta((p_3+q)^2)\delta((q^2))\frac{1}{(q+p_2)^2-m^2+i\epsilon}
$$</p>
<p>The $p$'s and $q$'s are all Minkowski 4-vectors and $\ p_3=p_1+p_2$. For simplicity one can work in the $p_3$ CM-Frame so that $\ p_3 =(M, \vec{0})$. </p>
<p>I have problem with the deltas (since they are coupled), <strong>I was thinking about the</strong> $q_0$ <strong>integration first</strong>, how would you guys proceed doing the $q_0$-integration?. I use the composition formula for the Dirac delta <a href="http://en.wikipedia.org/wiki/Dirac_delta_function#Composition_with_a_function" rel="nofollow">Dirac Delta Comp. Wikipedia</a>, but when I replace $q_0$ in the argument of the other delta I get something like $\delta(M)$ where $M$ is the mass of $p_3$. This delta makes no sense to me since we're not even integrating over $M$? Anyone know how to do this integral from the begining? Any hint appreciated, thanks. </p> | 6,700 | [
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<p>We consider a scalar theory in a $1+D$ dimensional flat Minkowski
space-time, with a general self-interaction
potential, whose action can be written as
\begin{equation}
A=\int dt\, d^D\! x \left[\frac12(\partial_t\phi)^2-\frac12(\partial_i\phi)^2
-U(\phi)\right] \,,
\end{equation}
where $\phi$ is a real scalar field, $\partial_t=\partial/\partial t$,
$\partial_i=\partial/\partial x^i$ and $i=1,2,\ldots,D$.
The equation of motion following from \eqref{action}
is a non-linear wave equation (NLWE) which is given as
$$
-\phi_{,tt} + \Delta \phi = U'(\phi)=\phi +\sum\limits_{k=2}^{\infty}g_k\phi^k \tag{1}$$
where $$\quad{\Delta}=\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}\,.
$$
I just want to know the logic, that how they transformed the equation (1)</p> | 6,701 | [
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<p>Suppose there is the following situation:</p>
<blockquote>
<p>Blocks $A$ and $B$, with masses $m_A$ and $m_B$, are connected by a
light spring on a horizontal, frictionless table. When block $A$ has
acceleration $a_A$, then block $B$ has, by Newton’s second and third
laws, acceleration $-a_A\frac{m_A}{m_B}$.</p>
</blockquote>
<p>What does it mean for block $B$ to have a negative acceleration? Since the blocks are connected by a spring, does not block $B$ move in the same direction as block $A$?</p> | 6,702 | [
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<p>What effects would occur if the earth's core goes cold? Would the planet stay liveable after this happens?</p> | 6,703 | [
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<p>So I collected current and voltage data from a simple circuit with a power source and a resistor, using a multimeter. I created a graph for this data using excel and got the y-intercept (which is basically the value of the current when the voltage was zero) and found it to be a small negative number:</p>
<p><img src="http://i.stack.imgur.com/jecTE.jpg" alt="enter image description here"></p>
<p>What I'm wondering about is what is the physical explanation of this value, which by ohm's law should have been zero. Is it simply experimental error in the data collected by the multimeter? Or does it have any deeper physical meaning?</p> | 6,704 | [
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<p>I'm specifically looking for Schrödinger's Cat states involving superpositions of two, or if it's been done more, coherent states, i.e. monomodal states of the form $$|\psi\rangle=a|\alpha\rangle+b|\beta\rangle.$$
What states of this form have been produced in experiment? How even can the weights be? What regions of the $\hat{a}$-eigenvalue $\alpha$ and $\beta$ are accessible? If more than two coherent states can be superposed, how many? What phase-space geometries are possible so far?</p>
<p>I'm also interested in what techniques are currently used to generate these states.</p> | 6,705 | [
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<p>In an intro to GR book the Ricci tensor is given as:</p>
<p>$$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu \lambda}^{\lambda}+\Gamma_{\nu \sigma}^{\lambda}\Gamma_{\mu \lambda}^{\sigma}]$$</p>
<p>I have gotten to the point where I can work out a given Christoffel symbol, but I am still having trouble working out the above tensor as a whole (just algebraically speaking). If I'm not mistaken, $R_{\mu\nu}$ should end up a $\mu$x$\nu$ (i.e. 4x4) matrix just like the energy-momentum tensor on the other side of the field equations. In the above rendering $\sigma$ is clearly a dummy index to be summed over, and I can see how $\lambda$ is also a dummy index in the first term. But the $\lambda$s in the other terms seem to be free indices, which would then introduce incompatible dimensions in the matrix operations. I appreciate it if someone can point out the error of my ways.</p> | 6,706 | [
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<p>As far as i know, microwave (used in microwave oven) and wifi all operate on the same frequency, but why microwave can heat the foods while wifi wave can't?</p> | 230 | [
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<p>I am trying to model a system in which cubes of about 2 cm in size are floating in a circular water thank of about 30 cm in diameter. The cubes move around under the influence of the fluid flow induced by four inlets that point toward the center of the tank, and are located at the positions $0$, $\pi/2$, $\pi$, and $3\pi/2$. The flow velocity ranges from 0 to 10 cm/s, with an average velocity around 6 cm/s.</p>
<p>My questions are the following:</p>
<ul>
<li>What would be the Reynolds number of the system? In particular, should I take as characteristic length the size of the cubes, or that of the tank?</li>
<li>For such a system, what is the limit Reynolds number for the turbulent regime?</li>
<li>What would be the correct form of the drag force, and do you intuitively think that the orientation of the blocks is negligible from a drag coefficient point of view?</li>
</ul>
<p>Thanks for your help!</p> | 6,707 | [
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<p>What is similar from dynamic point of view between book lying stationary on horizontal table and rain drop falling down with constant speed?</p>
<hr>
<p>I can find two similarities</p>
<ol>
<li>Force acting on them is equal</li>
<li>Both have horizontal velocities zero</li>
</ol>
<p>Is there any other similarities possible? It is not mentioned how many similarities we have to give that means we should give all possible similarities.</p>
<p>Answer should contain related basic physic like acceleration, velocity, displacement, force,etc. We can say stuff which can be understand by high school student. </p> | 6,708 | [
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<p>The optical pumping experiment of Rubidium requires the presence of magnetic field, but I don't understand why. </p>
<p>The basic principle of pumping is that the selection rule forbids transition from $m_F=2$ of the ground state of ${}^{87} \mathrm{Rb}$ to excited states, but not the other way around ($\vec{F}$ is the total angular momentum of electron and nucleus). After several round of absorption and spontaneous emission, all atoms will reach the state of $m_F=2$, hence the optical pumping effect.</p>
<p>But what does the Zeeman splitting have anything to do with optical pumping? Granted, the ground state, even after fine structure and hyperfine structure considered, is degenerate without Zeeman splitting, but the states with different $m_F$ still exists.</p>
<p>In addition, how is the strength of <a href="http://en.wikipedia.org/wiki/Optical_pumping" rel="nofollow">optical pumping</a> related to the intensity of magnetic field applied?</p> | 6,709 | [
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<p>I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation. </p>
<p>Here is the part of the derivation that I didn't follow: </p>
<p><img src="http://i.stack.imgur.com/7B0wm.jpg" alt="enter image description here"></p>
<p>I don't get how he goes from 6.9 to 6.10 by partial-differentiating the term inside the integral. If this is allowed, I was probably missed my calculus class the day it was covered. Can someone tell me more about this? Which part of calculus is this from? </p> | 6,710 | [
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<p>Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. </p>
<p>$$\sum _i ( \mathbf F _i - \dot{\mathbf p}_i) \cdot \delta \mathbf r _i = 0$$</p>
<p>$$S[q(t)] = \int ^{t_2} _{t_1} \mathcal L (q,\dot{q},t)dt$$</p>
<p>After playing with d'Alembert's principle we find that we can rewrite the whole thing as $$\sum _i \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i}-Q_i\right] \delta q_i$$</p>
<p>This can further be rewritten under certain conditions so we get the exact form of the E-L equation. </p>
<p>It seems to me that both ways of arriving at the result are fundamentally different. </p>
<p>A function must obey the E-L equations in order to minimize the action over a path, but when we look at the virtual work, it appears that they come from the fact that (quoting Goldstein) "particles in the system will be in equilibrium under a force equal to the actual force plus a 'reversed effective force'."</p>
<p>I think I understand the principle of stationary action, I can see how it leads to the E-L equations, but d'Alembert's Principle seems so arbitrary, I can't see any motivation for it. </p> | 6,711 | [
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<p>I would like to hear the best arguments for and against the <a href="http://en.wikipedia.org/wiki/Many-worlds_interpretation" rel="nofollow">Many Worlds interpretation</a> of QM.</p> | 6,712 | [
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<p>Recently in class we went over Newton's Third Law. In the book they put an example of hitting a punching-bag with your fist and hitting a piece of paper, or an object with much less mass. It's clear that you cannot exert more force on the paper than what the paper can exert on you, otherwise it might stop your fist in the process, just like the punching-bag stops your fist. Is there a limit, however, on much force you can exert on a paper? This might sound silly but it got me thinking for the past couple of days. </p>
<p>In order to make it a little more clear consider a Car with a paper in front of it. The car-system can continuously increase the acceleration, but this means that the paper will increase the acceleration as well. Ergo, by Newton 2nd Law, the force exerted on the paper has to increase. Right? That same force back to the car. Therefore, the car can continuously increase the force it exerts on the paper. </p> | 6,713 | [
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<p>An electrostatic field is characterized by the fact that it depends only by $r$, isn't it? If it is true, I don't understand why this expression, given in cylindrical coordinates,</p>
<p>$${\bf E(r)}=\frac{\alpha}{z^2}{\bf u_r}-2 \frac{\alpha r}{z^3}{\bf u_z} $$</p>
<p>rapresents an electrostatic field.</p> | 6,714 | [
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<p>According to <a href="http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html" rel="nofollow">NASA’s Saturn fact sheet</a>, Saturn has a density considerably less than water and the lowest density of the major planets. What compositional and/or structural mechanisms are theorised to be the cause of this low density?</p>
<p>Edited to clarify: I am after not only calculations of densities, but from this, a model of what the structure of Saturn would be like to accommodate this.</p> | 6,715 | [
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<p>The attenuation of a wave through a medium can be modeled by the <a href="http://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law" rel="nofollow">Beer-Lambert Law</a> using an <a href="http://en.wikipedia.org/wiki/Attenuation_coefficient" rel="nofollow">attenuation coefficient</a>. If $I$ is the intensity, and $I_r$ is a reference intensity, then what is the physical difference between modelling attenuation using the two functions below?</p>
<p>I am looking for a conceptual understanding of the difference between $I_1$ and $I_2$. What does the integral in $\beta_1(\tau,\omega)$ imply?</p>
<p>In the equations below, $\tau$ is the time, $\omega$ is angular frequency, and $Q(\tau)$ is a real-valued function of time $\tau$.</p>
<p>I have seen both $\beta_1$ and $\beta_2$ functions used in the context of the attenuation of seismic waves through <em>layered media</em>, and I am wondering if one model is perhaps better than the other.</p>
<p>${I_1} = {I_r}{\beta _1}(\tau ,\omega )$</p>
<p>${I_2} = {I_r}{\beta _2}(\tau ,\omega )$</p>
<p>${\beta _1}(\tau ,\omega ) = \exp \left[- {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$</p>
<p>${\beta _2}(\tau ,\omega ) = \exp \left[- {\frac{{\omega \tau }}{{2Q(\tau )}}} \right]$</p>
<p>A reference for these two equations is Y. Wang, Seismic Inverse Q Filtering (<a href="http://www.scribd.com/doc/45448335/SEISMIC-INVERSE-q-FILTERING#page=132" rel="nofollow">link</a>). See for example, pg. 121.</p> | 6,716 | [
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<p>How do you find the net <a href="http://en.wikipedia.org/wiki/Mechanical_advantage" rel="nofollow">Mechanical Advantage</a> (MA) of two joint machines. Do you add or multiply the individual MA?</p>
<p>Suppose I have two sets of wheel and axle connected by a fixed pulley. Each of the wheel has a radius of 100cm and each of the axle has a radius of 10cm. What will be their combined MA? I am quite sure that both of them has a MA of 10. But what will be their total MA? Is it 20 or 100? Should I add them or multiply them?</p> | 6,717 | [
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<p>There's another question on physics.SE whose <a href="http://physics.stackexchange.com/a/2140/5971">answer</a>, if I have understood it correctly, explains that the farther the points are in space the faster they are moving away from each other.</p>
<blockquote>
<p>Actually, the apparent speed with which two of the points on the
circle in a distance $D$ of each other would move relative to each
other will be $v = H_0 D$ where $H_0$ is the speed the balloon itself
is expanding.</p>
</blockquote>
<p>This means that even the ruler sitting on my desk is expanding, just that the expansion is very very slow owing to the ruler's small size relative to the cosmic scales. However, if everything is scaling then everything should apparently be of the same size to an observer in such an expanding space. The <em><a href="http://www.imdb.com/title/tt1609715/quotes?item=qt1284401" rel="nofollow">size million pants</a></em> quote from the show "The Big Bang Theory" comes to mind. </p>
<p>To be able to counter that, there're some phenomena that we can observe which are scale invariant. Red-shift would be one such phenomenon. But is red-shift the only way we can tell that the space is expanding? Is there a logical flaw in my thinking? I know that red-shift is a result of the Doppler effect - there's bound to be a decrease in the frequency of a signal, as perceived by an observer, emitted by a source which is moving away relative to the observer.</p> | 6,718 | [
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<p>We have a basic equation $F=ma$. Now can we change this equation to another variable and start a new era in physics? If so, then how? If not, then why?</p> | 6,719 | [
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<p>In terms of General relativity we have as a matter of principle that anything that has inertial mass contributes to gravity. All forms of potential energy have inertial mass, it follows that the potential gravitational energy of a star must contribute to the gravitational field of that star. Depending on the amount of matter that is present, (the remains of) a dying star may contract to a white dwarf, a neutron star, or a black hole.</p>
<p>Let me compare contraction to a white dwarf to contraction to a neutron star. </p>
<p>As a thought experiment, let's say a known amount of gas is distributed in a region of galactic space, and something triggers that amount of gas to contract to a star. From a thought experiment point of view, how much of that star's gravity must be attributed to arising from its internal gravitational potential energy?</p>
<p>Here's why I ask: <br>
In the case of ending up as a neutron star there is significantly more contraction, as gravity overcomes the electron degeneracy pressure. It seems to follow that in the case of contraction to a neutron star <em>more potential gravitational energy is released</em>. </p>
<p>Let's say that in this thought experiment the number of atoms (and types of atoms) of the star is known, and the flow of atoms/particles being ejected from the star is known. So theoretically you can give an expression for the amount of potential energy per unit of rest mass. </p>
<p>The critical case is the Chandrasekhar limit, of course. How much gravitational potential energy should be attributed to a star that is just at the Chandrasekhar limit?</p>
<p>In the real world you never know the amount of atoms in a star, of course. All you can assess is the strength of the gravitational field of that star (the amount of spacetime curvature.) </p>
<hr>
<p>[Later edit]</p>
<p>As pointed out by Trimok, I had failed to appreciate that there is no explicit expression for potential energy in the equations, <a href="http://physics.stackexchange.com/questions/45145/potential-energy-in-general-relativity?">explained by Stan Liou</a>. I have rewritten the title of the question accordingly.</p>
<p>So the question only touches interpretation of the theory. It would seem a star that is just above the Chandrasekhar limit will collapse deeper than an star just just below it. In the interpretation, does that make a difference as to the amount of potential gravitational energy that is to be attributed to the system?</p> | 6,720 | [
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<p>Since someone commented this on this question(<a href="http://physics.stackexchange.com/questions/93345/what-is-the-probability-of-ice-in-boiling-water?noredirect=1#comment190832_93345">What is the probability of ice in boiling water?</a>), I would like to ask what is the probability that all the air ends up in the upper right corner of the room and we suffocate, considering that brain death happens after 10 minutes, and an average room has 40 cubic meters of air.</p> | 6,721 | [
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<h3>Situation</h3>
<p>Police shooting a <a href="http://en.wikipedia.org/wiki/LIDAR_speed_gun" rel="nofollow">laser device</a> into his rearview mirror, bouncing off the oncoming car license plate. I want to know if this second surface mirror, and the symmetric system--laser gun to mirror to license plate of moving car to mirror to laser gun---has a built in error because of the wavelength shift due to the glass of the mirror. Velocity and wavelength change for the raypath inside the mirror's glass, but would revert to the original wavelength for the "in air" portion of the raypath.</p>
<h3>Question</h3>
<p>Is this symmetry perfect and creates no error in measuring the car's speed?</p> | 6,722 | [
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<p>I need to find the Lagrangian for charged particles in EM fields considering relativistic effects. Is action integral Lorentz invariant.
$$A = \int_{t_1}^{t_2} L (q_i, \dot q_i, t) dt $$</p>
<p>According to my note</p>
<blockquote>
<p>According to the first postulate of special relativity, the action integral $A$ must be invariant because the equation of motion is determined by extreme condition $\delta A = 0$. </p>
</blockquote>
<p>I do not understand how does this make $A$ invariant.</p> | 6,723 | [
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<p>Today one of my instructor told me that gases cannot have heat transfer thru conduction because molecules are far apart and so it cannot transfer heat,infact the diffusion process of gases transfers the heat.
my thot at this point was,even though molecules are far apart,they transfer momentum to next molecules and we can feel the pressure and temperature(Kinetic energy).But why not heat? if we compress the gas at very high pressure, when molecules are very closer to each other, it will start transfer of heat by conduction? can someone shed some light on this?</p> | 6,724 | [
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<p>Consider the symmetry $SU_L(2)\otimes U_Y(1)$. The entries of $SU_L(2)$ doublet will have same U(1)-charge. How can this be shown mathematically?</p> | 6,725 | [
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<p>Why the lepton number conservation is connected with the invariance of the lagrangian under global phase (U(1)) transformation of the wave function? How to distinguish global gauge phase and global "leptonic" phase? And how to build the operator of the leptonic number of particles if there isn't "local leptonic invariance"?</p> | 6,726 | [
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<p>In my opinion, the <a href="http://en.wikipedia.org/wiki/Grassmann_number">Grassmann number</a> "apparatus" is one of the least intuitive things in modern physics. </p>
<p>I remember that it took a lot of effort when I was studying this. The problem was not in the algebraic manipulations themselves -- it was rather psychological: <em>"Why would one want to consider such a crazy stuff!?"</em> </p>
<p>Later one just gets used to it and have no more such strong feelings for the anticommuting numbers. But this psychological barrier still exists for newbies. </p>
<p>Is there a way to explain or motivate one to learn the Grassmann numbers? Maybe there is a simple <strong>yet practical</strong> problem that demonstrates the utility of those? It would be great if this problem would provide the connection to some "not-so-advanced" areas of physics, like non-relativistic QM and/or statistical physics. </p> | 6,727 | [
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<p>I am a computer scientist interested in network theory. I have come across the Bose-Einstein Condensate (BEC) because of its connections to complex networks. What I know about condensation is the state changes in water; liquid-gas. Reading the wikipedia articles on the subject is difficult, not because of the maths, but the concept doesn't seem to be outlined simply. Other source share this approach going straight into the topic without a gentle introduction to set the scene for the reader. </p>
<p>I would appreciate a few paragraphs describing the problem at hand with BEC (dealing with gas particles right? which kind, any kind? only one kind? mixed types of particles? studying what exactly, their state changes?), what effects can occur (the particles can form bonds between them? which kind of bonds? covalent? ionic?), what do we observe in the BEC systems (some particles form many bonds to particles containing few bonds? The spatial configurations are not symmetric? etc), and what degrees of freedom exist to experiment with (temperature? types of particles? number of particles?) in these systems.</p>
<p>Best,</p> | 6,728 | [
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<p>When talking about neutron cross-sections, literature is usually investigating isolated cases of Neutron + Atom. Here, the abundance of hydrogen is dominating neutron fluxes through material.</p>
<p>I wonder whether the reflection or capture propability of neutron radiation changes when the flux is penetrating a system of (organic) molecules. I can imagine two effects potentially reducing the neutron cross-section for bounded hydrogen:</p>
<ol>
<li><p>The dense grid of organic material (folded proteins etc, imagine thick tree or a croud of humans) could be able to shield their hydrogen atoms from beeing struck by neutrons. Assuming organic material to be much denser (nuclei/volume) then other material like soil, rocks.</p></li>
<li><p>Neutrons are not able to transfer their whole energy to the hydrogen nucleus when it is electrically bounded (p is bounded to its electron, which is bounded to the electrons of the molecular system, which introduces inertia to the protons ability to move). Concluding that the capability of slowing down neutrons is reduced compared to free hydrogen.</p></li>
</ol>
<p>Is this complete non-sense or could there be a measurable influence to neutron fluxes when using (a) non-organic or (b) organic material, containing the same amount of hydrogen?</p> | 6,729 | [
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<p>So recently in physics class, we learned about the magnetism <a href="http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm">right hand rules</a>. </p>
<p>One of them states that the index finger points in the direction of the velocity of a particle, the middle finger points in the direction of the magnetic field, and the thumb points in the direction of the magnetic force.</p>
<p>I'm curious why the magnetic field vector is perpendicular to the magnetic force vector.</p> | 6,730 | [
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<p>I have tried to find in the literature a proper nice and beautiful Bloch sphere to describe the trajectory of a nuclear spin, starting in z-axis, using a pulse sequence of an initial 90º pulse with zero phase (which means that the pulse is launched in the x-axis), wait for a time $\tau$, and then launch another 90º pulse with zero phase, and wait the same amount of time tau to measure an echo. </p>
<p>Please notice that I am talking about Nuclear Magnetic resonance, and most of the books don't offer a Bloch sphere to explain this echo, which is supposed to be the first! echo, and only use a 90º and 180º pulse to explain that. </p>
<p>I have read the original article of Hahn echo, but I saw that his diagram is not clear. </p> | 6,731 | [
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<p>I'm trying to explain in simple terms what the weak interaction does, but I'm having trouble since it doesn't resemble other forces he's familiar with and I haven't been able to come up (or find on the web) with a good, simple visualization for it. </p> | 6,732 | [
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<p>What experiment would disprove <a href="http://en.wikipedia.org/wiki/Friedmann_equations" rel="nofollow">Friedmann model</a> of cosmology?</p>
<p>As a layman, I have read a lot of articles and threads in specialized forums. I am probably wrong, but I developed an impression that that theory is circular, 99.9% self- referential. The only hard experimental datum seems to be the redshift, and then most other parameters, distance, luminosity, age, scale factor, expansion rate, etc. are model-dependent and therefore confirm the theory,
a) - what are the other hard facts at the bottom of the theory, if any?</p>
<p>If I am not wrong, the prerequisite of a scientific theory is that it can be disproved.</p>
<p>b) - how do you disprove that redshift is determined by stretching of space, </p>
<p>c) - how do you disprove that space can expand and that just a metric devised by a theorist can make it expand?</p>
<blockquote>
<p>spacetime is not an object expanding like some rubber foam. It's a combination of a manifold and metric. The expansion means the metric is time dependant, and this is explained by the requirement that the metric be a solution to Einstein's equation.</p>
</blockquote> | 6,733 | [
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] |
<p>So I have a graph:</p>
<p><img src="http://i.stack.imgur.com/BL8Xn.png" alt="enter image description here"></p>
<p>The value of the gradient/slope is $1.6±0.4$ and the value of the intercept is $0.9±0.4$. But what are the units of the graph? Is the unit of the gradient $v^2M^{-1}$? What about the unit of the intercept?</p> | 6,734 | [
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<p>I saw a pretty simple homework question <a href="http://physics.stackexchange.com/questions/39281/needed-energy-for-lifting-200-kg-weight">here</a> that asked how much work it takes to lift a 200 kg weight, and while the math for a basic answer is simple the weightlifter in me instead wanted to actually approach this question with a critical and pragmatic eye. There are many factors that one could consider to easily get a value different from the answer in a conservative field (for example: 1960 N for 1 meter).</p>
<p>I figure exact answers are obviously impossible. I want to instead see what methodologies can be used to start formalizing my upcoming reasoning (and also take this opportunity to correct any misconceptions I have) in hopes of closing in on a reasonable method to create an answer with more data. I also would love to see similar bio-mechanical studies and related material. </p>
<p>I mainly broke this problem up into two levels:</p>
<p>Mechanical: </p>
<p>The human body is a collection of support structures (basically curved trusses), complex muscular tissue (non-linear springs with various attachment points), and connective tissue (constraints). Something I really could use clarification about here is the effect duration of the lift could have on the work exerted.</p>
<p>Thermodynamic:</p>
<p>Our body metabolizes chemicals to generate store energy in tissue, then later break down that tissue into work and heat. Let's ignore the energy lost due to digestion and other secondary factors, and instead rate the efficiency of the human "engine" as a function of how much work he creates from the energy he expends doing the exercise.</p>
<p>The Problem:</p>
<p>Let's say we have a (large) man who lifts a bar 1 meter up from the earth when he does a deadlift, squat, or bench press.</p>
<p>How much energy does this body use to move 200kg doing each motion?</p>
<p>How much energy does this body expend to move 200kg doing each motion?</p> | 6,735 | [
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<p>Why do cosmic bodies such as planets, stars, satellites revolve?<br>
What made them to revolve after the formation of universe?</p> | 231 | [
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<blockquote>
<p>A uniform rod of mass 1.2 kg and length 1.8 m is pivoted in the horizontal position as shown (black point).<img src="http://i.stack.imgur.com/cGmYy.png" alt="enter image description here">
The rod is at rest and then released. The acceleration due to gravity is $g = 9.8 m/s^2$.
1) Find the rotational inertia of the rod relative to the axis perpendicular to the screen and passing through the pivot point.
2) What is the angular speed (in rad/s, but do not include units) of the rod as it passes through the vertical position (when end marked B is at the bottom)?</p>
</blockquote>
<p>My attempt of solution:</p>
<p>To find moment of inertia, I use the moment of inertia of a uniform rod respect to it's center of mass (middle in this case), which is $$I_{cm}=mL^{2}/12$$
where $m$ is mass of the rod and $L$ its length. Then I use the parallel axis theorem to find the moment of inertia in this case: $$mL^{2}/12+0.45^{2}m=0.567$$</p>
<p>For the 2º part, since the net torque on the rod is not constant I use potential energy to find first the speed of center of mass at the vertical position. The initial height of center of mass is 0.45 m taking as reference point position of the middle point of the rod at vertical position. So:
$$mg0.45=1/2mv_{cm}^{2}+1/2I\omega ^{2}=1/2mv_{cm}^{2}+1/2I(v_{cm}^2/0.45^{2})$$</p>
<p>From here I get $v_{cm}$ and later the angular velocity.</p>
<p>Where am I wrong?</p> | 6,736 | [
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<p>A big argument by the nitrogen-in-the-tire crowd is that:</p>
<blockquote>
<p><em>Nitrogen atoms are bigger and thus less likely to escape the tire, bringing stability to your tire pressure.</em></p>
</blockquote>
<p>If <a href="http://en.wikipedia.org/wiki/Earth%27s_atmosphere#Composition">Earth's atmosphere is %78.084 percent nitrogen</a>, then the non-Nitrogen composition is ~22%. If this 22% is more likely to seep out of the tire, and Nitrogen doesn't seep out, then simply filling the tire up with air will maintain the original 78% N, plus the added nitrogen which would be 78% of the remaining 22% (assuming total seepage).</p>
<p>Simply, won't a tire that's been filled up 10 times in the course of its normal life already be disproportionately Nitrogen if this argument is true?</p> | 6,737 | [
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<p>Analogy: assume that I have constant rain fall and I have a water bucket to collect this rain. If I am rest relative to the earth, I will catch a certain amount of rain. However, if I now move towards the rain, I will increase the amount of water I collect. Now I want to apply this idea to a source emitting photons instead of rain drops.</p>
<p>I image a distance light source (distance star) that is emitting a constant number of photons such that I can ignore the $1/r^2$ fall off of the intensity (that is, the intensity does not change very much over some appreciate distance). So instead of a water bucket, I now have a light collector that measures light intensity. Here is my question: <strong>Does the light intensity measured by my light collector that is moving towards the distant star increase or stay the same when compared to a light collector at rest realtive to the distant star?</strong></p>
<p>My first thought was that the light intensity would increase because the light is blue shifted and higher energy photons will therefore produce a higher intensity. However, I believe that there might be another contribution due to length contraction. Since an observer moving towards a light source has to account for length contraction, does this mean that there is an increase in photon density? If so, this higher photon density in the frame of the light collector will also contribute to increasing the light intensity. </p>
<p>Can someone verify or correct my thinking.</p> | 6,738 | [
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] |
<p>In classical theory, for electrons or protons, charge and angular momentum combine to give the magnetic moment.</p>
<p>Does a similar consequence hold for the generalization of charge to other forces, like the strong force or weak force?</p> | 6,739 | [
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<blockquote>
<p><em>If a ball under the influence of gravity falls straight down from a height $h$, collides elastically with the floor, at the instant of collision, what forces does it experience?</em> </p>
</blockquote>
<p>Shouldn't the floor exert a force equal to the weight of the ball, on the ball itself? Then won't the forces cancel out?(since the only to forces on the ball are it's weight and the force of the floor, which are equal and Opposite) </p>
<p>I know that the better way to find resulting velocity, etc. Would be to use work-energy or kinematics, but I would like to reconcile those with the force treatment </p> | 6,740 | [
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<p>Please help me out, I’m missing something.</p>
<p>We know that, right now, space is expanding at roughly 73km/s/Mpc.</p>
<p>This means: two points in space 1Mpc away from each other “move” 73 km farther away every second. Of course they are not actually moving, it’s the space between them that’s expanding: that’s why two objects can drift apart faster that the speed of light if they are far away enough).</p>
<p>Now: scientists tell us that the universe is not just expanding, but that it’s expansion is accelerating.</p>
<p>Initially my understanding of the accelerating expansion was: if now it’s 73km/s/Mpc, at some point in the future it will be 74, 75, ….</p>
<p>But it turns out it’s not like that.
It turns out that that 73 is actually DECREASING (which is why the Hubble sphere is expanding) and that “accelerating expansion of space” means something different.</p>
<p>Apparently it means “accelerating growth of the scale factor” $a(t)=\frac{d(t)}{d0}$ (where $d(t)$ and $d0$ are the proper distance between two points at time t and time 0).</p>
<p>So I understand: “accelerating universe” just means that $a’’(t)>0$ (i.e. the proper distance between two points changes over time at in increasing rate), while the Hubble parameter $H(t)=a'(t)/a(t)$. decreases over time (so at some point in the future two points in space will “move” away from each other at, say, 65km/s/Mpc instead of 73).</p>
<p>In other words: the proper distance between the two objects increase faster and faster as the distance between them increases, BUT every single Mpc between them actually increases (slightly) SLOWER.</p>
<p>BUT: if everything I’ve said is correct (and please tell me if it’s not), I don’t get this:</p>
<p>If “accelerating universe” does NOT mean “increasing Hubble parameter”, but it just means that the proper distance between two objects increase at an increasing rate WHAT’S THE BIG DEAL?</p>
<p>I mean: the very basic fact that space is expanding everywhere implies that as time pass two objects will drift away faster and faster, since their recession velocity increase with their distance.</p>
<p>What am I missing?</p> | 6,741 | [
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<p>An air bubble was initially released from the bottom of an ocean. Assume that the air in the air bubble is an ideal gas and its <strong>temperature remains constant</strong> at 25 degrees Celsius. Is heat added or removed from the air bubble as the bubble rises?</p>
<p>My reasoning is that since total internal energy, $U \propto T$, the total internal energy must have remained constant. But from the laws of thermodynamics, $U = W + Q$. Since $pv = nRT$ remains constant, no work is done on the bubble and hence $W = 0$. It follows that $Q$ must be equals to $0$, i.e. heat is neither removed nor added to the air bubble.</p>
<p>Is my reasoning sound?</p> | 6,742 | [
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<p>I have a theory I think might explain why the universe has accelerated in a way that doesn't require dark energy. I'm wondering if someone has proposed this theory before (did some research and couldn't find anything).</p>
<p>The theory is that the big bang was essentially a huge super-nova-like eruption inside an even larger universe. Just like the visible universe has super novae that happen inside much much larger galaxies and dust clouds, so could the Big Bang have been a huge explosion in a much huger universe. The acceleration we have seen evidence of could, then, have been caused if the big bang was off center in the huge amount of matter in the outer universe - ie if one side exerted more gravity than the other, mass closer to that side would accelerate away from mass closer to the center of the big bang.</p>
<p>I imagine a huge black-hole sucking in a giant swath of matter from the greater universe, coalescing into a small area that leaves a large space around it rather empty. Then when it explodes (or perhaps it just emitted large amounts of matter out of it via something like hawking radiation), it fills this space again with what we now see as our universe.</p>
<p>This theory doesn't seem to require any sort of bizarre unknown physics, like dark energy, singularities or anything else that causes divide by 0 errors in established physical equations. Has it been proposed before? Also, I'd be interested to know if anyone has any concrete reasons why this theory wouldn't work.</p> | 6,743 | [
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<p>Reference request concerning the <a href="http://en.wikipedia.org/wiki/Hollow_Earth#Concave_hollow_Earths" rel="nofollow">Hollow concave Earth hypothesis</a>. I am searching for this paper: </p>
<blockquote>
<p><em>A Geocosmos: Mapping Outer Space Into a Hollow Earth</em> authored by M. Abdelkader and published in <a href="http://www.springer.com/physics/journal/11216" rel="nofollow">Speculations in Science & Technology</a> (vol. 6 pg 81–89) in 1983. </p>
</blockquote>
<p>I would be very grateful if anyone could provide me with a link to the paper. </p>
<p>NOTE: The paper discusses the mapping of all of space into a spherical body of earth's volume via non-Euclidean geometry. I have spent quite a bit of time searching for the paper. I heard that one of the AMS journals published between 1982-1983 contained the abstract. The paper has been cited once or twice, but there are no links available; vol. 6 has not been placed online as far as I can see.</p> | 6,744 | [
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<p>Is there such as thing as the viscosity of stars in a galaxy, along the lines of gravitational attraction between stars changing the dynamics. </p>
<p>If so, how is that put in terms of the Virial Theorem?</p> | 6,745 | [
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<p>Just hoping for some clarity regarding Hamilton's characteristic function (W). When we take a time independent Hamiltonian we can separate the Principle function (S) up into the characteristic function minus $ht$, yes I know its the Legendre transform but,</p>
<p>\begin{equation}
W=S+ht
\end{equation}</p>
<p>Meirovitch in his <em>Methods of Analytical Dynamics</em> p 356 gives $h$ as being the Jacobi energy function $h$ as defined in earlier chapters both his and Goldstien's texts. </p>
<p>This is the only time I have seen it being called $h$ rather than the Hamiltonian. I was just wondering if anyone had read through it and perhaps noticed something different in Meirovitch's definition that escapes me. Most authors define these integration constants as the Hamiltonian instead. I know the difference is subtle but it is intriguing as to why he chose $h$ not $H$! Is it just down to how you express the conjugate momenta?</p> | 6,746 | [
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<p>I have been looking at taking the continuum limit for a linear elastic rod of length $l$ modeled by a series of masses each of mass $m$ connected via massless springs of spring constant $k$. The distance between each mass is $\Delta x$ which we use to express the total length as $l=(n+1)\Delta x$. The displacement from the equilibrium position is given by $\phi(x,t)$. </p>
<p>The discrete Lagrangian in terms of the $i$th particle $\mathscr L$ is composed as follows, </p>
<p>\begin{equation}
\mathscr L=\frac{1}{2}\sum _{i=1}^{n}m\dot \phi _i^2-\frac{1}{2}\sum ^n _{i=0}k(\phi_{i+1}-\phi _i)^2
\end{equation} </p>
<p>At this point we take the continuum limit such that the number of masses in the fixed length of rod tends to infinity and correspondingly the inter-particle distance tends to zero. It is fruitful to multiply top and bottom by $\Delta x$ such that we can define two quantities that remain constant during this limit namely the linear density ($\mu=m/\Delta x$) and the elastic modulus ($\kappa=k\Delta x$).</p>
<p>\begin{equation}
\mathscr L=\frac {1}{2} \sum _{i=1}^{n}\Delta x\bigg(\frac{m}{\Delta x}\bigg)\dot {\phi} _i^2-\frac {1}{2} \sum _{i=0}^{n}\Delta x(k\Delta x)\bigg(\frac{\phi _{i+1}-\phi _i}{\Delta x}\bigg)^2
\end{equation}</p>
<p>It is easy to see why the linear density remains constant since both the number of masses per unit length increases while simultaneously the unit length decreases. </p>
<p>However my question is regarding the elastic modulus, I fail to see how it remains constant in this limit.</p>
<p>The argument goes as follows;
Since the extension of the rod per unit length is directly proportional to the force exerted on the rod the elastic modulus being the constant of proportionality. The force between two discreet particles is $F_i=k(\phi _{i+1}-\phi _i)$, the extension of the inter particle spacing per unit length is $(\phi _{i+1}-\phi _i)/\Delta x$. Therefore (HOW) $\kappa=k\Delta x$ is constant. Its easy to relate them but why is it constant!?! </p> | 6,747 | [
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<p>At first, I know the question sounds ambiguous and maybe pseudo-scientific, but it's a thing I've been arguing about with my colleage for quite some time and while neither of us knows much about quantum mechanics besides the popular statements you see in internet, it has been bugging me to finally end the argument.</p>
<p>Here's the point: let's say there's a coin lying on the ground. He says, that there is a very small chance, but still a chance, that the coin isn't really there, but, say, 5 meters away. That's because there's a probability of finding an electron 5 meters away and thus, a probability of finding all of the object's particles 5 meters away. I'm not really sure if there IS a probability of finding an electron that far away, since I've always imagined the electron cloud as something existing around the nucleus but certainly not reaching infinity. Bet even so, doesn't the fact that I SEE the coin determine that the coin is in fact there? I'll never know the exact position of all the particles, but the uncertainty is small, for sure not as big as 5 meters.</p>
<p>I'm sorry if I offended anyone by this question that probably seems stupid to many, that wasn't my intent.</p> | 6,748 | [
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<p>Is there a formula that can get sunrise at a particular latitude and longitude (and elevation from sea level)? If so, what it is? (Why does it work?)</p> | 6,749 | [
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<p>In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in space, will sum to 0 [1] (this is the reference frame in which the CMB anisotropies are minimal). Is this not a form of spontaneous Lorentz symmetry breaking ? While the underlying laws of nature remain Lorentz invariant, the actual physical system in study (in this case the whole universe) seems to have given special status to a certain frame.</p>
<p>I can understand this sort of symmetry breaking for something like say the Higgs field. In that situation, the field rolls down to one specific position and "settles" in a minima of the Mexican hat potential. While the overall potential $V(\phi)$ remains invariant under a $\phi \rightarrow \phi e^{i \theta}$ rotation, none of its solutions exhibit this invariance. Depending on the Higgs model of choice, one can write down this process of symmetry breaking quite rigorously. Does there exist such a formalism that would help elucidate how the universe can "settle" into one frame ? I have trouble imagining this, because in the case of the Higgs the minima exist along a finite path in $\phi$ space, so the spontaneous symmetry breaking can be intuitively understood as $\phi$ settling <strong>randomly</strong> into any value of $\phi$ where $V(\phi)$ is minimal. On the other hand, there seems to me to be no clear way of defining a formalism where the underlying physical system will randomly settle into some frame, as opposed to just some value of $\phi$ in a rotationally symmetric potential.</p>
<p>[1] The rigorous way of saying this is : There exists a reference frame S, such that for all points P that are immobile in S (i.e. $\vec{r_P}(t_1) = \vec{r_P}(t_2) \forall (t_1, t_2)$ where $\vec{r_P}(t)$ is the spatial position of P in S at a given time $t$), and any arbitrarily small $\epsilon$, there will exist a sufficiently large radius R such that the sphere of radius R centered on P will have total momenta less than $\epsilon c / E_k$ (where E is the total kinetic energy contained in the sphere).</p>
<hr>
<p>Ben Crowell gave an interesting response that goes somewhat like this :</p>
<p>Simply put then : Causally disconnected regions of space did not have this same "momentumless frame" (let's call it that unless you have a better idea), inflation brings them into contact, the boost differences result in violent collisions, the whole system eventually thermalises, and so today we have vast swaths of causally connected regions that share this momentumless frame.</p>
<p>Now for my interpretation of what this means. In this view, this seems to indeed be a case of spontaneous symmetry breaking, but only locally speaking, because there should be no reason to expect that a distant causally disconnected volume have this same momentumless frame. In other words the symmetry is spontaneously broken by the random outcome of asking "in what frame is the total momentum of these soon to be causally connected volumes 0?". If I'm understanding you correctly, this answer will be unique to each causally connected volume, which certainly helps explain how volumes can arbitrarily "settle" into one such frame. I'm not sure what the global distribution of boosts would be in this scenario though, and if it would require some sort of fractal distributions to avoid running into the problem again at larger scales (otherwise there would still be some big enough V to satisfy some arbitrarily small total momentum). </p> | 6,750 | [
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