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<p>To make a superfluid rotate in an annulus shaped container, we start with a normal fluid, rotate the container, then cool it to below critical temperature to get a rotating superfluid. </p>
<p>The allowed values of circulation in a superfluid rotating with non-zero velocity are n*h/m where n could be 1,2,3.. or -1,-2,-3.... If we impart an angular velocity less than 0.5*h/m to the normal fluid, it goes to rest when converted to the superfluid state. If we impart 0.8*h/m, it goes to quantized circulation of h/m when converted to superfluid state.
The explanation I got for this from my Professor's notes is that due to the high free energy barrier between the 2 nearest quantized circulations for the superfluid, it goes to the closest one. Is this correct ? Should the circulation not always fall to the nearest lower value of quantized circulation when converted to superfluid state ? How does free energy come into role for this ? </p>
| 3,364 |
<p>Is it really possible in the foreseeable future to create a gamma ray laser? I've read these two articles on Wikipedia:</p>
<p><a href="http://en.wikipedia.org/wiki/Hafnium_controversy" rel="nofollow">the Hafnium controversy</a></p>
<p><a href="http://en.wikipedia.org/wiki/Induced_gamma_emission" rel="nofollow">Induced gamma emission</a></p>
<p>It sounds pretty amazing, although apparently no one has reproduced the phenomenon since 1998...</p>
| 3,365 |
<p>After reading an recent news "<a href="http://www.dailymail.co.uk/sciencetech/article-2300651/Stargazers-capture-picture-planet-suns--just-like-Luke-Skywalker-s-home-planet-Tatooine-Star-Wars.html" rel="nofollow">Stargazers capture first picture of a planet with two suns – just like Luke Skywalker’s home planet of Tatooine in Star Wars</a>", I am thinking that: can we calculate the probability of extrasolar planets, binary stars and so on?</p>
<p>Several years ago, we were not sure of the existence of extrasolar planet. But now, we have found many of them.</p>
<p>I think we should be able to calculate the probability via first principles when we focus on galaxy formation. Then, this problem is different from the Drake-Sagan equation for estimating the number of detectable extraterrestrial civilizations.</p>
<p>And, the result should be useful as an guider for astronomical observations.</p>
| 3,366 |
<p>What will be the consequence (severe ones) on laws of physics if a particle that travels <a href="http://en.wikipedia.org/wiki/Faster-than-light">faster than light</a> is discovered?</p>
<p>I am looking for a more general answer so that a high school student would be able to understand. Or is it not possible to explain a high school student without bringing special relativity into context?</p>
| 3,367 |
<p>Suppose engineers built a large circular room in a rotating space station where if one looked directly up from any location, one could see the floor.</p>
<p>If one used a ladder to reach the center of the room, could they balance an object in the center of the room's rotation, such that the object floated unsupported? Would it be easy to place the object there or quite difficult?</p>
| 3,368 |
<p>If a laser beam is looked at from the side versus a dark background, a sparkling effect can be seen caused by dust particles in the air hit by the beam. </p>
<p>Is there any simple model or coarse estimations how often that would happen, how bright the particles flash and how long a single flash will be?</p>
| 3,369 |
<p><em>Let $x^\mu$ be the coordinates of a reference frame, $K$, where all bodies feel the same constant and uniform acceleration $\textbf{a}=\textbf{g}=-\nabla\varphi$; let $\xi^\mu$ be the coordinates of a Locally Inertial Frame, $LIF$. Using the <strong>Principle of Equivalence</strong>, show that the linear part in $\textbf{g}$ of the interval for $K$ is</em>
$$
ds^2=(1+2\varphi/c^2)c^2dt^2-d\textbf{x}^2+o(1/c^2).
$$ </p>
<p>My attempt:</p>
<p>P.E states the metric tensor, $g_{\mu\nu}$, respect to a generic system of coordinates its related to metric tensor of a flat Minkowskian $\eta_{\alpha\beta}=\text{diag(1,-1,-1,-1)}$ via the Jacobian of the diffeomorphism $x\mapsto\xi$. Obviously such a diffeo can be of the form
$$\xi^0=ct,\xi^i=x^i-1/2g^it^2,$$
and the P.E states that
$$
g_{\mu\nu}=\frac{\partial\xi^\alpha}{\partial x^\mu}\frac{\partial\xi^\beta}{\partial x^\nu}\eta_{\alpha\beta},
$$
so the 00 component is
$$
g_{00}=\frac{\partial\xi^\alpha}{\partial x^0}\frac{\partial\xi^\beta}{\partial x^0}\eta_{\alpha\beta}=\frac{\partial\xi^0}{\partial x^0}\frac{\partial\xi^0}{\partial x^0}-\frac{\partial\xi^i}{\partial x^0}\frac{\partial\xi^j}{\partial x^0}\delta_{ij}=1-\Big(\frac{v^i-g^it}{c}\Big)^2\simeq1-2\varphi/c^2++o(1/c^2,\textbf{g}^2),
$$
here i take $2v^itg^i/c^2=2x^ig^i/c^2=-2\varphi/c^2.$
Other problem arises becouse my $g_{0j}\neq0$.
What i wrong?</p>
| 3,370 |
<p>A Starship is going to accelerate from 0 to some final four-velocity, but it cannot accelerate faster than $g_M$, otherwise it will crush the astronauts.</p>
<p>what is the appropiate equation to constraint the movement so the astronauts never feel a gravity higher than $g_M$? for a moment i thought the appropiate relationship was</p>
<p>$$ \left\lvert \frac{d u}{d \tau}\right\rvert \le g_M $$</p>
<p>where the absolute value is of the spatial component of the four-acceleration</p>
<p>But going down this route i get the following:</p>
<p>$$ \lvert u_F \rvert = \int_0^{\tau_F} \left\lvert \frac{d u}{d \tau} \right\rvert\,d \tau \le g_M \int_0^{\tau_F} d \tau = g_M \tau_F $$</p>
<p>where $u_F$ is the spatial component of the final velocity, and $\tau_F$ is the proper time it takes to reach the final velocity. The above gives me:</p>
<p>$$ \tau_F = \frac{ \lvert u_F \rvert }{ g_M } $$</p>
<p>i'm doing some silly mistake, because there are no gamma factors, and i'm getting a finite proper time to reach $\lvert u_F \rvert = c$</p>
| 3,371 |
<p>In talking about production/decay processes, I've heard people speaking of decay modes or cross sections being 'phase space' suppressed. For example, a two body final state is more likely to occur than a three body final state, since in the later case the three particles must share the initial energy momentum whereas in the former case only two particles must share.</p>
<p>But, how do I quantify such an argument based purely on the phase space integration that appears in the cross section/decay rate formulas. The two-body phase space integration has dimensions [mass]$^0$ whereas the three-body phase space integration has dimensions [mass]$^2$. How then am I to compare the phase space 'volumes' available to a process if they have different dimensions?</p>
| 3,372 |
<p>Here is a <a href="https://www.youtube.com/watch?v=YnuDx8RsraU" rel="nofollow">Video</a> showing the honey accelerating in the hot water. As you can see, there are also dynamics. Since the water started stationary, I guess the dynamics arise because of convection flow. What is the explanation for the fast acceleration? It seems very counter-intuitive.</p>
<p>Is there a way to connect the heat equation and the Navier-Stokes equation, or is there an easier method? I also cannot say how to model the honey. Is it a liquid or a solid?</p>
<p><img src="http://i.stack.imgur.com/DIVqJ.png" alt="Honey flowing into hot water"></p>
| 3,373 |
<p>What i really want to ask how much has the Milky Way moved, relative to where it was "at the big bang" or the soonest time that makes sense (since i doubt "at the big bang" makes much sense in this question). I suppose the galaxies have non-zero impulse, otherwise we wouldn't see things like galaxy collisions. So, relative to where our galaxy or whatever was there (dust cloud?) "in the beginning", how much did we move?</p>
| 3,374 |
<p>I asked this question because I supposedly did last year, Stanfor Klein which belongs to the Solar Energy Laboratory of the University of Wisconsin says that "the color of a car does not affect its internal temperature".</p>
<p>I wonder why Metals with different colors perhaps do not absorb different doses of temperatures? and as a consequence, when different metals of different colors are exposed to strong radiation it is not so differently warmed?</p>
| 3,375 |
<p>Wikipedia says, </p>
<blockquote>
<p>A <a href="http://en.wikipedia.org/wiki/Black_hole" rel="nofollow">black hole</a> grows by absorbing everything nearby, during its life-cycle. By absorbing other stars, objects, and by merging with other black-holes, they could form <a href="http://en.wikipedia.org/wiki/Supermassive_black_hole" rel="nofollow">supermassive Black-holes</a></p>
</blockquote>
<ul>
<li>When two black-holes come to merge, don't they rotate with an <em>increasing</em> angular velocity as they come closer and closer (how does it from a neutron star? I mean, who's powerful?)</li>
</ul>
<p>And it also says,</p>
<blockquote>
<p>Inside of the event horizon, all paths bring the particle closer to the center of the black hole.</p>
</blockquote>
<ul>
<li>What happens to the objects that are absorbed into a black-hole? Which <strong>state</strong> are they really are <em>now</em>? They would've already been plasma during their accretion spin. Would they be on the surface (deposited), or would they still be attracted and moved towards the center? If so, then the surface of black-hole couldn't be a <strong>solid</strong>.</li>
</ul>
| 3,376 |
<p>We know that space cannot spread a sound wave as there is no "air" or a medium that would support the spread of a sound wave. However if we put ourselves in the vicinity of an exploding star, would it be possible to hear something?</p>
<p>The question arises from the idea that within the explosion of a star (first few seconds or less) you may hear a noise due to the explosion of the star...</p>
| 3,377 |
<p>It´s usual to read in QFT books of how it is "easier" to have a canonically normalized kinetic term. So, for instance:</p>
<p>$${\cal L} = {1 \over 2 }\partial_{\mu} \phi \partial^{\mu}\phi - {1 \over 2 } m^2 \phi \phi - {\lambda \over 4!} \phi^4$$</p>
<p>is canon. And:</p>
<p>$${\cal L}_2 = \partial_{\mu} \phi \partial^{\mu}\phi - m^2 \phi \phi - 2 {\lambda \over 4!} \phi^4$$</p>
<p>is not.</p>
<p>Now, both of them have the same classical equations of motion, since ${\cal L}_2 = 2 {\cal L}$. Supose I just carry on quantization on the $\phi$ field as usual. The free propagator is:</p>
<p>$$\langle 0| T\{\phi(x_1) \phi(x_2)\} |0 \rangle = {i \over 2} \Delta_F(x_1-x_2) $$
- the 1/2 factor comes from the fact that is is now the Green function of $ (\square+m^2)$ instead of ${1 \over 2 } (\square+m^2)$ </p>
<p>If I calculate the four point function at tree level:</p>
<p>$$\langle 0| T\{\phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4)\} |0 \rangle = \\ = -i 2 \lambda \int d^4z {\Delta_F(z-x_1) \over 2} {\Delta_F(z-x_2) \over 2} {\Delta_F(z-x_3) \over 2} {\Delta_F(z-x_4) \over 2} $$</p>
<p>All the "2" factors in the equation above disappear if I exchange ${\cal L}_2$ by ${\cal L}$. But, since they don't cancel (there's a $2^{-3}$ left) and this carries on to the cross section, I feel like I´m getting different results from equivalent Lagrangians.</p>
<p>What am I missing? Am I obliged to have the Kinetic term in a canonical normalization (so that ${\cal L}_2$ is "wrong")? If so, what conditions impose this normalization? Or, if not, and the two Lagrangians are really equivalent: how do this $2^{-3}$ disappears before becoming a catastrophic $2^6$ decrease in the cross section?</p>
<p><strong>POST ANSWER EDIT</strong></p>
<p>So, my take on the answer (please correct me if I got it wrong): starting from ${\cal L}$ above, if I do a field redefinition $\phi \rightarrow \sqrt{2}\phi$ I get:</p>
<p>$${\cal L}_Z = \partial_{\mu} \phi \partial^{\mu}\phi - m^2 \phi \phi - 4 {\lambda \over 4!} \phi^4$$</p>
<p>which is not ${\cal L}_2$ (in which I multiplied the whole ${\cal L}$ by 2).</p>
<p>In both ${\cal L}_Z$ and ${\cal L}_2$ I messed up with the normalization of the propagator, that means I will get $\langle p | \phi(0) | 0 \rangle = {1\over\sqrt{2}}$ instead of $\langle p | \phi(0) | 0 \rangle = 1$. The LSZ formula would then read:</p>
<p>$$\langle p_1 p_2 | S | p_3 ... p_n \rangle = ({1\over\sqrt{2}})^n (\mbox{amputaded diags.})$$ </p>
<p>In the case of ${\cal L}_Z$, all the factors in the four point function would be: </p>
<p>$$\langle 0| T\{\phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4)\} |0 \rangle = \\ = -i 4 \lambda \int d^4z {\Delta_F(z-x_1) \over 2} {\Delta_F(z-x_2) \over 2} {\Delta_F(z-x_3) \over 2} {\Delta_F(z-x_4) \over 2} $$</p>
<p>Which amputates to: $$-i 4 \lambda (2\pi)^4 \delta(\mbox{momentum})$$ and
$$\langle p_1 p_2 | S | p_3 ... p_n \rangle_{{\cal L}_Z} = -i \lambda (2\pi)^4 \delta(\mbox{momentum})$$ </p>
<p>Exactly the same as ${\cal L}$. Now, the same operation on ${\cal L}_2$ gives:</p>
<p>$$\langle p_1 p_2 | S | p_3 ... p_n \rangle_{{\cal L}_2} = -i {\lambda \over 2} (2\pi)^4 \delta(\mbox{momentum})$$</p>
<p>Showing that, field re-definitions are ok but multiplying the whole Lagrangian is not, the cross section will change by a factor 4. @user1631 said in his answer that means redefining $h$, I'll have to carry out this calculation without $\hbar =1$ to check that.</p>
| 3,378 |
<h2>Residual Resistivity</h2>
<hr>
<p>I saw that the graph of resistivity to temperature of alloys like nichrome is like so<img src="http://i.stack.imgur.com/KkiDU.png" alt="enter image description here"></p>
<p>Meaning that even at 0 K it has some resistivity just like copper :</p>
<p><img src="http://i.stack.imgur.com/GnQ1v.png" alt="enter image description here"></p>
<p>I read some where "It is the residual resistivity due to defect scattering" Is this related to the defects that i studied in solid state chemistry about lattice defects.Can some body elaborate?</p>
<ul>
<li>An alloy is a mixture of metals and a temperature coefficient of resistivity comparable to metals then why is its graph more linear than metals. </li>
<li>Is this because of the lattice structure of an alloy?</li>
</ul>
| 3,379 |
<h2>Superconductivity</h2>
<p>I read in a book "Physics - Resnik and Halliday" the explanation of Type-I Superconductors{cold ones} that:</p>
<blockquote>
<p>The Electrons that make up current at super-cool temperatures move in coordinated pairs.One of the electrons in a pair may electrically distort the molecular structure of the superconducting material as it moves through,creating nearby short-lived regions of positive charge.the other electron in the the pair may be attracted to the positive spot.According to the theory the coordination would prevent them from colliding with the molecules of the material and thus would eliminate electrical resistance</p>
</blockquote>
<p>Is this the only explanation or can somebody give me a more intuitive explanation that also takes into the problem of defect scattering as in the case of resistance and also explains the Type-II superconductors{hot ones}</p>
<p>P.S. What on earth are coordinated pairs?</p>
| 1,018 |
<p>Can we transfer energy from one place to another separated by arbitrarily large distances without any time lag?</p>
<p>For instance, if Alice and Bob are two observers making measurements having a singlet state, they synchronize their clocks and then go to sufficiently large distances both of them apply a magnetic field of equal magnitude (say B) in the positive Z direction. Now, if Alice makes a measurement in the X direction, he will measure a probability of a half for it to be up in that direction but he also changes the direction of the spin of the electron towards the right which implies that Bob's electron will face the left direction that is turning from positive Z direction ( a 90 degrees) which involves emission or absorption of energy. In this way, we are actually transferring energy very efficiently and very quickly (instantaneously).</p>
| 3,380 |
<p>Consider the following scenario:</p>
<ul>
<li><p>I get in a spaceship, and travel really close to the speed of light for a while, and then come back.</p></li>
<li><p>A lot of time has passed on the Earth, but since I was traveling so fast, I only experienced a few years passing.</p></li>
<li><p>So, my friends on Earth are dead, whereas I'm only a few years older.</p></li>
</ul>
<p>But what I'm having trouble wrapping my head around, is why is it <em>them</em> that's dead, and not <em>me</em>?</p>
<p>After all, given what I understand about relativity, it's just as fair to say that my spaceship stayed still, and it was actually the Earth that traveled really fast and then came back to my ship.</p>
<p>In that scenario though, the Earth being the fast-moving ship, and my ship being the stationary body, wouldn't it be that I am dead, and everyone on the Earth is just a few years older?</p>
<p>If there really is no preferred frame of reference, then why does the ship-traveler live while the people on the Earth die?</p>
| 84 |
<p>I remember once, as a child, thinking that objects do not really "move," but that at a very small scale they would have to "disappear" and then "appear" again at their newly shifted position, just the way computers render moving particles based on refresh rates. This relates to <a href="http://en.wikipedia.org/wiki/Zeno%27s_paradoxes" rel="nofollow">Zeno's paradox</a> which is solved by infinite sums. </p>
<p>Then I heard about quantum wave function collapse and the double slit experiment, and then thought: oh, maybe nature solved the problem by turning anything that wants to move into a wave instead of making a single particle "appear" and "disappear" in new positions as it moves. Waves is by the way a very elegant solution in comparison. </p>
<p>My question is: was my thinking correct? are waves (and wave collapse) nature's way to make particles move around?</p>
| 3,381 |
<p>Density of ice is much higher than air. Then how can hail stone remain in earth atmosphere before they fall down ? </p>
| 3,382 |
<p>Is <a href="http://en.wikipedia.org/wiki/Wormhole" rel="nofollow">wormhole</a> a practical concept? If not what did the scientists do, to theorize it? Does it have any limitations pertaining to speed of light?</p>
| 3,383 |
<p>I know that materials behave differently in the nanoscale. One of the reasons that I have heard as to why this happens is due to the increased surface area of the atoms. But, are there other factors? Are the potentials that act on particles different? Do other effects emerge/become insignificant that cause the different behavior at the nanoscale? </p>
| 3,384 |
<p>Inspired by the thunderstorm overhead, and after reading the question and answers <a href="http://physics.stackexchange.com/questions/28560/voltage-and-current-of-positive-lightning">Voltage and current of positive lightning</a> - what is an effective means to explain this phenomenon to a layman?</p>
<p>Also, as demonstrations are more often than not a better way of explanation, how could positive lightning be safely and meaningfully demonstrated?</p>
| 3,385 |
<p>I want to know why some semiconductors band gap decreases after doping with elements. <strong><em>Burstein-Moss band-filling effect</em></strong> can be useful to explain band gap widing in a semiconductor materials but i was unable to find any logical explanation for band narrowing effect. Can you please explain the mechanism of band gap narrowing.</p>
<p>This below quotes were taken from a research article.</p>
<blockquote>
<p>There is general agreement that two competing phenomena are dominant
in affecting the absorption edge in heavily doped semiconductors.
First, the well-known Burstein-Moss band-filling effect which shifts
positively the measured band-edge energy with increasing carrier
concentration. In this case the measured optical gap $E_{m}$ is the sum of
the optical gap of the lightly doped material $E_{0}$, plus that due to
filling of the conduction band due to Is $\Delta E_{BM}$, I. E. $E_{m}=E_{0}+E_{BM}$.
Thc second phenomenon which affects the optical absorption edge with
increasing donor density is due to a change in the nature and strength
of the interaction potentials between donors and the host crystal.
This latter effect gives rise to a band-gap shrinkage and to some
increased tailing of the absorption edge. In this case the measured
optical gap is $E_{m}=E_{0}+E_{BM}-\Delta E_{g}$., where $\Delta E_{g}$ is the gap shrinkage.</p>
</blockquote>
<p>I didnt understand the explanation given in second phenomenon. What is author meant by due <strong><em>to a change in the nature and strength of the interaction potentials between donors and the host crystal</em></strong>.</p>
<p>Advance thanks for your help</p>
| 3,386 |
<p>I ask this question, because at the end of this long day I'm just too dazed to derive the proofs myself (even though I know that I should feel ashamed for this). </p>
<p>So, the question: </p>
<p>Given two <a href="http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29#Simple_gravity_pendulum" rel="nofollow">simple gravity pendulums</a>, attached to the same hinge point. Both masses and lengths of the strings (or rods) are equal. The only difference between the two is that the motion of one pendulum takes place in a plane perpendicular to the motion of the other. </p>
<p>So, supposing that initial conditions are chosen such that these pendulums never collide, can the position of the center of mass of both pendulums be described by the motion of a single pendulum with twice the mass, the motion of which is not restricted to any single plane? </p>
<p>Intuitively, I'd say: perhaps only for small angles. For example, I can easily imagine one pendulum going one-way (spinning a full 2$\pi$ per period), while the other is at near-standstill. Obviously, the motion of the COM can <em>not</em> be described by a single pendulum without significant, non-physical changes to that model (the periods of both are unequal in that case, generally meaning the center of mass moves along non-periodic Lissajous patterns in $x$,$y$,$z$.). </p>
<p>So, what I'm really asking is: under what conditions is this possible, if at all? What are the constraints and/or adjustments you'd need to make to this "replacement pendulum" model for it to work? </p>
<p>And how does this all translate to <a href="http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29#Compound_pendulum" rel="nofollow">physical pendulums</a>?</p>
| 3,387 |
<p>I was wondering what is the group theoretic way to find the resulting charges of matter fields after a scalar field is given a vev.</p>
<p>In the case of the EW symmetry breaking, one can directly read the charges from the Lagrangian by setting the Higgs field $H=v+h'$ and going in the unitary gauge.</p>
<p>Given a gauge group $\mathcal{G}$, a set of field with their charges under that group. What is the way to find the charges if I give a vev to $n$ fields under the remnant group $\mathcal{G}_\text{br.}$. This is a priori totally unrelated to any Lagrangian and should have a purely group theoretic answer.</p>
<p>A simple example would be $\mathcal{G}=U(1)^k$ with $m$ fields. If I give a vev to $n$ of them, we'll have $U(1)^k\to U(1)^{k-n}$ (assuming the $n$ fields have linearly independent charges). My problem is that I can't find how to get the charges.</p>
<p>I would also be interested in the non-abelian case, and with not only scalar fields but other spin in the spectrum. Any references would also be very welcome!</p>
| 3,388 |
<p>Among the other ways, one way to prove the Earth is round is the <strong>triple-right triangle</strong>.</p>
<p>The idea is simple: </p>
<ol>
<li>Starting from point A you move in a straight line for a certain distance.</li>
<li>At point B, turn right 90° degrees, move along the line for the same distance.</li>
<li>At point C, turn again to the right and do the same.</li>
<li>We'll eventually get back at the starting point: point A and C are the same location, thus we just created a triangle with 90° degrees.</li>
</ol>
<p>This proves that that Earth has a spherical shape (not a perfect sphere), since these movements would only create a square with three sides if we were to do it on a flat surface.</p>
<p>However, the "problem" of this experiment is that it's not really doable on a small scale. The distance must be so much that the curve of our planet can be taken into consideration. Walking 1 meter, then one meter and then another meter won't create a triangle, since the curve of the planet is not that strong.</p>
<p>So my question is: what's the minimum distance we'd need to travel for this experiment to work?</p>
| 3,389 |
<p>When two balls of different masses, thrown from equal height they reaches the ground at the same time. Can anyone explain this in terms of laws of Physics(or with mathematical equations)?</p>
| 85 |
<p>What is the output of a CNOT gate if both inputs are in superposition?</p>
<p><img src="http://i.stack.imgur.com/GF1HJ.png" alt="CNOT gate"></p>
<p>For example, what happens if:
$\left|x\right>=\alpha_x\left|0\right>+\beta_x\left|1\right>$
and
$\left|y\right>=\alpha_y\left|0\right>+\beta_y\left|1\right>$. Note that the $\alpha$s and $\beta$s can have imaginary parts.</p>
<p>For another example, if:</p>
<p>$$\begin{gather}
\alpha_x=0.6\times e^{i\theta_1} \\
\beta_x=0.8\times e^{i\theta_2} \\
\alpha_y = \frac{\sqrt3}{3}\times e^{i\theta_3} \\
\beta_y = \frac{\sqrt6}{3}\times e^{i\theta_4}
\end{gather}$$</p>
<p>then what is $\left|x\oplus y\right>$?</p>
| 3,390 |
<p>I'm a high school student. I'll start self learning of physics and maths. I want to be theoretical physicist. I'm especially interested in gravitational physics. I wonder which website advice below is better to follow? Why? </p>
<p><a href="http://www.staff.science.uu.nl/~Gadda001/goodtheorist/index.html" rel="nofollow">http://www.staff.science.uu.nl/~Gadda001/goodtheorist/index.html</a></p>
<p><a href="http://math.ucr.edu/home/baez/books.html" rel="nofollow">http://math.ucr.edu/home/baez/books.html</a></p>
| 86 |
<p>I can't really imagine the way the magnetic field would be created due to electron flow in an inductor. We say that in a straight current carrying conductor the magnetic field follows the thumb rule. So according to that shouldn't the magnetic field get nullified inside the coil as<br>
every loop will of the coil will produce its own magnetic field. Also why does it take time to create a magnetic field once we supply current. I need a comprehensive and lucid explanation.</p>
| 3,391 |
<p>I tried to look up some facts regarding sunspots and its relation between the brightness of the sun, only to find information that are intriguing yet not what I am looking for.</p>
<p>My understanding is that sunspots are darker compared to the brightness around it, so I am thinking ... the more sunspots there are visible the brightness of the sun must be lowered. However, the book I am reading tells me that the brightest moment of the sun is related to having more sunspots. </p>
<p>I am not quite sure if this is even a worthy question, but can someone explain to me why this is true (or false) and how it works so that a middle schooler can understand? </p>
| 3,392 |
<p>Here is an example of Cassegrain telescope: Parallel rays from a distant object get reflected by the concave mirror forming an image at its focus behind the convex mirror. This image acts as a virtual object for the convex mirror, and it forms a real image in front. Let's apply mirror equation for this convex mirror to find the distance at which final image is formed.<img src="http://i.stack.imgur.com/e2yjI.gif" alt="enter image description here"></p>
<p>1/v + 1/u = 1/f</p>
<p>u = + 90; f= +70;</p>
<p>Therefore, 'v' comes out to be +315 cm. I am troubled to see this positive sign in image's distance, would not this mean that the image is formed to the left of convex mirror? Does sign convention fail in this case? Or have I made some fatal mistake?</p>
<p>The exact question statement goes like this:</p>
<p><img src="http://i.stack.imgur.com/c1JUa.jpg" alt="enter image description here"></p>
| 3,393 |
<p>I wanna become a scientist, so wanna improve my skill like problem solving etc. Could you suggest me some more skills which I need to improve.</p>
| 40 |
<p>I am not sure if I understand how forces work during hammerthrow. From my understanding athlete works with Fmuscle force on a ball and rotate it and a ball works with equal opposite force on an athlete causing rotation in a opposite direction. The mass of athlete is greater than the mass of ball so athlete rotates with ball in 1 direction(Fmuscle direction). Does it mean if ball would be heavier(more massive) than athlete, the athlete could not rotate it(even if strong enough to move it) ?
<img src="http://i.stack.imgur.com/VdTcY.png" alt="enter image description here"></p>
| 3,394 |
<p>Let's imagine a boat on a lake. Observer A is sitting on the shore. Observer B is sitting in the boat on the bow. Observer B has a ball attached to the end of a string which he holds in his hand.</p>
<p>Observer A sees that the boat is not moving. Boat B has some mass. So does the ball. Let's ignore water/air friction.</p>
<p>Then observer B slowly moves towards stern. The boat moves appropriately in the opposite direction, as A can see. Then observer B stops, so does the boat. </p>
<p>Finally, the observer B goes back to the bow. Because of the law of conservation of momentum, in the end observer A sees the situation exactly as it was in the beginning.</p>
<p>Now let's repeat the situation, but before Observer B is coming back from the stern to the bow he starts spinning the ball on the string with sub-light speed.</p>
<p>Because of the speed, according to the special relativity, both observers should observe increased mass of the ball.</p>
<p>For observer A on the shore it would be like the observer in the boat carries a lightweight ball in one direction and then a heavy ball in the other direction.</p>
<p>In result, he would observe that the boat moved as compared to the initial position. This in turn would violate the momentum conservation law, so I conclude it is not going to happen.</p>
<p>Why? Isn't the increased mass of the spinning ball be perceived by observer A and B?</p>
| 3,395 |
<p>As in most tight-binding atomic model, the Hamiltonian $H$ is sparse and in my problem, it is a <a href="http://en.wikipedia.org/wiki/Banded_matrix" rel="nofollow">band matrix</a>.</p>
<p>How to compute the partition function $Z = Tr\left( {{e^{ - \beta \hat H}}} \right)$ efficiently since $H$ is so large.</p>
<p>For the two method coming into my mind, which one is better for my Hamiltonian of sparse huge band matrix?</p>
<ol>
<li><p>Diagonalize $H$ first, then use the eigenvalues to calculate $Z$</p></li>
<li><p>Find the exponential of $H$ first, then trace</p></li>
</ol>
| 3,396 |
<p>When a muon decays from rest, typically what fraction of the energy is carried off by the electron? I tried looking into some papers, but I wasn't sure how to interpret the graphs they displayed. </p>
| 3,397 |
<p>I was reading a paper that described how the force a low-thrust torsion pendulum was measured. In it, the paper states a laser is bounced off a mirror and the displacement is "...based upon the beam reflection time." The paper states that the device can measure sub-micrometer displacements.</p>
<p>Conceptually as I understand it, this measurement device would have 4 major components. A laser emitter, a mirror, a detector, and a controller. The controller would power on the laser, then note how long it takes for the detector to respond to the signal.</p>
<p>However, the time of flight difference for ranges this size are vanishingly small. For instance, it'd take a photon roughly $3.336×10^{-12}$ seconds to travel an additional 1mm along the beam path.</p>
<p>If you flip that over to cycles per second, that would suggest the controller has to operate at around ~300 GHz. Only then could it check the sensor often enough to have the temporal resolution to resolve a 1mm change in the beam path length.</p>
<p>This seems like an absurd clock speed for any sort of computer controller. Is there another component, or concept that I'm missing?</p>
| 3,398 |
<p>I once read a paper, in which:</p>
<ul>
<li>a fluid in a container was heated from below,</li>
<li>after reaching temperature $T_1$, a circular motion (convection) was clearly distinguishable, in form of <strong>cylinder</strong>,</li>
<li>after reaching temperature $T_2$, the circular motion splitted into <strong>two</strong> circular convections, side by side two cylinders,</li>
<li>after reaching temperatures $T_3, T_4, \ldots$, etc. <strong>waves</strong> appeared on cyliders' flat-sides (where the height is measured),</li>
<li>the frequency of the waves doubled at temperatures $T_4, T_5, \ldots$.</li>
</ul>
<p>Ratio of every temperature pair $T_1/T_2, T_2/T_3, \ldots$ was the same as in logistic map ratios of parameter $r$, and in every other map - the Feigenbaum number, the <em>period-doubling route to chaos</em>.</p>
<p>I cannot find the paper again.. Does anyone remember such paper? Or maybe other?</p>
| 3,399 |
<p>I'm looking for an introduction to the treatment of <a href="http://en.wikipedia.org/wiki/Piezoelectricity" rel="nofollow">piezoelectricity</a>, specially in semiconductors emphasizing the dependence on the orientation of the unit cell and the interactive effects with the electron-holes pairs.</p>
| 3,400 |
<p>What is actually a resonating vibration and <a href="http://en.wikipedia.org/wiki/Resonance" rel="nofollow">resonance</a>?</p>
<p>I have searched many books and made Google search too but couldn't understand it clearly.</p>
| 3,401 |
<p>I have tried a google search and checked my condensed matter books but I can't find out what pulsed neutron diffraction is and how it differs from inelastic neutron scattering.</p>
| 3,402 |
<p><a href="https://en.wikipedia.org/wiki/Ionocraft" rel="nofollow">Ionocraft</a> or "lifters" are lightweight devices that produce thrust by ionizing the air around an electrode, and then accelerating the ions toward another electrode with an electric field, during which the ions push against neutral air molecules and produce thrust.</p>
<p><img src="http://i.stack.imgur.com/SO3MF.gif" alt="Ionocraft flying"></p>
<p>There <a href="https://en.wikipedia.org/w/index.php?title=Electrohydrodynamic_thruster&oldid=464338641" rel="nofollow">used to be a Wikipedia article for "Electrohydrodynamic thruster"</a>, which was deleted for not having any references. It said that the efficiency can be improved by separating the ionization and acceleration into stages, and using multiple acceleration stages:</p>
<blockquote>
<p>Ionocrafts form part of this category, but their energy conversion efficiency is severely limited to less than 1% by the fact that the ioniser and accelerating mechanisms are not independent. Unlike the ionocraft, within an EHD thruster, the air gap in its second stage is not restricted or related to the Corona discharge voltage of its ionising stage.</p>
<p>The first stage consists of a powerful air ioniser which, when supplied by high voltage in the kilovolt to megavolt range, ionises the intake air into ion clouds which flow into the second stage of the device. The second stage consists of one or multiple stages of ion accelerators, powered by voltages in the kilovolt or megavolt range, in which the ionised fluid is moved on a straight path along the length of the accelerating unit. </p>
</blockquote>
<p>Are the claims in this article realistic/true? Know any references for it? Is this well-known under a different name?</p>
<p>Sounds like the same principle as <a href="https://en.wikipedia.org/wiki/Electrostatic_fluid_accelerator" rel="nofollow">Electrostatic fluid accelerators</a>? But that article says "[Generating thrust] relies on the same electrodes and electric field as the corona [ionization] process."</p>
| 3,403 |
<p>In high pressure physics what is the difference between <a href="http://en.wikipedia.org/wiki/Diamond_anvil_cell" rel="nofollow">diamond anvil cells</a> (DAC) and Bridgman cells? My understanding is that they are both forms of anvil cell but in the scientific literature I'm reading they are referred to separately.</p>
| 3,404 |
<p>Just from reading <a href="http://rads.stackoverflow.com/amzn/click/0071636412" rel="nofollow">this</a> slightly technical introduction to supersymmetry and watching <a href="http://itunes.apple.com/itunes-u/supersymmetry-grand-unification/id384233338#ls=1" rel="nofollow">these</a> Lenny Susskind lectures, I thought that the Lagrangian of any "reasonable" supersymmetric theory can always by derived from the superfield formalism; such that the F term of the superpotential contains the mass and interaction terms and the D term of $\Phi^{\dagger}\Phi$ describes the kinetic terms or free part.</p>
<p>But then I read in <a href="http://motls.blogspot.com/2012/03/35-years-of-nnn4-yang-mills-theory.html" rel="nofollow">this</a> article (with the main topic about N=4 SYM theories) as an "aside note", that there exists no (known) superspace for D=10, N=1 YM theories described by the Lagrangian</p>
<p>\begin{equation}
L = \mathrm{tr} \left[ -\frac{1}{4}F_{\mu\nu} F^{\mu\nu} + i\bar{\Psi}D^{\mu}\gamma_{\mu}\Psi \right]
\end{equation}</p>
<p>for example.</p>
<p>My questions now are:</p>
<p>Is there an "easy" or "intuitive" (meaning such that I can get it :-P...) way to understand why in this case there is no (known) superspace for this theory (such that the Lagrangian can not be derived from the methods mentioned above?). Or more generally what are the limitations of the superspace formalism; for which kind of theories does it work and under what conditions is it not applicable? </p>
| 3,405 |
<p>When experiencing alpha decay, atoms shed alpha particles made of 2 protons and 2 neutrons. Why can't we have other types of particles made of more or less protons?</p>
| 3,406 |
<p>An electric field in a conductor causes charges to redistribute so as to cancel out the original field, bringing the field to zero. This is, I think, a common argument for why conductors are generally opaque to EM waves.</p>
<p>But some conductors are transparent, including various electrolytes and indium tin oxide in LCD displays. What are the mechanisms by which conductors can be transparent?</p>
| 3,407 |
<p>I was going through Mark Srednicki's book on QFT. It says in the relativistic limit the Schrodinger equation becomes something like :</p>
<p>$$ i\hbar\frac{\partial}{\partial t} \psi(\vec x,t) = \sqrt{-\hbar^2c^2\nabla^2+m^2c^4}\psi(\vec x,t) $$
Now he says that if I expand the square root (say binomially) it will have infinite no. of spatial derivatives acting on $\psi(x,t)$; this implies that
equation is not local in space.</p>
<p>What exactly does it mean to say the equation is not local in space?</p>
| 3,408 |
<p>Apart from other reasons, recently my interest in this area got piqued when I heard an awesome lecture by Seiberg on the idea of meta-stable-supersymmetry-breaking. </p>
<p>I am looking for references on learning about phase transitions/critical phenomenon in supersymmetric field theory - may be especially in the context of $\cal{N}=4$ SYM. </p>
<p>It would be great if along with the reference you can also drop in a few lines about what is <em>the</em> point about this line of research. </p>
<p>To start off,</p>
<ul>
<li><p><a href="http://arxiv.org/PS_cache/hep-th/pdf/0303/0303207v1.pdf">this one by Witten, Cachazo and Seiberg</a> and <a href="http://arxiv.org/PS_cache/hep-th/pdf/0612/0612073v2.pdf">this one by Gukov and Witten</a>. </p></li>
<li><p><a href="http://web.mit.edu/~mcgreevy/www/fall08/handouts/lecture25.pdf">this lecture by McGreevy</a> and the references at its end.</p></li>
<li><p>may be <a href="http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.0945v3.pdf">this paper by Alday, Gaiotto, Tachikawa, Gukov and Verlinde</a> too..</p></li>
</ul>
<p>I would be very happy to be pointed to may be some more pedagogical/expository references about this theme of supersymmetric phase transitions. </p>
| 3,409 |
<p>I am working on an model of a permanent magnet synchronous machine. Right now I am stuck with calculating the eddy current losses caused by the harmonics of the stator magnetic field in the electrical steel of the rotor. Or to put it differently. How do I calculate the eddy current in electric steel at high frequencies and low flux density? </p>
<p>I would like to use something very simple like
$$P_{ec} = \sum\limits_\nu \sigma_{ec} (f(\nu-1))^2 B^2_\nu m_\nu$$
were $P_{ec}$ are the eddy current losses in watts. $\sigma_{ec}$ are the specific eddy current losses for the material, but I don't know if they are any good for frequencies above 2000Hz and I would like to calculate the losses up to 100kHz if that is even possible. $f$ is the fundamental frequency an $\nu$ is the ordinal of the harmonics. $B_\nu$ is the respective flux density and $m_\nu$ is the mass. </p>
<p>Now there are two big questions. I know the amplitude of the flux density in the air gap (above the surface) but how do I calculate the flux density in the lamination (electric steel: M330-35A). Apparently I have to consider the skin depth, but I have no values for the permeability at such high frequencies and comparably low flux densities. And also, how do I calculate the mass?
$$m_\nu = A \delta \rho$$
($A$ - surface of the rotor , $\delta$ - skin depth, $\rho$ - density of the electric steel)
If I take the same flux density as in the air gap and calculate the masses like above, I obtain losses that are so low, that I am pretty sure they can't be right.</p>
<p>Does anyone have an idea how to solve this problem by adjusting the described or with another approach. I don't need a 100% accurate result. If I am 50% off that is still ok. Any references to text books or papers are also very much appreciated. </p>
| 3,410 |
<p>Is it only the spin of a particle that can be entangled with another particles spin?
Also is there any good physical interpretation of the spin of a particle? because the rotational invariance of entaglement almost literally blows my mind.</p>
| 3,411 |
<p>In their celebrated work, Capelli Itzykson and Zuber established an
<a href="http://arxiv.org/abs/0911.3242"><i>ADE</i>-classification of modular invariant <i>CFT</i>s</a> with chiral algebra $\mathfrak{su}(2)_k$.</p>
<blockquote>
<p>How much of that classification can one see using the tools of perturbative quantum field theory?</p>
</blockquote>
<p>Presumably, one can't see the exceptional $E_6$, $E_7$, $E_8$ family...<br> what about the $A_n$ versus $D_n$ families?</p>
| 3,412 |
<p>I'm just trying to understand this problem from a qualitative perspective. The Doppler effect is commonly explained in terms of how a siren sounds higher in pitch as it is approaching a particular observer. I understand this is because the velocity of the wave is constant and so the frequency of the waves increase as they are bunched together. What would happen if a siren was mounted on say a plane traveling at a supersonic speed? To clarify what would the observer observe/hear? Apologies if my question is not phrase very well my knowledge of physics is very rudimentary.</p>
| 3,413 |
<p>I'm looking for texts about topics in string theory that are "advanced" in the sense that they go beyond perturbative string theory. Specifically I'm interested in</p>
<ol>
<li>String field theory (including superstrings and closed strings)</li>
<li>D-branes and other branes (like the NS5)</li>
<li>Dualities</li>
<li>M-theory</li>
<li>AdS/CFT</li>
<li>Matrix theory</li>
<li>F-theory</li>
<li>Topological string theory</li>
<li>Little string theory</li>
</ol>
<p>I'm not interested (at the moment) at string phenomenology and cosmology</p>
<p>I prefer texts which are (in this order of importance)</p>
<ul>
<li>Top-down i.e. would give the conceptual reason for something first instead of presenting it as an ad-hoc construction, even if the conceptual reason is complicated</li>
<li>Use the most appropriate mathematical language / abstractions (I'm not afraid of highbrow mathematics)</li>
<li>Up to date with recent developments</li>
</ul>
| 3,414 |
<p>When considering observables and their corresponding operators, would it be correct to believe that discerning discrete values for an observable is possible ONLY when $\psi$ is an eigenfunction of the operator? Alternatively, would it also be correct to believe that the average value of an observable is ALWAYS obtainable regardless if $\psi$ is an eigenfunction of the operator? </p>
<p>Thanks for your help. </p>
| 3,415 |
<p>I am currently facing the problem of calculating integrals that take the general form</p>
<p>$\int_{R} P(\sigma)d\sigma$</p>
<p>where $P(\sigma)$ is a probability density over the space of mixed quantum states, $d\sigma$ is the Hilbert-Schmidt measure and $R$ is <em>some subregion</em> of state space, which in general can be quite complicated.</p>
<p>Effectively, this can be thought of as a multivariate integral for which Monte Carlo integration techniques are particularly well suited. However, I am new to this numerical technique and would like to have a better understanding of progress in this field before jumping in. So my question is:</p>
<p>Are there any algorithms for Monte Carlo integration that have been specifically constructed for functions of mixed quantum states? Ideally, have integrals of this form been studied before in any other context?</p>
| 3,416 |
<p>Right now, my understanding is that, a mixture of photons of many different frequencies is perceived as white by your eye. While no photons at all, is perceived as black. And photons with the blue frequency only cause you to see blue, etc.</p>
<p>My question is, how is the "brightness" controlled? I think it has to do with how much blue photons are coming at your eye, a low amount will be dark blue, a high amount will be... a lighter blue. But then I think, to get light blue, isn't it a mixture of mostly blue photons with white light (photons of all frequencies) to produce a blueish white or a light blue?</p>
<p>Also, when colors combine to produce different colors, is there any photon combining that exists or is it because your eyes see mixtures of photons and not photons themselves?</p>
| 3,417 |
<p>Graphene has two atoms in its primitive unit cell. This makes it intuitive to see that the tight binding Hamiltonian can be constructed as a $ 2 \times 2 $ matrix $H$ acting on a spinor $S$ that consists of the wavefunction from an atom in sublattice A and B.</p>
<p>$H_{monolayer}=\gamma \cdot \begin{pmatrix}
0 & k_x-ik_y \\
k_x+ik_y & 0
\end{pmatrix}$</p>
<p>$S_{monolayer}=\begin{pmatrix}
|\psi_A\rangle\\
|\psi_B\rangle
\end{pmatrix}$</p>
<p>Bilayer Graphene has four atoms in a primitive unit cell and its tight binding Hamiltonian is a 4x4 matrix whose matrix elements represent the hopping between said lattice sites (depending on how it is stacked and what hopping parameters you wish to involve in the calculation). An example might be as follows:</p>
<p>$H_{bilayer}=\begin{pmatrix}
0 & 0 & 0 & v(k_x-ik_y)\\
0 & 0 & v(k_x+ik_y) & 0\\
0 & v(k_x-ik_y) & 0 & \gamma'\\
v(k_x+ik_y) & 0 & \gamma' & 0
\end{pmatrix}$</p>
<p>$S_{bilayer}=\begin{pmatrix}
|\psi_{A1}\rangle\\
|\psi_{B2}\rangle\\
|\psi_{A2}\rangle\\
|\psi_{B1}\rangle
\end{pmatrix}$</p>
<p>Where the basis is chosen is in an arbitrary order (1 and 2 indices refer to layer number).</p>
<p><strong>How does one write this in a "two-component basis" and what does that mean?</strong> Also, what is this hamiltonian acting on in this case? The bilayer hamiltation in this basis (which I do not know what it represents) is written as follows:</p>
<p>$H'_{bilayer}=-\dfrac{\hbar^2}{2m}\begin{pmatrix}
0 & (k_x-ik_y)^2 \\
(k_x+ik_y)^2 & 0
\end{pmatrix}$</p>
| 3,418 |
<p>In <a href="https://archive.org/details/IntroductionToQuantumMechanics_718" rel="nofollow">Griffiths</a>' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, we have: $$a_{j+2}\approx\frac{2}{j}\,a_j.$$ Up to this point I totally agree (one just takes the limit). </p>
<p>However, the subsequent derivation of solution $a_{j}$ and $h(\xi)$ I attached from the Griffiths' textbook are very confusing. </p>
<ul>
<li>How did it go from $a_{j+2}\approx\frac{2}{j}a_j$ to the solution of $a_{j}$? </li>
<li>Also, how do the second and third approximations work in $h(\xi)$?</li>
</ul>
<p>My questions are mainly mathematical. I very much hope someone can provide a derivation or refer a link where these questions may already be answered.</p>
<p><img src="http://i.stack.imgur.com/qklld.jpg" alt="enter image description here">
<img src="http://i.stack.imgur.com/wBTkg.jpg" alt="enter image description here"> </p>
| 3,419 |
<p>I'm going mad about the problem.</p>
<p>I really don't understand why do electron have 1/2 spin number, why they are not actually spinning.</p>
<p>I can accept that the electrons have their own magnetic field, which is certain, but why do they have $\hbar\sqrt3/2$ of angular momentum, and I don't know what the heck is spin number.</p>
<p>I've read the definition of <a href="http://en.wikipedia.org/wiki/Spin_%28physics%29" rel="nofollow">spin</a> and <a href="http://en.wikipedia.org/wiki/Spin_quantum_number" rel="nofollow">spin quantum number</a> more than a hundred times but there is no betterment. I've smashed my head in my desk more than a hundred times either.</p>
<p>My question is the title. Why can't I just think the spin as rotating?</p>
<p>What I've saw recently,</p>
<blockquote>
<p><em>electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum.</em></p>
</blockquote>
| 3,420 |
<p><a href="http://physics.stackexchange.com/questions/87123/equation-for-relativistic-electron-and-two-component-spinor">Here</a> I asked about getting an equation for two-component spinor as the alternative for Dirac equation. It was found that it is called Majorana equation. It may be easily derived by using historical Dirac's method of deriving homonymous equation, but with using the requirement of the absence invariance of the Lagrangian under U(1) transformations. But in the linked question I also wrote about the absense of invariance of spinor $\left(\frac{1}{2}, 0 \right)$ representation of the Lorentz group under discrete transformations of the Lorentz group. If these symmetries is absent, why do we need to analyze this equation?</p>
| 3,421 |
<p>I'm not quite sure this question fits the format of this site but I try to word it the best I can to comply the rules.</p>
<p>The question is simple: How far can we go talking about the origin of the universe before admitting that the initial conditions cannot be explained without postulating some kind of god-like, physics-unexplainable, force/whatever?</p>
<p>I'm interested to know if there's a mainstream physics theory that aims to prove this. </p>
| 51 |
<p>How does this number get calculated?</p>
<blockquote>
<p>About 380,000 years after the Big Bang the temperature of the universe fell to the point where nuclei could combine with electrons to create neutral atoms.</p>
</blockquote>
<p><a href="http://en.wikipedia.org/wiki/Photon_epoch">http://en.wikipedia.org/wiki/Photon_epoch</a></p>
<p>I've seen it in many places (or something close to that), but there is never a citation or any explanation.</p>
| 3,422 |
<p>I am just starting to dig a little deeper into particle interactions, and just have an introductory college physics background (no quantum mechanics). But I am interested in the conditions of the early universe and am looking for a list of all the particles in the standard model, and the relationships between the particles, so you can see which particles turn into other particles under which conditions, and stuff like that.</p>
<p>Is there any such a thing? Or if not, it would be really cool to see a list of all the particles in the early universe and the different types of interactions they had with each other. Such as:</p>
<ul>
<li>What happens when a quark + antiquark come close together? What if they collide?</li>
<li>What happens when 2 gluons collide? Come close together?</li>
<li>What happens when 1 quark and 1 gluon come together?</li>
<li>What happens when 2 photons collide (can they collide?)?</li>
<li>What particles radiate photons? Under what conditions?</li>
<li>...</li>
</ul>
<p>Does such a resource exist? Or if not, where would you go about finding that?</p>
| 3,423 |
<p>When talking about the wave-particle duality, teachers and books say that when you send a single photon through a slit, it makes a wave pattern. But if you send that particle through the slit and "you observe it directly", then it appears as a single point (a particle).</p>
<p>What is meant by "observe"? Is that like with your eyeballs? Or is that with some measuring device? It's unclear what that word means, it could mean anything haha.</p>
<p>One reason I am wondering is because, it seems like the act of trying to measure it directly (if "measure with a device" is what is meant by observe) would mean sending out some sort of radiation or particle itself, so yes, that would mess with the experiment. But maybe I just don't quite understand yet, so looking forward to a bit of clarification.</p>
| 3,424 |
<p>I am quite puzzled with the problem that spectral analysis has been either applied to noisy dynamical systems or to chaotic ones. I was wondering why nobody makes analysis of non-linear dynamical systems based on their autocorrelation? At least for non-linear oscillators which are essentially periodic and seem to suit for such analyses.</p>
<p>For example I simulated Van der Pol oscillator for 500 seconds from initial condition $(1.1,0.1)$. The 2D ODE is as follows:
$$
\frac{d\textbf{x}}{dt}=
\begin{cases}
\mu(x_0-1/3 x_0^3-x_1)\\
\frac{x_0}{\mu}
\end{cases}
$$
Where I have set the $\mu$ to $5$. The plot of the oscillator and the autocorrelation function as defined by Wiener are as follows:
<img src="http://i.stack.imgur.com/i3AnX.png" alt="enter image description here">
<img src="http://i.stack.imgur.com/ffyf6.png" alt="enter image description here"></p>
<p>Sorry for the lack of labels. The $x$-label in the second plot is representing lag in seconds and the $y$ axis is $C(\tau)=\frac{1}{T}\lim\limits_{T\to \infty}\int^T_{-T}x_0(t)x_0(t+\tau)dt$.</p>
<p>What would be wrong with such analysis when the autocorrelation function exists? Is this a totally dumb question??</p>
| 3,425 |
<p>Does anybody know the status of the problem to define the wave function (non-relativistic Quantum Mechanics) of a particle localized at a definite point? </p>
<p>Landau-Lifshitz says in chapter 1 that this function is $\Psi(x)_{x_o} = \delta(x-x_0)$ and gives an explanation that it produces the correct probability density when it is used to span some other arbitrary wave function $\Psi(x)$. The problem is of course that the wave function given above squares to a non integrable function. As far as I know this problem is unsolved. My question is if anybody knows the status quo of this problem. I am sorry if this question may be duplicated, I could not find it amongst the answered questions. </p>
| 3,426 |
<p>I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.</p>
<p>Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by <a href="http://en.wikipedia.org/wiki/Fresnel_lens#Solar_power">this Wikipedia article</a>:</p>
<blockquote>
<p>New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells.</p>
</blockquote>
| 3,427 |
<p>I know that when electrons encounter photons, they become excited and move to an orbit farther away from the nucleus of an atom as a result. What I want to know is exactly why the photons <em>cause</em> the electrons to enter this state.</p>
<p>Edit: Sorry, I wasn't being very clear. What I mean is, why do photons interact with electrons?</p>
| 3,428 |
<p>Have you ever noticed that when you are filling a container with fluid. As it approaches the top, it makes a different sound? You can tell by listening when your about to reach the top. Why is this?</p>
| 3,429 |
<p>I was wondering whether there exists a theory that describes Mie Scattering for spheres that have a constant dipole moment. Since there are theories that describe Mie scattering in the case of a charge, this might exist too?</p>
| 3,430 |
<p>Just starting to learn physics. Now reading about constant motion and after this paragraph
I have a task:</p>
<p>Boy with growth of 1.5 m runs at a speed of 3 m / s in a straight line, passing under a street lamp, hanging at a height of 3 m. Show that the shadow of his head moves uniformly, and find the speed of the movement.</p>
<p>How can I know the height of his shadow to solve it?</p>
| 3,431 |
<p>The proof of the No-Cloning Theorem states "By the linearity of quantum mechanics, ..." -- Could someone please give me a rough sketch/outline of what this means. Does it have to do with the Hilbert Space that wave functions live in? </p>
<p>I apologize if this question isn't specific enough, I just wanted to fully understand this concept. </p>
| 3,432 |
<p>Consider a container with some fluid of density $\rho_l$ and volume $V_l$. This is kept on a measuring device and has weight $\rho_l V_lg$. Now, consider a block of density $\rho_b$ and volume $V_b$. This block is put into the fluid and here, its apparent weight equals $\rho_b V_bg - F_b$ , where $F_b$ equals the buoyant force which equals $V_b\rho_lg$. </p>
<p>Therefore, the apparent weight of block equals $gV_b(\rho_b - \rho_l)$. </p>
<p>What happens if you take the weight of this whole apparatus? Will it equal $$gV_b(\rho_b - \rho_l) + \rho_lV_lg,$$ or $$\rho_bV_bg + \rho_lV_lg?$$</p>
| 3,433 |
<p>Can anybody cast some physical insight into this? I've been studying differential equations on my own and don't understand how you can have a whole host of general solutions. It seems like a rather curious situation which we don't come across in other areas of mathematics. Is there anything more to the discussion that I'm missing? </p>
| 3,434 |
<p>According to wikipedia, here are the Cygnus X-1 vital stats:</p>
<pre><code>Mass 14-16[7] M☉
Radius 20–22[8] R☉
</code></pre>
<p>A radius of 10 R☉ means a volume of 10^3 = 1000 Sols. Divided by 16 M☉ that means that Cyg X-1 is 60 time <em>less</em> dense than Sol. So how could it be a black hole?</p>
| 3,435 |
<p>The equations of motions for a simple pendulum is given by </p>
<p>$$\ddot{\theta} ~=~ -\frac{g}{\ell}\sin(\theta),$$</p>
<p>where $g$ is acceleration due to gravity and $\ell$ is the length of the pendulum's string. Notice that the differential equation is of second order, does this mean that if I solve this equation numerically, the numbers that I get refers to the change in the velocity of the pendulum?</p>
| 3,436 |
<p>I had self-studied Griffiths(~ 4 chapters), and Sakurai (~2.5 chapters) for quantum mechanics some months ago. Now, I have to take a course in QM this sem, and I want to further my understanding of basic QM. I am looking for a textbook that is more advanced than the one mentioned above, possibly more mathematically inclined, and that will give me a different perspective. In particular I am looking for a book that is more modern and discusses QM, through examples in modern theoretical physics. Also, I have been trying to study QFT, but I think I lack in my understanding of QM, and don't have enough intuition (I don't know if it is possible to develop physical intuition in QM. But I at least want to develop some kind of mathematical intuition). </p>
<p>One book that comes to mind is <a href="http://rads.stackoverflow.com/amzn/click/9810241054" rel="nofollow">Ballentine - Modern QM</a>. But I went through it, and I didn't find it that detailed. So, it would be great if you could recommend some other book. </p>
<p>I also know about Cohen-tannoudji et al, and I don't quite like it. </p>
<p><strong>Some online concise notes, that give a different perspective would also do.</strong> </p>
| 87 |
<p>How can two seas not mix? I think this is commonly known and the explanation everyone gives is "because they have different densities".</p>
<p><img src="http://i.stack.imgur.com/KpmVf.jpg" alt="enter image description here"></p>
<p>What I get is that they eventually will mix, but this process takes a long time.</p>
<p>From what you see in this picture you can see that they have a clear separation line as if you would mix water and oil.</p>
<p>Basically what I'm skeptical about is the clear separation line between them. Putting highly salted water and normal water in the same bowl will cause almost instant mixing.
Can you get the same effect as shown in the picture in a bowl at home ?</p>
<p>I'm looking for a more complete answer than just that they have different densities.
Thanks.</p>
<p>EDIT: Looking more on the "density" hipothesis I also found <a href="http://www.wimp.com/chemicalreaction/">this</a> which I found interesting :)</p>
| 3,437 |
<p>Schwinger has on his grave (it seems) the relation between the g-factor of the electron and the fine structure constant:</p>
<p>$$g~=~2+\frac{\alpha}{\pi}+{\cal O}(\alpha^2)$$</p>
<p>Did Schwinger or somebody else ever give a simple explanation for the second term of the right hand side? The 2 appears from Dirac's equation. The second term is due to the emission and absorption of a photon. Is there a simple way to see that this process leads to the expression $\frac{\alpha}{\pi}$?</p>
| 3,438 |
<p>I've been thinking about refrigeration technology and am a bit confused about two common answers. Specifically, the part where the expansion valve releases the pressurized fluid and stuff gets real cold.</p>
<p>One is that refrigeration works by lower pressure = lower temperature. This makes sense to me because if there is lower pressure, I can imagine it as the opposite of higher pressure = higher temperature. Less pressure means particles are freer to move apart and thus eventually boil and lose energy as they travel further away from each other.</p>
<p>The other is the enthalpy of vaporization, which I understand as meaning that some amount of energy is required for a phase change. This also makes sense to me: when the refrigerant enters the low pressure side of the valve, the particles are more free to spread apart, begin to boil, and thus suck up surrounding energy to break apart from each other. Although, it seems a bit more magical than the lower pressure = lower temperature explanation (it seems odd that already hot liquid particles "suck up" more heat).</p>
<p>Could someone please help me understand this better? Thanks!</p>
<p>NOTE: As you can see, I think of this in very layman terms and am currently reading books like Feynman's lectures. My background is engineering, not physics, so I tend to understand things best in a much more physical-visualization, implementation-details kind of way.</p>
| 3,439 |
<p>I'm just curious highschooler beginning an interest in electronics and this concept of light detecting resistance component is really intriguing to me. I assume it's an application of the Photoelectric effect but how exactly does a CdS <a href="http://en.wikipedia.org/wiki/Photoresistor" rel="nofollow">photoresistor</a> (aka LDR) do what it does.</p>
<p><img src="http://i.stack.imgur.com/yIxZY.jpg" alt="Common CdS LDR"></p>
<p>How can two metal wires and and a zig zag of cadmium on silicon create a light sensitive resistor? What is happening here?</p>
<p>How does a common Photocell really work?</p>
<p>Edit: I've read the wikipedia page and a whole lot of other sources but I wish for a more technical and in depth of the mechanism. Thank you :D</p>
<p>My biggest question is why CdS... why not anything else?</p>
| 3,440 |
<p>Imagine driving in a straight line on a ice lake, when you hit the brakes, if your goal is to stay in straight path with no spinout, which wheels would you choose to have locked: front or rear? Assuming the steering wheel is kept fixed in both cases, I learned that it's better to have front wheels locked in this case. But can someone explain in accurate terms why? Thanks.</p>
| 3,441 |
<p>I'm confused about the physical interpretation of the four-velocity $U^\mu=\frac{dx^\mu}{d\tau}$ in General Relativity. I know that it is a tangent vector to a particle's "worldline", but what does this mean more physically?</p>
<p>For example, I am comfortable with what $U^\mu$ means in Special Relativity. In your inertial frame, you cover a distance $\Delta x^\mu$ and your clock says time $\Delta \tau$ has passed, and by taking the limit as $\Delta \tau \to 0$ this defines your $U^\mu$.</p>
<p>But I'm unsure about what $U^\mu$ means in curved space, or even in an accelerated reference frame. In either case the frame is no longer an inertial frame, which makes it confusing to interpret $\tau$, because it's no longer the "proper time in a frame", there is no one frame we are working in. </p>
| 3,442 |
<ol>
<li><p>First of all, why wasn't the sextant ever used for land navigation? The horizon is easier to see at sea, but land based sextants could be used in conjunction with artificial horizons (as at sea when horizon is hidden by fog).</p></li>
<li><p>Parallax has been used by both the US army and navy to measure distance to targets. The devices that used this principle were called <a href="http://en.wikipedia.org/wiki/Coincidence_rangefinder" rel="nofollow">coincidence rangefinders</a>. It seems this system was still used after the introduction of radar. Why was this system eventually phased out?</p></li>
<li><p>Finally my main question: How can one calculate the error for measurements made by sextants and parallax devices?</p></li>
</ol>
| 3,443 |
<p>The use of renormalisation constants often puzzles me. A good example is the use of $Z_2$ in the equation (7.58) of Peskin Schroeder. $Z_2$ is defined in equation (7.26).
as $Z_2^{-1} = 1-\frac{d\Sigma}{dp}$. Later in equation (7.31) it is said: </p>
<p>$Z_2-1 = d\Sigma/dp$ although this term is supposed to be infinite. But $d\Sigma/dp$ is treated of being smaller than 1. Okay, in this example the Pauli-Villars renormalisation is used where a rather high $\Lambda$ is needed to make $d\Sigma/dp$ larger than 1. But what would be if $Z_2$ were computed with dimensional regularization ?</p>
<p>Shouldn't be at least : $d\Sigma/dp + (d\Sigma/dp)^2 + (d\Sigma/dp)^3 + \dots $</p>
<p>I know that $d\Sigma/dp$ is of order $\alpha$ and there should be counter term to make the sum of both small (to order $\alpha$). On the other hand I am almost sure that when the counter terms of the next order $\alpha^2$ are calculated that it is already forgotten that there was also a term $(d\Sigma/dp)^2$ which also need a counter term and for $\alpha^3$ order again and so on.
Could somebody explain it to me ? Thank you. </p>
| 3,444 |
<p>We are trying to emulate <a href="http://www.dailymotion.com/video/xv1j0n_the-secret-life-of-chaos-2010_shortfilms&start=2091" rel="nofollow">the chaotic system Jim Al-Khalili demonstrate</a> (3 min video).</p>
<p>In our chaos lab, we are trying to research the chaotic system shown in the video.
We are using just a webcam and a regular PC monitor. Our goal is to build a bifurcation tree for this system and to show how by changing the parameters of the system (location and distance of the camera from the monitor, time delay (between capturing and showing what is captured on the screen)) we can see a transition from "order" to chaos.</p>
<p>The problem is we are not sure what exactly is the mathematical representation of the system, and what is the thing that is "doubling" and going to chaos (the y axis of a bifurcation tree - like voltage peaks on the diode in a chaotic RLD system).</p>
<p>How do we approach this subject (if it's even possible)?</p>
| 3,445 |
<p>Consider a curved space, e.g. Schwarzschild:
\begin{align*}
ds^2 = -\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2
\end{align*}
Now, the energy of a photon is $E = \hbar \omega$, and $|\mathbf{k}|= \frac{2\pi}{\lambda}$, but am I correct in assuming that $\omega \neq |\mathbf{k}|$?</p>
<p>Because if $k^\mu = (\omega,\mathbf{k})$ then $k_\mu k^\mu = 0$ implies that:
\begin{align*}
g_{tt} \omega^2 + g_{rr}(k^1)^2+g_{\theta\theta}(k^2)^2+g_{\phi\phi}(k^3)^2 =0
\end{align*}
So basically, is it correct that the relationship between $\omega$ and $|\mathbf{k}|$ will vary in curved space? (And so relationships like $E = \frac{h}{\lambda}$ no longer hold?)</p>
| 3,446 |
<p>If you think of a one-parameter group of transformations <a href="http://books.google.ie/books?id=2mc9e04FzMgC&lpg=PA36&ots=Pxfs8GcFvW&dq=bluman%20symmetry%20and%20integration%20%22if,%20in%20addition%20to%20satisfying%20axioms%22&hl=vi&pg=PA37#v=onepage&q=bluman%20symmetry%20and%20integration%20%22if,%20in%20addition%20to%20satisfying%20axioms%22&f=false" rel="nofollow">along a curve</a> in the plane as a (Lie) group, and the tangent vector to the curve as a generator of the curve we can intuitively understand <a href="http://en.wikipedia.org/wiki/Lie%27s_third_theorem" rel="nofollow">Lie's Third Theorem</a> on how a local Lie algebra generates a global Lie group.</p>
<p>Does this theorem apply for semi-groups? More specifically, does it apply to Wilson's Renormalization group, in other words do the <a href="http://journals.aps.org/prb/pdf/10.1103/PhysRevB.4.3174" rel="nofollow">renormalization group equations</a> generate a curve? </p>
<p>(It seems as thought they generate the flow of the coupling parameter</p>
<p><img src="http://i.stack.imgur.com/S9xLD.png" alt="enter image description here"></p>
<p>so if so - is that all it is, a mathematical irrelevancy or is there physical meaning?)</p>
<p>I ask to understand whether one can give Wilson's renormalization (semi) group, in the context of the statistical-mechanical partition function, any kind of similar interpretation. If it helps, refer to the <a href="http://xbeams.chem.yale.edu/~batista/vaa/node41.html" rel="nofollow">1-dimensional Ising model</a> as an example, thank you.</p>
<p>References </p>
<ol>
<li>"Renormalization Group Theory", Yale 430b/530b "Statistical Methods and Thermodynamics" Lecture Notes <a href="http://xbeams.chem.yale.edu/~batista/vaa/" rel="nofollow">http://xbeams.chem.yale.edu/~batista/vaa/</a></li>
<li>Bluman "Symmetry and Integration Methods for Differential Equations" P. 37</li>
<li>Wilson, "Renormalization Group and the Kadanoff Scaling Picture", P. 3177</li>
<li>Grossco, "Solid State Physics" 2nd Ed. P. 764 (For the Flow Diagram)</li>
</ol>
| 3,447 |
<p>we already know that if we plot speed vs time of free falling object it will be
y=gx graph because we know the acceleration is gravity.</p>
<p>If it have air-resistance acceleration will be change right?.</p>
<p>So how we can plot the speed vs time graph of free falling object with air resistance ?</p>
<p>It have some equations or not.</p>
<p>because i want to plot it by using python. </p>
<p>Thanks. </p>
<p>sorry about my english.</p>
<p><img src="http://i.stack.imgur.com/cvlE5.jpg" alt="example graph from google"></p>
<p>This picture i saw from google but i want to make it by coding</p>
| 3,448 |
<p>Knowing some about thermodynamics and reactions, I do understand how it can be shown that a change is reversible. But irreversible? Why can't it be that a change that was deemed irreversible thousands of years ago via new changes perhaps developed by physicists or changes, processes and reactions from some other part of the world or space, a change that was deemed irreversible in the future can be shown to be reversible?</p>
<p>I think that for instance diseases that were deemed irreversible as science progressed, we could make changes that were priorly said to be irreversible, in fact reversible.</p>
| 3,449 |
<p>I believe I am missing something simple here. My question concerns flywheel energy storage.</p>
<p>Say we have stored some amount of energy in a spinning flywheel. The flywheel is attached to a generator. So the flywheel (and therefore the generator) would be spinning rapidly at first, and then gradually slow down as the rotational energy is converted into electrical energy. </p>
<p>It is my understanding that the induced voltage decreases as the angular velocity of the generator coil decreases. So how is it that we can get electricity at a constant voltage from this setup? Am I misunderstanding how the generator works?</p>
<p>Thanks in advance.</p>
| 3,450 |
<p>Does anyone have any suggestion on a handbook of instrumentation?
In particular <a href="http://en.wikipedia.org/wiki/Nuclear_Instrumentation_Module" rel="nofollow">NIM</a> instrumentation...
I would like an approach which will explain the techniques and how to use NIM modules.</p>
<p>In particluar I would like an approach from the point of view of the user, not the developer.
For instance, ''what does the "veto" do in a dual timer?'', ''How to use a discriminator for a simple trigger?'' and stuff like that.</p>
| 3,451 |
<p>I think this is an interesting question, to which I don't really know the answer to. (Also, not a homework question.)</p>
<p>Say you have an uncharged metal sphere constrained to move in the z-axis. There is a charged ring lying in the x-y plane centered at the origin. Two cases: 1) the diameter of the ring is larger than the sphere, so that the sphere can pass through the ring, and 2) the diameter of the ring is smaller than the sphere, so the sphere cannot pass through the ring but can touch it.</p>
<p>What are the equilibrium position(s) of the ball?</p>
<p>For case 1) it is obvious that the center, by symmetry, is an equilibrium point. But are there more? The complication arises because of the finite size of the sphere. As the sphere starts to pass through the ring, the charge of opposite sign to the ring is induced near the ring, but the angle is very shallow, so there is not much attractive force. On the other hand the induced charge of the same sign gets pushed to the far end of the sphere, which causes a strong repulsive force, so the sphere might get repelled as it enters the ring. I'm not sure whether this is correct though. It might very well turn out that there is only 1 equilibrium point... </p>
<p>So aside from solving the system exactly to find the equilibrium points, is there a way to argue how many there are, for the 2 cases, and where they are?</p>
| 3,452 |
<p>Can someone give a qualitative description on why the density of states for a two dimensional free electron gas is independent of energy while it is not in one and three dimensions? In one dimension it goes as $E^{-1/2}$ and in three $E^{1/2}$.</p>
| 3,453 |
<p><a href="http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics" rel="nofollow">Quoting Wikipedia</a>:</p>
<blockquote>
<p>In <a href="http://en.wikipedia.org/wiki/Statistical_mechanics" rel="nofollow">statistical mechanics</a>, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.</p>
</blockquote>
<ol>
<li><p>Is it possible to apply Maxwell–Boltzmann statistics to objects as large as nebulae; globular clusters or galaxies, <em>that is, treating stars as Maxwell-Boltzmann particles</em>; or even the universe as as whole, <em>treating galaxies or clusters of galaxies as Maxwell-Boltzmann particles</em>?</p></li>
<li><p>Can the Universe be considered in thermal equilibrium? Or does an expanding Universe imply non-equilibrium?</p></li>
</ol>
| 3,454 |
<p>The question follows from <a href="http://en.wikipedia.org/wiki/Xkcd" rel="nofollow">xkcd</a> cartoon <a href="http://www.xkcd.com/941/" rel="nofollow">"Depth Perception (941)"</a>. I've isolated the frames that describe the concept here.</p>
<p><img src="http://i.stack.imgur.com/vNeDp.png" alt="Credit: Randall Munroe, xkcd"></p>
<p>In words, one could theoretically point two cameras at the sky, and displace them so that, if viewed as components of a projected 3D image, the starfield of the night sky would have a perceivable depth. That is, <a href="http://en.wikipedia.org/wiki/Sirius" rel="nofollow">Sirius</a> or <a href="http://en.wikipedia.org/wiki/Alpha_Centauri" rel="nofollow">Alpha Centauri</a> would appear closer than, say, <a href="http://en.wikipedia.org/wiki/Betelgeuse" rel="nofollow">Betelgeuse</a>.</p>
<p>The idea <em>sounds</em> interesting, but I was wondering whether it's actually possible. That is, how large would the displacement of the cameras need to be to create a perceivable depth? To quantify, let's say we're trying to reduce the scale from 5 light-years to 100 m. Would this require a displacement of 5 light-years / 100 m$\times$(separation of eyes)$\approx1/400$ light-years $\approx136$ AU or is it more complicated than that?</p>
<p>I guess the maximum we could achieve is by taking two images of the same field of stars, one year apart and combining them, to give a separation of the "eyes" of about 2 AU. I don't know enough astrometry myself to be sure.</p>
| 3,455 |
<p>A cone standing on its tip is considered to be in unstable equilibrium as a slightest force could topple it. So, if the cone is stood on its tip with no other force other than gravity (and the corresponding <a href="http://en.wikipedia.org/wiki/Ground_reaction_force" rel="nofollow">ground reaction force</a>), will it continue to stand without toppling? Has this been attempted experimentally?</p>
| 3,456 |
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