title: Calculating the speed of an object moving relative to another moving Object (Frame of reference)

text: > **Possible Duplicate:**   >  How to deduce the theorem of addition of velocities? Let's say that you are in a rocket speeding at 90% the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90% the speed of light. According to Newtonian physics, we add both velocities. Thus, the bullet should be going at 180% the speed of light. But we **now** know that according to Einstein the sum of these velocities is actually close to 99% the speed of light, because length contracts and time slows down. But how can you determine mathematically, that bullet is going at 99% the speed of light?   Additionally, can you also calculate, how much time is slowed down, if you know the velocity? Is there any formula?


title: Speed of light and relativity

text: Suppose there are two observers $A$ & $B$ both are in motion, now $A$ sees $B$ is moving with speed $'u'$. A says that another object $'d'$ is moving with speed $c+u$ w.r.t. A in the same direction as $B$. What stops him from saying this ? The formula $\frac {u+v}{1+\frac{uv}{c^2}}$ is such that $'v'$ is speed of the object $'d'$ w.r.t.$ B$ and not $A.$ and $u$ is the speed of $B$ w.r.t. $A$, now even though $B$ also sees that the object $'d'$ moves faster than the speed of light. He can still always see light to travel with the speed of light and also he can be an inertial observer? What is the problem here ?


title: Stretching a piece of foil with a hole

text: I have a 20x20cm piece of foil with a circular hole of diameter 1cm in the centre. If I slowly stretch the foil both horizontally and vertically, what happens to the size of the hole? Does the size of the hole change? If the foil doubles in size to 40x40cm, will the hole be of diameter 2cm? Do any factors affect the size of the hole? The temperature of the room, the speed at which I stretch the foil?


title: Can a revolving body self-gravitate?

text: If a body is revolving around a point at radius R with tangential velocity V, does General Relativity predict that at some tangential speed, the body will revolve around the point without any external forces acting on it? Since General Relativity models gravity as a curvature of spacetime, I was thinking that an observer at the center of rotation of the revolving body would measure some curvature of space as a result of the revolution. This is because the body would be contracted in the tangential direction relative to the observer, such that C<2*PI*R where 'C' is the circumference of the revolving body measured by the observer (the revolving body is thought of as a ring of particles). This implies a curvature of space, which would be equivalent to a gravitational force.


title: mechanical disadvantage of pulleys

text: I have a servo that is fixed and that can move a heavy load approximately one inch. I need the fixed servo to move a light load approximately 4 inches. How can I do this with a set of pulleys? Is it even possible. I don't have access to any sort of gears or timing belts either, just pulleys.


title: How do centripetal forces and gravity work for objects in a rotating cylinder?

text: The following is a question from a past exam paper that I'm working on, as I have an exam coming soon. I would appreciate any help. * * * A fairground ride takes the form of a hollow, cylinder of radius $R$ rotating about its axis. People stand against the cylinder wall when it is stationary and its axis is vertical. The rotation is then started, and once the cylinder has reached its operating angular velocity $\omega$, its axis, and the people, are gradually rotated to lie horizontally,as shown in the figure below. ![Figure](http://i.stack.imgur.com/1H3xO.jpg) A person called Alice who has mass $m$, is indicated by $A$ in the figure at the moment when she subtends angle $\theta$ relative to the downwards vertical direction, as seen from the cylinder’s axis. Assuming that Alice does not slide relative to the cylinder wall, and neglecting her size compared to the radius $R$.   1. Show that the component of force upon Alice from the cylinder wall normal to its surface is given by $N = mg\cos\theta +m\omega^{2}R$.     * The acceleration is towards the center of the cylinder, hence $$\sum{F}=ma \Rightarrow N - mg\cos\theta = ma \Rightarrow N = mg\cos\theta + m\omega^{2}R.$$   2. Show that the component of force upon Alice from the cylinder wall parallel to its surface is given by $F = mg\sin\theta$.     * This follows from the diagram.   3. Derive an expression for the minimum $\omega$ required to ensure that Alice remains in contact with the cylinder wall at all times.   4. Show that in order for Alice to avoid sliding along the cylinder wall, $\omega^{2} \geq \frac{g\sin\theta}{\mu_{S}R} - \frac{g\cos\theta}{R}$, where $\mu_{S}$ is the coefficient of static friction between her and the wall.     * $f_{S} \leq\mu_{S}N \Rightarrow mg\sin\theta \leq \mu_{S}(mg\cos\theta +m\omega^{2}R).$ Rearranging, we get $$\omega^{2} \geq \frac{g\sin\theta}{\mu_{S}R} - \frac{g\cos\theta}{R}.$$   5. Use the result of question 4 to show that Alice will not slip for any $\theta$ if $\omega^{2} \geq \frac{g}{R}\sqrt{1 + \frac{1}{\mu_{S}^{2}}}$. * * * **Question** I would like to know if the working for the questions above that I have answered are correct. I'm not sure what concept underlies question 3. For question 5, I put $\theta = \pi$, the angle that subtends when she is at the top but that doesn't work. I'm looking for hints to be able to answer questions 3 and 5. Thank you for your time.


title: Electromotive Force Definition for Induced Current

text: If in DC , Emf is the work done to carry a unit positive charge from -ve to +ve terminal . What is induced EMF then , in a similar sense ? To carry a unit charge around a loop once along a fixed circuit ? Where is a -ve to +ve place ?


title: Why on-shell vs. off-shell matters?

text: The definitions between on- and off-shell are given in Wikipedia. Why is it so important in QFT to distinguish these two notions ?


title: Operator norm directly from phase space representation of photonic quantum operator

text: I'm interested in calculating the operator norm of a Hermitian operator, say $B$, acting on the Hilbert space of square integrable functions. The context is I have an optical system in all its infinite dimensional glory. To be clear, by operator norm I mean the largest eigenvalue, or $||B||_{\infty}=\sup \\{ \langle \psi \vert B \vert \psi \rangle | \langle \psi \vert \psi \rangle =1 \\}$. I have a neat expression for $B$ in terms of its Characteristic function / Wigner function. My question is whether anyone knows a neat and easy way of calculating the operator norm within the phase space picture. Of course, I could truncate the operator to small Fock numbers (assuming that this is a good approximation), work out the matrix in the Fock basis and crunch the numbers. However, I'm just curious whether there is a more elegant solution. I have in mind such equalities as $tr ( A ) = \int W_{A}(r) dr $ and $tr ( A^{\dagger} B ) = \pi \int W_{A}(r)W_{B}(r)dr$, which give a useful correspondence between Wigner functions and the operators themselves.... are these any such expressions for norms? Being pessimistic I guess the answer might simply be NO! However, I thought it would be worth consulting the collective wisdom of TPSE.


title: Computing $\Delta m$ in $\beta^-$ decay

text: > $\require{mhchem}$What is the energy $Q$ released when $\ce{^131_53I}$ > decays and $\ce{^131_54 Xe}$ is formed? The atomic mass of $\ce{I}$ is > $130.906118~u$ and the atomic mass of $\ce{Xe}$ is $130.90508~u$. To compute $\Delta m$, I thought of $130.90508~u - 130.906118~u = 0.298962~u$ but answer is $0.001038~u$ way off. How do I actually compute $\Delta m$?


title: Baryon asymmetry

text: Baryon asymmetry refers to the observation that apparently there is matter in the Universe but not much antimatter. We don't see galaxies made of antimatter or observe gamma rays that would be produced if large chunks of antimatter would annihilate with matter. Hence at early times, when both were present, there must have been a little bit more matter than antimatter. This is quantified using the asymmetry parameter $\eta = \frac{n_{baryon} - n_{antibaryon}}{n_{photon}}$ From cosmological measurements such as WMAP, $\eta \approx (6 \pm 0.25) \times 10^{-10}$ However, the source of baryon asymmetry is said to be one of the Big Problems of Physics. What is currently the state of the art regarding this puzzle? What's the best fit we can get from the Standard Model? What do we get from lattice simulations?


title: What is known about the classification of N=4 SCFTs with central charge 6?

text: I was talking about K3 surfaces with some physicists, and one of them told me that the N=4 superconformal field theories with central charge 6 are expected to be relatively scarce. In particular, one should expect a lot of a priori different theories (e.g., those coming from sigma models whose targets are different hyperkähler surfaces, or the Gepner model) to be isomorphic. I have not found similar statements in the mathematical literature, but it sounds like a statement that, if suitably tweaked, could conceivably make sense to mathematicians. **Question:** Where can I find such a claim (and perhaps additional justification)? Also, I am curious to know if there are underlying physical principles behind such a claim, or if it was conjectured due to a scarcity of characters (i.e., the space of suitable modular/Jacobi forms is small), or perhaps some combination.


title: Why does the second law of thermodynmics only occur in the time dimension?

text: If I break an egg, how broken it is depends on "when" I am not "where" I am. Why is the time dimension special?


title: What does centre of lift depend on?

text: I've read in many places that centre of lift is about quarter chord of the wing and that post-stall lift (the part developed on lower surface) has centre midchord. The later makes sense; the pressure is distributed more or less uniformly (or at least symmetrically) on the lower surface. But what determines the centre of pressure on the upper surface? The streamlines normally look like this: ![streamlines around wing](http://upload.wikimedia.org/wikipedia/commons/b/bc/Airstreams_around_an_airfoil_in_a_wind_tunnel.jpg) and the pressure field like this: ![pressure field around wing](http://www.av8n.com/how/img48/90flat03p.png) But I've never seen explanation why the pressure should be lowest in the forward part (though it's the trailing edge that actually drives the circulation). Or can it only be explained by numeric calculation of the pressure and velocity field? Also is it possible to describe (at least approximately) how this depends on the shape of the airfoil (like just flat plate or supercritical airfoil with the thickest point further aft) or is it again only possible by numeric calculation?


title: On self-duality of N=4 super Yang Mills theories

text: I am looking at S-duality a bit, and was wondering if anyone had the answer to the following question. It is known that the supersymmetric Yang-Mills theory in 4 space dimensions is self-dual with gauge group or U(N) or SU(2). But is it known whether there are more Yang-Mills theories that are self-dual? On the other hand, is it known that some are not? Any help much appreciated!


title: Why doesn't a particle's velocity effect the strength exerted on it by an electric field?

text: Here is what I know: $F = E q = m a$ so $a = \frac{E q}{ m}$ and we can increase the acceleration ($a$) of a particle in an electric field ($E$) by either decreasing its mass ($m$) or increasing its charge ($q$). Here is where I am confused: $a = \frac{v}{ t}$, thus $E = \frac{m v }{q t}$ Yet, increasing the velocity ($v$) of a particle traveling through an electric field has no effect on the electric field. Why not?


title: bc CFT Energy-momentum tensor from Noether's theorem

text: Following Polchinski's book (String Theory 1), we have the bc action : $$S = \frac{1}{2 \pi}\int~d^2z ~b\bar \partial c\tag{2.5.4}$$ where $b$ and $c$ have holomorphic weights $\lambda$ and $1- \lambda$. From this action, it is said, that applying the Noether's theorem, we get the energy-momentum tensor : $$T(z) = :(\partial b)c: - \lambda \partial(:bc:)\tag{2.5.11a}$$ With $\lambda=2$ (and so a holomorphic central charge $c=-26$), one has the correct Operator Product Expansions for Fadeev-Popov ghosts : $T(z) b(z)$ and $T(z)c(z)$, with holomorphic weights $(2,-1)$ so it is coherent, but, in fact, I am not able to apply the Noether's theorem/ Ward identities formalism used in chapter 2.3. [So here, we may take $\lambda = 2$ for simplicity] (This question is related, while different, to a previous question :http://physics.stackexchange.com/a/69755/6316) > So, the question, is, how get $(2.5.11a)$ from $(2.5.4)$ ? [EDIT] I was able to get this: Starting from action : $$S = \int d^2z g^{ab} b_{bc} \partial_a c^c \tag{1}$$ We make a variation : $$\delta \sigma^d = \epsilon \rho(\sigma) v^d \tag{2}$$ The variation of the action, relatively to $\partial \rho$, is : $$\delta S = (\epsilon g^{ab} b_{bc} v^d \partial_d c^c)~\partial_a \rho\tag{3}$$ This corresponds to a current : $$j_a =i v^d(- b_{ac}\partial_d c^c) \tag{4}$$ The energy-momentum tensor is : $$T_{ad} =- b_{ac}\partial_d c^c = (\partial_d b_{ac}) c^c- \partial_d(b_{ac} c^c)\tag{5}$$ Because only the holomorphic part is not null (from equations of movement ?), that is $b=b_{zz}, c = c^z$ we have : $$T(z) = T_{zz} = (\partial_z b) c- \partial_z (b c)\tag{6}$$ [EDIT 2] There is a correct derivation in this reference (see formula 1.14), but it is using the standard derivation of the energy-momentum tensor from the action relatively to the metrics. So, it seems that it is the only way. Of corse, at the end of the variation of the action, we may always write $\delta g_{\mu\nu}=\epsilon (v_\mu \partial_\nu \rho + v_\nu \partial_\mu \rho)$, and then get the correct current $(4)$ and the current energy-momentum tensor $(5)$. We have to take care to that $b_{zz}$ and $c^z$ do not depend explicitely on the metrics.


title: Does gas spread out equally everywhere?

text: An excerpt from this page: > Gases can fill a container of any size or shape. It doesn't even matter how > big the container is. The molecules still spread out to fill the whole space > equally. That is one of their physical characteristics. If a fixed quantity of gas is let out in a limited space, will it spread out equally and maintain a fixed gas distribution throughout the space? Or, does it go where the gravitational pull is maximum? And what factors affect the distribution of gas in a given area?


title: Why does magnetic field increase when the number of coil turns increase?

text: Since we all know that to increase the magnetic field we can do these of the following things: (i) Increase the number of turns in a coil, (ii) Increase the current following through conductor, and there would be more but I know only these (upto my standard - Xth ) now if I use the first method to increase the magnetic field then the length of conductor will also increase because I am increasing the number of turns; say I am increasing the length of conductor by $x$ meters then as we all know that: $R = \rho l/a$ you know resistance is equal to resistivity multiplied by length of conductor and then divided by the area of cross section. According to this $R$ is directly propotional to length of conductor and by Ohm's law $R$ is inverously proportional to current Therefore if I increase the length by $x$ m of conductor then the resistance will also increase at a rate of $x \Omega $ and current will decrease at $x$ Ampere, then how will the magnetic field increase if the current is being reduced?


title: Is there any difference between a proper photon and a photon of thermal radiation?

text: Does a thermal photon radiate spherically whereas a photon radiates in a circle/beam? How do you tell a thermal photon of, say 3 eV, from a photon of light?


title: Relationship between slit size and wavelength in diffraction

text: Almost in every book on physics we can find a statement like "diffraction gets stronger when the size of the slit is comparable to the wavelength". Let's say we have a wall in a bathtub with a slit in it. When water waves reach the slit, the books usually come up with the Huygens' principle to explain that the points on the wavefront which are near the edges interfere in some fancy way so that the water waves spread out radially. However I do not see any connection between the size of a slit and the wavelength. I do understand that if we have a really tiny slit, there are very few points on the wavefront which produce radial waves, and if the slit gets bigger, there are more points that produce the same secondary waves which eventually interfere in an interesting way. But what it has to do with the wavelength? If we have a tiny slit and even more tiny wavelength, how does this change the game? So for example, if we have a big wavelength, the diffraction would be: ![enter image description here](http://i.stack.imgur.com/3vLXg.png) But if the wavelength gets smaller while the slit remains the same, I'd expect the very same diffraction with the only difference that the resultant wave will have a shorter wavelength: ![enter image description here](http://i.stack.imgur.com/AcZKO.png)


title: Non-coinciding event horizon and apparent horizon

text: > **Proposition:** the event horizon and the apparent horizon of a black hole > always coincide. As a reminder: the event horizon is defined as the boundary of the closure of the causal past of future null infinity, i.e. the boundary of the closure of the set of points which can reach infinity by a timelike path (so the boundary of the black hole). The apparent horizon is defined as the boundary of the set of trapped surfaces, where a trapped surface is defined as a surface for which all outgoing null rays converge. **Question:** The above proposition is known to be false. Can someone give a counterexample? As always for counterexamples, the simpler the better! I know the following:   * For a counterexample we necessarily need to consider a non-stationary black hole, as Hawking proved that the proposition is true for stationary ones.   * The apparent horizon is always inside the black hole. But doesn't this imply that there are points inside the black hole from which light rays can diverge?   * Apparently, one counterexample is given by stellar collapse, though I fail to understand this counterexample, i.e. I don't recognize the points which are outside the apparent horizon and inside the black hole.   * Unlike the event horizon, the apparent horizon is observer dependent. Wikipedia states: " _For example, it is possible to slice the Schwarzschild geometry in such a way that there is no apparent horizon, ever, despite the fact that there is certainly an event horizon._ " Clearly they don't coincide then, but doesn't this contradict the fact that they do coincide for stationary spacetimes? So basically, I know some differences between the two concepts, but I cannot think of a concrete example for which we can identify the apparent horizon and the event horizon, and see that they are different.


title: Simple Projectile Motion Question

text: A volcano erupts 50m below the sea level. A rock leaves the crater at 20 m/s at an angle 30 deg with the vertical line. The rock has a mass of 15kg. IGNORE WATER RESISTANCE. It gets out of the water, reaches a max height and falls back it. When does it reach 30 below sea for the first and second times? $a = <0, -9.8>$ $v = <20sin(30), -9.8t+20cos(30)> = <10, -9.8t+17.32>$ $s = <10t, -4.9t^2+17.32t>$ For the object to be 30m below sea level, y=20m, so $s_y=20 \implies -4.9t^2+17.32t-20=0$. However, no real solutions exist to this. This is so simple, I apologize for asking, but what am I overlooking?


title: Quasi 1D insulators with strong spin-orbital interaction

text: We know that the spin-1 chain realizes the Haldane phase which is an example of symmetry protected topological (SPT) phases (ie short-range entangled phases with symmetry). The Haldane phase is protected by the $SO(3)$ spin rotation symmetry. If we change the symmetry, we may obtain other possible SPT phases. This motivates us to ask the following question: **What are good examples of quasi 1D insulators with strong spin-orbital interaction?** There are large $U$ Mott insulators and there are small $U$ band insulators. Here, we are interested in both, and like to see examples for both cases.


title: B-meson naming convention

text: An unbarred $B$-meson contains $\bar{b}$ (an anti-bottom quark), whereas a barred $\bar{B}$-meson contains $b$ (a bottom quark). What is the historical reason for this hellish naming convention?


title: Special Relativity and $E = mc^2$

text: I read somewhere that $E=mc^2$ shows that if something was to travel faster than the speed of light then they would have infinite mass and would have used infinite energy. How does the equation show this? The reason I think this is because of this quote from Hawking (I may be misinterpreting it): > Because of the equivalence of energy and mass, the energy which an object > has due to its motion will add to its mass. This effect is only significant > to objects moving at speeds close to the speed of light. At 10 per cent of > the speed of light an objects mass is only 0.5 per cent more than normal, at > 90 per cent of the speed of light it would be twice its normal mass. As an > object approaches the speed of light its mass rises ever more quickly, so > takes more energy to speed it up further. It cannot therefore reach the > speed of light because its mass would be infinite, and by the equivalence of > mass and energy, it would have taken an infinite amount of energy to get > there. The reason I think he's saying that this is as a result of $E = mc^2$ is because he's talking about the equivalence of $E$ and $c$ from the equation.


title: Veritasium - Firing bullet in block - along center and away from center

text: This question is about this video on YouTube, in which a bullet is fired vertically into the center of a wooden block from below, sending the block up into the air. Next, a bullet is fired vertically but off-center into a similar block from below, again causing the block to rise into the air, but simultaneously to rotate. The video asks for a prediction as to which block will rise higher. Please post your guess/prediction/solution along with answer or in comments. My first guess was that air friction decreases when object is rotating, but now I think that's not the reason. Someone please shed some light on this. ### EDIT Is it possible that the first bullet went deeper inside the first block and thus the first block had lesser kinetic energy than expected?


title: Efficiency of Stirling engine and Carnot's theorem

text: I want to calculate the efficiency of this Stirling cycle for an ideal gas $pV = nRT$ ![taken from Nolting. Grundkurs Theoretische Physik 4. Spezielle Relativitätstheorie und Thermodynamik](http://i.stack.imgur.com/AhKqy.png) The mechanical work is $$ \Delta W_{12} = - \int_{V_1}^{V_2} p(V) \mathrm{d}V = -nRT_2 \ln \frac{V_2}{V_1}\\\ \Delta W_{23} = \Delta W_{41} = 0\\\ \Delta W_{34} = -nRT_1 \ln \frac{V_1}{V_2} $$ On the isothermal curves the change in inner energy $\Delta U = \Delta W + \Delta Q$ is zero. $$ \Delta Q_{12} = - \Delta W_{12} > 0\\\ \Delta Q_{34} = - \Delta W_{34} < 0 $$ On the isochoric (isovolumetric) curves the heat quantities are $$ \Delta Q_{23} = C_V (T_1 - T_2) < 0\\\ \Delta Q_{41} = C_V (T_2 - T_1) > 0 $$ The efficiency is then $$ \eta = \frac{-\Delta W}{\Delta Q} $$ $ \Delta Q$ is the input heat, i.e. sum of all the heat quantities $> 0$: $$ \Delta Q = Q_{12}+Q_{41} = n R T_2 \ln \frac{V_2}{V_1} + C_V (T_2 + T_1) $$ $\Delta W$ is the total mechanical work: $$ \Delta W = W_{12}+\Delta W_{34} = - nR(T_2 - T_1) \ln \frac{V_2}{V_1} $$ So finally the efficiency is $$ \eta = \frac{T_2 - T_1}{T_2 + \frac{C_V (T_2 - T_1)}{nR \ln V_2 / V_1}} < \eta_\text{C}. $$ It is smaller than the efficiency of the Carnot cycle. But it should be equal to it if all processes are done reversibly. The calculations are taken from a textbook (Nolting: Grundkurs Theoretische Physik 4) which actually points out this problem as a question to the reader. My only explanation is that this process is not reversible but I don't know how to tell without actually seeing how the isothermal and isochoric processes are realized. So my questions are:   * Is this a contradiction to Carnot's theorem that the efficiency $\eta_\text{C} = 1 - T_1/T_2$ is the same for all reversible heat engines between two heat baths?   * Is this cycle reversible?   * _Is_ there a way to say whether a process is reversible or irreversible only with a figure like the one above?


title: Angular momentum contained in E&M fields

text: I am trying to do this question as a prep for my quals and find my solution too simplistic. I fear I am missing something important. "Iron atoms (atomic mass 56) contain two free electron spins that can align with an external magnetic field. An iron wire 3 cm long and 1 mm in diameter is suspended vertically and is free to rotate about its axis. A strong magnetic field parallel to the wire's axis is applied. How large is the resulting change in its angular velocity? The density of iron is 8 $g/cm^3$." I assume the axis of the wire is in the z direction, so that $\vec{B} = B \hat{z}$. By using Faraday's law $ \oint \vec{E} \cdot \vec{dl} = -\frac{d}{dt} \int \vec{B} \cdot \vec{da}$ on a circular loop in the xy-plane, I get that the electric field produced by applying the external magnetic field is $\vec{E} \sim \hat{\phi}$. Hence the Poynting vector would be $\vec{S} \sim \hat{\phi} \times \hat{z} = \hat{r}$ and the angular momentum change in the fields would be 0 since $\vec{L} \sim \vec{r} \times \vec{S}$. I feel this approach is missing something, especially since I am not using any of the information that's given in the problem and all the other problems never give information that is not useful to solve them. Am I missing something?


title: Metric interpretation of self-adjoint extensions?

text: I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a geometric interpretation along the lines of non commutative geometry, Lipzchitz distance, etc. Particularly some extensions (such as Albeverio-Holden pseudodelta) seen as if we have just cut a segment of length $l$ from the free solution and then just pasted the half-lines. So in some sense they could be argued to be just the free solution over two half-lines separated a distance $l$. Still, the full set of extensions is four parametric, so it is not clear to me if the rest of the parameters have a geometric interpretation, or even if this one can be translated to a Lipzchitz distance. Also, in Non-Commutative Geometry (NCG) sometimes the separation is linked to Higgs potential, definitely not to self-adjoint extensions; Should I be suprised at the implied connection between both concepts?


title: Normalization of the Chern-Simons level in $SO(N)$ gauge theory

text: In a 3d SU(N) gauge theory with action $\frac{k}{4\pi} \int \mathrm{Tr} (A \wedge dA + \frac{2}{3} A \wedge A \wedge A)$, where the generators are normalized to $\mathrm{Tr}(T^a T^b)=\frac{1}{2}\delta^{ab}$, it is well known the Chern-Simons level $k$ is quantized to integer values, i.e. $k \in \mathbb{Z}$. My question is about the analogous quantization in $SO(N)$ gauge theories (A more standard normalization in this case would be $\mathrm{Tr}(T^a T^b)=2\delta^{ab}$ ). Some related subtleties are discussed in a (rather difficult) paper by Dijkgraaf and Witten _Topological Gauge Theories and Group Cohomology,_ but I am not sure about the bottom line. Does anyone know how to properly normalize the Chern-Simons term in $SO(N)$ gauge theories, or know a reference where this is explained?


title: Quantum mechanics as classical field theory

text: Can we view the normal, non-relativistic quantum mechanics as a classical fields? I know, that one can derive the Schrödinger equation from the Lagrangian density $${\cal L} ~=~ \frac{i\hbar}{2} (\psi^* \dot\psi - \dot\psi^* \psi) - \frac{\hbar^2}{2m}\nabla\psi^* \cdot \nabla \psi - V({\rm r},t) \psi^*\psi$$ and the principle of least action. But I also heard, that this is not the true quantum mechanical picture as one has no probabilistic interpretation. I hope the answer is not to obvious, but the topic is very hard to Google (as I get always results for QFT :)). So any pointers to the literature are welcome.


title: Is inflation deterministic?

text: In some theories inflation is supposed to be able to turn quantum fluctuations into macroscopic inhomogeneities. I don't understand how an isolated system such as the universe can undergo such a random transformation : if at the beginning the universe is in a state $A$, quantum mechanics says that $A$ will evolve to $B=UA$ with $U$ being a unitary operator, and general relativity is also a deterministic theory. So does inflation suppose that the universe is not isolated or does it use some modified theories which include randomness?


title: Questions about Type HE Matrix String Theory

text: I was reading the heterotic string section of this thesis desertation by Luboš Motl, since I _think_ I now understand the Type IIA Matrix String Theory. The only thing I knew about Type HE Matrix Strings, however, was: That It has a gauge-group of $O(N)$. From the paper, I can see that the Type HE Matrix Strings, in order to be consistent, need some sort of "extra terms" (wrt the BFSS Lagrangian Density/SuperYM Lagrangian Density ) in order to exist in 11-Dimensional spacetime. Now,   * What is the intuition behind this; intuitively, why does the absence of these "extra terms" cause such problems in 11-Dimensional spacetime. Given the amount of intuition behind Type IIA Matrix Strings, I expect that there may be some intuition in Type HE Matrix Strings too ?    * Where does the new gauge group of $O(N)$ come from when the gauge group of BFSS itself, was $U(N)$ all along.


title: Thermodynamics question

text: Here is a question I am trying to solve. Sorry for the highlighting, it is not relevant. ![enter image description here](http://i.stack.imgur.com/9OxTR.png) Here are my answers. For a), I would write $dE = dQ + dW$, following the first principle of thermodynamics. And for $dW$, I would write $dW=Fdx = 2l \sigma dx$. But I am a bit worried about the $T$ dependency of $\sigma$... Is my answer fine ? For b), I fix $E(0)$ to be equal to zero, and I compute $\Delta E = E(x) = Q + 2\sigma l x = Q + 2 (\sigma_0 - \alpha T_0) l x$. For c), I just have $W = 2l\sigma x$. I don't really get the point of the question and I have the feeling I miss something important, especially because some questions seems redundant. Can anybody please help me ? Thanks.


title: Uniqueness of the 5 string theories

text: This question combines several sub-questions, the common theme being: why the known 5 string theories are unique? Firstly, regarding heterotic theory. I understand the only allowed gauge groups are $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$. I wanted to ask why these are the only two options but then I found this question. The answer is modular invariance. However, there is something I don't understand here. Bosonic string theory is modular invariant on 26D Minkowski spacetime, but it also remains modulo invariant when any number of the dimensions are compactified on any lattice, as far as I understand. So why here modular invariance depends on the precise compactification? Now, two questions about type I theory Why $SO(32)$ is the only allowed gauge group? Is it some analogue of modular invariance for BCFT? Why are open strings needed to make a consistent theory? I understand you can't get rid of the closed strings in an interacting theory, roughly since an open string can become closed by "stitching" its ends. However, why can't we remain with closed strings only? Is it because of S-matrix unitarity? Finally, a broader question. Each of the 5 string theories involves a special construction, one might say a "trick". For example we need different right and left movers for the heterotic theory and projection to unorientable strings for type I theory. How do we know there aren't more "tricks" nobody thought about? Is it due to classification of SUGRAs? In particular, how do we know it's impossible to introduce fermions without requiring SUSY?


title: Electrostatics:2 concentric spheres

text: An external charge when brought close to a conductor induces bound charges on the nearer side and free charges on the further side so that the field induced in the conductor can counterbalance the field due to the external charge,right? Now in case of 2 concentric spheres, if the outer one is charged & the inner one is grounded, then the field inside the outer sphere is zero everywhere,then no charge should be induced on the inner sphere as well,then by Gauss' Law,field in between the spheres remains zero causing zero potential difference and thus no question of capacitance comes,right? How is that possible???


title: Inflation of universe and Information stored therein

text: In an inflationary model of our universe, the space-time expands within a finite boundary. Some other models are also proposed where the universe is not explained to have been expanding rather the masses of the objects are presented to be expanding with time. If so, what happens to the information scattered in space? Does it inflate too? And in literal sense, what is the meaning of inflation of information?


title: Why doesn't amplitude affect the speed of sound?

text: I understand why amplitude doesn't affect the speed of the sound AFTER the 'leading compression'. The extra force provided on one stage of the cycle is countered on the other stage. But shouldn't the 'leading compression' travel faster if amplitude is greater? Because wave motion hasn't been set up yet shouldn't all compressions face the same amount of resistance while moving forward?


title: Coupled Oscillators

text: This is an exercise of my last exam. Since I couldn't find anybody who solved it or knows how to, it would be really nice if somebody could tell me if my thoughts on it go into the right direction. Two harmonic Oscillators can be described by the Hamiltonian $\hat H_a + \hat H_b + \hat H_c$ where $\hat H_a = \hbar\omega(\hat a^\dagger \hat a + \frac{1}{2})$   $\hat H_b = \hbar\omega(\hat b^\dagger \hat b + \frac{1}{2})$ and the coupling Hamiltonian is $\hat H_c = \hbar\Omega(\hat a^\dagger \hat b + \hat b^\dagger \hat a)$ with $\Omega \ll \omega$ * * * a) Evaluate the commutation relation between $\hat H_c$ and the operators $\hat a^\dagger \hat a$ and $\hat b^\dagger \hat b$. * * * This is not a problem at all, at least when you know about the canonical commutation relation. $[\hat H_c,\hat a^\dagger \hat a] = -[\hat H_c,\hat b^\dagger \hat b] = \hbar\Omega(\hat b^\dagger \hat a - \hat a^\dagger \hat b)$ * * * b) Given that the two oscillators are degenerate, what is the significance of the four operators $\hat a^\dagger \hat a,\ \hat b^\dagger \hat b,\ \hat a^\dagger \hat a + \hat b^\dagger \hat b$ and $\hat a^\dagger \hat a - \hat b^\dagger \hat b$ in evaluation the energy level shift resulting from the coupling. * * * Ok, if things commute that means that they have simultaneous eigenstates. The operators do all commute with $\hat H_a$ and $\hat H_b$ which makes totally sense because they are just the solutions of two different oscillators. To evaluate the energy shift occurring due to the coupling Hamiltonian I need to use an operator that commutes with every Hamiltonian. In this case it would be the one for the total quantum number $n_{tot} = n_a + n_b$ which is $\hat a^\dagger \hat a + \hat b^\dagger \hat b$ because from (a) I know that it commutes. So, if this is right, with part (c) comes my problem. * * * c) Evaluate the first order energy shift for the states $\varphi_+ = (\varphi_{n_a}\varphi_{n_b - 1} + \varphi_{n_a-1}\varphi_{n_b})/\sqrt{2}$   $\varphi_- = (\varphi_{n_a}\varphi_{n_b - 1} - \varphi_{n_a-1}\varphi_{n_b})/\sqrt{2}$ where $\varphi_{n_a},\varphi_{n_b}$ are the eigenstates of $\hat H_a$ and $\hat H_b$ respectively. * * * To get the energyshift I need to send $\hat H_c$ onto my states. But to use these eigenstates with my coupling Hamiltonian I would have to rewrite it ($\hat H_c$) in terms of the total energy because this operator does commute and would have simultaneous eigenstates. But this doesn't work. I thought that it might be some expansion I need to do because one should calculate the 'first order' energy shift, but this could be because the coupling Hamiltonian is a simplified one and I didn't figure out how to expand the total energy operator. Did I get this right? Maybe someone could give me a hint how to proceed.   I got the ideas of how it could work from perturbation theory.


title: Mirrors into Infinity

text: Could someone please name the phenomenon regarding the stretch of reflections into infinity between two opposing mirrors, and also explain why the reflections curve away instead of meeting at a perspective point within the reflections?


title: Observed angular velocity and motion parallax

text: I would be highly obliged if someone would guide me in the right direction regarding this. I am trying to understand and mathematically explain the relation between angular velocities as observed by a moving observer of an otherwise, static object. For example- A tree passes by, as we travel in a bus. So what is the relation of the angular velocity of the tree as observed by an observer in the bus with the velocity of the bus and the distance, so involved? Also please point me towards any published material on this topic. Thanks.


title: A question about Poincare invariance of Polyakov action

text: I have a question the variation of the Polyakov action, related to this Phys.SE post. For Polyakov action $$ S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu}.$$ Consider the variation (p. 14 in Polchinski's string theory book) $$ \delta S_p~=~ \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \delta X_{\mu} \nabla^2X^{\mu}$$ $$ - \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau (-\gamma)^{1/2} \delta X_{\mu} \partial^{\sigma} X^{\mu} |^{\sigma=l}_{\sigma=0} \tag{1.2.27} $$ The open-string and closed-string boundary conditions $$\partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l)=0 \tag{1.2.28} $$ $$ X^{\mu}(\tau,l) = X^{\mu}(\tau,0), \partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l), \gamma_{ab}(\tau,l)= \gamma_{ab}(\tau,0) \tag{1.2.30} $$ will make the "surface term" (the second term in the RHS of Eq. (1.2.27)) vanishes. It is said (p. 14-15 of Polchinski) > _The open-string boundary condition (1.2.28) and closed string boundary > condition (1.2.30) are the only possibilities consistent with > $D-$dimensional Poincare invariance and the equation of motion._ The boundary condition (1.2.28)/(1.2.30) will make the surface term vanish. The resulting equation of motion is tensor equation.   $$(-\gamma)^{1/2} \nabla^2 X^{\mu}=0$$ My question is, why non-vanishing surface term will break the Poincare invariance? E.g. a Lorentz transformation $X_{\mu} \rightarrow \Lambda_{\mu}^{\nu} X_{\nu}$ will not affect $\sigma,\tau$ space


title: Spin Glass Transitions in Random Bond Ising Model (RBIM)

text: In brief, is there a list of spin glass transition properties for the RBIM on different lattices? Is there any know results about the relationships between these probabilities for a graph and its dual? In more detail: I'm interested in RBIMs on a multitude of lattices. Specifically, I'm interested in those models in which the magnitude of the coupling on all links is some value $J$, but the sign is +1 with probability $p$ and $-1$ with $(1-p)$. On any lattice there is a transition at $T=0$ from the ferromagnetic phase (at and around $p=0$) to a spin glass phase, that occurs at some critical probability $p_c$. The value of $p_c$ depends on the lattice. For the square it's around 11%. I can find various resources here and there reporting $p_c$ for various lattices. But they are very hard to find and not exhaustive. Does anyone know of a complete list of all such probabilities ever calculated? Also, for the square and triangular/hexagonal lattices, the $p_c$ for the primal lattice and p_c' for the dual satisfy $$H(p_c) + H(p_c') = 1$$ Where $$H(p) = -p \log_2(p) - (1-p)\log_2(1-p),$$ the binary Shannon entropy, where logs are taken base 2. Is this known to always hold true, or to hold true under certain conditions? Are there known results for which the rhs is > 1?


title: Qubit initial state

text: Suppose that a qubit in an initial state that we don't know was measured, and the result was 1. Is it possible to know the initial state of the qubit by the result measured? And if the result would have been 0?


title: What defines the maximum charge a capacitor can store?

text: The formula for a capacitor discharging is $Q=Q_0e^{-\frac{t}{RC}}$ Where $Q_0$ is the maximum charge. But what property defines the maximum charge a capacitor can store? If it depends on capacitance then that means it depends on the voltage you put across the capacitor, but how can any capacitor "cope" with any voltage?


title: How does liquid in a ball affect its rotational acceleration down a ramp?

text: Suppose we have a shell with mass $M$ and radius $R$. If we let that roll without slipping down a ramp of angle theta to the horizontal, we can easily find the acceleration of the shell the instant the ball is let go. Now, what if we completely fill the shell of mass $M$ and radius $R$ with a frictionless fluid of mass $M$ and let it roll down (without slipping) a ramp with angle theta? I have been thinking of this for a while but cannot seem to find what the instantaneous acceleration is as soon as the $\text{shell + liquid}$ is let go.


title: Fluid mechanics - rotating Bucket - euler fluid - symmetries

text: Rotating Bucket, gravity, euler fluid. E.g faber ch 2 or landau, if we start from the full 3 d equations in cylindrical coördinaties, we have symmetry over the azimuthal angle, but why is there symmetry for radius and z ? I.e why pressure and speed should not depend on r and z ? For example, I may have vortex in z and r, e.g what would happen at a very high speed / high Reynolds number? The most simple solution is the one with symmetry clear but other configurations with vortices are allowed by the equations ? Same line of questions is for the symmetry of the angular / radial speed, the most simple configuration is the one with constant angular speed (I.e. As a rigid body) but are configurations with different angular speeds allowed ? The pressure profile will also vary, but why not a speed that goes with $r^2$ or anything else ? Thanks


title: Why are coke bottles that much fizzier if you shake them?

text: We've all been there: you drop your bottle of soda at some point and when you try to open it, it bursts into foam. My question is, then: why does shaking a carbonated drink make the dissolved gas escape? (I can't think of appropriate tags for this so feel free to retag.)


Does Earth produce metallic elements in its core?


title: What does it mean to integrate out fields from a theory?

text: I've done a fair bit of reading on this subject and I'm still confused about the basic principle of integrating out fields in QFT. When we have a function of 2 fields a and b, f(a,b), and we integrate out the heavy b-fields to give f(a), by what mechanism does the b-field dependence disappear? Are we basically saying that integrating out the b-fields is tantamount to solving the probability amplitude of those fields appearing and that because they are heavy their contribution is vanishingly small? Also I've seen the contraction of fields talked about with regards to integrating out, what is the role of these contractions when integrating out? Contracting fields seems the best means of making a particular field disappear from the equations we consider!


title: Significance of the the Lagrange multipliers in statistical mechanics

text: In classic thermodynamics one can derive the Maxwell Boltzmann statistics by solving a Lagrange multipliers equation. In this process a new parameter $\beta$ is introduced to take account of the total energy constraint, $$L = \beta(E-\sum_i p_i \epsilon_i)+...$$ and it is later connected to $\frac{1}{k_B T}$ by inserting our knowledge about perfect gases. On a side note, about every book around seems to skip the passage with a frustrating "it can be shown that". My problem is that, given similar Lagrange multipliers equation, I'm not sure whether one can in general relate $E$ and $\beta$: it is evident that $\beta$ has no physical significance by itself, and it's always bounded to some parameter in the constraints that it is multiplying. For example, say that I have an array of cells that can be either occupied by 0, 1 or 2 dots. I have some constraints, one of which is the density of the dots $\rho$. The Lagrange multipliers equation has the form $$L = \gamma(\rho-p_1-2p_2)+...$$ where $p_i$ is the probability of having a cell with $i$ dots. Is there some way to eliminate the $\gamma$ parameter in the final explicit formula for $p_1$, and use $\rho$ instead?


title: Curie's principle in electromagnetic field theory

text: I am looking for some explanation and if possible also some references about the applications of Curie's principle in electromagnetic field Theory, precisely in the computation of magnetic (resp. electric) fields generated by different configurations, like the infinite wire, the cylindrical, the spherical or even more complicated configurations (in an undergraduate context), that is I'm looking for "laws"/"approaches" to handle symmetries and invariances in the many differents situations I'm faced with. EDIT : what I mean by that is how to exploit the symmetries and invariances of a certain configuration to determine in what composent the field depend and in what direction(s) it propagate.


title: How did Newton find out force has something to do with acceleration?

text: Its about Newton's second law of motion, $$F=ma.$$ It says the acceleration of an object is directly proportional to the net force and is inversely proportional to the object's mass. Yes I can imagine that. But there's still something I don't really understand but I don't know what it is.. I know its a stupid question but i am just confusing...   I mean, in math: $$\mathrm{acceleration=(final\ velocity-initial\ velocity)/time\ elapsed}$$ But, in physics, $a=F/m $ Does that mean $F/m = \mathrm{(final\ velocity-initial\ velocity)/time\ elapsed}$ ? and how did Newton find out forces has something to do with acceleration and mass?


title: Newton's third law of motion: An apple resting on the surface of earth

text: My friend asked me a question which was: > For an apple resting on the floor, what is the action and reaction force > pair? I said that the Earth exerts a gravitational force on the apple and the apple also pulls on the earth and both these forces are equal in magnitude and opposite in direction this is action and reaction pair. He said: > Okay, Earth is exerting a downward force on the apple, so the apple can not > accelerate downward because of the normal force of the floor, but what about > the Earth? Since the apple is exerting a force on the earth in an upward > direction. I said the Earth will not accelerate in an upward direction because the apple is in contact with the Earth we will treat the apple and the Earth as a single object and the force that the apple exerts on the Earth is internal to the object and the internal force can not cause the acceleration. This is my own logic; I have not read it anywhere. So tell me, am I right?


title: What is the optimal weight for a golf ball?

text: I am 32 now, and have forgotten the basic physics formulas we used in school. I am sitting with a question that is bugging me. If I had to hit two golf balls of different weight with the same force, which one travels further? I imagine that a $200\;\rm g$ golf ball travels farther than a $1\;\rm g$ ball, and a $200\;\rm g$ ball also travels farther than a $2\;\rm kg$ ball, when all balls are acted on with the same force. How can I use to work out the optimal mass to get the most distance?


title: Does irrotational imply inviscid?

text: Let us consider a 2D irrotational flow, such that $\nabla\times\boldsymbol u =\boldsymbol 0$. Defining the stream function such that $\boldsymbol u =\nabla\times\psi \boldsymbol n$ where $\boldsymbol n$ is the unit vector perpendicular to the plane of the motion. The incompressibility condition implies $\nabla^2\psi =0$, so $\nabla^2 \boldsymbol u= \nabla^2 \nabla\times\psi \boldsymbol n=\nabla\times\nabla^2\psi \boldsymbol n=\boldsymbol 0$. From this follows that in the Navier-Stokes equations the term $\nu\nabla^2\boldsymbol u=0$ vanishes, no matter the viscosity. Now, does this allow us to say that the fluid is inviscid? Or, after considering the nature of the inertia, we can say that:   * if inertia can be neglected, the equation of the motion is $\nabla p=\boldsymbol 0$.   * if inertia is important, the flow is at high Reynolds number. My problem is: Is every irrotational flow inviscid? This is kind of counter- intuitive. But I think my error is in the "no matter the viscosity"...


title: Is tea weaker if you make it in a half full cup of water?

text: Imagine you put a tea bag in a cup and half fill it with boiling water. Then after one minute you take the tea bag out and then fill the cup up to fill with hot water. Will the tea be weaker than if you had filled the water up to the normal level with boiling water in the first place and then left it for one minute before taking the tea bag out? Intuitively I can make an argument that it should be but is there a mathematical description for how much weaker it would be? (I am not sure about the right tags for this?)


title: A Book about the Bohr-Einstein debate?

text: A book about the Bohr-Einstein debate? Is there any book that details the correspondence between the two? The only books I could find are popular science books, I wonder if there is a book that lists the correspondence in a more 'raw' form.


title: Breaking the simultaneity

text: I have expressed my self in a detailed way. I've labeled the points to which I want an answer. * * * I have doubt in the Einstein's rail experiments which he mentioned in his book.   In this experiment we preassume events A and B are simultaneous with respect to the embankment. The observer $M$ observes the events $A$ and $B$ simultaneously.   The **point of view** of $M$ for the observer $M^{'}$ is:   "According to me $M^{'}$ will observe event $A$ sooner and event $B$ later,that is "dissimultaneously" due to motion towards right with respect to me(embankment)." > $3.$ What is the point of view of $M^{'}$ for himself,that is would the > events $A$ and $B$ as seen(observed) by himself be "dissimultaneous? My guess is that there exist oly one "reallity" of observer $M^{'}$. We know for sure this reallity as calculated by the observer $M$. If there is only one reallity of $M^{'}$ then the point of view of $M^{'}$ for himself will be:   "Today I was in atrain, I don't know if it was moving or not. I saw two flashes of light namely A and B. They were not simultaneous. Infact A occured earlier than B.   I ask why the reallity of $M^{'}$ should be the same?   One can answer: If the reallity of an observer is different for different observers then this will be a paradox. To explain this paradox consider this thought experiment:   **One's experiment**   A metallic ball $b_0$ is placed on a wedge inside the train as shown:   ![Image 1](http://i.imgur.com/xYKvXSJ.jpg)   Train is moving towards right w.r.t outside observer. Two sisters "Sheela" and "Munni" are standing near the wedge. Naturally when the centre of the wedge coincide with the outside standing observer say $O$ two fashes of light $P$ and $Q$ occurs simultaneously in train's frame. After some time when train stops $O$ goes to the train and meets "Sheela" and "Munni". He finds both of them alive.   He thinks "were $P$ and $Q$ simultaneous in my frame?" But if so then in my frame train is moving so one of the light rays would hit the ball earlier than the other and one of the two sisters will die with the heavy ball fallen. This can't be true because in reality there can't be paradoxes, either the girl should alive or not so reality for my frame and the train's frame should be same. And i've seen them alive so $P$ and $Q$ are not simultaneous in my frame.   As the train is moving towards right in my frame so $P$ should occur later than $Q$ so that both the rays of light would hit the ball simultaneously.   We see that the reality(fate) of "Sheela" and "Munni" is that they both are alive so we should accept the consequence of " _STR_ "(i.e. the breaking of simultaneity for different observers).   We also see that the demand for the reality to be same is that $P$ and $Q$ must not be simultaneous for $O$.   I argue to one's experiment:   "The reality for observer $M^{'}$ can be different from what the point of view of $M$ is for the reality of $M^{'}$. In other words if reality is same then $M$ s thinking is that $M^{'}$ has faced both A and B dissimultaneously so when $M$ will meet $M^{'}$ after stopping of train $M^{'}$ should tell him that A and B occurred dissimultaneisly but it can also be the other way round.   When $M$ will meet $M^{'}$ the reply of $M^{'}$ can be:" NO! both A and B were simultaneous".   If reality of $M^{'}$ is different from $M$s point of view we will be in a paradox. This paradox exists. A similar can be realized in practice as explained below:   **Anupam's experment**   Consider an apparatus managed by $M^{'}$ inside the train:   ![image 2](http://i.imgur.com/SnuDbti.jpg)   A long stick is placed on a wedge. Two metallic balls are placed at its two ends. Another wedge is placed at the mid of the stick upon which an egg is balanced. Two observers are there $R$ and $R^{'}$. $R$ is sitting inside the train near the wedge. $R^{'}$ is standing outside. Train is moving towards right w.r.t. $R^{'}$.. Naturally the wedge and $R^{'}$ coincides at this very moment two light rays are sent from the wedge in the opposite directions as shown towards $b_1$ and $b_2$.   The reality of $R$ for himself is:" The two rays hit the balls $b_1$ and $b_2$ simultaneously. The balls falls simultaneously. The stick remains balanced. The egg is alive!."   The reality of $R$ from the point of view of $R^{'}$ is:   "Since the train is moving towards right so the ball $b_1$ will be hit earlier than $b_2$. $b_1$ and $b_2$ do not fall simultaneously. The stick has lost its balance. The egg has fallen. The egg is dead!."   The paradox has appeared in reality.   The reality(fate) of $R$ is different from his point of view than $R^{'}$'s. > $4$ What will happen to the egg when $R^{'}$ will meet $R$ ?. Since there can be two different realities as I shown so "one's assumption" is wrong.   The **point of view** of $M^{'}$ for himself in Einstien's experiment can be different from the **point of view** of $M$ for $M^{'}$ i.e. A and B can be simultaneous in the train's frame.


title: Radiation pressure question

text: If radiation pressure is a pressure exerted upon any surface exposed to electromagnetic radiation then why do planets such as the ones in our solar system maintain consistent orbits? I'm guessing the pressure must be very slight but in the vacuum of space wouldn't even the slightest pressure effect the consistency of any planets orbit?


title: Damping ratio and damping coefficient of two objects connected by spring

text: There are two objects m1 and m2 connected by a spring and a viscous damper (e.g car body and wheel connected by spring) From wiki: $\zeta ={c \over 2{\sqrt {mk}}}$ ζ is damping ratio, c is damping coefficient, k is spring constant In wiki example, there is only one mass connected to ground. But here I have two masses m1 and m2. Can I use an equivalent m for m1 and m2? If yes, how to calculate equivalent m from m1 and m2 Edit: I need to calculate zeta to find zeta = 1 is critical damping, whether zeta !=1 is under or over damping sorry here is the wiki link about damping http://en.wikipedia.org/wiki/Damping#Example:_mass.E2.80.93spring.E2.80.93damper And here is the sketch ![enter image description here](http://i.stack.imgur.com/HBoa0.png)


title: Field strength vanishes iff $A_{\mu}$ is pure gauge

text: Is it true that the field strength $F_{\mu\nu}$ in a non-Abelian gauge theory with gauge group $G$ vanishes if, and only if, the gauge field $A_{\mu}$ is a pure gauge? I can show one implication. If $A_{\mu}=\frac{i}{g}U\partial_{\mu}U^{\dagger}$ where $U \in G$, then the field strength vanishes, but I am struggling with the other implication.


title: Calculating the Center of Mass

text: We have a homogeneous body that looks like this: ![Homogeneous body](http://i.stack.imgur.com/j9fuR.jpg) I have tried dividing the body into different parts using the following definition: **R** g * **A** = **R** 1 * **A** 1 + ... **R** n * **A** n I was thinking I could first divide it into two parts using the x axis, and then add those two together using the definition above. But how can I calculate the coordinates for the center of mass for the lower part?


Is there evidence of dark matter in our galaxy? How can we measure this, say, how many percent of the center of our galaxy is dark matter? I did not find the answer in the question What's Dark Matter and Who Discovered it.


title: What was the most distant supernova spotted by a amateur astronomer until today?

text: What type of commercial amateur telescope and what method ( _difference imaging_ ,...) did he use for identifying the supernova?


title: Impact of gravity on bone structure

text: Some suggest that if a human was born and raised in a gravity-less environment their bone structure/strength would be different that for children born on earth. If this is the case do you think this change would be gradual (over several generations) or would it be immediate? Also would these changes be permanent in the DNA (genetic mutation?). If the said human, born in a gravity-less environment comes to earth, surely they would have problems dealing with the effect of gravity but would they be able to acclimatize gradually ?


title: When the Sun goes red giant, will it "vaporize" away a significant fraction of Jupiter and Saturn moons?

text: Since Jupiter and Saturn's moons are composed of a rock+ice mix. For example, Callisto is 50% rock and 50% ice. When the Sun finally goes red giant, could it melt a significant portion of the 50% of Callisto that is ice? And over what timescale?


title: Dimensional Regularization involving $\epsilon^{\mu\nu\alpha\beta}$

text: Is it possible to dimensionally regularize an amplitude which contains the totally antisymmetric Levi-Civita tensor $\epsilon^{\mu\nu\alpha\beta}$? I don't know if it's possible to define $\epsilon^{\mu\nu\alpha\beta}$ in e.g. $$d-\eta$$-dimensions where $\eta$ is considered small and which we set to zero in the end. So what are your thoughts?


If I have a rectangular loop moving through a magnetic field, which way will it go? A uniform magnetic field (say it's pointing into the page) A square/rectangular loop of wire moving into the field from left to right with constant speed. The induced current should be counterclockwise right?


title: What does velocity dispersion (sigma) reveal about a galaxy?

text: I'm getting hung up on this term. In studying SMBHs, I see that velocity dispersion strongly correlates with mass. Just what is the velocity dispersion? How can the velocity dispersion of the galaxy be expressed in one figure (sigma) if it has to be measured all over the galaxy? I can imagine the velocity dispersion changes with radius, so "which" point is used? Why exactly is it that higher velocity dispersion is correlated with higher Mass? And is all this referring to the bulge only or to the entire galaxy? So to get velocity dispersion, is "one slit" at the center enough or do you scan across? I hope someone can answer this without relying heavily on the maths. I am really just trying to understand the concept, because I think that velocity dispersion is real, not just a mathematical construct.


title: What is the largest observed celestial body?

text: I'm not referring to phenomena such as galaxies or clusters but rather, what is the largest discrete celestial body that has been observed in the universe?


title: Classical results proved using quantum mechanics

text: Are there any results in classical mechanics that are easier to show by deriving a corresponding result in quantum mechanics and then taking the limit as $\hbar\rightarrow0$? (Are there classical results that were first discovered through taking the classical limit of quantum mechanics even if they are now easier to demonstrate, in hindsight, using classical mechanics?) Update: One semi-example is the derivation of classical adiabatic invariants from the quantum adiabatic theorem in some textbooks. But the quantum adiabatic theorem isn't really any easier.


title: Jupiter: Zonal Jets in opposite directions

text: I recently read that the Great Red Spot arises from the shear instability of zonal flow from zonal jets travelling in opposite directions. Is there any reason why such zonal jets travel in opposite directions anyway? I thought it would be more intuitive if such zonal jets, over their billions of years of existence, would be overcome by the greatest group of zonal jets that collectively travel in a single/similar direction, and eventually all join together to form one in one single direction.


title: Discrete sum over a gaussian function

text: I have a sum of the form $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $α>0$ is some constant, and I don't mind if the limit $N\rightarrow\infty$ is taken. I know there is a possibility of exchanging the "variable of summation" by introducing some "relative coordinates" $\tau=n−m$ and $\eta=m+n$, but I don't know how to continue from there.


title: Proof of renormalizability based on analyzing the symmetry of effective action: isn't regulator also important?

text: In QFT Vol2 written by Weinberg(Chap 16-17), or very much similarly in Adel Bilal's notes(Chap 7), a powerful way of proving renormalizability is presented: Analyze the symmetries of the quantum effective action (QEA), and since QEA generates all 1-particle irreducible diagrams, the symmetry of it tells us what kind of counterterms are needed. If the symmetry is strong enough to limit the form of counterterms such that they are already contained in the original Lagrangian, the renormalizability is proved. But it seems that, for the terms in their intermediate derivations to make sense at all, they must be understood as already regularized. However, it is not obvious that there exists a regularization scheme that respects all the utilized symmetries(In the QCD case they are Lorentz invariance, global gauge invariance, antighost translation invariance, ghost number conservation). Is this a missing step in their proof?


title: Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?

text: Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which $x^0=\varphi$, how to prove that the metric components $g_{0i}=0$ for $i=1,2,3$? My try is as follows. Because $W$ is perpendicular to the surface $x^0=c$, let $U=(c,0,0,0)$, then for every vector $V=(c,x^1,x^2,x^3)$ inside $x^0=c$, we must have $$ W_a(V^a-U^a)=0 $$ which leads to $$ W_i=0,\;\;i=1,2,3 $$ Now because $W$ is HSO, we must have $$ W_{[a}\nabla_bW_{c]}=0 $$ Then I am stuck. Since when $W_i=0,i=1,2,3$, the above equation holds automatically, thus can give me nothing. Could anyone help?


title: Ice throw from industrial wind turbine

text: I was watching a documentary where the residents didn't want to have a wind turbine erected nearby. One of the comments from the company was that an ice throw would have very little importance since it would not travel far from the tower.               Description of turbine:     400 foot tower     150 ft length of blade     175 mph tip speed      I converted into SI units and calculated a worst case throw of 700 m (without any air resistance). My question: how far would the ice go with air resistance? Assume a football shape of 1 kg. What is the final velocity?


title: How can the potential on a cube's surface replicate the potential of a point particle?

text: If i have a cube (either hollow, or an insulating solid) and i want its surface have a potential such that it looks like a point particle outside of the box does that mean the exact potential on the cube itself is the same as the potential of a point particle?


title: How should I regularize this integral?

text: I need to calculate the following integral (which is divergent): \begin{equation} I(m,C)=\int_{-\infty}^\infty {\rm d}\omega\int_{\rm space}{\rm d^3 k}\ln\left(\omega-\frac{k^2}{2m}+C+i\epsilon\right) \end{equation} where $m$ and $C$ are constants, and $\epsilon$ is an infinitesimal positive number. How should I regularize this integral to get $I$ as a meaningful function of $m$ and $C$?


title: What would it be like if we had a planet very close to us?

text: My desktop background rotates through a bunch of space and nature scenes, and this one came up. ![image of planet](http://i.imgur.com/DWQYy.jpg) What would be the effect on people walking around down here, if another earth was looming overhead instead of our moon.


title: Is there any way to judge whether a bolide might produce a meteorite?

text: If you see a bolide, is there any way to judge whether it's worthwhile to try to find an impact? (I mean, short of a big "bang" and a debris cloud rising from the remains of your car...) I imagine it's a once-in-a-lifetime thing at best, but it seems like the meteor hounds have a few leads per year.


title: What makes a supermassive black hole move through space?

text: In this example of a Neutrino jet, it seems that the **supermassive black hole** moves around the space and meets up with a interstellar object sometimes. ![enter image description here](http://i.stack.imgur.com/AinGR.jpg) Source of the above image and a larger version. What force gives a supermassive black hole the ability to move around the space ?


title: Galaxies seen from Earth

text: If we observe two galaxies from the Earth that are diametrically opposed and each 10,000 light-years from the Earth, will the separation distance between the galaxies be 20,000 light-years? Really, the question is: if the galaxies instead were separated from Earth by 10 billion light-years, would the separation distance between the galaxies then be 20 billion light-years?


title: Existing photometry data of the open star cluster Cepheus OB3?

text: I am doing research with my teacher on photometry of the open star cluster Cepheus OB3, and we are trying to find research papers so we can target stars in the photos we took. Are there any existing resources available on this matter?


title: Io - Turned inside out

text: In a lot of the descriptions of the moon Io it frequently states that the moon, because of volcanic activity, has been literally "Turned inside out". What exactly is meant by "Turned inside out" and why does this not destroy the moon, or have an affect on its orbit? **Update:** I do assume _turned inside out_ means that, over time, the lavas from below become the new surface. I do not know much about volcanoes, and I am making the assumption that would make the lower surfaces more brittle.


title: Are telescopes with a concave lens useful for astronomy?

text: Are telescopes with a concave lens (instead of convex ones) ever useful for astronomy? And if so, where are they used? Do they ever affect resolving power?


title: Recovering all of Maxwell's equations from the variational principle

text: Whether you can get the first couple of Maxwell equations from a variational principle? In the second volume of the Landau theoretical physics said that it is impossible.


title: Will changing amplitude change the frequency?

text: Will changing the amplitude change the frequency of a wave, or is it possible for a specific frequency (50 Hz. for example) to generate from shifting amplitude patterns?


title: Non-linear dynamics vs Chaos

text: I am confusing between non linear dynamics and chaos. Chaos is also a non- linear dynamics right? then what is the difference between chaos and non- linear dynamics? What I understood about chaos is that, it is some errors or small deviations which repeat itself in space-time and with time it amplifies. Also, this depends on the initial condition as well. This is also a non-linearity right?


title: Lightning on my Car

text: Could anyone clarify to me the following:   1. Lightning gives off high current when it hits a car/vehicle, and since the electrical components of the car is grounded on the chassis, are there possibilities that these high currents can destroy or damage the electrical components,battery, antenna etc through the 'grounding'?   2. what are the 'areas' or 'lightning zones' on my car which have the possibility of getting struck by lightning?   3. My friend says the car is being 'positive' with respect to the ground(Earth) hence the lightning which is mostly negative flash tends to strike the car, How can the car be positive when the entire exterior of the car is mounted on the chassis which is negative?


title: Variations of double slit experiment with their results to look up

text: Are there variations of double slit experiment with their results to look up? for example besided the two parallel slits, having two holes or two slits at 90 degree angles?


title: Time evolution of wave spectrum

text: A useful way of thinking (not only) oceanic waves is to consider them as a superimposition of linear modes: the elevation η of the sea surface is given by: 1: $\eta({\bf x}, t) = \int_{-\infty}^{\infty} \hat{\eta}({\bf k}) e^{i({\bf k \cdot x} - \omega t)} d{\bf k}$ where $\omega$ and ${\bf k}$ are related by a dispersion relation, and $\hat{\eta}({\bf k})$ is a sort of time independent Fourier transform. $\hat{\eta}$ is given by: 2: $\hat{\eta}({\bf k}) = const \int_{-\infty}^{\infty} e^{-i{\bf k \cdot x}}(\eta({\bf x}, 0) + \frac{i}{\omega({\bf k})} \frac{\partial \eta({\bf x}, 0)}{\partial t}) d{\bf x}$ meaning that if we know $\eta({\bf x}, 0)$ and its time derivative we can determine the amplitude of all the normal modes, and neglecting non linear interactions between the modes we know the evolution of the wave. Clear and smart. What's the simples way to obtain relation 2 from relation 1? Thanks!


title: Primary constraints for Hamiltonian field theories

text: I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures on quantum mechanics". My question is the following: I have momentum variables that depend on the spatial derivatives of the generalised coordinates, but not on the time derivatives of the generalised coordinates. Is this a primary constraint or not? I have conflicting thoughts on this. On the one hand, there are texts that say a primary constraint occurs when the definition of a momentum variable is not invertible for the corresponding velocity. By this criteria, I do have a primary constraint because the momentum does not depend on the time derivative of the generalised coordinates. On the other hand, Dirac for example says that a primary constraint is a function of the form $\chi(p,q)=0$ that comes from the definition of the momenta. This is not the case for me, since I have a function that also depends on the spatial derivatives of the q's. By this criteria, I don't have a primary constraint. Any help much appreciated.


title: Ring theory in physics

text: Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics problems?


title: Is there a fully quantum-field theoretic treatise of Planck's Law for black-body radiation?

text: I recall from my undergraduate statistical mechanics and QM classes that Planck's Law may be derived fairly straight-forwardly by considering the density of states of EM radiation in thermal equilibrium with a particle inside a cubic box. Indeed the derivation on this wiki page is just the same and appears (IMO) to be fairly concise and accurate. I wondered therefore, that if we invoke all the magic of QFT (such as the fields being promoted to operators and higher (than tree-level) order particle interactions taken into account,) can a similar derivation still be carried out? (Or indeed a dissimilar one?) Associated to which, would such a derivation (if it is even possible) require consideration of QED alone or would the effects of the broken electro-weak symmetry have a significant effect on recreating the correct radiation spectrum?


title: Why can't fermions be affected by effective gravity in non-linear quantum electrodynamics?

text: Quantum electrodynamics based upon Euler-Heisenberg or Born-Infeld Lagrangians predict photons to move according to an effective metric which is dependent on the background electromagnetic field. In other words, photon trajectories are curved in presence of electromagnetic fields, meaning that an effective gravity is acting upon. If part of fermion masses is allegedly of electromagnetic origin, the question why their trajectories are not affected by this effective gravity naturally comes to mind.


title: Snell's law starting from qft?

text: Can one "interpret" Snell's law in terms of QED and the photon picture? How would one justifiy this interpretation with some degree of mathematical rigour? At the end I would like to have a direct path from qft to snell's law as an approximation which is mathematically exact to some degree and gives a deeper physical insight (i.e. from a microscopic = qft perspective) to Snell's law.


title: Vapor Pressure and Container

text: The following comes from my text and concerns vapor pressure: Consider a liquid in a closed, cylindrical container that is half full. The molecules at the surface of the container and along the edges have the fewest neighbors; therefore, they will evaporate more quickly than other molecules. This intuitively makes sense. The text also says that vapor pressure is independent of the shape and volume of a container. Now consider the above situation in two cylinders of the same volume. The first has a much smaller radius than the second. Since the smaller radius cylinder has a greater proportion of its molecules along the edges, I would think that it would have a slightly higher vapor pressure. Is this correct? Perhaps I'm just over-complicating things. Any help is appreciated!


Why does an armature of a generator rotate? a) like poles repel b) unlike poles attract c) current flows up d) an outside force turns it I am thinking that (d) is the correct answer but still not sure of it


title: Does the universe have an edge/boundary/barrier? If yes, what is at the edge?

text: My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they began? ie space itself is curved...or would they hit some invisible barrier?


title: Shape of the universe?

text: 1. What is the exact shape of the universe? I know of the balloon analogy, and the bread with raisins in it. These clarify some points, like how the universe can have no centre, and how it can expand equally everywhere in all directions.   2. But they also raise some questions, like if you are on the surface of a balloon and travel in 1 direction, you will eventually return to your starting point, is it possible our universe have this feature?    3. If it has, or had, would this be a symmetry of sorts ($\psi(x)=\psi(x+R)$), and as such have a conserved quantity associated with it (by Noether)?   4. Assuming "small curled up dimensions" wouldn't these dimensions have this type of symmetry, what are the associated conserved quantities?   5. Is it known exactly what the geometrical shape of the universe is? (on a large scale) (I am not talking about only the observable universe).   6. How does one define the "size" of a dimension, is this scale only applicable to curled up ones?   7. Is it possible to describe to a layman the shape of the universe without resorting to inept analogies?


title: If the universe has a shape, what is the surface?

text: Since the idea of the universe was that it was expanding, is there a way to know the surface area of the universe like the growing 3D object it is illustrated? The idea was that the universe was expanding, and then I think that the surface would expand, but my concept is limited about such a metric. I think I read about the mass of the universe, but is there an agreed measure of the surface of the (bounds of the) universe? Since the universe is assumed to have a shape, then what the surface be?


title: Capacitance of a capacitor

text: Why does the capacitance of a capacitor increase if the distance between the two plates of a parallel plate capacitor is decreased? I think, with decreasing distance between the two plates, the force of attraction between the charges on the two plates will increase, and as a result more charge will be stored.


title: Initial separation of neutron star/black hole binaries?

text: How would I go about finding the distribution of initial separations (i.e. the lengths between the centres of mass) of stars that make up binary systems. I am interested in neutron stars and stellar black holes. This paper uses an assumption that > "The initial separation of a binary is chosen from a distribution that is > assumed to be logarithmically flat: $\Gamma(A) \propto 1/A$." but it neither defines $\Gamma$ or $A$, so yeah... helpful.


title: Rate of Cooling

text: I was given the following problem: > Two objects have a temperature difference of 20 degrees C at time 0. Two > minutes later, they have a temperature difference of 18 degrees. How long > does it take for them to have a difference of 10 degrees. Initially, I tried to approach the problem with Newton's Law of Cooling. However, I believe my solution is flawed: $$\frac{dT}{dt} = -kT$$ letting $T$ be the temperature difference between the two objects. The solution to the above equation is $T(t) = ce^{-kt}$. Inserting the provided values, I found that $c = T(0) = 20$ and $T(2) = 18 = 20e^{-2k}$, so $k = 0.05268$. Setting $20e^{-0.05268t} = 10$, I found that $t = 13.2$ minutes. **Is Newton's Law of Cooling applicable if I simply let $T$ be the difference in temperatures between two objects?** Also, the problem implies that both objects are changing temperature. **What do I need to do to accommodate for both objects changing temperatures?**


title: Can anyone explain what a superleak is?

text: In the context of Helium can anyone explain what a superleak is and why it could be useful?


title: What is the "direction" of the transition dipole moment? (Understanding Eq. 9.29, Charge and Energy Transfer 3rd Ed, May & Kuhn)

text: For a real vector $\mathbf{r}$, the direction is given by: $\hat{\mathbf{n}}=\mathbf{r}/\left|\mathbf{r}\right|$. The transition dipole moment is a complex vector. How do you define its direction? The point of this question is that I am trying to understand the meaning of equation 9.29 of the book Charge and Energy Transfer 3rd Ed by May & Kuhn, which reads: $$J_{mn}=\frac{\left|\mathbf{d}_{m}\right|\left|\mathbf{d}_{n}^{*}\right|}{R_{mn}^3}\left[\mathbf{n}_{m}\cdot\mathbf{n}_{n}-3\left(\mathbf{e}_{mn}\cdot\mathbf{n}_{m}\right)\left(\mathbf{e}_{mn}\cdot\mathbf{n}_{n}\right)\right]$$ In the next paragraph I attempt to explain the notation in this equation. I'll point out the parts that confuse me. Here we are considering two molecules labeled $m$ and $n$, and we only consider two electronic levels in each, the ground state $g$ and one excited state $e$. $\mathbf{d}_{m}$ is the transition dipole moment for the transition $g\rightarrow e$ of the $m$th molecule, and similarly for $\mathbf{d}_{n}$. $\mathbf{n}_{m}$ is a unit vector pointing in the direction of $\mathbf{d}_{n}$ ( **I don't understand this** ), and similarly for $\mathbf{n}_{n}$. Finally, $R_{mn}$ is the distance between the centers of mass of the molecules. The quantity we are calculating here, $J_{mn}$, is the excitonic coupling between the two molecules. This can be seen as the rate at which an exciton at a molecule $n$ will be transfered to a molecule $m$ initially in its ground state. Here we calculate $J_{mn}$ using the _dipole- dipole interaction_ approximation, valid when the molecules are sufficiently far apart. Note also the factor $\left|\mathbf{d}_{n}^{*}\right|$ in the equation. **I don't understand this bit of notation either**. I am not sure how to interpret the complex conjugation sign ($*$) as it appears inside absolute-value brackets ($|\square |$).


title: Why exactly does current carrying two current wires attract/repel?

text: When to parallel wires carrying currents in same direction I1 & I2. http://www.youtube.com/watch?v=43AeuDvWc0k this video demonstrates that effect. My question is, why **exactly** does this happen? I know the reason, but I'm not convinced with it. One wire generates the magnetic field which is into the plane at another wire. Electrons are moving in the wire which experience Lorentz force F = q( **V X B** ). My arguments are,   1. this force is experienced by the electrons, not the nucleus. And these electrons that are in motion are the "Free electrons". So, when the experience force just they alone should be drifted towards/away from the wire but not the entire atom(s).    2. The only force binding an electron to the material/matter is the coulombic attraction force from the nucleus. If the Lorentz force is sufficiently large, then it should be able to remove electrons from atoms. I other words, they should come out of the material. But I never heard/read of any thing like that happening. Why doesn't this happen? In any case, atoms must not experience any force, then why is it that entire wire is experiencing a force of i( **L X B** )?


title: How is earth a magnet while it's volume is mostly above curie temperature?

text: I was assuming the earth's magnetism is similar to a permanent magnet; But below a depth of somewhere near the Mohorovičić discontinuity (Moho) between earth's crust and mantle it should be above curie temperature. That depth is pretty shallow compared to earth's radius, something like 40km deep.   Is earth roughly like a very thin hollow magnet?   Or is the magnetism caused by some completely different mechanism?


title: How do I properly use Newton's Third Law and Free Body Diagrams?

text: **A force of 14.0N is applied to a block as shown in the diagram below** ![](http://img14.imageshack.us/img14/9337/diagramzw.jpg)! If the coefficient of friction is between the surface and the blocks is 0.35 a) What is the acceleration of the two blocks? b) What is the force that the 3.0kg exerts on the 1.0kg block? **Solutions Given** a) $\sum F = 14.0N - 13.7N = 0.28N$ $\frac{0.28N}{4.0kg} = 0.07m/s^2 = a$ b) $\sum F = (1.0kg)(0.07m/s^2) = 0.070N$ $F_{fric} = (0.35)(1.0kg)(9.8m/s^2) = 3.43N$ $F = 0.07N + 3.43 = 3.5N$ **Confusion** Here is the thing, but my never show (probably due to print cost?) the free- body diagrams. I want to understand a **systematic way** of solving this, that is, draw FBD and then apply Newton's Second law. I already know that they first considered the whole system as "one mass" and they applied the 2nd Law and then the third law on each. I want to do this _with_ a Free-Body Diagram. Here is my attempt. I am going to denote the masses as 1 and 3 in the subscripts. Here are my freebody diagrams. For the block with mass 3kg ![](http://img197.imageshack.us/img197/1871/force3.jpg)! For the block with mass 1kg ![](http://img507.imageshack.us/img507/8589/force1x.jpg)! My equations should be $\sum F = F - (\mu m_3 g + F_{13}) = m_3 a$ $\sum F = F_{31} - \mu m_1 g = m_1 a$ Here is the problem, if I attempt to solve this I get (add the two equations together) $F - \mu (m_1 + m_3)g - F_{13} + F_{31} = (m_1 + m_3)a $ If I were to go ahead and plug in the values I get $14N - 13.72N - F_{13} + F_{31} = 4a $ $0.28N - F_{13} + F_{31} = 4a $ How come I can't get the same answer this way? Am I missing another equation? Perhaps a constraint? It would seem to make sense if it happens that $|F_{13}| - |F_{31}| = 0$ But when I broke it into components like this, I am dealing with the vectorial components, not the magnitiude. So really vectorically, I need $\vec{F}_{13} = -\vec{F}_{31}$ So I actually get if I plug in the vector force Thank you


title: Horsepower at certain RPM point without knowing torque?

text: I want to know the horsepower produced by an engine at certain RPM by knowing another certain RPM point? Let's suppose that an engine produces 200 hp at 4000 RPM, how many horsepower is produced by that engine at 5000 RPM? What is the process?, Do I miss another factors?


title: Is Dark Energy A Constant?

text: My understanding of dark energy is that it's like hot air bouncing around in a balloon, except the air is tiny subatomic particles that can't be seen or even detected in any way except for their apparent effect on the universe's expansion, and the balloon is gravity. I question whether they can really tell if the universe is expanding faster now than it was a billion years ago, there are so many variables and margins of error, but if it is, wouldn't it slow down later? The propulsion effect of dark energy would taper off like a rocket running out of steam as it cools down and gets sucked into black holes, and then no matter how far or fast the universe had expanded, over an infinite amount of time, gravity would be all that was left, wouldn't it? I don't believe in black hole radiation, I don't think anything could ever escape the huge gravity of a massive black hole as everything has gravity, because everything distorts space by it's existence, but even if you believe that the universe ended with nothing but photons flying around, then wouldn't their gravity be the same anyway, and enough to cause the big crunch, the only logical cause for the big bang?


title: Why are planetary systems so rare?

text: According to this site there are 258 know planetary systems and 302 planets. Mostly each of the listed system has only 1 planet of Mercury's or Mars' size, while our system has up to 8 planets. From what I know our sun is not a "big" star, so theoretically other bigger stars probably threw out more planetary material, so they should have many more planets. Humans know about billions of stars and since planets are made from what remained from star's birth then why aren't there billions and billions of planets?


title: Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

text: Towards the end of the paragraph with the title **String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras** of this article, it is explained that the symmetry U(1)$_L$ x U(1)$_R$ is enhanced to SU(2)$_L$ x SU(2)$_R$ for a string theory at the self dual radius. The text further explains that to complete the U(1) symmetries to SU(2), new generators J$^{\pm}$ of the SU(2) group with corresponding charge densities $\rho^{\pm} = :exp(iX^5):$ are needed. To see that the old U(1) generators ($\partial_{\tau}X^5$ and $\partial_{\sigma}X^5$) together with the generators corresponding to the new charge densities combine to SU(2) one would do an operator product expansion (OPE) to retrieve the commutator of the SU(2) algebra. I'd like to see some details (or at least how to get started) of this calculation.


title: Why can I move through air?

text: Forgive me if this a simple question with an obvious answer I can't see. I was thinking about pressure and air pressure and how we have thousands of pounds of air pressing down on us. What I'm wondering is why I can move through it so easily. Because of all this air, shouldn't the pressure and the compression of all this air make it really difficult to move through? I think this because when I'm in a swimming pool, the deeper I go, the more difficult it is to move my arms in non-streamline movements. > How come it's so easy to move through it? I also have a second question which is related to but is different which is > When I'm moving through air, am I increasing the air pressure by pushing air > together say when I move my arm?


title: Calculating the magnetic flux density of a thin, flat conductor

text: I am currently preparing for a physics exam and I have troubles understanding how to solve the following exercise: > A current that has the constant current density $\vec{j}$ flows through a > thin, flat conducter of width $w$, height $h$ ($h \ll w$) and very large > length. Calculate the magnetic flux density $\vec{B}$ in a point $P$ which > has distance $d$ to the midline of the conductor (see this figure). The standard solution suggests to divide the conductor into infinitesimal current filaments and then to integrate. They find that a infinitesimal current filament with current $\mathrm dI$ has the magnitude: $$\mathrm dB = \frac{\mu_0 \mathrm dI}{2 \pi r} \cos(\alpha),$$ where $r = \sqrt{x^2 + d^2}$ is the distance between current filament and $P$ and $\alpha$ is the angle between $r$ and the $y$-axis. I actually thought that starting with Biot-Savart's law would be a good idea, but this doesn't look like Biot-Savart's law. How do I arrive at this formula? If it is Biot- Savart's law, where should I get the factor $2$ and the cosine from? Thanks for any help in advance.


title: Where 2 comes from in formula for Schwarzschild radius?

text: In general theory of relativity I've seen several times this factor: $$(1-\frac{2GM}{rc^2}),$$ e.g. in the Schwarzschild metric for a black hole, but I still don't know in this factor where 2 comes from?


title: Ricci scalar for a diagonal metric tensor.

text: I was wondering if there is a general formula for calculating Ricci scalar for any diagonal n by n metric tensor?


title: What is the refractive index of air at high temperatures (> 200 °C)

text: Is there a method/formula to calculate the refractive index of air at high temperature (e.g. 400 °C - 800 °C)? I could only find formulas specified up to 100 °C and our physicist at university couldn't help me out, so I bother you now..


title: What might be a satisfying criterion to qualify something as having physical existence?

text: I'm a philosophy undergraduate student who has an interest in, and respect for (as most people do) what physicists know about the world. In philosophy, the traditional criterion of physical existence has been extension in some dimension of space (not necessarily the three most familiar dimensions). For example, we might say that no instance of my experience of the colour red (not the wavelengths of light that correspond to red) physically exists since no instance of my experience of red has extension in space. However, I wouldn't be surprised to learn that there are some things that do not have extension in some dimension of space, and if that's the case, then the traditional criterion fails. I could have asked this question on the philosophy stack, but I'm more curious about the properties that a physicist might say physical things (the objects of study in physics) have that non-physical things (ideas, experiences, etc) do not have.


title: Basic understanding of stress tensors in a fluid

text: So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are these: what precisely is a stress tensor in the context of fluid dynamics, and why is it the case that stress tensors are necessarily symmetric in this context? One of the resources I found referred to the entries of the stress tensor matrix as being the representations of the directional stresses on an infinitesimal cube in a flow, supposing that this cube of fluid is static. This would explain the symmetry of the tensor (since the fluid cube is not rotating, the shear stresses would have to be equal in magnitude at the corners), but it would also indicate to me that the perpendicular forces (the diagonal elements of the tensor) must all be zero, since the particle is also not translating in the fluid. I know this is incorrect, but why? Disclaimer: I have never actually taken a course in physics. I'm in a graduate mathematics program, and this is my first time encountering any of these concepts.