Calculate quadrotor propeller torque due to aerodynamic drag

I'd like to know the right formula to calculate torque of a quadrotor propeller (propeller pitch is constant) due to aerodynamic drag in terms of angular velocity $ \omega $. For the sake of my simulation I'm assuming `APC 10x4,7` propellers. I'd like to refer to UIUC Propeller Data Site, so please assume only the set of data given there is available while answering. I know really little of aerodynamics, I'm planning to design a control system in Simulink later on, but would like to derive a non-linear mathematical model first. Please note I need to use data from UIUC Propeller Data Site for APC 10x4,7, which means the propeller below. I'm guessing all is there to derive a formula $\tau_d(\omega)$, I just don't understand how to do it. ![Top view of APC 10x4,7 propeller, taken from \[UIUC Propeller Data Site\]\(http://www.ae.illinois.edu/m-selig/props/propDB.html#APC\)](http://i.stack.imgur.com/hAWT6.jpg) ![Side view of APC 10x4,7 propeller, taken from \[UIUC Propeller Data Site\]\(http://www.ae.illinois.edu/m-selig/props/propDB.html#APC\)](http://i.stack.imgur.com/02YnN.jpg) **EDIT:** By now I figured out, lift force of a wing can be calculated as: $ F_T = C_T \rho A \frac{V^2}{2} $, where $C_T$ is lift coefficient, $\rho$ means air density, $A$ is wing total area and $V$ is relative air speed. I also asked a friend who told me, for a helicopter propeller this transforms into: $F_T(\omega) = C_T \rho A r^2 \omega^2 = C_T \rho \pi r^4 \omega^2 $, with $A = \pi r^2$ being the area of the rotor, $r$ being its radius and $\omega$ being the propeller's rotational velocity. I guess, assuming the latter formula is true, one could derive one for drag force from it substituting $C_D$ for $C_T$ and rotor blade's frontal area (its projection, rather) for $A$. Could this be right? Thrust and drag coefficients from UIUC:               RPM    CT       CP     2377   0.1039   0.0431     2676   0.1058   0.0437     2947   0.1059   0.0437     3234   0.1083   0.0444     3494   0.1096   0.0450     3762   0.1121   0.0460     4029   0.1136   0.0466     4319   0.1155   0.0474     4590   0.1177   0.0484     4880   0.1199   0.0494     5147   0.1213   0.0500     5417   0.1228   0.0508     5715   0.1239   0.0513     5960   0.1253   0.0520     6226   0.1261   0.0524     6528   0.1274   0.0531      Geometry data from UIUC below. I believe, $c$ is aerodynamic chord length, $R = 0.127m$ (propeller radius), $\beta$ is propeller's pitch at the respective point. I believe I can calculate propeller's side area using simple quadratic integration rule. But first I need to know my reasoning is right. ![taken from \[UIUC Propeller Data Site\]\(http://www.ae.illinois.edu/m-selig/props/propDB.html#APC\)](http://i.stack.imgur.com/nQUsv.png)               r/R    c/R     beta     0.15   0.109   21.11     0.20   0.132   23.90     0.25   0.156   24.65     0.30   0.176   24.11     0.35   0.193   22.78     0.40   0.206   21.01     0.45   0.216   19.00     0.50   0.223   17.06     0.55   0.226   15.33     0.60   0.225   13.82     0.65   0.219   12.51     0.70   0.210   11.36     0.75   0.197   10.27     0.80   0.179   9.32     0.85   0.157   8.36     0.90   0.130   7.27     0.95   0.087   6.15     1.00   0.042   5.04


Is the number 1 a unit?

In dimensionless analysis, coefficients of quantities which have the same unit for numerator and denominator are said to be _dimensionless_. I feel the word _dimensionless_ is actually wrong and should be replaced by "of dimension number". For example, the Mach number is of dimension one. Many people write, for this case: > Mach-Number | Dimension: "-" | Unit: "1" As mentioned before, I would say 'Dimension: "1"' in this place. But what about the unit? $\text m/\text s$ divided by $\text m/\text s$ is equal to one. But is the number one a unit by definition? Or should one say that the Mach number has no unit and therefore 'Unit: "-"'?


Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?

Consider the Schwinger-fermion approach $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ to spin-$\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the enlarged Hilbert space and gauge redundancy complicate our symmetry analyses. Now let's take the translation-symmetry as an example. The unitary translation-symmetry operator $D$ is defined as $D\psi_iD^{-1}=\psi_{i+a}$, where $\psi_i=(f_{i\uparrow},f_{i\downarrow}^\dagger)^T$ and $a$ is the lattice vector. As we know, the transformation $\psi_i\rightarrow \widetilde{\psi_i}=G_i\psi_i(G_i\in SU(2))$ doesn't change the spin operators and the projective opearator $P=\prod_{i}(2\hat{n}_i-\hat{n}_i^2)$(Note here $P\neq \prod _i(1-\hat{n}_{i\uparrow}\hat{n}_{i\downarrow})$). Similarly, in the new basis $\widetilde{\psi_i}$, we can define another translation-symmetry operator $\widetilde{D}$ as: $\widetilde{D}\widetilde\psi_i\widetilde{D}^{-1}=\widetilde\psi_{i+a}$. But $D\widetilde\psi_iD^{-1}=G_i\psi_{i+a}\neq \widetilde\psi_{i+a}$, which means that **$\widetilde{D}\neq D$, the translation operators depend on the 'fermion basis' we choose**. Does this imply the translation operators unphysical? But the translation operators should be physical, so are they equivalent in the physical subspace, say for any physical spin-state $\phi=P\phi$, does $\widetilde{D}\phi=D\phi$? If this is true, then how to prove it? Thanks in advance.


A commutation problem in Hubbard model

Does the Hubbard Hamiltonian $$H=-t\sum_{\langle ij\rangle \sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.+U\sum_{i}n_{i\uparrow}n_{i\downarrow}$$ commute with $\sum_{i}\mathbf{S}_i^2$? where $\mathbf{S}$ is the spin angular momentum.


How the car keys prevent the shock caused by Static Electricity

I was reading around about the Radio Frequency and its effect on the GAS/Petrol forecourts, suddenly i came across something posted by someone, "I personally went on a trip with some friends, one who received a nasty shock every time he tried to enter the car. Since he was the only one receiving these shocks I theorize that the source of his shocks were the shoes he was wearing. I told him to hold his key out toward the door and a spark jumped between the key and the body of the car. This prevented him from receiving the painful shock. But I could see how this could also ignite a gas fire." I couldn't understand how the static electricity could cause the shock from the car ? and how could be the source of this is Shoes ? and how holding the key could prevent someone from this shock ?


How do I recover the 1D wave equation from the Lagrangian?

Consider small displacements, $y(x,t)$, of an element of a string (circled in red and shown below) from equilibrium. ![enter image description here](http://i.stack.imgur.com/T8tnU.png) The force balance in the vertical direction yields: $$+\uparrow \Sigma F: -T\sin(\theta_1)+T\sin(\theta_2)=(mdx)\frac{\partial^2 y}{\partial t ^2}$$ Where we (well.. my prof) have equated the tension on either side of the element because we are considering the case in which there is no horizontal acceleration and because we are making a small angle approximation $$T_{left}\cos(\theta_1)=T_{right}\cos(\theta_2).$$ After linearizing and simplifying, we end up with the 1D wave equation: $$c^2 \frac{\partial^2 y}{\partial x ^2} = \frac{\partial^2 y}{\partial t ^2}$$ where $c^2 = \frac{T}{m}$. For the Lagrangian approach, I am able to write down the kinetic energy of the element and of the entire string (I think): For the string: $$T=\frac{1}{2}\int_{_0}^{^L} \rho \dot y(x)^2 dx.$$ Where $y$ is a function of $x$ because we can have a different velocity in the $y$ direction at every point $x$. For an element of the string: $$T=\frac{1}{2}m(dx)\dot y^2.$$ I am stumped at the potential energy for both the element and the string as a whole and I can't seem to figure out how to recover this wave equation from first principles. The potential energy should be proportional to the displacement $y$ from equilibrium but from what I'm seeing online it doesn't look anything like the potential energy due to a spring. What I've seen here is that "for a taut string, the stored energy of system described by the height function $u(x)$ is assumed to be proportional to the change in length": $$\int_a^b (\sqrt{1+u'^2}-1)dx$$ Which is a complete mystery to me. So my question is how do I recover the EOMs for a taught string using the Lagrangian? I am also interested in how this particular case can be recovered from a more general solution but I may not have the background to understand that.


How can you have positive electricity

In a circuit, you have two opposite charges. One would be negative, which would consist of electrons. The other would be positive charge, and protons carry this. But protons aren't mobile, so how does this work? This might be tagged as a duplicate, as it is similar to a question posted here, but that post didn't have a very clear answer.


Looking For The Derivation Of Gellmann-Nishijima Relation

How is the Gellmann-Nishijima relation $Q=I_3+\frac{Y}{2}$ derived? Either a derivation or a link will be helpful.


Why can you touch two positively charged rods?

For a physics lab on the Triboelectric effect, we rubbed two rods with fur which gave both of them a positive charge. We then brought them close together, and they obviously repelled. We then held one rod down firmly and touched the rods together. We were able to do so. Why? Coloumb's Law says that as the distance between two like charges become zero, the force becomes infinite. Using superposition on all the charges in the rod, shouldn't the infinite force have prevented us from touching the rods together?


Has a system of natural units been designed for human-like scales?

A long time ago I was thinking about how the Imperial system of measurements is arbitrary and annoying, and I decided to design the best system of units ever (I wasn't very old then). I worked on this idea occasionally for years without making any progress. When I finally got serious about it, I discovered Planck Units and that seemed to settle the issue. Now my problem is that Planck Units are so small that they can't be used for "normal" things without huge exponents, most of these being very different depending on the quantity being measured. Solving human-scale equations with these units by computer would thus either be very inaccurate due to floating- point errors (for numbers that are even within range) or very slow due to the need for extended precision. They also go "up" but not "down", which renders almost half of the possibile signed floating-point values as useless. I considered the idea of creating a new system by raising each Planck Unit with a standard exponent or multiplier, but I think this would bring some into an acceptable range but not others. So what I want to know is whether a system of natural units exists that uses units that are appropriate for work with technology that humans can interact with, or can be made appropriate without introducing too many arbitrary elements.


Why the speed of light is represented by $c$?

In almost every textbook, I've found that the speed of light is $c \approx 3 \times 10^8\: \mathrm{m/s}$. I wonder why it's just $c$ ?


Is it possible that matter/antimatter collisions emit resonance frequencies of H/He? I am researching a high energy phenomenon that occurs between the frequencies of 1200-1580 MHz. This come from a very very long way away though, so the signal does change a little bit. I'm trying to find out if matter/antimatter collisions could be the cause, and my mass estimates are asteroid sized masses of matter and anti-matter colliding, and it all seems very promising. Other than the fact that such collisions probably emit high energy Gamma Ray Bursts, not Radio Bursts. However, I found this to be interestingly near the resonance frequency of hydrogen (1420MHz) and I saw a figure that said Helium was 1400, but I'm not sure if that is correct. Is it possible for Matter-Antimatter collisions to emit the resonance frequency of there respected elements? Such as Hydrogen-antiHydrogen collision, emitting a radio signal at 1420 MHz?


Baryogenesis only at the Planck scale, or none at all?

I can think of three general ways of explaining why the universe contains more matter than antimatter: (1) Near the Planck time, the universe had zero baryon asymmetry, but at some later time, determined by some GUT energy scale, the Sakharov conditions were satisfied, and the baryon asymmetry became nonzero. (2) Nonconservation of baryon number occurs only at Planckian energies. Near the Planck time, the baryon asymmetry evolved from zero to some nonzero value. (3) The Sakharov conditions have never been satisfied. The baryon asymmetry has always been nonzero, and has simply scaled as expected. (Apparently one expects $\Delta n=n_B-n_\bar{B}\propto s$, where $n$ is number density and $s$ is entropy density). It seems like most theorists are interested in #1, but is there any reason why 2 and 3 aren't possible? 2 seems pretty reasonable, since for the reasons given in this answer, we have good reasons to think that baryon number is not conserved under Planckian conditions. 3 also seems reasonable to me, since even if baryon number is nonconserved at Planckian energies, that's only one of the three Sakharov conditions. I don't see any obvious fine-tuning objections to #3, since the scaling of baryon asymmetry with cosmological expansion isn't particularly drastic (not an exponential decay or anything). Is there something unphysical about maintaining $\Delta n\propto s$ all the way back to the Planckian era? Some people might object to #3 on aesthetic grounds, since we "expect" the initial conditions of the universe to be symmetric, but that seems weak to me. After all, we don't object aesthetically to the fact that homogeneity was an imperfect symmetry of the early universe, and we even accept that the early universe was in a thermodynamically unlikely state.


How do we know that $F = ma$, not $F = k \cdot ma$

It seems intuitive that $a\; \propto \frac{F}{m}$, as the greater the force that is applied on an object, the greater its acceleration will be. Inversely, the greater the mass of the object, the slower the acceleration will be. However, when rewriting proportions as equations, you must introduce a constant of proportionality, and in this case of a direct proportion, if $a \propto \frac{F}{m}$ then when rewriting as an equation you will have $$a = k\cdot\frac{F}{m}$$ In order to get the standard formula $F = ma$ this constant must be $1$. However, how do we know that this is the case? How do we know that the constant isn't $2$ and the formula $F = \frac{1}{2}ma$, for instance?


Transformer: primary side & secondary side current 180 degree out of phase

I am a novice in electrical engineering. I notice that in transformer the secondary side current & current referred to as primary are 180 degree out of phase from each other. But why it is so, I don't know.I did some theoretical analysis on this topic but no phasor analysis(why 180 degree). So, please give me suggestions. I also thought a phasor analysis but I need 10 reputations.


Finding the position of a planet between two other planets of known mass and distance

Here is the question: > A planet with mass $m$ and a second with mass $M$ are separated by a > distance $d$. A third planet with mass $m_3$ happens to be midway between > $M$ and $m$. Where could the third planet be positioned (distance from the > larger planet $M$ in meters) so that the net gravitational force is zero? My confusion lies with how to solve for the position of the third planet. I am given this equation to find the force between two planets at a given distance $r$: $$F = {GMm}/{r^2}$$ With this I can then set the sum of the two forces to zero: $$0 = {GMm_3}/r_1^2 + {Gmm_3}/r_2^2$$ My confusion lies in the fact that both $r_1$ and $r_2$ are unknown. However, we do know that $r_1 + r_2 = d$. But I am confused with how to solve for either $r_1$ or $r_2$.


decay with initial velocity:

decay with initial velocity: All static particle decay to particles with momentum distribution in the same shape like ![enter image description here](http://i.stack.imgur.com/i8gD5.jpg) that's implied by structure function and calculated by MG. but the decay of W emits e like (similar to) ![enter image description here](http://i.stack.imgur.com/mXogV.jpg) Why that? The only explanation is the W generated by p--p has initial velocity. The superficial explaination of resonance are contradictory to the logical deduce. This will modified the mass of it to 15 Gev. Is it reasonable?


Which pressure to use in the $T~ds$ equation?

Let's say I have an adiabatic, rigid, open container that has an amount of air at some pressure, $P_\text{cv}$, and some temperature, $T_\text{cv}$. I have heated pressurized air coming into the container. I am trying to calculate the change in specific entropy of the air in the container. I use the $T~ds$ equation, but I am not sure which pressure to use. The equation used is $$T\frac{ds_\text{cv}}{dt}=\frac{du_\text{cv}}{dt}+P\frac{dv}{dt}.$$ Would the pressure term in the equation be the pressure of the incoming air, or would it be the pressure of the air in the control volume?


Math or Physics degree?

I am hoping to become a physicist focusing mainly on the theoretical side in the future. I am trying to decide whether to go for a physics or math undergrad course. Assuming that I am capable of doing either, what are the pros and cons of either route? I know that mathematics is essential to doing physics, and in most math courses, there are applied math modules that are very much related to physics. Also that many research physicists have math degrees. But surely there is a reason why people choose the physics course over the math course and vice versa? Thank you.


273 + degree Celsius = Kelvin. Why 273?

Temperature conversion: 273 + degree Celsius = Kelvin Actually why is that 273? How does one come up with this? My teacher mentioned Gann's law (not sure if this is the one) but I couldn't find anything related to this, which law should it be?


Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian

The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's _Classical Mechanics._ Adding the total time derivative of a function of $q_i$ and t to the Lagrangian will not change the the Euler-Lagrangian equation. So if we make the following change to Lagrangian, $$L'(q,\dot{q},t) = L(q,\dot{q},t) + \frac{dF(q_1,q_2,...,q_n,t)}{dt}$$ we can get $$ \frac{d}{dt}\frac{\partial{L'}}{\partial{\dot{q_i}}} - \frac{\partial{L'}}{\partial{q_i}} = 0 $$ from $$ \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q_i}}} - \frac{\partial{L}}{\partial{q_i}} = 0 $$ How can we get the corresponding Hamiltonian equation part? This is to prove $$ \dot{p'_i} = \frac{\partial{H'}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H'}}{\partial p'_i} $$ from $$ \dot{p_i} = \frac{\partial{H}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H}}{\partial p_i} $$ where $p'_i = \frac{\partial L'}{\partial \dot q_i}$. **Edit** The corresponding $H'$ is $$ H' = \sum_k{p'_k \dot{q_k}} - L' $$ where $p'_k = \frac{\partial L'}{\partial \dot q_k}$.


running of coupling constant as a function of distance?

There are many papers about the running of coupling strength as a function of momentum/energy scale, but are there any experimental papers about coupling strength as function of distance? Also, are there any good books about this topic? oops ,really sorry i forgot to specify "experimental"paper


QCD mass gap finite temperature

QCD has a mass gap. Does heating up QCD to a finite temperature change this fact, or is it a property which is independent of temperature?


Entanglement entropy at finite temperature

The entanglement entropy is computed on a constant time slice. At finite temperature the time is periodic. How then can the entanglement entropy be calculated on a constant time slice? For example this is done in Eq. (3.34) of the paper 0605073.


Why do batteries become weaker

I was thinking about my phone (Sony Xperia Z2) and when I put it to charge and remove it, the phone says "Please remove the charger, to save battery life," which is very intuitive question for a middle school student because of my lack of scientific knowledge Why do batteries become weaker as I use it more and charge it more?


Interference of EM Waves with Orbital Angular Momentum

If you have two coherent collinear e-m beams of same frequency and polarization, but 180 degrees out of phase, they will destructively interfere. If you introduce orbital angular momentum of L=3 (three revolutions per wavelength) to one beam and L=1 to the other beam, ceteris paribus, will they still destructively interfere in the exact same manner?


Energy definition in special relativity

I'm going through the early homework assignments for my special relativity course and I've got myself a little confused about energy. I've got a basic understanding of what the 4-momentum is, having defined it as $m\dfrac{dx^{\mu}}{d\tau}$, and shown that this is equal to $m\gamma (|\vec{v} |)(c,\vec{v})$ where $\vec{v}$ is the classical velocity in the inertial reference frame associated to the above Cartesian coordinates. Now one of my assignments has began by saying > Denote the components of $p^{\mu}$ as $(E/c, \vec{p})$... I don't understand why the time component of the 4-vector is being _denoted_ as $E/c$. You can't just go arbitrarily denoting quantities by other quantities! So I'm left wondering whether this is a definition of relativistic energy, or if it follows from some other result in special relativity? Thanks for any clarification.


Fighting a black hole: Could a strong spherical shell inside an event horizon resist falling in to the singularity?

As a thought experiment imagine an incredibly strong spherical shell with a diameter a bit smaller than the event horizon of a particular large black hole. The shell is split into two hemispheres, and the two hemispheres are released from either side of the black hole such that they will meet and join inside the event horizon, forming a stationary uniform shell around the singularity. Of course, we consider that the shell is flawless, and exactly centred around a stationary singularity - clearly this is an unstable equilibrium but we imagine it is possible. Is is possible that the structural forces provided by such a shell could be strong enough to resist the extreme gravity conditions inside the event horizon, and hence avoid being sucked into the singularity? (For the purposes of this question we can consider taking the shell to extreme levels of strength, e.g. consider that the shell is made from a theoretical maximally densely packed neutronium-like material that is strongly resistant to any further compression due to the Pauli exclusion principle).


The Moon is slowly moving away from the earth. Does this mean that a total solar eclipse wasn't possible at some point in earth's history?

When the moon was closer to earth, was it still possible to witness a total solar eclipse millions of years ago? Or was the view-able space so small that it was impractical to even witness it?


What do massive photons have to do with superconductivity?

I keep reading that the idea of massive photons leads to an explanation of the Meissner effect but I fail to see how photons are involved with the repulsion of fields inside a superconductor. How can a photon effect the electromagnetic fields inside a superconductor?


Parallel universe and Infinite monkey theorem

Is the Infinite monkey theorem helpful for determining the existence of the very same our universe somewhere else?


Complete list of nuclear fusion reactions

Could anyone suggest good source of all fusion reactions and their cross section vs energy graphs, including ones which were ruled out for nuclear fusion reactors (i.e. including endothermic and reactions with only 1 product)?


Are Newton's three laws of motion correct?

New technology brings new ideas with these new ideas we have to look at the old ones. Where else is a better place to start then Newton's three laws of motion! With our common age of technology do we apply these "old laws of physics" into our new age?


Does the length of the sidereal day vary systematically?

I'm confused about some properties of the sidereal day, in particular whether its duration varies systematically over the course of the year.1 It seems to me that that must be the case, but the details are a bit confusing to me. I understand that length of a sidereal day is the time interval between successive transits of the line of equatorial longitude corresponding to a specified right ascension, $R$ (usually $0^\text{h}$). Each sidereal day, this line shifts against the earth's direction of rotation by some amount, $\Delta\alpha_{SID}$, to a new position, so that the sidereal day is always shorter than the earth's rotational period on its axis (the "stellar day"). What's confusing me is that the length of the sidereal day is often described as if it were constant; but it can't be, nor can the corresponding changes in the equatorial longitude of $R$, $\Delta\lambda_{SID}$, even assuming a constant rate of precession:2 If the sidereal day has a constant length, $\Delta\alpha_{SID}$ must be constant, but then $\Delta\lambda_{SID}$ will vary (between $\Delta\alpha_{SID}/\cos\varepsilon$ and $\Delta\alpha_{SID}\cdot\cos\varepsilon$) as a consequence of the varying equatorial latitude at which the the line of equatorial longitude corresponding to $R$ intersects the ecliptic. But this is impossible, since equatorial longitude should change uniformly, so that $\Delta\lambda_{SID}$ should be the same for each (same-length) sidereal day. The reverse can't be the case either: that each sidereal day, the position of the line corresponding to $R$ shifts by the same fixed amount, $\Delta\lambda_{SID}$, along the ecliptic. If that were the case, $\Delta\alpha_{SID}$ and thus the length of the sidereal day would vary (between $\Delta\lambda_{SID}\cdot\cos\varepsilon$ and $\Delta\lambda_{SID}/\cos\varepsilon$) resulting in a sidereal day of varying length; but if the length of the sidereal day varies $\Delta\lambda_{SID}$ can't be constant. So the only conclusion I can come to is that _both_ $\Delta\lambda_{SID}$ and the duration of the sidereal day, and thus $\Delta\alpha_{SID}$, vary, subject to the constraints imposed by spherical trigonometry: $$\Delta\lambda_{SID}\cdot\cos\varepsilon<\Delta\alpha_{SID}<\Delta\lambda_{SID}/\cos\varepsilon$$ and $$\langle\Delta\lambda_{SID}\rangle=\langle\Delta\alpha_{SID}\rangle$$ Is this correct?3 Do both of these quantities vary systematically in this way, with these properties.4 Is "the" sidereal day really a _mean_ sidereal day?5 * * * (1) Though a distinction between a "mean sidereal day" and an "apparent sidereal day" is sometimes mentioned, the terminology I've come across is confusing, and descriptions that do discuss variations seem to be referring to different, unspecified, phenomena. Here, I am asking about the properties of an idealized "mean sidereal day", specifically whether even this "mean" day has systematic variation in duration. (2) My thinking went like this. Take the plane of the ecliptic as reference. Then precession is just the a rotation of the equatorial coordinate system around an axis perpendicular to that plain. The earth's rotational axis is not collinear with that axis, so that the great circle of the equator intersects the ecliptic plane at two points, one of which can be used as a reference to "track the progress" of precession. Clearly, the progress of that point along the ecliptic — i.e., the change in the intersection's ecliptic longitude, $\Delta\lambda$ — has a direct one-to-one relationship with the rate of precession. What's not clear to me is how this relates to the corresponding change in right ascension, $\Delta\alpha$. But what's being measured in that case is the distance between the reference point and some initial point on the ecliptic, within the equatorial system, an arc that passes through different declinations and thus $\Delta\alpha$ values that vary from $\Delta\lambda$ as described above. (Moreover, this seems to me the only way that $\langle\Delta\lambda\rangle$ and $\langle\Delta\lambda\alpha\rangle$ to be the same, which must be the case for the method described in (3) to work.) (3) I'm reasonably confident that the first is true, but I'm not sure how to arrive at the second; though it must also be true, since otherwise it would not be possible to determine the precessional period (in sidereal days) from $24^\text{h}/\langle\Delta\alpha_{SID}\rangle$, as is commonly done. (4) The corresponding (by definition, assuming constant rotational rate for the earth) _constant_ quantities for the stellar day must thus be $$\Delta\lambda_{ST}=\langle\Delta\lambda_{SID}\rangle\cdot\frac{d_{ST}}{d_{SID}}=\Delta\alpha_{ST}=\langle\Delta\alpha_{SID}\rangle\cdot\frac{d_{ST}}{d_{SID}}$$ where ${d_{SID}}$ is the length of a mean sidereal day, and ${d_{ST}}$ the length of a stellar day. (5) Or perhaps a _mean_ mean sidereal day. (A very mean sidereal day?)


Can we transport energy over infinite distances through vacuum using light?

I know that light (or electromagnetic radiation in general) attenuates in intensity as the square of the distance it travels.   * Why does it attenuate?    * Are the photons being scattered by the medium they pass through? I also know that the energy carried by light _quanta_ is a function of their frequency only.   * If the answer to B is yes, then can we possibly transmit light energy (and information) over infinite distances through a perfect vacuum? I may be wrong on some assumptions, so please enlighten me.


Deriving expression for motional emf

I'm having a trouble deriving the expression for emf caused by the Lorentz forces in a moving conductor. Consider the following classical example of a slidewire generator which consists of U-shaped conductor and a rod which is able to slide on the conductor. The system is placed in a uniform non-varying magnetic field. ![Illustration](http://i.stack.imgur.com/FifpA.jpg) I know that because the rod is moving with a certain speed, the charge carriers (i.e. electrons) are also moving, and because of that they experience Lorentz force, which eventually causes them to concentrate on one end of the rod and leaving the positive ones at the other. The force itself is equal: $$|\mathbf{F}|=q |\mathbf{v} \times \mathbf{B}|= q v B \sin \theta$$ (I include the sine on purpose, for the general case) We can describe a field: $$|\mathbf{M}|=\frac{|\mathbf{F}|}{q}=|\mathbf{v} \times \mathbf{B}|= v B \sin \theta$$ The work done by the non-electrostatic Lorentz force on unit of charge is: $$\mathcal{E}=|\mathbf{M}| L=L |\mathbf{v} \times \mathbf{B}|= L v B \sin \theta$$ (where $L$ is the length of the rod) We can multiply this expression by $\frac{\mathrm{d}t }{\mathrm{d} t}$: $$\mathcal{E}=|\mathbf{M}| L \frac{\mathrm{d}t }{\mathrm{d} t}=L |\mathbf{v} \times \mathbf{B}|\frac{\mathrm{d}t }{\mathrm{d} t}= L v B \sin \theta\frac{\mathrm{d}t }{\mathrm{d} t}=B \sin \theta \frac{L v~\mathrm{d}t }{\mathrm{d} t}$$ Because $L v ~ \mathrm{d} t = \mathrm{d}S$ we can rewrite it as: $$\mathcal{E}=B \sin \theta \frac{\mathrm{d}S }{\mathrm{d} t}$$ So because of the Lorentz force we have a sine instead of a cosine like in Faraday's law $$\frac{\mathrm{d} \Phi}{\mathrm{d} t}=\frac{\mathrm{d} (B S \cos \theta)}{\mathrm{d} t}$$ as we would expect. Where is my mistake here?


A Global Measure on a Gravitational System

I am a computer science engineer and I am trying to simulate the gravity on my computer; several objects with different masses randomly distributed in a Three-dimensional space. I would like to compute a global measure (scalar) that describes the dynamic of that system at each instant T. My question is what kind of measure can I use?


Does quantum mechanics require classical measurement apparatus?

I am trying to learn quantum mechanics and I have a question. Landau, in his quantum mechanics book says that it is in principle impossible to formulate basic concepts of quantum mechanics without using classical mechanics. But I learned a bit about consistent histories and I think it makes sense to talk about closed quantum mechanical systems using this language if I didn't understand something wrong. Then I have listened to a lecture video of a very famous physicist (Milner prize winner) who says that quantum mechanics does not make sense if you can't go out of the system and measure it using an infinite apparatus (he considers decoherence and when you calculate expectation value of an observable you see that you can talk about classical probabilities, that is non-diagonal terms disappear). Hence wavefunction of universe does not make sense. I believe what he says is actually same as Landau. Is it meaningful to talk about closed systems which does not interact with outside such as universe? What does framework of consistent histories says about this?


AdS3 soliton of Witten - for Hawking-Page transition

> Are there explicit AdS$_3$ soliton solution? in the sense of Witten's Anti De Sitter Space And Holography and Hawking-page transition paper, by doing a $$\tau_E, y ,r \to y, \tau_E ,r$$ from a geometry with a Euclidean time $\tau_E$ cigar space and a compactified $y$ circle ![enter image description here](http://i.stack.imgur.com/UshaH.png) to a geometry with a $y$ cigar space and a compactified Euclidean time $\tau_E$ circle ![enter image description here](http://i.stack.imgur.com/JmbBC.png) > If yes, with a AdS3 soliton solution, what are the form of solutions? > > If not, without a AdS3 soliton solution, what are the obstructions?


Does high entropy means low symmetry?

According to Bogolubov postulate (various texts name it differently) in Non- equilibrium thermodynamics, the number of needed parameters to describe our system is decreasing with time, and finally at infinity, once our system reaches its equilibrium state, we will need only the usual three thermodynamic parameters $T,V,N$ to define the state. Ok, this seems to be very logical, however, how does entropy fit in there? Generally speaking we can say that entropy describes the disorder in our system, and the higher the entropy, the less order we have, and more disorder means less symmetry in the system. (Or is this statement wrong?)   And when a system has high symmetry, we need less parameters to describe it, which uncovers a contradiction than I can't understand: A system has reached equilibrium state, when the system's entropy reaches its maximum (we are talking here about isolated systems only), but maximum entropy means maximum disorder, maximum disorder means the need for a lot of parameters to describe it due to luck of order. But that is the opposite of what the postulate above states. So where is the mistake in this statement? (I asked my professor and he answered that defining entropy in a non-equilibrium state is, for now, an open question but I think that the answer should lie somewhere else...) (in case there is really some relation between Entropy and Symmetry, please provide some math how this is accomplished)


Understanding Work and the conservation of energy

_We have a car with a mass of $780 kg$ with travels with a speed of $50 km/h$. The car brakes and after $4,2m$ is stops completely. Warmth is created. Calculate the friction._ I solved this easily, by simply filling in the data like a headless chicken (the solution I'm showing is from the correction model, I got the same answers but I did it without writing anything down, so): $E_{total 1} = E_{total 2} $ $0.5mv_1^2 = 0.5mv_2^2 + Q = 0+ F_w . s $ $0.5 . 780 . (50/3.6)^2 = F_w . 4.2 $ $F_w = 1,8.10^4 N$ I didn't have any trouble with this, I got the same answer, but than I started thinking about it, and I am increasingly finding the solution illogical. The LHS is completely logical to me, but the right hand side. Normally, the formula of work is the resultant force times the distance d. But, after you've travelled the 4,2 meters, your $F_w$ is $0$. So how can you say that $F_w \times d$ = LHS, because by the time the total $d$ (4.2) is reached, the resistance has already turned into 0. What is the logic behind this? I know this is high school level so it is simplified, but even then, knowing it's simplified a lot, I don't understand the logic. Can someone explain?


Equipartition theorem for flowing gas

If an ideal gas is flowing with a velocity $v$, how is the equipartition theorem applied. Normally, we can say that $\frac{1}{2}mv_{x,rms}^2=\frac{1}{2}k_BT$. We can do the same thing for $v_y$ &c. But, when a gas is flowing in the x direction, I don't think that $v_{x,rms}=v_{y,rms}$. I'm not too sure of this, the distribution may be such that the rms velocities are preserved. If the rms velocities aren't preserved, obviously we cannot use $\frac{1}{2}k_BT$ as the temperature is the same (or is it?). So how does one analyse such a situation with the equipartition theorem? I'm not very good at Hamiltonian mechanics, so Wikipedia isn't helping. Question sparked off by Air velocity in a double-skin facade


Derivation of the Hartree-Fock equations. Functional varitation

I asked this question at chemistry.stackexchange.com, but the attendance of that source is a little bit lower than here. I would like to ask a question about mathematical derivation of the HF equations. Some moments related to the functional variation are not clear for me. I read a book of Szabo and Ostlund "Modern quantum chemistry". There the procedure of the Single determinant energy minimization is presented. Omitting the whole procedure i present the most obscure moment. Given the single determinant $| \Psi_0 \rangle = | \chi_1 \chi_2 \ldots \chi_a \chi_b \ldots \chi_N \rangle$, the energy $E_0 = \langle \Psi_0 | \mathscr{H} | \Psi_0 \rangle$ is a functional of the spin orbitals $\\{ \chi_a \\}$. $E_0$ is the expectation value of the single determinant $\vert \Psi_0 \rangle$, $E_0[\\{ \chi_a \\} ] = \sum\limits_{a=1}^N [a\vert h \vert a] + \frac{1}{2} \sum\limits_{a=1}^N \sum\limits_{b=1}^N [aa \vert bb] - [ab \vert ba]$, Here $[ij|kl]$ is the chemist's notation for a two-electron integral involving spin orbitals: $[ij|kl] = [\chi_i \chi_j | \chi_k \chi_l] = \int d\mathbf{x_1} d\mathbf{x_2} \chi_i^{*}(\mathbf{x_1}) \chi_j(\mathbf{x_1}) r_{12}^{-1} \chi_k^{*}(\mathbf{x_2}) \chi_l(\mathbf{x_2}) $ And one-electron integral is $[i|h|j] = \int d \mathbf{x_1} \chi_i^{*}(\mathbf{x_1}) h(\mathbf{r_1}) \chi_j(\mathbf{x_1})$ This equation can be varied (1) $\delta E_0 = \sum \limits_{a=1}^N [\delta \chi_a | h | \chi_a] + [\chi_a | h | \delta \chi_a] $ $+ \frac{1}{2} \sum \limits_{a=1}^N \sum \limits_{b=1}^N [\delta \chi_a \chi_a | \chi_b \chi_b] + [ \chi_a \delta \chi_a | \chi_b \chi_b] + [ \chi_a \chi_a | \delta \chi_b \chi_b] + [\chi_a \chi_a | \chi_b \delta \chi_b]$ $-\frac{1}{2} \sum \limits_{a=1}^N \sum \limits_{b=1}^N [\delta \chi_a \chi_b | \chi_b \chi_a] + [ \chi_a \delta \chi_b | \chi_b \chi_a] + [ \chi_a \chi_b | \delta \chi_b \chi_a] + [\chi_a \chi_b | \chi_b \delta \chi_a]$ Authors suggest a reader as an exercise to manipulate this equation for $\delta E_0$ to show that (2) $\delta E_0 = \sum \limits_{a=1}^N [\delta \chi_a | h |\chi_a] + \sum \limits_{a=1}^N \sum \limits_{b=1}^N [\delta \chi_a \chi_a | \chi_b \chi_b] - [\delta \chi_a \chi_b | \chi_b \chi_a] + \text{c.c.}$ It is clear that the first sum in (1) can be easily converted to that in (2) because $[\delta \chi_a | h | \chi_a]^* = [\chi_a | h | \delta \chi_a]$ Analogously for the second sum in (1) one can show that $[\delta \chi_a \chi_a | \chi_b \chi_b]^* = [\chi_a \delta \chi_a | \chi_b \chi_b]$ and $[\chi_a \chi_a | \delta \chi_b \chi_b]^* = [\chi_a \chi_a | \chi_b \delta\chi_b]$ I did the same manipulations with the third sum in (1) and obtained (3) $\delta E_0 = \sum \limits_{a=1}^N [\delta \chi_a | h | \chi_a] + $ $+ \frac{1}{2} \sum \limits_{a=1}^N \sum \limits_{b=1}^N [\delta \chi_a \chi_a | \chi_b \chi_b] + [ \chi_a \chi_a | \delta \chi_b \chi_b] $ $-\frac{1}{2} \sum \limits_{a=1}^N \sum \limits_{b=1}^N [\delta \chi_a \chi_b | \chi_b \chi_a] + [ \chi_a \delta \chi_b | \chi_b \chi_a] + \text{c.c.}$ So, comparing (3) and (2) i suppose that one should show that both terms in each sum are equal, i.e. $[\delta \chi_a \chi_a | \chi_b \chi_b] = [ \chi_a \chi_a | \delta \chi_b \chi_b]$ and $[\delta \chi_a \chi_b | \chi_b \chi_a] = [ \chi_a \delta \chi_b | \chi_b \chi_a]$ So, how to do that?


Finding optimal angle for projectile, taking into account linear (Stokes) drag

If you throw a projectile from the ground at a certain angle, it's not hard to see that, assuming we're in a vacuum, throwing it at 45º from the ground will always make it go farthest before it hits the ground again. I wanted to find out what's the optimal angle if we take into account air resistance. Before I go on, let me say that there's really no point to this question. I'm not trying to solve a practical problem. I just thought it would be fun to try to solve the equations. This is the model I'll use: We'll assume that air resistance is roughly proportional to velocity (aka. Stokes drag; this fails for high speeds/Reynolds numbers, so let's not go there); in symbols, $$\mathbf{F} = -k\mathbf{v}.$$ We'll throw the projectile with mass $m$ from $(0,0)$ with initial speed $v_0$, forming an angle $\alpha$ with the ground, and let's say $\gamma = k/m$ because it will turn up frequently. The differential equations are: $$\begin{align} \ddot{x} + \gamma \dot{x} &= 0 \\\ \ddot{y} + \gamma \dot{y} &= -g \end{align}$$ with initial conditions $(x(0), y(0)) = (0,0)$ and $(\dot{x}(0), \dot{y}(0) )= v_0(\cos \alpha, \sin \alpha)$. Solving (assuming I haven't made any mistakes), we get: $$ \begin{align} x &= \frac{v_0}{\gamma}\cos \alpha (1- e^{-\gamma t}) \\\ y &= (\frac{v_0}{\gamma} \sin \alpha + \frac{g}{\gamma^2})(1-e^{-\gamma t}) - \frac{g}{\gamma}t. \end{align} $$ Now the thing to do would be to solve $y(t) = 0$, substitute that into $x$, differentiate with respect to $\alpha$ and set it equal to $0$. However, that gets messy quickly, because the solution involves the Lambert-W function and it's all a mess. I haven't even tried to substitute into $x(t)$. So finally, my question is: is there a simpler or numerical way to solve this? Is there even a single angle which will always work, or does it depend on the conditions? Can we find that out without actually solving?


What will happen to a compressible fluid filled inside a sealed container when the container begins to accelerate?

Assume the fluid to be air and the container to be a car, will there be a pressure gradient which will result the fluid to stack at one end of the car or there is something else that I'm missing?


General wavefunction and Schrödinger Equation

I'm starting with quantum mechanics and the book I follow (Griffiths) first introduces the wavefunction as the probability density of the position of a 0-spin single particle. Later on I've realized that there is a bigger picture: the state of a system can be expressed as a superposition, which is a linear combination of the vectors of a particular basis, (depending on the observable we choose), with a (maybe continuous) set of coefficients that are probability amplitudes of the eigenvalues. If the chosen observable happens to be the position we obtain the wavefunction. So now I'm wondering the following: is this the definition of a wavefunction or is there a wavefunction for every observable, being this wavefuntion just a map from the possible values of the observable to their probability amplitudes? If so: is the time-dependence of this wavefunction always restricted by the Schrödinger Equation? For example, suppose that in the case of a 2-dimensional system with eigenstates |0> and |1> for a certain observable, with respective eigencalues 0 and 1. We have the superposed state f(0,t)|0> + f(1,t)|1>, where f(state, time) is the "wavefunction" that assigns probability amplitude to each eigenstates. Does this f function have to satisfy the Schrödinger Equation or other constraints?


What is the status of gravitational wave searches?

What is the status of gravitational wave searches such as LIGO?


link between real particles, representation of algebra and Young tableau

I know that different representations of this algebra correspond to different spin. One can sort the representation according to the casimir. For any simple Lie algebra, the operator $$ T^2 = T^{\alpha}T^{\alpha} $$ comutes with all the group generators such that $$ [T^2,T^{\alpha}]=0 $$ So $T^2$ takes a constant value on each irreductible representation. Thanks to Shur's lemma, one knows that $T^2$ is proportional to the identity for all representation. If one considers the rotation algebra $o(3)$, the Casimir takes the form $$ C=\delta^{ij}J^iJ^j=s(s+1) $$ where $J^i$ are the generators of $o(3)$ and $s$ is a half integer that labels each representation of dimension $$ d = 2s + 1 $$ For instance :   * $s=0 \Rightarrow d=1$ and $S_i=0$   * $s=1 \Rightarrow d=2$ and $S_i=\frac{1}{2}\sigma_i$   * $s=2 \Rightarrow d=3$ and $S_i=\Sigma_i$   * ... where the $S_i$ are matrix representations of $J^i$ and $\sigma_i$ are the Pauli matrix. Moreover there is an isomorphism between $su(2)$ and $o(3)$. $\\\$ Now if one looks at the Lorentz algebra, $o(1,3)$, one can do the same work but this time, there are two Casimirs that can be written $$ C_{\pm}=s_\pm(s_\pm+1) $$ Thus one has :   * $(s_+,s_-)=(0,0)\Rightarrow d=1$ scalar   * $(s_+,s_-)=(1/2,0)\Rightarrow d=2$ spinorial chiral (not sure of the english terminology)   * $(s_+,s_-)=(0,1/2)\Rightarrow d=2$ spinorial antichiral    * $(s_+,s_-)=(1/2,1/2)\Rightarrow d=2$ vectorial    * ... And again there is an isomorphism between $o(1,3)$ and $sl(2,\mathbb{C})$. $\\\$ **So my question is :** what is the correct algebra and representation to describe a spin one-half fermion (an electron for instance) ? From what I understand I could choose between $su(2)$, $o(3)$, $o(1,3)$ or $sl(2,\mathbb{C})$ but with a given representation. Same question for a spin one boson ? higher spins... ? $\\\$ My guess would be that :   * spin 1/2 fermion is described by the $su(2)$ in the fundamental representation (labeled by a single box within the Young tableau formalism)   * spin 1 boson is described by the $su(2)$ with the symmetrical combination of two fundamental representation (two box in a single row)   * spin 3/2 is described by the $su(2)$ with the symmetrical combination of two fundamental representation and one antisymmetrical one(two boxes in the first row and one box in the second) $\\\$ I know that this question is confused... and so am I... My goal is to undersand the link between the physical particles, the $su(N)$ algebra (for any $N$), the representation of the algebra and the Young tableaus. So instead of answering the question above, it might be easier to explain my those links... $\\\$ Thanks !


Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian

This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient space so I hope it will get more general mathematical physics interest. We start by casting LQG into the spinor language. Most of the details are not necessary, but we have a set of $N$ spinors $$|z_i\rangle =\left(\begin{array}{c} z^0_i\\\ z^1_i\end{array}\right).$$ In addition to the usual spinor relations, they satisfy _the closure relation_ , which is $$\tag{23} \sum_i |z_i\rangle\langle z_i| =A(z)\mathbb{1}, \qquad A(z):=\frac{1}{2}\sum_i\langle z_i|z_i\rangle.$$ Now the states in the coherent intertwiner framework are holomorphic functions of $\mathbb{C}^{2N}$ that are homogenous functions under $GL(2,\mathbb{C})$ transformations. Thus, they are elements of a Grassmanian $$\tag{81} Gr_{2,N}=\mathbb{C}^{2N}/GL(2,\mathbb{C}).$$ Now, we want a measure on this Hilbert space (and a resolution of unity) so we start by relating the measures on the three spaces above (Eq (93) in that paper): $$\int_{\mathbb{C}^{2N}}\left(\prod_jd^4z_j \right)e^{-\sum_k\langle z_k|z_k\rangle}F(z)=\int_{GL(2,\mathbb{C})}\\!\\!\\!\\!\\!\\! d\lambda~ e^{-Tr(\lambda^\dagger\lambda)}|\det(\lambda)|^{2(N-2)}\int_{Gr_{2,N}}\\!\\!\Omega(z)F(z) .$$ 1) Here is my first question; **Why does the measure on $\mathbb{C}^{2N}$ have the exponential?** Is it there because we are looking at coherent states? I am also a bit confused about the measure on $GL(2,\mathbb{C})$, but for the moment I'm ok just thinking it probably comes from the Haar measure and not the main point of my question. The next thing they do in that paper is to say the measure on $Gr_{2,N}$can be represented as $$\tag{94} \Omega(z)=\left(\prod_j d^4z_j\right) \delta^{(4)}\left(\sum_i|z_i\rangle \langle z_i|\right).$$ Then they go on to find the volume of $Gr_{2,N}$ as $\frac{\pi^{2(N-2)}}{(N-1)!(N-2)!}$ which gives a normalized measure on the Grassmanian. 2) So the next question is **How does this measure jive with the closure constraints?** If we perform the integration over the Grassmanian we enforce the constraint $$\sum_{i}|z_i\rangle\langle z_i|=0\rightarrow A(z)=0.$$ While that doesn't directly contradict the closure constraints, it does seem like it is adding an extra condition which is not implied by anything else. Can someone help me understand this measure? (and extra points for anything about that $GL(2,\mathbb{C})$ part.)


why do lightbulbs sometimes unscrew by themselves?

I've seen some light fixtures which exhibit the behavior of lightbulbs gradually unscrewing by themselves.


Why do we need high energy to explore small dimensions?

I am taking a quantum physics class, and for the life of me, I can not remember why we would need a vast amount of energy to understand the microscopic universe.


Does opening or closing the window in a non-AC car in the summer affect how much heat is felt inside?

This will perhaps look like a very basic and trivial question. But I find it confusing. As an experience, when you are travelling in non-AC car in summer, have people felt if putting on or putting off the windowpane affect how much heat it feels inside to passengers. On one hand, I feel that heat can still come inside thru glass particles (This makes me ask if heat needs media to travel?) and so putting on windowpane should not reduce heat felt inside. But psychologically it feels like by closing the window, you are blocking heat. Also the windowpane will at least block the heated air from coming inside. But does this heated air really brings much heat apart from the heat coming through anyway or it actually in practical acts more like a ventillator thus reducing heat felt inside. I realize that this question may appear strange, but it pops up to my head everytime I travel in non-AC car in summer. I have tried to articulate it as well as I could. Do insulator windowpane glasses reduce heat felt inside? because they will transmit lesser heat inside. How do black shade glasses work?


I have a two-part question:   1. First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for the Chern-Simons action for a general $3$-manifold $M$. My question is if anyone knows of any follow-up to this, or notes about their paper?   2. To those who know the paper: They say that they have no problem defining the action modulo $1/n$ (for a bundle of order $n$) as $n\cdot S = \int_B Tr(F\wedge F)$ $(mod 1)$, but that this has an $n$-fold ambiguity consisting of the ability to add a multiple of $1/n$ to the action - What do they mean here? Also, later on they re-define the action as $S = 1/n\left(\int_B Tr(F\wedge F) - \langle \gamma^\ast(\omega),B\rangle\right)$ $(mod 1)$ - How does this get rid of the so-called ambiguity?  Basically my question is if anyone can further explain the info between equations 3.4 and 3.5 in their paper. Thanks.


Half-massive sphere in space

A sphere contains all the mass on one side while the other side is empty of matter. Both sides are separated by a strong material, you name it, that will keep the sphere together. Given that the mass side is more massive than the sun or at least very massive, how would it behave in respect to gravity? Would it spin?


Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$

I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$. Is it necessary that these two vector fields commute? Would you give me an example for these two vector fields?


Electric potential energy and signs

I know that electric potential is negative near a negative charge and positive near a positive charge. But does this mean a small positive 'test' charge has a negative electric potential energy near a negative charge and a small negative charge would have a positive?


Reflection of light

> **Possible Duplicate:**   >  What is the difference between a white object and a mirror? When I look at a red object under white light, I see it as red because it absorbs the other colours and reflects the red light. When I wear black clothes in summer, I get hotter than if I wear white clothes because the black absorbs all colours of light and the energy associated with it. So my question is, what makes the difference between a white piece of paper and a mirror? The white paper doesn't absorb any colours, that's why it looks white. But what makes that different from a mirror, which also reflects the light that hits it? Or is my basic understanding of light and colour wrong? I'm not exactly experienced in Physics, I just like knowing how some things work.


Why is the ground state of the ferromagnetic tetrahedron threefold degenerate?

I'm preparing a presentation on Spin-Ice, but something's been bugging me for a while. On the Wikipedia page for Geometrical Frustration, it says the following about easy spins on a tetrahedron with ferromagnetic interactions: > There are three different equivalent arrangements with two spins out and two > in, so the ground state is three-fold degenerate. ![](http://i.stack.imgur.com/7CRYL.png) I just don't understand why.   * We're considering easy spins with ferromagnetic interactions, so any ground state needs 2 spins pointing towards the center (in) and 2 spins pointing away from the center (out).    * Given a tetrahedron, I can think of 6 different ways to distribute 2 spins in and 2 spins out on the vertexes. $(iioo,ioio,oiio,iooi,oioi,ooii)$    * You can rotate any of the above states to reproduce another, which might mean they are not different.  **But then, why does wiki state there are three _different_ arrengements?** **Note** Each of the above configurations has a total magnetic moment in a different direction. We can place our $(x,y,z)$ axis so that each one of the first three cases has a total moment in the positive direction of one of the axis, and each of the last three has a total moment in the negative direction of one of the axis.


How a spacecraft travelling near light speed avoid asteroids?

How would a spacecraft traveling near light speed avoid a (relatively dense group of) asteroids? Or suppose such spacecraft is designed, how would the physics work for steering it inside such a "cloud" of asteroids? If it simply by designing the spacecraft to have huge acceleration, wouldn't that mean lots of constraints on the material the spacecraft can be made of?


Vacuum to vacuum transition amplitude

I have two questions about Vacuum to vacuum transition amplitude.   1. Can any particle stay in $|0\rangle$?   2. I was studying this topic from Srednicki's QFT book. He writes in eq.$(6.22)$ $$\langle0|0 \rangle_{f,h} = \int \mathcal{D}p \mathcal{D}q \, \exp \Big[i\int_{-\infty}^\infty dt (p \dot{q} - H_0(p,q) - H_1(p,q) + fq + hp \Big]$$ $$ =\exp \Big[-i \int_{-\infty}^\infty dt H_1 \Big(\frac{1}{i} \frac{\delta}{\delta h(t)}, \frac{1}{i} \frac{\delta}{\delta f(t)} \Big) \Big] \times \int \mathcal{D}p \mathcal{D}q \,\exp \Big[i\int_{-\infty}^\infty dt (p \dot{q} - H_0(p,q) + fq + hp \Big] $$ Here in the second line the arguments of $H_1$ suddenly changes from $p$,$q$ to $\frac{1}{i} \frac{\delta}{\delta h(t)}, \frac{1}{i} \frac{\delta}{\delta f(t)}$. How can I derive this line from the first line?


Level of calculus required for physics

First time for me here so kindly let me know if I violate the rules - especially if this is a duplicate. After reading the page how to become a good theoretical phycist, I started a serious revision of calculus. For exercises, Math.SE is a good place. I was attempting exercises from this user (question s/he asked and answered) and realized I couldn't solve the majority of them. So the question is what is the level of mastery of calculus required for physics? What is(are) the best book(s) for that? I'm interested in limits, integration and infinite series for now. Background: standard mathematics with limits, integration, infinite series, differential equations (ODEs and PDEs) and numerical analysis and optimization. Everything was fast paced because professors wanted to finish the program as fast as possible ( during 3 semesters ) so I really never got good at any of the listed above.   Note: self-study kind of books will be most helpful.


Physical Interpretation of Poynting Vector

I'm looking for a physical interpretation of the Poynting Vector. I understand that it should be thought of as an energy flow due to the electromagnetic field, but would I be correct in saying that in some sense it's equivalent to a photon flow/packet? After all, for an electromagnetic wave, the poyntig vector yields the direction of propogation, or the direction in which the photon is moving. Does this analogy apply in general?


Real world usage; pump flow vs pressure vs pump head, what is more useful in this situation?

I am currently making a water cooling system for my computer and decided to see if I could get some better help here than what people think they know on the computer forums. So for my cooling system I have three, what are called, water blocks. They have a inlet and an outlet and are the biggest source of restriction. I specifically have three, and they each have 1/4in input and output barbs, I then connect them all with two manifolds. The manifold is made of 3 PVC T fittings and an end cap. So all the T's are setup such to look like this TTT . One of the ends of the tops on the T's is capped and the other is the input/output, while the bottom of the T's goes to and from the water blocks. (I hope this is very easy to visualize, if you cannot ask me and I will try to explain more) The input for each manifold is 1/2in tubing and the T's run to x3 1/4in tubing lines. So the water enters in a 1/2in tube, is split to x3 1/4in tubes, then combines back into a 1/2in output tube. Each waterblock is responsible for, at max, 190watts of heat to dissipate. So the entire loop must remove ~600watts of heat + the pumps heat = ~625 watts. This is where the radiator comes in, I found a nice chart that tells me the size I need to remove X watts from a system so I know I have the right size radiator. The radiator has 1/2in tubing inputs and outputs (but it is restrictive cause of the inside geometry. So my question is, what should I consider in a pump when trying to find one for this system? People talk of flow rate, vs system pressure, vs pump head. I know the head is how hard the pump can push with no restriction in its flow. So I figure pressure is combined in the pump head figure. What no one talks about though is how to find out how much flow and pressure is needed for the water cooling loop to work as it should. Even if I have the right size radiator to remove the amount of heat I have it is worthless if the water is not moving in the right manner. So can I possibly buy a pump that has too much flow; how can I figure the minimum amount of flow needed for my loop to work? EDIT: Ok I have found out a little bit more.. So I figure part of what I need to know is the bernoulli equation, which will describe the flow and pressure of the water at a certain point in the system. What I guess I need to do is equate the amount of heat water can remove at a specific velocity. So each water block gives off 190watts, and water has a thermal conductivity of .6Watts/m*K at 0C. How do I equate these two figures to figure out how much flow and head my pump should have?


Does a ski racer with a larger mass have an advantage?

Does a ski racer with a greater mass have an advantage over a racer with a lesser mass? If mass of one racer is 54 kg and the mass of a more slender racer is 44 kg I know the speed at which they descend should be equal if they were to fall in a vacuum. How does friction force, air resistance and momentum play a part in determining the advantage or disadvantage of larger mass in ski racing?


Skiing downhill

The other day on skiing holiday we've been arguing about whether an adult has weight advantage over a child when skiing downhill. I was claiming that gravity is a constant regardless of object's weight. I was claiming that weight doesn't affect skier's speed. I can still remember primary school experiment where we dropped same sized objects of different matter in vacuum. Styrofoam and iron ball of the same size and they both fell equally. It is true that skier's maximum speed is determined by his air drag (and snow/ski resistance) and heavier objects have higher persistence making them move through resistive medium easier. But compared to a child an adult would have more drag due to larger _surface_ so I suppose this somehow evens out. I was basically claiming that weight doesn't matter at all but that other factors are much more important when it comes to skier's speed like: ski stiffness (weight force much better distributed over the whole ski surface than with softer skis) and ski sliding surface smoothness and especially skier's position (to reduce drag as much as possible). I also got a bit confused when I started to think about force vectors where greater weight results in longer vectors resulting in stronger movement pull. ## Question If we approximate is it safe to say that a child can ski just as fast as an adult? Or that a thinner adult would ski faster than his same high companion that is overweight due to larger drag surface? Or does his weight benefits his speed? **What is the correct answer here?**


How do I find angular and linear velocity after normal force and "infinite" friction force?

Look at this picture: http://i.imgur.com/mY8ShkV.png Here we have a ball resting on top of a platform (the contact point is marked red). I know the normal vector (marked green). I also know the mass, radius, and current angular and linear velocity of the ball. Let's say I set the new angular velocity after friction which is always big enough ("infinite") has been applied to: $V_a = V_l \times rN$ where $V_l$ is the linear velocity, $r$ is the radius of the ball and $N$ is the normal vector. Does this math check out, is this correct? Because this model doesn't work if I leave the ball leaning on the side of the platform where it should obviously roll of the platform and fall. But it doesn't. What is the correct mathematical model for simulating "infinite" friction? I suppose you will also have this problem when you want to simulate finite friction and the friction doesn't give it's max force.


About free quarks and confinement

I simply know that a single free quark does not exist. What is the reason that we can not get a free quark? If we can't get a free quark then what is single-top-quark?


How "to take" this integral?

When I learned anharmonic model of crystal, I read that considering anharmonic oscillations and Boltzmann distribution for the "atoms" of crystal we can get the dependence of distance between the "atoms" from a temperature as $$ \langle r \rangle = r_{0} + \alpha T. $$ As I understood the words below, it's like $$ \langle r \rangle = \frac{\int \limits_{0}^{\infty}re^{-\frac{U}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{U}{kT}}dr} \approx \frac{\int \limits_{0}^{\infty}re^{-\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr} = |x = r - r_{0}| = \frac{\int \limits_{0}^{\infty}(x + r_{0})e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}, $$ and then - $$ \langle r \rangle \approx r_{0} + \frac{\int \limits_{0}^{\infty}xe^{-\frac{ax^{2} + bx^{3}}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}. $$ What can I do on the next step?


Is the speed of light the ultimate speed limit?

As we all know nothing can go faster than the speed of light as mentioned by most of our pioneer's in physics. But as I was listening to one of the statements of Sir. Stephen Hawkins he stated that at the time of big bang the universe expanded to about a thousand light years in just few trillionth of a second, how it's possible. Are our laws of physics wrong as if they are correct they should be applicable everywhere in every conditions. Please answer thihs question as answer in above link was not about it.


'Warp'- type travel; is it possible?

I just re-read an article about space travel, and the same article claimed that Star Trek-type 'warp' nuclear engines ARE theoretically possible, but that it will take around 500 years for humans to develop and adapt them for use in the engines of space craft. How true is this statement?


Why do rockets accelerate fastest horizontally?

I've heard that rockets accelerate fastest when travelling horizontally to the ground, not downwards or upwards. Is that true, and why?


Wavelength and resolution

I'm reading some texts that seam to assume knowledge of light that I'm not too familiar with. How does wavelength of light relate to the minimum distance span that can be observed (i.e. you cannot make a lens big enough to resolve individual atoms), and is this a light phenomena or an intrinsic wave phenomena?


We know that hydrogen and helium are lighter that air. Therefore it is certain that they would create a lifting force when enclosed in a balloon. How do we describe the relationship of balloon size and related lift?


Calculation of the G-force

I have a formula which is $\text{G-force} = \frac{v\omega}{9.8}$, where $v$ is speed and $\omega$ is the angular velocity. I've seen on the internet that G-force is actually $\text{acceleration}/9.8$. I'm confused as to which formula is correct. For simulating the motion of particle taking a turn, would omega simply be velocity divided by radius of turn? Assuming Cartesian coordinates.   Another funny thing I noticed is that while simulating particle motion, a 7G turn showed up as an almost straight line (while using a constant turn motion model) with a velocity of 900m/s and time interval of 1second. Am I simulating wrong or is my use of the first equation wrong?


Why isn't the force modelled which confines excess charge to remain inside a conductor?

Excess charge placed on a conductor distributes itself on the surface of a conductor. But why isn't the force which confines the charge to remain there, modelled in some way?


Problem on speed of light

As I was reading special relativity, my book says the speed of light is $c$ with respect to any other thing. Does that mean the speed of an individual photon is $c$ even with respect to another photon? I mean, shouldn't the relative velocity be zero?


How does quantization solve the ultraviolet catastrophe?

I understand how classical physics leads to the UV catastrophe. But I cannot understand how quantization solves it.   * How can quantization prevent the body from radiating a lot of energy? I know this question is similar to many other questions but I need a layman's explanation (without too much mathematical details).   * Finally why does the peak reach zero intensity after a certain frequency? ![enter image description here](http://i.stack.imgur.com/2vACu.png) Image source:- Wikipedia


Are there 2 kinds of photons, one that mediate the electromagnetic interaction and the other the quanta of light?

It is usually said that photons are the force carriers or the mediators of the electromagnetic forces between electric charges. At the same time we know also that electromagnetic waves on the quantum level are described in terms of photons. It seems to me that we have 2 types of photons:   * the photons which mediate the electromagnetic interactions which are offshell hence their masses do not necessarily have to be zero, I suppose    * and the photons which are the quanta of electromagnetic waves which are massless. Am I correct, or am I confused and missing something?


Force when acceleration is zero

i tried to find it in books but i was unable to do so and i am not able to understand this and I couldn't think of a better example. i repeat. its just an example. if i punch someone's face with my arm having constant velocity, it will mean that the acceleration of my arm is zero. F=ma, therefore force exerted by my arm is zero. Now if the force applied is zero the why does the person whom i punched experiences pain. please help. thanks.


How can a non-conservative field be a scalar multiple of a conservative field?

Okay so I was reading this from University Physics by Freeman and Young and on the topic of inductors as circuit element, they wrote that $\mathbf{E_c} + \mathbf{E_n} = 0$ which makes no sense to me Here is an excerpt of the text and underlined the confusion with green ![enter image description here](http://i.stack.imgur.com/Ae1lO.jpg)


Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter

The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations. Say I consider an Euclidean space and a time parameter. How does the Euclidean metric (which in canoncial coordinates is $\delta_{ij}$, with $i,j\in \\{1,2,3\\}$), transform under a Galilean transformation, more specifically a velocity boost $t\mapsto t,x\mapsto x-vt$? In a way, from the three dimensional pov, which doesn't see $t$, this is just a translation. So here is what I'd like a derivation for: > $$(t\mapsto t'=t,\ x\mapsto x'=x-vt)\ \Longrightarrow\ > h_{ij}\equiv\delta_{ij}\mapsto\ ?$$


Surface Area Relative to Element Distribution

We tend to find heavier elements like iron and others toward the center of celestial bodies. Lighter elements like carbon, hydrogen, and oxygen are almost always towards the surface. That makes sense because heavier elements have a greater gravitational effect and would naturally want to move toward the center. Now think about the chaos that ensued during initial formation of the celestial body. Heavier elements could get jumbled in with lighter elements throughout the depths. Even to this day, we have lighter elements being pushed to the surface through volcanic gases and ocean fissures. **Question** : If elements were not always arranged from heaviest to lightest during creation, would this in anyway introduce volatility in the overall surface area of the celestial body? Imagine the worst case scenario that would be utterly unimaginable: gold and iron wrapped around the surface with oxygen and hydrogen trapped deep within the depths. Would the overall surface area be any different over time as heavier elements pushed their way in and lighter elements rise to the surface? Intuitively, I would say it would change over time, but to what degree and at what rate of change depends on how elements were initially distributed.


Increase the signal-to-noise ratio of photomultiplier

It is a GRE physics problem. A technique would most likely increase the signal- to-noise ratio of a photomultiplier tube is to   1. operate the tube at a lower temperature   2. operate the tube at higher voltage per dynode than usually specified    3. use a radioactive source to saturate the noise level   4. use a retarding potential on the first dynode   5. use a photocathode of lower work function I tried searching for information regarding photomultiplier, but none seems to fit any of the above options.


Why doesn't this model plane fly?

I have been designing a model plane for Design Technology for the past month or so, and today I laser cut my final design and assembled, it then tested it. Upon testing the plane does not get any lift, whereas the previous testing model which was virtually the same did. The plane is built using Balsa Wood, and Assembled with hot glue (I used as little glue as possible to reduce weight :) ) Any Ideas? Images: http://imgur.com/a/fdKxZ


How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system?

I have a material specified by a permittivity tensor written in Cartesian coordiantes: $$\begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz}\\\ \epsilon_{yx} &\epsilon_{yy} & \epsilon_{yz}\\\ \epsilon_{zx} & \epsilon_{zy} &\epsilon_{zz} \end{pmatrix}$$ I want to write all my equations in cylindrical coordinates and so I have to write the permittivity tensor in cylindrical coordinates too. How can I do that?


Why is there an emission of gamma rays?

When a spontaneous radioactive reaction happens, there is an emission of gamma rays (in most cases) What causes this emission?


Covariant derivative-Differential

I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be $\dfrac{\partial}{\partial t^\prime}=\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial t^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial t^\prime}$ $\dfrac{\partial}{\partial x_1^\prime}=\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial x_1^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial x_1^\prime}$ I was trying to understand but I'm afraid I can't.. if oyou for instance add the terms in the second part of the equations you get $\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial t^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial t^\prime}=\dfrac{\partial}{\partial t^\prime} + \dfrac{\partial}{\partial t^\prime} =2 \dfrac{\partial}{\partial t^\prime}\neq \dfrac{\partial}{\partial t^\prime}$ How is that possible?


Classification of small extra-solar system bodies

A Small Solar System Body (SSSB) is an object in the Solar System that is neither a planet, nor a dwarf planet, nor a satellite. This encompasses all comets and all minor planets, as well as all asteroids (that are classed as minor planets). I want to know if there is a term for classifying Small Extra Solar System Bodies. What about comets and minor planets in systems other than our own. Do they have a classification? Is there also a recognised categorisation for Small Bodies regardless of whether they are in the solar system or not? In the same way the 'solar system' pertains to our system but a 'planetary system' means any system including but not limited to our own.


A question on the doped Kitaev-Heisenberg model?

Recently, some groups have studied the effects of doping the Kitaev model on honeycomb lattice(e.g.,http://arxiv.org/abs/1109.6681 and http://arxiv.org/abs/1109.4155) and their calculations show the existence of a topological p-wave superconducting phase. My question is as follows: As we know,one of the amazing properties of the Kitaev model is that the low energy physics(ground states properties and excitations above it) of the model does not depend on the signs of the coupling constants in the Hamiltonian. So for the (hole)doped case, is the existence of topological p-wave superconducting ground states still insensitive to the signs of the couplings we choose? Thanks in advance.


What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?

Ehrenfest 1931 gives an argument to the effect that the application of the spin-statistics theorem to composite systems is valid, but only as an approximation and under certain conditions. Unfortunately this article is behind a paywall. The abstract reads as follows: > From Pauli's exclusion principle we derive the rule for the symmetry of the > wave functions in the coordinates of the center of gravity of two similar > stable clusters of electrons and protons, and justify the assumption that > the clusters satisfy the Einstein-Bose or Fermi-Dirac statistics according > to whether the number of particles in each cluster is even or odd. The rule > is shown to become invalid only when the interaction between the clusters is > large enough to disturb their internal motion. The quaintness of the abstract is enhanced by the mysterious failure to mention neutrons -- this being explained by the fact that the neutron was not discovered experimentally until the following year. Pretty amusing that the American Physical Society still owns the copyright on a paper this old, one of whose authors was probably paid through my grandparents' and great- grandparents' income taxes. Googling quickly turns up various loose verbal characterizations of this result, without proof. Can anyone supply an outline of how the argument works and possibly a more precise statement of what the result really is? Some sources seem to refer to this as the Wigner-Ehrenfest-Oppenheimer (WEO) result or Ehrenfest-Oppenheimer-Bethe (EOB) rule. Historically, Feynman seems to have used this result to show that Cooper pairs were bosonic. Although I've tagged this question as quantum field theory, and the spin- statistics theorem is relativistic, the 1931 date seems to indicate that this would have been a result in nonrelativistic quantum mechanics. This answer by akhmeteli sketches what seems to be a similar relativistic result from a book by Lipkin. In the limit of low-energy experiments, the EO rule says the usual spin- statistics connection applies, and this seems extremely plausible. If not, then it would be too good to be true: we would be able to probe the composite structure of a system to arbitrarily high energies without having to build particle accelerators. Bethe 1997 has the following argument: > Consider now tightly bound composite objects like nuclei. It then makes > sense to ask for the symmetry of the wavefunction of a system containing > many identical objects of the same type, e.g., many He4 nuclei. This > symmetry can be deduced by imagining that the interchange of two composites > is carried out particle by particle. Each interchange of fermions changes > the sign of the wavefunction. Hence the composite will be a fermion if and > only if it contains an odd number of fermions[...] Note the qualifier "tightly bound," which implies a restriction to low-energy experiments. The abstract of Ehrenfest 1931 seems to say that this is a necessary assumption, but it is never used in Bethe and Jackiw's argument, and this suggests that their argument is an oversimplification. Fujita 2009 recaps the Bethe-Jackiw argument, but then says: > We shall see later that these arguments are incomplete. We note that Feynman > used these arguments to deduce that Cooper pairs [5] (pairons) are bosonic > [6]. The symmetry of the many-particle wavefunction and the quantum > statistics for elementary particles are one-to-one [1]. [...] But no one-to- > one correspondence exists for composites since composites, by construction, > have extra degrees of freedom. Wavefunctions and second-quantized operators > are important auxiliary quantum variables, but they are not observables in > Dirac's sense [1]. We must examine the observable occupation numbers for the > study of the quantum statistics of composites. In the present chapter we > shall show that EOB's rule applies to the Center-of-Mass (CM) motion of > composites. Unfortunately I only have access to Fujita through a keyhole, so I can't see what the references are or any of the details that they promise in this beginning-of-chapter preview. Related: Huge confusion with Fermions and Bosons and how they relate to total spin of atom Bethe and Jackiw, Intermediate Quantum Mechanics P. Ehrenfest and J. R. Oppenheimer, "Note on the Statistics of Nuclei," Phys. Rev. 37 (1931) 333, http://link.aps.org/doi/10.1103/PhysRev.37.333 , DOI: 10.1103/PhysRev.37.333 Fujita, Ito, and Godoy, Quantum Theory of Conducting Matter: Superconductivity, 2009


Nuclear Weapon Effects: Cloud Rise

I am interested in modeling the rise of a nuclear cloud to stabilization height post detonation. I am interested in any ideas you may have on the approach. I have collected some research on the topic but I am curious as to how someone might think through the problem without being exposed to one possible solution.


Do bifacial solar panels operate at lower temperature than normal solar panels?

Bifacial solar panels have a glass backplane instead of the usual aluminum and are designed to absorb and convert to electricity light incident on either face. Many manufacturers claim that their bifacial panels operate at lower temperature than normal (monofacial) panels because bifacial panels transmit the 20% of the solar spectrum that is too low-energy to be absorbed by silicon. See for example this manufacturer's datasheet. Meanwhile, the only reputable study I could find (apologies for the paywall) was of a lab-scale cell and not a commercial module. That study argues that convection is the dominant heat transfer mechanism and so the differences in optical absorption and emission aren't relevant. However, temperature differences of a few degrees centigrade can have a noticeable impact on solar cell efficiency and useful life. Anybody have a compelling argument or know of other relevant experimental work on this topic?


Kinetic energy when $F(t)$ graph is given

![enter image description here](http://i.stack.imgur.com/jBPTL.png) The very basic problem here is that the point where this graph ends is ambiguous. I mean, what is the corresponding $y$-value for the point whose $x$-valu is $4.5s$? Then I think we should find the area, then that will be the momentum, so use $\frac{p^2}{2m}$, as mass is given. So if theres any error in this, pls let me know...


What will the universe look like for anyone falling into a black hole?

I've heard that, from the perspective of an external observer, something falling into a black hole will eventually look "frozen": light waves will move to the infrared and further into lower frequencies. But what about that unfortunate individual falling into such a black hole? What will he/she see when looking back at our universe? For instance, I wonder, will it be the ultimate end of the Universe? Can someone who just crossed events horizon of the black hole still receive messages from our side?


Smoothed Particle Hydrodynamics, Chemical Reactions and Stiffness

Dear people from stackexchange physics. I have been using SPH for quite a while to simulate free boundary flow, and just recently we tried to include some kind of (simple) chemical reaction in our code. We are still doing some tests, but one curious thing that we found was that we needed much smaller time steps than we usually used to be necessary. I have never dealed with chemical reactions, but I don know that some of them fall on the category of 'Siff Equations', and as far as the wikipedia description goes, it seems to be my case, but I would like to ask if anyone tried anything similar and also found that their equations were stiff. I never studied stiff equations, I just knew that they existed and recently read a bit more about them, but that's it. Last but not least, do anyone know how to check if a set of SPH equations is stiff or not? The wikipedia recomendation seemed at least complicated to calculate for SPH.


Work done in non-uniform circular motion

In non-uniform circular motion, the work done by force should not be zero because tangential force acts on body. Is this correct? If I am wrong, please give a proper explanation.


Thomas-Fermin approximation of cold atoms in a harmonic trap

In Bose-Hubbard model of cold atoms, one can use Thomas-Fermi approximation to get a rough number of total particle number. In Thomas-Fermi approximation, the site density is $n(R_i)=（\mu-\kappa R_i^2）/U$,where $\mu$ is the chemical potential and $\kappa$ is the trapping potential. The atomic cloud extends to the Thomas-Fermi radius $r_{TF}=\sqrt{\mu/\kappa}$. Why in Thomas-Fermi approximation, one can neglect the kinetic energy. Does it mean it only work for strong interaction regime? When I use Gutzwiller mean field method to calculate the total particle number in a trap, I find the total particle number varies a lot as the hopping amplitude varies.


Magnetic field of uniformly charged rotating hollow sphere

I want to compute the magnetic field due to a homogeneously charged, rotating sphere with radius $R$, angular velocity $\vec{\omega} = \omega\hat{z}$ and total charge $Q$. I want to use Biot-Savart law, so I first computed the current density $\vec{j}(r,\theta,\phi) = \frac{\omega Q}{4 \pi R} \delta(R-r) \cdot (-\sin\theta \sin\phi,\sin\theta\cos\phi,0)^T$. After taking out all the constants with respect to integration and removing the integral over the radial component with the $\delta$ function, what stays in the Biot-Savart integral is $$\vec{B}(\vec{r}) \propto \iint_{\theta,\phi}^{\pi,2\pi}\frac{\vec{j}(\vec{y}) \times (\vec{r} - \vec{y})}{|\vec{r}-\vec{y}|^3}\sin{\theta}\;\text{d}\phi\;\text{d}\theta$$ where $\vec{r} = |\vec{r}|\cdot(\sin\theta_r\cos\phi_r, \sin\theta_r\sin\phi_r,\cos\theta_r)^T$ is the point where the $B$-field is evaluated, $\vec{y} = R\cdot(\sin\theta\cos\phi, \sin\theta\sin\phi,\cos\theta)$ is a point on the sphere. and the $\vec{j}$ is now missing the $\delta$ part, and some constants. The problem I have is, that the vector inside the integral is $$\begin{pmatrix} \sin\theta\cos\phi(|\vec{r}|\cos{\theta_r} - R\cos\theta )\\\\\sin\theta\sin\phi(|\vec{r}|\cos\theta_r-R\cos\theta)\\\\-|\vec{r}|\sin\theta\sin\phi\sin\theta_r\sin\phi_r-|\vec{r}|\sin\theta\cos\phi\sin\theta_r\cos\phi_r+R\sin^2\theta\end{pmatrix}.$$ Since all terms involving $\sin\phi$ or $\cos\phi$ will integrate to zero, this implies that I will get a $B$-field that is along the $z$ axis, no matter where it's evaluated. This can't be, because in the next part I'll show that far away the field looks like that of a dipole. I have been sitting over this for hours, can someone help me clear this up? Where am I going wrong?


Streamlines for steady flow problem

I need to calculate the streamlines for the steady flow given by the velocity field $$\mathbf{u}=(u,v,w) = \left( 0,\frac{y}{y^2 +z^2} , \frac{z}{y^2 +z^2}\right)$$ Now I've been able to notice that $\dfrac{ \partial y}{ \partial z} = \dfrac{y}{z}$ therefore $y = ze^{c}$ but I'm not sure how I should continue. I've tried to solve $\dfrac{ \partial y}{ \partial s} = \dfrac{y}{y^2+z^2}$ but I get a $y^{z^2}$ term in my answer which makes me doubt whether I'm doing it correctly.


Nonequilibrium thermodynamics in a Boltzmann picture

The Boltzman approach to statistical mechanics explains the fact that systems equilibriate by the idea that the equillibrium macrostate is associated with an overwhelming number of microstates, so that, given sufficiently ergotic dynamics, the system is overwhelmingly likely to move into a microstate associated with equilibrium. To what extent is it possible to extend the story to nonequilibrium dynamics? Can I make concrete predictions about the approach to equilibrium as passing from less likely macrostates to more likely macrostates? (Wouldn't this require us to say something about the geometric positioning of macrostate regions in phase space, rather then just measuring their area? Otherwise you'd think the system would immediately equilibriate rather than passing through intermediate states.) Can the fluctuation-dissipation theorem be explained in this way? **Edit:** Upon a bit more poking around, it looks like the fluctuation- dissipation theorem _cannot_ be explained in this way. The reason is that this theorem discusses the _time-independent distribution_ of fluctuations in some macroscopic parameter (e.g. energy of a subsystem) but, as far as I understand, it does not describe the _time dependence_ of such a parameter. In particular, I'd really like to understand is if it's possible to explain things like Fourier's Law of thermal conduction (that the rate of heat transfer through a material is proportional to the negative temperature gradient and to the cross-sectional area) with a Boltzman story. According to these slides, it's surprisingly hard.


Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian operator which, among the others, also has real ($\mathbb{R}$) eigenvalues? Would they correspond to observables? If no, why not?


Magnetism due to relativity?

So I have been reading in some books that magnetism does not have to be assumed a priori, but can be obtained from the electric field + special relativity. And I have seen how this leads to the common formula for the magnetic field of a current carrying wire. Fine. What about materials that are inherently magnetic? Such as iron, or magnetite? Surely their magnetic field is not a consequence of relativity? (if yes, who's moving and with respect to whom?)


Time dilation in special relativity

Suppose a star ship is moving with some velocity. Two light pulses one in direction similar to star ship another opposite to it is shot towards the space ship. Then how time inside space ship adjust to make velocity of light constant?


Spin with Stern gerlach Experiment

Within a week, I am trying to conduct a seminar on the topic " Spin with Stern gerlach Experiment" at my University. It would be great if you guys could provide me some inputs regarding the things I should say about the experiment during the seminar. I am still unsure about the outlines I should draw in my mind about what I should present serially.......Since it would be the first seminar of mine infront of more than 100 people, I am really confused as well as afraid.


How does the focal length of the lens change the distance from the camera to an object?

I am reviewing some old image processing notes and have the following problem: > I have a camera with a projection plane of $2cm\ by\ 2cm$ and a $15mm$ lens. > If you want an $1.5m$ object to take up half of the resulting image, how far > away should it be? It seems like the standard $x = \frac{fX}{Z}$ equation should be useful here, but I am not sure if this information gives focal length at all. How should this be solved?


Relativity - time dilation

I'm learning about relativity and I'm having some issues with it and the twin paradox. I found many questions and answers on this subject but they did not answer my specific problem. In my thought experiment I added a third twin - triplet. So here it goes: Let's have triplets $A$,$B$ and $C$. sitting still at same point in space. They want to test relativity so they do this experiment.. They all reset their clocks, and $B$ and $C$ start moving together away from $A$ at some ridiculous speed $v_{365}$ which coincidentaly has lorentz factor $365$. they move at uniform motion in only one direction (let's say acceleration time nears 0). State 1: $$\begin{align}t(A)=t(B)=t(C)&=0 \\\ v(A,B) = v(A,C) &= v_{365}; \\\ v(B,C) &= 0\end{align}$$ $t(A)$ being A's clock, $v(A,B)$ being velocity at which $A$ and $B$ are moving away from each other, etc.. Now after some time, let's say when $t(A) = 1\text{ year}$, brother $B$ starts to decellerate to full stop. It also shows up that brother $C$ was sleeping through whole first part of the experiment. But now he wakes up and sees that his brother $B$ is accelerating away from him in the opposite direction than was planned but that doesn't bother him. State 2: $$\begin{align}t(A)&=1\text{ year} \\\ t(B)=t(C) &\stackrel{?}{=} 1\text{ day} \\\ v(A,B) &= 0 \\\ v(A,C)=v(B,C) &= v_{365}\end{align}$$ Now after another while lets say when $t(A)=2\text{ years}$, brother $C$ sees that brother $B$ isnt stopping any time soon so he decides to match his speed.. (brother $C$ stops now from point of $A$) State 3: $$\begin{align}t(A)&=2\text{ years} \\\ t(B)&= 1\text{ year }1\text{ day} \\\ t(C)&= 2\text{ days} \\\ v(A,B)=v(A,C)=v(B,C)&=0\end{align}$$ That's it.. now they stand still some light year away from each other, but they can communicate and compare their clocks.. here are my questions: 1] Is my assesment of time dilation effects correct in this scenario? if not which state contains first error and why? 2] If my assesment is correct, then from point of brother $C$ it was brother $B$ who was accelerating away, yet it was also brother $B$ who aged when they got in same speed - therefore it seems to me its always the one who is matching speed who doesnt age - is this a general rule? 3] If answer to questions 1 and 2 is yes, then can this effect be used to gather infomation that would take much longer to extract in normal speeds? Regards


What is the highest speed time dilation has been tested? How close to the Special Relativity prediction did it get?


Kirchoff Loop and current

The pic of the circuit/problem: ![enter image description here](http://i.stack.imgur.com/kS5Pm.jpg) I got the problem right, but i want to make sure my reasoning is correct for the current left of the 2.1 V battery. Its the same current as $I_1$ because the charge carriers do not split in anyway, and the battery only does work on them moving the carriers from a low potential to a high potential. However, if $I= \frac{dQ}{dt}$, wouldn't supplying energy via moving the charges from low potential to high potential cause the charges to move more quickly, i.e. increase the current? A little fuzzy on why it is the same current, if someone could enlighten me I would very much so appreciate it.


Projective Transformations in GR

A Thought Experiment: We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made to vary upto some final state. During this process of experimentation the physical separations change while the coordinate labels remain fixed to their own positions. [Coordinate separations remain unchanged]. The length and the orientation of a vector changes in this process both in the 3D and in the 4D sense. We are passing through different/distinct manifolds in our thought experiment and if the 4D arc length does not change we are simply having a transition between manifolds for which ds^2 is not changing but the metric coefficients are changing.We consider the option of $ds^2$ changing in this posting. Query: Our experiment indicates at projective transformations operating in the physical sense[considering changes in the metric and in the value $ds^2$].A time dependent field is being observed where the metric coefficients are not being preserved. Is it important to include projective transformations[concerned with the non-preservation of the metric] in the mathematical framework of GR?


**Question: Is the Avogadro's constant equal to one?** I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the `mole` and to `Avogadro's constant`. And, to be honest, it doesn't make any mathematical sense to me.               1 mole = 6.022 * 10^23     Avogadro's constant = 6.022 * 10^23 * mole^(-1)      What? This hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!


Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: 1-Are matrices and second rank tensors the same thing? 2-If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)?


String total cross sections at asymptotically high energy

I only have a vague understanding of string theory, but a solid understanding of particle physics. At asymptotically high energy (Regge limit), the string cross section is dominated by the exchange of the leading trajectory -- the graviton trajectory, corresponding to an intercept $\alpha(0)=2$. Regge phenomenology dictates that the string-string total cross should rise linearly with energy: $$\sigma_\text{tot} \sim s^{\alpha(0)-1} = s$$ which is startling to me. Given that gravitons are massless, they are not subject to the Froissart-Martin bound $\sigma_\text{tot}<\log(s)^2$. So, is it really true that string-string total cross sections exhibit this growth?


Two charges in a medium where different permittivity is adjacent

I was thinking about the following: Assuming that you have NON SPHERICALLY SYMMETRIC charges and they are both in medium 1. Does it make a difference(whereby I am referring to the total force) if there is a second medium 2 in the local area of both, such that both charges are NOT in the second medium, but very close to it to the case where only medium 1 is around them?