title: How much heat needed to heat 400g of ice at 0°C to 20°C in a 200g aluminium pot?

text: An aluminium pot has a mass of 200g and contains 400g of ice at 0°C. How much heat would be needed to melt that ice and then raise the temperature of the resulting water to 20°C. The specific heat capacity of water and aluminium are 4200 J Kg ^-1 and 910 J Kg ^-1 respectively, and the latent heat of fusion of ice is 330 KJ/Kg. My attempt so far is as follows: Heat from ice to water - Q = mL = .4Kg x 330x10^3J = 132x10^3J Heat for water from (0°C to 20°C) - Q = mcΔT = .4Kg x 4200 x (20-0) = 33.6x10^3J Heat for pot from (0°C to 20°C) - Q = mcΔT = .2Kg x 910 x (20-0) = 3.64x10^3J Thus the answer is 3.64x10^3J + 33.6x10^3J + 132x10^3J = 169.24x10^3J I don't know how to do this question but would try go this way about it, however it looks terribly wrong to me and would like some help please.


title: Creation of the Electromagnetic Spectrum

text: After seeing this image: http://mynasadata.larc.nasa.gov/images/EM_Spectrum3-new.jpg And reading this: > "The long wavelength limit is the size of the universe itself, while it is > thought that the short wavelength limit is in the vicinity of the Planck > length, although in principle the spectrum is infinite and continuous." The question is, how did the EMS evolve? Was it weak at first, i.e. at Planck time, which allowed certain reactions to modify, shape or add properties to it? What really defined the position and order of everything that is the EMS? E.g. colors. What defined or added something as colors to be inserted in that specific range of 400nm to 700nm. Hope this is not to philosophical and thanks in advance.


title: Helium-4 superfluidity and gauge symmetry breaking

text: Is there an accessible account of superfluidity in Helium-4 as a manifestation of "global gauge symmetry" breaking? And what is meant by "global gauge symmetry"? I was taught that gauge symmetries were by definition local. Is it just a different terminology in condensed matter?


title: What is the curvature of the universe?

text: What is currently the most plausible model of the universe regarding curvature, positive, negative or flat? (I'm sorry if the answer is already out there, but I just can't seem to find it...)


title: How powerful would an explosion have to be for its effects to be felt worldwide?

text: If an explosion were to occur at any point on earth, how powerful would that explosion have to be for it to be audible or otherwise detectable by every person on the planet? Detection could mean either seeing or hearing the blast or feeling the tremors created by the shock wave. Bonus question: is any such explosion possible without it destroying the planet, the atmosphere or wiping out all life on earth? * * * A rough estimate puts the average distance between most antipodes on land at just shy of 20 000 kilometres. The largest nuclear bomb ever detonated, Tsar Bomba, had a yield of 50-58 megatons of TNT and was detectable almost a 1000 km away, according to Wikipedia: > The heat from the explosion could have caused third-degree burns 100 km (62 > mi) away from ground zero. A shock wave was observed in the air at Dikson > settlement 700 kilometres (430 mi) away; windowpanes were partially broken > to distances of 900 kilometres (560 mi). Atmospheric focusing caused blast > damage at even greater distances, breaking windows in Norway and Finland. > The seismic shock created by the detonation was measurable even on its third > passage around the Earth. The most powerful volcanic eruption known was that of Mount Tambora in 1815. Classified as Volcanic Explosivity Index 7 (note that it goes up to 8) with an estimated yield of 800 Mt, it was heard about 2 600 km away. > On 5 April 1815, a moderate-sized eruption occurred, followed by thunderous > detonation sounds, heard in Makassar on Sulawesi 380 km (240 mi) away, > Batavia (now Jakarta) on Java 1,260 km (780 mi) away, and Ternate on the > Maluku Islands 1,400 km (870 mi) away. On the morning of 6 April, volcanic > ash began to fall in East Java with faint detonation sounds lasting until 10 > April. What was first thought to be the sound of firing guns was heard on 10 > April on Sumatra island more than 2,600 km (1,600 mi) away. The Wikipedia page on TNT equivalents list some interesting events but branches off into seismic and cosmic events after the entry for the Tsar Bomba. So, is it possible to go bigger?


title: Lagrangian Systems

text: Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function $L$ is called the Lagrangian. In the case when $M$ is a Riemannian manifold and a particle in $M$ is moving under some conservative force field, taking the Lagrangian to be the kinetic minus the potential energy we recover Newton's second law. I know that one of the main advantages to the Euler-Lagrange equations over Newtons is the way in which they simplify constrained systems. I know another is the coordinate independence of the equations. However, in all applications I've seen the manifold is always Riemannian and the Lagrangian is always $K-U$. My questions are:   1. Why do we have this abstract definition of a Lagrangian system and of an abstract Lagrangian?    2. What are some of the cases in which $L$ is not $K-U$ that gives interesting results?    3. Or cases in which the manifold is not Riemannian?


title: How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

text: many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial \dot{q}}-L$$ but none of the textbooks explain how this is done. My specific problem is that I have Lagrangians that do not depend on $\dot{q}$ and therefore should have $\frac{\partial L}{\partial \dot{q}}=0$, hence $H=-L$. But my impression from the clues I have is that it is not that simple. Let's say the Lagrangian is $$L(q)=\ln(q)-(2q-10)\lambda$$ Now as far as I know the Legendre transformation should give a function $f^*(p)=\sup(pq-L(q))$ (this implies $p=\frac{\partial L}{\partial q}$) which is obtained by substituting the stationary point $q_s$ of $\sup(pq-L(q))$ into $pq-L(q)$ thus getting $f^*(p)=pq_s-L(q_s)$ (for instance wikipedia's Legendre Transformation page explains this). Doing this for the example above: $$\frac{\partial (pq-L(q))}{\partial q}=\frac{\partial (pq-\ln(q)+(2q-10)\lambda)}{\partial q}=p-\frac{1}{q}+2\lambda$$ must be 0 for a stationary point, thus $q_s=1/(p+2\lambda)$. And hence the transformation should be $$f^*(p)=p\frac{1}{p+2\lambda}-\ln(\frac{1}{p+2\lambda})+(2\frac{1}{p+2\lambda}-10)\lambda$$ and this should be the Hamiltonian. But this equation does obviously have nothing to do with the textbook Hamiltonian. Rather, an answer to another question has in a similar case treated $L(q)$ as being dependent on $\dot{q}$ implicitly (Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation ... answer by Qmechanic). It mentions using the Dirac-Bergmann method for obtaining the Legendre transform. Trying something along the lines of this other question the above example seems to give $$p=\frac{\partial L}{\partial \dot{q}}=0$$ and $$p \approx 0$$ (an equality modulo constraint as the answer to the question linked above says). And then $H=\dot{q}p-L$. The difference seems to be that the Legendre transform is done with respect to two different variables, $q$ and $\dot{q}$ - but it was my understanding that it had to be done with respect to all variables the Lagrangian depends on. So how does the $qp_q$ term vanish if we have only the $\dot{q}p_{\dot{q}}$ term left? Thanks. edit: changed ln to \ln as Plane Waves suggested, and sup to \sup. And yes, sup is the supremum over all q, as Vibert said, sorry for forgetting to mention that.


title: Dissolving oxygen into water

text: I was wondering how one would actually calculate how much oxygen would dissolve into water given the necessary initial conditions, and what those initial conditions would need to be. I assume they would be pressure, and initial concentration, but I really don't know where I would go from there. Clearly air and water have different concentrations of gases and liquids, despite having been in contact for thousands of years. And once in water, is oxygen still considered gaseous? I assume it is, but why is it called gaseous- what quality of it deems it a gas despite being surrounded by liquid?


title: Measuring Entangled Qubits

text: Suppose we have a pair of entangled qubits. $$ |\psi\rangle = \frac{1}{ \sqrt{2} } ( |00\rangle + |11\rangle ) $$ Now we give one qubit to Alice and other to Bob. Alice measure the her qubit to know the spin in z-direction and bob simultaneously measures the spin in x direction. Since alice and bob both know that these qubits are entangled, so whatever is the value of spin for first qubit in z direction same will be the value of spin in z direction for the second qubit. similarly the spin value in x direction, which bob measures, will be same for both the qubits. Hence we have known the spin in both x and z direction simultaneously. But this would violate heisenberg uncertainty principle. Can somebody explain that why cant we measure the entangled qubits simultaneously in both bit and sign basis, what i mean to say is that Alice and Bob both have their instrument and they can process their qubit to get an answer ? Please try to explain in layman language.


title: Effect of slits and a lens

text: We have a screen with two slits (Young style) separated by a distance $d$, one of them receives a planar wave of $600nm$, the other receives a planar wave of $400nm$. Behind the slits there's a screen we observe. Only with this, as the waves have different wavelengths, I guess there can't be any interference, we will only see the difraction pattern, the two functions of the form $\sin^2(x)/x^2$, with the principal maximums separated a distance $d$. Am I right here? Now, we put a convex lens behind the slits so the screen in which we observe is in the focal plane. Ok, I have three configurations: 1.- In the first one, the system is configured such that the slits are far away from the lens. Here, we can approximate the wave that arrives as a planar wave, and therefore the lens will perform the Fourier transform in the focal plane of the screen. The diffraction of the slits also performs the Fourier transform, so this configuration should lead to having only two bars of light in the screen, centered in the focus. Am I right? 2.- The slits are in the focal plane on the lens, such that the lens is in the middle of slits-screen. Here, the same thing should happen, right? As the light comes from the focal plane, the lens must do the fourier transform with no extra things, and we should get the two bars, again both of them in the same line (center of the screen). Am I right here? 3.- The last one, I can't see... the lens is just behind the slits so the distance between slits and lens is $\approx 0$. My guess here is that the two centers of the intesity distributions $\sin^2(x)/x^2$ will go to the center of the screen (the focus) because they go perpendicular to the lens, but the rest of the pattern will just be compressed a little. Again the wave don't interfere, so the intensities just sum up, and the resulting will be: $$I=I_1\frac{\sin^2(\alpha x)}{(\alpha x)^2}+I_2\frac{\sin^2(\beta x')}{(\beta x')^2}$$ Being $\alpha$ and $\beta$ the some factor of compression due to the lens, that could actually be a function of $x$. Am I right here? Am I completely wrong? What would happen?


title: Electromagnetic waves and boundaries

text: I am reading about dielectric boundaries and electromagnetic plane waves following griffiths ch7. When considering a boundary at z=0 with electric fields perpendicular to the plane of incidence it states that the boundary condition, $$ \left[ \frac{1}{\mu}(\textbf{k}\times \textbf{E}+\textbf{k}_{r}\times \textbf{E}_{r}) - \frac{1}{\mu_{t}}(\textbf{k}_{t}\times \textbf{E}_{t})\right] \times \textbf{n}=0 $$ can be rewritten using Snells law as, \begin{equation} \sqrt{\frac{\epsilon}{\mu}}(E-E_{r})\cos \theta-\sqrt{\frac{\epsilon_{t}}{\mu_{t}}}E_{t}\cos \theta_{t}=0. \end{equation} My questions are; 1) Can someone clarify what a plane of incidence is and how it differs from a boundary? 2) Can anyone explain how to get from eq 1 to eq 2.


title: why do some orbiting objects have a changing position?

text: I know that the reason why they stay in orbit but why do some move away from the earth(the moon) or come closer and eventually fall? And why does the moon move away from earth?


title: Best way to affect a compass in 2 meter distance

text: Visitors of two public exhibitions A and B are equipped with compasses. **Exhibition A:** I want to guide these visitors along a certain path throughout the exhibition by "overriding" the earth magnetic field so that they can follow there compasses "north" along the path. The path where this deviation should work on is no wider than 1.5 meters. This deviation can be turned on and off by a switch (thus can not be accomplished by permanent magnets). **Exhibition B:** I want to notify the visitors of events by generating a short magnetic pulse that will shortly deviate their compasses. This should work over a ground circle area of at least 1 meter radius. **Assumptions:**   * Local earth magnetic field strength is below 65 uT.   * There are no other electromagnetic influences.   * Height of compasses above ground are 2 meters maximum. **Research so far:** I built a small solenoid magnet with about 400 windings, 4.8V and iron core. Although getting up to 1400 uT at the magnet itself, it rapidly drops in the distance (between ~1/r^2 and ~1/r^3) to a level where it cannot be distinguished from the earth magnetic field; reaching 70uT as soon as 20cm. In order to reach compasses in 2 meters height (200cm distance, 20cm*10) I would have to amplify the strength by a factor of 1000 (10^3, because of 1/r^3) in case I assume a Dipole (Wikipedia). I doubt this can be accomplished with my small self-built magnet. What would be the most cost- and time-efficient way to accomplish exhibition A and B? In particular I'm interested in:   * how to simulate such an environment on a computer,   * what kind of magnet type/layout is suitable,   * can this be accomplished with off-the-shelf equipment,   * are there any dangers for the visitors. I'm happy to provide more details if needed.


title: Tight Binding Model in Graphene

text: I'm following a calculation done by a guy who's done it a bit different than what I've done before (used nearest neighbour vectors and a DFT instead of what I will show below), I'm not quite sure how to invert this expression he gives. We are looking at formulating the tight binding picture for graphene, using the convention $ \mathbf{r} = l \mathbf{a_1} + j \mathbf{a_2}$ for the position of the unit cell, and the number $s = 1,2$ for the intra-cell atom. State 1 is located at position $\mathbf{r}$. On-site potential is $\epsilon_0$ and hopping potential is $t_0$ as usual. The Hamiltonian of this picture is written in a basis of localised orbitals $\left| \mathbf{r} , s\right>$. $$\begin{align} \hat{H} &= \sum_\mathbf{r} \biggl(\sum_{s=1,2} \left|\mathbf{r},s\right> \epsilon_0 \left<\mathbf{r},s\right|\biggr) \\\ &\quad + \left|\mathbf{r},1\right> t_0 \bigl(\left<\mathbf{r},2\right| + \left<\mathbf{r} - \mathbf{a_1},2\right| + \left<\mathbf{r} - a_2 , 2\right|\bigr) \\\ &\quad + \left|\mathbf{r},2\right> t_0 \bigl(\left<\mathbf{r},1\right| + \left<\mathbf{r} + \mathbf{a_1},1\right| + \left<\mathbf{r} + a_2 , 1\right|\bigr)\end{align}$$ To make this a bit clearer without drawing a diagram, the top line obviously refers to on-site potential and the second and third to the nearest neighbour hopping (i.e. electron at atom 1 hops to atom 2 either in same unit-cell or the one at $r-a_1$ or $r-a_2$). This is where it starts to get hazy for me, the next step is that using Bloch's Theorem the eigenvectors for the Hamiltonian are given by: $$ \left|\mathbf{k}, \alpha\right> = \frac{1}{\sqrt{N_C}} \sum_\mathbf{r} \sum^2_{s=1} e^{i \mathbf{k} \cdot \mathbf{r}} A_s \left|\mathbf{r},s\right>$$ where $A_s$ are coefficients, $N_C$ normalising factor. When I have done the TBM method for graphene in the past, I used a discrete Fourier Transform to get an expression for the state in momentum space, inverted it and put it back into the original Hamiltonian, giving an expression for the Hamiltonian in k-space. I would guess it is a similar technique in this case and my main question would be how to invert the above expression if this is the case? The sum of s is confusing me when I try to do this! Additionally 1) How exactly do we arrive at this using Bloch's theorem? I know in the TBM we look for eigenfunctions which are a linear combination of orbitals, and that the expression does look kind of similar to that. Is the above expression in reciprocal space? 2) The $\alpha$ is later used to refer to the positive or negative (conduction or valence) bands, is it included here for convenience and continuity to later or is there a way to infer that from the Hamiltonian or unit cell already? It is not doing anything in the expression at the moment, but is it used somehow when we invert it (maybe like a k-space equivalent of s=1 or 2)? 3) I guess the result of applying that equation to the Hamiltonian is that you get a 2x2 matrix of the Hamiltonian in k-space, diagonalise this and find the energy eigenvalues? Would really appreciate some help on any of this, I have been looking online for stuff all day but everyone does it differently, leaves stuff out and uses different notation!


title: $\hat{\imath}$ component of force exerted on an electron by a magnetic field?

text: > The magnetic field over a certain range is given by $\vec{B} = > B_x\hat{\imath} + B_y\hat{\jmath}$, where $B_x= 4\: \mathrm{T}$ and $B_y= > 2\: \mathrm{T}$. An electron moves into the field with a velocity $\vec{v} = > v_x\hat{\imath}+v_y\hat{\jmath}+v_z\hat{k}$, where $v_x= 5\: \mathrm{m/s}$, > $v_y= 8\: \mathrm{m/s}$ and $v_z= 9\: \mathrm{m/s}$. The charge on the > electron is $-1.602 \times 10^{-19}\: \mathrm{C}$. What is the > $\hat{\imath}$ component of the force exerted on the electron by the > magnetic field? Answer in units of $\mathrm{N}$. I know that $\vec{F}=q\vec{v} \times \vec{B}$, so plugging in I have: $$\vec{F}=(-1.602 \times 10^{-19})<5,8,9> \times <4,2,0>$$ My confusion is to whether or not multiply my velocity vector components by charge (the scalar) or if I take the cross product between $\vec{B}$ and $\vec{v}$ first? I'd also like to know why whichever operation comes first does in fact come first.


title: Is it possible to make a light beam act like a stream of water from a spining hose?

text: If we had the ability to make an actuator that could turn around at or past the speed of light and I attached a high miliwatt laser to it, then spun the laser around on the actuator at the speed of light, would the light beam bend act like a the stream of water from a spinning hose as the laser went around it?


title: Is there a general physics simulator for learning purposes?

text: Is there a complete physics simulator that I can use to do general simulations for learning purposes? For example:   1. Create a sandbox.   2. Fill with a gas.   3. Load a 3d solid model like this (but 3d).   4. Fill it with a dense liquid.   5. Load gravity.   6. Watch, measure and understand how a barymeter works. It doesn't need to be precise, just usable, so I guess it is not impossible. The point would be to simulate and visualize any kind of exercise you would find in your physics book. It would be the mother of the learning tools. If it doesn't exist, is anybody interested in programming it?


title: No closed orbits for a Newtonian gravitational field in 4 spatial dimensions

text: We are supposed to show that orbits in 4D are not closed. Therefore I derived a Lagrangian in hyperspherical coordinates $$L=\frac{m}{2}(\dot{r}^2+\sin^2(\gamma)(\sin^2(\theta)r^2 \dot{\phi}^2+r^2 \dot{\theta}^2)+r^2 \dot{\gamma}^2)-V(r).$$ But we are supposed to express the Lagrangian in terms of constant generalized momenta and the variables $r,\dot{r}$. But as $\phi$ is the only cyclic coordinate after what I derived there, this seems to be fairly impossible. Does anybody of you know to calculate further constant momenta?


title: Hamiltonian reduction having constant of the motion

text: I have this $2^n*2^n$ matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but 0100 it is) $s_+$ is defined with $s_x+is_y$ and $s_-$ is defined with $s_x-is_y$ (creation and annichilation operators) I've created the hamiltonian composing pauli matrices with kronecker product. I now that I have only few operation in my algebra: $I, s_z, s_+[k].s_-[k-1]$ where k is the spin number where this operation is made. This algebra sends valid state to valid states, preserving the constant of my motion. How can i reduce the dimension of my matrix deleting row and columns to remain in a proper subspace? How can I have a $n*n$ matrix?? Thanks!


title: Can the vanishing of the Riemann tensor be determined from causal relations?

text: Given a _Lorentzian manifold and metric tensor_ , "$( M, g )$", the corresponding _causal relations_ between its elements ( _events_ ) may be derived; i.e. for every pair (in general) of distinct events in set $M$ an assignment is obtained whether it is _timelike_ separated, or _lightlike_ separated, or neither ( _spacelike_ separated). In turn, I'd like to better understand whether causal separation relations, given abstractly as "$( M, s )$", allow to characterize the corresponding Lorentzian manifold/metric. As an exemplary and surely relevant characteristic (cmp. answer here) let's consider whether _the Riemann curvature tensor vanishes_ , or not, at each event of the whole set $M$ (or perhaps suitable subsets of $M$). Are there particular causal separation relations which would be indicative, or counter-indicative, of the Riemann curvature tensor vanishing at all events of set $M$ (or if this may simplify considerations: at all events of a _chart_ of the manifold); or on some subset of $M$? To put my question still more concretely, consider as possible illustration of "counter-indication": (a)   Can ~~any _chart of a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor_ (or, at least, a whole such manifold) contain ~~ [Edit in consideration of 1st comment (by twistor59): -- _the Riemann curvature tensor vanish at least in one event of a 3+1 dimensional Lorentzian manifold_ if each of its _charts_ contains -- ]   * fifteen events (conveniently organized as five triples): $A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q\,\,\,\,$, and $\,\,\,\, U, V, W$,   * where (to specify the causal separation relations among all corresponding one-hundred-and-five event pairs): $s[ A, B ]$ and $s[ A, C ]$ and $s[ B, C ]$ are _timelike_ ,   $s[ F, G ]$ and $s[ F, H ]$ and $s[ G, H ]$ are _timelike_ ,   $s[ J, K ]$ and $s[ J, L ]$ and $s[ K, L ]$ are _timelike_ ,   $s[ N, P ]$ and $s[ N, Q ]$ and $s[ P, Q ]$ are _timelike_ ,   $s[ U, V ]$ and $s[ U, W ]$ and $s[ V, W ]$ are _timelike_ , $s[ A, G ]$ and $s[ G, C ]$ and $s[ A, K ]$ and $s[ K, C ]$ and   $s[ A, P ]$ and $s[ P, C ]$ and $s[ A, V ]$ and $s[ V, C ]$ are _lightlike_ , $s[ F, B ]$ and $s[ B, H ]$ and $s[ F, K ]$ and $s[ K, H ]$ and   $s[ F, P ]$ and $s[ P, H ]$ and $s[ F, V ]$ and $s[ V, H ]$ are _lightlike_ , $s[ J, B ]$ and $s[ B, L ]$ and $s[ J, G ]$ and $s[ G, L ]$ and   $s[ J, P ]$ and $s[ P, L ]$ and $s[ J, V ]$ and $s[ V, L ]$ are _lightlike_ , $s[ N, B ]$ and $s[ B, Q ]$ and $s[ N, G ]$ and $s[ G, Q ]$ and   $s[ N, K ]$ and $s[ K, Q ]$ and $s[ N, V ]$ and $s[ V, Q ]$ are _lightlike_ , $s[ U, B ]$ and $s[ B, W ]$ and $s[ U, G ]$ and $s[ G, W ]$ and   $s[ U, K ]$ and $s[ K, W ]$ and $s[ U, P ]$ and $s[ P, W ]$ are _lightlike_ , the separations of all ten pairs among the events $A, F, J, N, U$ are _spacelike_ ,   the separations of all ten pairs among the events $B, G, K, P, V$ are _spacelike_ ,   the separations of all ten pairs among the events $C, H, L, Q, W$ are _spacelike_ , and finally the separations of all twenty remaining event pairs are _timelike_   ? Conversely, consider as possible illustration of "indication": (b)   Is there a _3+1 dimensional Lorentzian manifold with ~~everywhere vanishing Riemann curvature tensor_ (or, at least, one of its _charts_ ) which doesn't ~~ [Edit in consideration of 1st comment (by twistor59): -- _nowhere vanishing Riemann curvature tensor_ such that all of its _charts_ \-- ] contain   * twenty-four events, conveniently organized as four triples ($A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q$) and six pairs ($D, E$; $\,\,\,\, S, T$; $\,\,\,\, U, V$; $\,\,\,\, W, X$; $\,\,\,\, Y, Z$; $\,\,\,\, {\it\unicode{xA3}}, {\it\unicode{x20AC}\,}$),   * where (again explicitly, please bear with me$\, \\!^*$): the sixty-six separations among the twelve events belonging to the four triples are exactly as in question part (a), each of the six pairs is _timelike_ separated, the separations of all fifteen pairs among the events $D, S, U, W, Y, {\it\unicode{xA3}}$ are _spacelike_ ,   the separations of all fifteen pairs among the events $E, T, V, X, Z, {\it\unicode{x20AC}\,}$ are _spacelike_ , $s[ D, {\it\unicode{x20AC}\,} ]$ and $s[ S, Z ]$ and $s[ U, X ]$ are _spacelike_ ,   $s[ E, {\it\unicode{xA3}} ]$ and $s[ T, Y ]$ and $s[ V, W ]$ are _spacelike_ , $s[ A, {\it\unicode{xA3}} ]$ and $s[ A, Y ]$ and $s[ A, W ]$ are _spacelike_ ,   $s[ A, {\it\unicode{x20AC}\,} ]$ and $s[ A, Z ]$ and $s[ A, X ]$ are _timelike_ ,   $s[ A, E ]$ and $s[ A, T ]$ and $s[ A, V ]$ are _timelike_ , $s[ C, {\it\unicode{x20AC}\,} ]$ and $s[ C, Z ]$ and $s[ C, X ]$ are _spacelike_ ,   $s[ C, {\it\unicode{xA3}} ]$ and $s[ C, Y ]$ and $s[ C, W ]$ are _timelike_ ,   $s[ C, D ]$ and $s[ C, S ]$ and $s[ C, U ]$ are _timelike_ , $s[ F, {\it\unicode{xA3}} ]$ and $s[ F, D ]$ and $s[ F, S ]$ are _spacelike_ ,   $s[ F, {\it\unicode{x20AC}\,} ]$ and $s[ F, E ]$ and $s[ F, T ]$ are _timelike_ ,   $s[ F, V ]$ and $s[ F, X ]$ and $s[ F, Z ]$ are _timelike_ , $s[ H, {\it\unicode{x20AC}\,} ]$ and $s[ H, E ]$ and $s[ H, T ]$ are _spacelike_ ,   $s[ H, {\it\unicode{xA3}} ]$ and $s[ H, D ]$ and $s[ H, S ]$ are _timelike_ ,   $s[ H, U ]$ and $s[ H, W ]$ and $s[ H, Y ]$ are _timelike_ , $s[ J, D ]$ and $s[ J, U ]$ and $s[ J, Y ]$ are _spacelike_ ,   $s[ J, E ]$ and $s[ J, V ]$ and $s[ J, Z ]$ are _timelike_ ,   $s[ J, T ]$ and $s[ J, X ]$ and $s[ J, {\it\unicode{x20AC}\,} ]$ are _timelike_ , $s[ L, E ]$ and $s[ L, V ]$ and $s[ L, Z ]$ are _spacelike_ ,   $s[ L, D ]$ and $s[ L, U ]$ and $s[ L, Y ]$ are _timelike_ ,   $s[ L, S ]$ and $s[ L, W ]$ and $s[ L, {\it\unicode{xA3}} ]$ are _timelike_ , $s[ N, D ]$ and $s[ N, S ]$ and $s[ N, W ]$ are _spacelike_ ,   $s[ N, E ]$ and $s[ N, T ]$ and $s[ N, X ]$ are _timelike_ ,   $s[ N, V ]$ and $s[ N, Z ]$ and $s[ N, {\it\unicode{x20AC}\,} ]$ are _timelike_ , $s[ Q, E ]$ and $s[ Q, T ]$ and $s[ Q, X ]$ are _spacelike_ ,   $s[ Q, D ]$ and $s[ Q, S ]$ and $s[ Q, W ]$ are _timelike_ ,   $s[ Q, U ]$ and $s[ Q, Y ]$ and $s[ Q, {\it\unicode{xA3}} ]$ are _timelike_ , and finally the separations of all ninety-six remaining event pairs are _lightlike_   ? (*: The two sets of causal separation relations stated explicitly in question part (a) and part (b) are of course not arbitrary, but have motivations that are somewhat outside the immediate scope of my question -- considering Lorentzian manifolds -- itself. It may nevertheless be helpful, if not overly suggestive, to attribute the relations of part (a) to "five participants, each finding coincident pings from the four others", and the relations of part (b) to "ten participants -- four as vertices of a regular tetrahedron and six as middles between these vertices -- pinging among each other".)


title: Electrostatics with Yukawa mass

text: I seem to have forgotten how to solve Laplace's equation when there is a Yukawa mass $\mu$. I want to find the Yukawa potential due to a homogeneously charged sphere of radius $R$ and charge density $\rho$. **My attempt** I need to solve $(\nabla^2-\mu^2)\phi = -\rho$ inside and outside the sphere. Since there is spherical symmetry, I can convert to spherical coordinates, and assume the potential $\phi$ has no angular dependence: $$ \frac{1}{r}\frac{\partial^2}{\partial r^2} (r\phi) - \mu^2 \phi = -\rho $$ Outside the sphere, there is no charge, so I solve the homogenous equation giving the solution: $$\phi_\text{out}(r) = A \frac{e^{-\mu r}}{r}+B \frac{e^{\mu r}}{r}$$ and inside the sphere, where there is charge density, I solve the inhomogenous equation, giving the solution: $$\phi_\text{in}(r) = \frac{\rho}{\mu^2}+C \frac{e^{-\mu r}}{r}+D \frac{e^{\mu r}}{r}$$ **Problem** I have four constants of integration ($A$, $B$, $C$, $D$) but only three pieces of information to fix them.   1. The potential should not blow up as $r\rightarrow\infty$: I must have $B=0$   2. It should not be singular as $r\rightarrow 0$: I must have $D=-C$. And therefore, $$\phi_\text{in}(r) = \frac{\rho}{\mu^2} + 2 C \frac{\sinh( \mu r)}{r}$$   3. Also, I must have continuity at the surface of the sphere $r=R$: Then I have $$A=\big(\frac{\rho R}{\mu^2}+ 2 C \sinh \mu R\big)e^{\mu R}$$ My potential has an undetermined constant $C$; What is the final piece of information that will fix $C$?


title: Quantum Field Theory from a mathematical point of view

text: I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view. Are there any good books or other reference material which can help in learning about quantum field theory? What areas of mathematics should I be familiar with before reading about Quantum field theory?


title: If human energy use were to increase by 2.3% per year, would we use all solar energy by 1400 years?

text: Assuming 2.3% per year exponential growth of human energy consumption (so, roughly speaking, that corresponds to multiplying consumption tenfold every century), it's argued that human annual energy consumption would be equal to annual solar energy output in 1400 years time. Is that correct? For reference, human energy consumption is currently about $2 \times 10^{13} W$ (source: http://physics.ucsd.edu/do-the-math/2011/07/galactic-scale-energy/ )


title: An example which contradict to Newton's 3rd law?

text: Let a,b be two charged particles. $$\vec{r}_a(0)=\vec{0}$$ $$\vec{r}_b(0)=r\hat{j}$$ $$\vec{v}_a(t)=v_a \hat{i}$$ $$\vec{v}_b(t)=v_b\hat{j}$$ In which both $v_a$ and $v_b$ $<<c$. Then $$\vec{E}_{ab}(0)=\frac{q_a}{4\pi \epsilon r^2}\hat{j}$$ $$\vec{B}_{ab}(0)=\frac{\mu q_av_a}{4\pi r^2} \hat{k}$$ $$\vec{E}_{ba}(0)=-\frac{q_b}{4\pi \epsilon r^2}\hat{j}$$ $$\vec{B}_{ba}(0)=\vec{0}$$ Note that $v_a$ and $v_b$ $<<c$ thus a and b almost obey Coulomb's law. Moreover, $\vec{j_i}(\vec{r})=q_i\delta(\vec{r}-\vec{r}_i)\vec{v}_i$ hence BS law can be applied. Hence $$\vec{F}_{ab}(0)=q_b(\vec{E}_{ab}+\vec{v}_b \times \vec{B}_{ab})$$ $$=\frac{q_a q_b}{4\pi \epsilon r^2}\hat{j}-\frac{\mu q_av_a v_b}{4\pi r_b^2} \hat{i}$$ But $$\vec{F}_{ba}(0)=-\frac{q_aq_b}{4\pi \epsilon r^2}\hat{j}$$ Consequently $$\vec{F}_{ab} \ne -\vec{F}_{ba}$$ This result contradict to Newton's 3rd law!! But I cannot find any error... It troubled me.


title: Hamiltonian mechanics and conservation of energy?

text: Can anyone explain to me Hamiltonian mechanics relation to conservation of energy? I'm not very good at mathematics, and I know it's important into understanding Hamiltonian mechanics. However, can it be explained in a simple way?


title: Magnetic field of a moving point charge

text: I'm having a hard time to grasp one simple idea. Let us say we have a moving point charge, q, with velocity v, in the x direction. Now calculating both the Magnetic and the Electric fields is easy: We know that B=0 in the charge's frame, hence we obtain B in the lab's frame. Now, let us say we would want to know the same fields, but with a stationary charge, positioned above the moving charge (perpendicular). I could not just surmise that there is no Magnetic field in the moving charge's frame since, in turn, the other one is now moving in the other one's frame, so in both frames there is Magnetic field. I would be more than grateful if someone could settle this, thanks.


title: Why it is difficult to drink tea with a straw?

text: It is easy to drink cold drinks, soft drinks with straw but when I tried to drink tea with straw then that was very unpleasant and even for time I lost the sense from my tongue. My question is why it is difficult to drink hot liquids with straw?(Drinking cold liquids is very pleasant!!!)


title: Ferromagnets and magnets tend to align in the center. Why is that?

text: When you bring a large iron plate and a magnet, the magnet attracts the iron plate and it tends to slide itself to the center. When I place it on the edge, it always aligns at the center why is that?


What's the significance of large-scale anomalies in CMB that are confirmed by Planck? I've read somewhere that the cold spots can provide support for string theory or it may be due to a parallel universe . To what extent is that true ? Are there models that can explain these anomalies?


title: blood fluidics - can you determine the force exerted on a bound red blood cell under shear stress?

text: Is it possible to calculate the force exerted on a bound, infected red blood cell under various shear stresses? The strength of an adhesive interaction is normally measured by SPR or AFM, however using microfluidics, you can increase the shear stress until the cell dislodges, therefore, can you use this data to calculate force?


title: How to charge a field?

text: In a previous post [ Noether theorem, gauge symmetry and conservation of charge ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied to a global $U(1)$ gauge symmetry, or via the covariant (or minimal, or Weyl, or ...) substitution with a $U(1)$ local gauge symmetry and antisymmetry of the electro-magnetic field applied to the equations of motion. I nevertheless still get confused about something: what is a charge ? More precisely, how to charge a field ? Indeed, if one believes that only the global gauge construction is able to demonstrate the conservation of the charge, one has to admit that the charge is not defined that way ! The same demonstration applies for the conservation of the particle number. Indeed, I used in [ Noether theorem, gauge symmetry and conservation of charge ] a Lagrangien for _uncharged_ particles to show the conservation of a particular current through Noether theorem. **So the question becomes: how to charge this particle ?** In particular: how to charge this particle and still conserve the global gauge ? NB: In the local gauge construction, it seems to be easier to give a charge to the system, but then only the second Noether theorem applies, and people get annoyed by that. **To conclude, a** (possibly important) **remark about my point of view:** I would like to understand the problem of "charging the field" in a condensed matter perspective, when I believe we have no a priori idea of the charge of the quasi-particles appearing in our effective theories. Can we overcome this problem ? Of course, any point of view (especially the opinion of QFT physicists who might have already tackled this problem) is warmly welcome :-) And the question "how to charge a field ?" is just a warm-up for this more complicated one "how to define the charge of an effective, emergent field ?" which is entirely subsidiary question for the moment. **IMPORTANT EDIT** I just became aware of this question [ How does non-Abelian gauge symmetry imply the quantization of the corresponding charges? ] which is strongly related, and well answered. I nevertheless think one can continue to discuss here about the condensed matter problem. Indeed, there are some people who believe that a superconductor (among other emergent phases of matter) have charge neutral excitations [see e.g. http://dx.doi.org/10.1103/PhysRevB.41.11693 which is too old to be on arXiv, but this one: http://arxiv.org/abs/cond-mat/0404327 rephrases it in a different way.]. So the question follows: how can these excitations at low temperatures become again charged at higher temperature ? How can we measure charge current from an uncharged excitation ? How can we "charge" a field after all ?


title: Minimum $E$ of $p\bar{p}$-collision for $q\bar{q}$ pair with mass $m_q$

text: I am currently working out the energy required to create a particle anti- particle pair from a collision of a proton travelling along the x-direction with an anti-proton which is at rest. The particle has a mass $m_q$. Conservation of 4-momentum in the rest frame of the target anti-proton ($c=1$): $p_1=(m_\text{p},0,0,0)$ The moving proton has this minimum energy, the quantity we are trying to find: $p_2=\left(E, \sqrt{E^2-m_p^2 },0,0\right)$ Then what I find confusing is the terminology, it says that $p_{q\bar{q}}=\left(\sqrt{p^2+4 m_q^2 },p,0,0\right)$ Where has this $p$ come from and what is it? Is it the momentum of the center of mass of the $q\bar{q}$ system? Also, we don't seem to have taken much care about reference frames here, or rather I havn't thought about them really which is worrying. Finally, it says that the minimum energy, E, the $q\bar{q}$-pair will be at rest in the CMF frame. What is this frame referring to? The center of mass of the particle/anti-particle pair? I understand that the threshold energy to produce these will be a combination of the energy of the proton and their rest masses but I don't know how we have the equation (above)


title: Calculating the field of an infinite flat sheet of charge using the superposition principle

text: I am trying to calculate the field of an infinite flat sheet of charge (a plain with uniform charge density $\sigma$) using the superposition principle. I know that the field of an infinite line charged with uniform charge density $\lambda$ is $E(r)=\frac{2\lambda}{r}$. I want to consider the plain as infinite number of lines from $-\infty$ to $\infty$. I wrote that $$ dE=\frac{2\lambda}{r} $$ and since $\lambda=\sigma dl$ we get $$ dE=\frac{2\sigma dl}{r} $$ $$ E(r)=\int_{-\infty}^{\infty}\frac{2\sigma dl}{r} $$ Which is clearly wrong: First, I still have $r$, secondly I get that the above equals to $$ \frac{2\sigma}{r}\int_{-\infty}^{\infty}dl $$ which does not converge. What are my mistakes, and how do I get the correct result: $2\pi\sigma$ ?


title: Relation between statistical mechanics and quantum field theory

text: I was talking with a friend of mine, he is a student of theoretical particle physics, and he told me that lots of his topics have their foundations in statistical mechanics. However I thought that the modern methods of statistical mechanics, for example the renormalization group or the Parisi- Sourlas theorem, come from the methods of quantum field theory or many-body techniques (Feynman diagrams and so on). I notice that books also regarding modern concepts, such as spin glasses, don't require any other knowledge then basic calculus. Can someone explain which is the relation between these subjects? What topics should I study of field theory or similar to have a deep understanding in statistical mechanics?


title: Charge density of turning object

text: A charged disk or sphere will create currents around its turning axis if a rotation is added. The total current can then be calculated by adding all concentric currents together. Every current (I take the example of a thin charged disk, as it is the most easiest casus) is as big as: $$ dI = dQ / dt = f*\sigma*2 \pi *rdr $$ So the total charge would be something like: $$ I = \omega\int_0^R \sigma r dr $$ With $R$ the radius of the disk,$f$ the frequency, $\omega$ the angular speed (so we can leave the $2\pi$ out) and $\sigma$ the charge density. In my physics book they consider $\sigma$ as a constant. But I was wondering: as electrons (who determine the charge) have mass, they also have a centrifugal force, which would imply that the charge density is NOT the same at every distance from the centre, so we would have to define a $\sigma$ in function of $r$ (so $\sigma$ is **not** just total charge on total surface). Would this centrifugal force give considerable different results and if so, could anyone show me a calculation of the difference in total current? Also if the difference is not considerable for a small disk, would it be for a very large disk, because the centrifugal force is bigger at a bigger radius (for example a disk with radius 1000km)?


title: Physics of the electric hot plate

text: For an electronics experiment, I began wondering about the **electric hot- plates** (specifically the temperature dependence over time). If I were to measure the given temperature over time, I assume that the temperature would _not_ increase linearly. In fact, I suspect the initial heating rate will only decrease. Do anyone have some intuition regarding a model for this? An answer giving an ODE is fine too...


title: Is a scaled-up aircraft carrier 'ski-ramp' a viable system to impart enough velocity to significantly assist a spacecraft to orbit?

text: Not a boost straight up to escape velocity, just sufficient added momentum to make a significant economic saving on fuel cost, mass and complexity of a standard launch. It seems ludicrous to me how much fuel in a standard launch appears to be wasted pushing thousands of tonnes of rocket fuel in a vertical direction from a standing start. The system I envisage would have a horizontal track of possibly 2-3km (distance to be debated), acceleration via external systems (electromag or JATO?), continue up a ramp of slowly increasing gradient, with craft's own engines now running, and exit the ramp at between Mach 1 or 2. Safety run-off track if craft's own engines fail to ignite properly. Winged style craft for stability from ramp & shuttle-style landing. Similar I know to the Phys.SE 'Rail gun' question here, but acceleration could be more survivable. Had a longer script but it appeared to over-run the site limit. Acknowledge AdamRedwine's railgun question -would have added my contribution to that but my Reputation today is only 1. Acknowledge Gerry Anderson's Fireball XL5 puppet show on British TV in the '60s - the launch concept has always stayed with me! Submitted for criticism.


title: What causes stress concentration (aka stress risers/raisers) at corners?

text: I've read a few explanations about why stress concentration occurs at sharp corners but I don't find the explanations intuitive. Can anyone explain it perhaps using an analogy such as atoms "holding hands" with neighbor atoms or a similar easy to understand analogy? Most explanations I've shown just show the result of a finite-element-analysis based stress analysis. This shows the emergent property but doesn't explain the underlying mechanism that causes the stress to concentrate at corners. Better understanding the cause of the crashes of the De Havilland Comet jet aircraft due to metal fatigue at the corners of their rectangular windows is the context for this question. See: http://www.cracked.com/article_19623_6-small-math-errors-that-caused-huge- disasters.html The answer I'm looking for should not contain any math.


title: Can inertia be explained by Bremsstrahlung?

text: Considering that on the atomic level objects consists of densely spaced positively and negatively charged particles, does not the acceleration of those objects lead to Bremsstrahlung of those particles? And although the monopole field is zero, couldn't higher order multipole radiation escape and cause the inertia? I would think at least a small effect like this must happen, even if it doesn't explain all of the inertia.


title: Estimate quarter mile time

text: I need to estimate a drag race quarter mile time given the car's weight, bhp and preferably the drive (FWD, RWD, 4WD). I know $v(t) = ds/dt$ and $a(t) = dv/dt = d^2s/dt^2$, but how can I get the function $s(t)$ to calculate $v(t)$ and $a(t)$? I thought maybe Power to Weight ratio would help for solving the acceleration but I have no idea what to do with it and maybe the drive would just be a constant which is then subtracted from the calculated time. Can you help me?


title: Gravitational Potential of a Sphere vs Gravitational Binding Energy of a Sphere

text: My question is about two equations regarding uniform spheres that I've run into: $V=\frac{GM}{r}$ ... and ... $U = \frac{3}{5}\frac{GM^2}{r^2}$ $V$ is unknown to me, and is described (in Solved Problems in Geophysics) as "the gravitational potential of a sphere of mass M." I also found it online called "the potential due to a uniform sphere." $U$ is what I've seen before and I know it by the descriptions "sphere gravitational potential energy" or "gravitational binding energy." My understanding is that $U$ is the amount of energy required to build the sphere piece by piece from infinity. I also recognize $GMm/r$ as the gravitational potential between two masses. Can someone explain the difference between these concepts? How can $GM/r$ be the "gravitational potential of a sphere"? Isn't that what $U$ is? Thank you very much.


title: Why does vibration loosen screws?

text: I am trying to figure out why vibrations (say, from an engine) loosen screws. It seems to me that there is evident symmetry between loosening and tightening a screw. I am wondering what breaks this symmetry.


title: Uniform constant magnetic field and traditional attractive force

text: Why uniform constant magnetic fields can not exert net force on a piece of iron whatever strong it might get?


title: Is the search for a Simple-group-based Electro-Weak theory over?

text: **Just wondering:** We know that, in its current form of the $SU(2)_L\times U(1)$, the electroweak theroy rides a wave of huge success. However, is it not possible that the correct **simple** group unification could reveal physics that the current model doesn’t? For example, it could reveal the existence of a unique gauge boson, like the photon for EM ('EM-photon') produced by $U(1)$ invariance, which might be called the 'EW-photon.’ This EW-photon, with the right group structure, could then produce the $W^+,W^-,Z_0$ and the EM-photon and the known physics at the E-W scale? Does anybody know whether there is any activity on this, other than Technicolour? Links will be greately appreciated.


title: Effects of a non-Lorentz-invariant vacuum state

text: I'm here asking about real or though experiments (i.e., physical effects) where, at least in principle, one can see some consequence of a non-Lorentz- invariant vacuum state in an otherwise Poincare invariant theory. Let me develop the question. Assume a theory in which the Hamiltonian (that closes Poincare algebra with the rest of generators) $H$ acts non-trivially on the vacuum state $|0>$, where 'non-trivially' simply means: $$H|0> = E \, |0>$$ with $E$ a positive constant. Since the Hamiltonian transforms as the temporal component of a 4-vector, the vacuum state is not Lorentz invariant. Therefore, the theory is not Lorentz invariant (it is usually claimed that this is an additional condition besides the Poincare algebra). However, I'm not able to see any consequence of this fact. I think that this does not affect any cross- section or decay rate. I think that one can redefine the Hamiltonian so that $H'=H-E$ and $H'|0>=0 \,$ (this does not affect the Poincare algebra if one also redefines the boost generators properly). I know this seems obvious (this redefinition is usually done in canonical quantization when one doesn't adopt normal ordering), but I've just read this in this forum: http://physics.stackexchange.com/a/8360/10522 > However, in special relativity, energy is the time component of a 4-vector > and it matters a great deal whether it is zero or nonzero. In particular, > the energy of the empty Minkowski space has to be exactly zero because if it > were nonzero, the state wouldn't be Lorentz-invariant: Lorentz > transformations would transform the nonzero energy (time component of a > vector) to a nonzero momentum (spatial components). One has a family of vacuum states related by Poincare transformations which are unitary, I don't think this is a problem... what do you think? Added: 1) I'm not thinking about a Poincare invariant Lagrangian with a potential of the form $(A^2(x)-v)^2$, where $A_{\mu}(x)$ is a vectorial field that acquires a vacuum expectation value $v$. Assume that every field has zero vev. 2) I'm looking for Lorentz violating effects instead of vacuum energy effects unless you argue that these vacuum energy effects (Lamb shift, spontaneous emission, etc.) break Lorentz symmetry.


title: How many fundamental fields / constraints are in Maxwell's Equations?

text: I've seen electromagnetics formulated in the potentials ($\bf A$, $\varphi$) (and their magnetic counterparts) in the Lorenz gauge for a long time. Justifications for using these include that they are less singular than the fields ($\bf E$, $\bf H$) and reduce the redundancy of Maxwell's equations. All the fields derive from two vector and two scalar potentials (total of 8 potentials). In older engineering works (1950s-1969), I've found they don't use the ($\bf A$, $\varphi$) potentials at all, and instead use a set of two vector potentials called the Hertz vectors by the authors, usually noted by ${\bf\Pi}$, from which all the fields derive. They seem to have an additional level of regularity to them because the ($\bf A$, $\varphi$) potentials derive from them. There are also a total of only 6 scalar quantities between the two, from which all the fields derive. Modern authors seem to prefer the dyadic Green's functions. I haven't studied these in any depth, but it appears to me that in this formulation, all the fields derive from 5 scalar quantities? Not sure about this one so feel free to correct me. But it got me thinking about my questions: Is 3D classical EM, how many fundamental constraints are there in Maxwell's equations? What is the minimum number of quantities that can be defined, from which all the fields can be said to derive? I suspect the differential forms / tensor formulation will be necessary to get to the bottom of this, but if there is an interpretation in terms of 3D vector calculus, that'd be great.


How can we determine during a lunar eclipse whether the earth moves faster or the moon using minimum instruments?


title: Explanation Needed: Thermodynamics of a hot/cold water jet machine

text: I didn't know where to begin with this problem. I eventually found a solution online, which is why I'm reposting this question with an answer. I was wondering if anyone can explain the one question I have. A self-contained machine only inputs two equal steady streams of hot and cold water at temperatures $T_1$ and $T_2$. Its only output is a single high-speed jet of water. The heat capacity per unit mass of water, $C$, may be assumed to be independent of temperature. The machine is in a steady state, and the energy in the incoming jets is negligible. $(a)$ What is the speed of the outgoing jet in terms of $T_1$, $T_2$ and $T$, where $T$ is the temperature of the outgoing jet? $(b)$ What is the maximum possible speed of the jet? $(a)$ The heat intake per unit mass is $\Delta Q = (C(T_1 - T) - C(T - T_2))/2$ Since the system is in a steady state, $\frac{v^2}{2}=\Delta Q$, therefore $v=\sqrt{C(T_1 + T_2 - 2T)}$ **_why is that the case? Why does all the heat go towards its kinetic energy per unit mass? That is my one question_** $(b)$ $\Delta S = \frac{C}{2}[ln(\frac{T}{T_1})+ln(\frac{T}{T_2})] \ge 0$ So $T\ge \sqrt{T_1 T_2}$ and $v \le v_{max} = \sqrt{C(T_1 + T_2 - 2\sqrt{T_1 T_2})}$ What an insidious problem, no?


title: What can justify the decrease of the electrical conductivity with the increase of light intensity?

text: I have currently been working with a sample that "appears to" decrease its resistance when I cover it and protect it from light. Basically it presents the opposite behaviour of a photoresistor. What kind of phenomenon may cause this? Or it has to be a measurement error? EDIT: I am talking about typical room light from fluorescent lamps on the ceiling. Experimental Setup: Sample: A dense, sintered, ceramic pellet (with a disc shape, 1 cm in diamater and ~3mm thick). The faces of the disc were carefully painted with conductive silver ink to make the electrical contacts. An Agilent 4294A impedance analyzer was used to measure the electrical properties with a 0.5 V AC signal. This was done at air, in a room with the ceiling lights turned on. Approaching my hands to the sample, blocking the light, the resistance increases.


title: Gravitation not instantaneous = non elliptic orbits?

text: When I studied physics some time ago my teacher explained that if we consider the gravitational atraction not instantaneous, such as the General Relativity says, the planets would be attracted towards the position were other planets were before and not the current one, getting spiral orbits instead of ellipses. What was the solution to this problem? Because we do observe elliptic orbits. I guess when we use speed "c" we can't continue using newton laws and we need the realtivity. Many articles, even in the Wikipedia, say that the speed of the gravitons must be much higher than "c".   Is it a complet nonsense or some academics believe it? regards


title: The border of the System

text: In General Relativity, if the system accelerates, the inside of the system and the outside of the system will have different speed of time. Where is the boundary of the system? If a human accelerates closer to the speed of light, does that mean the human exist inside of their specific system?


title: Derivation of relativistic energy

text: The concept of relativistic energy comes from it's conservation in relativistic mechanics for an elastic collision. It seems to me that another possible derivation could equate the energy of a single particle before, with the kinetic energy after of N particles after the energy is distributed amongst them in the form of the Boltzman distribution, as well as individually being of the form $mv^2$. Are there any problems in doing this?


title: Distance between two light rays of a beam in a material.

text: Why is it that the distance between two light rays changes as they pass from one material to another? It must have something to do with the change in refractive index, but to me it seems the ''ends'' of the beam are refracted by equal amounts. Can you explain this? See the picture below. ![enter image description here](http://i.stack.imgur.com/DxMtq.png)


title: What reflective media do laser shows use?

text: I am having a hard time in finding out what exact light media laser shows use. I am trying to build a laser show myself. I know that the laser light is reflected off these particles in such a way that that it makes the laser line "viewable" in _all directions_ Can somebody explain to me how exactly do the collection of particles make it viewable in all directions and what exact conditions are necessary? Does the angle of the incoming laser light matter? Does the size of the particles matter? Does the uniformity of how the particles are dispersed matter? Would water vapor work? I have tried using a fog machine, but the red laser that I am using only reflects off of the fog particles in a way that makes it viewable _only from a certain perspective_. This would not be a good show to the people standing in one side of the room vs. another.


title: Are there examples in classical mechanics where D'Alembert's principle fails?

text: D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero. This is obviously true for the constraint of a rigid body where all the particles maintain a constant distance from one another. It's also true for constraining force where the virtual displacement is normal to it. Can anyone think of a case where the virtual displacements are in harmony with the constraints of a mechanical system, yet the total work done by the internal forces is non-zero, making D'Alembert's principle false?


title: Is there an equation for the strong nuclear force?

text: The equation describing the force due to gravity is $$F = G \frac{m_1 m_2}{r^2}.$$ Similarly the force due to the electrostatic force is $$F = k \frac{q_1 q_2}{r^2}.$$   1. Is there a similar equation that describes the force due to the strong nuclear force?    2. What are the equivalent of masses/charges if there is?    3. Is it still inverse square or something more complicated?


title: What are the constituents of communicated time differences between moving objects?

text: In "The Elegant Universe," two individuals, George and Gracie, are noted to be moving relative to one another in space and wearing clocks. They sync their clocks upon passing each other. Each observes the other's clock to be ticking more slowly. Gracie communicates with George via cell. > Gracie communicates her time to George as he recedes into the distance. > Initially, Gracie thinks that George will hear her reported time before his > clock reaches the same time. But then, she takes into account travel time > and realizes the travel time will more than compensate, and George will > actually hear her time after his clock has passed that time. Then, the author notes that > Gracie realizes that even if George takes the travel time into account, he > will conclude from Gracie's communication that her clock is running slower > than his. By this last statement, does it mean that the time difference will be greater than what can be accounted for by mere communication travel time? I would think that some of the time difference is due to Gracie (from George's perspective) transmitting the time after that time has already elapsed on his clock. After all, Gracie's clock is moving more slowly (due to time dilation) relative to his clock and that accounts for the extra time difference (beyond the communication travel time). Do I have that correct?


title: Converging mirrors and the transition between inverted and non-inverted images

text: I was talking with a friend recently about concave mirrors, which frequently invert the reflected image - I think we were playing with a spoon. I raised the question, if you had a mirror whose shape could be easily controlled so you could move it from convex through being flat to being concave, at what point does the image flip from being the right way up to being inverted - is there some sort of discontinuity? If not how would this transition occur?


title: Why zero modes of the internal Dirac operator must be in representations of the isometry group of the compact space

text: Imagine a manifold $\mathbb{R}^{1,3}\times{}B$ where $B$ is a compact group- manifold with isometry group $U(1)\times{}SU(2)\times{}SU(3)$. Let's consider the Dirac equation for a massless Spinor field. $iD_{1+n}\Psi=0$ and separating the compact space part with the 4-space part $i(D_4+D_{compact})\Psi=0$ we see that the compact dirac operator might be regarded as a mass operator for the 4 dimensional spinor. Imagine we are interested in solutions whose mass is zero, that is, solutions whose eigenvalues of the internal Dirac operator is zero. I am trying to understand the impossibility of arranging these zero mode fermions on complex representations of the isometry group, but for that first I need to understand why the fermions obtained by dimensional reduction must form a representation (be it real or complex) of my isometry group. This is asserted in this paper (Witten 1981). On the third paragraph of the ninth page it is said > If the ground state is a product of four dimensional Minkowski space with > one of the $M^{pqr}$ (for our purposes this just means a compact group > manidold with the desired isometry group), then the zero modes of the Dirac > operator in the internal sace will automatucally form multiplets of > $U(1)\times{}SU(2)\times{}SU(3)$ since this is the symmetry of the internal > space. Could someone please be more explicit of why this is so?


title: physically modelling the Saltatory nerve impulse transmission?

text: The nerve impulse transmission is specifically a biophysical process. Under a _resting_ stage, the membrane is already polarised (presence of charge on either side leading to a potential difference across it, due to its finite capacitance). Changes in this polarisations mediated by several agents, cause some part of this membrane to be depolarised (inversion of the charges in that localised area). This changes the potential difference across that region and starts ion flow from the depolarised region of the membrane to the adjacent polarised region, inducing the same depolarisation there. Myelinating a neuron is similar to lowering the capacitance of the neuron. Myelinating the neuron causes subsequent action potentials to be spatially separated, and the conduction of voltage between them occurs primarily through ion flow along the neuron. Here are some nerve conduction basics. > Now my actual question. **Modelling a neuron as a simple one dimensional > _membrane/cable_ , how can a lowered capacitance lead to a faster conduction > of the voltage perturbation i.e a change in voltage (depolarisation), which > is initially limited to a small localised region? This conduction of the > voltage change can occur through the ion flow along the inner side of the > membrane and also due to long distance changes in potential due to the > altered distribution of charges along the membrane**. A very good physical > modelling of the neuron is given here and here, but because of the quite > complicated mathematical nature of the modified telegrapher's equation which > appears as the final answer, I am unable to understand how lowered > capacitance increases the speed of voltage conduction? Tell me if this question is off-topic or too biological to be answered here.


title: Determine laser cavity modal content from beating / interference?

text: I'm trying to better understand the modal content and dynamics of laser cavities. I'm specifically interested in mode-locking, although this question is pretty general and applies to CW lasers too. I understand that the modal spacing, Δf, in a cavity is set by cavity geometry and the gain bandwidth, ΔF, is set by the gain medium. Therefore, the maximum number of modes that can oscillate is ΔF/Δf. However, what determines how many modes will actually be excited, and when some are (i.e. standard lasing operation), which modes dominate? To quantify the modal content of a laser, I appreciate you can using a scanning Fabry perot interferometer connected to an oscilloscope and visualise each mode, providing the resolution and free spectral range are selected carefully. However, I wonder if there are any easier ways of interfering this, say electrically? For example, I wonder if shining the laser onto a photodetector (with bandwidth ~ 1 GHz say) will cause interference between modes, creating difference frequencies, which can then be visualised on an electrical spectrum analzyer. So, if I had 3 excited modes in my cavity, spaced by 10 MHz, then would I see 2 peaks on the electrical signal analyzer: one at 10 MHz (for the difference 30-20 MHz and 20-10 MHz) and a lower amplitude peak at 20 MHz (for the difference 30-10 MHz)? And similarly, if my laser were single-frequency, could we deduce that there is no interference on the photodetector as there is only one incident mode, so the only signal generated is in the THz regime (due to the wavelength of visile light, say), which can't be detected by the photodetector. Hence, we'd see no signal on the electrical spectrum analyzer? Any guidance / references much appreciated. Thanks


title: Wavefunction collapse to a state that seems impossible

text: Let $\Omega$ be an observable and let $\Psi = \sum a_i\phi_i$ be a decomposition of a state of a system (satisfying the Schrödinger equation) in eigenfunctions of $\Omega$ (assume for simplicity it is a finite linear combination and there is no degeneracy). Measuring the observable $\Omega$ will give the value $\lambda_i$, the eigenvalue of state $\phi_i$, with probability $|a_i|^2$, in which case the wavefunction will collapse to $\phi_i$. However, there is no reason why $\phi_i$ should be a state of the system. How is that possible? A few ways out occur to me:   1. the state instantanously "evolves" to a new, valid state   2. such a measurement is somehow not possible Concretely, consider a particle in a 1-D box, and let $\Omega$ be the momentum operator. Then the state can be written as an "linear combination" (in this case an integral combination) but no state of definite momentum is a solution. If we imagine the walls of the box, the infinite potential barriers, to be finitely wide, then if 1) is the case, it looks like the measurement of the momentum could cause the particle to tunnel through an infinite barrier (it would collapse to a state with uniform position probability, and evolve to a state that is nonzero on both sides of the barrier). What is actually going on? Or did I just totally misunderstand something? **EDIT:** Strictly speaking such a $\phi_i$ always is a state of the system (only the time-evolution is governed by the Schrödinger equation). To be more precise would be to remark that it seems that measurement could make it go into a state of (extremely) much higher potential without any other effort than making that measurement.


title: Spacetime and the conservation laws

text: I'm reading Peter Atkins' book, _Galileo's Finger_ , and in the chapter on energy, he makes the points that the conservation of momentum stems from the shape of space (that it's smooth and not lumpy) and that the conservation of energy stems from the shape of time (that it's smooth and not lumpy). I'm not totally clear on how the shape of spacetime leads to the conservation laws. Could someone elucidate the relationship, in layman's terms?


title: Is a real life electric shield possible?

text: I got this question from playing games like Halo and Borderlands (I know kinda dumb but raised a good question) in which the primary protection is an electric shield. Now I'm wondering if it would be possible to use energy to create a barrier in which physical objects either get disintegrated or deflected. Id imagine magnetism would come into play somewhere. This kind of technology would be arguably the most useful invention because it would enable us to protect ourselves, not in war, but in situations that normal protection couldn't. Such as on spacecraft and rockets, common collisions and asteroids wouldn't be an issue. I would imagine the amount of protection, size, and duration would depend on the amount of energy available but if you had a system in which it's self propelling and generating more than it's using(such as a car batter and multiple alternator) then that wouldn't be an issue. Also I'd imagine the device would be small because the shield would appear and disappear with the addition and removal of the energy source. So basically, is it theoretically possible to make a shield or barrier out of energy and if so, are we close or very far from this kind of breakthrough? thanks in advance


title: Anomalies from a Renormaization Group Equation (RGE)

text: This is an approach to anomalies which seems unfamiliar to me..   * Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu \nu} \rangle $. Is there any standard such $W$?    * Now one says that this $W$ must satisfy the following R.G equation,  $(\mu \partial_\mu + 2\int d^dx g^{\mu\nu}\frac{ \delta }{\delta g^{\mu \nu} } )W = 0$ Where does this come from?   * So substituting the first equation into the second and defining $A_{anomaly} = \langle T^\mu_\mu \rangle$ we get, $\mu \partial_\mu W = -2\int d^dx A_{anomaly}$. And one one wants to take two derivatives of the L.H.S w.r.t the metric to get the LHS to look like, $\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle $.  But I don't know what is this theorem which is now being used to say that this derivative of the 2-point function of the stress-tensor will necessarily have the following form, $\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle = \frac { C_T} { 4(d-2)^2 (d-1)}\Delta^T_{abcd} \mu \partial_\mu (1/x^{2d-4} ) $ where $C_T$ is some number and $\Delta^T_{abcd} = \frac{1}{2}[S_{ac}S_{bd} + S_{ad}S_{bc}] - \frac{S_{ab}S_{cd} }{d-1} $ , where $S_{ab} = \partial_a \partial_b - \delta_{ab} \partial^2 $ Can someone kindly help as to what is the motivation/proof of the above structure? Is something similar known for derivatives of higher point correlations of the stress-tensor?


title: Help with a solid sphere sliding and then rolling

text: Initially at time $t=0$, a solid sphere slides with velocity $v$ along a horizontal surface. The coefficient of friction is u. Find the required time for the sphere to stop sliding (the sphere will then be rolling). Ive tried using Newtons equations of motion but I'm not sure what the final velocity will be? At first I thought it may be zero because it will be rolling but obviously it will still have some translational velocity. I've also found the work done by the friction on the sphere and taking this figure away from the initial kinetic energy leaving (what I thought) would be the rotational kinetic energy that the ball has. However I can't get to the answer. I'm almost certain I'm just missing a small trick but I have no idea where to go with it.


title: Force applied perpendicular to direction of motion, static or kinetic friction?

text: Let's assume we have a 1 kg cube on a flat uniform surface. The coefficients of friction between them are $\mu_k = 0.25$ and $\mu_s = 0.50$. This cube is moving at 1 m/s in the y direction, with whatever force is needed to overcome friction being applied. There is no force or motion in the x direction. Assume $g = 10\, {\rm m / \rm s^2}$ in the z direction. Now, if a force is applied in the x direction which type of friction applies, static or kinetic? Edit: To clarify, if the cube is moving forward, and gravity is acting down, the new force is being applied to the right. My rudimentary model of friction is that it acts like two pieces of sandpaper where the bumps can interlock when there is not motion, but when they are moving over top of each other they don't have time to settle in. In that model, I would think kinetic friction would apply, as the direction of movement shouldn't matter. However, I realize that in reality friction is a result of molecular attractive forces, and probably a lot of other more complicated things I'm unaware of. Because of this, I suspect the answer may lie somewhere between static and kinetic friction.


title: Block and Pulley Problem with Friction

text: ![HW Problem](http://i.imgur.com/EAVEwED.png) **1.) What is the acceleration of the hanging mass? (Indicate the direction with the sign of your answer.)** **2.) Determine the tension in the cord. (Indicate the direction with the sign of your answer.)** Seems like the simplest problem but I cant seem to get the right answer. Here's my approach: * * * $n $ = normal force on $m_1 $ $f_s $ = friction force on $m_1 $ $T $ = tension in the cord $\mu_k $ = coefficient of kinetic friction * * * **Problem 1.)** $(1)\quad n = m_1 g = (40kg)(9.8m/s^2)$ $(2)\quad n = 392 N$ $(3)\quad f_s = \mu_k n = (0.22)(392N)$ $(4)\quad f_s = 86.24N$ $(5)\quad T = m_2 g - f_s$ $(6)\quad T = (13.4kg)(9.8m/s^2) - 86.24N$ $(7)\quad T = 131.32N - 86.24N = 45.08N$ $(8)\quad m_2 a_{m_2} = m_2 g - T$ $(9)\quad (13.4kg) a_{m_2} = 131.32N - 45.08N$ $(10)\quad (13.4kg) a_{m_2} = 86.24N$ $(11)\quad a_{m_2} = 86.24N/13.4kg = 6.44m/s^2$ $6.44m/s^2$ is wrong, though. Could someone point me in the right direction? Any feedback is welcome, and thanks in advance **SOLUTION:** Equation (5) should be $T-f_s = m_1 a$ $T-f_s = m_1 a$ becomes (after rearranging) $T-(40kg)a = 86.24N$ and $m_2 a = m_2 g - T$ becomes (after rearranging) $T + (13.4kg)a = 131.32N$. Then we throw those two equations into a matrix: \begin{bmatrix} 1 & -40 & 86.24 \\\ 1 & 13.4 & 131.32 \end{bmatrix} Solve, and our answers will be: $a = 0.84m/s^2$ $T = 120.01N$


title: Geodesic Deviation between Test Particles from Gravitational Wave

text: I'm having trouble understanding how Carroll (Spacetime and Geometry p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, $\vec{U'}$ that begin parallel and near each other, and $\vec{S}$ is a vector connecting one geodesic to another at equal affine parameter values, then the equation of geodesic deviation is: \begin{align*} \frac{D^2}{d\tau^2}S^\mu = R_{\ \ \nu\rho\sigma}^\mu U^\nu U^\rho S^\sigma \end{align*} We work in the weak- field limit and the transverse-traceless gauge. If we assume our particles on the geodesics are moving slowly, then $\vec{U} \approx (1,0,0,0)$, so: \begin{align*} \frac{D^2}{d\tau^2}S^\mu = R_{\ \ 00\sigma}^\mu S^\sigma \end{align*} Now the bit I don't understand is how Carroll is able to turn the double covariant derivative on the left into a simple double-derivative with respect to $t$: \begin{align*} \frac{\partial^2}{\partial t^2}S^\mu = R_{\ \ 00\sigma}^\mu S^\sigma \end{align*} Carroll's reasoning is that _"for our slowly moving particles we have $\tau = x^0 = t$ to lowest order"_ , but I don't know what he means. I just don't understand why the Christoffel symbols vanish in the covariant derivatives. I have read several books about this. Some say the Christoffel symbols vanish because we work in a local inertial frame... but then why doesn't the Riemann tensor on the RHS also vanish?


title: quick question about degeneracy

text: For two non-interacting particles, with eigenfunctions $\phi_{n1}(x1)$ and $\phi_{n2}(x2)$ in a one-dimensional potential well $V_{(x)}$ with n = 1,2,.... Consider two spinless non-identical particles: What is the degeneracy of the ground state and first excited state? I'm thinking it should be 0 for both. Now Consider two spinless identical particles, I'm thinking degeneracy for ground state is 0, and first excited state is 2. (since ground state both n = 0, and first excited state it can either be 1,0 or 0,1)


title: What limits of integration do Landau and Lifshitz use here?

text: In "Mechanics" book by Landau and Lifshitz, $\S18$ "Scattering", problem 7 on deducing the form of scattering field given the effective cross-section as a function of scattering angle an equation is obtained: $$-\pi\text{d}\log w=\text{d}\left(\frac s w\right)\int_0^{s^2/w^2}\frac{\chi'(x)\text{d}x}{\sqrt{\frac{s^2}{w^2}-x}}.$$ Before this, several variables have been introduced: $s=1/r, x=1/\rho^2, w=\sqrt{1-\frac U E}$, where $U>0, U(\infty)=0$. They then say: > This equation can be integrated immediately if the order of integration on > the right-hand side is inverted. Since for $s=0$ (i.e. $r\to\infty$) we must > have $w=1$ (i.e. $U=0$), we have, on returning to the original variables $r$ > and $\rho$, the following two equivalent forms of the final result: > $$w=\exp\left\\{\frac1\pi\int_{rw}^\infty\cosh^{-1}\left(\frac{\rho}{rw}\right)\frac{\text{d}\chi}{\text{d}\rho}\text{d}\rho\right\\}=\exp\left\\{-\frac1\pi\int_{rw}^\infty\frac{\chi(\rho)\text{d}\rho}{\sqrt{\rho^2-r^2w^2}}\right\\}.$$ As I understand, they integrate taking $\frac s w$ as variable of integration. This seems OK, and I tried to take integral with the seemingly sensible limits from $0$ to $\frac s w$ (i.e. variable upper limit), but the result for RHS appears quite strange: denoting $\frac s w=\beta$, here's what I get after switching order of integration: $$\int_0^\beta \text{d}x\chi'(x)\int_0^\beta\frac{\text{d}\beta}{\sqrt{\beta^2-x}}=\int_0^\beta\text{d}x\chi'(x)\left(\log\left(\beta+\sqrt{\beta^2-x}\right)-\log\sqrt{-x}\right).$$ Given that $x=1/\rho^2$, the RHS seems to become complex, which makes me think I do something wrongly. I suspect that I'm taking wrong limits of integration, but L&L don't specify which ones are to be taken! Supposedly the readers _could_ determine this themselves, but I'm not as smart. So, what are the right limits of integration here?


title: What made up light photons?

text: mass is energy per c square $m=E/c^2$ energy is made up of photons but what made up photon itself? what made up a single photon? * * * **Replay to comment:** but as we can see in history early phyisicists though that atom in non-fissionable (thats why they called it atom), now we call photon elementary but it may be a composite particle!, why not?


title: Importance of the study of the diatomic molecule

text: I wanted to know the importance and applications of the study of the diatomic molecule in physics I know that the question may not make much sense to you, but while exposing the solution of relative motion of the diatomic molecule, some students asked me about their applications and could not answer. On the internet, I have not found anything useful, so I turn to you. Thanks for your attention.


title: Phase functions non-existent in original (J. Strutt) Lord Rayleigh's work?

text: This is related to my previous Phys.SE question on the derivation of the phase function - upon sifting and scanning through 600 pages of John Strutt's collected work, there is **absolutely no mention** of Rayleigh's phase function (related to Rayleigh scattering) quoted by a million other sources with no derivation or justification. Rayleigh's scientific papers (1899.) @ archive.org I know it is some sort of godforsaken approximation for which nobody on the Web has a justification or derivation for and it's really bothering me. I have no problem with empirical data reconstruction, but I want to see this "empirical data" which everyone just throws around. Why do people say something is "defined". Nothing in physics is defined, it's not a recipe for an exotic top-of-the-head meal, throw three cosines, five g and you're there. It has to be constrained by nature in some way. Please, assist!


title: How do I intuit viscosity in a rotating fluid?

text: Suppose I have two plates with a viscous fluid in between. I slide them in the same direction (a direction in their own plane), one at $5 \,\text{m/s}$ and the other at $6 \,\text{m/s}$. Due to the relative velocity, there will be a viscous damping force tending to equalize the velocities of the plates. Now suppose I have two concentric cylindrical shells, the second with a radius $6/5$ that of the first. I put viscous fluid between them and rotate the entire system such that the cylinders have tangential velocities $5 \,\text{m/s}$ and $6 \, \text{m/s}$. Because the system is executing rigid rotation, there will be no viscous damping. Suppose I take a small cube of the viscous fluid in either scenario. In both cases, there's a similar velocity gradient across the cube, but only in one case does the cube dissipate energy. Why? I can work this out by writing the Navier-Stokes equation and transforming to cylindrical coordinates, but I am still having a hard time visualizing the kinematics such that this becomes clear to me.


title: Earth's rotation isn't that orbit?

text: If the earth is rotating at some $465~\text{m}/\text{s}$ at the equator and that's really fast.   1. Shouldn't we in that case be in orbit with the earth just not fast enough?    2. How fast do we need to move?   3. If the earth stopped rotating (gradually), shouldn't gravity increase significantly?   4. Do satellites need extra speed when they are released or does it just depend on their inertia from the earth's rotation?   5. If the satellite decelerated against the earth direction of rotation will it fall?    6. If the earth rotated extremely fast, shouldn't the earth's crust along with everything else get ejected away by the centrifugal force? I'm sorry if the questions cover many matters but I preferred to ask them all at once instead of asking each separately.


title: Problem solving moments in equilibrium

text: > _Two straight ladders, AC and BC, have equal weights $W$. They stand in > equilibrium with their feet, A and B, on rough horizontal ground and their > tops freely hinged at C. Angle CAB = 60$ ^o$, angle CBA = 30$^o$ and AB = > $l$ (see diagram). Find the vertical reactions at A and B._ ![illustration of mechanical problem](http://i.stack.imgur.com/dTUt5.png) Resolving moments about A: $$\frac12l\sin30^o\cdot\sin30^o\cdot W + (l-\frac12l\sin60^o\cdot\sin60^o)W = lN$$ $$\implies N = \frac{3W}4$$ Resolving moments about B: $$\frac12l\sin60^o\cdot\sin60^o\cdot W + (l-\frac12l\sin30^o\cdot\sin30^o)W = lR$$ $$\implies R = \frac{5W}4$$ > Find the magnitude and direction of the force exerted on BC by AC. **I couldn't work out how to solve this?** The required answer is $\frac12W$, 30$^o$ above horizontal.


title: Maximizing Multiplicity of Einstein Solid == (Temperature = $\infty$)?

text: If I have a system consisting of 2 Einstein solids (`A` and `B`) is it equivalent to say that maximizing the multiplicity of the system is the same as setting the temperature to $\infty$? I have two reasons to support this Idea. 1) The total multiplicity of the system can be expressed as: $$\Omega_{tot} = \Omega_A(q_A,N_A) \Omega_B(q_B,N_B)$$ Which is the product of the individual multiplicities, and $q = q_A + q_B = $ Total quanta of energy shared between the system, and $N = N_A + N_B =$ total number of quantum harmonic oscillators. I wish to maximize the multiplicity with respect to oscillator `A`'s total internal energy ($U_A)$ $$\left( \frac{ \partial \Omega_{tot} }{ \partial U_A } \right) = 0 \implies \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = \frac{ \Omega_{tot}'}{ \Omega_{tot}} = 0$$ This is true because Ln(x) is a monotonic function. But it should be noted that the definition of temperature is such that: $$\frac{1}{kT} = \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = 0 \implies T= \infty$$ `k` here is the boltzman constant. 2) This makes some intuitive sense. On the Gaussian distribution as a function of Internal energy, the maximum multiplicity lies on the peak. The change in temperature gives a quantification of the direction of heat flow. If the maximum multiplicity is attained, then heat cannot flow any higher, it can only flow downward. This corresponds to infinite temperature. I hope my intuition here is correct.


title: Time-reversal symmery and topological insulators

text: I have a very naive question about the notion of time-reversal symmetry applied to topological insulators that are studied in experiments. If I understand correctly, the exsistance of time-reversal symmetry is crucial for the occurence of topologically protected surface states in topological insulators. At least this is what the theory of topological insulators tells us. But I have a problem with applying this theory to real materials. In real world there is always dissipation and therefore there is an "arrow of time" and time-reversal symmetry does not hold (right?). So how can we talk about time-reversal symmetry in real materials that we call topological insulators? Please help me find where I went wrong in my argument. I realize that there must be something wrong but can't figure out what exactly.


title: Different values of same physical quantity in time domain and space domain OR Physical explanation of Cauchy Schwarz result

text: I read this example in a popular math book I was browsing, and the author does not give any physical explanation. The example: Consider a rock falling freely under gravity. There are no viscous or dissipative effects. We will find the average speed at the end of 5 seconds. **Time-domain.** $v = 32 t \mbox{feet/sec}$ Average over $5$ seconds $$\langle v\rangle_\text{time} = \frac{1}{5} \int_0^5 32 t \mathrm{d}t = 80\text{ feet/sec}$$ **Space Domain.** At $t =5 $ sec, $y = 400$ feet. $$\langle v\rangle_\text{space} = \frac{1}{400}\int_0^{400} 8\sqrt{y}\; \mathrm{d}y \approx 107\text{ feet/sec}$$ Sorry about the use of imperial units. But I just dont understand the physics of this. The source uses this only as a demonstration of Cauchy Schwarz inequality. Is there something elementary I am missing out here? Essentially, he derives $$\frac{1}{T}\int_0^T v(t) \mathrm{d}t \leq \frac{1}{L}\int_0^L v(y)\mathrm{d}y$$ The math is ok to follow, but my physical intuition just cant get it somehow.


title: Where's the energy in a boosted capacitor?

text: Suppose I look at a parallel plate capacitor in its rest frame and calculate the electrostatic energy, $E$. Next, I look at the same capacitor in a primed frame boosted in the direction perpendicular to the plane of the plates. In this frame, the $E$-field is the same strength, there is no magnetic field, and the volume over which the $E$-field extends is less by a factor $1/\gamma$. This suggests $E' = \frac{1}{\gamma} E$, but relativity states that energy transforms as $E' = \gamma E$. Where is the missing energy?


title: What happens to this potential energy?

text: Let's say I turn on a Van de Graaff which creates a large positive charge. Now let's say I have an object with a positive charge in my hand and I start walking toward the Van de Graaff from $x$ meters away. If I walk an arbitrary distance towards the Van de Graaff, I am doing work and the charge in my hand is gaining potential energy. Now somebody standing next to the Van de Graaff suddenly turns it off. What happens to that potential energy of the charge? By the conservation of energy it should be conserved somehow, but it appears not to be. Any help is appreciated.


title: When does $T^{ij} = T_{ij}$?

text: Suppose we have some tensor with components $T^{ij}$. Then suppose that we also have $T_{ij}$. When would $T^{ij}T_{ij} = (T^{ij})^2 = (T_{ij})^2$?


title: Non-uniform lens flares

text: I assume -- and maybe wrong -- that lens flares, in photography, are caused by light getting "trapped" inside the curved lens and bouncing around (due to refraction and non-perfect total internal reflection) causing bright spots at periodic intervals, when they partially escape on the imaging side. This would give the classic circular(ish) spots when a camera pans up towards, say, the sun, where the shape of the spots mimics that of the lens/lenses and aperture. However, I was looking at the Hubble Deep Field image and noticed that a few of the brighter stars have _only_ horizontal and vertical flares. Similarly, stars (as in the light source) are often drawn as _n_ -pointed polygons, implying distinct flares at regular angles. What is it about the lens, in these cases, that causes flares to propagate seemingly non-uniformly or directionally?


title: Calculating the illumination of the moon

text: I am trying to find a formula that will enable me to calculate the illumination of the moon down to one thousandth of a percent, given that the Gregorian year, month, day, and hour is known. Can anyone help me with this formula?


title: How did the radiative flux of each gas giant planet change with respect to time (since their formation)?

text: We know that each gas giant planet was warmest when it was young. This warmth came from internal heating from both radioactive decay and from gravitational potential energy. This warmth, in turn, means that the planet emits blackbody flux. So this raises some interesting questions. Among them - how much of this early daytime flux affected the day-to-day temperatures of their moons?


title: Calculating vector components from other vectors

text: I'm busy studying for my Physics exam this evening and I've come across a vector problem that I cant quite solve. Any help would be appreciated! We are given the following vectors:               A = 5i - 6.5j     B = -3.5i + 7j      We are also told that a vector C lies in the xy-plane and is perpendicular to A. The scalar product of B and C is equal to 15. With this information we have to calculate the components of the vector C. I have tried calculating the magnitude of B and then trying to solve the equation:               |B||C|cos{theta} = 15      But I didn't get very far. I have a feeling I'm just not seeing something and making a careless error. Thanks in advance!


title: How dark does it have to be to see the gegenschein?

text: I know I've seen zodiacal light, especially at my old club's best dark-site. However, I'm pretty sure I've yet to see the gegenschein. How good does the seeing have to be to be able to see that?


title: What is the typical career path to become a professional Astronomer?

text: Here is a typical question which I have been asked many times while giving public lectures in various places. While I know one of the paths like Diploma, Masters and PhD but sometimes this is not so obvious for that eager young person who thinks to go on that path. So what that would be?


title: A method to estimate the relative magnitude of a star using nearby stars

text: I remember a method to make a reasonable estimate of the magnitude of a given star X by using two other stars of known magnitude as references. The method used evaluation phrases like "the star X and the reference star appear to have the same brightness, and this sensation remains even after prolonged observation" or "X and the other star appear to be the same, but after some time, X appears slightly brighter" or "X and A are completely different in brightness". Each of these phrases had a value (IIRC from 1 to 5) and then with a formula you could estimate the magnitude of X from the magnitude of the reference stars and the values obtained from the comparison phrases. Do you know the name of the method, and the exact protocol ?


title: How does an accelerating charge radiate electromagnetic waves?

text: When a charged particle gets accelerated it emits electromagnetic waves. In reality, when a charged particle gets accelerated the electric field around the charge remains unchanged, but the magnetic field changes. Here the energy provided to the particle - by the energy source accelerating it - gets divided into the energy contained in the increased magnetic field and EMWs radiated. But I don't understand the mechanism by which these EMWs are emitted. I have read that when a charged particle gets accelerated the electromagnetic field associated with it gets detached from it and this detached electromagnetic field is that we call electromagnetic waves. So is it that, when the charged particle gets accelerated, some part of the magnetic field associated with it gets detached from it and simultaneously (but apart from the effect of the event described above) the magnetic field associated with the charged particle gets changed according to the amount of energy being supplied or being taken away by the external source accelerating the charge particle? Thus, in the entire process the electric field around the charge particle remains constant. Is it so? If this is true, then do we have to assume that the magnetic field being detached itself and instantaneously gets converted into EMW of the corresponding energy? But then I get again the same question: By which mechanism? Also I thought of an entirely different mechanism: When the charged particle gets accelerated some part of its electric field as well as some part of its magnetic field gets detached from the charged particle and that itself is the EMW. And simultaneously (but apart from the effect of the event described above) both the magnetic field & electric field associated with the charged particle gets changed according to the amount of energy being supplied or being taken away by the external source accelerating the charge particle? But here is one more problem: If we are assuming that the electric field is decaying into the EMWs, we will have to consider the contribution of the externally accelerating source to increase the electric field of the charge particle regardless the charge is being accelerated or decelerated. As far as I think, this cannot be done directly. We should assume the contribution of the external source to increase the magnetic field associated with the charge and its instantaneous conversion into electric field. Now again, same question: By which mechanism? So overall I have complete unclarity about the mechanism of the production of EMWs when a charge particle gets accelerated.


title: Peltier heating/cooling of small surface area

text: I am looking at peltier heaters/coolers, and all of them are thin wafers like the one pictured below. ![Peltier device](http://i.stack.imgur.com/zZBXg.jpg) This works fine for some things (ie. a microchip or a beer coaster) but not for others (ie. anything elongated). Assume, for the sake of argument, that I wish to heat a screwdriver to melt the ice off of frozen screws while I unscrew them. I want to put a peltier device inside of the handle of the screwdriver (as this is the most practical place to conceal it). The problem is that the only point of contact will be with the roughly 0.5cm diameter of the 'bit'. So my question now becomes whether I should somehow use a thermal conductor to connect the peltier device to the screw bit (if so, how inefficient is this), or whether to use a multi level peltier device (in an attempt to elongate the design). Alternatively, is there another solution I have not thought of? I have included a crude drawing to further clarify my question. Also, ignore for the moment that there is no heat sink in this design. **Option 1: Using a thermal conductor** ![enter image description here](http://i.stack.imgur.com/tDCMy.jpg) **Option 2: Using a multi-level peltier device** ![enter image description here](http://i.stack.imgur.com/979q8.jpg)


title: Kinematics with non constant acceleration

text: A particle experiences an acceleration described by $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant. To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0? If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$ = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$ = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$ How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration) Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0?


title: Why does boiling water in the microwave make a cup of tea go weird?

text: When I boil water in the kettle, it makes a nice cup of tea. Sometimes I need to use a microwave because a kettle isn't available. I boil the water in the mug and it looks pretty normal, but when I drop in the teabag the water froths up and looks foamy. I don't see what the chemical difference is here, so I assume it must be some physical difference. I have noticed this with multiple types of tea and multiple microwaves, the results being consistent so it's not just a weird microwave or something like that. What is the reaction here and how/why does it occur? Here is a photo of the 'fizzy' looking tea just after dunking in the teabag. ![enter image description here](http://i.stack.imgur.com/DxXfZ.jpg)


title: How is time not a constant?

text: I am a visual person, so it's hard to imagine the information I keep getting, but shouldn't time be a constant? If you were traveling at the speed of light and your able to cover $299{,}792{,}458$ meters in one second, aren't you just going faster than anything else? Why does time slow down as you approach high speeds? Think about it on a $x-y$ `2D` grid. I start off from point $(0,0)$. Then I travel at $299{,}792{,}458\ \mathrm{m/s}$ horizontally, so my $x$ after traveling one second should be at $299{,}792{,}458$. If each $x$ point is one meter apart, wouldn't this mean you are just traveling so fast that your eyes are all blurred up until the point where you slow down to a speed where your eyes could distinctively see whats going on around you, yet nothing is slowed down, it's just you were able to physically reach that speed within a second. So time doesn't stop its always constant in the sense that it never stops or increases in speed, it's just ticks a constant pace. If I throw a ball at a speed and it will travel at that same speed, but if I threw the ball at the speed of light, I can pretty much make the ball reach the location faster than the time it should of reached mathematical wise, which that doesn't even make any sense.


title: Does the earth gets closer to the sun?

text: We know that the sun loses an amount of it's mass equivalent to the amount of energy it produces, according to the E=mc² equation. so the sun is losing mass every second. Does this affect the space-time curvature it creates. Or does this affect the distance between the sun and the Earth. Does losing mass affects the gravity of sun or other planets?


title: Determining a person's sight from his speech

text: Suppose we meet a person wearing glasses. _Can we determine whether the person is short-sighted or long-sighted_? However,due to courtesy,we are not allowed to ask him to try the glasses and in general we can't even make any mention of them. **Attempt at the solution:** I tried using the image of the eyes but since practically for normal glasses we are at infinity.So the images appear almost the same.