title: How do animal perceive distances with their eyes and ears

text: I am studying how animals (including the human beings) can perceive distances thanks to their eyes and their ears. I am focusing on the fact that they always go in pairs: two eyes, two ears, etc. About the sight, I think our eyes use the parallax. But is this enough? Is this the only way we can perceive distances with them? About the sounds, I think it is merely by interferometry, but I am not convinced, because the distances between our ears and the wavelength are not always comparable to one another. What do you think about it? Thanks in advance, Isaac


title: Translational and Rotational Energy in Special Relativity

text: In non-relativistic classical mechanics: $$\tag{1} \begin{eqnarray} \text{KE}&=&\frac{1}{2}\sum_i m_i v_i^2=\frac{1}{2}\sum_i m_i (\vec{v}_{cm}+\vec{v}_{i,rel})^2 \\\ &=&\frac{1}{2}\sum_i m_i (v_{cm}^2+2\vec{v}_{cm}\cdot\vec{v}_{i,rel}+v_{i,rel}^2) \end{eqnarray} $$ where $i$ represents a particle in the system of mass $m_i$ and velocity $\vec{v}_i$. Here, $\vec{v}_{cm}$ is the center of mass velocity, and $\vec{v}_{i,rel}$ is the velocity of $i$ relative to the center of mass. Since $\sum_i \vec{v}_{i,rel}=0$, $$\tag{2} \text{KE}=\frac{1}{2} \left(\sum_i m_i \right) v_{cm}^2+\frac{1}{2}\sum_im_iv_{i,rel}^2 $$ Thus, almost trivially, we get the famous statement that the total kinetic energy of a system can be decomposed into a translational kinetic energy of the center of mass and a rotational kinetic energy around the center of mass. However, notice that this all depended on the kinetic energy of particle $i$ being linearly proportional to $v_i^2$, which is not the case in special relativity. My question is this: Can a similar statement like "the total kinetic energy of a system can be decomposed into a translational kinetic energy of the center of mass and a rotational kinetic energy around the center of mass" be made in special relativity? Of course, there are no rigid bodies in special relativity, and there is no 4-vector analog to center of mass, but that is of no importance here. Additional question: It can be shown (in non-relativistic classical mechanics) that if you had masses $m_1$ and $m_2$ that if an explosion releasing energy $U$ blasted these masses away from each other, the number $U$ is invariant under a Galilean transformation. Is it invariant under a Lorentz transformation?


title: Determining diameter of effective image circle of large lens in a Camera Obscura

text: I am in the process of gathering materials and designing a large lensed Camera Obscura. I would like to estimate the approximate diameter of the image circle and the approximate field of view. in degrees, but all the formulae I have encountered require parameters I do not yet know. I am under the impression that a free lens has an intrinsic image circle diameter/FoV. **—The lens** **Diameter:** 304.8mm (12.0"in) **Focal Length:** 1143mm (45.0"in) **Type:** Plano-convex glass **—The Room:** An inflatable inhabitable sphere with a hanging translucent screen about a meter from the lens mounted on the wall. I am using 'clear' Polyethylene sheeting as my ground-glass surface. I have built a lensed Camera Obscura before with a 150mm diameter 2000mm FL glass lens. Documentation of project: https://vimeo.com/94692036 **—The Question:** How does one go about determining the Image circle/field of view of a single lens?


title: What if the kinetic energy of a particle was some other function $f(v)$?

text: This is a "what if this was how the universe worked" kind of question. I don't know if those belong in Physics StackExchange, and I apologize if they don't. Suppose we have two reference frames connected by some linear transformation $x'^\mu=\Lambda^\mu_\nu x^\nu$. In a modern language, we thought the correct $\Lambda^\mu_\nu$ was the Galilean transformation, and we concluded that kinetic energy was $\frac{1}{2} m v^2$. Later, some smart people (Einstein and others) realized that $\Lambda^\mu_\nu$ should be the Lorentz transformation. Then, if one assumes momentum conservation and that kinetic energy is a line integral of $\frac{dp}{dt}$, one can reach the conclusion that kinetic energy is $(\gamma -1)mc^2$, which like $\frac{1}{2} m v^2$ is a function of $v^2$. Now, lets work backwards. Suppose we lived in an alternate universe, and suppose we knew that the kinetic energy of a particle of mass $m$ is $$ \text{KE} = m f(v)$$ where $f(v)$ is a smooth function of $v=|\vec{v}|$. In the case of non- relativistic classical mechanics, $f(v)=\frac{1}{2}v^2$. In our alternate universe (with its own symmetries, not necessarily Galilean or Lorentz), with an arbitrary $f(v)$, can we reconstruct the transformation $\Lambda^\mu_\nu$ that connect different frames of reference in our alternate universe? For every $f(v)$, does an associated $\Lambda^\mu_\nu$ exist? If so, is it unique?


title: Where is a closed form also exact?

text: I'm not very familiar with exterior derivatives. I've some trouble following argument (which is a part of a proof that if the Riemann tensor vanishes, $R^{\,\rho}_{\;\,\sigma \mu \nu}=0$, iff there exists a coordinate system where the components of the metric are constant.) One finally get to the equation: $$\nabla_\mu \omega_\nu =0$$ Taking the anti-symmetrization of the last equation: $$\nabla_{[\mu} \omega_{\nu]} =\text{d}\omega=0 $$ Which means that $\omega$ is closed. However in general this does not mean that $\omega$ is exact, i.e. $\omega=\text{d}\alpha$, for some scalar function $\alpha$. Then **'since we have restricted the topology of the region in which we are working'** the one- form must also be exact. I don't see how the vanishing of the Riemann tensor directly leads to the bold-texted conclusion. More generally in what space is a closed form also always exact ?


title: Most particles that have ever been modeled

text: Two billiard balls collide. Consider the two balls only, how many particles are in that system? Have particle systems of that size been modeled with each individual particle before? What is the largest object ever been modeled with pure particle physics?


title: What would happen if an incredibly high energy photon passed through a human body?

text: What would happen if an incredibly high energy photon passed through a human body (or any other material)? When I say incredibly high energy photon, I mean higher than any known energy scales... higher than electronic energy scales, higher than nuclear transition energies, etc... In my mind, I pictured the photon tearing apart whatever it passed through. However, browsing Wikipedia's article on the Photoelectric Effect, I noticed that the cross section went as $1/E^p$, where I think $E$ is energy and $p$ is some number. So, the probability of an atom being ionized by a high energy photon actually drops with the photon energy (although I don't know what approximations went into this calculation... first order perturbation theory?). If the other light-matter interaction processes also have cross sections of the form $1/E^p$, then an incredibly high energy photon would pass through us without harming us! Or maybe this question is unanswerable because we don't yet understand physics at such high energy scales. What are your thoughts on this? What do you think would happen if an incredibly high energy photon passed through a human body?


title: Why is the $\langle v_{x}^{2} \rangle=\frac{1}{3} \langle v^2 \rangle$?

text: For a randomly moving particle. Or, I suppose that 1/3 could generalise to 1/n, where n is the non rotational degrees of freedom for that particle. Related reference Kinetic Theory of Gasses.


title: Discarded by Relativity

text: Both the General & Special Relativity discarded Newtonian mechanics of absoluteness. According to Einstein's view, Time, Mass, Length and Space are interdependent. So, Did Relativity discarded only _absoluteness_ in space and mentioned that all motions are **relative**... What else suffocated in Physics due to Relativity..?


title: Does Heisenberg's energy-time uncertainty principle imply that quantum computing is no more efficient than classical computing?

text: See http://arxiv.org/abs/quant-ph/0006080v1 "On Non Efficiency of Quantum Computer", by Robert Alicki. In this paper, the author argues using Heisenberg's energy-time uncertainty principle, that quantum computing is no more efficient than classical computing. This paper convinced me, but I'm just an amateur and also biased toward toward this view. I'm curious why the experts still believe that quantum computing is more efficient than classical computing, given the argument made in this paper in 2000.


title: Beginning with an arbitrary classical equation for energy, how do I get the QM Hamiltonian?

text: For linear momentum I can use the de Broglie equation, but what about energy in terms of moment of inertia or some other form?


title: Can diffusion produce energy?

text: A friend and I recently got into a silly argument where I stated > pure diffusion can't produce energy since **diffusion are a part of passive > transport**. He stated if we > If we have tinny turbine which allow to transfer only one molecule at a time > due to the different concentration the molecule will make turbine rotate and > produce energy (which I thought this assembly will be a non-permeable since > direct contact is must in diffusion ) On the other hand we encounter some research trying to produce energy by differentiating level of water separated by a semi-permeable. And they accomplish producing energy how that happen (I think they give an energy in another way by differentiating concentration and they collect this energy. It is only an engineering)


title: Why do we always consider the area of inner solenoid while calculating the mutual inductance of two coaxial solenoids?

text: While we know that flux linked to an area is the dot product of the area vector of the loop(and not the actual area encountering the field lines) and the magnetic field linked to it, why then do we consider the area of inner solenoid while deriving the expression of mutual inductance of the outer solenoid. Most of the textbooks say that since magnetic field lines produced by inner solenoid are totally confined within that solenoid hence the effective area of outer solenoid linked to this flux is numerically equal to the area of inner solenoid. This sounds really contradictory to the definition of magnetic flux.![enter image description here](http://i.stack.imgur.com/FlIc0.png)


title: Q range calculation

text: I did a neutron diffraction experiment and I need to figure out which Q range we used. How should I calculate it from the incident wavelength and the diffraction angle?


title: Does the future already exist? If so, which one?

text: In the NOVA Fabric of the Cosmos program, Brian Greene explains a theory in which there is no "now", or more specifically, now is relative. He describes an alien riding a bicycle on a far off planet who sees our future now if he bicycles towards us. If he sees our future now, which one does he see? Does it change depending on what we do in our "now"? If it doesn't, does that mean everything is preordained to happen a certain way? Or did I completely misunderstand Greene's explanation?


title: Finding deceleration and velocity using distance and time

text: A car is moving down a street with no brakes or gas. The car is slowing due to wind resistance and the effect of friction. The road is flat and straight. The only data I have are timings taken at 100m intervals along its path. I have the following measurements:               Marker (100m intervals)     Time (seconds)     Start                       N/A     100m                        6.49          200m                        7.66     300m                        9.15     400m                        10.71      Is this enough information to calculate how much time until the car reaches a specific speed of 2 m/sec? **What I'm Trying:** If I could get the deceleration rate, I could use: $$\ D=vt+(1/2)at^2 $$ I can solve for v to get a velocity, and then use the velocity and deceleration to figure out when the car will reach 2m/s. But the deceleration is not constant. The best I can do is get average deceleration over an interval. It seems like there should be a way to get the deceleration curve based only on time across a distance. I'm lost.. any help is greatly appreciated.


title: Discreteness and Determinism in Superstrings?

text: So Gerard 't Hooft has a brand new paper (thanks to Mitchell Porter for making me aware of it) so this is somewhat of a expansion to the question I posed on this site a month or so ago regarding 't Hoofts work. Now he has taken it quite a big step further: http://arxiv.org/abs/1207.3612 Does anyone here consider the ideas put forth in this paper plausible? And if not, could you explain exactly why not?


title: Work being done without distance

text: I have this issue: If you push a 40.0 kg crate at a constant speed of 1.40 m/s across a horizontal floor (µk=0.25 ), at what rate (a) is work being done on the crate by you and (b) is the energy dissipated by the frictional force? For starters, here is what I have so far: RN = mg = 40*9.8 = 392 N Ff = muRN = 0.25*392 = 98 N KE = 0.5mv^2 = 0.5*40*(1.40)^2 = 39.2 J At this point, I don't know how to answer part (a). If W = FD , how do I find the work being done without a distance?


title: Question about entropy and "smart materials" (that remember shape)

text: We all know that entropy is the measure of chaos in the system, and it's always increasing in the system. Now comes my question - how does entropy work in smart materials? Here is a youtube video about smart material that I'm talking about, it remembers its shape - you can distort it, but when you heat it, it returns back to its original shape. So effectively it reverses from more chaotic state to a more uniform state: http://www.youtube.com/watch?v=fsBHF_j2FJ4 Therefore my question - how come the system became less chaotic just by adding heat? It looks like entropy decreased, but the 2nd law of thermodynamics says that entropy can only increase.


title: How does infrared light 'erase' phosphorescence on zinc sulfide?

text: I found some sheets of zinc sulfide in my basement that phosphoresce green for up to 24 hours or so after exposure to bright light in the violet range or shorter. One of the first things I tried was drawing on it with a violet laser pointer (405nm, 5mW) and as expected it draws bright lines. What I found more surprising was that I can 'erase' the phosphorescent lines if I focus my green laser onto them. I've determined with some color filters that it's infrared at 1064nm leaking from the green laser causing the erasure. This erasure does not appear to damage the panel in any way, once erased the erased spot can be lit back up to full brightness normally. How is this infrared light erasing the panel, and how effective would other infrared wavelengths be at doing this?


title: Calculating the launch angle of a horizontal launch (mechanics)

text: I need some help with the following question: A smooth spherical object (the first object) is projected horizontally from a point of vertical height H = 26.38 metres above horizontal ground with a launch speed (initial velocity) of u = 21.05m/s A second identical object is projected from ground level with launch speed (initial velocity) v and at an angle A above the horizontal. Calculate the value of this launch angle A in radians, if the second object is to have the same horizontal range and the same time of flight as the first object. Take acceleration at gravity to be $g = 9.81m/s^2$.


title: Temperature of ideal gas after volume increases in piston

text: A three part question, all related ## Part 1. My question is about an ideal gas in a rigid container with two (equal volume) compartments separated by a rigid wall. One compartment has an ideal gas at some P, V, T. The other side is vacuum. If suddenly the wall is removed, what happens to temperature? Would this be any different if it were a piston, where we pulled the piston to double the volume? ## Part 2. I found a similar question that had already been asked. From John Rennie's answer there it appears that the temperature should not change in both cases, but I am not sure. What is most surprising to me is that everyone seems to be happily agreeing that this can be understood by $PV=nRT$ with $P$ and $N$ constant, and that somehow this equation implies that if you decrease $V$ then $T$ must increase! Isn't that just the opposite of what the formula says? $V$ and $T$ are directly proportional, so if $V$ increases, $T$ must increase, at least according to the ideal gas law. ## Part 3. Finally, cooling and air-conditioning seems to work on the principle of expanding volume (going from a thin pipe to a wider one) causing a decrease in pressure, and hence temperature. But $V$ increased, and $P$ decreased, so why did $T$ change?


title: Can a the volume flow through a positive displacement pump be reliably measured by tracking revolutions of the drive shaft?

text: The question refers to non-compressible fluids. I'm mostly thinking about monopumps (excentric screw pumps), but this also should apply to any type of gear pumps as well, possibly also to piston pumps. Theoretically, it should be possible to measure the flow by simply counting the revolutions and multiplying by displacement volume per revolution. The one source of error (I can think of) is the flux those pumps experience when operating against high heads and/or after some wear near the sealing line. Are there other factors to be taken into consideration? Are there real world examples?


title: Terminal velocity?

text: I am having a problem with a particular concept. Here is where I have gotten, since the ball never loses contact with the stair, it will rotate around through the edges, the edges being the pivot, (used parallel axis theorem and all that), and since all collisions are inelastic,i figured the ball will lose the normal component of its velocity.But I still couldn't figure out how to find the terminal velocity(average). How and why? I have posted an image for clarity.


title: How do I calculate how something should stay in position under gravity?

text: Excuse me if this too simple a question - I'm not a regular (consciously) user of physics. I have a ship in space and it is under the effect of three moons. The ship can rotate and thrust to move within the 2D world. My problem is: how do I maintain the ship's position if I know some details about the moons? Here is what I've gotten so far (see the edit for a revised example): $F$ = $F_1$ + $F_2$ + $F_3$ is the total forces of all three moons; $\small M_1,M_2,M_3$, exertet on the object, $S$. Then $F_1 = G \times m_{M_{1}} \times m_{S} / R^2$ where $R = \text{distance}(S, M_1)$. Once I find $F$ I want to find my acceleration, so I use the formula $a = F / m_S$. With $a$, I want to find my new velocity, so I use the formula $V_f = V_i + a \times t$. I then find my new position using $P_n = P_o + V_f$. Two questions:   1. Is this correct up until this point?   2. From this new position, how do I determine the amount of thrust (a scalar) to apply? Is it the length of $V_f$? And how do I determine the new angle, if I have the current rotation of the ship? * * * **EDIT** : If the above is too hard to understand, here is a revised version that may follow a bit better. Given a spaceship's angle $\theta$, position $[x_s,y_s]$, mass $m_s$, and initial velocity $v_s$ and three moons with mass $m_i$ and position [$x_i, y_i]$ exerting a gravitational force on the ship. How do I find the necessary velocity and angle of the ship to counter act the moons and maintain position? * * *


title: How can dynamics be reversible if inflation-style baby universe spawning is allowed?

text: I just finished readying Sean Carroll's book, "From Eternity to Here", and have a question about reversibility and inflation: Assume inflation allows random quantum fluctuations to produce high energy density regions that pinch off from the original spacetime and grow into disconnected baby universes. If we slice spacetime a while after this disconnection has occurred, how would the time-reversed dynamics know to spontaneously connect the two separated regions of spacetime together? Would this situation necessarily violate time reversibility (or render it undefinable)? In particular, assuming that the entire multiverse is infinitely large, and each baby universe spawning can be described by a finite number of variables, there would be an infinite number of completely indistinguishable spawning events. Slicing spacetime a short while after each one, they could be reconnected arbitrarily without violating the dynamics.


title: Is there an established standard for naming exoplanets?

text: I understand that exoplanets are named by adding a lowercase letter to the a designation of the planet's parent star or stellar system, beginning with 'b' (the star itself is 'a') in order of discovery (and orbital distance from star where discoveries of more than one planet around a star are announced together). My question is whether there is an established standard for using a space between the stellar designation and the planet's letter. Is it _Kepler-22 b_ or _Kepler-22b_? Journals seem to have different editorial policies that determine how they handle this, but the comprehensive databases available online (e.g., the Extrasolar Planets Encyclopaedia, the Visual Exoplanet Catalogue, and NASA's Planet Quest) seem to all use a space.


title: How can the derivative of this trace be constrained?

text: I am studying for my exam on relativity and I am going through some problems sets including ones where I was not very successful in so I want to know how to do this problem. > (Convergence of particles): Consider a freely falling reference particle in > vacuum as in Exercise 1. Assume now in addition that the Hessian matrix of > the gravitational potential does not vanish initially in the vicinity of the > reference particle. Recall that the displacement of a nearby particle $y(t)$ > is given in terms of its initial displacement $y(0)$ (we assume that the > particle is intitially at rest relativie to the reference particle) by > $$y(t) = A(t)y(0)$$ NOTE: Just to give some context we say that $y(t)$ is a displacement between two freely falling particles positioned $x_{0}(t)$ and $x(t)$ so $y(t) = x(t)-x_{0}(t)$. We can say that a nearby freely falling particle satisfies to first order in y the tidal equations, mainly we can see that: $$\frac{\mathrm{d}^{2}y}{\mathrm{d}t^2}=-\nabla^2\psi(x_0(t))\cdot{y(t)}.$$ Back to the problem. > We call $A$ the deformation matrix. Show that there is an initial > displacement $y(0)\neq0$ and a time $t^*>0$ such that $y(t^*)=0$. > > Hint. Recall the definition of the strain matrix: > $$\theta=\frac{\mathrm{d}A}{\mathrm{d}t}A^{-1}.$$ Now to solve this the question breaks it down to several small parts. I was able to solve the first part so I will skip it. But here is the part I am having trouble with: > (2) Deduce from the tidal equation the following ordinary differential > inequality: > $$\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}\theta\leq-\frac{1}{3}(\operatorname{tr}\theta)^2$$ > Hint: Decompose $\theta$ into its diagonal and its traceless part. NOTE: We deduced from lecture a long time ago that $\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}\theta=-|\theta|^2-\operatorname{tr}M$ where $M=\nabla^2\psi(x_0(t))$ or in other words $M$ is the hessian of the gravitational potential of the particles that are freely falling. We also know that $\operatorname{tr}M=\Delta\psi(x_0(t))$ and in a vacuum this is equal to 0. So, concerning this problem I figure if we have to break down $\theta$ to a diagonal part and a traceless part that means we can call the diagonal $\sigma$ and the traceless part $\omega$, such that $\theta=\sigma+\omega$. That means that: \begin{align} &\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}(\sigma+\omega) \\\ &\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}(\sigma) \end{align} Since the trace of the omega is 0 but honestly I am not sure what more to do. I think I actually have to move that derivative and maybe I will be able to get that inequality.


title: Is it believed that all UV completions have "Maldacena duals"?

text: I have heard occasional rumors that effective field theories have gravity duals. For example, I've been told that UV momentum cutoffs in N=4 SYM become finite radii in AdS. I've heard speculations about AdS duals of plain old QCD. And I know it's thought that CFTs always have gravity duals. Is it believed that any UV completion of an effective field theory must have a Maldacena-style gravity dual (in Witten's sense, meaning that currents become boundary conditions)? Can QFTs with gravitational sectors have gravity duals? What about string theory? Are there any necessary reasonableness conditions? (Maybe insist that the renormalization flow is a gradient flow?) Is locality necessary? (String theory seems to think that gravity must be holographic. Can you have a holographic description of a holographic description?)


title: what's the center of mass for triatomic-molecule system

text: My text use the following example to explain the center of mass. There are three balls (mass $m$) sitting in the origin, at $x=l$ and $x=2l$, each two mass are connected with a spring of constant $k$. The system can only move along $x$ direction. To find the center of mass, I setup the coordinate system with first ball placed at $x=0$, the second ball placed at $x=l$ and the third ball placed at $x=2l$. Set $x_1$, $x_2$ and $x_3$ to be the offset from the corresponding equilibrium positions. To find the center of mass, I do the following $$ x_{com} = \frac{mx_1 + m(x_2+l) + m(x_3+2l)}{m+m+m} = l + \frac{x_1+x_2+x_3}{3} $$ The text said since all the ball have the same mass and they separated equally, so the center of the mass will be at the geometrical center of the system, that is, $$ x_{com} = l $$ But from the math, we have the last term, I know the conclusion of the text is correct but what's the physical point that we have $x_1+x_2+x_3 = 0$?


title: What is the minimum optical power detectable by human eye?

text: If one is in complete darkness, what is the minimum optical power that the eye can "see" (let's say in 500-600 nm range). I found that for 510 nm, 90 photons can be detected (http://en.wikipedia.org/wiki/Absolute_threshold). Calculating the energy for these photons and considering 150 fs pulses with 80 Mhz repetition rate, results in an average power of ~ 3 nW. Does anyone has other sources of information about this ?


title: Fourier Transform on a Riemannian Manifold

text: The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but that question was asking for a FT on a Lorentzian Manifold. The answer given is actually very good but then it gets a little too technical and quite cryptic to me... The first comment on that question actually points at the right direction but could someone give me an example on a specific Riemannian Manifold (aside Euclidean), for example on a 2-cylinder with metric $g=dz^2+r^2d\theta^2$? **PS:** Maybe I should have posted the question on Mathematics.SE but I was hopping for the physicists more direct approach to these kind of problems...


title: Friction in circular motion

text: > A circular horizontal table spins around its axis of rotation with a > constant angular speed. It completes one revolution in 2 seconds. An object > whose mass is $M = 0.8 \ \text{kg}$ is placed on the table and is connected > with a string to some weight with the mass $m = 0.16 \ \text{kg}$. > Coefficient of friction between the object and the table is $0.1$. Determine > for the following radii whether the body will remain at rest with respect to > the table, or will move towards the center of the table or will move to the > edge of the table: $40\text{cm}$, $25\text{cm}$, $15\text{cm}$, > $8\text{cm}$. ![Image](http://i.stack.imgur.com/ne5HZ.png) I thought to solve it like that: The only force acting on the weight is gravity, and it is the same force which is pulling the object (string tension, if we neglect string's mass) - $mg \approx 1.6\ \text{N}$. This force together with friction are the components of the centripetal force. My intuition says that the centripetal force which is needed to keep the object at rest with respect to the table should be: $MV^2/R$. Knowing the period, we can calculate the speed: $2\pi R/T=V$. So the centripetal force needed is $MR 4\pi ^2/T^2 = MR \pi ^2 \approx 8R$. The friction here may act on the object in any direction, as I understand. It depends on the radius. But the max friction value is: $f_{s, max} = \mu N = \mu Mg = 0.1 M \approx 0.8\ \text{N}$. **Case 1:** ($R=0.4\ \text{m}$): The centripetal force needed is $\approx 3.2\ \text{N}$. However the pull force of the little weight is $1.6\ \text{N}$ and the max friction is $0.8\ \text{N}$ which is not enough to keep the object spinning with the table, and therefore, it will just move towards the edge of the table and will eventually fall ( _Pull force and friction force are directed to the center of the table_ ). **Case 2:** ($R = 0.25\ \text{m}$): The centripetal force needed is $\approx 2\ \text{N}$. The pull force ($1.6\ \text{N}$) and the friction ($0\leq f \leq f_{s,max}$) which equals to $0.4\ \text{N}$ are enough to keep the object at rest with respect to the table ( _Pull force and friction force are directed to the center of the table_ ). **Case 3:** ($R = 0.15\ \text{m}$): The centripetal force needed is $\approx 1.2\ \text{N}$. The pull force ($1.6\ \text{N}$) is much bigger than the needed centripetal force, but the friction in this case is opposing the body movement towards the center of the table. Therefore, the friction force vector is in the opposite direction ($\vec f = -0.4\ \text{N}$). So the object will rest with respect to the table here as well. **Case 4:** ($R = 0.08\ \text{m}$): The centripetal force needed is $\approx 0.64\ \text{N}$. The pull force ($1.6\ \text{N}$) is much bigger and the friction force acting on the object in the opposite direction is not enough to stop it from moving towards the center as a result of the string pull ($\vec F_{pull} - \vec f = 1.6 - 0.8 = 0.8\ \text{N}$). Therefore, the string will move towards the center. To sum up, the friction is what helps expanding the "rest radius range". If there was no friction, then the "rest radius range" will become narrow and will basically be some explicit radius ($F_{pull} / F_{needed\ centripetal} = 1.6 / 8 R = 1 \Rightarrow R \approx 0.2$). Any other radius which will be bigger than that will cause the body to fly off the table. Any radius smaller will cause the body to move towards the center of the table. I wanted to make sure that I'm right in my answers.


title: Are Newton's law of Cooling and Stefan's law related?

text: Many of Indian school textbooks claim a proof of Newton's law of cooling from Stefan's law of black-body radiation. As far as I am aware of, Newton's law is based on cooling from convection currents and Stefan's law on radiation. There is not supposed to be any relation between them. **Question:** Is there any relation between them, and can Newton's law of cooling be derived from Stefan's law? I found many answers and resources on Google, but an answer referring to a well-established paper, book or resource of same kind will be highly appreciated. **UPDATE** There's a question in the syllabus they teach us such as "Derive Newton's law of Cooling from Stefan's Law." And here's one of the links which show the solution. They are approximating Stefan's law to Newton's law(mathematically) by considering $T-T_0$ very small. They even claim that Newton's law is applicable for even small temperature differences whereas in reality Newton's law is applicable for all temperature ranges. Please help. Is there any strong reference which can help shut down this misconception. I understand why are they different but a reference might help much.


title: Dark Matter 'Stars'

text: I'm aware that the Milky Way has a dark matter 'halo' around it, presumably a spherically symmetric distribution. But I'm completely ignorant regarding the theories explaining dark matter... Is there any reason to not expect a star-sized object to also be made of dark matter? I know they'll be extremely difficult to detect, but I'm wondering if it's even physically possible to exist.


title: Lego Blender and gear ratios

text: I bought the Lego Kit LEGO Crazy Contraptions. It allows the learner to build a blender. My son, the engineer, said something to our grandson, his son, about a gear ratio. Can someone translate?


title: How many of the 230 crystallographic groups are realized in nature?

text: All of them or only a subset? This is a famous and fundamental result in solid state physics.


title: If two events separated by a spacetime interval of zero, can they both be said to be happening "now"?

text: Can the concept of c be validly expressed as "the rate at which an event propagates through space"? There was a television program last year featuring Prof Brian Cox. The presenter asked him "Is it true that when we look at this particular star, we are seeing it as it was 18 million years ago?" And Professor Cox replied "Or 'now'." This was a turning point in my comprehension of relativity and I want to make sure I understood it correctly. Was his point that with simultaneity and time not being absolute, the idea of "now" is in effect meaningless? This method of expression makes sense to me in that it explains length contraction and thought experiments such as the barn / ladder paradox: from the POV of the front of the barn, the event of the rear of the ladder crossing the threshold of the rear barn door hasn't reached it yet, while from the rear of the barn the same is true for the front of the ladder. Is this a valid analogy, or is it over-simplistic and will lead to failure of comprehension elsewhere?


title: Question about the proof that heat capacity goes to zero if temperature approaches $0K$

text: I don't completely understand the proof that is given for the claim that the heat capacity goes to zero, if the temperature approaches $0K$. They do it as follows, if $C_x$ is the heat capacity where the parameter $x$ is taken constant, and we heat the system from $0K$ to $T_0K$, then : $$S(T_0) = \int_0^{T_0}{dS} = \int_0^{T_0}{\frac{dQ}{T}} = \int_0^{T_0}{\frac{C_xdT}{T}} $$ (where the differential in the second step is not a real differential) The only way for this integral to converge, is when $C_x$ is zero in the limit of $T\to 0K$. But I have 2 problems with this. First they claim also that it is impossible to approach zero kelvin, so in fact there doesn't exist a path from $0K$ to $T_0 K$, so it should be impossible to heat the system from $0K$ to $T_0K$. This is worrying me, because in the proof of the claim that it is impossible to reach $0K$, they use a very similar reasoning showing that the integral can not exist. Secondly the path must be reversible, so even if there exists a path from $0K$ to $T_0K$, how do we know there also exists a reversible one?


title: What's the corresponding symmetry of enstrophy conservation?

text: In fluid mechanics, especially 2D turbulence study, people talk about conservation of enstrophy. But I can't really understand enstrophy very well, and what's the corresponding symmetry of enstrophy conservation?


title: Induced current using a reference system bound with a moving charge

text: Suppose we have a charge moving at velocity $\mathbf{v}$ in the same plane of a square wire. If I sit in a reference frame where the square wire is still, since the charge is moving with velocity $\textbf{v}$ in this coordinate system, I will see an induced current in the wire. $$\textbf{B} = \frac{\textbf{v}}{c^2} \times \textbf{E} $$ $$ \frac{d\phi_B}{dt} \neq 0 $$ Now, what If I choose a reference system where the charged particle is at its origin? According to this frame, since the the electric charge(and its electric field) is static, $\text{rot}\,\textbf{E}$ will be zero. $$\nabla \times\mathbf{E} = 0$$ But this means that there is no induced current.   Are my assumptions right? If not, how should I estimate the induced current in a reference system bound with a moving charge at its origin?


title: Physical representation of volume to surface area

text: I was looking at this XKCD what-if question (the gas mileage part), and started to wonder about the concept of unit cancellation. If we have a shape and try to figure out the ratio between the volume and the surface area, the result is a length. For example, a sphere of radius 10cm has the volume of $\approx 4118 cm^3$ and an area of $\approx 1256 cm^2$. Therefore, the volume : surface area is $\approx 3.3 cm$. My question is: what is the physical representation of length in this ratio?


title: Why does smashing a TV remote load its batteries?

text: Many times I had the same problem: my TV remote's batteries were too weak for it to operate, but then, when I shook the remote a bit and smashed it on my palm, somehow the remote regained its ability. I'm pretty sure that many people faced this phenomenon. Why does this happen? PS: Contacts are OK.


title: Understanding the mathematical formulation of the Faraday effect

text: The Faraday Effect is a magneto-optical effect which results in the polarisation of light being rotated I would really want to understand the theory of this, but I only have a basic EM knowledge, not really covering EM field in materials. Can anyone shed some light on the origin of the matrix term in the wikipedia article? I've tried to read the reference article but it's a bit too old and I'm afraid he's using old-fashioned conventions...


title: Are there more distinctive names of "null curves" with certain additional properties (absence of "chord curves")?

text: In this answer (to the question " _In general relativity, are light-like curves light-like geodesics?_ ", PSE/q/76170) a particular example of a curve is discussed whose " _tangent is everywhere null_ " and which is therefore called a " _null curve_ ". I'll restate the example curve explicitly as $$\nu : \mathbb R \rightarrow \cal M,$$ together with a coordinate function $$\mathbf r : \cal M \rightarrow \mathbb R^{1,2}; \qquad \mathbf r := \\{~t,~x,~y~\\}$$ such that $$\mathbf r \circ \nu[~\lambda~] := \\{~t_{\nu}[~\lambda~],~x_{\nu}[~\lambda~],~y_{\nu}[~\lambda~]~\\} = \\{~\lambda,~\text{Cos}[~\lambda~],~\text{Sin}[~\lambda~]~\\}.$$ The calculation of the corresponding " _tangent_ " value proceeds (roughly) via $$\begin{align} \left( \frac{d}{d\lambda}[~t_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\nu}[~\lambda~]~] \right)^2 & = & \\\ 1 - \left( -\text{Sin}[~\lambda~] \right)^2 - \left( \text{Cos}[~\lambda~] \right)^2 & = 0\end{align}$$ for all values $\lambda$ in the domain of " _null curve_ " $\nu$. Now, interestingly, for any two distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, there exist curves (which in the following are suggestively called "chord curves") $$\kappa_{ab} : [~a, ~b~] \subset \mathbb R \rightarrow \cal M,$$ such that $$\kappa_{ab}[~a~] = \nu[~a~], \qquad \kappa_{ab}[~b~] = \nu[~b~]$$ and: the " _tangent_ of $\kappa_{ab}$" is everywhere **positive**. As one concrete case consider $$\mathbf r \circ \kappa[~\lambda~] := \\{~t_{\kappa}[~\lambda~],~x_{\kappa}[~\lambda~],~y_{\kappa}[~\lambda~]~\\} = $$ $$\\{~\small{\lambda,~\text{Cos}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Cos}[~b~] - \text{Cos}[~a~] \right),~\text{Sin}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Sin}[~b~] - \text{Sin}[~a~] \right)~}\\},$$ with the corresponding " _tangent_ " value $$\left( \frac{d}{d\lambda}[~t_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\kappa}[~\lambda~]~] \right)^2 = $$ $$\small{1 - \left( \frac{\text{Cos}[~b~] - \text{Cos}[~a~]}{b - a} \right)^2 - \left( \frac{\text{Sin}[~b~] - \text{Sin}[~a~]}{b - a} \right)^2 = 1 - 4 \left( \frac{\text{Sin}[~(b - a)/2~]}{b - a} \right)^2 \gt 0.}$$ (As a sidenote: similarly, for given distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, one may ask for "chord curves" whose " _tangent_ " value should be everywhere **negative** ; but I have not been able to construct a corresponding concrete example. In the following, it is not necessary to distinguish "positive chord curves" from "negative" ones, if cases of the latter exist at all. Relevant is only that the " _tangent_ " value of a "chord curve" exists everywhere and is **not "null" anywhere**.) **My question is:**   Is there some specific name (i.e. more distinctive than "just another _null curve_ ") referring to " _null curves_ " which do **not have any** "chord curves" at all? (Are such special cases of " _null curves_ " perhaps called " _null geodesics_ "?)


title: Solving the source free Maxwell equations for plane waves

text: I've been trying to solve the maxwell equations: $$\nabla\cdot\vec{D}=0,\quad \nabla\cdot\vec{B}=0$$ $$\nabla\times\vec{E}=-\frac{\partial \vec{B}}{\partial t},\quad \nabla\times\vec{H}=\frac{\partial \vec{D}}{\partial t}$$ Where $\vec{D}=\epsilon \vec{E}$, $\vec{B}=\mu_0 \vec{H}$ Here $\epsilon$ is the three by three matrix: $$\epsilon=\epsilon_0A$$ Where $A$ is symmetric and $\mu_0$ and $\epsilon_0$ are the usual constants. I've been looking for plane wave solutions, real part implied, of the form $$\vec{D}=\vec{D_0}e^{i(\vec{k}\cdot\vec{x}-\omega t)},\quad \vec{B}=\vec{B_0}e^{i(\vec{k}\cdot\vec{x}-\omega t)}$$ By solve here I mean find conditions on $\vec{D_0},\vec{B_0},\vec{k}$ only in terms of $\omega$. By substitution I have found these relations: $$\vec{k}\cdot\vec{D_0}=0,\quad\vec{k}\cdot\vec{B_0}=0$$ $$\vec{k}\times\frac1\epsilon_0A^{-1}\vec{D_0}=\omega\vec{B_0},\quad-\vec{k}\times\vec{B_0}=\mu_0\omega\vec{D_0}$$ I want to figure this out on my own, but I've been playing with these things for a long time now and I believe to get this in the form $\vec{D_0}=\vec{D_0}(\omega),\vec{B_0}=\vec{B_0}(\omega),\vec{k}=\vec{k}(\omega)$, requires some knowledge outside of my skill set, I would much appreciate it if anyone could point me in the right direction here.


title: Spinor irreducible reps of the Lorentz group and their algebra

text: Antisymmetric tensor of rank two can be connected with spinor formalism by the formula $$ M_{\mu \nu} = \frac{1}{2}(\sigma_{\mu \nu})^{\alpha \beta}h_{(\alpha \beta )} - \frac{1}{2}(\sigma_{\mu \nu})^{\dot {\alpha} \dot {\beta} }h_{(\dot {\alpha} \dot {\beta} )}, $$ where $$ h_{(\alpha \beta )} = (\sigma^{\mu \nu})_{\alpha \beta}M_{\mu \nu}, \quad h_{(\dot {\alpha} \dot {\beta} )} = -(\tilde {\sigma}^{\mu \nu})_{\dot {\alpha }\dot {\beta }}M_{\mu \nu} \qquad (.1) $$ are an irreducible spinor representations (for other definitions look here). With generator $J_{\mu \nu}$ of the Lorentz group and corresponding irreducible representation $T(g) = e^{\frac{i}{2}\omega^{\mu \nu}J_{\mu \nu}}$, by rewriting antisymmetric tensor $\omega^{\mu \nu}$ with using spinor formalism we can get $$ T(g) = e^{\frac{i}{2}\left(\omega^{(ab)}J_{(ab)} + \omega^{(\dot {a}\dot {b})}J_{(\dot {a}\dot {b})}\right)}, $$ where (compare with $(.1)$) $$ J_{(ab)} = \frac{1}{2}(\sigma^{\mu \nu})_{a b}J_{\mu \nu}, \quad J_{(\dot {a} \dot {b} )} = -\frac{1}{2}(\tilde {\sigma}^{\mu \nu})_{\dot {a }\dot {b}}J_{\mu \nu}, $$ so the Lorentz group is generated by two symmetrical spinor tensors. I got commutation relations of these tensors: $$ [J_{(\dot {a} \dot {b})}, J_{(\dot {c} \dot {d})}] = \frac{i}{2}\left( \varepsilon_{\dot {a}\dot {c}}J_{(\dot {b} \dot {d})} + \varepsilon_{\dot {b} \dot {d}}J_{(\dot {a} \dot {c})} + \varepsilon_{\dot {a} \dot {d}}J_{(\dot {b} \dot {c})} + \varepsilon_{\dot {b} \dot {c}}J_{(\dot {a} \dot {d})}\right), $$ $$ [J_{(a b)}, J_{(c d)}] = \frac{i}{2}\left( \varepsilon_{ac}J_{(bd)} + \varepsilon_{bd}J_{(ac)} + \varepsilon_{ad}J_{(bc)} + \varepsilon_{bc}J_{(ad)}\right). $$ But commutator $[J_{(a b)}, J_{(\dot {c} \dot {d})}]$ isn't equal to zero, against expectations. It's equal to $$ [J_{(a b)}, J_{(\dot {c} \dot {d})}] = -\frac{i}{8}\left( (\sigma^{\beta})_{b\dot {c}}(\sigma^{\nu})_{a\dot {d}} + (\sigma^{\beta })_{b \dot {d}}(\sigma^{\nu})_{a \dot {c}} + (\sigma^{\beta })_{a \dot {c}}(\sigma^{\nu})_{b \dot {d}} + (\sigma^{\beta})_{a \dot {d}}(\sigma^{\nu})_{b \dot {c}}\right)J_{\beta \nu}, $$ which isn't zero (look here). Should it be so?


title: Work on ideal gas by piston

text: Imagine a thermally insulated cylinder containing a ideal gas closed at one end by a piston. If the piston is moved rapidly, so the gas expands from $V_i$ to $V_f$. The expanding gas will do work on the piston, how much work is done if the piston is moved away at a speed $w$ ?


title: How to theoretically determine the angular momentum of an atom?

text: To determine if an atom is a boson or a fermion I have to count the fermions that constitute the atom (protons, neutrons and electrons). My question is: How to theoretically (as opposed to experimentally) determine the total angular momentum of an atom? Can I just sum the spin of all the electrons and the nucleus spin, and the orbital angular momunta of the electron orbitals? For example: A Hydrogen atom in the ground state of can be considered like a spin 1 particle?


title: Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically

text: Consider a system of the two identical point positive charges situated in the space (isolated from influence of any other external fields) as shown in the figure.Particle1 is at (a,a,0) and Particle2 is at (0,a,0).Velocity of them ( at the instant considered ) are as shown in the figure. Now by applying Biot- Savart law we find that magnetic field due to Particle2 at the position of Particle1 is along +ve Y axis. And thus the force acting on Particle 1 is along +ve Z axis.(By Fleming’s left hand rule) Now similarly analyzing the force on particle 2 we find that no force is there on Particle 2. Now if we check net torque on the system of the two particles about Y axis then it is non zero and along the –ve Y axis…We have no external forces or torques acting on the two particle system considered , still net external torque on the system is non zero…WHY ? also NLM3 seems to be violted ...WHY? I know that the electromagnetic field itself takes away some momentum and moment of momentum about the considered axis but I think that if I consider only two charged particle as my system than the Abraham Lorentz force can be assumed to be acting upon the system and that is sufficient to make sure that we have considered the momentum being carried away by the electromagnetic field itself. Even after considering the action of the AbrahamLorentz force for the two particle system, both the Newton's third law, the angular momentum conservation rule and also the linear momentum conservation rule seem to be violated..... Because the Abraham Lorentz forces do not exactly counter balances the force and the torques on the considered two particle system due to simple magnetic field..... ![enter image description here](http://i.stack.imgur.com/0weBR.jpg)


title: How can things be chaotic on a quantum level, yet tangible on a classical level?

text: This may seem basic, but I am wondering if anyone has any input on this topic. It doesn't make any sense to me (I mean I don't need to use the Schrödinger equation to find my cell phone...). I just do not understand how things can appear to be highly statistical when observing the rudiments of matter, yet when observing the large scale result, very consistent (for all practical reasons). How can things with no position make up something with a position? Is it possible that our detection methods are flawed beyond what Heisenberg suggested? Is there some intuition here that I could be missing?


title: Degrees of freedom in physical equations

text: Say we have the field equation: \begin{equation} f^{\prime}(R)R+3\square f^{\prime}(R)-2f(R)={\kappa}^{2}T, \end{equation} why is the non-vanishing of the second term means that there is an extra degree freedom $\phi\equiv f^{\prime}(R)$


title: Wall stress of a hexagonal pressure vessel

text: **Problem:** I want to calculate the stress in the walls of a hexagonal pressure vessel but I can't manage to get coherent results. For long vessels, cylinders are supposed to have the lowest hoop stress but I obtain out-of-wall stresses with close or lower magnitudes with my method, which is wrong... **Wrong method:** 1) I cut the cross section where I assume the max stress will be 2) drew a free body diagram where $\sigma$ is the unknown and $F_p$ is the outward force that the pressure generates on each wall. I considered 2 options, one where the pressure is always normal to the walls (not represented), and one where it's omnidirectional coming from the center (represented). I am not sure about which one applies. 3) Respectively, $F_p=p*h*a$ and $F_p = 2\int^{30°}_{0°}p*h\frac{a}{2}cos(\theta)d\theta$ ; $F_{\sigma}=\sigma *h*t$ 4) When projecting on X and Y, $\sigma$ appears to be out-of-wall (no Y component), and considering only $2*\sigma_x*cos(30°)*h*t=2F_p$, I end up with 87MPa and 38MPa at the corners (lower elsewhere) compared to 75MPa for a cylinder (t=1mm,a=375mm)) - which does not sound right at all. a is the length of a side, h the height of the vessel (flat ends), p the internal pressure, t the thickness of the walls and $\theta$ the angle between the pressure vector and the normal to the considered face. ![enter image description here](http://i.stack.imgur.com/fKQUL.png) **Question** What is the right method?


title: Will a sound composed of the frequencies 450Hz, 650Hz 850Hz have a clearly defined musical pitch? Why?

text: According to my lecturer, the perceived pitch of a sound composed of the following harmonics: 750Hz, 1000Hz, 1250Hz is equal to the fundamental frequency which is the highest common factor of the harmonic frequencies; so 250Hz. He also said that the harmonic frequencies 450Hz, 650Hz 850Hz do not have a clearly defined musical pitch. How can this be true when the harmonic frequencies have a highest common factor and therefore a fundamental frequency of 50Hz?


title: Why do steam bubbles increase in size as they rise?

text: In the following video (a customer's review of a glass kettle), we can observe water boiling: http://youtu.be/jByY5I7Xk7w?t=2m55s As the kettle starts to boil at around 2:55, we can see large steam bubbles being formed at the bottom, where the heating element is, and these bubbles shrink as they rise. Presumably this is because they are coming into contact with cooler water. Then we get a crazy convection current for a bit before the element switches off again. After the chaotic motion has died down (and the fluid is presumably very well mixed) we see small steam bubbles being forming at the bottom, which grow as they rise. I can think of two possible explanations for this, and I'm curious as to which is correct:   1. The water is superheated. Nucleation sites exist on the bottom of the kettle, so that's where steam bubbles form. Steam is produced at the interface between steam and water, which causes the bubbles to grow as they rise.   2. The pressure at the bottom is slightly higher than at the top. Assuming a depth of 15cm, the boiling point at the bottom of the water is about $100.3^\circ \mathrm{C}$, compared to $100.0^\circ \mathrm{C}$ at the top. Bubbles form at the bottom because the heating element is still slightly hotter than $100.3^\circ \mathrm{C}$, and as they rise they drag hot water up into the slightly lower-pressure area, where it turns to steam because its boiling point lowers, and this increases the size of the bubble. In particular I'm interested in whether the second of these explanations plays a role. If it doesn't happen in a boiling kettle, is there any situation in which it does?


title: Pushing with a lorentz contracting stick

text: If I use a stick to push and accelerate an object, my hand pushes one end of the stick distance $x$, while the other end of the stick pushes the object distance $y$. Distance $y$ is smaller than distance $x$, because of Lorentz contraction of the stick. My hand does work $Fx$. Work $Fy$ is done on the object. Energy $F \cdot(\text{Lorentz contraction of the stick})$ seems to disappear. So I'm asking, what happens to the "missing" energy? EDIT: In this thought experiment pushing causes the object and the stick to accelerate, which causes the stick to Lorentz-contract. In extreme case the length of the stick becomes zero, which means my hand moved a distance of the stick's length kind of unnecessarily. Shorter stick saves energy. EDIT2: I noticed that "lost" energy approaches zero, when force approaches zero. This suggests the energy loss is linked to deformation of the stick. EDIT3: This very simple problem may be very difficult to understand, so I ask this way: A good push rod is rigid. Relativity says rigid push rods don't exist. So what kind of energy goes into a push rod, that is as rigid as relativity allows, when we use the push rod, using moderate force, and the speed that the push rod is accelerated to, is relativistic?


title: The transactional interpretation of quantum mechanics

text: John Cramer’s transactional interpretation of quantum mechanics (TIQM) is billed as resolving the fuzzy agnosticism of the Copenhagen interpretation while avoiding the alleged ontological excesses of the Many Worlds Interpretation. Yet it has a low profile. Is this because no-one care anymore about ontology in physics, or is there something about TIQM which undermines belief in it?


title: Why is the unit of hydraulic impedance/resistance kilogram per second and hypermeter?

text: I'm trying to figure out the unit of impedance in the hydraulic analogy of electronic networks. Assuming $$Z=\frac{P}{Q},$$ with $P$ as pressure difference, $Q$ as volumetric flow rate and $Z$ as impedance, the unit would be $$\frac{Pa}{m^3/s} = \frac{N/m^2}{m^3/s} = \frac{kg/ms^2}{m^3/s} = \frac{kg}{m^4s}$$ Intuitively, this unit doesn't make sense. Did I get some part of the unit conversion wrong? If not, could someone help me understand the unit?


title: How many times has the "stuff" in our solar system been recycled from previous stars?

text: Is there a cosmologist in the house? I've got a basic understanding (with some degree of error) of some simple facts:   * The Universe is a little over 13 billion years old. Our galaxy is almost that old.   * Our solar system is roughly 4.6 billion years old.   * The heavier elements (Carbon, Oxygen, etc..) are only produced by stars. When our solar system coalesced, the stuff that makes up the planets came from a previous generation of stars. (Meanwhile our Sun is busy making its own slightly-heavier elements, but those aren't ours. Those belong to the next "generation" after our star dies.) So we're all sitting on a rock whose matter was spewed out of a star (or stars) that preceded this solar system. Got it. Don't need that lesson. But what I want to know is... How many cycles of coalescing, fusing, doing the nucleosynthesis thing, exploding -- and then repeating the cycle -- have this atoms in this ball of rock I'm standing on been through? How do you know this? Can you tell by the make-up of the matter in the solar system? Looking around the Orion arm to see remnants of our former home? Wild ass guess?


title: Calculating force of a car

text: I'm simulating car cruise system and my results are not making any sense. I have been told that my force formula is wrong but I don’t understand why it is wrong and how I can fix this. Does anybody know how I can fix this problem?               double force = 5*throttle - (speed > 0.0 ? 1.0 : -1.0) * (0.1 * sqrt(fabs(speed           acceleration = force / mass;             distance = speed * time_step + 0.5*(acceleration) * (time_step*time_step);          speed += (acceleration * time_step);


title: objects distorted by the earths curvature

text: We are doing a project where we hope to make people think about the fact that the earth is round. We want to imagine if we in London could see others places in the world and how the earths curvature would distort them. We want to make chairs. How would we make this exact ? We are trying to find the degree of curvature of certain places in relation to london. We hope to distort a model of a normal chair in software like CAD by inserting a 'degree of curvature' . Can you help with this ???


title: Can I Demagnetize a ferromagnet with a hammer?

text: If an iron is exposed to an external magnetic field, the iron's domains are all aligned with that ext-B . If I strike it with a hammer, will it be demagnetized? Comparing it with heat, I know the Tc point, but with vibrations what is the point like in heat? When would I expect the iron to be temporarily demagnetized? How much force must I apply to move those domains apart by hammering it?


title: Notation for Propability Amplitudes

text: I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude scattering, and then he wrote the following on the board: $$[1-,\,1+,\,3-,\,3+]$$ I assume these are probability amplitudes, but I don't quite what the integers and the signs mean.


title: Scanning Electron Microscopy: Explanation of Numerical aperture and Extraction Field

text: I am trying to understand what numerical aperture and extraction field means in a scanning electron microscope. From what I understand numerical aperture is used to set the Depth of Focus which is generally the permitted focus range. Is Extraction field is voltage difference between the Objective Lens and the specimen? I am not sure how extraction field plays a part in resolution or the focus. Thanks for helping me out!


title: Flow in parallel paths after pressure regulator

text: Consider the following:                                              /‾‾ Valve --- Watering hose     Pump --- Pressure regulator ------ Valve --- Watering hose                                    \__ Valve --- Watering hose      A pump generating an arbitrary pressure and a pressure regulator to curb it at 1.5 bar. After the regulator we have three valves in parallel. If I close one of the valves partially or completely will the volume flowing through the other two valves change? I hope that the pressure regulator would compensate closing/opening valves by keeping the pressure constant but I'm not sure. Background is a watering solution in gardening. I aim to decouple several strands of watering hose so they can be adjusted independently.


title: Does the 1-D poisson's equation have monotonic potentials if $\rho=\rho(\phi(z))$?

text: I am solving the 1-D poisson equation: $$\frac{d^2 \phi}{dz^2}=-4\pi\rho(\phi)$$ with the additional requirement that $\rho(\phi(z=0))=0$. If I start by multiplying each side by $\frac{d\phi}{d z}$ and integrate from 0 to $\phi '$ I get $$\frac{1}{2}\left(\frac{d\phi}{dz}\right)^2=-4\pi\int_0^{\phi '}\rho(\phi) \frac{d\phi}{d z} dz$$ so I have $$\frac{1}{2}\left(\frac{d\phi}{dz}\right)^2=-4\pi\int_0^{\phi '}\rho(\phi) d\phi$$ Because of the square on the left hand side I will only get solutions when the right hand side is positive. I find this requirement odd because you will always get that the derivative of the potential never changes sign and is always monotonic when the charge density is written in the form $\rho(\phi)$. Am I doing something wrong here or have I misinterpreted the result? More specifically If I provide a charge density that changes sign with $\phi$ will I be unsuccessful in finding a solution? Also for anyone who is familiar with it, this first series of steps is used in the derivation of the Child-Langmuir law for the thickness of a plasma sheath, but in the ion sheath problem $\rho$ does not change sign (See the Bohm Criterion if interested).


title: Decomposing a Tensor Product of $SU(3)$ Representations in Irreps

text: Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$? Here $3$ and $\bar{3}$ are the fundamental and anti-fundamental of $SU(3)$, in this case.


title: Will freely rotatable polarizers align?

text: **Will two freely rotatable linearpolarizers (placed in sequence and at some angular offset less than, say, 45 degrees) eventually align if you shine (plenty of) unpolarized light at the first one?** **If so, will the second polarizer align itself with the first one, or will _both of them_ start to rotate towards alignment (although perhaps not necessarily equally quick)?** _Edit_. There are over 400 views and five answers now (one at +1, the others at 0), which are all highly appreciated. However, I thought it might be useful to provide a bit of my initial intuition with regards to the first part of the question. I am assuming that the light is polarized vertically after passing the first polarizer and that the second polarizer (pictured) is slightly offset. To represent the vertical polarization I give a single photon some vertical extent (possibly in terms of probability amplitude). This the segment A-B. _I am not at all sure that such representation is allowed or misguided._ ![enter image description here](http://i.stack.imgur.com/RNriy.jpg) Now the observation is that a collision (absorption or reflection) at B will produce more angular momentum (of the polarizer) than a collision at A.


title: Effective Mass and Fermi Velocity of Electrons in Graphene:

text: In graphene, we have (in the low energy limit) the linear energy-momentum dispersion relation: $E=\hbar v_{\rm{F}}|k|$. This expression arises from a tight-binding model, in fact $E =\frac{3\hbar ta}{2}|k|$ where $t$ is the nearest-neighbor hopping energy and $a$ the interatomic distance. But how does one know that the Fermi velocity is given by $v_{\rm{F}} = \frac{3ta}{2}$? Normally one would use $m_{\rm{eff}}^{-1} =\frac{1}{\hbar^{2}} \frac{\partial^{2}E}{\partial k^{2}}$ and $v_{\rm{F}} = \frac{\hbar k_{\rm{F}}}{m_{\rm{eff}}}$, but in this case $m_{\rm{eff}} = \infty$.


title: How feasible is Laser Plasma Acceleration for the post-LHC world?

text: I was wondering how feasible Laser Plasma Acceleration for the post-LHC world is. It seems very promising.


title: Spring extensions

text: > A certain spring has attached to it a mass of 25 units: on increasing the > load by 6 units it extends 2.5 cm. > > a) What is the time of oscillation under the original load? > > b) What will be the velocity and acceleration when it is midway between its > lowest and mean positions if it is loaded as at first, pulled down 5 cm and > let go? I am guessing the spring is hung vertically. Do I work out the modulus of elasticity first? I guess the equations $\omega=\sqrt{k/m}$ will be helpful.


title: Does this type of phase transition exist?

text: The short version of this question is:   * Is there, or could there be, a system with a phase transition where adding a small amount of heat causes a discontinuous jump in its temperature? Below are my reasons for thinking there might be. At a first-order phase transition there is a discontinuity in the first derivative of $\log Z(\beta)$, where $\log Z$ is essentially the free energy and $\beta=1/k_B T$ is the inverse temperature. As a consequence, its Legendre transform $S(E)$ has a segment of zero second derivative. (Here $S$ is the entropy and $E$ the expected value of the energy. The two functions are related by $S(E) = \log Z(\beta) + \beta E$.) This means that the function $E(\beta)$ has a discontinuity. These basic properties of first-order phase transitions are illustrated below: ![enter image description here](http://i.stack.imgur.com/nGueV.png) The slope of the third plot, $d E/d\beta$, is related to the heat capacity, which becomes infinite when $\beta$ is at the critical value. However, it seems to me that the opposite phenomenon could also happen, where the discontinuity is in the first derivative of $S(E)$, and consequently $\log Z(\beta)$ has a straight line segment, and $E(\beta)$ has a section of zero rather than infinite slope, like this: ![enter image description here](http://i.stack.imgur.com/2rYY9.png) Rather than having an infinite heat capacity for a critical value of $\beta$, such a material would have a zero heat capacity for a critical value of its energy density, meaning that at the critical point, adding a small amount of energy would cause a discontinuous change in temperature. It seems that it wouldn't be too hard to construct a toy model that exhibits this "dual" type of phase transition. All you really need is a very high density of states at the critical energy value. (However, I have not explicitly constructed such a model yet.) In a similar way, one could construct the dual of a continuous phase transition. Here the second derivative of $S(E)$ would diverge at the critical point, and the heat capacity would smoothly approach zero around the transition. I have never seen anyone refer to these types of transition, but I don't know whether this is because (a) they don't happen, (b) they're not considered very interesting, or (c) I just don't know the correct term for this phenomenon. Therefore my questions are   * Does this type of transition occur in physical systems? If so, does this transition type have a name, and is there a well-studied example?   * If not, is there a fundamental reason why it can't happen? What assumptions are needed to prove that it can't?


title: Electrostatic adhesion instead of glue. Is it possible?

text: I am thinking about the way to attach the printed photographs to the wall but not using the frame. And the most interesting idea for me is the use of electrostatics. In addition I have found the following video that shows this process in action: http://youtu.be/r0u85k5EpcE Also I think that something like this is used in some kinds of wall-climbing robots. So does somebody know how it works? I suppose that the materials of the objects are very important here. Probably it works because the wall and photo are made of paper, and plastic between them gets charged and consequently attracts paper. What other materials or combinations of materials can provide this effect? And how is the electrostatic charge transferred to the material (plastic foil and paper)? I know only about objects gaining electrostatic charge after rubbing them. But in the stick from this video it must be done by some simple electric circuit...


title: what is use of relativistic action?

text: this is relativistic action: $$S=\int_C \mathcal {L}dt$$ where the $\mathcal{L}$ is $-m_oc^2\gamma^{-1}$ what is use of relativistic action!?


title: What causes the triplet state in Helium?

text: I am not familiar with the notation used on wikipedia (http://en.wikipedia.org/wiki/Triplet_state). Is there a more physcial way to explain the cause of the triplet state (maybe without referencing the wave function)? In my book it says the folllowing: > 'For the overall spin of the electrons exist the possibilities S = 0 and S = > 1, which is >why the term scheme of the Helium atom splits in a singulet and > a triplet system.' I also found that J (total angular momentum quantum number) = L+-S can be 0,1 and 2 but I can't see why. On a related note: does the intercombination prohibition mean that one helium atom can only be singulet or triplet forever? Edit1: Thanks for your quick answer. But aren't you implying in your solution that L=0 (due to L-S <= J <= L+S)?


title: Kirchoff's rules and RC circuits

text: ![enter image description here](http://i.stack.imgur.com/4moUP.png) You can view the image bigger: http://i.stack.imgur.com/4moUP.png I'm studying RC circuits and I can't understand the part when we use Kirchoff's rule. When working with batteries, when we're going through a branch, when we hit on the negative plate first we take it's voltage to be positive. How does that work with capacitors ? How do we know which plate will be positive? Also, the person who solved this wrote: $Vc + I1 * 1 Ω- I10 * 10 Ω = 0$ Wouldn't it be : $Vc - I1 * 1 Ω + I10 * 10 Ω = 0$ Because, if we are going clockwise, when we're going through the $1Ω$ resistor we're going in the opposite direction of the current and for the $10 Ω$ resistor we're going in the same direction as the current. Also what if I went counterclockwise would the $Vc$ be negative? And finally is this solved correctly? ![enter image description here](http://i.stack.imgur.com/TXbNs.png)


title: Explosion of a Neutron Bomb

text: Watching "The Dark Knight Rises", Bane announces halfway into the movie at the stadium that what they have is a neutron bomb. But then at the end of the movie there is an actual nuclear explosion shown as if it was a fission bomb. My understanding was that a neutron bomb only releases highly energetic particles like neutrons so that all organic matter is destroyed by the radiation. I thought the point was to keep buildings/cities/material wealth intact so that at first you deploy a neutron bomb, kill the people in the city, wait for radiation to settle down, and then move in and take over the city. So a neutron bomb shouldn't blast like that with the ability to level a city and everything in it. Is a neutron bomb supposed to "explode" like that? Did I have the wrong idea? Or is it just another case of Hollywood getting the physics wrong? How much of an "explosion" is a neutron bomb supposed to have? Thanks.


title: Pascal's law: pressure of fluid at different locations

text: I know that's stupid question, but I'm really confused what my teachers says, so I need to check that theory. Here are just two ordinary connected containers, which are full of water. ![Image NO 1](http://i.stack.imgur.com/Z7WI1.png) On grounds of theory of hydrostatics we can say that :   * `p3` is greater than `p1`   * `p4` is greater than `p2`   * `p1` equals `p2`   * `p3` equals `p4` Let's say:   * `p1` = 2Pa   * `p2` = 2Pa   * `p3` = 5Pa   * `p4` = 5Pa But what if _piston 1_ goes down, like here: ![Image NO 2](http://i.stack.imgur.com/mX3MX.png) What pressure will be at places `p1`, `p2`, `p3`, and `p4`? Will they increase by the same number with maintaining amounts based on hydrostatic theory, or they will be same?


title: Construction of Observer Coordinates in General Relativity

text: In _A First Course in General Relativity,_ Second Edition, by Bernard Schutz, on the bottom of Page 8, he is expanding on the spacetime diagrams in Figure 1.5. He states _"[...] the four angles are equal [...]",_ referring to angles the moving frame axes have with respect to the non moving frame axes and vice versa. However, nowhere does he demonstrate or prove that the $\bar{x}$-axis makes and angle of $\theta$ with $x$-axis. How is this proved or demonstrated?


title: Is it feasible to convert the energy of gamma-photons from gamma-emitting nuclear isomers into electrical power?

text: It is known that gamma rays emitted by nuclear isomers, naturally occurring or manufactured, carry huge amounts of energy. Co-60 for example decays into the isomeric state of Ni-60 by $\beta ^-$ emission (plus the neutrino). Ni-60 then emits two $\gamma-$photons of energy $E_{\gamma_1}=1.333$ MeV and $E_{\gamma_2}=1.173$ MeV, a total of about 2.5 MeV, and attains its ground state. This amount of energy from a single decay event, is about $10^6$ times the energy in a single event in a conventional battery! **The question is** : Can we utilise this amount of energy, by combining appropriate scintillators and photovoltaic cells in order to convert that energy into electrical power? Arguments for and against will be appreciated.


title: I understand the Big Bang Theory (BBT), but how was the matter in the BBT created?

text: I understand the Big Bang Theory to consist of all of the matter being pulled into one great gravitational pull. such a great force that it expelled the matter out causing the idea of Red-Shift and everything is moving away from us. But I always understood the BBT as how the universes and everything was created. How can this be so if there was matter to be pulled in and expelled during the BBT? Oh dear... My mind is soon to reenact that of the Big Bang. So yeah, how was the matter created BEFORE the BBT for the BBT to take place? Please correct me if my understanding of the BBT is incorrect.


title: Cavity dynamics from traveling waves to standing waves

text: This is a question on mathematical physics. The conventional reasoning about the characteristic wave modes in a cavity is to apply the boundary conditions, namely, wave amplitude is equal to 0, and obtain discrete cavity-mode frequencies. My question here is, supposing we start by generating some arbitrarily shaped traveling wave inside the cavity. If this traveling wave contains both cavity-mode and non-cavity-mode frequencies, after bouncing around inside the cavity for while, how are those cavity-mode "standing" waves kept and other non-cavity-mode waves disappeared? In a math-physics setting, the dynamics obey a usual wave eq, with proper boundary (decaying) conditions. This is solved and the full dynamics in the solution should give us the answer. Has this problem already been fully treated and discussed in some math-physics books? if anyone knows such references, please kindly point me to them. Thanks! [Modification] \--- Below is the revised question --- Let's consider a Fabry-Perot type wave filter, in which the input waves enter from one side and leave on the other end. Due to the constructing and destructing superposition at the output end, the output waves only take certain resonant frequencies. For example, $\omega_f = n\pi c/d$, where d is the distance between the plates. In this set-up, input waves would be expanded over various non-resonant modes, which are in the functional space larger than the cavity-mode space. My question is: as the decay rate $\gamma \rightarrow 0$, how do non-resonant modes "disappear" at the output end? I have assumed the decay is so weak that it wouldn't be responsible for the "disappearance" of non-resonant modes. So my original intention of the question, was to look into a detailed mathematical solution to this process. By doing that, I might better understand the dynamics of "disappearance" of non-resonant modes. I believe the actual mechanism must be the conversion from non-resonant modes to resonant modes. Another interesting thing is, this dynamic process should be independent of the specific material that the reflecting plates are made of.


title: How did Copernicus establish the relative distance to the superior planets?

text: I understand that the relative distances to the planets had been calculated using various methods since ancient times, and, in particular, that the assumptions of the Copernican model of the Solar System allow one to easily establish the relative distances, $d$ (in AU), from the Sun to the inferior planets, Mercury and Venus, from their (easily measured) greatest elongations, $\varepsilon$ and simple trigonometry: $$d=\sin\varepsilon$$ Hoever it is not clear to me what methods Copernicus or his contemporaries used to determine the relative distances to the superior planets. My texts say only that these were found by "slightly different geometrical methods" (p.44) or that "the analysis for a superior planet is more involved" (p.37). What methods were those?


title: What is the significance of Planck force?

text: I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4/G$, where $c$ is the speed of light; $G$ is the gravitational constant. Thus (the speed of light^4) / gravitational constant = 1.21031359 $\times 10^{44}$ newtons.


title: Under what condition charges do not flow in closed circuit?

text: I wanted to ask under what conditions will charges not flow in a closed circuit. Or when is current through the circuit zero even when an EMF is applied? Like in the case of potentiometer, we say that we are measuring the emf of the battery because current through the secondary circuit is 0? So what condition are we fulfilling here so that current is 0?


title: Direction of friction for object that rolls with slipping

text: Let's say you hit a cue ball with a pool stick that causes an impulse. If the pool stick hits the ball above the ball's center of mass, is the direction of friction different than if the pool stick hits the ball below the center of mass? What determines the direction? Assume the ball is rolling with slipping.


title: Representations and transformations under an $SU(n)$ Lie groups?

text: I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the $\mathbf{\bar{3}}$ with Dynkin indices $(0,1)$, a state transforms like : $ϕ→ϕg^{−1}$. And for the adjoint representation $(1,1)$: $\mathcal{O}→g\mathcal{O}g^{−1}$. But then, if I take $SU(2)$, because the $\mathbf{2}$ and $\mathbf{\bar{2}}$ are equivalent, they should transform in the same way? And what about a representation labeled by $(2,0)$ in $SU(3)$? Should a state in this representation transforms like $\Psi→g^2\Psi$?


title: Laplacian in 4 spatial dimensions; 4th dimension warped

text: How can I prove the form of the Laplacian in four spatial dimensions, using the identification $y = y + 2\pi R$ for the fourth dimension and assuming the others as the usual Cartesian ones? I want to do this by letting the covariant Laplacian work on the 4-dimensional space, so my problem reduces to: what form should the 4-dim metric assume?


title: Raman Scattering and the Kramers-Heisenberg Formula

text: Using the words of the wikipedia article Raman Scattering: > The Raman effect corresponds, in perturbation theory, to the absorption and > subsequent emission of a photon via an intermediate vibrational state, > having a virtual energy level. From this, we can derive the Kramers-Heisenberg Formula for the differential cross section, which takes account of this intermediate state, but, from what I understood, excludes the absortion of more than one photon. Well, my question is: the others processes where more than a photon is absorbed, like Hyper Raman Scattering or second Hyper Raman Scattering, are described by the Kramers-Heisenberg Formula as well?


title: Is the continuous transformation of energy from one form to another one free? Or it consumes some quantity of who knows what?

text: Sorry for the unclarity. I probably created some bias in your mind having tagged my question with energy-conservation and conservation-laws. I simply consider that, relatively independently of matematical models, energy is everyday changing its "form" both "spontaneously" or triggered by us using our machines. So I ask: why the universe seems to be designed in a way that it "seems" free (at no cost) to change cinetic to thermal energy, for instance? Why we do not need to pay a fee to move energy from "light" to heat? Like exchanging currency without having to pay for that service. I stress once again that I am not referring to the basic consideration of losses. I do not discuss the conservation of energy itself.


title: Quick question concerning the Heisenberg model

text: I got a small, rather technical question concerning the Heisenberg model. (It is technical indeed.) Consider the Heisenberg Hamiltonian: $H = \sum_{(i,j)} S_{i} S_{j}$ = $- \frac{J}{V} \sum_{q} \gamma_{q} S_{q} S_{-q}$. with $\gamma_{q} = 2 \sum_{\alpha=x,y,...}cos(q_{\alpha})$ and the Fourier transform $S_{i}=1/V \cdot \sum_{q} S_{q} e^{iqr_{i}}$ Where we assume a lattice constant of a=1 and impose periodic boundary conditions on a hypercube of edge length L and respective volume V. Here comes my question: I don't see how one arrives at the equation for the fourier transformed H in detail. I mean...i see where the cosines are coming from if you only have -q and q left and suspect that one has to apply an identity like $\delta_{ij} = \sum_{q} e^{iq(r_{i}-r_{j}}$ but somehow it doesn't work out for me and even though it has nothing to do with the physics it leaves a bad feeling behind. I'd be really thankful if someone could help me. Best regards and thanks in advance.


title: Why should optical fibers be thin?

text: What are the reasons that optical fibers have to be thin (small radius of the fiber)? Is there a good picture which explains this in detail?


title: References on the physics of anyons

text: Anyone know some good introductory references on the physics of anyons?


title: Integration of Dirac propagator

text: I've got confused with Dirac (or Spinor) Propagator. Everywhere in books I have examples of integration Klein-Gordon propagators, which are quite easy. But I don't understand how to integrate 4-component function. My second question - how to deal with a mix of Dirac and KG propagators in one equation? I know that if we have many scalar vertices we have to regularize our integral. And examples are shown in lot of books - Peskin, Maggiore, many more. But I couldn't find case with Dirac propagators. My equations were with: $$S_a(x_1-x)D(x-y)S_b(x-y)S_a(y-z)S_b(z-y_1)D(z-y_2)$$ Where $x_1$ were position of initial particle (it's 1->2 decay process) and $y_i$ are positions of final particles. $x,y,z$ coordinates are for internal vertices. We started from a-fermion particle (that's why there is $S_a$ propagator) and we ended with b-fermion and $\phi$ - scalar particles. Interaction Hamiltonian contained only these 3 particles. If there is required, I can even post Feynman diagram here for process. EDIT: Here are the two Feynman Diagrams for decay process, in third order of perturbation. Described problem was for the first one: ![enter image description here](https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak- prn2/v/t34.0-12/10363750_801409816536818_1693252894_n.jpg?oh=ef70b424d50f193856bc928c48909a19&oe=537F435A&__gda__=1400883615_ce6af1cbc461f014e8453ea50db92632) Ah: our theory is $\mathcal{H}_I=k\varphi\bar{\psi_b}\psi_a$


title: Why does strong interaction increase with distance?

text: I read numerous times that strong interaction increases with distance. But how can one actually derive the force-distance relation from the lagrangian (quark field + gluon field + gauge coupling)? Since I can't use perturbation theory, I don't know what to do with it.


title: $\gamma^5$ factor in Quantum Field Theory

text: I have a problem with interpretation of $\gamma^5$ factor in the interaction Hamiltonian. I know that $\frac{1\pm\gamma^5}{2}$ is a helicity projection and it requires helicity conservation in scattering process. But what does single $\gamma^5$ mean? My interaction Hamiltonian: $$\mathcal{H}_I=\varphi\bar{\psi_b}A\psi_a$$ I have two cases, one when $A=1$ and one when $A=i\gamma^5$. After long time I've got quite simple first-order perturbation theory for $A=1$. But what about $\gamma^5$? (I have to describe a decay of a particle into b and $\varphi$ particles. $a$ and $b$ are fermions, $\varphi$ is spinless). I will appreciate any help, or even better, any title of handbook with similar case.


title: How does one refute a Machian mechanism for inertial emergence?

text: **Introduction:** Consider the diagrams representing the duality between the weak and strong principles of equivalence. Now based on how these diagrams were taught to us (at least how they were taught to me); each of the two graphics are displayed for the sake of demonstrating how two separate situations are actually physically indistinguishable for the inside observer (Einstein himself). ![enter image description here](http://i.stack.imgur.com/QwjKc.png) After careful consideration and analyzation, I claim that these diagrams in no way admits any information which can be used to suggest that in each diagram the two scenarios are in any way physically or _situation-ally nonequivalent_. That is; free fall may not be described as any sort of new class of inertial motion as the Wikipedia article on "General Relativity" suggests. Equivalently gravitational forces may not be thought of as a new class of accelerated motion. In my opinion this idea that free fall represents a new class of inertial motion is simply an artifact of the assumption that gravitational fields are not present at a distance far removed from a gravitational source. Essentially I believe all frames in which we don't feel inertial forces _must be free falling frames_ (such as orbit or flat space drift) Even though it initially doesn't seem that way (Since the gravitational source cannot be seen from a far distance). Perhaps this is already considered obvious to some but I know it is not to everyone (Since so much work is being done to understand inertia by other means...such as say radiation reaction forces(1)) I don't want anyone to misinterpret my meaning; I COMPLETELY agree with the principle of equivalence, in fact I believe it is far more powerful than most physicists give it credit for. I only claim that the diagrams are _misleading_ because each diagram uses two seemingly different yet _already fully equivalent_ scenarios as a means to describe how they are actually (surprise!) physically equivalent from the point of view of the inside of an elevator _but makes no claims about the equivalence on the outside as well_ ; In each diagram there is an indirect suggestion being made; that the two situations are _actually distinguishable_. It is as if Einstein pops his head out the door and says "Oh I thought I was free falling, turns out my elevator was just drifting in locally Minkowskian space-time, silly me". I say he was always free falling, just not in the traditional way. In my opinion all Minkowskian frames are free falling ones. The geodesic equation says that mass-energy follows its natural geodesic path through space-time but mass can only be measured via force probe if there is a resistance to that natural geodesic path. When you accelerate an object in a way which admits an inertial property of matter(One that is resisting its natural geodesic path through space-time) such as... the pushing force on our backsides as we accelerate in car, it MUST be the result of an objects coupling to the gravitational sources throughout the universe (and Earth too!) being resisted. This is essentially the meaning of a Machian principle of equivalence. $\frac{d^{2}x^{\lambda }}{d\tau^{2} }+\Gamma ^{\lambda }_{\mu \nu }\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=0$ For the sake of clarity I will augment the geodesic equation to demonstrate how inertia may be calculated as an Machian effect of changing the natural geodesic path of an object. (The point here is to show that the connection coefficient is BUILT from all mass energy in the universe.) $m\frac{d^{2}x^{\lambda }}{d\tau^{2} }+m\Gamma ^{\lambda }_{\mu \nu }\left ( \left ( T_{\mu \nu } \right )_{Universe} \right )\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=F^{\lambda }_{inertial}$ Also when most of us speak of the geodesic equation, we refer to it's description of an object in an inertial frame since it is always equated to zero but the equation is equivalent to the four force equation which is not necessarily zero. This is precisely why inertia manifests. When an object is free falling. $F^{\lambda }_{inertial}=0$ When an object is feeling inertial forces. $F^{\lambda }_{inertial}\neq 0$ (1) I intend no disrespect to the legitimacy of radiation reaction forces or to those undertaking research in radiation reaction forces, only that if there are radiation reaction forces then they are available here, raining upon us on the surface of the Earth all the time since we are always standing in an accelerated frame. I would even go as far to say that radiation reaction research may serve as a means to understand inertia and gravitation far better than we currently do. **Conclusions:** 1a) Weight and inertia are one-in-the-same as told by the strong principle of equivalence. 1b) Applied forces and gravitational normal forces are one-in-the-same as told by the strong principle of equivalence. 2) Free fall, free drift and orbit are one in-the-same as told by the weak equivalence principle. 3) An objects mass-energy content can only be defined by effective measurement of an obstruction (by a force probe) to the object's natural geodesic path through space-time. For this reason mass-energy MUST be defined this way. 4) The principle of equivalence is Machian by virtue of the augmented geodesic equation. (NOT IN THE SENSE THAT MACH ORIGINALLY PROPOSED ~ SE USERS: PLEASE DO NOT REFERENCE THIS VERSION OF MACH'S PRINCIPLE WHEN COMMENTING OR ANSWERING BECAUSE IT IS NOT THE CONCERN HERE) Yet most physicist claim any sort of Machian principle is erroneous. Additional note: When I approached my professor with this information; I originally suggested that if all objects with mass-energy in the universe (which share time-like separations) contributes to the natural geodesic paths of every other object in the universe. Then inertia can be described in terms of a Machian framework due to the principle of equivalence itself. He claimed that my argument was useless since that no one has been able to explain WHY mass energy contributes to space-time curvature. While I agree that the mechanism which somehow curves space-time due to the presence of mass energy is not well understood, I disagree with him that the information is pointless since these ideas must remain valid once we finally do figure out the mystery behind gravitational mechanism. The certainty of a Machian process for inertial emergence should not change once we understand gravitational mechanism. In my opinion, the understanding of gravitation in terms of a Machian principle of equivalence seems beneficial for solving new problems or setting new boundary conditions. **Questions:** 1) Where is my logic/thought process flawed? 2) How can a Machian principle of equivalence of this type be refuted? (Since it is explicitly described by the geodesic equation and the principle of equivalence.) 3) If an object's mass energy content can only be defined by a measure of an obstruction to its natural geodesic path through space-time, then how can two separate interactions such as electromagnetism and gravitation both generate by experiment the exact same values for an objects mass-energy if the two interactions are not fundamentally one in the same? (Let me say that the lack of measurable gravitational forces and the availability of measurable electromagnetic forces at microscopic scales compared to macroscopic scales where the opposite is true seems all too artificial.)


title: Virial of a system

text: I had obtained $$\overline{E_{kin}} = -\frac{1}{2}\overline{\sum_j\mathbf{r}_j\cdot\mathbf{F}_j}$$ and was asked to show that if the forces are conservative then $$\overline{E_{kin}} = \frac{1}{2}\overline{\sum_j\mathbf{r}_j\cdot\mathbf{\nabla}_jE_{pot}}$$ where $E_{pot} = \sum_{1\leq j,k\leq n}^{j<k} V_{jk}(\mathbf{r}_j-\mathbf{r}_k)$ is the total potential energy. I had tried to proceed by noticing that $\mathbf{F}_j = \sum_{k(\neq j)}\mathbf{F}_{jk}$ $$\overline{E_{kin}} = -\frac{1}{2}\overline{\sum_j\mathbf{r}_j\cdot\mathbf{F}_j}\quad\quad\\\ \quad \quad= -\frac{1}{2}\overline{\sum_j \mathbf{r}_j\cdot \sum_{k(\neq j)}\mathbf{F}_{jk}}$$ Substituting $\mathbf{F}_{jk} = -\nabla _j V_{jk}(\mathbf{r}_j-\mathbf{r}_k)$, $$-\frac{1}{2}\overline{\sum_j \mathbf{r}_j\cdot \sum_{k(\neq j)}\mathbf{F}_{jk}}\qquad\qquad\\\ \quad=\frac{1}{2}\overline{\sum_j \mathbf{r}_j\cdot \sum_{k(\neq j)}\nabla _j V_{jk}(\mathbf{r}_j-\mathbf{r}_k)} $$ $$=\frac{1}{2}\overline{\sum_j \mathbf{r}_j\cdot \nabla _j\sum_{k(\neq j)} V_{jk}(\mathbf{r}_j-\mathbf{r}_k)}$$ How do you show that $\sum_{k(\neq j)} V_{jk}(\mathbf{r}_j-\mathbf{r}_k) = \sum_{1\leq j,k\leq n}^{j<k} V_{jk}(\mathbf{r}_j-\mathbf{r}_k)$?


title: Does it matter if we are made of matter or antimatter?

text: Is there any difference between matter and antimatter other than the charge? Could we (theoretically of course) find in space a civilization made of antimatter?


title: Interpretation Klein-Gordon equation

text: What is the problem if we try to interpret KG equation as a single particle equation? Also I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum mechanics.


title: Classical Rutherford scattering (partial) derivation

text: I am having trouble answering the following question, please could you help! Thank you in advance for any assistance you can give. Consider classical Rutherford scattering of a particle with mass $m$ and initial speed $v_0$ from a potential \begin{equation} V(r) = \frac{\alpha}{r} \end{equation} where $\alpha$ is some constant and $r$ is the location of the particle from the origin. Starting from Newton's second law, show that \begin{equation} |\Delta {\bf{p}}| = \frac{2 \alpha}{v_0 b} \cos \Big(\frac{\Theta}{2} \Big). \end{equation} Note that $b$ is the impact parameter and $\Theta$ is the scattering angle. Please note that I have already shown that "from geometry" the change in momentum is $|\Delta {\bf{p}}| = 2 p \sin(\Theta / 2)$, and that $b v_0 = r^2 \frac{d \theta}{dt}$ where $t$ is time and $\theta$ is the angle $\angle({\bf{r}},{\bf{r^*}})$ where ${\bf{r^*}}$ is the point of closest approach. I am unsure however if the the above two equations will be of assistance.


title: Apparent color of flame as superposition of spectral lines?

text: In some cases one can identify a substance by the color emitted as it burns in a flame. A green flame might indicate the presence of copper. So here is the question. If we know wavelength and intensity of emission spectra for an element, isn't what we "see" as it burns simply a superposition of these? If I try to reconstruct the _apparent_ color of copper from its spectrum as $\sum f_n \cdot i_n /\sum i_n$ in which $f_n$ and $i_n$ are the frequencies and relative intensities respectively, it seems to work. But trying this with, say, helium, using approximations because there are a lot of lines, I consistently get something in the green range. This site gives a simplified spectrum for helium in a tube. Approximating the intensities I get something in the green range. I tried a more detailed calculation using selected NIST data and obtained a similar result. But helium is consistently shown as having a red appearance in a flame or in a tube (link). One guess is that there are invisible emission lines, and helium has many. If these are included in the sum they might drag the apparent color toward the red. Is my basic understanding incorrect? Any insights appreciated.


title: Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

text: I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for this claim, and the basis for this claim was not immediately apparent after quite a long time of consideration on my behalf. Furthermore, while trying to deduce the answer, I was lead to the complement of my first question: why does a continuous eigenvalue spectrum of an operator lead to non-normalizable eigenfunctions (i.e. the inner products don't exist)?


title: Do you feel more impact force when falling straight down or tipping over?

text: Suppose you fall from the top of a ladder straight down. You will hit the ground with an amount of force. Now suppose that you fall over while holding onto the ladder, tipping over in an arc instead of falling straight down. You will hit the ground with another amount of force. Neglecting the mass of the ladder and air resistance, which impact will have the most force? Falling or tipping? This has been a bit of a debate between myself, my father, and my grandfather. I believe that they would fall with the same force because, relative to the ground, you start with the same amount of potential energy in both situations. My grandfather and father, however, guess that it would fall with around half the force because the force is dissipated somewhat by the forward motion and the support of the ladder. We would appreciate an answer from a third party to help us find the solution.


title: What is the fundamental differences between bound and entangled states

text: Specifically, are all entangled states considered bound?


title: Why search for theoretical particle?

text: Just out of curiosity, is there any special purpose in human life, the search for some theoretical particle for its uncertain benefit, to prove theory of everything which doesn't serve any advancement towards humanity (yet), using huge amount of resources which is better of allocated to fund research for alternative renewable energy? To enable warp drive? Anti-matter as alternate fuel? To know the ultimate answer of universe, life and everything?


title: Simple rotation of an atomic orbital wavefunction

text: We know that an atomic orbital wavefunction may be written in terms of polar coordinates, $$\psi (r, \theta, \phi) = R(r) A(\theta, \phi)$$ where $R(r)$ is the radial component and $A(\theta, \phi)$ is the angular part. There are many orbitals of interest, but in particular I'm interested in applying a symmetry operation on $d_{xy}$. We know that the angular part of this orbital is $\sin^2 (\theta) \sin(2 \phi)$. Suppose I want to rotate only along one axis, let's say $\theta$, so I fix $\phi$ and rotate $\theta$ by $\pi/2$. This means that the rotated orbital is now written as $A(\theta + \pi/2, \phi)=\sin^2(\theta + \pi/2) \sin(2 \phi)$. We know from trig. that $\sin(a + \pi/2) = \cos(a)$, therefore the angular part of our transformed orbital may now be written as $$A(\theta+\pi/2, \phi)=\cos^2 (\theta) \sin(2\phi)$$ however, graphical approaches to this problem for the real orbital suggest this $\hat C_4$ rotation should give back $-d_{xy}$. What am I missing?