The Spinning Log "Perpetual Motion" problem, and my attempt at a solution

So I was introduced to this "perpetual motion" riddle a few weeks ago. The problem goes like this: we all know perpetual motion machines are not possible, but this riddle _seems_ like it should work as a perpetual motion machine - the problem is to explain _why it doesn't work_. **Here's the situation:** (1) You take a room, and put a wall down the middle of the room - splitting it in to two equal half-rooms. (2) Take a perfectly cylindrical log whose length is the same as the length of the wall we just put in to make two half-sized rooms, and cut out a place halfway up the wall so that the log slips right in, length wise (if you were looking at the middle wall from the inside of one of the rooms, you would not see the ends of the log at all, you would a half-cylinder running the length of the wall - if you looked from the other room, you would see the other side of the cylinders length). (3) Make sure the space between the log and the wall is air-tight (the log is treated to absorb no water, there is no friction between it and the wall, etc), and fill one of the half-sized rooms all the way to the top with water. Now you have one side of the log exposed to air, and one side totally surrounded by water. The idea is that, since everyone has experience that logs rise in water, the side exposed to the water should "rise" and the log should spin, creating a perpetual motion machine. Now, I have two explanations of why it _doesn't_ work. One I came up with myself, and one that is proposed in the solution on the page in which I found the problem. **My Solution:** Water pressure increases as you increase in depth, so the water molecules hitting the log deeper in the water would have more force that the water molecules hitting the log less deep, but at each infinitesimal depth increase, the water is hitting the log at all angles, and the force from the molecules coming on from all directions would cancel out to leave a force pointing toward the center, at every place the log is exposed to water. Since all of the net forces from the water at each point exposed to water is pointing toward the center, there can be no net torque and the log cannot spin. The **net** force would be a slight force directly upwards (due to the difference in size of the force vectors pointing inwards between the less dense water up top and the more dense water on the bottom), but since the log is held in place, it cant move upwards. **The Proposed Solution** "To understand why this would not work, we need to look at how buoyancy itself works. When a lighter than water object (lets say a hollow ball) is submerged it is pushed up by the water apparently in spite of gravity. In fact the opposite is true. It is gravity pulling down on the water that pushes the ball up. The kinetic energy to lift the ball comes from an equal volume of water falling to occupy the space where the ball just was. As the ball moves up, the evacuated area is filled with water from above it; therefore, this water is falling. When the water was above the ball it had potential energy that is exchanged for kinetic as it falls to fill the void left by the ball thus providing energy to lift the ball. The opposite is true for a heavier than water object (lets say a brick), but it's still the same principle. As the brick falls through the water, it is filling space that was once filled by water. As the brick falls, it is providing energy to lift the water to fill the space it just evacuated. Now back to our spinning log example. If the log were to spin, it would not be evacuating any space for the water to fill. Even though a different part of the log would be filling that space, it would still be the same space it was filling before. There would be not downward movement (falling) of water to convert potential energy into kinetic, so there is no energy to cause the log to spin." **Here's my problem** I've never understood this explanation of buoyancy. Why, just because an object is less dense than water, is the water "rushing below it" to fill the space evacuated by the object as it rises? Why doesn't the object just stay where it is? It seems like you're using buoyancy to explain buoyancy - the less dense object floats upward due to buoyancy and the water rushes in below to fill the evacuated space, which causes the buoyant force upwards... what? Why is the water rushing below less dense objects, and not rushing below more dense objects? Am I missing something in the explanation? I understand that when you submerge an object in water, it displaces an equal volume of water, and the weight of the water displaced equals the bouyant force upwards, but I just don't understand why the water is "rushing underneath and pushing the object up".


What is the closest general-relativistic equivalent of a "time slice"?

In a newtonian universe, one can talk of a "time slice", that is, the state of the universe at a given point in (global) time. In a "typical" classical universe, a time slice would contain enough information to fully compute the state of the universe at any other point in time, backwards or forwards. So, disregarding what we know of quantum effects and talking purely of a general-relativistic universe, which concept is closest to a time slice?


Is it possible for one side of the universe to "meet" the other?

I've variously heard the shape of the universe being described as multi- dimensional, like a helix or mobius strip, and super string theorem seems to say there are lots of universes all piled up next to each other in vibrating planes. My question is, can the edges of our universe ever join up? Are they already all joined up? as in is our universe really in the shape of a giant donut (yum). or is this shear lunacy? I guess I am assuming a model of the universe where the "edge" is not necessarily expanding constantly "out" into infinite amounts of empty vacuum, but rather one where the edges are the boundaries to that expansion (so that it seems from an observable position that the universe is expanding, when really its flexing inside these boundaries). As indicated in the comments no doubt an inaccurate description, but I was imagining it like a balloon.


Neutrino mass and energy question

If a neutrino has mass then it travels less than the speed of light. Suppose I boost myself to the rest frame; i.e. bring it to rest in the laboratory. Now if it oscillates between different states and masses sitting there, where does the oscillating excess/loss energy and mass reside? In some internal state? I have a very limited knowledge of QM but in SR it would seem strange that "mass" would move into another place. A pointer to where I should start studying would (I hope) be sufficient.


Scale invariance Vs Conformal invariance

> **Possible Duplicate:**   >  Why does dilation invariance often imply proper conformal invariance? What exactly is the difference between the two? Can someone give an example of a theory which is scale invariant but not conformally invariant?


How could this person have discovered the resonant frequency from this string of magnets?

I stumbled onto this page http://mylifeisaverage.com/story/1364811/ and the post states that they were > all making strings and shapes with these sets of 216 really small spherical > earth magnets. What did I do? I strung them together and found the string's > resonant frequency. It was 15 Hz so I'm wondering how the poster could have figured this out?


How would an observer perceive movement on a train that's travelling near the speed of light?

Person A is on Earth and a train (or whatever you want to imagine) travels past him at near the speed of light. How would person A perceive movement on the ship? If time is slowed on the ship from the perspective of Person A, then if Person B were washing the windows or something (moving his hands up and down), would they look like they were going really fast, slow, or normal? If time is slowed from the perspective of Person A, then Person B's hands would be moving a certain distance (let's say the window is a meter) in a really short amount of time, no? And that would mean it looks like it is going fast. But then since Person A is perceiving the time as moving slower, it would look slower, no? Would they cancel and movement on the ship looks normal? I tried to see if the Wikipedia page could help, but it didn't seem to say anything about a stationary observer and a fast-moving vessel.


Why doesn't a neon sign seem that hot?

I heard that neon signs contain plasma, why aren't they hot? is it because the electrons and ions do not hit the lamp's wall? Is it because it is non thermal plasma and electrons and ions are not in thermal equilibrium? If that is the case do the electrons and ions and neutral atoms (all of them) hit the lamps wall?


Applying angular velocity to a rotation matrix

I have a very simple question. In our project we store an object's orientation as a 3x3 matrix which holds the orthonormal base of that object's local space. For instance if the object is aligned with the axis of the world, then its orientation matrix is :               1 0 0     0 1 0     0 0 1      Now we store the angular velocity as a vector that contains values in rad/s around each of the world's axis. For example if the object should rotate around the world's y axis at 20 rad/s, then the angular velocity is               0 20 0      My question is then, say we want to apply this velocity over 1 second to the orientation matrix, how would we do this?


A commercial airplane travels at a speed which is 85% of the speed of sound. The wings of the airplane are designed such that the bottoms of the wings are flat and the tops of the wings are curved so that air travelling above the wings follows a path which is 15% longer than the straight path of the air below the wings. The wings are approximately rectangular in shape with a length of 35 m and a width of 8 m. The thickness of the wings is negligible. What is the maximum mass that the airplane can have and still maintain enough lift to fly?


Gravitational slingshot maximum

I have recently read an article about gravitation slingshot assist used by Voyagers 1-2, and was thinking on why this hasn't been used for travel between solar and other systems.   I mean sligshot can be done as many times as it is necessary to get speed of lets say half the speed of light that would allow to travel to Alpha Centauri in ~10-20 years can it not? There must be a flaw in my thinking that 3 or 4 planets can be re-used to get to necessary speed otherwise it would already have been done (drawing below). Even if planets would align differently I should always be able 'find' the planet that would allow me to jump to one that is closer to the sun, and repeat the acceleration again and again. ![enter image description here](http://i.stack.imgur.com/t52Vs.png) **What maximum (theoretical) speed could be achieved using planets of solar system as sligshot and how much would this speed wary from planetary alignment and what realistic speed could be achieved?** UPDATE: To be more specific on the second part of the question Lets say craft weight's 500kg at starting speed of 30,000 km/h initially it slings around Mercury (`radius 2440km`), Venus (`radius 6052 - 300 (atmosphere) = 5750 km`), and Earth (`radius 6378 - 300(atmosphere) = 6050km`) until diameter of planets is to wide to not to crash craft on surface. Then it flies to the moons of Saturn - Titan (`radius 5150km`), Rhea (`1527km`), Lapetus (`1470km`), Dione (`1123km`), Tethys (`1062km`), Enceladus(`504km`), Mimas (`396km`) and starts slinging there until diameter is to wide too. What approximate maximum speed could it get to leave the solar system?


Has the possibility of Higgs boson being a composite particle excluded already?

I heard some theory, such as technicolor, predicts the Higgs-like particle discovered at LHC should be a composite particle (correct me if I am wrong). Has this possible been completely excluded already?


How much electricity could be generated by cloths made of thermocouples?

Lets say that we made a glove, shirt, pants and a hat out of the most effective thermocouple material available today. How much electricity would be generated by each, by a healthy person, on a cool day (98 degrees F vrs 79 degrees F?)


How does a magnet work?

I'm having trouble understanding how a magnet (not the field that is generated as a result but the material itself) work. The particles are aligned in a specific direction to give rise to force but I don't see how this alignment gives rise to "attraction" or force.


Spin in relativity

Mass and spin of the particle are used in classification of elementary particles. The mass is defined to be a Lorentz invariant quantity. On the other hand, the spin is a spacelike 4-vector and cannot be defined as an invariant quantity. My question is, what would be a convenient definition of a relativistic spin 3-vector? As is known, 3-vectors get contracted/dilated under Lorentz transformations. So, is the 3-spin defined to be equal to the rest frame 3-spin, or do we allow it to contract/dilate? It seems to me that the first option is generally used, especially when considering Thomas precession and similar effects. Although, I haven't seen any relevant discussion in textbooks on this topic.


Finding the current in a parallel circuit

Two resistors are joined as shown. The top resistor receives a current of 3 A. What is the current in the other resistor? What is the current that enters at junction A? I'm confused on how the two currents are 3A and 1A because I thought in a series circuit all the currents are the same. Why is the answer to the current entering junction A 4 A and not 1 A? ![](http://i.stack.imgur.com/XviUa.jpg)


Infinite heat capacity or susceptibility means fluctuation on all scales

I remember reading in an introductory text to phase transition (sorry I don't remember the name) that at a second order phase transition the specific heat and the magnetic susceptibility become infinite and that this implies fluctuations on all length scales. I understand how this implies strong fluctuations as they are respectively linked to the variance of the energy and of the magnetization, or that being infinite it's not possible to construct a preferred length scale with them; but I'm not sure to understand why it forces the fluctuation to happen at ALL the scales. Do you have an intuitive argument or example for that? Thank you in advance.


Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where $\hat{U}$ is the (unitary, linear) evolution operator and $\hat{H}$ the Hamiltonian (the most general version of which; i.e. time-dependent with instances at different times non- commuting). In S-picture, one can easily show that (1) is **equivalent** to $$ i\hbar \frac{d}{dt}|\psi_S(t)\rangle=\hat{H}_S|\psi_S(t)\rangle\tag{2} $$ where $\psi$ is a state, $|\psi_S(t)\rangle = \hat{U}(t)|\psi_S(0)\rangle \equiv \hat{U}(t)|\psi_H\rangle$ and $\hat{H}_S:=\hat{H}$. In H-picture, it is straightforward to show that (1) **implies** $$ \frac{d\hat{A}_H}{dt} = \frac{\partial\hat{A}_H}{\partial t}+\frac{1}{i\hbar}[\hat{A}_H,\hat{H}_H]\tag{3} $$ where $A$ is an observable and $\hat{A}_H(t)=\hat{U}(t)^\dagger \hat{A}_H(0)\hat{U}(t)\equiv\hat{U}(t)^\dagger \hat{A}_S\hat{U}(t)$ (and also $[\hat{H}_{H},\hat{H}_H]=0$ implying that the time dependence of $\hat{H}_H$ is purely explicit, i.e. $\hat{H}_H=\hat{H}_S\equiv \hat{H}$ with $[\hat{H},\hat{U}]=0$). **My question:** is it possible to obtain (1) from (3), i.e. to show that (1) is **equivalent** to (3)? Some thoughts on this: it is extensively mentioned in literature that both pictures yield same answers. Therefore, it should be possible to obtain (1) from (3) since (1) and (2) are equivalent. Assuming (3), the best I can get is that given an observable $A$, the operator $$ \hat{C}:= \hat{A}_S\left(\frac{d\hat{U}}{dt} \hat{U}^\dagger - \frac{\hat{H}}{i\hbar}\right) $$ must be skew-Hermitian.


What is a good reference for the quantum mechanical description of lasers?

I am currently taking a graduate level class on lasers. The primary focus of the class is on the design and engineering aspects of lasers, e.g. resonator design. However the first portion of the class is an overview of the quantum mechanics of laser processes in semi-classical terms: transition probabilities for a 2-level dipole atom in an external (classical) field, perturbation theory, and matrix QM. I have an undergraduate degree in physics, so the basics of QM are familiar to me, but it has been years since I seriously applied any of that knowledge. In addition, the professor is moving through this material very rapidly, as it is not the primary focus of the class. I can follow well enough that I'm not worried about my grade, but I'm feeling a bit cheated out of a more thorough understanding of this topic, which I would really like to have. Not only is this subject matter an appropriate level of advancement over my existing knowledge, it is also the basis for something (lasers) which I will likely spend a career dealing with. Can anybody recommend a text which covers this topic, and would be appropriate for somebody with my background? I would value a clearly written description over absolute rigor, but I am by no means looking for a laypersons description of QM. For example, I am quite comfortable already with the notation and concepts in the textbook "Quantum Mechanics of atoms, molecules, solids, nuclei, and particles" By R. Eisberg and R. Resnick. The ideal reference would be at a similar level of sophistication, but provide a more in-depth look at laser processes in particular.


Reason for different type of energy transfer for two kinds of collisions

According to my physics book, if an electron were accelerated with 15 MeV of (kinetic?) energy and collided into a 100g thermally insulated copper block (not sure if the fact it is thermally insulated is relevant) the energy would transfer into thermal energy on the copper block and the copper block would heat up by a few degrees. However, if I were to collide into that same copper block with a bowling ball travelling with 15 MeV of kinetic energy then that energy would be transferred into kinetic energy on the copper block and the copper block would move. It seems to me, that in both cases, there is a thing smashing into another thing with 15 MeV of energy. When the first thing is small (e.g. electron) the energy is transferred into thermal energy. When the first thing is large the energy is transferred into kinetic energy. Why doesn't the electron's energy get transferred into kinetic energy as well?


Why does the light travel slower in denser medium?

Wikipedia says that "in general, the refractive index of a glass increases with its density." And the refraction index of water vapor is less than ice, and even less than liquid water. Is there any simple explanation to that?


What is the mechanism behind the slowdown of light/photons in a transparent medium?

So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the "vacuum" between the atoms?


Difference in velocity of light in change in medium

It is often seen that according to physics the light changes it's velocity according to the medium through which it is traveling. So can it be explained that why so happen?


Dna Fingerprint

I recently came across an article indicating that the half life of DNA in the most ideal situations is 521 years (http://www.nature.com/news/dna- has-a-521-year-half-life-1.11555). However, since human's DNA is 99.9% identical (and 98% identical to chimps), I can't quite understand how DNA evidence is often used in old samples. There has been DNA extracted from 100,000 year old Neanderthals, and DNA evidence has been used in cases over 50 years old. However, with a 521 year half life, after only 0.75 years, there would only be 99.9% of the DNA left. Surely if 99.9% of a human's DNA is identical after only this short amount of time wouldn't it be impossible to distinguish any sample from any other? After 50 years only 93% would remain the same, meaning it would be much less similar to a human than just any chimp I would think. And the 100,000 year old Neanderthal would only have 1.6*10^-57 % of the original sample left. How is there possibly any sort of useful information left? All in all, how can DNA be distinguished from other samples when they are all so similar and the half life is so low?


Newton's Third Law Clarification

Assuming you place an object so heavy on a table that it breaks it, then according to newtons third law the forces must cancel out (equal magnitude and opposite direction), but if this is true, then how can the object break the table in the first place?


Two suns, one moon, and one planet?

I have a question about how would seasons and the moon cycle be affected in a system where one planet orbits Sun #1, and Sun #1 orbits a second sun. Online I found this description: > "Type II: "Close Stars" - The suns may occupy different parts of the sky, > indicating that one sun orbits the other farther out than the planet." (I > had a sketch, but didn't know how to upload it) To make it easier to explain I'm going to assume this system's Sun #1, planet, and moon are equal mass and size of Earth, it's moon and it's sun. The planet is in a habitable zone of Sun #1 but outside the gravitational pull of the second sun. So I guess my questions are:   * How would the moon cycle be affected when the planet orbits between both suns? Would there be times when it seemed like the sun never set or would days be twice as short, because there is a sun on both sides of the planet for 3-4 months out of the year?   * Would the same situation described above cause a significant shift in seasons(summer, winter,spring) when the planet is between two suns?   * Does the size of a sun, or it's mass more dictate it's gravitational pull? Is it even possible for a system such as this to survive or would it eventually get stuck in between both suns gravitational pull, or switch to a figure 8 orbit, and eventually be thrown out of both suns orbit.


Minimal coupling of an atom to the EM field

The Hamiltonian of an atom coupled to an EM field, both described quantum mechanically is: $$H = \frac{1}{2m}(\hat{p}-q\hat{A})^2 = \frac{\hat{p}^2}{2m} -\frac{q}{m}\hat{p}\hat{A}+\frac{q^2}{2m}\hat{A}^2$$ Under the condition of transversality ($\hat{p}$ and $\hat{A}$ are vectors). I have only seen it treated in the dipole approximation where the last term becomes a constant and so unphysical; but what is its general interpretation? Is it shifting the energy of the free photons, it doesn't look diagonal in number of occupation basis.


(Earth's) magnetic field

Always when we want to represent the magnetic field of the earth we see a similar image: ![enter image description here](http://i.stack.imgur.com/BRFM7.gif) My question is, what exactly does this show? What are the blue and orange lines, what do they represent? Why are they curved like that? I have a vague idea about field lines, but I do not fully comprehend this. An explanation would be appreciated


Quantum Scales and the Flatness of Space-time

I know that on the smallest scales, general relativity predicts that space- time is flat. But I've also read that space-time can be described as a sort of "quantum foam" for distances smaller than the Planck length. Isn't this something of a contradiction (one more way in which quantum mechanics and general relativity disagree)? To make matters worse, theories about quantized space-time means that space-time could be "discrete", and not smooth at all. So which principle does the physics community think is right? The idea of smooth, locally flat space-time, or the idea of "block-like", discrete space- time?


Very basic question: When to use $s=vt$, $s=1/2vt$, $s=at$ and $s=a/t^2$?

Very basic question: When to use $s=vt$, $s=\frac{1}{2}vt$, $s=at$ and $s=\frac{a}{t^2}$? What was the difference between those?


CFT and the Coleman-Mandula Theorem

The Coleman-Mandula theorem states that under certain seemingly-mild assumptions on the properties of the S matrix (roughly: one particle states are left invariant and the amplitudes are analytic in external momenta) the largest possible Lie algebra of symmetries of a (non-trivial) S matrix is given by Poincaré times an internal symmetry.   On the other hand, there are (interacting) field theories whose Lagrangians are symmetric under the conformal extension of the Poincaré group, and in some rare case this property is retained even at the quantum level.   Why (interacting) conformal invariant QFTs do not contradict the theorem? Is it possible to define an S matrix in these theories? I have read somewhere that they do not admit a particle interpretation, what does it mean exactly?


Can the Metropolis-Hastings algorithm be generalized to quantum systems?

The Metropolis-Hastings algorithm is an efficient way of simulating classical ensembles using the Monte Carlo method. Is there a generalization of this algorithm to quantum systems? What I DON'T have in mind is Wick rotation to a classical Euclidean system.


Conversion formula from spectrophotometer readings to any standard color space profile?

I have readings of colors from a spectrophotometer that records across the human visible color spectrum 390nm to 790nm intervals. I'd like to convert this into any color space, could be CIE XYZ, HSL, CIE-Lab, etc. Any color temperature is okay.


A sonar continuously emits x(t), a general but known waveform and is reflected by a target. hypothetical question

A sonar continuously emits x(t), a general but known waveform that is reflected by a target and received by the sonar. Both the sonar and the target move in the 3-dimensional space in a general but known manner. That is the positions of the sonar and the target are known. What would be the relation between and the transmitted and received signasl? What would happen if the sound speed, sonar speed and the target speed are all comparable? Is it possible that the received signal contains something like multipath? that is some past signal points of different times are received at the same time? Like receiving multiple balls thrown at different times being received at the same time.


Schrödinger operator with a potential defined implicitly

let be the problem $$ -\frac{d^{2}}{dx^{2}}y(x)+f(x)y(x)=E_{n}y(x)$$ however we have a problem, we do not know the potential but its inverse $$ f^{-1}(x)=g(x) $$ we know $ g(x) $ but not $ f(x) $ what happense then since a) the function $ f(x) $ may be multi-valued b) the function $ f(x) $ may not exists even if we know $ g(x)$ howe can we analytically solve $ f^{-1}(x)=g(x)$ so we get $ f(x) $ ?? in the case $ f(x)=f(-x) $ is even.


Where is a good place to learn (classical) optics?

I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue.


Good books about waves and optics

> **Possible Duplicate:**   >  What’s a good textbook to learn about waves and oscillations?   >  Where is a good place to learn (classical) optics? I'm looking for a good book about waves and optics, only basics things and a lot of tricks to solve exercises.


What's a good textbook to learn about waves and oscillations?

I'm taking a course on waves and oscillations using Crawford from the Berkeley series (out of print excluding international copies), and would like to know if anyone has any suggestions for a better book.   We cover:   * coupled oscillators,   * forced oscillators,    * Fourier analysis,    * traveling waves,  and some other topics. Any help with the topic would be appreciated.


Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?

In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable $\xi$ for the system in the state corresponding to $|x\rangle$ is made a large number of times, the average of all the results obtained will be $\langle x|\xi|x\rangle$, provided $|x\rangle$ is normalized." Then he states that the average value of any function of $\xi$, $f(\xi)$ is $\langle x|f(\xi)|x\rangle$. Now, taking $f(\xi)$ to be the function that is equal to unity when $\xi=a$ and to be zero otherwise, noted by $\delta_{\xi a}$, the average value of this function is the probability $P_a$ of obtaining the result $a$ when we perform a single measurement of the observable $\xi$. Now I know that this really gives the probability, because $\langle x|\delta_{\xi a}|x\rangle=|c_a|^2$, where $c_a$ is the coefficient of the eigenstate of $\xi$ corresponding to the eigenvalue $a$ in the expansion of $|x\rangle$. But Dirac seems to "know" that this is the probability without "knowing" that $|c_a|^2$ is the probability of obtaining a certain result. So my question is: how do we know that this expression $\langle x|\delta_{\xi a}|x\rangle$ is really the probability $P_a$ without using the expansion coefficients?


Question about torque and center of mass

If a yo-yo's string is not attached to anything and the yo-yo is dropped, it is obvious it will fall at $g$. In this scenario, Tension of string = 0. If a yo-yo's string is attached to a cieling and it is dropped, what is $R$ (path of center of mass)? Well, we know from experience that the yo-yo is going to fall at least, but is it going to fall at $g$? We know it is rotating as well. What would $T$ (tension of string) be? Would $T$ = yo-yo's mass * gravity? Assume that there is no friction. A related question is: if there is a stick in space and a force is applied to the center of mass, and for another stick in space an equivalent force is applied to the edge of the stick, will both sticks' center of mass move the same? Would the 2nd stick be rotating?


Pathria's "Statistical Mechanics" first edition

Does anyone know where I could find and purchase the book "Statistical Mechanics" by R. Pathria, in 1st edition (the 2nd and the 3rd are readily available, but I really need the first). I believe the first edition was published in 1972. I have searched high and low for it, but can't seem to find it even on Amazon.


Symmetry of the stress tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a surface), and returns a vector $t_n$ which is the traction vector. It is then shown that conservation of angular momentum leads to symmetry of the matrix. However, tensors are more more naturally presented a multilinear functions. I wonder:   * What type of tensor is the stress tensor? Is $n$ a vector or a co-vector? What about $t_n$?   * Is there a way to understand the symmetry when thinking of the stress tensor as a function of two vectors (or two co-vectors), under which it will seems intuitive why $\sigma(A,B) = \sigma(B,A)$? _Edit_ : To clarify, let's look, for example, at the 1st coordinate of the traction vector $t_n$ of an arbitrary normal $n$: This is $\left<e_1, \sigma(n)\right>$. From symmetry, this is equivalent to $\left<n, \sigma(e_1)\right>$ - the inner product of $n$ with the traction vector of a surface orthogonal to $e_1$. Mathematically, I understand why this is correct. But is there any intuitive meaning as to why these two quantities are the same?


Can resistance of wire be ignored

I was doing some physics homework involving direct current circuits and resistors in series, and I started to question the accuracy of the following property of resistors in series, namely that the voltage across the battery is equal to voltage drop across the resistors. But what about the resistance in the wire itself. I wouldn't think that you could just ignore that, especially for very long wires or wires with a significant amount of resistivity. So my question is, why don't text books write $\Delta V=IR_{1} + IR_{2}+...+IR_{n} + IR_{w}$ where $R_{w}=$ The resistance in the wire


Diff(M) and requirements on GR observables

This question is kind of inspired in this one: Diff(M) as a gauge group and local observables in theories with gravity The conundrum i'm trying to understand is how is derived the (quite) extraordinary statement that in GR there are no local observables. I just want to stress that this is indeed an extraordinarily counter-intuitive assertion (with extraordinarily dramatic consequences for any compatible theory of quantum reality), and it deserves an extraordinarily robust explanation. Among the statements i see are mostly involved in this argument are:   * _**It is usually argued that Diff(M) is a gauge transformation._** This point i don't have any sort of issue; the atlas of reference frames we use to describe space-time are indeed a human convention and physical laws should not depend on such conventions   * _**A physical observable should be invariant under any gauge transformation._** whaaa? i mean, this is clearly preposterous non-sense; actually i think this is more dangerous than simple non-sense, is just circular self-justification. This is actually saying that observables should be scalars? I can provide you a proof that this is non-sense by saying that the momentum of a particle is an observable in my reference-frame, and in other Diff(M) gauge (that is, another reference frame) i will see a different momentum of the same observable. All eigenvalues and eigenvectors transform according to the vector representation of the Poincare group. You are now of course free of dismiss such 'proof' as non-sense because my assumption that the momentum of a local particle is an observable is non-sense, but then, how are you sustaining that part of the argument, without actually saying again that Diff(M) is a gauge transformation? how do you avoid the circularity in this argument? U(1) gauge is a bad example because this is an internal symmetry; Diff(M) are space-time symmetries and i don't think that what is true in there (A eletromagnetic vector potential being unobservable, but B and E being observable, hence observables are invariant under U(1) )   * _**It is usually argued that observables in GR formally exists in the asymptotic boundary of the space-time._** Is there an argument for this that is unrelated/independent to the previous point?


can we PHYSCALLY (not by mathematics) justify that $ \zeta (-s)= 1+2^{s}+3^{s}+4^{s}+... $

the values $ \zeta (-1)= -1/12 $ and $ \zeta (-3)= 1/120 $ give accurate results for casimir and to evaluate the dimension in bosonic string theory so is there a physcial JUSTIFICATION to justify that in phsyics (not mahthematics) every time we see a divergent series like $ 1+2^{s}+3^{s}+......... $ otherwise how it would be possible that zeta regularization gave only correct results for the series $1+2+3+4+5+..... $ and $1+8+27+64+.. $ but not for example for $ 1+4+9+16+25+..=0 $ why a matheamtical fucntion would be useful only for certain values but not for others :(


Understanding the forces involved in a bounce from a collision?

> A rock is thrown towards a window, and hits it. The window does not break > and the rock reverses its direction and falls under the window. My question is, under this circumstance since the rock reversed it's direction of movement I want to say that the force exerted by the window was larger than that of the rock because if they were equal, the rock would simply stick to the window and slide down until it reaches the bottom. However, the answer to this problem claims that the window and the rock had the same force. Can someone explain me what is going on?


Is QFT mathematically self-consistent?

After recently going through a short program of self-study in quantum mechanics, I was surprised to find a quote attributed to Feynman essentially saying he was extremely bothered by the computational process of renormalization. It's "dippy" that anyone should have to subtract one infinity by another in order to arrive at a finite answer. What's he referring to there, in rough terms? And what's the latest in attempts to replace this computational procedure with something more physically plausible, so that the theory can have more meaning and less dippiness?


Plotting a bandstructure along High-symmetry points when kx,ky,kz is known

Suppose you know kx,ky,kz points along with the corresponding energies. Basically, you know about the 4-D E(k) dispersion. How you do then convert that data into the bandstructure plots you commonly see which show E(k) along a closed path through high-symmetry points through the Brillouin zone?


Which is the axis of rotation?

This should be simple, but it keeps bothering me. If a rigid body has no fixed axis, and a torque (defined relative to a point $A$) is applied, it will rotate around $A$. But often I can also calculate the torque relative to another point $B$ (which often seems to be non-zero too). So does this mean that the rigid body will have an angular acceleration about both axises? This seems a bit strange to me. (For a fixed axis I assume that a rotation around any axis (other than the fixed axis) is impossible, because there will always be zero torque around those axes.)


Differential Equation in Spherical Harmonics Derivation

I've been reviewing derivations of the spherical harmonics in quantum mechanics; mostly as review but also to make sure I understand where the concepts arise from. However, every derivation I've seen makes use of the following differential equation/ identity that seemingly comes from nowhere. I've not yet been able to figure out where it comes from in looking through various texts. So, here I am to ask: where does this come from? $$\frac{d}{d\theta}+l cot(\theta) \equiv \frac{1}{(sin(\theta))^l} \frac{d}{d\theta}(sin(\theta)^l)$$


Electric field due to nonconducting sphere

For calculating electric field outside a nonconducting sphere with a hollow spherical cavity. When I use the rule (Charge density= $dQ/dV$), I don't know exactly what is $dV$, is the volume here refers to the volume of the Gaussian surface ($V= 4/3 \pi r^3$) so that $dV$ will be = $ \pi r^2 dr$, or the $V$ is the volume containing the charges only, so it will be =$ V_0 – V_1 = 4/3 \pi r_0^3 - 4/3 \pi r_1^3 $. Thus, since $ r_0$ and $r_1$ are constants, therefore $dV$ will be = 0? Note: $r_0$ is the radius for the whole sphere, $r_1$ is the radius for the cavity, and $r$ is for the Gaussian surface.


How to calculate heat exchange/deltaT in a moving volume of fluid?

I am building a preheater for a maple syrup evaporator and am going to use the steam generated by the heating process to pre-heat the incoming sap from, say 5 degrees C to (hopefully) something on the order of 80-90 C. (the goal is to heat the sap/fluid before it gets into the evaporating pan so that we are more efficient - the steam used is "free" - it is a waste product of the process and capturing the heat and subsequent condensation for hot water is desirable) My flow rate through the copper tubes will be on the order of 35 to 56 liters per hour (10 to 15 gallons per hour) (this is for the end points of the assembly) I only know the outside diameter of the copper pipes - they will be 1/2" OD.   There will be 5 parallel lines inside a hood and at each end a manifold - the incoming cold sap comes into the low side manifold and makes its way up to the high side manifold in one of the 5 pipes. The manifolds are 20 inches long and are perpendicular to the other pipes which are 40 inches long. The heat exchanger will sit in a hood to provide a little back pressure for the steam and to collect the steam and force it to contact the pipes. The boiling surface area of the sap underneath the heat exchanger is larger than the heat exchanger and the hood but most of the steam rises through the hood and will transfer some heat to the copper which in turn heats the fluid. So how can I calculate the some likely heat exchange/output temperature? It seems to me that the issues are:   * efficiency of heat transfer from steam to copper   * efficiency of heat transfer from copper to fluid   * how much time the fluid is in the lines I guess I am looking for an upper bounds and lower bounds (I know I won't get the fluid over 100 C - that's about all I know for sure - well - I guess I can also be sure that the temperature will rise some - but not sure how much.) Many people report output temps on the order of 95C but I am skeptical. What can I do to improve the heat transfer to maximize the scavenged heat? I can try to find fins from a baseboard radiator, however this is for food product - so I am unsure if I will do that. Some pictures of something I would like to do: http://s662.photobucket.com/albums/uu349/bkm_photos/Oil%20Tank%20Evaporator/?action=view&current=evap-096.jpg&newest=1#!oZZ80QQcurrentZZhttp%3A%2F%2Fs662.photobucket.com%2Falbums%2Fuu349%2Fbkm_photos%2FOil%2520Tank%2520Evaporator%2F%3Faction%3Dview%26current%3Devap-087.jpg%26newest%3D1 and http://www.leaderevaporator.com/p-286-parallel-flow-sap-pre-heater.aspx Wood is used as the heat source and one can assume a constant fire/heat source that will be providing 10 to 15 gallons of evaporation per hour from a 24" by 60" pan. **EDIT (answering a comment)** The wall thickness of the copper pipe is 0.569 inches. There is only one fluid being heated - that is the sap from a maple tree. It is water with sugar in it - and by boiling it we can remove most of the water until it turns to maple syrup. The level of the fluid (sap) in the evaporator pans is kept constant by a float valve. The sap in the pan boils - that is where the steam comes from. So the rate of evaporation will be the same (approximately) as the flow through the heating tubes. The heat is generated by a wood fueled fire in a firebox just below the evaporator pans that hold the sap. Maple sap is held in a storage tank above the whole contraption. Gravity feeds it into the evaporator (a 5'x2' pan) through the preheater/heat exchanger. The fluid flow/level is regulated by a float valve. Steam is a by-product of the process - and using the wasted heat from the steam (allowing it to condense on the copper pips and transferring heat then drawing that condensation away so as not to have to re-heat it) makes the process a little more efficient. One study that was done shows that there is approximately 15% improvement in boiling rate (best case) for preheating the sap before it goes directly into the pan that gets heated.


potential difference problem?

As far as i know potential difference between 2 points is defined as the negative line integral of electric field between those 2 points. I also know that when magnetic field changes curl of electric field is not 0 and potential difference makes no sense. But when we have an inductor in a rlc circuit then people always say that there is a potential drop across the ends of inductor,but since magnetic field is changing potential difference should make no sense. please help.


What is known about the Magnetic North Pole's location before 1800?

I was recently startled to find that the Earth's North Magnetic Pole is moving at upwards of 40 km per year, with an additional ~80 km daily elliptical drift about its mean position due to variations in solar wind. The Wikipedia article on the NMP has a wealth of information, including this image: ![enter image description here](http://i.stack.imgur.com/OQtk0.png) I'm particularly surprised at the existence of "modelled" data, showing a lot of detail, about the NMP's location for centuries before anyone went there. Neither that Wikipedia article nor this one on the Earth's magnetic field, which include the image, mention where the modelled data come from. I get the impression from sources like this that the model is derived from historical measurements of magnetic declination. If that is the case I would be disinclined to trust the model much: the Earth's magnetic field has a complex enough structure that the magnetic north and south are not antipodal, with nontrivial spatial variations in magnetic declination at any given time. Has anyone seen references that explain this model and what measurements they are based on?


I need help with the following Physics' questions

I need to know which formulas and the step-by-step answers for the following physics questions: > _1\. You throw a ball upward with an initial speed of 7.0 m/s, and it > returns to your hand o.92 seconds later, ( assuming air resistance can be > ignored)._ > >   * _(a) What is the average acceleration vector of the ball?_ >   * _(b) Find the highest up ward distance that ball can get._ > > > _2\. A ball was thrown at an angle of 45.0 degrees above the horizontal, > that it traveled a horizontal distance of 496ft, and that it was caught at > the same level from which it was thrown._ > >   * _(a) What was the ball's initial speed?_ >   * _(b) How long was the ball in the air?_ >


Calculating the dimensional wall-normal coordinate for a self-similar compressible boundary layer using Levy-Lees transformation

How can I convert my self-similar boundary layer solution that is a function of the nondimensional wall-normal coordinate $\eta$ to be a function of dimensional $y$? For instance, if I determine from my boundary layer solution that $\delta_{99}$ occurs at $\eta = 5$, for a given set of dimensional parameters how do I determine the corresponding physical coordinate? This is very straight forward for Blasius flow but is not as obvious to me for compressible flow. The definition of $\eta$ is shown below. The integral dependence on $y$ is what is throwing me off. $$\eta = \frac{U_e}{\sqrt{\int_0^x\rho_eU_e\mu_edx}}\int_0^y\rho dy$$


Determining the length of a Torsional Pendulum

Currently working on this question, however I'm not sure how to solve it. > As a pendulum swings in simple harmonic motion at the surface of the Earth, > the angle the pendulum makes relative to its equilibrium position is given > by $\theta(t) = (0.140 \:\mathrm{rad})\cos(5.72t)$ where t is in seconds. > What is the length of this pendulum? > > a) 0.140 m > > b) 0.250 m > > c) 0.300 m > > d) 0.439 m > > e) 0.801 m The equation provided indicates that this is a torsional pendulum, correct? Therefore the length of the string with torsion supporting the disc at the bottom does not matter at all? If so, why?


Poincare group vs Galilean group

* One can define the Poincare group as the group of isometries of the Minkowski space. Is its Lie algebra given either by the equations 2.4.12 to 2.4.14 (..as also given in this page - http://en.wikipedia.org/wiki/Poincaré_group..) or equations 2.4.18 to 2.4.24 of Weinberg's volume 1 of his QFT books? What confuses me is that in deriving the commutation relations between $J^{\mu \nu}$ and $P^\mu$ he did use quantum theoretic arguments about the Hilbert space operator $U$ but I guess there is nothing quantum about the Lie algebra he derives in the aforementioned equations. Is that right?   * This quantum confusion steepens when one looks at the $K_i$ (..relatvistic boost along the $i^{th}$ spatial direction..), $P_j$ (..linear momentum along the $j^{th}$ spatial direction..) commutator being non-zero. This is justified by saying that the exponential action of the boosts and the translations on the Hilbert space states do not commute and that is being reflected here. (..they pick up an extra phase proportional to the mass and the dot product of the boost velocity and the displacement vector..)  But if the afore mentioned equations are really the Lie algebra of the isometry group of the Minkowski spacetime then in the Galilean limit shouldn't they be instead reflecting the fact that Galilean boosts and translations when acting on the spacetime coordinates do infact commute. But the $K_i$ and $P_j$ commutation continues to be non-zero even when the Galilean limit is taken on page 62. This makes me strongly suspicious that the equations 2.4.12 to 2.4.14 are not the Lie algebra of the isometry group of the Minkowski spacetime but are the Lie algebra of the group whose elements are $U(\Lambda, a)$ (..using Weinberg's notation..)…right?   * So is the "low velocity" limit taken on page 62 recovering non-relativistic quantum theory ? (and not Newtonian physics)   * On page 89 of the same book he derives the topology of the inhomogeneous Lorentz group as being $R^4 \times R^3 \times S^3/Z_2$. Since this is a connected manifold, I guess that by the term ``inhomogeneous Lorentz group" he is meaning only the proper orthochronous component of the full relativistic symmetry group. right?    * I can't see how the above topology matches with the semidirect product structure for posibly the same thing as given on this Wikipedia page -http://en.wikipedia.org/wiki/Poincaré_group ?  When people talk of the Poincare group is the full symmetry group of relativity is what is being referred to or is it just its proper orthochronous component (and not the other 3 components) ?   * I am familiar with the notion of "central charge" as in the "first" term on the RHS of the TT OPE of CFT…what also has the interpretation as the zero-point energy when doing the plane-minus-point<->cylinder conformal transformation. In this light it is not clear to me as to what is meant when one says that one can add in the "mass" as a central charge to the Galilean group..an extra generator which commutes with all the rest so that with this "central extension" the free particles will lie in the unitary representations of the Galilean group rather than the projective representations before the extension. I would be grateful if someone can shed light on this issue and help reconcile the two "different" notions of central charge.   * When one takes a "low velocity" limit of the Poincare algebra to get the Galilean algebra then is one taking just a non-reltivistic limit or is one also taking a non-quantum limit ?  (..I guess this will depend on my first query about whether what Weinberg calls as the Poincare algebra in the quoted equations has any quantum effect encoded in it {as it seems to be!} or is it just the Lie algebra of the isometry group of the Minkowski spacetime..)


Ion Drive Propulsion Top Speed

I would like to know if there is some formula / graph which would provide / show the efficiency of a certain type of propeller in space. Specifically, I'm interested in the acceleration attainable at certain speeds. I'm writing a science fiction book and I'm trying to make it as correct as possible, fact wise. The propeller I'm talking about is the ion drive Now, my book takes place in a world where fusion power is finally ours. So, please, let us assume that we have unlimited energy so you could power a dozen huge ion drives non stop. OK, there's the question of argon/xenon fuel, let's assume we have 1 year of that. **So... the question is... what speed could you reach?** If a continuous acceleration of $10\frac{\mathrm{m}}{\mathrm{s}}$ is applied (I put that number because it would also constitute an advantage for my crew - living in Earth's gravity), that would mean that a ship would reach the speed of light in just 347 **DAYS** But I know that's impossible because the EFFICIENCY of the ion drive would DECREASE as the ship's speed would approach the exhaust speed of the drive's "nozzle" (well, it doesn't have a nozzle per-se, as you can see in Wiki, but anyway...) * * * Please do not fear to elaborate on top of my question. Let's suppose for example that maybe the ion drives of the future have a much higher thrust/efficiency/nozzle exhaust speed. This isn't only about currently POSSIBLE facts but also about THEORETICAL limitations which might be overcome in the future (such as fusion energy).


Where did the energy of the charge go?

Suppose there is a positron and an electron, and they both collide, and we get $E=2mc^2$ of energy from the collision. Now, the charge also got vanished. Now suppose, I create neutrinos from the collision's energy, which are neutral and that doesn't violate conservation of charge, now the mass-energy of the neutrinos is same as the mass energy of the positron and electron destroyed. But instead if I make electron and positron pair again from that energy, it doesn't violate conservation of charge either, but this implies that creating one kind of charge requires positive energy and creating the other requires negative energy in some form. Isn't it ?


Are group representations possible when the solution space is not a vector space?

As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector space. Then fields satisfying the theory will have to transform under some representation of that symmetry group, by construction. What happens if we have some internal or external symmetry structure that is no longer acting on a vector space? The gauge group diffeomorphisms of general relativity spring to mind. Is there some more general 'representation' type theory which comes to our aid? And are there any examples of internal symmetries where this viewpoint is needed? Apologies if this question is imprecise or flawed - I'm just starting to get my head around the foundations of the subject! Many thanks in advance!


How does sound travel in space?

In relation to this question: How can a black hole produce sound? Which notes that the hole "produces" sound. The top answer states that: > What you think of as the hard vacuum of outer space could just as well be > seen as a very, very, very diffuse, somewhat ionized gas. That gas can > support sound waves as long as the wavelength is considerably longer than > the mean free path of the atoms on the gas. I get that there is "stuff" in space, what I don't get is how sound travels it. I learnt in high school that sound was a wave - as an example, you could fix a shoelace at one end and vibrate the other - voila, a wave forms. And then sound kind of moves in the same way through, say, air, because of the slight molecular attraction between individual molecules is enough to create a similar waveform. And then you get to space, and there aren't any molecules nearby to irritate each other, so there's no sound. If that's the case, why does it matter _how_ long the wavelength is, if there isn't a molecule nearby for the initial molecule to affect, how can sound travel? I'd understand a _little_ better if the particle a just _hit_ particle b (like a pool ball) and b carried the sound - but that has little to do with waves and wavelengths. Or is it that the "jet" that is thrown out travels along as a contiguous blob, with a sound wave embedded within it?


What is the first paper to report observations of polaritons?

I am seeking references to the first articles regarding the observation of polaritons.


Is there a non-technical way to understand why proving confinement is difficult?

As far as I understand, the basic reason why we are not able to solve QCD at low energies is because it is a strongly coupled quantum field theory, and a part from exceptional situations in simplified models (involving for instance supersymmetry), we have no tools to fully analyze such regimes. Correct me if I am wrong, but this is a rather technical obstruction. Is there thus a simple, non-technical reason why proving confinement is such a difficult problem?


Imposing anti-commutation relations on fermionic quasi-particles

In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi- particles to be fermionic in case of a interacting fermion system we began with and impose anti-commutation relations accordingly. Like in BCS theory, while using the Bogoliubov-Valatin transformation to diagonalize the Hamiltonian, we assume that the new operators are also fermionic in nature. Please explain more on this step and how is it justified.


Thermal Penetration Depth discrepancy

I've been working on a project that involves thermoacoustics, and one of the commonly-used values in this field is know as the thermal penetration depth. It is calculated as follows: $$\delta_k = \sqrt{2K / (C_p 2 \pi f \rho)}$$ where $K$ is the thermal conductivity of the fluid in question (helium) $C_p$ is the isobaric specific heat of the fluid $f$ is the frequency of vibration $\rho$ is the density of the fluid For my project, I've been using this document as a reference guide In it, the given $\delta_k$ is 0.1 millimeters. The given frequency is 400 $Hz$. The fluid is helium at 10 bar. Based on this information, I found the density of helium (~1.6 $g/mL$), the thermal conductivity of helium (~0.15 $W/mK$), and its isobaric specific heat (5.193 $J/gK$). However, when I plug these values into the equation, I get delta_k = 0.00379 mm, which is considerably off. Am I doing something wrong here? Is this widely-cited paper incorrect on such a basic fact?


Momentum And A Car Collision

I am studying an example problem, concerning the very topic mentioned in the title. In this example problem, a car has a head-on collision with the wall; the initial and final velocity are known, as well as the mass. My question is, why does the car rebound off of the wall? Why doesn't the normal force of the wall reduce the car's velocity to zero, with the car remaining there, instead of reducing the car's velocity to zero, and then giving it a velocity in the opposite direction?


Calculations of apparent magnitude

I was attempting to do some calculations of apparent magnitude to help solidify my understanding of the topic, but have been running into some confusion. According to Wikipedia, the apparent magnitude can be given as: $m_x = -2.5\log_{10}(F_x/F^0_x)$ where $F_x$ is the observed flux and $F^0_x$ is a reference flux (in other words, this equation provides the _difference_ of apparent magnitude between two observed values). Also, this is assuming that the same wavelength band is used in both flux measurements. Flux, in turn, can be calculated as: $F = \frac{L}{A}$ where $L$ is the star's luminosity and $A$ is the flux density. Since stars act as point sources, this can be simplified to: $F = \frac{L}{4\pi r^2}$ where $r$ is the distance to the star. Since, historically, Vega has been used as the reference zero-point (having an apparent magnitude around 0.03), I tried doing a simple calculation to find out the apparent magnitude of Fomalhaut using the values for luminosity and distance given in Wikipedia for both of them. First, the flux of Vega: $F_{Vega} = \frac{37\,L_\odot}{4\pi (25.3\,ly)^2}$ $F_{Vega} = 4.5999\times 10^{-3}\,L_\odot/ly^2$ Next, the flux of Fomalhaut: $F_{Fomalhaut} = \frac{17.66\,L_\odot}{4\pi (25\,ly)^2}$ $F_{Fomalhaut} = 2.2485\times 10^{-3}\,L_\odot/ly^2$ Now, to calculate the apparent magnitude: $m_{Fomalhaut} = -2.5\log_{10}(\frac{2.2485\times 10^{-3}\,L_\odot/ly^2}{4.5999\times 10^{-3}\,L_\odot/ly^2})$ $m_{Fomalhaut} = 0.7777$ Huh?? Fomalhaut's apparent magnitude is supposed to be **1.16**. Even correcting for Vega's offset of 0.03, we still come up with **0.8077**. Why are the calculations failing? I don't think I've made a mistake in the mathematics. Am I using the wrong values?


Formation on image on screen?

When we first learn elementary geometrical optics, the first thing taught to us is the two broad divisions of the type of reflection.-Regular and Irregular or Diffused.   The difference in cause for both of these is that in case of regular reflection an incoming ray is perfectly reflected in a specific direction but in diffused reflection, the reflected ray are diffused in several direction probably owing to minor irregularities of the surface. Hence the diffuse reflection does not form our image as a mirror does but instead produces general luminance. But then in case of multi-media-projectors or when we use a paper or a wall as a screen for real images, an otherwise diffuse surface (wall, paper, etc) produces an image which can be seen by us. Is this not only possible when the incoming rays are REGULARLY reflected on to our eyes?


How to explain spin of electron?

How can we explain spin of electron, or the spin of other fundamental particles? If we think the spin of electron is similar to the spin of a ball or planet we make a mistake. We say it is an intrinsic property. However, in calculating magnetic momenta and other cases we consider it as spinning entity. It's too difficult to abandon the model which we see. Without which to explain abstract one becomes challenging.


Knots and strengh of a rope

I read a few times that a knot can reduce the strenght of a rope, but I can't understand why this happens. Can someone explain me what happens to a rope tied with a generic knot and stretched? Is there a way to calculate the reduction of resistance from the form of the knot?


What would happen to Earth if the Sun suddenly disappeared?

> **Possible Duplicate:**   >  If the earth left the solar system for interstellar space. How long would > it take for atmosphere to freeze Suppose the Sun simply vanished, what would be the consequences to Earth?


How can we describe the electrons of multi-electron atoms (i.e. not Hydrogen) when equations/analytic solutions only exist for Hydrogen?

I've been digging into emission spectra of different elements and found that such things as the Rydberg equation, Bohr's model, and quantum mechanics can only fully describe the single electron in the Hydrogen atom. How did we then make the leap to s,p,d,f shells of multi-electron atoms? How accurate is our analysis of these more complicated elements? Rydberg Equation (side-note: Is this an empirical 'data-fitting' equation? What's the significance of that?) $$\frac{1}{\lambda}=R_H\left( \frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$ Hydrogen: ![enter image description here](http://i.stack.imgur.com/9yWL5.jpg) Helium: ![enter image description here](http://i.stack.imgur.com/EyQi5.jpg) Iron: ![enter image description here](http://i.stack.imgur.com/Tx0Vp.jpg) Potassium: ![enter image description here](http://i.stack.imgur.com/g23gw.jpg)


metric signature explanation

Can anyone explain what metric signature is? I have a basic knowledge regarding tensors, btw. Also, how is it related to fundamental understanding of general relativity? Thanks.


Real and imaginary parts of dielectric constant vs refractive index?

So for a complex dielectric constant $\epsilon = \epsilon_a + i\epsilon_b$, the wave vector and index of refraction are related to it through $k = \frac{\omega}{c}n$ and $n = \sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}}$. According to Jackson, the real part of the dielectric is related to polarization and anomalous dispersion, while the imaginary part is associated with dissipation of energy into the medium. If you write the wavevector as $k = \beta + i \alpha/2$ and plug it in the general wave formula (just in 1D right now) of $e^{ikr} = e^{-\alpha r/2}e^{i\beta r}$, the intensity drops as $e^{-\alpha r}$, so $\alpha$ is the attenuation constant, which tells you how quickly the wave dies out in the medium. But, if you plug that form of $k$ into the above equations to solve for $\alpha$ and $\beta$ as a function of $\epsilon_a$ and $\epsilon_b$, you find that $\alpha$ and $\beta$ are both a function of both $\epsilon_a$ and $\epsilon_b$. This is counterintuitive to me, because intuitively I'd think that the attenuation constant $\alpha$ would only be based on $\epsilon_b$, due to dissipation, and the same with $\beta$ and $\epsilon_a$. Can anyone give a good physical explanation for this "mixing"?


Why is the phase velocity used in the definition of the refractive index?

I'm aware of the so-called group index but why is the phase velocity used in the standard definition of the index of refraction? What advantage does this offer?


What would be measured, under current technology, of a particle moving faster than light? (Somewhat duplicate)

Sorry, since the question has been posed, but I insist that others posed not from this position. Can any listener deduce whether we could measure, by analyzing electromagnetic disturbance, a particle$^1$ traveling faster than light? (or current measuring device, for the sake of assuming for questioning) Studying to understand cosmic rays, inductance, and capture technologies by the artifacts of Tesla.. The name most studied by us uneducated electrical folk. \-- $^1$ For the 2011 OPERA neutrino experiment, see this Phys.SE post.


Dimensional regularization and IR divergences and scale invariance

I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant.   * Does dimensional regularization see "all" kinds of divergences?  I mean - what does it _exactly_ mean when one says that power law divergences and IR divergences disappear in the dimensional regularization. So is more regularization needed in general over and above dimensional regularization?   * Does anything about the divergences get specially constrained if the theory is scale invariant?  I have often heard it being said that dimensional regularization "preserves" scale invariance.


Kinetic energy conservation in a collision

Kinetic energy conservation in collision. In any type of collision type of collision if we take the system to comprised of both bodies then the net external force is zero. So the work done by external force is also zero. By the relation $W=K_f-K_i$ as $W$ is zero hence kinetic energy is conserved. But this not the case. Can you tell me what is the flaw in my argument? Some people say that the argument has neglected the fact that some kinetic energy is transformed into heat. But heat is created due to the interactions between bodies during collision which is thus an internal force and thus shouldn't affect kinetic energy.


Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?


Is the collapse of the wave function inherently time asymmetric?

Schroedinger's equation, as we all know, is time symmetric. In quantum field theory, we have to come up with a more sophisticated CPT reversal, but the essential point remains unchanged. However, the collapse of the wave function in the Copenhagen interpretation is manifestly time asymmetric. Correct me if I'm wrong, but can you uncollapse a wave function, converting it from an eigenfunction to a superposition of eigenfunctions? Is this asymmetry connected with the thermodynamic arrow of time and the second law in statistical mechanics, or are they independent? How would an uncollapse look like, and can we experimentally arrange for an uncollapse? Why are there more collapses than uncollapses? If an observer unobserves a quantum thingie, does that thingey uncollapse?


What if an asteroid the size of the moon hitting earth

Would we all parish due to excessive heat? Or would that be only limited to the area near the impact while as the people on rest of the earth would die from other phenomenons such as mega earthquakes, volcanic activities, tsunamis etc. Does it matter where the impact is - if it landed in antarctica we would have massive floods but if it landed in the middle of eurasian continent, the effect would be different?


Effect of variable permittivity

If I immerse a rod vertically in a liquid with a relative permittivity gradient (the permittivity decreases with depth), will the rod stretch (will the spacing of the atoms in the rod be affected by the varying permittivity)?


What is it called when two particles are associated so that what happens to one happens to the other?

There was some experiment that I read about some time back in which two particles (or the same particle, but split into two) were sent in opposite directions, but when something happened to one, it happened to the other at the same moment, as if they were the same particle. I thought the word for this was "entrainment", but looking that up does not lead back to this experiment. What is the term for this, assuming that I am not remembering something that never happened.


Yet another Young Double Slit Experiment

If I have a laser light incident on the double slit at an angle O < 90 what kind of interference pattern will I see? Will i see the same with less intensity or different pattern?


Is current related to the length of the conductor?

Ohm's Law tells us that $V = IR$. This implies that $I \propto \frac{1}{R}$. But, $R \propto l$, where l is the length of the conductor. This would mean that $I \propto \frac{1}{l}$. But this does not fit with the definition of current that says that current is the amount of charge $Q$ passing through a given point per unit time. Clearly, the amount of charge passing through a given point per second does not vary with the length of the conductor. How can this contradiction exist?


Divergent issue of Madelung's constant

This is a question triggered by this post Madelung's constant is defined to the coefficient of electrostatic potential energy in a ionic crystal. In the example of $NaCl$, \begin{equation} M = \sum_{ijk}{}^{'}\frac{(-1)^{i+j+k}}{\sqrt{i^2+j^2+k^2}} \end{equation} is conditionally convergent. > _Since this sum is conditionally convergent it is not suitable as definition > of Madelung's constant unless the order of summation is also specified. > There are two "obvious" methods of summing this series, by expanding cubes > or expanding spheres. The latter, though devoid of a meaningful physical > interpretation (there are no spherical crystals) is rather popular because > of its simplicity. Thus, the following expansion is often found in the > literature:[2]_ $$ M = -6 +12/ \sqrt{2} -8/ \sqrt{3} +6/2 - 24/ \sqrt{5} + > \dotsb = -1.74756\dots. $$ _However, this is wrong as this series diverges > as was shown by Emersleben in 1951.[3][4] The summation over expanding cubes > converges to the correct value. An unambiguous mathematical definition is > given by Borwein, Borwein and Taylor by means of analytic continuation of an > absolutely convergent series._ I have the following questions. 1) Expanding sphere leads to a divergent series. OK, what if I make a perfect spherical sample of $NaCl$, will the experimentally measured Madelung's constant to be infinity? ( **EDIT** : I maybe have some misunderstanding of divergence: it could be the case that the series is bounded but don't have a definite limit. So is the series for expanding sphere bounded? and if there is no definite limit, what's the experimentally measured value for a perfect spherical crystal?) 2) What physical principle dictates the order of summation? or why does finite number obtained by analytic continuation should be consistent with the observed value? 3) What is the role of charge neutrality here? I ever programed to compute Madelung's constant using the fractional charge idea((assign $\frac{1}{8}$ charge to the corner, $\frac{1}{4}$ to the edge, $\frac{1}{2}$ to the face ), which makes the expanding cube charge neutral. Does that mean charge neutrality is one of conditions that _must_ be enforced? or it is just for the sake of computational efficiency?


Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?

In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and isotropy of tha IF. By inertial frame Rindler means an ideal infinity extended rigid body moving freely in a world without gravity. This is distinct from an inertial coordinate system, that should be understood as an IF plus, in it, a choice of standard coordinates $x$, $y$, $z$ and $t$. As he says, the RP concerns inertial coordinate systems: the laws of physics are invariant under a change of inertial coordinate systems. I can't understand why this imply homogeneity and isotropy of an IF. If I suppose the existence of an special direction in some inertial reference frame, I could imagine some physical law governig the propagation of some signal (it can be light if you want, but it's not necessary), and if by measuring the velocity of this signal in two different directions and I get two different results, this would violate the isotropy of the IF and at the same time I could write the physical law in an invariant way under coordinate changes inside de IF (sure, it would depend on the special direction) and this would be in accordance with the RP as stated above. What is wrong with my reasoning?


How is Doppler redshift of distant galaxies established?

Doppler redshift of distant galaxies gave first hint that the universe is expanding. I am curious to know how this redshift is actually measured and interpreted from observation. Suppose I observe some visible line say red with wavelength \lambda_1. This is not enough to tell that this line is redshifted, i.e., the original line emitted from that galaxy had a smaller wavelength. How do I know the value of that particular line when it was emitted from that galaxy? It appears that mere observation of a line is not enough. So how do we proceed?


How is matter stored in black hole?

A black hole engulfs the matter nearby, how does it store the mass inside? We know that matter is composed of particles, does a black hole store the mass in massive particles? Or can we assume that it's composed of extremal black holes? what are theories about it? EDIT: I checked out Black-holes are in which state of matter? it doesn't provide a definite answer.


maximum distance between accelerating objects started at different times

Let there be two objects that have zero relative velocity with respect to each other in an inertial frame. If they both undergo identical accelerations, but one starts the acceleration at t = T1 and the other starts the acceleration at t = T2, then per special relativity the maximum distance that rest frame can measure between the two objects is (T1-T2) * c What acceleration equations show that the separation between these two objects as t approaches infinity approaches c * (T1-T2)


Standing sound wave tube

If there was a standing sound wave tube and a flammable gas was introduced then ignited, would the combustion be more forceful and more efficient since its following a standing wave, than just a gas ignited within a tube?


What is weak coupling of photon polarization to a pointer?

This question is refered to those who are familiar with the concept of weak measurement. **In short:** How can the polarization of a photon be coupled to the position of a pointer state? What is the pointer state? The position of some particle? How to realise that in a lab? **In detail:** Let's say we want to weakly measure the polarization of a photon. We write the observable $A=|H\rangle\langle H| - |V\rangle\langle V|$ and the initial state shall be $|\psi\rangle = \alpha |H\rangle + \beta |V\rangle$. Now we weakly couple it to a pointer by sending it through a birefringent element, i.e. depending on the polarization the photon needs more time to pass the birefringent element. Formally: The Hamiltonian whichs represents this coupling is given by $H=\lambda A \otimes P$ where $\lambda$ is the coupling strenght and $P$ the momentum operator of the pointer. Now $\exp{(-\text i H t)}$ will act as follows: $|\psi\rangle \otimes |g\rangle = (\alpha |H\rangle + \beta |V\rangle) \otimes |g\rangle \;\longrightarrow\; ( \alpha |H\rangle \otimes|g_+\rangle + \beta |V\rangle\otimes|g_-\rangle)$ where the first degree of freedome is clearly the polarization of the photon and the second is the pointer state (a gaussian say). Both $g_+$ and $g_-$ refere to small shifts due to the birefringent element. Then there is this well known fact that a post-selection whichs leads to an imaginary weak value $A_w$ causes a shift in the momentum-space of the pointer beeing proportional to $\text{Im} A_w$. The weak value is defined as $A_w = \frac{\langle \phi|A|\psi\rangle }{\langle \phi|\psi\rangle}$ where $|\phi\rangle$ is the post-selected polarization state of the photon. (reference: http://arxiv.org/abs/0706.4207 or http://arxiv.org/abs/0911.5139) Mathematically this is easy to understand, but I hace a very conceptual problem with all that: How do I have to understand the pointer state? Is it a position of a particle? If yes, how can its position be coupled (in a physically understandable way) to a photon? And how can a post-selection affect its momentum?


Why are rockets so big?

I'm curious why rockets are so big in their size. Since both the gravitational potential one need to overcome in order to put thing into orbit, and the chemical energy burned from the fuel, are proportional to the mass, so if we shrink the rocket size, it would seem to be fine to launch satellites. So why not build small rocket say the size of human? I can imagine small rocket would be easier to manufacture in large quantities and easier to transport. And maybe someone can make a business out of small rocket, carrying one's own satellite.


Moduli spaces in string theory vs. soliton theory

In both string theory and soliton theory, moduli spaces are frequently used. As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for string theory, moduli spaces is the spaces of all metrices divided by all conformal rescalings and diffeomorphisms. It seems like these two definitions(?) of moduli spaces are quite different, but the same terminology is used in both cases. I also learned that the name 'moduli spaces' comes from abstract geometry, but I don't know if that's any help here. My question is the following: Could anyone provide an intuitive connection between the two uses of moduli spaces, or highlight the differences?


What is a phrase for testing for a certain result?

Is there a word or phrase for when someone is testing for a certain result thereby skewing his findings?


Is there a relativistic (quantum) thermodynamics?

Does a relativistic version of quantum thermodynamics exist? I.e. in a non- inertial frame of reference, can I, an external observer, calculate quantities like magnetisation within the non-inertial frame? I'd be interested to know if there's a difference between how to treat thermodynamics in a uniformly accelerated reference frame and in a non- uniformly accelerated reference frame. Thanks!


Two hanging masses connected by springs

I had this problem for a candidacy exam, but wasn't able to get the complete answer. Their spring constants and masses are not the same, find the equilibrium position and frequencies of the system. Apparently once you find their equilibrium position, i.e. when gravity and the spring forces cancel out, you can transform to about that position and ignore gravity. Why do you get to ignore gravity? This is a normal modes problem, but I couldn't justify why gravity just disappears, but it does. Anyone have a simple reason why?


Why is the binding energy per nucleon of helium-3 less than that of helium-4?

I'd guess it has to do with the structure of helium-3 allowing for greater Coulomb repulsion between the protons, but I'm unsure.


What cools a drink?

When you stick ice in a drink, AFAICT (the last physics I took was in high school) two things cool the drink:   * The ice, being cooler than the drink, gets heat transferred to it from the drink (Newton's law of cooling). (This continues after it melts, too.)   * The ice melts, and the cool liquid mixes with the drink, which makes the mixture feel cooler. My first question is, to what extent does each of these cool the drink: which has a greater effect, and how much greater?      Secondly, how much of the cooling by the first method (heat transfer) is without melting the ice? That is, is there any significant amount of heat transfer to each speck of ice before it's melted, and how much does that cumulatively affect the drink's temperature?      I suppose all this will depend on a bunch of variables, like the shape and number of ice cubes and various temperatures and volumes. But any light that can be shed, let's say for "typical" situations, would be appreciated.


Why is classical mechanics determinism based on position and momentum only and not forces and scattering rules?

Consider a closed system (say a box) of $n$ particles. There is a well-known idiom/meme/law in classical mechanics that says that the position and momentum of those $n$ particles is all that is needed to determine the future and past configuration of the system. (A minor question, does this idiom have a name?) Why do we not consider two other pieces of information to be as important as the position and momentum of the $n$ particles?   * The force field (force as a function of space, and preferably of time too).   * The particle interaction rules (When two particles touch each other, do they whiz by or do they scatter? if they scatter what are the rules for determining angles?)


Quantum Expectation Values

I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi \hspace{0.2cm} dx$$ where $\psi(x,t)$ is a normalised state. I can understand the expectation of the position operator in terms of basic probability: one of the assumptions of QM being that $|\psi|^2$ is the probability to find the particle at $x$. The expected value is just the sum of the positions multiplied by their probabilities: $$\langle x \rangle _\psi=\int_{\mathbb{R}}x \hspace{0.1cm} |\psi(x,t)|^2 dx=\int_{\mathbb{R}}\psi^* \hspace{0.2cm} x \hspace{0.1cm} \psi \hspace{0.1cm}dx$$ by commutativity. I don't understand how this might work with the momentum operator, for example. The lecture notes for my course say that the expectation of self adjoint operators are defined in analogy to this, so that $$\langle p \rangle _\psi=\int_{\mathbb{R}}\psi^*\hspace{0.1cm}p \hspace{0.1cm}\psi\hspace{0.1cm} dx$$ I can't satisfy myself with this explanation; $p$ is a differential operator and so can't be moved about within the integral like $x$ can. For example $$\int_{\mathbb{R}}\psi^*\hspace{0.1cm}\psi\hspace{0.1cm} p\hspace{0.1cm} dx$$ makes no sense to me, as the operator hasn't been applied to anything. In any case, $\int_{\mathbb{R}}|\psi(x,t)|^2 p dx$ doesn't mean anything to me probability-wise. So basically I'm looking for an explanation as to why the expectation of self- adjoint operators are so defined. Thanks for any replies!


Restrained expansion of a piston/cylinder

Suppose an ideal gas in a piston cylinder has some initial pressure $p_1$, volume $V_1$, and temperature $T_1$. A pin, which holds the piston in place, is suddenly removed, and the gas quickly expands. Once equilibrium is reached, the final pressure of the gas is $p_2$. I am looking at a textbook solution which claims that the work done during this process is $$W = \int p\,dV = p_2\left(V_2-V_1\right),$$ which implies that the pressure is constant ($p=p_2$) throughout the whole process, as soon as the pin is released. Is this really the case, and are there not irreversibilities in this process that must be considered?


Truck driven from a small motor that can carry a heavy load, yet can travel fast.

I am building a truck from trash as materials. I have one small motor and a a few small gears, but no other engineered materials are allowed. The truck must carry a load for a distance of 3m. The winning truck is the truck with the greatest ratio of load carried:time. What do you think would be the best approach to this project in terms of the gears and the radius of the driven wheels? I don't know what would be the best approach in terms of torque and angular velocity. Any other tips or inputs would be greatly appreciated! Thanks!


Why must the gravitational wave components be much less than unity?

We start with the metric tensor \begin{equation} g_{\mu\nu}(x) = \eta_{\mu\nu} + h_{\mu\nu}(x) \end{equation} in the linearised theory, or \begin{equation} g_{\mu\nu}(x) = \bar{g}_{\mu\nu}(x) + h_{\mu\nu}(x) \end{equation} in the more general case with a curved background. However, in both cases, we apparently need \begin{equation} \vert h_{\mu\nu}\vert \ll 1. \end{equation} This condition demands that the components of the perturbation are much less than 1, in the coordinate system where the components of the background metric tensor are of order 1. Why do we need this condition?


What will be the shape states of water in a centrifuge in 0G

Consider the concept of these videos http://www.youtube.com/watch?v=Zip9ft1PgV0&sns=em and http://www.youtube.com/watch?v=cUhgKFV5Ri4&sns=em but set in a zero gravity environment and in a fully encapsulated and high pressure container. My questions are as follows: What shapes will the water conform to at increasing speeds? Will the water conform to a parabola and then ultimately a hollow cylinder as it does on earth? Is it possible that without an opening to spit and by continuously adding more water to the centrifuge, that the water could ultimately fold in on itself and conform to various states of torus, by creating vortices within itself? [Edit] Also, taking into account 0G, will the water even conform to its bounding container in a manner in which it receives traction and can be spun?


electron pathways

if there are two identically equal resistance paths, and you have a large group of electrons, will half of the electrons travel lets say left and half right? because I've been thinking that maybe the electrons travel down lets say the left one till there is a large enough amount of electrons that causes the resistance to change by just a fraction, thus forcing the remainder to switch and go right, and so on till all the electrons have traveled.. is this correct?


Empty batteries magically resurrect after reinserting them

It happened to me quite often (most recently with my wireless keyboard) that a battery stopped to work and then, if I unplug it and then plug it back in it works again, not just for a couple of minutes but even for a day or two, then you can repeat the process but this time it will last less and so on until it finally dies. The same trick works for my tv control. I can't really understand why, can somebody help?


Is it possible for a black hole to form for an observer at spatial infinity?

To my knowledge if you calculate the coordinate time (time experienced by an observer at spatial infinity) it takes an infinite amount of time for an object to fall past the horizon of a Schwarzschild black hole. Doesn't this imply that it takes an infinite amount of coordinate time for a Schwarzschild Black Hole to form since the last bit of in-falling matter won't ever ross the horizon as observed by someone at spatial infinity? If so, is it possible for other types of black holes (Kerr etc.) to form in finite coordinate time?


Ice and liquid water interacting across a boundary

Imagine we have two thermodynamic systems, one a mass of ice and the other an equal mass of liquid water, with both at 273.16K. Each system is isolated, except that they can interact with each other across a boundary that permits the exchange of heat but not matter or work. What will the two systems look like at equilibrium? Somehow I want to automatically imagine that each system will be identical, a combination of liquid water, ice, and water vapor at 273.16K. But if this is true then the two systems were initially at the same temperature but not in thermodynamic equilibrium, an apparent violation of the zeroth law of thermodynamics.


Dingle vs. Bondi: Twin Paradox Debate on BBC radio?

Herbert Dingle and H. Bondi debated the twin paradox on BBC radio before 1971. Does anyone have a link to the audio of this debate? thanks


Infinite degeneracy

Is something special for a quantum system with infinite degeneracy like free particle levels? $E=\frac{\hbar^2 \vec{k}.\vec{k}}{2m}$ Edit: I mean what is physical (or mathematical) significance of infinite degeneracy?


Quantum Entanglement Versus Inflation in the Early Universe?

**Quantum entanglement** is one of the most fascinating and mysterious phenomena in nature. It needs no interactions, or any sort of exchange for it to take place. It is possible, not against any rules of physics as far as we know, that all matter that was created in the early universe was in an entangled state. **The question is:** Is it possible to explain the uniformity and isotropy of matter in the universe, by means of quantum entanglement in the early 'days' of universe? If that could be possible, would it mean that there would be no need for the inflationary model any more? If this problem has been researched in detail, any references posted will be appreciated.


Physics history book with some math

I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something. I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?). I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed. I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations.


What does "clockwise" mean, exactly?

I am in the middle of a discussion with a friend about the meaning of the term "clockwise". Wikipedia indicates that a clockwise rotation goes as **top-right-down-left**. However, my friend argues that "the clock is facing opposite to you. So if you rotate top-right-down-left facing the side the clock is facing then it is anti-clockwise." Our question, therefore, is > does "clockwise" mean **top-right-down-left** or **left-down-right-top**?


Invariant spacetime - distance - Circular Motion

I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?


A question about the relativity of time

> **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?


Paradox of the Relativistic Record Player

> **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion This is a question that I thought up a few years ago when I was taking mechanics. I asked the professor but didn't really get a straight answer. Imagine a record spinning at relativistic speeds. In the lab frame, the circumference of the record should decrease according to the Lorentz contraction. However the radius of the record should remain fixed since it is orthogonal to the direction of motion. So does the shape of the record appear distorted since the circumference is smaller but the radius is the same? What would it look like? I have a feeling the answer is obvious and/or well known since it would seem to be similar to the situation of relativistic particles traveling around a circular accelerator but I can't think of how it can be solved.


What are the relativistic effects of expanding spacetime?

This is a question I've been mulling over for a while and I'm hoping someone here can point me in the right direction. Sorry if it's a bit of a novice question. For the record, I don't fully know GR, but don't let that stop you from using it in the answer. Since the universe is expanding - that is, the spacetime metric is expanding by way of a near-exponentially increasing scale factor - we can say that the distance between any two non-bound objects is increasing over time. Herein lays my dilemma; if there were two objects separated by a large distance that had no relative velocities initially, after a long time, the effects of expansion would cause them to have large apparent velocities away from each other. Given that there hasn't been any acceleration to cause these velocities, are there still relativistic effects in play? That is, is there time dilation between the two frames? Furthermore, given long enough time, the rate of increasing distance between the two objects could place them outside of their visible horizon (ie they are travelling away from each other at superluminal velocities). Since there was still no acceleration to achieve this feat, what can one say about the relativistic effects in this case? At first I thought this was an easy question. I thought of course there would be relativistic effects and when the objects go superluminal, the visible horizon is there to ensure there can never be causal contact and thus preserve physics. But then I thought what if spacetime stopped expanding abruptly (seems crazy but as far as I know, nothing makes this completely impossible)? Since there was no initial relative velocities, wouldn't the two objects return to being in the same inertial frame? And seeing as none of them experienced any sort of acceleration, how then could we describe their two final states? By which I mean, if we were to assume there were relativistic effects during transit, how would we overcome such simple paradoxes like the twin paradox, or other relevant ones? At this point, I'm stumped. I even attended a lecture by Miguel Alcubierre since he would have had to consider these types of effects in his design... No help. Equations are great to illustrate a point, but I'm really going to need a conceptual answer as well to fully understand this.


Einstein's mass-energy relation

Suppose we have 1 kg of wood and 1 kg of uranium and if we need to find out how much energy would each of the substance give, we'd have to use Einstein's mass-energy relation as follows: In the case of wood, $E_{wood} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ In case of Uranium, $E_{uranium} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ _Question:_ According to the equation, both give the same amount of energy. But in reality, 1 kg of uranium will give a lot of energy when compared to 1 kg of wood. What have I missed? Any explanation regarding the $E=mc^2$ _problem_ would be helpful...


positive energies

if the potential is bounded below $ V(x)=V(-x) \ge a $ for some real number a and we can be sure that as $ x\rightarrow \infty $ we know that $ V(x) \ge 0 $ then does it mean that the energies for our one dimensional system will be positive ?? my idea WKB quntization $ N(E)= 2\int_{0}^{a} \sqrt {E_{n}-V(x)} $ here $ V(a)=E $ is a turnign point so if the potential is positive for big 'x' then the energies should be also positive otherwise the epxresion inside the integral would be complex am i right ?


Is the superposition principle universal?

In David J. Griffiths' _Introduction to Electrodynamics_ , he claims that the superposition principle is not obvious but has always been found to be consistent with the experiments. So I was wondering have we found some physics quantities which do not follow superposition principle? If we have not till now why can't we generalize and make it into a law? More specifically: Griffiths was talking about electromagnetic force. My question is about the existence of something like mass or charge and which doesn't follow this superposition principle.


Dominos vs. Conservation of Energy

In this video a single flick of a finger tips 116000 dominos. Domino video I understand the work that needs to be done to move 116000 pieces (at least 100 kilos) of plastic is greater then that expended by flick of a finger. How does the energy conservation law apply here?


how must i understand this 2-loop integral?

let be the 2-loop integral... $$ \int d^{d}l\int d^{d}k \frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \int d^{d}l\frac{1}{(k+l)^{2}}=G(k,d)$$ this is the regularizñation of the subdivergence assumed to exist by dimensional rgularizaton then the final step would be $$ \int d^{d}k \frac{G(k,d)}{k^{4}(k+p)^{2}}$$ where $ G(k,d) $ must be taken in finte part to eliminate the pole when $ d=4-\epsilon $


Integration over $S^2$ in electrostatics

I'm studying for a test in electrostatics and I'm always failing on putting up the correct integrals. In one problem I have the surface of a sphere with radius $a$ and an opening angle of $2\theta$. There is a total charge $Q$ evenly distributed on the surface. I want to calculate the potential. In order to calculate the potential I first have to calculate the surface charge distribution, $\sigma(r)$, and this is where I always fail. I've done like this: $Q=\sigma A=\sigma 4\pi r^2$ Then I tried to convert this into spherical coordinates by changing $r$ into $r=a \sin\theta \cos\phi +a \sin\theta \sin\phi +a \cos\theta$ The correct integral should be $Q=\sigma a^2\int_0^\theta \sin \theta d\theta \int_0^{2\pi}\\!d\phi$ Can someone explain how to put up integrals like these?