darklight03/StackMIAsub huggingface dataset:
Consider a circular horizontal plane (like a round tabletop) rotating around its center. Consider a body A resting on this tabletop. Since the tabletop is rotating around its center, the body is moving along with it in a circular motion around the center of the circle. I understand that in circular motion, there's always a centripetal force, pushing the object towards the center of the circle. And in this case, this force is the static friction force, operating on the radial axis towards the center. I have two questions regarding this: I was under the impression that a static friction force, or any friction force, always operates in the opposite direction of some other force which operates on the same body. Is this correct? If so, what is the additional force (opposite to the friction force) that I'm missing in the above scenario? Is the static friction force operating only on the radial axis towards the center? Or is there also static friction along another axis?
I'm wearing glasses with a sunglass clip-on. This means I have my regular glasses and, on top of them, I have a second pair of lenses that work as sunglasses and attach to my regular glasses using tiny magnets. I'm also in a car, and I'm looking through a window layered with Insulfilm. Curiously, when I look through the Insulfilm with the sunglasses on, I see a rainbow pattern. If I keep my glasses on, but remove the sunglass clip-on, the rainbow is gone. If I look through a window without Insulfilm, the rainbow also seems to be gone. If I look through the Insulfilm with the sunglasses clip-on, but without my regular glasses, the rainbow is there. I believe my sunglasses are polarized, and the rainbow fades away when I tilt my head a little. Why exactly is this phenomenon happening? I guess it has something to do with polarization due to the rainbow fading when I tilt my head, but I can't figure out why it is a rainbow and why I can't see it without the sunglasses.
This popular question got me wondering if there is a simpler way to reduce radon in homes. Our house has a fairly standard radon mitigation system. It has a lot of parts. There is a hole cut in the basement floor. There is a thick pipe which goes up through all the floors with lots of bends and turns, like the old Microsoft "pipes" screensaver. There is a powerful, always-running fan in the attic pulling the air up and out through a roof vent. There are also other components, such as a manometer and other sensors. It all has to be serviced by technicians. Like any system with a lot of components and a lot of puncture points through surfaces, there are many points of potential failure. Why not have a standalone unit in the basement with a fan that blows the air through an electrostatically-charged baffle of gills or fins or whatever, collecting the radon until it decays? At least then it won't end up in the lungs, where it causes the damage. We could put lead shielding around the unit, if necessary.
I found a few questions that are similar to mine, but I do not think that either of them answers exactly what I want: Is there a Birkhoff-like theorem for stationary axisymmetric metrics? or Gravitational Collapse: Kerr solution is a vacuum solution but not for any rotating body? My question is this: When we look at simple models of gravitational collapse for spherically symmetric matter distribution (such as Oppenheimer-Snyder) we always exploit the fact that the solution to the Einstein field equations just outside any spherically symmetric collapsing matter is Schwarzschild and that this is the only possible solution. This is thanks to Birkhoff's theorem. An analogous theorem exists also for charged matter. Unfortunately, no such theorem exists for rotating matter, as the spherical symmetry is broken. This means that we can't approach the gravitational collapse of rotating matter the same way we would for spherically symmetric charged or uncharged matter collapse. So how do theorists actually study rotating black holes? Do we need some sort of computational or numerical methods?
My friend was sitting in a wheelchair, with a cup of coffee in one hand. That made me wonder, how would the wheelchair move if he turned only one rear wheel? He tried it, using his right hand to turn the right rear wheel. The wheelchair starting moving in a circle. No surprise. But I was interested in the radius of the circle. The circular track of the right rear wheel on the ground, had a radius that was slightly larger than the distance between the rear wheels (i.e. the distance between the two points of contact between the rear wheels and the ground). What determines the radius of the circular track of the left rear wheel (the unturned rear wheel)? Possibilities that come to my mind are: Location of all four wheels The size and shape of the contact patch between the wheels and the ground The coefficient of friction between the wheels and the ground, and the weight of the system The diameter of the wheels Angular speed of the right rear wheel How the person is sitting within the chair (centred, or leaning to the left, leaning to the right) (Assume the wheels are in parallel planes.)
I apologize if this question has been asked before, but I haven't been able to find a post which covers this exact question. Let's say that we have a driving car which is moving forward. At some point, one of the rotating tires hit a stone on the road. Which way does the stone move, in relationship with the angle in which the tire hits the stone? Can the stone ever be thrown backward onto another car? Does the general direction (forward, backward, sideways) depend furthermore on the shape of the stone? Whether it is for example perfectly spherical or asymmetric? I am kind of struggling how I would answer such a basic question, since there seems to be many factors at play here. Can you even answer this question without fully knowing all the nuances and details?
Suppose we have an isolated n-type semiconductor having varying concentrations of carriers over x. After some finite time, the concentration becomes uniform for a given constant temperature. We always say that this is because of diffusion which allows the movement of charge carriers to move from higher to lower concentrations. But shouldn't this diffusion occur because of the repulsive forces between electrons? why do we say it's because of the random motion? (the random motion allowing for the carriers to rearrange should be because of the electric field produced by them due to the gradient of charges). If we see some excess charges added to an isolated conductor, then the electric field produced by these excess charges allows the charges in the conductor to redistribute to settle on the outer surface where the net force experienced by the excess charges is zero. Shouldn't the same principle apply to semiconductors? What exactly is happening inside? I hope to get a clarified answer. Diagrams will be highly appreciated.
I'm trying to deepen my understanding of Von Neumann entropy (out of interest in the quantum Bekenstein bound) by learning more about the reduced density matrix (aka local density matrix), which is given by the modular Hamiltonians (aka entanglement Hamiltonian). As I understand it, modular Hamiltonians are only known to be local (i.e. an integral of a local operator) under three specific situations. When the state is the vacuum state in a half-plane (the only case where the modular Hamiltonian is given by the traditional Hamiltonian) When the state is the vacuum state and is limited to a ball and the field theory is a CFT When the state is the vacuum state and is limited to an arbitrary piece of a null plane I could not, however, find any explanation for how or why these are the three known cases where it is local (and additionally why only the first one is given by the traditional Hamiltonian). I'd appreciate any insight you could provide on this matter.
Adjectives and adverbs can be formed by adding "y" or "ly" to e.g. a noun, such as: heart -> hearty, heartily However, sometimes these words are not in use, or make no logical sense, such as: chair -> chairy, chairly Although you could come up with a situation where you'd use "chairy", e.g. "this block of wood looks quite chairy", you'd probably be more likely to use "chairish" or "chair-like". And if you read all the books in the world, you'd probably never come across the word "chairly" or "chairily". What is the status of words like these? Do they exist because they are technically correct and their meaning is obvious? Do they not exist until someone (notable) uses them in a (notable) text? Or, to make the question more specific: if I were to create a spell checker, should I allow or disallow these words?
I am teaching students at a high school about bending of a beam (the beam is clamped at one end and then an applied force acts downwards close to the end of the beam to bend it). I thought that it did not require further explanation that the deflection of the beam due to the force at the end is inversely proportional to the stiffness of the beam, proportional to the size of the applied force, and that it is dependent on the distance from the fixed point where the beam is clamped. However, I have been asked to provide justification for these statements and not to appeal to physical intuition which is not obvious to students who are not seasoned physicists. What type of justification would be enough to use these statements at an elementary level? How could I explain that these statements are true?
Compressive strength is how much inward force a given area of material can withstand before failure. The force tries to compress the atoms closer together. Mohs hardness is the difficulty of a material to be scratched. Abrasion resistance. However, when an object scratches another, there first is a point of contact where the harder material is forced into the surface of the other, and then lateral forces shear off a grove from the softer material. But at this point of contact, the forces are compressive, trying to force the atoms together, until one of the materials give way. As such, I would expect materials with a high Mohs hardness to have a high compressive strength and vice versa. This holds for ceramics such as silicon carbide, aluminum oxide, and others which have both a high compressive strength and a high Mohs hardness. Does this trend hold for most/all materials and to be expected for the above reasons? Are there any counterexamples? If so, what could explain a high Mohs hardness and low compressive strength or vice versa?
I really don't like the phrase "excited for" which seems to have become very common in recent years, as in "I'm excited for the weekend...". My sarcastic reply would be "I don't think the weekend cares or appreciates you being excited on its behalf". Anyway, I'm aware I have to accept that language changes, even though here I think it's just bad grammar. The question is ... are there any studies looking at how (the wrong use of) this phrase came about? I have a theory based on absolutely nothing, that perhaps it came from Spanish where 'por' can be used in this sense, e.g. Google translates "excited about this weekend" to "emocionad(o/a) por este fin de semana". Is it possible that this usage came via speakers of Spanish, perhaps in the US?
"Longingly" is the adverb form of "longing". Depending on where you look, "longing" is described as an adjective, a verb, or a noun: Oxford defines it as either a noun or an adjective. Cambridge, Collins, and Merriam-Webster seem to define it exclusively as a noun. Wiktionary says it's either a noun or the present participle verb form of "long". Despite the disparate definitions, I can see justification for all three forms. This raises a question: which distinct form of "longing" is "longingly" derived from? Or is the adverb formed indistinctly? I know that adverbs ending -ly are typically formed from adjectives, but English is full of exceptions. I checked a few etymology sources but they either only had information on "longing" itself, or only described longingly as being formed from longing as a general word.
My question is closely related to the answer of this question: Why is general relativity background independent and electromagnetism is background dependent? General Relativity is often stated to be "background independent", because it calculates how spacetime is curved. That's in contrast to other theories like classical electrodynamics which act on the manifold without interacting with it. I understand that GR interacts with spacetime (and that that's a great advancement in comparison to the theories before) - however, I do not understand, why it's called "background independent". Matter curves spacetime. That means to me that there has to be a spacetime first which can be curved by matter. If the universe is empty (in GR), Minkowski spacetime is still there. If spacetime were produced by matter like the electromagnetic field is produced by its sources I would understand the term "background independence". But to me it's only interacting with the background, not background independent. (Like an artist who is forming any possible object out of clay is not independent of clay... Just acting on it in every possible way) How shall I understand the term "background independence"? Is the term not precise? Did I get something wrong?
Quite naturally, the observable Universe is the only bit of the Universe we can extract information from, as light from farther away has not reached us yet, and there are zones from which we'll never even be able to extract information, as they are causally disconnected from us due to their high speed. Therefore, how do we know the cosmological principle holds at larger scales if we can't observe them? It just seems to me a quite bold assumption to make that the entire Universe is homogeneous and isotropic. Perhaps it presents irregularities which we'll never know of, and which would be necessary for us to not mess up with our cosmological models. Frankly, I don't quite see the point of building cosmological models if we only know what seems to be a small portion of the Universe, just like it makes no sense to try to see the picture of a jigsaw puzzle just by examining one random piece of it.
Recently I came across Normal matrices and their properties, one of which states that their eigenvectors are the same as their adjoint and are orthogonal. I've gone across some proofs and I understand it but when I tried to prove the same using the inner product, It somehow states that the above is true for any arbitrary matrix. Can someone possibly help me point out where I'm going wrong? Let's say A is a matrix and B is its adjoint. If x is an eigenvector of A with eigenvalue k, then, < x | A | x > = < x | kx > = k< x | x > also < x | A | x > = < Bx | x > hence < Bx | x > = k< x | x > = < kx | x > So, x is also an eigenvector of adjoint of A.
In the English language there are many words where the letter "a" is pronounced as a short (continental European) "e". Or at least very close to it. However dictionaries point out that in these cases the correct pronunciation is more an "ae" sound. Several websites on English pronunciation take the same view. So I decided to test this assumption. Over a period of weeks, whenever I watched an English or American television program, I payed attention how these "a"-words where actually pronounced by native English speakers, and made notes of my observations. I found out that in fact there are many words that have the pure "e" sound. A few examples: back, cash, cat, fact, relax, track, crash. So what is going on here? Are this words pronounced incorrectly, or are the dictionaries and websites promoting viewpoints about pronunciation that are outdated? Or has it to do with regional variations (dialects)?
In Photoshop (and many other computer graphics applications) there are are various ways that images/layers can be composited with their underlying background. These are sometimes called blending modes. One blending mode in particular is multiply, which has the convenient feature that it eliminates white backgrounds from black line illustrations while properly dealing with anti-aliased edges (see the example below). Is there a way to do this in LaTeX? I have a large number of images with a white background that need including in two documents with different coloured backgrounds. I'm looking for a quick way to remove the white background from the images without having to manually add an alpha mask to every image in Photoshop. Normal blending mode: Multiply blending mode: Close up of anti-aliased edge, properly feathered:
I found this MC question that asks about a soft iron bar entering a solenoid: At first, my answer was (B) because the iron bar should increase the magnetic field induced by coil M and, according to Lenz's law, this magnetic field should oppose the original current, making the lamp dimmer. Also, the magnetic field created by M should then induce a current in N in the opposite direction to the current in M (also Lenz's law), so since the current in M is right to left, then it should be left to right in N. I wasn't too confident in my answer though. Then, I noticed that they had drawn a DC power source in M. This would indicate no changing magnetic field, thus no current induced into either M or N. This would mean that the lamp doesn't change brightness and there is no direction of current in N. Right? Is this just a mistake in the question, or am I missing something? Thanks to anyone who can help.
Given a path, how do the magnitude of the velocity and acceleration vector along the path correlate? I am confused due to the fact that the acceleration is the change of velocity over time and in general if you plot the velocity and acceleration over time, you can see a correlation (e.g. constant velocity means zero acceleration). So I expected the same relationship for the magnitude of the velocity and acceleration vector. Moreover, I got confused because the magnitude of a vector is always positive and in the general plots of the velocity and acceleration over time it is not the case (e.g. linear decrease in velocity means constant negative acceleration). Therefore, can someone clarify if there is a clear relationship between the magnitudes of the velocity and acceleration vector and how this is defined? And if not, is there a relationship between the velocity and acceleration vector, so you can verify if your velocity and acceleration vector are correctly calculated?
Relativistically, the electric field of a moving charge is not purely radially directed, but is instead concentrated perpendicular to the line of motion. So, a current loop consisting of electrons as charge carriers should generate a "charge separation" effect, where the negative electric field is concentrated perpendicular to the loop, and is not perfectly cancelled by the stationary positive ions in the (electrically neutral) conductor. Conversely, in the plane of the loop, the positive electric field should dominate. This "charge separation" effect is separate from the magnetic field produced by the current loop, as it should have the effect of accelerating stationary charges in the surrounding space, which are unaffected by a magnetic field, because their velocity is zero. I have never heard of such an effect, which should be small but measurable, hence I am asking the question.
A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms in the Lagrangian using a 'bare' gauge field, but one can use the associated field strength or act on a gauge field with a covariant derivative. On the other hand, if one writes a term using a 'bare' gauge field (for example, a combination of the gauge field and the associated field strength tensor), although it will not be gauge invariant, will this term still be invariant under diffeomorphisms of the spacetime on which the field theory is defined? Since diffeomorphism invariance is sometimes taken informally to be ''background independence of the field theory'', I guess that this term should be invariant under diffeomorphisms? Or does diffeomorphism invariance fail because it is a specific example of a gauge symmetry?
This may be an amateur question, however here goes. I understand the double slit experiment creates a situation where detecting which slit the particle/wave goes through causes a collapse of the wave function and for the particle to act like a particle and not cause an interference pattern. What I want to know is this: If say for example, the detector emits a signal to a speaker if it detects the particle and the speaker emits an audible sound, my understanding is that the wave function would collapse - we measured the outcome. However, what if we left the detector doing its thing, but turned off the speaker so that we didn't know? Would this still be considered "measurement" or does measurement require recording/knowledge (future included) of the outcome?
I'm considering to choose between "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, and Yurii Nesterov's "Lectures on Convex Optimization" to supplement for my university course. I have had Calculus and Linear Algebra, my course covers constrained and unconstrained convex optimization, linear programming and constrain programming so i want to know which one of these two books will be good for me. The lecture i'm having mostly just focus on the implementation of the method but not the mathematical proof behind it so i want to learn more about the math behind optimization algorithms. I've read a few section from each book and personally i think Yurii Nesterov's book is more rigorous and easy to understand, but i still want to ask because i don't have enough time to study both and i want a strong foundation on this subject for my AI major. P/S: I'm also really enjoy rigorous books like Apostol's Calculus I,II so i want to know which of the two books above have similar style.
I often see "functional" used as an adjective in situations where I think that "function" would actually be the better choice. Specifically, I am referring to translations of German compound nouns. Here is an example: Funktionsarchitektur. This is the architecture of the function/functions. To be specific, we are talking about a technical function in a machine or vehicle. This is the architecture of the devices, control units, etc. which perform the specific function. ABS would be one example. In this case I would use "function architecture", translating it as a compound noun in English as well. For me, "functional architecture" is an architecture which functions and not an architecture of/for functions. I see a lot of cases in which the adjective "functional" is used instead of expanding the compound noun. Any thoughts on this?
while working on problems related to fluid mechanics i came across a problem in which we were asked to find the kinetic energy of a sphere which is under pure rolling and has non viscous fluid filled inside it. here you can see the figure : so my initial thoughts were to write the moment of inertia of the whole body about center of mass and then proceed as normal. But to my surprise this was not correct so i thought more about this and made a conclusion that if the fluid is non viscous then maybe we don't have to consider its rolling motion. This assumption made me do this question but i am not sure whether it's correct or not. I want to know what you guys think about this assumption.