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| Physics Question - ma = mg |
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| Nominal Animal:
You can measure mass (as opposed to weight) using angular momentum and torsion springs. By 1785, those were already so sensitive that Charles-Augustin de Coulomb could use one to measure electrostatic forces, and came up with Coulomb's law. By 1798, Henry Cavendish used one to directly measure the gravitational attraction between two masses in a laboratory experiment, the famous Cavendish experiment, although geologist John Michell designed the experiment (Coulomb following his plans), sometime before 1783, but alas, kicked the bucket too early. (Michell was 69 when he died, though.) With commercially available load cells, a turbo pump for pumping a good vacuum, some glass jars and such, you can definitely measure for example the gravitational attraction between say two marbles, in a freshman physics experiment. The most expensive thing is getting precise mechanics nowadays. A couple of hundred years ago there were watchmakers and others happy to help produce interesting scientific apparatuses, but now, you're better off with old mechanisms and new sensors! --- Quote from: TimFox on June 28, 2021, 03:30:39 pm ---1. Please stop calling g a physical constant, in the sense that G (universal gravitational constant) is a constant. --- End quote --- Hey, I blame the language! We need a word describing a variable that is not being varied in this particular context. For differential calculus, we use "derivative" and "partial derivative"; so would "partial constant" work? Yesterday, I spent a couple of hours admiring the concept and word geodesic. Let me waffle on a bit once again, but this time, as a hopefully entertaining but informative musing on how precise, exact terms make life interesting, and mushy overloaded ones a misery. (I am thinking of making a T-shirt with "Why don't you just use pastel tones to reduce gravity?, too.) As a background, geodesic is when you extend the concept of "straight line distance between two points" to surfaces; and from its name, one of the most used one is the "straight line distance" between two points on the surface of a sphere or geoid, like the Earth. On a perfect sphere, that distance is the shorter arc of the great circle – a circle passing through those two points, with its center at the center of the sphere. What happens, if you ask your Californian friend what the distance between LA and Las Vegas is? My bet is that the most common answer is something like "Oh, about four hours, at this time of day." Bitch, I asked you distance, and you gave a time interval. What sort of mindfuck is this? (If you happen to have the cultural context, you'd know that because primary transport in that region is via personal automobiles, and the physical distance is less relevant than the actual travel speeds achievable, it makes more sense to describe distances using the typical time taken to drive that distance, than anything else. Just like we include the completely unrelated inertial term in our "average acceleration on Earth due to gravity", because we don't have any better place to put it, and keeping it there takes care of it very nicely, thank you.) In a different cultural context, that friend might ask for clarification, say "As the crow flies, or?". Bitch, I ain't an ornithologist or an ornithopter; how the hell would I know how corvids fly anyway? They may do so ass first for all I care. Now, let's get to "geodesic". Let's say you're driving from LA to Las Vegas with your friends' kid on the back seat, thinking about normal kid stuff, like how many hydrogen bombs in the 50 to 100 kiloton range would they need to carve a canal from Los Angeles to Las Vegas. The kid asks, "Do you know what the geodesic between Los Angeles and Las Vegas is?" Why, you just looked that up a few days ago for this very thread, so you answer "About 368 km, but there is also an elevation difference of about 525 meters." SEE? No confusion. Straight, unambiguous answer unrelated to ornithology, free of oddities like trying to measure distances using time units, and so on. The kid happily takes the added cratering depth into account, you both smile, and have a nice car drive. Everybody wins. If you try to be physically or geometrically correct but do not know the term, the mistakes and confusion one gets mired in are endless. For example, let's say you ask "What is the distance between Los Angeles and Las Vegas?", I ask for clarification, for which you say "in a direct line, but ignoring the curvature of Earth and the difference in elevation". My answer, not to be snarky but maximum helpfulness, could likely be "Either 368 km or 39707 km, depending on which way you measure it." I know, I'd get bitch-slapped for that (if I wasn't driving, and I don't). But I lack the context necessary to give the "straight answer"; I either have to make assumptions (which are often wrong), keep asking for clarifications, or give an answer that outlines the possible set so the asker can determine what the hell they were trying to find out in the first place. And no, I don't really like "partial constant". But I'm used to it, because I use const every day, and it does not mean that something is constant, just that in that particular context I (or the C code I write that uses that) promises to the compiler to not try and modify that value. And volatile does not mean "flames or vapours are imminent, get your fire extinguisher ready!", it means "hey compiler, that value may change at any point, so make no assumptions about it okay?". And I have to talk about stuff like "immutable string literals" to get new programmers to understand you really don't get to modify them; that if you do, you broke a promise, and the kernel gives you a segmentation fault in confusion. But I perfectly understand the frustration with mushy and misused concepts and terminology. If we'd have a better set, one with say posters or web pages intended for different levels of complexity/understanding, that I could point others to and say "using these" ..., I would do exactly that. And I'd be darn happy; I'd buy it as a dozen posters and lovingly put them on my walls and give out as gifts; they'd make that much a real-world measurable difference. But scientists and their insistence of attaching peoples surnames to things yielding undescriptive (or in the case of original discoverer being someone else, sheer misleading history-twisting self-aggrandizement) words one has to learn by rote by the dozen and use "correctly" (read differently) in every single little domain, and science becomes a Black Magic only those versed in the Old Powerful Names of Power and History and Power have the sheer single-mindedness to delve into. I mean, I can say something like "My mother in law was being a sheer Cheney this week visiting us. Thank Dog her Trumpy sister could not come! Dealing with those two is like trying to watch The Kardashians and Dr. Phil at the same time. I don't know... I think I might be getting a bit Biden in my old age or something; I've been acting like a complete Dubya." and people would just get it. But use "mass" and "constant", and people start getting more confused and even angry... No, this is not sane or right. Yet, it is the world we have. So, what is one to do? Even writing about this like I've decided to do in this thread gets me flak about wasting others time, and gets me added to new ignore lists. I don't know. I just pick a topic at a time, think hard about it, and especially all the ways I know and think of it can be misunderstood, and if I decide it seems possible/worthwhile/useful, construct something that hopefully helps others wade through the word salad but with new mental tools and understanding at their disposal. |
| TimFox:
In technical English, a "parameter" is either a constant that can be varied, or a variable whose value is held constant during the calculation. The popular media keep confusing "parameter" with "perimeter", the boundary around an area. "Geodesic" is a very interesting concept. In geometry, it is the distance between two points on the manifold in question with the shortest length. In three-dimensional Euclidean geometry, it is merely the straight line between the two points. On the surface of a sphere, it is the great circle (arc with center at the center of the sphere) that connects the two points. Traditional ocean navigation plots the great circle on a globe or appropriate projection flat map, and then transcribes the points onto a piecewise-linear route on a Mercator projection, where any line segment is a line of constant compass heading, appropriate to the helmsman. In a classic textbook on general relativity, one example is an ant crawling on the surface of an apple (not really spherical), where the claim is made that sensing only the spatial derivatives along its short body, the ant will follow a geodesic path. That is a good parable, but I don't know if anyone has done the experiment with a real ant, or if it is literally applicable, since I don't know if the ant has a definite endpoint in its tiny mind (analogous to LA and LV). The traditional measurement of mass is to use a gravitational balance (such as held up by Lady Justice at the courthouse) and compare the mass in question to a standard mass, defined by the King. As a practical matter, a "steelyard" balance (q.v.) uses leverage to compare the mass in question to one or more standards. Technically, this equates the gravitational masses on both sides of the balance, but standard theory and Roland, Baron von Eötvös tell us that gravitational mass and inertial mass are equivalent, hence can be set equal. |
| Nominal Animal:
--- Quote from: TimFox on June 28, 2021, 04:44:48 pm ---In technical English, a "parameter" is either a constant that can be varied, or a variable whose value is held constant during the calculation. The popular media keep confusing "parameter" with "perimeter", the boundary around an area. --- End quote --- That works. Before I encountered the term eccentric angle, I consistently called \$\theta\$ "the angular parameter" in $$\left\lbrace\begin{aligned} x &= a \cos \theta \\ y &= b \sin\theta \end{aligned}\right. \quad \iff \quad \left\lbrace\begin{aligned} \varphi &= \arctan\left(\frac{b}{a}\tan\theta\right) \\ r &= \frac{ab}{\sqrt{(b \cos\varphi)^2 + (a\sin\varphi)^2}} \\ \end{aligned}\right. \quad \iff \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where the leftmost set gives the Cartesian coordinates for the points on the axis-aligned ellipse as a function of the angular parameter, or eccentric angle \$\theta\$; the middle one is the same in polar coordinates where \$\varphi\$ is the polar angle and \$r\$ the distance to the ellipse from origin in that direction; and the right side being the familiar implicit form in Cartesian coordinates. (Oh wait, it isn't an implicit form, because I left the one on the right side. Whatchamacall this form then? Pre-implicit form? Explicit form without any nekkid stuff?) It is so easy to accidentally forget that just because you have a \$\theta\$ or \$\sin\theta\$ or \$\cos\theta\$ term, does not mean that \$\theta\$ necessarily refers to a particular angle. Eccentric angle is illustrative, because you immediately intuitively know that "Ha! I bet it means it is not exactly the polar angle!". I like "angular parameter" too, because for e.g. Lissajous curves (\$x = A \cos(a \theta + a_0)\$, \$y = B \sin(b \theta)\$) \$\theta\$ really is just a parameter you need to vary to sweep the entire curve. --- Quote from: TimFox on June 28, 2021, 04:44:48 pm ---"Geodesic" is a very interesting concept. In geometry, it is the distance between two points on the manifold in question with the shortest length. --- End quote --- Yes; I didn't want to write that myself, because the concept of manifold would have to be explored first. Case in point: A couple of years ago, I helped a CS or SE major with a raytracer visualizing non-Euclidean space. It was piecewise Euclidean; essentially modeled using simplices (tetrahedra) whose faces are discontinuities yielding an overall non-Euclidean geometry. If you've seen the Cube movies, then you probably know the idea: each simplex is a "room", with arbitrary other "rooms" sharing walls in arbitrary orientations. Aside from the discontinuity (a coordinate transform we used barycentric coordinates for, yielding a pretty useful geodesic) at the faces of each simplex, the "rooms" themselves were Euclidean. It modeled pretty much perfectly the case where you're looking at the back of your own head, except with just simple geometric primitives. I still don't know whether you can consider that a manifold, because of the discontinuities at the tetrahedral mesh faces. "Piecewise Euclidean" suited me better; I don't know what they used in their report/paper/whatever. (In Cube, the walls are always perpendicular, so there are no discontinuities of the sort the tetrahedral mesh has; that I do believe is a proper manifold.) I also tried to avoid dealing with terms like "embedding", because defined the way it was, it is darned near impossible to give a nontrivial simulated model any specific dimensionality. If you take a cube, and "glue" its faces together permuting their orientation, what is its dimensionality? I think there might even be pathological cases (choices of mating face pairs and their relative orientations) where you'd need infinite number of dimensions to embed that into a higher-dimensional Euclidean space. If you consider how the tetrahedral meshes can be constructed, it is easier to define the meshed topology in terms of its internal consistency and structure, and completely ignore any efforts to see say where the tetrahedral lattice nodes would be if you wanted to assign them Euclidean higher-dimensional exact coordinates. I preferred to just show the tools, and leave it to mathematicians to define and help with the correct terminology. But I know for a fact that the darned thing worked, because I've seen the rendered images and fully understand how the math works and why. But put me among a gaggle of mathematicians, and they'll laugh me out of the room as I fumble trying to find the correct terms Words of Wisdom and Power and I Even Have This Hat You See to describe what math it is based on. If you think about that, it is kinda sad. Reminds me of the joke about a man who wanted to see the Circus Director. The director was busy, so snapped at the man to tell what he could do, and to be quick about it. The man proudly exclaimed he could emulate any bird! The director laughed at the man, saying whistlers and ventriloquists are a dime a dozen, and to get out; he was just wasting the Director's valuable time. Utterly dejected, the man sighed and flew away. (The reason I helped with that, is that I once had a dream test-playing a game I purportedly designed, somewhere between Marathon and Doom, but with exactly the sort of non-Euclidean geometry as described above. Basically, any enclosed tubular region of the game world, the world isn't described in absolute coordinates at all, but as piecewise Euclidean spaces glued together using minimal shear discontinuities. A particular example is of a slightly sloping tunnel/stairwell upwards, where you walk "around the corner" (in a circular arc) some 700-800 degrees, without returning to your original location at all. Not just "bigger on the inside", but no turn or bend ever goes the way you think it does. This is trivial to implement with suitably chosen data structures in almost any 3D graphics/physics engine, but at least in my dream the "uneasiness" of going one way, even coming back and going again the same way, but arriving at a completely different place you think, and not being able to find a discontinuity where the "error" or "trick" occurs (there really is none, they're within rounding error; it's just not really Euclidean, only piecewise Euclidean), was fun-kyy. I don't even like that sort of games, although I've played a little at parties for the social aspect; I have even less interest in designing or creating such games. I see Odd dreams, I guess.) |
| RJSV:
Yes, I agree, having the context of using 'g' in a varied environment, not just at Earth surface, and simplifying to be a smooth ball, uniform density, etc. So then "...local acceleration conditions..." for lack of a better word. Sounding better and better. |
| TimFox:
I think the better words are "g is the acceleration of gravity in a region of uniform gravitational field". |
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