General > General Technical Chat
Physics Question - ma = mg
Nominal Animal:
I forget to use the correct (and useful) subscripts too, sorry.
On this forum, you can write say abp, using [sub]..[/sub] and [sub]..[/sub]; these will show the same way in preview as they will in final posts. abp is written as a[sub]b[/sub][sup]b[/sup] in the editor. To get the teletype look, I use [tt]..[/tt]. To stop a bracketed term to be interpreted as markup, you can insert a [tt][/tt] or a [i][/i] (which literally means nothing in teletype, or nothing in italics, respectively) in a suitable place (after each [ the forum software considers the beginning of markup, changing each [foo into [[tt][/tt]foo), to break up a markup-looking expression without visual side effects. It's like adding zero-width spaces: the sequence the server sees is different, but the end result stays the same visually.
This forum also has MathJax support. For inline MathJax, for example \$E = m c^2\$, you bracket it with \$ on both sides. (That snippet being written as \$ E = m c^2 \$ in the editor.) For block expressions, use $$, for example $$ E = m c^2 \tag{1}\label{NA1} $$, to get $$ E = m c^2 \tag{1}\label{NA1} $$where the added stuff lets you create links to it using \$\eqref{NA1}\$ (rendering to \$\eqref{NA1}\$). The equation label namespace is usually the entire HTML page, so I recommend using labels with a prefix unlikely to be used in other posts. That way the visible tags can be the same, but the labels keep them separated.
Unfortunately, the forum code lacks the trigger to apply the MathJax translation on preview, so what you see is the bracketed MathJax expressions instead of the rendered results. At minimum, it'd only need a simple JavaScript call to a library function tacked on to the AJAX completion of the content preview load; for normal posts and thread view, a related call (that does not render source markup, but the intermediate form the forum software munges the source markup into) is automatically triggered by page load completion. (Implemented this way, it would not exactly match the way the MathJax module in the forum software does it, but the visual result would be the same for the vast majority of users. The difference would be that preview would lack the MathML translation step done on the server end. For full support, the MathJax server-side munging would need to be added to the preview server-side processing, plus the rendering trigger added to the AJAX completion. Here, MathJax is so rare I guess Dave and cohorts don't see it worth the effort. I'm not familiar enough with SMF innards to have an opinion, but I do want to point out that any such modification is at risk for adding new bugs, so it isn't just laziness; there is always a risk in modifying forum software.)
bostonman:
These last few posts about how to write 'g' are interesting.
First it shows that smart people need to be precise about how they write, speak, and dislike others who are wrong. It also connects with "discussions" I have with others.
I've noticed keeping my mouth shut is very difficult when I hear in correct statements, or feel the need to critique others.
No matter what value we call 'g', ma can still equal mg. If I accelerated my car from 0 to 9.8m in one-second, then the Force is the mass times 9.8m/s^2 thus making ma = mg.
Nominal Animal:
On the magnitude and intuition of forces.
Consider Prince Rupert's drops. Toughened teardrop-shaped glass beads, created by dropping molten glass into water.
Because the glass shrinks as it cools, energy gets stored in internal stresses – internal forces pushing each atom of the glass (mostly silicon and oxygen; silica being SiO2 and glass is silica plus impurities) – are immense. On the bulbous end, the pressure due to this is on the order of hundreds of megapascals; 700 MPa according to a 2016 paper published in Applied Physics Letters. Again, that force is not exerted by anything external. There is potential energy stored in the structure, yes, but no dynamic forces involved; only the same static force that say the ground exerts on the soles of your foot when you're standing put and not sinking into the ground, just pushing the atoms in the drop against each other and nothing external.
That is, if you put a Prince Rupert's drop in a padded box, it won't slowly release its structural tension, no more than standing on a granite floor makes it slowly mellow into marshmallow. Nor will it explode if you put it in vacuum. It'll just stay as it is, barring stuff like annealing due to temperature changes, structural changes due to ion or cosmic ray bombardment, and so on.
How much is 700 MPa? Well, 0.1 MPa = 100 kPa = 100,000 Pa = 1 bar by definition, and 1 bar is just slightly under the standard pressure of air at sea level (1.013 bar); or just about exactly the air pressure at 111m elevation at a 15°C temperature. So, we're talking about seven thousand atmospheres of pressure, give or take.
Such a glass bead certainly does not look like it has anything of the sort locked in its structure in static forces now, does it? That's because the internal forces, the internal stresses, are perfectly balanced. They have to be, or the bead would change shape. The reason cutting the tail makes the drop explode, is that the fracture thus generated (which propagates in the glass bead at five or six times the speed of sound in air, or 5200 - 6800 km/h), unbalances those forces; and no longer in balance the atoms start moving –– in the human scale, in explosive disintegration.
Glass, or even its easier version, silica or silicon dioxide, SiO2, is not hard to simulate on the atom level, but being amorphic, the structure is difficult to get "right" (the way it is in real-world glass or silica). I don't know if anyone has modeled Prince Rupert's Drops, but their macroscopic size makes it a bit daunting; you'd need serious (distributed) computing power to do it properly. Smaller simulations can obviously show details (say, a model of a small region of the surface, extending towards the center of the drop), and those have surely been done.
Anyway, on-the-envelope rough estimates (say, 160pm interatomic distance) means that force per atom based on the pressure is on the order of dozen or two picoNewtons (10-12 N). Doesn't sound much, until you realize a silicon atom only weighs 28 u ≃ 28 × 1.6605 × 10-27 kg ≃ 4.6×10-26 kg. If that force was applied to a single atom without any opposing forces, then the instant acceleration of that atom would be some forty million million standard gravities, or about 4×1014 m/s2.
"Stupendous" comes to mind, even if there is a typo of a few orders of magnitude in there; but it "feels about right" to me, so I won't re-check – it's just the crudest possible napkin math. (Oh, and 1 standard gravity ≝ 9.80 m/s². It's used in some science fiction to describe spaceship accelerations (usually high, as in hundreds of standard gravities of acceleration, since they need inertial dampeners to stop humans from pancaking if you go beyond single-digit standard gravity accelerations, and such high accelerations to get any place or be able to match velocities in any time frame dramatic enough for us humans), and I like how it sounds.)
The amount of energy stored as potential energy in the structure of matter –– and note that this is just the static, structural forces that don't do work since matter generally stays the same; we're not talking about the utterly ridiculous amounts of energy involved in the matter itself due to \$E = m c^2\$ –– and the static forces in it, are really completely outside our intuitive scales.
TimFox:
A brief comment about the mass-energy E=mc2 inherent in ordinary matter:
In nuclear reactions and similar problems, only a small part of this energy is available to the consumer, basically due to the small difference between neutron and proton masses, since "baryon number" is conserved. For nuclei, the protons and neutrons are baryons, so there can be transformations between them. The proton is the lightest baryon and things stop there, pending discovery of very long decay times that have been theorized but never observed for proton decay.
However, I have a vague recollection from an interesting lecture on black holes I attended in the mid-70's, that if one were in a safe orbit around a black hole, outside the event horizon, and very carefully (adiabatically) and slowly lowered a mass into the black hole on a very strong fishing line, the total energy transferred to the reel would, in fact, be mc2 .
Nominal Animal:
The one field where we do already get \$E = m c^2\$ out, is matter-antimatter reactions; specifically, electron-positron annihilation, as used in e.g. positron emission tomography and positron annihilation spectroscopy.
The positrons are generated by nuclear fission (in a radioactive material, called tracer or radiotracer in PET and injected into the patient at very small quantities, and just placed next to the material or structure to be analyzed for PAS).
It just happens that when a positron and an electron annihilate each other at low energies (meaning have low velocities if measured in the frame where their linear momentums cancel out), the end result is two gamma photons with very easily identified energies, and we can use those gamma photons for useful stuff.
At higher energies, by controlling the amount of kinetic energy, you can generate interesting exotic particles like mesons and W and Z bosons, which all tend to decay very quickly (as in within 10-18s or sooner, much much faster than we for example do time steps in atomic simulations) into leptons and hadrons and neutrinos and such. Interesting stuff for particle physicists, but it's those gamma photons that are very useful in practical applications.
The CERN Antiproton Decelerator, among other thinds, produces antihydrogen: atoms consisting of an antiproton and a positron. The numbers are small; in 2011, the ALPHA experiment captured 309 antihydrogen atoms for over fifteen minutes (1000 seconds).
In certain Science Fiction settings, antihydrogen is stored as a fuel, and generated in huge accelerator rings, sometimes surrounding planets or stars. It is likely unfeasible (and something like just sieving positrons and antiprotons from highly energetic environments would work better), but not out of the realm of possibility; not realistic, but close enough to be tantalizing.
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