Supersonic Man

May 30, 2016

the dimensionality of torque

Filed under: Hobbyism and Nerdry,thoughtful handwaving — Supersonic Man @ 10:33 am

Just clearing up a little thing that’s always bugged me…

“Torque” is rotational force: the measure of how hard you’re twisting something.  We don’t measure it directly; we can only take a measurement of it by gauging the amount of linear force that it exerts at a given distance from the axis of rotation.  Because of how levers work, this force is high if you’re close to the axis and low if you’re a long ways off.  The linear force times the length of the lever-arm from the center equals the torque.  So we measure it in foot-pounds or newton-meters. If you double the length, you halve the force, and the product is the same, so the particular force and length numbers don’t matter — only the combined value does.

But wait — force times length equals work.  Is torque in some sense the same thing as energy?  Hell no.  Torque is static; you aren’t doing work until you make it move.  If you exert force at the end of the lever through some distance, thereby rotating something while exerting that torque, that’s work.  If you continue to do so over more distance, that’s more work.  Now doubling the lever length halves the force at the end, but doubles the distance it has to travel for a given angle of turn, meaning that the particular lengths don’t matter — the work is the same for a given angle of turn.

Work equals torque times amount of rotation.  The correct SI unit of torque is not newton-meters, but newton-meters per radian.

So why don’t people mention the radians?  Because radians are defined as arc-length over radius: a distance divided by a distance.  This cancels out to a pure number, a dimensionless ratio.  Or so it is traditionally argued.

But we’ve clearly lost something there.  The rotation is a very real physical thing, and its presence or absence makes all the difference, as we’ve seen, when relating torque to work.  For that matter, just trying to solve for how far a projectile will fly, given its speed and angle, falls apart dimensionally if you say the angle is just a number.  I think it’s time to acknowledge that when we divide a curved distance by a straight distance, what’s left is the curvature, and this should be treated as an independent dimension in its own right.

Traditional analysis regards mass, distance, and time as the only fundamental dimensions; since then we’ve added quantities such as electric charge, and (when in a generous mood) various arcane quantum properties for which the universe seems to have conservation laws, such as quark type and “color”, and whatever it is that is shared by electrons and electron-neutrinos, but distinguishes them from muons and muon-neutrinos.  But these “material” values, as we might call them, don’t cover everything we need.

Some dimensional analysis enthusiasts have been saying that angles may not be the only neglected metric.  A minority even argue that inertial mass, gravitational mass, and “substantial” mass (whatever that is) are at least two and maybe three subtly distinct dimensions which just happen to always coincide in value.  (And indeed, the close relationship between energy, gravity, and inertia is still a mystery, despite apparent confirmation of the Higgs boson.)  But even without that, we all too easily forget that length is not a single dimension when dealing with vectors in space — each axis is separate, just as you’d think from hearing the word “dimension” in the first place.

Donald Siano has shown that if you get sophisticated enough with vector dimensions, it can express the concept of a measure of angle or rotation as a derived quantity.  He even shows that mathematical functions that we think of as pure numbers are incommensurable: for instance, sine and cosine are dimensionally distinct, and it makes no sense to add a cosine value to a sine, or (in further extensions by others) a logarithm to anything not logarithmic.  Such mathematical expressions cannot correspond to anything physical, and indeed, you should not normally ever see them when doing even the most abstract math.  So if we go for the full potato on using his “orientational” analysis, we don’t need a separate dimension just for angles.  But that’s quite cumbersome for everyday use — if we don’t want to be constantly doing 4×4 matrix arithmetic to keep our dimensions straight, we can just treat angularity as a dimension.

On the opposite side from those who want to nitpick subtly distinct properties of mass, some argue that the universe’s fundamental constants mean that quantities such as time and distance and mass and charge really are all commensurable, and at bottom there’s only one fundamental dimension (probably best expressed in our terms as energy), and all natural laws are dimensionless.  If so, so what, I say — that approach makes dimensional analysis less useful rather than more so. And if you think about vectors, space and time are still incommensurable, especially since in relativistic geometry, lengths on a timelike axis are imaginary numbers.

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May 21, 2016

a suggestion I intend to send to my legislators

Filed under: fun,Hobbyism and Nerdry,life,Rantation and Politicizing — Supersonic Man @ 12:28 pm

Senator Lois Wolk, Assemblymember Bill Dodd, Senate candidate Mariko Yamada, Assembly candidate Dan Wolk, and Assembly candidate Don Saylor,

Are you tired of dealing with pennies? I sure am. They take time and effort out of one’s day even if all you want to do is get rid of them. I don’t think any other economy keeps such a worthless coin in circulation — in Mexico, for instance, you never see anything smaller than a half peso. The US Treasury has been considering eliminating the penny from our coinage for twenty years, but hasn’t been able to move forward due to pointless obstructionism from assorted directions.

But fortunately, we don’t have to wait for the federal government to act. We can solve the problem right here in California. We can make it so people can use pennies if they want to, but nobody will need to. How can we do this? With a minor adjustment of the sales tax code.

All we have to do is make a rule that when buying retail at a location which accepts cash, the tax amount is rounded up or down by a cent or two, so that the total purchase price including tax is always a multiple of five cents. Note that this applies to noncash purchases as well, as long as they’re made at cash registers, so the amount remains consistent. But it would not apply to mail order purchases as they don’t offer a cash option. This means that we would not burden merchants in other states with adjusting to any new complexity.

The result would be that nobody who pays cash would need to either bring pennies, or receive them as change. People would become accustomed to nickel prices and before long, merchants might get into the habit of advertising nickel prices also. The other states would envy our penniless lifestyle and start copying us, and eventually the Treasury will stop minting pennies. And California will once again be seen as taking a leadership role.

But before that, we need someone to lead this idea in Sacramento. I’m hoping that among you, the legislators and candidates to represent me in Napa County, are the ones to do so.

I hope that this change can be accomplished by simple legislation, without requiring a ballot measure. If one is needed, I am confident that would pass, without requiring any substantial campaign effort.

Thank you for your attention, and I hope this idea appeals to you.

May 18, 2016

what we need in a president

Filed under: Rantation and Politicizing,thoughtful handwaving — Supersonic Man @ 1:44 pm

When we’re electing a president, the quality we need to look for is not experience or knowledge.  It’s not intelligence or education.  It’s not imagination or vision.  It’s not inflexible ideology or even steadfast ideals.  It certainly isn’t friendliness or folksiness or personal charm.

What we need from a president is wisdom.  The choices a president faces are often ones where an unwise decision can have calamitous consequences. What is needed above all else is good judgment in considering all the consequences of each major choice, and in discerning good reasons for action from bad ones.

This is why candidates always have to exaggerate their religious faith: because many voters believe religion to be a source of wisdom.

And this is why most voters dread the prospect of an egotistical blowhard in the White House: because we know that such a person is unlikely to act wisely.  But it’s also why an egotistical blowhard is winning the nomination: because the other candidates have tied themselves to continuing a whole range of policies which have been shown by their consequences to have been very unwise, thereby negating the advantages that should have been theirs by virtue of knowledge and experience.  The untried blowhard has a chance, at least, of turning out to be wiser than they were.

And it’s why the blowhard might even win: because the presumptive opponent, despite having every other quality a president ought to possess, has in the past supported many of those same dreadfully unwise policies.  Repudiating those bad choices today fails to reassure us, because the doubts are not about what side a candidate is on, but about how wisely the candidate would make the next difficult decision.

May 17, 2016

let’s try something easier: special relativity

Filed under: Hobbyism and Nerdry — Supersonic Man @ 5:21 pm

Here’s a way of explaining special relativity which I wish I had run across at a much younger age.  Thanks to author Greg Egan for finally getting this through my dense headbone.  For the first time, I kind of feel like all that stuff about time dilation and increased mass and so forth makes some sense.
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May 3, 2016

a possibly inept attempt to understand the hidden variable problem in quantum mechanics

Filed under: Hobbyism and Nerdry — Supersonic Man @ 10:40 pm

I’m trying to wrap my head partway around Bell’s Theorem, and what implies for the quantum interpretation conundrum.  I’m sort of hoping that I can explain it to myself in a halfway coherent way here.
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top science fiction writers

Filed under: fun,Hobbyism and Nerdry — Supersonic Man @ 1:33 pm

Who would be my picks for the top ten or so science fiction writers of all time?  Let’s take an initial stab:

(the inarguable immortals)
H.G. Wells
Olaf Stapledon
Philip K. Dick
Ursula K. LeGuin

(maybes)
Arthur C. Clarke
Frederik Pohl
Alfred Bester
Cordwainer Smith
Kurt Vonnegut
Frank Herbert
John Brunner
Roger Zelazny
Greg Bear
Octavia Butler
Vernor Vinge
Kim Stanley Robinson

(tempting, but probably not justifiable)
R.A. Lafferty
Greg Egan

(ought to read more before rejecting)
Bruce Sterling
Doris Lessing

Names I will definitely not be listing include Asimov and Heinlein… and also Sturgeon and Bradbury.

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