One. OK. We understand that.
Every human who ever suffered through that innocent-looking difficulty where you’re provided the dimension of a record and data about how vast the groove is and also asked how countless are on the next of the record and also then after a lot of confused algebra handed in solution and discovered it was a cheat question has actually that melted into your brain, and also maybe still resents the teacher or publication of mathematics puzzles the presented them through the an obstacle only to have actually the disappointing answer revealed.
This might be a generational frustration. I think but don’t recognize that compact discs and DVDs actually have actually concentric ring so the the how-many-grooves identical would it is in a meaningful, non-trick question; to inspect would need I make the slightest initiative so I’ll just trust the if I’m dorn someone will complain. In another thirty years the word difficulty may have actually disappeared indigenous the inventory. But it excited me, and also my Dearly Beloved, and also I’m certain irritated other civilization too. And, yes, we’ve all heard the those novelty documents where there’s 2 or 3 grooves on a side and also you don’t recognize until reasonably well right into the power which version you’re listening to, however I’ve never ever actually organized one in mine hand, and also neither have actually you. Because that the benefits of this discussion we may disregard them.
But the concern we plunge right into answering prior to we’ve i found it the trick is more like this: If we attracted a heat from the hole in the facility straight out, a radial line if I desire to do this sound mathematical, then it crosses some variety of grooves; how many? Or perhaps like this: how many times walk the groove go about the center of the record? and also that’s interesting. And I want to explain how I’d occupational out the trouble — in fact, just how I did work it out a couple of nights earlier — including a significant false start and also how that got me to a satisfactory answer.
Since I’m pretty good at calculus, my very first instinct is to shot using calculus techniques to settle the problem. How?
Well, imagine acquisition a small ring part out of the document — geometers might speak to this one “annulus”; my dad might call it a “washer” due to the fact that it’s the shape of that point you placed under screw heads for some factor that my father knows. That segment will have some component of the record’s groove top top it. If the inside and also the outside rings space far enough apart the groove will wind its method around numerous times. If they’re close sufficient together the groove winds its way around not even a full loop. If they’re just the right distance apart, though, they’ll wind about exactly once.
Now here’s the calculus-approach thought to things: the area the the document — the playable area, anyway — is comprised of some variety of these annuluses the one loop each. It’s straightforward to number out the area of an annulus (really: it’s the difference of the area of the outer rim’s circle and the inside rim’s circle; holding increase a washer, it’s the quantity of metal you have to make the whole disc, minus the amount you punch the end to make the hole). Therefore the number of grooves you cross in one line is the playable area (again, the area of an annulus) separated by the area of each one-loop annulus. If you monitor the reasoning here then you know integral calculus. (This part might need to be increased on prior to you take an exam on it.)
This method is madness, and attempting to follow it conveniently falls apart. There are several factors for this. The most obvious is the the width of this one-loop annulus is really just how much any allude on the groove is native its neighbors, walking in toward the center or out towards the edge. Yes, yes, it’s the very same groove, however you understand what ns mean: if we imagined being tiny sufficient to stand in the groove top top the radial line, and stared in the direction the the needle seems to go, there’s the allude on the left whereby the groove the cross the radial line, and also the point on the best where the groove the cross the radial line, and how far is it to the intersection? and also I’ll contact that the neighboring-groove distance even though the is really the very same groove because you think of a means to say that which isn’t as well cumbersome to transaction with. Anyway, if us knew that neighboring-groove distance we wouldn’t need to fiddle through the ring any. We might just division the diameter that the playable area through the neighboring-groove distance and be done through it.
Also, it’s conceivable the the neighboring-groove street is different closer come the center of the document than that is far away. A record turns in ~ a consistent rotational speed, after ~ all; the linear speed or the radial distance in between one component of the groove and the following isn’t obviously vital to the design. In ~ the rim of the record, where one fall the needle prior to starting, and where it gets stuck in ~ the finish if the auto-return of the arm isn’t working, the neighboring-groove distance absolutely does vary. I don’t believe that it different too much in the main section the the record, apart from the small stretches most records need to mark wherein one tune ends and also the following begins. But obviously, ns can’t say through certainty that it doesn’t differ enough. Assuming the doesn’t differ too much simplifies the problem, though, and also it appears reasonable enough, however that doesn’t assist me to speak what the street is.
So calculus stops working us, a update reversal. Yet how deserve to we still deal with the problem?
That thinking about the neighboring-groove distance gave me an idea, though. Think about that radial heat from the center of the disc come the edge. We desire to understand how plenty of times the groove crosses that line. That’s the same number of times the record turns, if we play the complete side: on each complete turn the needle will have actually moved in just a tiny bit, and crosses the radial line at a various point. So if we know the variety of rotations the disc renders to beat one side, climate we recognize the number of times the groove crosses that radial line.
And then among the bits assumed up throughout my false begin comes right into play: the record turns in ~ a constant rotational price per minute. Because that an LP, that’s 33 1/3 changes per minute. How numerous minutes go a document take come play? (This exact same sort of reasoning can be placed to 45’s or 78’s or even freaky various other records you can find.)
Well, that’s different for every record, that course. Yet one side takes generally something prefer 20 minutes, which have the right to be surprisingly brief if you’ve obtained so provided to CDs or MP3 players or internet radio whereby music just seems to sprawl the end without limit. 33 1/3 is one awfully suggestive number, though: it’s 100 split by three. Let’s say the a document takes 18 or 21 minute on the side, because those room multiples the three. Then an 18-minute record side would turn 600 times over the food of playing; a 21-minute record side would rotate 700 times. So i would call it something like six to seven hundred time the groove the cross the radial line.
That sounds satisfying as an answer. It just leaves one nagging question: am i right?
Solving a problem is fine, yet it’s not complete. We need to verify the solution somehow.
A fast search the the net reveals that the question around grooves top top a record side space overwhelmed by world answering “one”. Thanks heaps, guys. A few sources will offer the mentioned dimensions that a record, through the surprisingly large number of requirements out there. Groove width, because that mono or stereo, is pretty fine specified, yet sources gain vaguer around the mean separation between nearby grooves.
There are some fair factors for this. The width of a groove itself counts on peculiarities that the recording, such as the volume that the sound and also the noise compression in ~ play. Exactly how close with each other grooves can be depends on the high quality of the material. The record player is an extremely forgiving that groove distance, so, probably it’s better not specifying as well tightly what the typical should be. Permit the recording staff figure that out.
It’s much easier to uncover information around the gross physical dimensions of a record. Taking the playable exterior of a record to it is in 11.5 inches in diameter, and the within playing surface 4.75 customs in diameter, says it’s 3.375 inches from the external surface to the innermost. That appears to be around the appropriate size if i actually organize up and also look in ~ a record, return I would certainly probably have actually rounded that to roughly three and a half inches. Installation 600 come 700 grooves in the … contact it around 200 per inch … and I can’t find a source that claims what the typical variety of grooves per inch should be. 200 per inch sounds plausible, but I can’t really tell that by eyeball, and I’m fresh the end of diffraction gratings come check against the records I deserve to lay my hands on.
Let’s plunge into the mess of people saying points on the Internet. The Genesis the Vinyl Stereo record page, from Badenhausen.com, looks nicely authoritative what with showing up in both English and German or possibly Dutch, and also asserts “the radial distance from groove come groove will be about 120 micrometers at an mean signal length for both channels” top top a stereotype record. 120 micrometers native groove come groove, if I’m doing department correctly, implies just over 200 grooves to the inch, so ns seem to it is in pretty dead on, provided how few significant digits I have in play.
Since every net site has actually an equal and opposite internet site, though, here’s a comment thread from a internet site I never ever heard of prior to where a man I don’t know anything around cites two internet pages that don’t exist no longer to say a long-playing document averages around 225 lines per inch, yet could be together high as 300. Well, the lower end of 225 is close enough to my number that ns can’t controversy it as well much.
And in other I assumed would never ever happen, ns ran across a Yahoo! Answer web page with a clear, direct and useful answer come the question, “what is the typical groove spacing ~ above a CD-R and LP record?” (It to be Yahoo! Canada Answers; maybe that makes the difference.) Canadian answerer billrussell42 asserted the groove spacing to be variable but around 150 micrometers, which works out to 170 grooves per inch. I’m quiet comfortable with that answer and also delighted to uncover an actual price on a Yahoo! price page. For the curious, billrussell42 likewise says the spacing on a CD is 1.6 micrometers, top top a DVD 0.74 micrometers, and on BluRay 0.32 micrometers.
I’m not really happy that ns can’t verify my answer through a straight measurement. I have some authorities who may well it is in qualified endorsing the answer i reasoned out, but part of why I find them believable is that they execute agree through my answer. Yet my thinking is quite straightforward and the presumptions which go into it seem basic to check. If I have it really wrong, I’d like to know simply where ns go wrong.
So that’s my answer: the groove on an LP loops about the facility something like 700 times, or roughly 200 times per inch.
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Those disputing my reasons are welcome to say so, and I will respond with an essay ~ above what the Muzak corporation go to assist win people War II.