3047: Rotary Tool

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Apples

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Title text: The experimental math department's budget is under scrutiny for how much they've been spending on trains leaving Chicago at 9:00pm traveling at 45 mph.

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Explanation

This is one of 54 incomplete explanations: This page was created BY A CAR HEADING WEST AT 70MPH. Don't remove this notice too soon. If you can fix this issue, edit the page!

In the comic, a group of three "experimental mathematicians" has experimentally confirmed the answer to a math story problem that might normally appear in elementary school: "If Cueball has seven apples and Hairbun has five, how many apples are there?" Cueball counts the two groups of apples and states that the total is twelve. Blondie agrees that this is noteworthy.

Most people with a basic level of math would represent this as 7 + 5 = 12 and be confident of the answer without needing to count groups of physical objects. However, the title text states that there is an entire experimental math department dedicated to testing out common story problems in the real world, as if there was some doubt that the theories were sound.

It may also be an allusion to the most basic step of human mathematics, that of realising that seven of any conceived item plus five more of it will be twelve such items in total, and that numbers alone can therefore represent items without there being actual items to prove their own totals. Early accounting methods initially used proxy representations of the items, in a form of hybrid literal/symbolic manner, which meant that the combining of numbers of apples and combining numbers of livestock could be considered almost as different concepts, even though they had the same total sum applied only to different products.

This particular kind of abstraction sometimes fails in the real world when combining different things. For example, when measured volumes of two different substances are combined to make a solution, the volume of that solution is usually not exactly equal to the sum of the volumes of the original substances. Even in the case of combining equal volumes of nearly-freezing and nearly-boiling water, the result will not exactly equal the sum of the two volumes, since density-vs.-temperature curve of water isn't a straight line (and the specific heat capacity of water varies with temperature).

It is possible that this Experimental Mathematics department has been working on this particular level of problem, as part of a mostly pre-mathematical culture. They are just now checking that 7 apples plus 5 apples equals 12 apples, after perhaps extrapolating from the recently confirmed fact that (e.g.) 7 sheep plus 5 sheep equals 12 sheep. Their theory that this extends to apples (and any other items they have tested before this point) has so far not managed to support the null hypothesis in which it might not.

Many branches of science have a known division between the empirical approach (gathering direct evidence or practically demonstrating that something works) and the theoretical (developing abstract models that fit the available information without fully testing them). High-quality experiments tend to be difficult and expensive, so rigorous testing is normally reserved for problems that someone considers sufficiently important or interesting. Math often deals with numbers and situations that cannot be reliably reproduced. The department's focus on confirming what most people already know may face difficulties when applying for grant funding. In reality, experimental mathematics is the branch of mathematics which uses computation as opposed to "pure" deductive proof methods. This does not involve "verifying" simple arithmetic, but could encompass e.g. calculating long runs of the digits of pi in search of patterns that may not be 'obvious' from known principles but which could be proven once identified as a candidate for proof.

On top of the simple problem that requires simple addition (and possibly subtraction) to fully understand the answer of, the title text goes on to cover a slightly more complicated schoolroom mathematical problem, one which generally requires at least some understanding of multiplication and division (though more advanced problems of this type might require moving into the realms of algebra, and the nature of simultaneous equations in particular). These may take the analogous form of a train (or other vehicle) setting off at a given time and constant speed along a given hypothetical route, and comparing that against other trips made to/from the same location. As with the hyper-practical experimentations with the number of apples, these more advanced queries are being investigated by directly examining the real-world incarnations of the terms of the problem. It seems that enough identical repetitions have been attempted, at least of a particular Chicago-departing rail service, to have worried those who oversee the financial accounts. (Presumably the accountants at least know enough about numbers to know that the acceptable number of purchased train tickets plus yet more purchased train tickets is adding up to more train tickets purchased than the accountants can consider to be justified.)

As every regular train calling at Chicago Union Station either originates or terminates there, a train has to accelerate first before reaching 45 mph. To leave the station at this speed at 9:00pm, the department has to rent a train using one of only two through tracks, and resolve possible conflicts with other scheduled trains.

A flaw in the system is that with irrational numbers and infinitesimals. Those cannot be represented with physical objects easily and will probably need very precise things or are just impossible.

Transcript

This is one of 30 incomplete transcripts: Don't remove this notice too soon. If you can fix this issue, edit the page!

[Hairbun and Cueball stand at the left of the panel. Blondie stands at the right. Between them are two piles of apples, one of seven apples (stacked four on the bottom, two in the middle row, and one on top) and the other of five apples (stacked three on the bottom, and two on top).]

Cueball: Okay, with my seven apples added to your five, we have ... let's see ... twelve apples!

Blondie: Incredible!

Blondie: Perfect agreement with the theory!

[Caption below the panel:]

Experimental mathematicians

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Discussion

How come it's at 0.017 RPM for a minute?? and yet 1 RPM for a second? pls fix this randall Midnightvortigaunt (talk) 18:01, 5 February 2025 (UTC)

Its 0.017 RPM for the minute hand. The minute hand revolves once per hour or at 1/60 RPM ≈ 0,017 RPM --172.71.148.59 18:14, 5 February 2025 (UTC)

Ohhh that makes sense I didn't think about it like that Midnightvortigaunt (talk) 19:27, 5 February 2025 (UTC)

Mr.Dude (talk) 17:20, 7 February 2025 (UTC) I wonder what torque is needed to launch the average backyard telescope worthy of a tracking mount at Mach 8 given standard state pressures and temperatures of perhaps average conditions found in Randall’s back yard.

How come the comment above is invisible to me? 172.68.245.229 18:03, 5 February 2025 (UTC)

Possibly because people indented with spaces rather than with colons? 162.158.79.77 19:40, 5 February 2025 (UTC)

72 RPM for a record player...? 162.158.74.25 18:08, 5 February 2025 (UTC)

I could only find 78 RPM disks in the german wikipedia. 172.70.114.56 18:41, 5 February 2025 (UTC)

I came here to make the same comment: 72 is most probably a typo. The old records (at this date, very old, since the transition to vinyl records was 1948 to 1958 (in the US)) were 78 rpm, not 72 rpm https://en.wikipedia.org/wiki/Phonograph_record Rps (talk) 19:30, 5 February 2025 (UTC)

72 is (for example) relevent to font sizes (size 1 = 1/72 of an inch, size 72 = 1 inch), which might therefore have envaigled Randall's head for numbers by a different route, and got him confused. Conceivably he has had to deal with playing old 78s, but probably not for a long time... even the retro-revival of vinyl, recently, has probably not had quite so many old old records released to fill such nostalgic needs. So an easy brain-fudge/thinko to trip over on. 162.158.74.48 00:54, 6 February 2025 (UTC)

There used to be a record label call 72RPM records. 172.69.229.146 (talk) 19:07, 5 February 2025 (UTC) (please sign your comments with ~~~~)

We need one of those tables in here. DollarStoreBa'al (talk) 18:37, 5 February 2025 (UTC)

I made a change to the explanation that all of these numbers are realistic because, I checked out the speed of dental drills and they really do rotate that fast. I haven't checked out all of the other tools, but I suspect that they are also accurate. If you find that any of them are misstated, please correct my correction. Rtanenbaum (talk) 22:38, 5 February 2025 (UTC)

TABLE REQUEST When someone uploads a table, I'd like to recommend a second column for the frequency / reciprocal of the speed. "0.000000000073 minutes" is one every 13.7 billion minutes, or ~26,000 years. Thanks! 172.70.46.107 20:20, 5 February 2025 (UTC)

Me again. Should the column header "revolution time" be "rotation time"? In every instance, the axis of motion is within the object itself; even the second/minute/hour hands go around the axis. 141.101.76.73 16:41, 6 February 2025 (UTC)

TRIVIA 16 2/3 RPM phonographs were used for some voice-recorings back in the day. 172.68.26.24 21:01, 5 February 2025 (UTC)

My parent's old record player (60's, probably) had 4 possible speeds: 16, 33, 45, 78. By the early 80's the current ones only had 33 and 45. Rps (talk) 16:59, 7 February 2025 (UTC)

Album goes back to stacks of 78s. A symphony or opera would be 2, 3, 4 or more disks. They were bound like a photo-album with a leaf for each disk. "78" wasn't "standardized" until the format was fading. 3600-rpm motor and 46-tooth gear is incomplete (one tooth gear??) Early discs were from 60 to 130 rpm. Users would adjust speed by ear (also to ease pitch-matching for karaoke). Only as LPs arrived did someone invent the number "78.26 rpm" (no recordplayer and few lathes of the period were near that accurate). --PRR (talk) 02:34, 6 February 2025 (UTC)

Indeed, my parents had a large collection of old records and at least one had a speed marking of 80rpm.--172.68.186.43 09:17, 6 February 2025 (UTC)

With wind-up players, a lot of them started off playing at one speed and ended playing at a completely different one anyway...172.68.186.50 09:43, 6 February 2025 (UTC)

I suspect there's not many consumers needing a Uranium Enrichment Centrifuge... at least outside of a few countries in the Middle East. --172.70.58.6 08:50, 6 February 2025 (UTC)

Might face some regulatory / export license issues too.172.70.86.129 11:34, 6 February 2025 (UTC)

I feel like there was a lost opportunity to have Dr. Who's Sonic Screwdriver on the list. Maybe the rpms are unknown.162.158.159.107 13:05, 6 February 2025 (UTC)

The table says that 0.00070 "seems off; a sidereal day is 23.93 hours". That's just because (like all of the other settings) 0.00070 is quoted with only 2 significant digits. Every period between 23.64 and 23.98 hours would round to 0.00070 RPM. 162.158.134.199 13:58, 6 February 2025 (UTC)

The question I have is: why are dental drill speeds so high? 172.70.247.92 17:21, 6 February 2025 (UTC)

"why are dental drill speeds so high?" It hurts less. (Are you old enough to remember routine use of belt-driven dental drills?) You can cut a given amount of material (wood, steel, tooth) quickly with heavy force or high speed. Neither is really fun, but hi-speed is generally preferred. --PRR (talk) 19:08, 6 February 2025 (UTC)

Although some materials behave badly to heat (either work-hardening, for some alloys, or melting/burning, like plastics) and that's why variable-speed hand-drills/etc usefully have low speeds (for essentially the same force, when that's done via reostat rather than an actual gearbox). On the few occasions I've had my teeth drilled, I'm pretty sure I've detected the pungent smell of fried tooth-fragments, but it was nothing like as strong as smelling my own nose-flesh being burnt one of the times I had it cauterised to try (and fail) to prevent excessive nosebleeds. 172.69.79.139 21:15, 6 February 2025 (UTC)

The latest NMR CPMAS probes send their rotors to go at 9.6 Mrpm, M=mega. [1] --172.69.109.172 21:56, 7 February 2025 (UTC)

Should we list the rotor diameters to achieve the mach 8 speed mentioned in the title text in the table? I don't think that we should. guess who (if you desire conversing | what i have done) 06:01, 24 February 2025 (UTC)

I (obviously since I worked it all out) think it is in the spirit of the ridiculous idea of the comic and XKCD generally to do these calculations. That said, I'm getting different numbers than your update to make it Mach 8. Denver87 (talk) 16:21, 24 February 2025 (UTC)

I get the following: 4,799au, 74,866km, 37,733km, 3,144km, 52.4km, 1,588m, 1,165m, 728m, 175m, 34.9m, 21.0m, 149.7cm, 87.3cm, 174.7mm. Denver87 (talk) 16:21, 24 February 2025 (UTC)

Happy to share calculation notes, but here's the example for the dental drill: 300,000rpm = 5,000 rps; diameter of: 174.7mm --> circumference of: pi * 174.7mm = 548.8mm; 548.8mm * 5000rps = 2,744,000mm/sec = 2744m/sec; Mach 8 = 8 * 343m/sec = 2744m/sec. Denver87 (talk) 16:21, 24 February 2025 (UTC)

If you agree with the calculations, one of us can at least update it. Denver87 (talk) 16:21, 24 February 2025 (UTC)
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