3047: Rotary Tool

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Conic Sections

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Title text: They're not generally used for crewed spacecraft because astronauts HATE going around the corners.

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Explanation

This is one of 63 incomplete explanations: This page was created by a section through a cone. Don't remove this notice too soon. If you can fix this issue, edit the page!

A Kepler orbit describes the simplified motion of one celestial object relative to another. Such an orbit will form a conic section. A conic section is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a fourth type, while intersections of the plane with the point of the cone (just that point, a straight line through that point or else four converging lines that all meet at the point) are possible constructions that are usually excluded.

In reality, this model is based only on the most simple modeling of two point masses, and ignores any other factors such as the gravity of other objects, atmospheric drag, each object being a non-spherical(/non-point) body of non-uniform density and any relativistic effects, but it serves as a good basis for most orbital calculations before needing further refinements to cover the most relevant additional perturbations for a given scenario.

How conic sections emerge from various planar intersections with bidirectional cones, which technically continue beyond the 'top' and 'bottom' of each diagram.
In this comic, the shape is similar to the one in figure 3 (a parabolic trajectory that does not technically 'orbit' the focal mass), or (given the implication of this being based upon a mostly standard non-circular orbit) may be more that of figure 2 except for the correctly-angled plane for the ellipsoidal intersection being sent through the respective cone too close to the nominal 'end' of it.

In real conic sections, the cone extends to infinity. In the comic, however, the "conic section" representing the satellite's orbit (with its unseen point pointing generally to the left of the image) has been assumed to have its circular base (somewhere close to vertical, towards the right of the image) set at a distance that inconveniently crosses the indicated orbital path (that might be assumed to be fully elliptical, otherwise), resulting in sharp corners where the angled planar intersection through the cone meets that base. As alluded to in the title text, these corners would be extremely uncomfortable for an astronaut in a crewed spacecraft. Such an extreme and sudden change in direction would require a very large, potentially dangerous, G-force.

Being in a free orbit necessarily means following an ellipse (or very similar, outside of the mathematically strict two-body problem) in which there is net zero acceleration, combining the pull of gravity and the forces that would be felt due to the continually changing direction alone. Being forced off this ellipse to move across the totally imaginary and arbitrary conic-base would force an instantaneous acute change of direction for no other reason than to follow the imperfectly understood mathematical 'model' at two arbitrary inflection points (reaching the baseline and rejoining the true curve again), which would technically require infinite acceleration each time.

Even if the weird nature of space-time in the locality means that point-masses are seemlessly redirected via the off-elliptic part of the route, spacecraft are not dimensionless points. Even if only briefly, different parts of the spacecraft are likely to encounter the inexorable redirection of motion at different times (and maybe in subtly different manners). Even if that is somehow not a problem, anything (or anyone) sufficiently loose within the spacecraft hull may be considered to be charting their own more subtly individual 'conic-based' version of the orbit, the freefalling drift either suddenly sending them off into the side of the vessel and/or tending to continue onwards as the vessel effortlessly navigates its own sudden change of direction.

Alternately, deliberately 'cutting the corner' of an orbit (without the fabric of reality conspiring to enforce this) would require the best effort of the spacecraft to stick to the truncated-orbital path, requiring as much thrust by the craft as it can muster (which any occupants would have to endure), including along the less uncomfortable but still forceful passage along the 'straight' bit of the orbit through the curved space-time of the gravitational field of the parent body. We also aren't given any indication of how the 'radial' velocity might be intended to change during the 'flat' phase, such as if it obeys the same constant 'area sweeping' rule as for the elliptic part of the path.

Transcript

This is one of 34 incomplete transcripts: Don't remove this notice too soon. If you can fix this issue, edit the page!

[A view of the Earth, focused on Asia and the Indian Ocean with East Africa at left and the Western Pacific and Australia at right. A satellite is shown in an unusual orbit around the planet. This orbit is similar in shape to an ellipse, except it has two corners and a straight edge on one side, giving it a hill-like appearance.]

[Caption below the panel:]

All Keplerian orbits are conic sections. For example, this one uses the base of the cone.

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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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