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 62 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, non-gravitational forces (e.g. 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.

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1) Plane intersects perpendicular to conic axis, results in a circular line (often counted as an ellipse of zero eccentricity) around one cone.
2) Plane intersects at a small angle away from the perpendicular, results in an elliptic line around one cone.
3) Plane intersects exactly at the angle of the (opposite) slope of the cone, results in an open-ended line that continues parabolically to infinity at an ever-increasing width (by decreasing degree) but at constant offset from the parallel slope of that cone.
4) Plane intersects at an angle closer to the axis than the cone slope (including being exactly parallel to the axis, as here), resulting in two open-ended hyperbolic lines to infinity (eventually tending to diverge at the rate of the conic slope itself), one upon each cone.

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In this comic, the orbital shape is similar to the one in figure 3 (a parabolic trajectory that does not technically 'orbit' the focal mass) with the 'end' of the lower cone included. Or, given the implication of this being based upon a mostly standard non-circular orbit, it might be a version of figure 2 but with the angled plane being lower so that the ellipse is cut off by the nominal 'bottom' of the diagrammitic cone.

This comic shows an orbit as a cross-section through the center of an ice cream cone - the round part of the orbit traces the outline of a scoop of ice cream above the cone, and the straight sides converging at a point show the sides of the cone. Thus, it is a "conic section".

The comic does not indicate why or how this orbit involves the 'base' of the cone. In real conic sections, the cone effectively extends to infinity (whether or not the useful section of the intersecting curve does). 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 (presumed to be 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.

This could be due to anything from a distortion/discontinuity in space-time to the spacecraft itself being commanded to change trajectory. Being in a free orbit normally 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.

A sudden change in absolute direction could be due to some alteration in the fabric of space, but even very similar orbits rarely trace the exact same conic sections. Though there are at least two imaginary cones that could intersect the orbital plane exactly along any given orbital ellipse, the dimensions and directions of different orbital cones will be unlikely to have coincident 'bases' — i.e. not parallel, even discounting the question of what their distances must be from their respectively chosen conic points. If the point of orbital discontinuity was thus different for every individual orbit that was taken, then any component not firmly connected to the satellite (and not positioned exactly at its centre-of-gravity) would be required to experience (at the very least) a slightly different moment at which it is suddenly expected to drift in a different direction.

If the change in direction is instead due to a commanded manoeuvre, the applied thrust necessary to change orbit (and, for a time, maintain a straight trajectory even through the curved gravity well) is both wasteful of resources (compared to the normally completed orbit) and requires a rather sudden and obvious change of momentum to the whole craft.

Whatever the reason behind the diversion, the result would be extremely uncomfortable for an astronaut in a crewed spacecraft. The transition from experiencing freefall/microgravity to suddenly being out-of-synch with the ship's momentum (whether just momentarily, twice each orbit, or for extended periods as continual corrections are made) would be disruptive. Such an extreme and sudden change of direction would require a very large G-force, to a degree that may not merely be uncomfortable but potentially dangerous.

We also aren't given any indication of how the 'radial' velocity of the craft might be intended to change during the 'flat' phase, such as if it obeys a suitably modified version of the constant 'area sweeping' rule, as for the elliptic part of the path, or instead perhaps attempts to maintain a constant relative velocity to take the same time to cross the new path as it otherwise would. The consequences of any of these might add further difficulties to the operability of a satellite and/or discomfort to any occupants.

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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