I just love it when a bit of cool mathematics kicks the bejesus out of intuition. I first ran into this a few days ago in a book titled “The bedside book of geometry,” but there are many pages on the web that also explain it:
As an aside, I just noticed that this board has some family-friendly software installed. I used a, erm, slightly stronger f-word, which the software, instead of replacing with asterisks, actually changed into a milder form! Not sure if I should laugh or be infuriated…
There are many such cases. Another suprising one is the so-called “Birthday Problem”. (The startling answer of 23 people first occurs about halfway into the entry. The problem exposes a strong bias in our thinking towards fixating on a specific birthday, rather than any birthday.)
It’s an individual user setting you can change in your account profile. Under the “Modify Profile” items on the left of the page, select “Look and Layout Preferences”. Check or uncheck the “Leave words uncensored” box, according to your liking.
This whole thing about intuition is interesting. I ran into a neat example the other day, while reading one of the Bad Astronomy film reviews.
Phil Plait reviewed a film titled, if memory serves, “Gravity.” He explained how some of the orbital mechanics in the film was all wrong, but noted that he very much liked the film anyway, for its spectacular visuals and well performed drama. It occurred to me that we shouldn’t really like the film: the orbital mechanics was every bit as absurd as, say, a James Bond movie in which a central plot development hinged upon 007 being thrown off a roof and then floating away instead of falling. That is to say, in a standard Earth-bound film, we would never forgive such a ridiculous thing. But here we have an astronomer forgiving pretty much the same thing simply because it is set in space.
Why would this be? My guess is that it is simply because while Plait may be an astronomer, he is nevertheless human, i.e. his physics intuition evolved in the same place as mine. I have not seen the film, but from what he describes, it sounds to me as if, in the film, things move as we intuitively expect them to. And thus, even though his rational mind told him it is all wrong, his intuition was perfectly happy with it in a way that would not have been the case had the film been set in the more familiar terrestrial environment. That is perhaps also why few of us mind when explosions in space make a lot of noise, etc. etc.
A favourite peeve of mine is when someone falls from a tall building, and just before they hit the ground, Superman comes blasting from the side, snatching them away at close to the speed of sound, but they somehow don’t get injured from that! We seem to have specific intuitions about what causes injury. You see the same thing in disaster movies featuring volcanoes: a person is okay as long as they don’t touch the lava, even when they are suspended just inches above it and in reality would have burst into flame from the heat.
We can be thoroughly entertained by dodgy and downright absurd physics in films, even in cases where it’s a distraction for being patently and painfully absurd. It’s called “suspension of disbelief” and all fiction relies on it to a greater or lesser extent. Still, film directors are often the biggest offenders here. You’d think that somewhere in their frequently mind-bending budgets, they’d find space to pay an expert technical advisor or three.
However, moviegoers don’t like too much reality, either. For example, the first few episodes of the 60s TV broadcast Star Trek quite accurately showed the USS Enterprise blasting through space in complete silence. People didn’t like that and the producers insisted that an appropriate soundtrack be added, something that persists in most space scenes to this day.
Yup, the killer is huge acceleration, a sudden change in velocity. Since velocity is a vector, it has both magnitude and direction, and so acceleration can be a change in speed or its direction, or both. A body rotating around an axis at a fixed distance and a constant rate experiences no change in speed, yet it accelerates because its velocity is constantly changing direction.
Two of my peeves in this vein are that everything inexplicably goes slow motion in a gravity-free environment, and where artificial gravity is produced at the periphery of a rotating space station, they invariably forget to coach the actors on the Coriolis effect…
Speaking of artificial gravity. I rather liked the film “2010” (which was the sequel to Space Odyssey), and it seemed to me that the got much of it right as far as the science was concerned. But there was one scene that bothered me. The spacecraft has artificial gravity by being spinned. In one scene, a character lets go of a pencil in mid-air, and there it floats. But as far as I can work out, such an artificial gravity field, created via acceleration, would be indistinguishable from “real” gravity, and the pencil should drop to the floor. Isn’t that what Uncle Albert said?
I tend to be pretty hopeless with physics, or perhaps I just have too much intuition and too little brains. Being trained mostly in biology, of course I have my own list of pet peeves in movies - sharks portrayed as vengeful monsters, alien parasites that are somehow adapted to infect any life form whatever, and don’t even get me started on Hollywood genetics…
You’re thinking of Einstein’s “Equivalence Principle” of General Relativity, which pertains to rectilinear acceleration. (However, it should be noted that rotating gravitational fields are entirely possible, in which case such a field and artificial gravity through rotation would again be indistinguishable, but we are entirely unfamiliar with such fields.)
In a situation where artificial gravity is produced by rotation, the aforesaid “Coriolis effect” comes into play. Unless the pencil is exactly at the axis of rotation, it would appear to accelerate towards the “floor” — but following an apparently curved path, not the rectilinear path our experience suggests! The link has a tidy little animated graphic to illustrate the effect.
There was a similar situation in Arthur C. Clarke’s “Rendevouz with Rama,” in which humans explore a huge alien spacecraft. It was cylindrical, and had an artificial gravity by being rotated along its long axis. At one point, a character had to jump off a cliff into a lake, and he accelerated all the way.
At the time (this was years ago) I found it all but impossible to see why this would be so. What on Earth would make him accelerate once he wasn’t touching anything anymore? Someone on a mailing list tried to explain, and I had to spend days and days ruminating over the problem before it finally clicked. At least the acceleration bit. Still can’t quite work out all the Coriolis implications. I try to visualize such problems, and my ability to do so is limited. And then I also have to try working out whether the person’s acceleration is an acceleration relative to the cylinder itself, or to the space outside.
Never could get my flat head around even Newtonian physics - with quantum stuff I stand no chance whatever.
At the school where I teach I have now run into a problem. I used to be primary school science teacher, but they are adding a high school (up to grade ten, for the moment), so next year they need me to teach the high school kids. All very well if I taught them biology. But they need me to teach physics and chemistry, because the other high school science teacher they have available cannot teach anything other than biology - she has a B. Ed, with specialization in life sciences, and as such never studied anything else at tertiary level. I did study some other subjects (seeing as they are required for a B.Sc., even if you major in life sciences), so there you have it. But I scraped through that stuff without ever properly understanding it, and it was way back in the early 14th century or thereabouts. Nowadays I cannot even remember what calculus IS anymore, let alone apply it.
Thus I will very quickly have to upgrade my knowledge of these things - expect a slew of very dumb questions here…
Time and circumstances permitting, I’ll help where I can — and, being an inveterate sceptic, I doubt the questions will be dumb. A failure of understanding is more often than not the result of inadequate explanation.
At the risk of sounding hackneyed, the best way to learn is to teach.
I have noticed this from personal experience, even teaching at primary school level - once I have to begin thinking about how to explain something to kids, I begin to clearly see where my own knowledge or understanding is lacking, and then I can go read up on it.
I have considered doing some refresher courses through Unisa. but I don’t know if they still allow one to register for courses for non-degree purposes, as they used to. I inquired, but they never replied - apparently this is typical of their administration nowadays.
I inquired, but they never replied - apparently this is typical of their administration nowadays.
I think Rigel will agree with you, but my wife is quite happy with them, but then she’s been studying with them for years, so I think she by-pass the front office. She does studying as a hobby, is doing law at the moment.
For a bit of a “me too!”, since I’m a software developer, it’s the computer stuff that drives me crazy.
CSI: NameYourCityHere drives me crazy, as does “Bones”. I have no idea how biologically/athropologically/forensically accurate Bones is, but the supposed holographic computer simulations they run at the drop of a hat to create difinitive “evidence” is snort-worthy, and doesn’t inspire confidence that they get anything else correct. Their computer expert, Angela, should also be put behind bars for repeated and systematic violation of privacy laws immediately. Who needs a warrant if you can hack? Surely no court would frown on evidence gathered in that way…
I did my degree with them, many moons ago, and at the time, their administration was brilliant, as was the content and presentation of the courses. A year or two ago I started doing a post-grad teaching certificate, but let it go within a few months, because the contents were utter crap and the presentation (by supposed experts in how to present study material!) not much better.
Lucky for me, we now have such a shortage of teachers that one can get a job in the field without a formal qualification, albeit not at government schools. But I don’t want to work for a government school anyway.
Not so because the pencil will follow a genuine straight line trajectory once it is released (in accordance with Newton’s first law), as seen from an inertial frame of reference. But because it is seen from the non-inertial frame of reference of the rotating spaceship and astronauts, it will therefore appear to them to follow a curved path from its point of release to the “floor” of the spaceship. Draw an external set of axes and plot snapshots of the points where the pencil and astronauts will be at successive points in time. Now plot the pencil’s positions as seen from the astronauts’ position. They’ll see a curved trajectory, the curvature of which depends on the angular speed of the spaceship and its radius.
It’s a worthwhile exercise because it shows how intuition can fail.
No, I didn’t. Have a careful look at who wrote what.
Intuitively I actually expect the object to fall on a curved path, the intuitive reasoning I used (probably incorrect) was that of swinging something in a circle on a rope then releasing it, which results in instant outward movement (it won’t float), but along a curved path… in this case your hand is playing the part of the rope up until you release. This allowed me to clearly visualise what’s happening while assuming “my” frame of reference is standing still, but still knowing it’s rotating. (EDIT: On second thought this sounds completely incorrect)
I do have a caveat though: The reasoning here seems to assume, as many physics problems do, a vacuum. I would expect the air in the vessel, having been there for an appreciable amount of time, to be moving more-or-less at the same speed as the vessel thanks to what I recall about fluid dynamics. It would, intuitively, seem to me that the air in the vessel combined with the aerodynamics of the object will counteract some of this “swing” of the object as it falls. IOW: I’m not sure by how much, but I’d expect the effect to be counter-acted to some degree given a breathable atmosphere.
Assuming that the object’s density is significantly greater than that of air, the effect on the object’s motion of the atmosphere being dragged along by the spaceship is negligible, just as it is here on Earth where the atmosphere is also being dragged along both the Earth’s rotation and its revolution around the Sun.
It seems to me that the key point here keeps getting missed. As viewed from an external inertial frame of reference (pay special attention to the second paragraph of that entry), the pencil describes a circular motion (or, more generally, a helical one if the ship also has a component of motion along its axis of spin) until the moment it is released.
Upon its release, the pencil follows a straight line path as viewed from that external frame of reference. This is what Newton’s first law of motion guarantees because there are no longer any unbalanced forces acting on it. The pencil’s speed can be calculated from the ship’s rate of rotation in conjunction with the perpendicular distance of the pencil from the ship’s axis of rotation, while its direction will be tangent to the circle (or helix) it was following at the time of its release. That straight path will inevitably carry the pencil to the outside periphery of the spaceship (i.e., to its “floor”).
Now here’s the kicker: As seen from our external frame of reference, while the pencil is travelling along its rectilinear path, the rest of the ship and the astronauts are still following their previous circular (or helical) trajectory. Ergo, from their perspective, the pencil will appear to follow a curved path towards the floor.
If the point still isn’t clear to anyone interested, I strongly urge them to go through the exercise I suggested earlier.