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Ribbon around the Earth

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brianvds
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« on: November 23, 2014, 08:03:15 AM »

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:

http://mathforum.org/mathimages/index.php/Rope_around_the_Earth

The thing that really cooked my noodle is that it does not mater what size the ball is. I simply couldn't believe it, and so decided to test it empirically with a piece of string and a tennis ball.

Well freck me.

Grin
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brianvds
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« Reply #1 on: November 23, 2014, 08:05:38 AM »

Well freck me.

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... :-)
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Mefiante
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« Reply #2 on: November 23, 2014, 10:10:13 AM »

I just love it when a bit of cool mathematics kicks the bejesus out of intuition.
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.)



I used a, erm, slightly stronger f-word, which the software, instead of replacing with asterisks, actually changed into a milder form!
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.

'Luthon64
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brianvds
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« Reply #3 on: November 23, 2014, 11:29:55 AM »

Well, let's see: freck, freck fuuu-uck!
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brianvds
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« Reply #4 on: November 23, 2014, 11:33:24 AM »

Much better, thanks. :-)
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brianvds
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« Reply #5 on: November 23, 2014, 11:44:33 AM »

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.
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Mefiante
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« Reply #6 on: November 23, 2014, 12:18:11 PM »

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.

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!
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…

'Luthon64
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brianvds
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« Reply #7 on: November 23, 2014, 12:30:25 PM »

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... :-)
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Mefiante
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« Reply #8 on: November 23, 2014, 12:48:04 PM »

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?
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.

'Luthon64
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brianvds
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« Reply #9 on: November 23, 2014, 13:03:55 PM »

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... :-)
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Mefiante
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« Reply #10 on: November 23, 2014, 13:31:06 PM »

… 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.

'Luthon64
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brianvds
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« Reply #11 on: November 23, 2014, 14:12:09 PM »

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.
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« Reply #12 on: November 23, 2014, 14:57:13 PM »


[/quote]
I inquired, but they never replied - apparently this is typical of their administration nowadays.
[/quote]
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.
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BoogieMonster
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« Reply #13 on: November 23, 2014, 16:38:35 PM »

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...
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brianvds
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« Reply #14 on: November 24, 2014, 05:35:36 AM »

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.

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.
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cr1t
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« Reply #15 on: November 24, 2014, 10:39:27 AM »

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?
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.

'Luthon64



I'm a bit confused to what the issue you have,
 since the camera actors and pencil is in the spinning ship,
 we should see the pencil drop down to the floor in a straight line.

I think you mentioned the floating pencil which sounds wrong.
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Mefiante
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« Reply #16 on: November 24, 2014, 12:03:00 PM »

… we should see the pencil drop down to the floor in a straight line.
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.

I think you mentioned the floating pencil which sounds wrong.
No, I didn’t.  Have a careful look at who wrote what.

'Luthon64
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BoogieMonster
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« Reply #17 on: November 24, 2014, 13:13:32 PM »

It’s a worthwhile exercise because it shows how intuition can fail.

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.
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Mefiante
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« Reply #18 on: November 24, 2014, 13:53:50 PM »

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.

'Luthon64
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BoogieMonster
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« Reply #19 on: November 24, 2014, 14:11:14 PM »

I understand.  Grin
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Brian
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« Reply #20 on: November 24, 2014, 14:34:04 PM »

Quote
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”).

Fok...dis lekker as jy so dirty praat!
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brianvds
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« Reply #21 on: November 24, 2014, 14:45:12 PM »

I think I more or less have a handle on it now.

Next up: Plot the path of the object relative to a person running up the cylinder while time traveling. :-)
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« Reply #22 on: November 24, 2014, 14:47:31 PM »

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.

'Luthon64


I'm still not convinced. Because the pencil does not loose its forward momentum.

If you are in a plane and you drop a pencil, the pencil will appear to go straight down.
If you are outside said plane you would see the curved trajectory, and that would be the same for our spinning space station
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Rigil Kent
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« Reply #23 on: November 24, 2014, 15:46:15 PM »

If you are in a plane and you drop a pencil, the pencil will appear to go straight down.
The plane flies in a straight line at a constant speed, say eastwards. While the pencil is released, it continues moving eastward at the same rate as the plane. So if you are inside the plane, you notice only its earth-ward motion, and not its eastward motion. This explains why the pencil appears to be falling straight down to an observer inside the plane. But if the same exercise was repeated while the plane briskly accelerated, it would look as if the pencil described a curved path while falling.

Since the space station is rotating it is accelerating continuously in the sense that it changes direction the whole time. When the pencil is released, it loses the centripetal force gluing it to a circular path around the hub of the space station. So it can no longer accelerate along with the rotating space station anymore and it flies off in a straight line, and eventually crashes into the floor somewhere different to where it was released. From the inside of the space station, it's path looks curved, just like from inside of the accelerating plane.

Rigil
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Mefiante
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« Reply #24 on: November 24, 2014, 15:50:59 PM »

I'm still not convinced.
Then you really should do the exercise I suggested.  Holler if you need help with that.

'Luthon64
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« Reply #25 on: November 25, 2014, 04:30:19 AM »

I'm still not convinced.
Then you really should do the exercise I suggested.  Holler if you need help with that.

'Luthon64

When I first got my head around this years ago, I also spent much time drawing diagrams (or at least imagining them), in between pulling out my hair and so on. :-)

I have always found mechanics and relative motion the most difficult aspect of physics, and indeed any science. Seems you need a particular kind of mind for it, that I apparently don't have. Which is not to say you can't understand it if you don't have that kind of mind, but you will have far more difficulty understanding it.
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Rigil Kent
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« Reply #26 on: November 25, 2014, 06:48:40 AM »

Being trained mostly in biology, of course I have my own list of pet peeves in movies
What was your take on The Human Centipede?

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« Reply #27 on: November 25, 2014, 08:59:17 AM »

I'm still not convinced.
Then you really should do the exercise I suggested.  Holler if you need help with that.

'Luthon64


Ok so from the web.

http://askthephysicist.com/ask_phys_q&a.html

Quote
When the ball is simply dropped from say h=1 m, the Coriolis force is relatively small because the velocity is small for most of the time of the fall. So, the deflection should be modest. I will try to estimate the amount of deflection. The centrifugal force is mg and is always radially out, choosing +y radially out, ay=-g+ayCor; I will argue that the ball does not acquire enough speed for the Coriolis force to have a significant radial component, so ayCor≈0, and  y≈h-½gt2, vy≈-gt. So, the time to fall is approximately t≈√(2h/g)=0.45 s and vy≈4.4 m/s. The Coriolis acceleration, if the velocity is purely radial, points in the backward direction (to the left in my figure above) and would have a maximum magnitude of about 2v√(g/R)≈3.9 m/s2. If the ball drops almost vertically the Coriolis acceleration would be approximately ax≈2m





So the pencil would not have gained enough Coriolis effect to be deflected much.

In conclusion i don't think you can berate the film makers to much.

A little more on it
http://en.wikipedia.org/wiki/Artificial_gravity#Rotation

Quote
To reduce Coriolis forces to livable levels, a rate of spin of 2 rpm or less would be needed. To produce 1g, the radius of rotation would have to be 224 m (735 ft) or greater, which would make for a very large spaceship
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« Reply #28 on: November 25, 2014, 09:33:25 AM »

I have always found mechanics and relative motion the most difficult aspect of physics, and indeed any science. Seems you need a particular kind of mind for it, that I apparently don't have.
You may well be right.  I’ve always found it useful to pick an appropriate frame of reference that allows the various motions to be described as simply as possible.  Of course, it takes some practice to recognise such, and one could still argue that there’s the “right kind” of intuition involved in selecting such a frame to begin with.



Ok so from the web.
My goodness, that physicist certainly knows how to complicate things inordinately!   That’s not to say those answers are wrong, merely far more confusing and convoluted than they need to be.  There is a far simpler and clearer approach.  Once again, I urge you to do the exercise yourself, rather than relying on vague hand-waving such as “the Coriolis force is relatively small” and “the deflection should be modest”, followed by an approximation of dubious worth.  Nothing beats a precise formulation because the Coriolis effect isn’t limited to letting go of a pencil close to the outer edge of a rotating spaceship.  Its reach is considerably wider than that.

ETA:  The ISS is a bit smaller than that 440+ metre diameter suggested by Wikipedia.

'Luthon64
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« Reply #29 on: November 25, 2014, 14:47:16 PM »

Being trained mostly in biology, of course I have my own list of pet peeves in movies
What was your take on The Human Centipede?

Rigil


Thankfully, I haven't seen it. :-)
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brianvds
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« Reply #30 on: November 25, 2014, 14:49:27 PM »

Quote
To reduce Coriolis forces to livable levels, a rate of spin of 2 rpm or less would be needed. To produce 1g, the radius of rotation would have to be 224 m (735 ft) or greater, which would make for a very large spaceship

Perhaps, once you have lived in such an environment for a while, you'll get used to the Coriolis effect. Back on Earth, you'd need to adjust all over to the "normal" world again.
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Mefiante
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« Reply #31 on: November 26, 2014, 11:11:26 AM »

Perhaps, once you have lived in such an environment for a while, you'll get used to the Coriolis effect. Back on Earth, you'd need to adjust all over to the "normal" world again.
That’s no doubt true, but it wouldn’t obviate the appearance of weird counterintuitive motions that prevail in a Coriolis environment.

In any case, the interested reader can get a sense of this weirdness the next time s/he encounters a merry-go-round:  Set it spinning at a good clip, hold on facing the centre post, and swing one leg as if attempting to kick the centre post.

'Luthon64
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brianvds
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« Reply #32 on: November 26, 2014, 14:40:51 PM »

Perhaps, once you have lived in such an environment for a while, you'll get used to the Coriolis effect. Back on Earth, you'd need to adjust all over to the "normal" world again.
That’s no doubt true, but it wouldn’t obviate the appearance of weird counterintuitive motions that prevail in a Coriolis environment.

One can imagine that in the future, people may grow up in such an environment, and perhaps end up with very different physics intuitions than ours.

Quote
In any case, the interested reader can get a sense of this weirdness the next time s/he encounters a merry-go-round:  Set it spinning at a good clip, hold on facing the centre post, and swing one leg as if attempting to kick the centre post.

'Luthon64

Or let one person stand in the middle, and another on the edge, and then try tossing a ball to each other.
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Rigil Kent
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« Reply #33 on: November 26, 2014, 14:47:25 PM »

Set it spinning at a good clip, hold on facing the centre post, and swing one leg as if attempting to kick the centre post.
I'll make a point of going on my next merry-go-round ride sober. As a rule, I can totally relate to Tolla's analysis of the centripital trajectory of vomit.  Lips Sealed

https://www.youtube.com/watch?v=JPrHM-Jfprg


Rigil
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