The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. The problem is usually framed like so: If you wrapped a rope around the earth, how much more rope would you need to add so that it would be 1 meter above the ground at all points? The answer is only 2π meters!
-> 2pi * (n + 1 - n)
-> 2pi * 1
-> 2pi
If I remember my algebra correctly. Someone else check my work I'm a dropout
x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
If, instead we approximate it as a fractal... then the distance is infinite, or at least highly dependent on the thickness of the rope!
The error in the original is assuming that the radius is proportional to the height above the earth (Earthradius=0?).
The latter. But that's only if it's not somewhat taut. Some tension brings it closer to a circle and makes the actual thickness pretty unimportant.
But I like the idea overall. It means that lifting up the string makes it smoother and it actually needs less length. How's that for being unintuitive?
P.S. Not a physicist, but my child is studying maths and physics at Uni at present, so I have it on good authority that this is still going on. They told me in their first week one of their classes had a worked example where the lecturer used the phrase "Assume the penguin's beak is a cone".
Do you mean making simplifying assumptions to make a problem tractable? Of course it’s still going on. It has to be, otherwise you just cannot do anything.
> Assume the penguin's beak is a cone
It is impossible to consider the true shape of a penguin’s beak for several reasons:
- you’d need to go all the way down to the electron clouds of the atoms of the beak, at which point the very concept of shape is shaky
- every penguin has a different beak so even if you describe perfectly one of them, it does not necessarily make your calculation more realistic in general.
There is a spectrum of approximations one can make, but a cone is a sensible shape at a first order. It’s also simple enough that students can actually do it without years of experience and very advanced tools.
What do you think they should do instead?
I totally understand why simplifying assumptions are helpful in modelling and definitely don’t need you to explain that. It also is a bit ridiculous if you think literally about it which makes it something that is fun to laugh about as here.
And I don’t talk about work at parties anyway :)
Imagine a world where every bit of humor is interpreted literally and then refuted pedantically! What kind of a world would that be?
Without forcefully dumping the geometric "intuition", this would still feel counterintuitive to me!
Maybe it’s because I’m a pilot and we never account for altitude when measuring distance, my intuition puts the difference at “effectively zero”. I also have it internalized that the earth’s atmosphere is very thin.
If you’re talking about friction… oooh that’s an interesting one. Intuitively yes. But is it also negligible?
...and indeed it does. Here is a discussion https://aviation.stackexchange.com/questions/24641/what-is-t...
For a particular model, flying above the model's cruising altitude should lead to lower fuel efficiency.
https://en.m.wikipedia.org/wiki/Turbofan
Did you mean turboprop?
Thrust itself decreases because there are also fewer molecules to push against, so it can get quite complicated if you want to account for everything. But overall it is easier to fly faster higher up in the atmosphere. Also, atmospheric currents are important.
There is a useful discussion here: https://aviation.stackexchange.com/questions/57209/how-does-...
As a meta point, our intuition often fails us hilariously when we are dealing with stuff that is out of the scale we have commonly seen in our lives. We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data".
Say the earth is disc-shaped. Then the center of gravity is only directly beneath you if you're standing at the exact center. You get ever-so-slightly not parallel lines, just like on a round earth.
The fun part of a disc-shaped earth comes as you move towards the sides, and gravity, still pointing towards the center, makes you stand at an increasingly acute angle to the surface. The ground beneath you will then appear like one big endless mountainside, with an increasingly steep slope the further away from the center that you get.
Edit: come to think of it, maybe that effect would let you adjust the direction of the force, too. Thinking about center of mass can be treacherous with more complex shapes...
I am just joking with you, I know what you mean, however the fruit was hanging too low not to pick.
I think you could construct a curve such that the mass's gravitational pull on the right cancels out the pull on the left, for any point on the surface.
I feel like there are better things to do with my time than be as fascinated by it as some people.
I guess it must be a pushing force from below.
So, who's doing the pushing? I'm thinking a big turtle.
My math might be wrong, but if we were accelerating at 9.8m/s/s for at least 4000 years (roughly as long as we have continuously recorded history and the minimum time “gravity” has been observed) then we ought to currently be traveling through space at over 1,000,000,000,000m/s.
Now I’m no physicist, but I reckon that might end up violating causality.
So I guess what I'm saying is I see absolutely no problem with the flat earth arguments?
The problem with trying to "explain" this is that fundamentally, flat-earthers, to the extent that they could be said to have a coherent world view at all, are usually a kind of occasionalist[1]. They don't _believe_ in natural laws or cause and effect. For the most part, they believe that god is in complete control of all events, and things go down because god wants them to go down. There's no required explanation for _anything_. The sun moves across the sky because god wants it to, and he could stop it or make it go backwards if he wanted it to, etc.
Indeed, that a flat earth is incompatible with physics is part of the appeal of believing in it to begin with. They _want_ to overthrow Newton, because a clockwork universe is incompatible with their belief system.
It's also sort of immune to any kind of argumentation. The result of any experiment is simply that god wants it that way, that they're predictable and testable doesn't _prove_ anything, because you can do an experiment a million times, and god could still cause it to fail any time he wants to. God just doesn't want to argue with Netwon right now, for his own reasons, you see.
It's not a bad way to look at it for a start. Things move down because it is their nature to move downwards. And this kind of empirical law is what we rely upon for most thought.
It takes a lot of work to get to a theory that makes more general predictions.
And even after having that, 98% of the time my thought is effectively just "things move down." Another 1.5% it's "things move down at 9.8 m/s/s". It's an extreme edge/special case when I'm thinking "massive things are attracted to each other, with a force proportional to their masses and inversely proportional to the square of their distances".
And even with "massive things are attracted to each other, with a force proportional to their masses and inversely proportional to the square of their distances" ... if you ask me why, it's because "uh, they just do that?"
To be fair, that is somewhat the current scientific consensus on gravity. It just is. We can meassure it and determine a general constant and calculate with it (and even though some people claim to have understood way more, it is highly debated terrain as far as I know)
And in general, I was actually arguing with flat earthers recently a lot, I even met a flat earther in real life. It is an interesting intellectual challenge debating them. Basically rebasing all the physical theory I have. (Main summary is, they have a high ego, but lack understanding of everything and make up for it with make believe.)
If I found a school one day, one of the lessons will be the teacher telling the students: "The earth is flat! Proof me otherwise." Or more advanced, model a flat earth on a computer. Flat earthers try that for real - it gets weird very quickly, so much that I could not believe anyone taking it serious and it all is just satire. But they are for real (but with a very different concept of reality).
Sure, and if we come up with some fancy unified theory, and ask "why" once more, the answer will still be "uh, because they do?."
> But they are for real (but with a very different concept of reality).
We think ourselves so advanced. I wonder what big counterfactual scientists believed in the 1900s and 2000s will be laughed at a few hundred years from now.
And, of course, some of that will be libel; e.g. that we thought the world was flat "just like Christopher Columbus's compatriots" [who didn't].
That’s why you never hear of people who went to the edge of that dis: they slid down that mountainside, and dropped off :-)
Alternatively, you can postulate that disc to be arbitrarily thick.
That will decrease the deviations. If that’s not enough to make them immeasurable, postulate that the stuff “deeper down” has higher density.
In the limit, just postulate that there’s an enormous black hole millions of light years below the center of the earth.
Flat-earthers probably won’t accept Newton’s theory of gravity, however, so you can make up anything.
It's actually measurable on a human scale:
https://www.mathscinotes.com/2017/01/effect-of-earths-curvat...
1 5/8" difference over 693', or slightly less than 1 part in 5 thousand --- definitely measurable on a smaller scale with accurate machinists' tools.
Every time I see the phrase "common sense", I expect to see an example of the human failing you describe.
In all seriousness one of the things about LLMs that most impress me is how close they get to human-style hallucination of facts. Previous generations of things were often egregiously and obviously wrong. Modern LLMs are much more plausible.
It's also why they are correspondingly more dangerous in a lot of ways, but it really is a legitimate advance in the field.
I observe that when humans fix this problem, we do not fix it by massive hypertrophy of our language centers, which is the rough equivalent of "just make the LLM bigger and hope it becomes accurate". We do other things. I await some AI equivalent of those "other things" with interest; I think that generation of AI will actually be capable of most of the things we are foolishly trying to press hypertrophied language centers into doing today.
As in, the illustration would be less wrong if it only used 22/7 for Pi and correctly portrayed dimensions of Earth and flight heights.
Implied in the caption is that the speed is the same at all heights (given that an increase in distance is implied as an increase in time.)
This is again obvious nonsense - speed is a function of thrust versus drag, and it's safe to say that both of those are affected by air density.
It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.
Can you expand on this? My brain is not connecting the dots.
Whereas if you slow down, you drop to a lower, shorter, higher-speed orbit.
> The mean radius of the earth is actually 3,459 miles or over 18 million feet.
That’s off by 500 miles; the correct figure is 3,959 miles. That makes it almost 21 million feet, and yields a ratio of about 1.0013378, even smaller than the quoted 1.0015.
So, when flying towards the east, catching these currents can significantly reduce flying time. When flying towards the west, we want to avoid them by flying below or elsewhere.
But it still seems to me that there might be a gravitational/inertial effects at play as well. At a (hypothetical) infinite altitude, it can no longer be said the the plane is moving perfectly in lock-step with the gravity/rotational acceleration of the earth. This implies the inertia of the plane relative to the rotation of the earth still has an effect at lower altitudes.
The effect might be tiny, but would be interesting to learn more about it nonetheless.
Something like this does exist under general relativity :
https://en.m.wikipedia.org/wiki/Frame-dragging
However,
> This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.
A more nuisance would be that earth rotating generate all sorts of things in the atmosphere, including winds and Coriolis effect on the winds, and you can account for that considering the winds. Btw a flight from Chile to France and back, will have a leg significantly shorter (up to 2 hs in a 13hs flight) and which leg it is, depends on the time of the year.
I get that what really matters is the relative motion, but it still seems to me that there might be a gravitational/inertial effects at play, even if tiny.
Consider this thought experiment: Planes cannot really fly into space, but assume they can. At a certain altitude, it cannot be said the the plane is moving perfectly in step with the gravity of the earth. At infinite altitude, that certainly cannot be the case.
So that tells me there is some deviation due to the inertia of the plane, even at low altitudes. Like I said, the effect might be tiny, but would be interesting to learn more about it nonetheless.
However, rotation of the earth imparts a coriolis force on the air, which results in jetstream winds. Aircraft routes are optimized to use/avoid jetstreams for shorter travel times.
Approximate Earth as a flat line. (The 5000ft path is close enough that it is also represented by the flat line. This is the 5000ft path.)
Then make the 33000ft path which is a slightly looser line on top of this line.
This new path is not 4 times longer. Just a little bit raised, because 33000 ft is “nothing” compared to Earth. (To become 4x longer we would go deep into outer space and back.)
these kind of things are intentionally wrong "puzzles" that are designed to get hundreds of people mad and post rebuttals to "drive engagement" or whatever. the pictures of a wheel with sledgehammers and chains and jacks with lugnuts plainly still in place and a post "how can i get this off it's stuck and i've tried everything". sigh... it's just another form of trolling.
sure enough, notice the sibling comments here. how many nice people took the time to patiently explain the fallacy(ies) for the 1000th time. then the pedants who correct the grammar/math/etc. in the 98% correct explanations. then the "true believers"/trolls who don't get it and argue back. and so on.
https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-...
“Bloomberg spent $500 million on ads. The U.S. population is 327 million. He could have given each American $1 million and still have money left over.”
(~42M feet diameter shown in ~134 pixels).
