I have seen a very similar (incorrect) argument used to justify the idea of a flat earth. A builder on youtube made the argument (with a similar out of scale drawing of the earth) that if he drops a plumb bob and makes a right angle so he has a straight horizontal line and then goes across that line for a bit and drops another plumb bob, the two lines he has dropped are parallel, "proving" that the surface of the earth must be parallel to the horizontal line and therefore flat and not curved. If the earth's surface was actually curved he argued then the two lines he has dropped should tilt slightly inward towards each other. Which of course they do. The earth is just much much much bigger than in the diagram so the effect is within the margin of error for the measurement he was taking.
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".
Ah, but would they actually be parallel on a flat earth?
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.
Depends what causes things to stick to the flat Earth. IIRC flat earthers have various explanations for gravity, including the disc continuously accelerating upward; in that case you'd experience the same force everywhere on it.
Standard flat-earther response is to scornfully deny the existence of gravity. It's all density/buoyancy you see... Gravity is a hoax promulgated by the notorious cabalist Newton, in service to his Illuminati/Papal masters, etc, etc.
I'm considering what flat-surfaced shape you could construct with equal gravitational pull at all points. Maybe something where the center is thin as a point, the edges have a lot of depth, and they curve towards the center either convex or concave. Might run some calculus to figure it out.
That way you should be able design a disc-shaped earth with constant strength of the gravitational force on the whole surface. But it would still have a center of mass (likely lying outside the shape you're describing, in the void beneath the center point), and the direction of the force should still be pointing towards that center, no? So the problem the GP has described, that you're starting to tilt as you move towards the edge, should remain in principle.
Related Wikipedia article: https://en.wikipedia.org/wiki/String_girdling_Earth#Implicat....
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!
And the text about the airplane problem was added on 2024-11-26: https://en.wikipedia.org/w/index.php?title=String_girdling_E...
This could be why dimensional analysis is one of the few things from physics class that can't be drilled enough..
Without forcefully dumping the geometric "intuition", this would still feel counterintuitive to me!
(2pi * (n + 1)) - (2pi * n)
-> 2pi * (n + 1 - n)
-> 2pi * 1
-> 2pi
If I remember my algebra correctly. Someone else check my work I'm a dropout
For convenience, we set τ=2pi. :-)
x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
How do you pronounce that symbol?
Tau. cf vihart: https://www.youtube.com/watch?v=jG7vhMMXagQ&t=0s
The only issue I see with this is that as a classic physics trope, we've approximated the earth as a sphere.
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?).
We actually model the earth as a very large spherical cow. This is approximately the same for most purposes but ends up being more convenient.
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".
A spherical cow /in vacuum/
> infinite, or at least highly dependent on the thickness of the rope
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?
Exactly, if you're only 1cm off the surface you follow every nook and cranny. If you're 10km off the surface only Everest is a blip.
Just because your initial fractal path is infinite does not imply that a line offset from it is also infinite (even for an infinitely thin rope), at least if the offset version is not self intersecting.
Even if the math of the arc length was correct (and you don't need to be a math professor to figure out it isn't) there's another logic misstep.
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.
> 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.
He is talking about orbital mechanics, rather than free space. When you are in an orbit, the shape of the orbit is determined by your speed. At every distance from the center of the object you are orbiting (such as the Earth), there is a speed that makes your orbit a circle. If you are going at any other speed then your orbit will be an ellipse instead. Too fast and your orbit rises higher above the Earth. Too slow and it dips back down closer to it. If you try to “catch up” with an object ahead of you in your orbit by speeding up you will only turn your orbit into an ellipse that gets further away from the Earth, and thus further away from the object you were trying to catch. Instead of catching it you’ll go up and over it. As Niven wrote, “forward is up, up is back, back is down, and down is forward”. It’s rather counterintuitive at first. Playing KSP can help you get a feel for it, especially once you start docking multiple craft together.
It’s even worse than that. By speeding up you end up actually getting further behind your target because in your new higher orbit you actually move slower on average, and as your average orbital radius gets longer, so does the circumference, so you end up on a "detour" trajectory compared to your target!
Whereas if you slow down, you drop to a lower, shorter, higher-speed orbit.
Just to point out here what's different between "space" and "not space": "Space" assumes no "height control",i.e. ways to exert force "down or up" along the earth-object direction. That's obviously not true for a plane. If you can exert force in that direction, you can change speed and keep the shape of the trajectory around earth constant.
This is called Kepler's Third Law, right? Radius^1.5 :: Period
One question I've always had with this: How does the rotation of the earth affect an airplane's flight time, if any? And how does this change with altitude?
I think the funny thing about this article is this numeric error (though not so egregious as the one that caused the article!):
> 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.
just 2piR and then extra h change the result very little fraction. How is that counter-intuitive :)
Curious, does the air being thinner affect flight time?
There's no straightforward answer because there are many factors that affect flight time and fuel economy, including the aerodynamics of the plane and the engine technology. I hazard a guess that for commercial airplanes these are chosen primarily for reasons of fuel economy per seat and then that determines the model's designated cruising altitude.
For a particular model, flying above the model's cruising altitude should lead to lower fuel efficiency.
From what I've learned reading AdmiralCloudberg's plane crashes analysis [1]: altitude heavily matters in fuel consumption. Jet planes use a lot less fuel at a higher altitude, up to the point that a plane on the verge of running out of fuel at a medium altitude might manage to squeeze in 50 or 100 more miles of flight by climbing 5000 feet, even accounting for the increased fuel consumption during climb. I guess that correlates with speed as well. Turbofan engines, on the other hand, are more fuel efficient than jet engines at lower altitudes, hence they remain common for interstate transit. The difference seems to be directly caused by the effect of air "thickness" on the engines.
Presumably this would affect drag significantly. Here are the equations of motion of an aircraft https://eaglepubs.erau.edu/introductiontoaerospaceflightvehi...
...and indeed it does. Here is a discussion https://aviation.stackexchange.com/questions/24641/what-is-t...
I think in a way that’s why planes fly so much further up than you’d think they’d need to. They want more consistent and minimal atmospheric conditions. Less air means less energy means less turbulence, I think?
If you’re talking about friction… oooh that’s an interesting one. Intuitively yes. But is it also negligible?
And of course pi = 22/7! ;)
That's actually a lot closer to the actual value of pi than the implied difference in the article vs the 0.15% actual difference in path length.
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.
Are we talking spacetime?