Sorry for the forum/blog downtime today. Many things went wrong during davean’s heroic upgrade. (I blame the LHC.)
Feynman used to tell a story about a simple lawn-sprinkler physics problem. The nifty thing about the problem was that the answer was immediately obvious, but to some people it was immediately obvious one way and to some it was immediately obvious the other. (For the record, the answer to Feynman problem, which he never tells you in his book, was that the sprinkler doesn’t move at all. Moreover, he only brought it up to start an argument to act as a diversion while he seduced your mother in the other room.)
The airplane/treadmill problem is similar. It contains a basic ambiguity, and people resolve it one of a couple different ways. The tricky thing is, each group thinks the other is making a very simple physics mistake. So you get two groups each condescendingly explaining basic physics and math to the other. This is why, for example, the airplane/treadmill problem is a banned topic on the xkcd forums (along with argument about whether 0.999… = 1).
The problem is as follows:
Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?
The practical answer is “yes”. A 747’s engines produce a quarter of a million pounds of thrust. That is, each engine is powerful enough to launch a brachiosaurus straight up (see diagram). With that kind of force, no matter what’s happening to the treadmill and wheels, the plane is going to move forward and take off.
But there’s a problem. Let’s take a look at the statement “The conveyor belt is designed to exactly match the speed of the wheels”. What does that mean?
Well, as I see it, there are three possible interpretations. Let’s consider each one based on this diagram:
1. vB=vC: The belt always moves at the same speed as the bottom of the wheel. This is always true if the wheels aren’t sliding, and could simply describe a treadmill with no motor. I haven’t seen many people subscribe to this interpretation.
2. vC=vW: That is, if the axle is moving forward (relative to the ground, not the treadmill) at 5 m/s, the treadmill moves backward at 5 m/s. This is physically plausible. All it means is that the wheels will spin twice as fast as normal, but that won’t stop the plane from taking off. People who subscribe to this interpretation tend to assume the people who disagree with them think airplanes are powered by their wheels.
3. vC=vW+vB: What if we hook up a speedometer to the wheel, and make the treadmill spin backward as fast as the speedometer says the plane is going forward? Then the “speedometer speed” would be vW+vB — the relative speed of the wheel over the treadmill. This is, for example, how a car-on–a-treadmill setup would work. This is the assumption that most of the ‘stationary plane’ people subscribe to. The problem with this is that it’s an ill-defined system. For non-slip tires, vB=vC. So vC=vW+vC. If we make vW positive, there is no value vC can take to make the equation true. (For those stubbornly clinging to vestiges of reality, in a system where the treadmill responds via a PID controller, the result would be the treadmill quickly spinning up to infinity.) So, in this system, the plane cannot have a nonzero speed. (We’ll call this the “JetBlue” scenario.)
But if we push with the engines, what happens? The terms of the problem tell us that the plane cannot have a nonzero speed, but there’s no physical mechanism that would plausibly make this happen. The treadmill could spin the wheels, but the acceleration would destroy them before it stopped the plane. The problem is basically asking “what happens if you take a plane that can’t move and move it?” It might intrigue literary critics, but it’s a poor physics question.
So, people who go with interpretation #3 notice immediately that the plane cannot move and keep trying to condescendingly explain to the #2 crowd that nothing they say changes the basic facts of the problem. The #2 crowd is busy explaining to the #3 crowd that planes aren’t driven by their wheels. Of course, this being the internet, there’s also a #4 crowd loudly arguing that even if the plane was able to move, it couldn’t have been what hit the Pentagon.
All in all, it’s a lovely recipe for an internet argument, and it’s been had too many times. So let’s see if we can avoid that. I suggest posting stories about something that happened to you recently, and post nice things about other peoples’ stories. If you’re desperate to tell me that I’m wrong on the internet, don’t bother. I’ve snuck onto the plane into first class with the #5 crowd and we’re busy finding out how many cocktails they’ll serve while we’re waiting for the treadmill to start. God help us if, after the fourth round of drinks, someone brings up the two envelopes paradox.
xkcd 
With an aeroplane…surely velocity is measured in relation to the air and not the ground?
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Once again – consider the basic math.
Ignore the intricate complexities of the mechanics and just assume:
a) The treadmill and wheels can spin to infinite velocity
b) the axles of the plane can take such speeds
c) the belt constantly matches the speed of the wheels
d) the wheel/treadmill friction is enough that they never slip regardless the force involved
Now consider that the wheels are NOT driven by the engine but by the forward motion of the plane.
The plane is stationary on the belt and the engines are started. The plane starts to roll forward and the wheels spin up – as does the treadmill. The forward thrust of the engines continues to move the plane forward acellerating the wheels. The treadmill accelerates also. Because the plane *must* move faorward (ok the friction in the axels *will* slow it somewhat but the fact the wheels turn means that it doesnt just push it backwards) the wheels and treadmill will continue acellerating infinitely.
Regardless of what happens to the wheels/treadmill the engines will push the plane to the correct speed and it will take off.
This has been proven over and over and over again by physcist with much more important things to do 😛
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RE: Today’s comic. Might I recommend doggdot.us? It aggregates digg, slashdot, and del.icio.us into one convenient feed. Also you can add and subtract news feeds. Handy.
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Thanks for the replies.
I suppose I can understand this being proposed as a physics question that shows the uniqueness of a vehicle that is propelled with a jet engine rather than its wheels, and the way in which the treadmill can’t possibly stop it from moving forward.
That being said, the question “will it take off” seems to imply (to me) that we’re asking whether the airplane will generate any lift before it rolls off the end of the treadmill. As in Randall’s picture, I’m imagining a treadmill roughly the same length as the aircraft, not an infinitely long treadmill that the aircraft can accelerate to ~200 mph on. So, in my mind, even accepting the scenario that this is a physics question that assumes everyone knows the airplane can’t generate lift without airflow, the “will it take off” part still doesn’t make any sense within the bounds of common sense.
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Except, of course, for the first sentence in the problem statement.
Which says, “Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway.”
[Captcha: truly supposition]
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Pwned.
My mistake, I missed that part.
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Well /I/ don’t get it anyway. The plane’s on a treadmill, but there’s .. aw crap. Those bleedin’ jet engines, they force the air currents over the wings, don’t they. And here I was thinking 747s secretly were VSTOL and I never knew.
Oh, and I think the use of “was” is correct in the context of the Neil Armstrong thing, because it’s referencing a fact, not something that may be. If it were to reference anyone else, I think it should be “were”. Should doesn’t sound right..
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“Because the wheels on an airplane are free spinning — that is, the wheel turning is not what propels the plane, such as on a bicycle — they are unaffected by any motion that might be going on beneath them. Therefore, as long as the treadmill is the actual length of a runway, the plane will get up to speed and take off just like it would on a stationary surface.”
EXACTLY! The wheels may experience twice as much acceleration, however, most likely causing their failure, so it wouldn’t matter.
EXCEPT the friction between the wheels and the ground (which would slow up the plane considerably, in addition to lighting it on fire) and the fact that the airplane’s load rests upon the wheels preceeding takeoff!
Therefore something would happen that would resemble if a plane tried to take off on a 50 metre runway or before it could gain significant speed– the speed would be a lot less than normal, but it should still produce forward motion…
UNLESS the plane is light enough to take off anyways— planes can take off at a lot less of the speed than normally on commercial flights. Without passengers to worry about (safety or mass wise) it would still be able to take off.
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“Why’s the fuel light blinking?”
“Tanks are empty – we spent all the money on the giant conveyor belt.”
“Oh.”
“Drink?”
“I thought you’d never ask.”
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Scium has nailed it 🙂
THe problem is simply workign out if the extra amount of *friction* and other forces slow the plane enough that it cant take off in the required space.
And when you consider the power of those engines I think it’s easy to guess the answer 🙂
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This image shows the truly intuitive solution to the ‘plane and conveyor belt’ problem. As you can see, the plane won’t take off.
That said, the force of friction on the plane is going to be uk*Fn with very little difference based on the speed of the two surfaces(The mech. engineer I asked said you wouldn’t even bother considering anything besides uk*Fn). The net force on the plane, therefore, will effectively be:
Fnet = Fengine – uk*Fn – Fair
Fengine is incredibly massive compared to uk*Fn, and Fair would be nothing at rest, so at rest you could approximate as F = Fengine, and the plane will move forward at full speed.
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I think it’s a great problem since it introduces relativity (moving ground) into a situation that usually calls for basic Newtonian equations.
The absolute Newtonian speed of the plane is zero.
The speed of the plane relative to the ground is the same as it ever was.
Now if it the engine thrust doesn’t have enough vertical component on it’s own to initiate take off, then you’d need an airplane on a treadmill in a windtunnel, but at no point is “absolute movement” a requirement for take-off. Math can be deceptive.
I think we can move onto the Two Envelopes Problem. The initial paradox results from calculating the expected value of the second envelope without calculating the value of the first, unopened envelope (both are in the same condition). The “harder” problem resulting from opening the first envelope is not a paradox at all. It basically boils down to the fact that it’s better to make double your money than to loose half. By trading the envelope, you’re taking part in an advantageous gamble. Even if you involve two twins in the equation, and one must loose, both are making the rational choice by trading, the fact that each has to win at the other’s expense doesn’t affect the situation, since most gambling happens this way.
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I realize I’m falling into Randall’s trap. And I’m doing it anyway.
@Marcus,
Why must you be so wrong?
In a single sentence you say “Now if the engine thrust doesn’t have enough vertical component…”
No one is seriously considering the vertical component of the thrust. Even if it does happen to be non-zero, it’s still way too small. The only way this plane is getting off the ground is by moving forward along the belt.
And you go on to say “then you’d need an airplane on a treadmill in a wind tunnel.”
Where did this wind tunnel come from? Why do people keep suggesting a wind tunnel? If you got a big enough wind tunnel, you could make a 747 leave the ground with its engines off and the wheel brakes on. Treadmill or no.
But that’s entirely beside the point and outside the scope of the original question. What we’re saying is, a normal jet aircraft, on a “normal” treadmill (as long and wide as a runway) will be perfectly capable of moving down the belt at sufficient speed to achieve takeoff.
[Or, if you subscribe to group #3, believe that friction should be accounted for, believe in infinite speed treadmills, and believe that this is the correct mathematical interpretation of the problem, despite there being a much more rational one suggested by group #2, then the math says the plane cannot have a non-zero speed. And so its stuck in place by infinity.]
Please leave your giant wind tunnel at home until we move on to the next lab, where we will be levitating giant squid with it.
Next: I think you’ve misunderstood the paradox part of the Two Envelopes Problem.
You say: “The ‘harder’ problem resulting from opening the first envelope is not a paradox at all. It basically boils down to the fact that it?s better to make double your money than to loose half.”
But you see, when you first chose an envelope, the odds were 50/50 that you’d get the one with more money in it. Any swapping thereafter will not change those odds. Most rational people can see right away (in a “common sense” way) that there’s no reason to swap after your first choice.
Looking in one envelope hasn’t changed that.
And yet by your (apparently correct) reasoning, if you’re offered two envelopes, choose one, and peek in it, you’ll want to swap for the other one EVERY SINGLE TIME.
That doesn’t seem like a paradox to you?
Let’s try this. Let’s play the game (with pennies) 10,000 times each. We’ll get someone fair and consistent to randomize how much money is in each game, and keep track of how much each of us has won.
I’ll pick an envelope at random and keep the contents.
You pick one at random, look in it, and decide to swap it for the other envelope every time.
After 10,000 games, will you have substantially more money than I?
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@Spike
infinity times zero undefined? hmm, am noit mathematician so mthat may be true, I prefer to use a thought experiment:
Consider the universe to be infinite (at least in space if not time), if there is any probability at all of a civilisation forming with the ability to travel faster than light then logically in such an infinite universe there must be one, or, more importantly, an infinite number of them. with an infinite number of civilisations able to travel anywhere instantly (OK am taking faster than light travel to mean instantaneous to anywhere) then it is also logical that one should show up on my doorstep this morning, or in fact an infinite number, everywhere, all the time. I’m not trying to be silly, so instead am prepared to accept that in a nearly infinite universe there may be slight limitations on this, in which case I’ll only ask for a single alien, sometime in my life. as one hasn’t appeared thus far I’m prepared to consider infinity (number of civilisations) times zero (probability of one constructing faster than light device and visiting me) to be zero.
Yes I know the argument is very flawed simply in that the universe isn’t infinite (Olber’s paradox), but I just can’t accept that infinity times zero is anything but zero
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There’s going to be a maximum force that the treadmill can exert upon the mass of the airplane, especially given that the wheels themselves are free-wheeling.
Once the airplane overcomes the coefficient of static friction, it’s just going to slide, acting with the force of kinetic friction. If the treadmill is travelling at a speed approaching infinity, I’d actually argue you’d be looking at the coefficent of kinetic friction falling to approach zero. The time for the electro-chemical reaction which causes friction between two masses would fall, and soon approach zero, and the frequency of the two depths on the tread would approach infinity, approximating a smooth surface with no coefficient of static friction..
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THE GODDAMN MYTHBUSTERS ALREADY SOLVED THIS. HERE’S THE FUCKING VIDEO.
Now shut the fuck up about it already. You can’t refute experimental evidence.
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What if you put the Large Hadron Collider on a treadmill?
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Dear poster named “MYTHBUSTERS”,
I find your amazing gift for combining capitol letters and profanity quite compelling. You’ve totally swayed my opinion with your profound arguments.
I saw the episode you’ve linked when it first aired. What, its existence is supposed to prevent people from discussing the topic?
I feel perfectly capable of refuting experimental evidence. Especially when said evidence is based on poor science and an unclear statement of the original problem.
Not that, in this case, I dispute their conclusion. Just some fine points regarding their assumptions and methods.
The continued existence of people who think they can tell the internet to “shut up” amazes me. What gave you, all-caps-boy, the idea that you have any control whatsoever over what people say or don’t say in response to someone else’s blag post? Srsly!
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that was longggg….. I like #2 (:
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@anglave… don’t bother, he is only the living embodiment of an internet archetype. if you strike him down, a thousand more will rise up in his place. they can never be silenced, only ignored.
@everyone… if the treadmill is allowed to accelerate to infinity, and we assume that friction is not absent, the drag against the wheels will eventually match the thrust produced by the engines. I’m just saying. The plane CAN be stopped.
couldn’t “the plane cannot have a nonzero speed” be interpreted to mean “the plane cannot move?”
that’s essentially what the equation is saying… unless friction brings the plane to a standstill, the terms of the equation will not be satisfied.
@SJ Zero… wtf are you talking about? seriously, it sounds interesting, but i just don’t understand it.
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I’ll keep saying it until everyone gets the message:
The plane’s engines push against the AIR, not the CONVEYOR BELT.
Nothing the conveyor belt does, backwards or forwards, will prevent the engines pushing against the AIR and pushing plane forward until it reaches takeoff speed.
Now everyone repeat after me: AIR isn’t CONVEYOR BELT
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PI IS EXACTLY 3.
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The plane would hit the handle bars at the end.
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@says: I get it, i get it… the plane’s engines push against the air. i understand that. on any realistic conveyor belt, i’m sure the plane would take off just fine. what i’m saying is that at an extremely high (ridiculous, unattainable) speed, the friction between the treadmill/wheels/axles would start to drag the plane down, eventually reaching a state of equilibrium where the treadmill, moving at about a hundred thousand miles per hour, would push backwards just as hard as the plane’s engines could move it forwards. then, after a couple of seconds, the plane’s wheels would disintegrate, the landing struts would come into direct contact with the treadmill, the plane would be hurled backwards, do somersaults down the runway, hit the asphalt, and explode.
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Ah – that last point is where the Conveyor-Belt Fairies (TM) come in…
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If its wheels are aligned at tangents to the spindle then yes it will.
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and that Wikipedia article on the 2-envelope paradox could be improved – the reasoning fails at step 6, not step 7. Someone is wrong on the internet!
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What we need is a great big conveyor belt,
Big enough to take a plane and all it’s got.
Keep it rolling for a hundred years or more –
Turn out multi-coloured cocktails by the score…
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Sorry – it’s Friday afternoon in Australia, I’m bored, & I’ve been eating sugar-coated peanuts.
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Dude, if you want to see a close race where the reds are neck-and-neck with the blues, a battle of intelligence against stupidity, check out this Gogle trends graph of how XKCD has been doing against its nearest competition:
http://www.google.com/trends?q=%22XKCD%22%2C+%22retard%22%2C+%22your+mom%22&ctab=0&geo=all&date=mtd&sort=1
It’s especially good when you consider that ‘retard’ is also a word in French . . .
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Oh, god. Did this really turn into the argument that Randall specifically intended to stop? This is so ironic I could crap.
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they did an ep of mythbusters about this matter
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I love the warning label at the bottom of your main page. No where does my Bachelors of Art become more obvious then when you start putting physics equations on stuff.
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Today’s comic fits well with the mouseover caption from comic 262 http://xkcd.com/262/
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I figure you guys have probably seen this, owing to your completely understandable velociraptor phobia. Just in case you haven’t, I’m curious to see what your results would be.
http://www.bunkbeds.net/velociraptor/index.php
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The brachiosaurus looks familiar. Is that Gertie?
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Well, the plane’s speed may depend on if it’s listening to Smack My Bitch Up on its iPod.
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Well, the plane’s speed would also depend on whether or not it’s listening to Smack My Bitch Up on its iPod.
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oh.
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@ET: i would last 51 seconds btw
@everyone: Hi!! *wave*
@Randall: My nice story – Today is Friday. I do not have to work tomorrow. Time to get drunk.
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Haha, some friends and I were just talking about this one on Monday night. Two engineers and a retired computer programmer with a sport pilot license (none of whom were me) came to conclusion #2.
I’m a wet-lab biologist. I can do arithmetic and logs but not dynamics.
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Listen, Anglave, the point of my post was not to try to silence everyone on the internet. It was to point out the profound idiocy of trying to debate a topic that has already been put to rest.
The question of whether or not an airplane on a treadmill will take off has been answered: Yes. Yes, it will.
Did we see an answer in the form of an equation? No, we saw it in the form of an experiment. An experiment fully capable of being reproduced. We don’t need to check the math or hypothesize further once the experimental data has been confirmed. We can continue to debate HOW it might work, yes, but not WHETHER it will work, as we know that it will.
Really, I was trying to shut up all the moronic “it can’t take off because it’s remaining stationary” nerds here who think they understand physics. If you or your fellow blag comment posters want to keep yourselves in the dark, be my guest. But isn’t the point of debate to arrive at the TRUTH? You act as if the only reason argument exists is to determine who has the bigger e-penis. That’s just asinine.
Yeah, fuck me for writing such a lengthy response, but it deserved to be said.
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Regardless of whether or not it moves, the plane is never going to take off. In order to get up to take-off speed, there’s some taxiing where it builds up inertia on the ground and the engines build up power by increased intake of air. Since the plane’s not accelerating (or at least, not quickly) it’s not going to get the airflow required to make the engines produce enough thrust to get off the ground.
Right?
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@William L: Wrong. A plane can take off in a slight wind, which would have much more effect on the things you’re talking about than its wheels being turned backwards would. Its movement in relation to the ground is mostly irrelevant, and its engines are pretty good at accelerating, or increasing airflow through them. That’s how it takes off in the first place. We’re not talking about scramjets here.
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Pi = 3
And furthermore, someone please ban Mr. Hentaifapper.
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Motion seconded. Pi is exactly equals 3.
Oh, and that other part, too.
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More specifically, if the experiment began with the treadmill moving the airplane backward at seven miles per hour, and then the plane turned on its engines, it could still take off safely (otherwise known as a 6 knot tailwind).
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Hmm seems a lot of this argument revolves around another unspecified point:
Friction.
If we consider an idealised system where no friction to do with the wheels exists, then the plane can take off, else, if the rotation of the wheels is able to exert a frictional forec on the plane, then it can’t
Further, yes, we all get the idea, mythbusters di an episode on this, but by saying lack is like doing an experiment with a cat, a box and some nerve gas and observing the result and thinking this then means nobody should ever talk about it again (or tell Schrodinger, come on, like the guy’s going to notice mittens isn’t around?)
(incidentally the human test below has the words “Italy and worth-while”… oxymoron?)
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