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  • This video contains the answers to my four revolutionary riddles,

  • so if you haven't seen the riddles yet, you should probably watch them before you watch the answers.

  • It's OK; I'll wait. Just click this card up here.

  • [Ticking clock sound]

  • Now, when I filmed the riddles, I also filmed the solutions at the same time,

  • but that was before I received your 15,000 comments and dozens of video responses,

  • so I'm re-shooting parts of this solutions video to incorporate the results I saw in the comments

  • and to use some of your video responses to help explain the solutions.

  • Let's get to it!

  • OK, by looking at the comments, for #1, over 15% of you said "a cylinder containing sand",

  • Now, this contains some powder.

  • [Chuckles] See how it rolls...

  • It, uhh, it doesn't seem to roll...

  • ...very far before it stops, and then it won't roll again

  • because I think the sand just kind of levels off in there.

  • Maybe you were thinking a bigger kind of sand,

  • like these small gravely stones.

  • Let's try that.

  • That actually rolls pretty well.

  • Nearly 25% of you said, "a cylinder half-full of water,"

  • so let's try that.

  • This rolls very well,

  • so it's not water...

  • and nearly 45% of you said, "a cylinder half-full of a viscous liquid."

  • Here, I have a half-full container of honey,

  • so let's see how it rolls...

  • That's not bad; it's rolling and it's stopping,

  • and it's rolling some more...

  • This is a pretty good guess,

  • and I think the behavior is not exactly the same as the mystery cylinder,

  • but it definitely is similar,

  • and that's no coincidence.

  • The mystery cylinder actually contains

  • honey...

  • and ping pong balls.

  • There are two ping pong balls submerged in this honey.

  • So if I place that on the ramp,

  • the center of gravity is not above the point of contact with the ramp, and so it rolls forward,

  • but now, because those ping pong balls are in the front,

  • they change the center of gravity, and so it's exactly over the point of contact,

  • and so it stops briefly,

  • but then, as the viscosity of the honey allows those ping pong balls to move up,

  • the center of gravity shifts forwards again, allowing this little container to roll.

  • So that is the trick of the mystery cylinder.

  • Pretty easy if you want to try it out at home.

  • Now, I challenged you to run two laps of this track, where the first lap, you could go as slowly as you like,

  • but the second lap, you had to go much faster,

  • such that your total average speed was twice the speed of your first lap.

  • Now, when I was first asked this question by Simon Pampena,

  • it took me a long time and scribbling on paper, and just something didn't seem to work out,

  • and that's because...

  • you can't actually do this.

  • It's impossible.

  • I mean, you might think I could run 3V₁ for my 2nd lap, and that would mean my total average speed is 2V₁.

  • The problem is you can't just add the two speeds together and divide by two,

  • because you spent much more time in your first lap, so that speed is weighted more heavily into the average,

  • so you'd have to run, well, impossibly fast.

  • Let me explain.

  • The velocity of the first lap was the distance around the track divided by t₁, the time it took you.

  • Now, if you want your total average speed to be 2V₁,

  • well then it needs to be 2d ÷ t₁.

  • You need to run twice the distance in the same amount of time it took you to run the first lap,

  • but you've already run that first lap, and so you have no time remaining to run the second lap!

  • Even if you went the speed of light,

  • you would not be able to increase your total average speed up to twice the velocity of your first lap.

  • It is just mathematically impossible!

  • So this may seem like a bit of a trick question,

  • but the point to me is how doable it sounds, how it seems like something you should be able to do,

  • but you can't. It's actually impossible.

  • Riddle #4, the question about the train,

  • was actually answered pretty well, with most people mentioning something to do with the wheels,

  • but of course, that makes sense, in a series of riddles which are about rotation, rotational motion.

  • Some people though did point out that maybe it was the steam that was going backwards,

  • or maybe air molecules in the train, and that is actually a pretty clever point,

  • however I wouldn't really consider the air in the train part of the train,

  • so indeed, the part of the train that is moving backwards

  • is the flange part of the wheel, which is below the rail.

  • That is the part of the train which is moving backwards.

  • To understand why, you just think about a spinning wheel.

  • The top of the wheel is moving forwards at speed 2V,

  • and the bottom of the wheel is not moving forward at all; it is stationary with respect to the track,

  • and that is what we call "rolling without slipping," and that's how most wheels work.

  • At least, that's how they're designed to work.

  • Now, in the case of trains, they have to have flanges so the train doesn't fall off its rails,

  • but of course, when these pieces come around during the rotation of the wheel,

  • they actually extend beyond the rail, and therefore, they are going backwards with respect to the ground,

  • so the part of the train that's moving backwards is always changing,

  • but it's always that part of the flange,

  • that part that extends beyond the wheel that is below the level of the track.

  • So what happens when you pull the bottom pedal of a bike backwards?

  • Well, about 45% of you thought that the bike would move backwards,

  • about a quarter said it would move forwards, and a quarter said the bike wouldn't move at all,

  • and 5% said it depends on something...

  • So let's give it a shot and see what happens.

  • I'm going to pull backwards on the bike pedal in 3, 2, 1...

  • Woah!

  • The bike did indeed move back,

  • and for virtually all bikes, this is what you will find,

  • but the explanation is not just as simple as "well, the net force on the bike is back,

  • so therefore, it has to accelerate backwards,"

  • and to prove that that logic doesn't work, well, just have a look at this video by George Hart.

  • [George] Watch this. Again, I pull the same pedal backward, but now...

  • the bike moves forward!

  • [Derek] I'll put a link to the full video here and a link to his website in the description.

  • So the reason the bike moves backwards is because of the way these gears are set up, the diameter of the tire,

  • and also, the distance from this crank to the pedal itself.

  • Because as a bike moves forwards,

  • the pedal, even when you're pushing back on it, never actually moves backwards with respect to the ground;

  • it's always moving forwards.

  • [George] So if you drag a string behind the pedal of a bike moving forward,

  • the string is always moving forward.

  • Now just play that movie backward in your mind,

  • and it may be clear how pulling the string backward could make the bike move backward;

  • They move forward together, so they move backward together.

  • [Derek] Another way to think about this

  • is to consider the path traced out by the pedal as the bike moves forward.

  • This is called the trochoid.

  • For all ordinary bikes, the pedals are moving much slower than the tires,

  • so the pedal is always going forward with respect to the ground,

  • but George modified his bike

  • so the ratio of the pedal to the wheel radius was greater than the ratio of the front sprocket to the back sprocket,

  • and this ultra-low gear changes the trochoid so the pedal DOES actually go backwards

  • with respect to the ground as the bike goes forwards,

  • and that's why he could pull back on the pedal

  • and make the bike go forwards.

  • This is the same reason why if you were to pull backwards on the flange of a train wheel,

  • you could actually get the train to move forwards, if you pulled backwards with enough force.

  • For all normal bicycles, pulling back like this on the bottom pedal will cause the bicycle to move backwards,

  • but, depending on the gear ratio, you can get the bike to move forwards,

  • so it was those people who said

  • it depends on the ratio of these gears and the size of this crank to the radius of the back wheel

  • that were actually the most correct.

This video contains the answers to my four revolutionary riddles,

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4つの革命的な謎が解決! (4 Revolutionary Riddles Resolved!)

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    林宜悉 に公開 2021 年 01 月 14 日
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