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  • Hi.

  • My name is Michael, and on this episode of Michael's Toys, we are going to be playing with:

  • Blocks.

  • I'm sure you've played with blocks before, and have noticed that it's quite fun to put one block on top of another.

  • It stays. You can keep doing this and build a tower as tall as you want.

  • But what if you don't want the tower to just go up? What if you also want the tower to go to the side?

  • How far can it reach over to the side...

  • ...without falling over? Well, this question is known as the Block Stacking Problem

  • and its solution is The Leaning Tower of Lire.

  • You can actually mechanically build a Leaning Tower of Lire, just by feel, by taking a number of blocks.

  • Here I've got five, and notice that when I put one block on top of another, that top block can be pushed out...

  • ...but only to a certain point,

  • beyond which its center of gravity:

  • the point from which gravity appears to be pulling it down; is no longer above the support, and a torque is produced,

  • and the object rotates off. So, if I make sure the center of gravity is

  • just above the support, it will stay.

  • But now, I can treat both of these blocks like a single object, and balance them on top of a third block.

  • Now, just by feel -not using math or engineering- I'm just gonna see how far out...

  • ...both blocks...

  • ...can overhang this third bottom- [block drops] Well, okay, that's too far, but you can rebuild.

  • Whoa.

  • Perfect. Fourth block.

  • Okay, it's actually not heavy; I'm just really weak. Okay. Now, this is...

  • Can it go farther- Nah, it's about...

  • Whoa, okay, fifth block. Here we go, fifth block.

  • Again, I am pushing this to the limit, the extreme, at every step of the way...

  • ...but it's rough, cause of course; I'm doing this...

  • ...in real life. Now let's see...

  • ...if...

  • Okay, this one needs to come in...

  • Nice!

  • So, we have built here the beginning of a Leaning Tower of Lire:

  • I say beginning, because this tower will have no end.

  • You can keep doing this forever; and your tower of single blocks can reach out to the side as far as you want:

  • but there are diminishing returns, because the amount of overhang we get with each new block goes down,

  • and it goes down by a specific amount.

  • Now again, I did this with blocks that aren't really perfect: they've got holes in them,

  • they're not completely homogeneous, and I'm not that great at balancing things; but if you look at it-

  • If you look at the gaps closely: you'll notice that here at the top, the top block can overhang the second block

  • by about 1/2 of its length, but then the second block overhangs the third by about 1/4, and then we have 1/6,

  • and then we have 1/8.

  • 1/2, 1/4, 1/6, 1/8...

  • Next would come 1/10, 1/12, 1/14, 1/16...

  • This is

  • a Harmonic

  • Series.

  • The numbers: the amount of overhang, becomes smaller and smaller for every new block we add.

  • In fact, it turns out to be 1/(2n), where n is the number of blocks.

  • Here, we have 4 blocks, and the overhang is 1/(2n). 2 * 4 = 8, so it's 1/8.

  • BUT;

  • Even though the amount of overhang we can get keeps getting smaller,

  • it never reaches 0. So, these blocks can overhang as far as we want; as long as we have enough of them.

  • I brought this concept up to Adam Savage, and in his workshop, we built a Leaning Tower of Lire...

  • ...with more than five blocks.

  • ADAM: Michael, you want to build something or demonstrate a thing.

  • MICHAEL: I want to build a Leaning Tower of Lire.

  • ADAM: A Leaning Tower of Lire?

  • ADAM: A Leaning Tower of Lire? MICHAEL: A Leaning [Tower of Lire], yeah.

  • MICHAEL: It's all about hangover.

  • MICHAEL: Not- not the not the bad kind, the interesting kind. ADAM: Not the bad kind, yeah. Okay.

  • MICHAEL: I have a playing card here. ADAM: Yeah.

  • MICHAEL: And it's pretty obvious that it's gonna balance on its center of mass, right? ADAM: M-hm, yeah.

  • MICHAEL: Well, I can overhang the card on a table by...

  • MICHAEL: ...lining up so that exactly half of the card is off the table and half is on. It's balanced. ADAM: Right.

  • MICHAEL: You can go out a full card length after using only, I believe, four cards.

  • ADAM: Okay, what about- what about two card lengths?

  • MICHAEL: Two card lengths, you're gonna need 31 cards.

  • ADAM: Three card lengths? MICHAEL: 227.

  • ADAM: [laughter]

  • MICHAEL: Now, I know what you're thinking: what about six?

  • MICHAEL: Now, I know what you're thinking: what about six? ADAM: Yeah, it's like a million.

  • MICHAEL: Six card length: a hundred thousand. ADAM: Yeah, it's like a million.

  • MICHAEL: Six card length: a hundred thousand.

  • MICHAEL: Six card length: a hundred thousand. ADAM: A hundred thousand!?

  • MICHAEL: Because each next overhang is smaller than the last...

  • MICHAEL: Because each next overhang is smaller than the last... ADAM: Yeah. I see.

  • MICHAEL: ...and the order is simply 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, ...

  • MICHAEL: Playing cards are great because they're so thin that, you know, 31 of them is like not even as thick as a deck...

  • MICHAEL: ...and 200 of them is only 4 decks. ADAM: 4 decks, yeah.

  • MICHAEL: So, it's actually not that tall; but they also are usually built with this kind of air cushion...

  • MICHAEL: So, it's actually not that tall; but they also are usually built with this kind of air cushion... ADAM: Right, they- they slide...

  • ADAM: Right, they- they slide... MICHAEL: And- and they slide across each other,

  • MICHAEL: and they're also hard to measure, these- these fractions, because they don't have...

  • ADAM: So, we could do this out of wood; I have some...

  • ADAM: I have some cheap plywood that, um, would be-

  • ADAM: We could cut out two hundred and forty some odd pieces in a few minutes.

  • MICHAEL: That would be incredible.

  • ADAM: All right, so... By your measure, we should be able to hang out...

  • ADAM: ...two full lengths of these... MICHAEL: Yeah.

  • ADAM: ...within 31 of these bricks. MICHAEL: That's right.

  • ADAM: I'm- I'm so dubious about that.

  • ADAM: Okay.

  • ADAM: Do you want to work on the- these twelve top ones while I play around with the...

  • MICHAEL: Well yeah, the top ones...

  • MICHAEL: Okay, and this one... ADAM: Yup- yup, I see it...

  • ADAM: Back this way, back this way- Oh, perfect.

  • ADAM: This doesn't strike me as it's gonna work.

  • ADAM: O-Oh! Okay... Hold on.

  • ADAM: [mumbling] Let's see here...

  • ADAM: Let go- Whoa! Huh.

  • ADAM: [Let's] See here... MICHAEL: Here's the one that you drew on.

  • ADAM: Yeah, but I'm just, uh... Oh!

  • MICHAEL: Look how close... ADAM: I'm 3/4 of an inch away from two full [lengths]...!

  • ADAM: I didn't think that was possible! MICHAEL: So, there are four slats that are not even above the table.

  • ADAM: That's mind-blowing!

  • ADAM: Okay, we come- protect it again. MICHAEL: Yes.

  • ADAM: I wanna get that last 3/4 of an inch, and I can do it.

  • ADAM: I'm pivoting on the back here...

  • ADAM: So, I think this is the most we're gonna get; and I'm about to mark it.

  • ADAM: Bottom one is there...

  • ADAM: ...top one...

  • ADAM: Is that right? Yup. MICHAEL: Yeah.

  • ADAM: We're within 1/2 of an inch of two complete lengths of this hanging out over the edge of the table.

  • ADAM: I am frickin' blown away by that.

  • MICHAEL: This is a structure we see in super ancient buildings. Like before... ADAM: Really?

  • MICHAEL: ...um, more, you know, permanent solutions were found for stretching things up and across; this...

  • MICHAEL: ...worked. ADAM: Wow.

  • ADAM: I really dig that.

  • ADAM: I'm gonna get on the other side. I want to, um...

  • ADAM: You know, we tried...

  • ADAM: We tried for years, on Mythbusters, to think about a proper way to do "the straw that broke the camel's back".

  • MICHAEL: Ooh, yeah.

  • ADAM: [laughter] And it strikes me as we're...

  • MICHAEL: How touchy is it? ADAM: I don't- I feel like...

  • MICHAEL: I feel like a card placed there would- would collapse it.

  • ADAM: I feel like a card placed here might actually do it.

  • ADAM: Ready? MICHAEL: Well, yeah.

  • ADAM: Okay, here we go.

  • MICHAEL: Oh, oh!

  • ADAM: [laughter] MICHAEL: So...

  • MICHAEL: So, we were really on that center of mass, right here. ADAM: That was...

  • MICHAEL: There's a little more mass on that end? No. That's why this is not a great bridge.

  • MICHAEL: I mean, it's a cool-looking bridge until someone walks across it. ADAM: Yeah.

  • MICHAEL: Once they pass the center of mass... ADAM: *boops* Yeah.

  • MICHAEL: Oops! It's not balanced anymore. ADAM: *boops* Yeah.

  • ADAM: That was... deeply, deeply satisfying. MICHAEL: That was really, really fun.

  • ADAM: Thank you, sir. MICHAEL: Thank you.

  • It's fun to watch things topple over, so let's talk about toppling...

  • When you have an object, and its center of gravity is above a support, it stays.

  • It's quite stable, but if I tilt this a little bit...

  • Ah! It falls over.

  • Because at a certain point, the center of gravity is no longer over a support and that gravity force-

  • -actually, it's space-time curvature, but we can think of it as a force-

  • -causes the object to rotate around a pivot point: which in this case, is right where it contacts the ground.

  • Falls over.

  • But not all objects have that property: take a look at this toy.

  • This toy is inflated. It's full of my air, in fact: my own breath; but if I tilt it over...

  • ...it never reaches a point at which it falls over. In fact, if I let go...

  • ...it rights itself.

  • You cannot beat it.

  • This is a self righting toy, and the reason it can always find its way back up is that its center of gravity is not, say, in the middle.

  • This is full of air, but there's also a bit of sand down at the bottom, and the sand is very dense.

  • It's more dense than air, more dense than water; so, the gravitational attraction it has to the Earth

  • is not located in the middle of the object geometrically,

  • but it's located quite far down: very near the bottom of the object.

  • So, when it tilts, the center of gravity is always somewhere out here,

  • and this kind of a torque causes it to right itself.

  • I can actually draw this out for you using a Sharpie: this might make it a bit more easy.

  • Here is our little self-righting shape. If the center of gravity is low enough: like, if I put it down here...

  • ...it'll be stable, standing up like this.

  • But, if I turn it, the center of gravity is now here,

  • and down is like this; so the torque actually pulls the object back up.

  • This is called an equilibrium state, and it's a stable one:

  • because anything that moves the object away from this state has a tendency to bring it back.

  • BUT;

  • This is technically also an equilibrium state: because the center of gravity is right above the support.

  • But; it's unstable, because in order to make a toy like this balance upside down,

  • I have to be extremely precise...

  • ...but I'm not.

  • The tiniest change: a little vibration, a little air current, or any mistake in my balancing; gets magnified.

  • There are a lot of really fun toys that exploit this property.

  • Here's a super cute, little, fun one that is a fox, and you just can't knock this fox over.

  • It'll roll, and spin, if you put it up on its head...

  • ...it always wants its butt on the ground. It is endlessly fun; especially if you're a baby.

  • A really classic center of gravity toy is the classic balancing bird.

  • Now, this is a bird: it clearly looks like a bird, but it balances quite surprisingly.

  • It'll balance right there, on my fingertip.

  • And they often come with stands: this one has a pyramid stand.

  • The reason it can balance right on its beak is that the wingtips are weighted; there's something heavy.

  • Maybe it's metal, maybe it's clay; I don't know, but it's very heavy.

  • So, the center of gravity of this object isn't somewhere near the middle, the average location of all of its material,

  • instead, it's drawn out here.

  • Because gravity is attracting the heavier, more massive parts towards Earth more strongly.

  • So, the beak is the exact center of gravity, and as long as the center of gravity is above a support,

  • the object

  • the object doesn't

  • the object doesn't fall.

  • Quite

  • Quite beautiful.

  • Now, I know what you're thinking: "Michael, self-righting toys are really fun."

  • "They're a great way to learn and demonstrate the center of gravity, geometry, torque; but they're just not creepy enough."

  • Well, luckily, I have an answer for you.

  • This is a Russian doll.

  • It's quite popular to give to little children, and it has bells in it, so it makes a noise as it moves.

  • I'd like to leave

  • two of you alone.

  • And as always,

  • thanks for watching.

  • Do you live in Australia? Are you going to be in Australia this month, January?

  • Well, so will I, along with Adam Savage.

  • We are bringing Brain Candy Live to Australia.

  • We're going to Perth, Melbourne, Brisbane, Sydney, Adelaide; It's going to be a blast.

  • Tickets are running out, but you can get them in the description down below: check that one out.

  • I hope to see you there.

  • [Michael gives a gentle, embracing kiss]

Hi.

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B1 INT US adam michael gravity overhang tower block

The Leaning Tower of Lire

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    Amy.Lin   posted on 2018/02/23
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