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]
