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The Leaning Tower of Lire

May 04, 2020
Hello. My name is Michael and in this episode of Michael's Toys we are going to play with: Blocks. I'm sure you've played with blocks before and noticed that it's quite fun to put one block on top of another. It keeps. You can continue doing this and build a

tower

as tall as you want. But what if you don't want the

tower

to just rise? What if you also want the tower to go to one side? How far can you go to the side... ...without falling? Well, this question is known as the block stacking problem and its solution is The Leaning Tower of Lire.
the leaning tower of lire
In fact, you can mechanically build a Leaning Tower of Lire, simply by touch, taking several blocks. Here I have five, and notice that when I place one block on top of another, that top block can be pushed outward... ...but only up to a certain point, beyond which its center of gravity: the point from which gravity appears be pulling it down; is no longer above the support, torque is produced and the object rotates. So if I make sure the center of gravity is right above the stand, it will stay. But now I can treat both blocks as a single object and balance them on a third block.
the leaning tower of lire

More Interesting Facts About,

the leaning tower of lire...

Now, just by feeling - without using math or engineering - I'm going to see how far... ...both blocks... ...can stick out from this third bottom. Well, okay, that's too much, but you can rebuild. Oh. Perfect. Fourth block. Well, it's not really heavy; I am very weak. Well. Now, this is... Can you go further? No, it's about... Wow, okay, the fifth block. Here we go, fifth block. Again, I'm taking this to the limit, to the extreme, every step of the way... ...but it's hard, because of course; I'm doing this... ...in real life. Now let's see... ...if... Well, this one needs to come in...
the leaning tower of lire
Good! 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 individual blocks can extend to one side as much as you like: but there are diminishing returns, because the amount of overhang we get with each new block decreases, and it decreases by a specific amount. Now, I did this with blocks that aren't really perfect: they have holes, they're not completely homogeneous, and I'm not very good at balancing things; but if you look at it, if you look at the spaces closely: you'll notice that here at the top, the top block may stick out from the second block by about half its length, but then the second block sticks out from 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...
the leaning tower of lire
Then would come 1/10, 1/12, 1/14, 1/16... This is a Harmonic Series. The numbers: the amount of overhang, gets smaller and smaller with each 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 is 1/8. BUT; Although the amount of protrusion we can get is getting smaller and smaller, it never reaches 0. Therefore, these blocks can protrude as much as we want; as long as we have enough. I pitched this concept to Adam Savage and in his workshop we built a Leaning Lire Tower... ...with more than five blocks.
ADAM: Michael, you want to build something or prove something. MICHAEL: I want to build a Leaning Lire Tower. ADAM: A

leaning

tower of Lire? ADAM: A

leaning

tower of Lire? MICHAEL: An inclination, yes. MICHAEL: It's all about the hangover. MICHAEL: No, not the bad kind, but the interesting kind. ADAM: Not the bad kind, yeah. Well. MICHAEL: I have a playing card here. ADAM: Yes. MICHAEL: And it's pretty obvious that it will balance on its center of mass, right? ADAM: Mmm, yes. MICHAEL: Well, I can hang the card on a table... MICHAEL: ...by lining it up so that exactly half of the card is off the table and the other half is on top.
It is balanced. ADAM: Right. MICHAEL: You can use a full length card after using, I think, only four cards. ADAM: Well, how about... how about two card lengths? MICHAEL: Two card lengths, you'll need 31 cards. ADAM: Three card lengths? MICHAEL: 227. ADAM: MICHAEL: Now, I know what you're thinking: how about six? MICHAEL: Now, I know what you're thinking: how about six? ADAM: Yeah, it's like a million. MICHAEL: Length of six cards: one hundred thousand. ADAM: Yeah, it's like a million. MICHAEL: Length of six cards: one hundred thousand. MICHAEL: Length of six cards: one hundred thousand. ADAM: One hundred thousand!?
MICHAEL: Because each next ledge is smaller than the last... MICHAEL: Because each next ledge is smaller than the last... ADAM: Yes. 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 are so thin that, already You know, 31 of them aren't even as thick as a deck of cards... MICHAEL: ...and 200 of them are only 4 decks. ADAM: 4 decks, yes. MICHAEL: So it's actually not that high; but they're also usually constructed with this type of air cushion... MICHAEL: So it's actually not that high; but they are also usually constructed with this type of air cushion...
ADAM: Well, they slide... ADAM: Well, they slide... MICHAEL: And- and they slide over each other, MICHAEL: and they are also difficult to measure, these fractions, because they don't have... ADAM: So, we could do this with wood; I have some... ADAM: I have some cheap plywood that, um, would be... ADAM: We could cut two hundred and forty pieces in a few minutes. MICHAEL: That would be amazing. ADAM: Okay, then... 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... I have a lot of doubts about that.
ADAM: Okay. ADAM: Do you want to work on these top twelve while I play with the... MICHAEL: Well, yeah, the top ones... MICHAEL: Okay, and this one... ADAM: Yeah, yeah, I see... ADAM: Back this way, back this way... Oh, perfect. ADAM: I don't think this is going to work. ADAM: O-Oh! Agree, hold. ADAM: Let's see here... ADAM: Let it go- Wow! Hey. ADAM: Look here... MICHAEL: Here's the one you drew. ADAM: Yeah, but I'm just, uh... Oh! MICHAEL: Look how close... ADAM: I'm 3/4 of an inch away from two full...! ADAM: I didn't think that was possible!
MICHAEL: So there are four ribbons that aren't even above the table. ADAM: That's amazing! ADAM: Okay, come on, protect him again. MIGUEL: Yes. ADAM: I want to get that last 3/4 inch and I can do it. ADAM: I'm pivoting backwards here... ADAM: So, I think this is the most we're going to get; and I'm about to mark it. ADAM: The one below is there... ADAM: ...the one above... ADAM: Is that so? Yes. MIGUEL: Yes. ADAM: We are 1/2 inch away from two full lengths of this hanging over the edge of the table. ADAM: I'm fucking impressed by that.
MICHAEL: This is a structure that we see in super old buildings. As before... ADAM: Really? MICHAEL: ...um, what's more, you know, permanent solutions were found to expand things; this... MICHAEL: ...worked. ADAM: Wow. ADAM: I really like that. ADAM: I'm going to get to the other side. I want to, um... ADAM: You know, we tried... ADAM: We tried for years, at Mythbusters, to come up with a good way to do "the straw that broke the camel's back." MICHAEL: Oh, yeah. ADAM: And it strikes me that we are... MICHAEL: How delicate is it? ADAM: No... I feel like...
MICHAEL: I feel like a card placed there would collapse it. ADAM: I feel like a card placed here could really help. ADAM: Ready? MIGUEL: Well, yes. ADAM: Well, here we go. MIGUEL: Oh, oh! ADAM: MICHAEL: So... MICHAEL: So, we were really at that center of mass, right here. ADAM: That was... MICHAEL: Is there a little more mass on that end? No. That's why this is not a great bridge. MICHAEL: I mean, it's an attractive bridge until someone crosses it. ADAM: Yes. MICHAEL: Once they pass the center of mass... ADAM: *boops* Yes. MIGUEL: Whoops! It is no longer balanced.
ADAM: *boops* Yes. ADAM: That was... deeply, deeply satisfying. MICHAEL: That was very, very fun. ADAM: Thank you, sir. MICHAEL: Thank you. It's fun to watch things fall, so let's talk about how they fall... When you have an object and its center of gravity is above a support, it stays. It's pretty stable, but if I tilt it a little... Ah! It falls. Because at a certain point, the center of gravity is no longer on a support and that force of gravity (actually, it is 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 makes contact with the ground.
Fall down. But not all objects have that property: look at this toy. This toy is inflated. In fact, it is filled with my air: my own breath; but if I tilt it... ...it never gets to the point where it falls over. In fact, if I let it go... ...it right itself. You can't beat it. This is a self-righting toy, and the reason it can always find its way back is that its center of gravity is not, say, in the middle. It is full of air, but there is also some sand at the bottom, and the sand is very dense.
It is denser than air, denser than water; So, the gravitational attraction it has towards the Earth is not geometrically located in the middle of the object, but rather it is located quite low: very close to the bottom of the object. So when you tilt, the center of gravity is always somewhere out here, and this kind of twisting causes it to straighten out. I can actually draw this for you using a Sharpie: This might make it a little easier. Here is our little self-righting shape. If the center of gravity is low enough: If I leave it down here... ...it will be stable, standing like this.
But if I rotate it, the center of gravity is now here and down it is like this; so the torque actually pulls the object up. This is called a state of equilibrium and is stable: because anything that moves the object away from this state has a tendency to bring it back. BUT; Technically, this is also a state of balance: because the center of gravity is just above the support. But; It's unstable, because to make a toy like this balance upside down, I have to be extremely precise... ...but I'm not. The slightest change: a small vibration, a small draft or any error in my balance; is magnified.
There are a lot of really fun toys that exploit this property. Here's a super cute, little, fun one that's a fox, and you just can't take it down. He will roll and turn if you put him on his stomach... ...he always wants his butt on the ground. It's endlessly fun; especially if you are a baby. A truly classic center of gravity toy is the classic balancing bird. Now this is a bird: it clearly looks like a bird, but its balance is quite surprising. It will balance right there on the tip of my finger. And they usually come with supports: this one has a pyramidal support.
The reason it can balance on its beak is because the tips of its wings are weighted; There is something heavy. Maybe it's metal, maybe it's clay; I don't know, but it weighs a lot. So the center of gravity of this object is not near the middle, the average location of all its material, but is drawn here. Because gravity attracts the heaviest and most massive parts more strongly towards the Earth. So the peak is the exact center of gravity, and as long as the center of gravity is above a support, the object does not fall. Quite quite beautiful.
Now I know what you're thinking: "Michael, self-righting toys are so much fun." "They are a great way to learn and demonstrate center of gravity, geometry and torque, but they are not scary enough." Well, luckily I have an answer for you. This is a Russian doll. It is very popular as a gift for young children and has bells so it makes noise when it moves. I would like to leave two of you alone. And as always, thanks for watching. You live in Australia? Will you be in Australia this January? Well, me too, along with Adam Savage. We're bringing Brain Candy Live to Australia.
We go to Perth, Melbourne, Brisbane, Sydney, Adelaide; It's going to be wonderful. Tickets are selling out, but you can get them in the description below – check it out. I hope to see you there.

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