Name: 傾斜的里爾塔 (The Leaning Tower of Lire)
Uploaded: 2021-01-14T07:23:53.000Z
Duration: 15 min 9 s
Description: 【看影片學英語】數萬部 YouTube 影片，搭配英漢字典即點即查，輕鬆掌握單字發音與用法，長久累積看電影不必再看字幕。

未來進行式

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

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?

...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...

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

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...

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

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

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...

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,

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,

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

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.

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...

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

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.

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.

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

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.

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傾斜的里爾塔 (The Leaning Tower of Lire)

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