Let's build bridges.
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Normally I'd build a bridge say across a river.
So suppose that this blue area here is a river.
And what would it mean to build a bridge across this river?
Well I'd probably put pylon on either side, on either of the two banks.
And then I'd build the bridge across those those two pylons.
Here's a diagram.
Alright I've got the earth, I've got the
river, here are the two banks of the river.
And then I can build a bridge that goes from
one bank to the other, but it's attached to both banks.
But I want to think about a one-sided bridge.
Instead of being connected to both banks, I just
want to be connected to one of the banks, sure.
So instead of building a bridge that's attached to both banks,
I want to build a bridge that's just attached to one bank.
And then I want to know,
how long can I make this bridge before it collapses?
Well, I mean look.
If, if you allow me to build a bridge from
some super strong metal like unobtanium, well no problem right?
I mean, I'll just build a really, really long bridge, there's
going to be no limit to how long that bridge could be.
And that's not really what I mean, right? What I'm really going to ask you to do.
Is to build the bridge out of these blocks.
All right, so you're allowed
just to stock blocks on top of each other.
And I want the thing to be stable, I don't want it to fall over.
And I want to know, if you start stocking blocks,
how long of an overhang can you get alright?
What's the maximum possible overhang that you can achieve if I give you n blocks.
I could try to get started just with some of these foam blocks.
I could try to build a
stack of these blocks and see how far I can get these blocks to overhang.
Before they fall over. But why did my block tower just fall over.
Well the issue has to do with center of mass.
A block tower is going to fall over if the center of mass isn't supported.
So what I mean by that, that the center of mass of the first block isn't
supported by the second block, then that first block will fall off.
If the center of mass of just the first two
blocks together, which is maybe over here somewhere isn't supported.
If it isn't above the third block, then those two blocks are going to fall.
My tower's going to collapse.
So that's the issue, I want to build a
really tall tower of blocks with a really long overhang.
And yet I want it to be stable.
Of course the easiest way to make it stable is just to make my
block tower perfectly vertical, but then I don't have any over hang at all.
So these two forces are really working against each other.
All right, my desire to have a really long over hang should
be playing against my desire to have my block tower not collapse.
Well let me, let me propose a specific configuration.
Here's the configuration I'm proposing.
So imagine that all these blocks are exactly the same.
They're all one block long and I've staggered them like this.
So this first block is offset by half a block width, from the second block.
The second block is a quarter of a block width pushed in.
The next block is a sixth pushed in.
The next block is an eighth, this distance here is an eighth of a block.
The next block here is a tenth of a block.
If I put another block under there, I'd offset it by twelfth of a block.
The next block will be by fourteenth of a
block, then sixteenth of a block and so on.
I need to check that that configuration is stable.
So the center of mass of the top block is smack in the middle of the top block.
And since
I pushed the top block over half a block from the second block, that puts the
center mass of the top block right above
the edge of the second block so it's stable.
What of the second block?
Well the center of mass of the second block is right
in the middle of the second block, but that's not relevant.
I mean yes the second block isn't tipping over but what I really need to know.
Is what's the combined center of mass of
the first block and the second block together?
So figure that out, I just need to remember that I pushed the second
block over a quarter of a block from the edge of the third block.
And then I could compute the center of
mass of the first and the second block together.
And I find out that in that case the center of mass is right there.
It puts it right above the third block
which means the first two blocks together are stable.
The first block is stable, the first two blocks are stable, but that's just the
top two blocks. I need to know this general.
So I need to compute the center of mass, the top
n blocks relative to the right edge of the next block.
And to do this I'll start by averaging some center of masses.
And I want to average the center of masses of the top end blocks.
I don't really need to know the y coordinate of the center of masses.
So I'm just going to add together
the x coordinates of the center of masses. Where I put the origin
at the right-hand edge of the block right under the stack.
Alright so I'm going to add together these x coordinates.
And then once I've added together these x coordinates
to average them, I need to divide by n.
Okay, so just got to figure out where this blocks really are in in space.
So first of all, where is block number one?
Where is the top block relative to the
next block under this collection of the top
end blocks.
Well that blocks center of mass is right here at
one half plus a half, plus a fourth, plus dot dot dot plus one over two n, right.
The, the center of the mass is just a block by itself is at
one half, and then this records how far over I've pushed the top block.
Relative to the next block in the stack after the top n
blocks, okay. What about block number two?
Well that looks very similar right, again it's one half, because that's
where the center of mass is just in the block by itself.
But the second block doesn't get pushed over half, it gets pushed over a fourth.
And then a sixth and so on until it's 1 over 2n.
So a fourth plus a sixth plus until I get to 1 over 2n.
That's how far over I pushed it relative to the next block.
And a half then moves me over to the middle of block number two.
Then block number three has a similar looking formula.
It's a half plus now a sixth plus an eighth until I get to 1 over 2n.
And finally I get to the nth block, which
is right above the block that I'm measuring everything from.
So its center of mass is at a half plus just how
much I pushed over the nth block which is 1 over 2n.
So now I gotta look at this and see
if there's anything I can say about this complicated sum.
Well, I've got a half, I've got a half, I've got a half, I've got a half.
Every single one of these n terms has a half, so that gives me n over 2.
I've got n halves all together. I've also got a one half here and no extra
one halves, so I can add just this one half coming from right here.
I've got a quarter here and a quarter here and then no more quarters.
So I've got two quarters, and I can add those.
How many sixths do I have?
Well, I've got a sixth in here inside the dot
dot dot, I've got sixth here, I've got sixth here.
The next term doesn't have any sixths, so I've got 3 6ths altogether.
And then
it was going to keep on going right, I could count how many eighths
I have, I could count how many tenths I have and so on.
And eventually I'll notice that I've got a 1 over 2n, a 1 over
2n, a 1 over 2n and a 1 over 2n, I've got n 2ns.
And
that's it. Alright, that's all the terms in a sum.
So then I'm dividing this whole thing by n.
Well this is a half, this is also a half, this is a half and this is a half.
Here I've got n halves.
So instead of writing a half plus 2 4ths plus
3 6ths and everything, I could just write n halves.
So I've got n over 2 plus n over 2 over n or altogether right, that's just
n over n. That's just one.
So relative to the right hand edge of the next
block, the center of mass of the top n blocks.
Is right on the left hand edge of the next block, which means it's stable.
So it's stable, but now what kind
of overhang can I get with that configuration?
So this case, I just used six blocks, but the total
overhang is easy to calculate, right, I want to count what the total overhang.
It's the distance from the left edge of the
top block to the left edge of the bottom block.
And it's just a half plus a forth plus a sixth plus a eighth
plus tenth, it's the total amount that I shifted all the blocks over by.
Now if I build the same kind of configuration, but instead of 6 blocks,
I built it with some large number, call it big N number of blocks.
Then this total overhang
is the sum little n from 1 to big N minus 1, over 1 over 2 n.
Cause this is the total amount that I'd be shifting all the locks over by.
That looks like the harmonic series, and indeed the harmonic series diverges.
Well, that means the sum of 1 over 2 to
the n, n goes from 1 to infinity, also diverges.
But that means that by choosing n big enough, I can make this overhang
as large as I desire. Well, here it is in, in the real world.
I've built one of these harmonic towers, and you can
see that I've got a pretty significant amount of overhang here.
I'll rotate it a little bit, and I've got this river here.
Anyway, I could make the
overhang as long as I'd like, as long as I'm willing to use more slabs.
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