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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