Compound interest and e (part 3)

WEBVTT
Kind: captions
Language: en

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These videos were supposed to
be about e, but compound

00:00:03.430 --> 00:00:05.380
interest was the natural
way to introduce it.

00:00:05.380 --> 00:00:07.450
And since I've already gone
down this compound interest

00:00:07.450 --> 00:00:10.860
track, let's just keep going so
that I can put these videos in

00:00:10.860 --> 00:00:15.920
both my mathematics and
my finance playlists.

00:00:15.920 --> 00:00:17.910
And actually, before I
continue, I just took the Excel

00:00:17.910 --> 00:00:20.030
and I just wanted to show you
how it converges to e.

00:00:20.030 --> 00:00:23.000
So this is how many
periods I'm compounding.

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So in this formula
right here, this is n.

00:00:27.070 --> 00:00:29.220
This column a that I
have, right here.

00:00:29.220 --> 00:00:31.710
And then this is essentially
what happens when I evaluate 1

00:00:31.710 --> 00:00:34.060
plus 1 over n to the n power.

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You can actually look
at the formula.

00:00:37.130 --> 00:00:37.840
My Excel formula.

00:00:37.840 --> 00:00:42.750
It's 1 plus 1 over this blue
cell, to that blue power.

00:00:42.750 --> 00:00:43.810
That's just that formula.

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And I did it for a bunch of
numbers, and I just double

00:00:45.800 --> 00:00:46.540
the numbers every time.

00:00:46.540 --> 00:00:48.780
So I go to a very high
number very quickly.

00:00:48.780 --> 00:00:51.970
And you see that very quickly
it converges to this

00:00:51.970 --> 00:00:53.610
number: 2.7183.

00:00:53.610 --> 00:00:56.180
And this number just keeps
going on and going on.

00:00:56.180 --> 00:00:57.160
And that number is e.

00:00:57.160 --> 00:01:01.500
And what's interesting is, in
fact, you go to Google, and

00:01:01.500 --> 00:01:03.750
you type in search on e.

00:01:03.750 --> 00:01:04.630
They give you the number.

00:01:04.630 --> 00:01:05.910
Because Google is
actually a calculator.

00:01:05.910 --> 00:01:07.250
And you could look
up e other places.

00:01:07.250 --> 00:01:09.545
And I think there are
sites that calculate e

00:01:09.545 --> 00:01:11.590
to arbitrary decimals.

00:01:11.590 --> 00:01:15.460
There's actually some people
who, for whatever reason, they

00:01:15.460 --> 00:01:17.570
see numbers like pi and e.

00:01:17.570 --> 00:01:18.430
They can see them.

00:01:18.430 --> 00:01:22.020
And they can recite the digits
to arbitrary decimal places.

00:01:22.020 --> 00:01:24.800
And I think the more you
realize-- the more you see

00:01:24.800 --> 00:01:30.500
where e pops up in the world,
and pi, and imaginary numbers--

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I think you'll realize that
these numbers, I think they're

00:01:33.200 --> 00:01:36.780
somehow scratching the surface
of something very deep.

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I mean we're just touching on
them because they pop up

00:01:38.890 --> 00:01:41.340
everywhere in completely
different places

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in the universe.

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And they're all almost
magically related.

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And I'll show you that over the
course of the next videos.

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I think this will really give
you motivation if you're in

00:01:52.770 --> 00:01:56.290
the mood for starting
a new cult, perhaps.

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Well, anyway, let's continue
with the compound interest

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before, because we need to be
able to finance our cult.

00:02:04.490 --> 00:02:07.200
Or maybe our cult is
financed by giving

00:02:07.200 --> 00:02:09.260
other people financing.

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By being loan sharks.

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Well, anyway, the example I
just gave was the situation

00:02:13.575 --> 00:02:16.430
in which I'm charging
100% interest.

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So let's generalize it a little
bit, to the situation where

00:02:19.280 --> 00:02:22.400
I'm charging some other
percentage of interest.

00:02:22.400 --> 00:02:24.515
So let's say I'm
charging r percent.

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Oh, my rate-- right.

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My rate is r percent.

00:02:33.580 --> 00:02:35.205
That's essentially what
I'm going to charge.

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The rate will be
r, as a decimal.

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So it'll be 10r%, if
I were to write it.

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But as a decimal-- so, for
example, if I'm charging 25%,

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my rate would be .25, and
I would write that as 25%.

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Just to clarify.

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So what would you owe me at
the end of the year depending

00:02:57.880 --> 00:03:01.030
on how often I compounded?

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Well, just going back to what
we said before, you have your

00:03:03.200 --> 00:03:06.510
initial principal, which in
every example we've done so far

00:03:06.510 --> 00:03:10.410
was $1.00, but I'll just write
p so we can get general.

00:03:10.410 --> 00:03:13.660
And then the amount that you
owe me after one compounding

00:03:13.660 --> 00:03:19.760
period is 1 times the
principal, plus my annual

00:03:19.760 --> 00:03:24.800
interest rate-- so in this case
it's r-- divided by the number

00:03:24.800 --> 00:03:26.240
of times I'm compounding.

00:03:26.240 --> 00:03:28.270
So that's n again.

00:03:28.270 --> 00:03:30.610
And I'm raising that
to the nth power.

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And just so this makes sense to
you in the terms we thought

00:03:33.380 --> 00:03:40.020
about, when n is-- let's say, r
is equal to 10% and

00:03:40.020 --> 00:03:42.520
n is equal to 2.

00:03:42.520 --> 00:03:45.635
And this is what you owe me at
the end of a year-- so if n is

00:03:45.635 --> 00:03:48.030
equal to 2, that means we're
compounding twice over a year.

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Or that we're charging
essentially half of this

00:03:50.760 --> 00:03:52.680
rate every six months.

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So if you were to borrow,
let's say, p is equal to,

00:03:56.510 --> 00:03:58.370
I don't know, $50.00.

00:03:58.370 --> 00:04:00.660
That's how much you
initially borrow from me.

00:04:00.660 --> 00:04:04.532
So all this formula says is,
after every period you will

00:04:04.532 --> 00:04:08.390
owe-- so, after one period,
how much will you owe?

00:04:08.390 --> 00:04:09.185
This is how much you borrowed.

00:04:09.185 --> 00:04:15.590
Then after the next period,
after six months, you'll owe

00:04:15.590 --> 00:04:21.840
me this p, $50.00, plus the
interest rate divided by the

00:04:21.840 --> 00:04:23.120
number of periods in the year.

00:04:23.120 --> 00:04:26.410
So this is essentially kind of
an annual interest rate, but if

00:04:26.410 --> 00:04:27.750
I'm charging you every
six months, I'm going

00:04:27.750 --> 00:04:29.140
to divide it by 2.

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So it's 10% over
2 times $50.00.

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And this is the same
thing as what?

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This is the same thing as 50
times 1 plus our rate divided

00:04:52.850 --> 00:04:56.830
by the number of times
we compound, right?

00:04:56.830 --> 00:04:59.650
And this is after 6 months,
as I highlight right here.

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And then after another 6
months, I'm going to take this

00:05:02.130 --> 00:05:06.960
number, and then I-- you know,
let's call this x-- and

00:05:06.960 --> 00:05:15.650
I'm going to charge you x
plus 10% over 2 times x.

00:05:15.650 --> 00:05:22.570
Or I'm going to charge you
x times 1 plus 10% over 2.

00:05:22.570 --> 00:05:23.670
And this was x.

00:05:23.670 --> 00:05:27.220
So after a full year, I'm
charging you $50.00 times

00:05:27.220 --> 00:05:35.260
1 plus 10% over 2, times
1 plus 10% over 2.

00:05:35.260 --> 00:05:39.660
And that's the same thing as--
well, going in the opposite

00:05:39.660 --> 00:05:45.070
direction-- as $50.00 times 1
plus-- we could write this is

00:05:45.070 --> 00:05:50.790
a decimal-- .1 over 2
to the second power.

00:05:50.790 --> 00:05:51.000
Right?

00:05:51.000 --> 00:05:52.830
I'm just multiplying
this times itself.

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So, in general, when I
compound-- and now I think

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you'll see the relationship
between what I just wrote out

00:05:56.980 --> 00:05:59.100
there-- and experiment with
some numbers on your own if

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you're getting a little bit
confused, or if I'm going

00:06:01.020 --> 00:06:03.260
a little bit too fast.

00:06:03.260 --> 00:06:06.470
Hopefully you see that this
is the same thing as this.

00:06:08.990 --> 00:06:13.070
So let's see what happens as I
try to compound continuously,

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or as n approaches infinity.

00:06:18.020 --> 00:06:20.210
Let me get some space.

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So the amount that you owe me--
so we can call that final

00:06:23.330 --> 00:06:32.420
payment after a year-- payment
is equal to the amount you're

00:06:32.420 --> 00:06:37.740
borrowing-- I don't like this
color-- times 1 plus the

00:06:37.740 --> 00:06:44.290
interest rate over n
to the nth power.

00:06:44.290 --> 00:06:48.170
Well, let's just make
a substitution.

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Let's say that-- and I think
you'll understand why I'm doing

00:06:51.220 --> 00:06:56.890
the substitution-- let's say
that r over n-- and let's say I

00:06:56.890 --> 00:06:59.790
want to find the limit as n
approaches infinity, as I

00:06:59.790 --> 00:07:01.420
compound continuously.

00:07:01.420 --> 00:07:08.260
So the limit as n approaches
infinity of 1 plus r

00:07:08.260 --> 00:07:11.890
over n to the nth power.

00:07:11.890 --> 00:07:12.920
Let's make a substitution.

00:07:12.920 --> 00:07:19.970
Let's say that 1 over x
is equal to r over n.

00:07:19.970 --> 00:07:23.360
If 1 over x is equal to
r over n, what is this?

00:07:23.360 --> 00:07:27.290
Let's see, that means
that n is equal to xr.

00:07:27.290 --> 00:07:27.540
Right?

00:07:27.540 --> 00:07:29.750
I just crossed multiplied.

00:07:29.750 --> 00:07:36.030
And then, if n is equal to xr,
what's-- n approaching infinity

00:07:36.030 --> 00:07:40.640
is the same thing, assuming
that r is constant, that's

00:07:40.640 --> 00:07:43.190
the same thing as x
approaching infinity.

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Or we could view it the same
way the other way around. x

00:07:44.990 --> 00:07:46.610
approaching infinity is
the same thing as n

00:07:46.610 --> 00:07:48.190
approaching infinity.

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And so we can make the
substitution here, and we get--

00:07:53.740 --> 00:07:58.610
this is the same thing as the
limit as x approaches

00:07:58.610 --> 00:08:01.210
infinity of what?

00:08:01.210 --> 00:08:04.240
1 plus-- we said r over n is
the same thing as 1 over x,

00:08:04.240 --> 00:08:06.460
we just defined it that way.

00:08:06.460 --> 00:08:07.610
To the nth power.

00:08:07.610 --> 00:08:11.100
But we said n-- this
substitution comes into this.

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So n is just equal to xr.

00:08:14.510 --> 00:08:16.690
Remember, r is just
a constant, right?

00:08:16.690 --> 00:08:22.100
And this is the same thing as
the limit, as x approaches

00:08:22.100 --> 00:08:28.990
infinity, of 1 plus
1 over x to the x.

00:08:28.990 --> 00:08:30.800
And then when you multiply
exponents like that, that's

00:08:30.800 --> 00:08:34.780
the same thing as that
whole expression to

00:08:34.780 --> 00:08:37.140
the r power, right?

00:08:37.140 --> 00:08:38.950
And this r is a
constant, right?

00:08:38.950 --> 00:08:41.020
We're not taking the limit on
r, or anything like that.

00:08:41.020 --> 00:08:46.570
So this is the same thing as
the limit, as x approaches

00:08:46.570 --> 00:08:55.350
infinity, of 1 plus 1 over x
to the x, and all of

00:08:55.350 --> 00:08:57.790
that to the r power.

00:08:57.790 --> 00:09:02.240
And then what did we figure out
that this was in the previous

00:09:02.240 --> 00:09:03.530
two or three videos?

00:09:03.530 --> 00:09:05.630
Well, this is equal to e.

00:09:05.630 --> 00:09:10.170
So this is equal to
e to the r power.

00:09:10.170 --> 00:09:18.155
So if I charge an interest rate
of 10%, and I want to compound

00:09:18.155 --> 00:09:22.210
it continuously over one year,
at the end of one year, you're

00:09:22.210 --> 00:09:26.970
going to owe me e to the
10% power times the

00:09:26.970 --> 00:09:28.730
original principal.

00:09:28.730 --> 00:09:31.720
So we said that this is equal
to e to the r, so p times

00:09:31.720 --> 00:09:33.490
this is equal to p.

00:09:33.490 --> 00:09:34.580
There's a p the whole time.

00:09:34.580 --> 00:09:37.590
I'll do it in the blue
so you remember.

00:09:37.590 --> 00:09:39.870
I dropped the p somewhere
along the way.

00:09:39.870 --> 00:09:41.500
There should have
been a p here.

00:09:41.500 --> 00:09:42.820
But it's just a scaling factor.

00:09:42.820 --> 00:09:44.040
Should be a p here.

00:09:44.040 --> 00:09:46.430
I could have taken the p out,
because it's a constant, put

00:09:46.430 --> 00:09:48.520
the p here, and it would
have stayed there.

00:09:48.520 --> 00:09:50.790
But anyway, I'm about to run
out of time and I will see you

00:09:50.790 --> 00:09:54.450
in the next video where we know
how much we're going to

00:09:54.450 --> 00:09:56.920
pay if we continuously
compound for a year.

00:09:56.920 --> 00:09:59.630
Let's see what we'll pay if we
continuously compound for

00:09:59.630 --> 00:10:03.480
multiple years, at a rate
of r percent per year.


Introduction to compound interest and e

WEBVTT
Kind: captions
Language: en

00:00:01.240 --> 00:00:04.580
Let's, just for the sake of our
imaginations, assume that I'm

00:00:04.580 --> 00:00:07.940
the local loan shark, and you
need a dollar for whatever

00:00:07.940 --> 00:00:10.510
purposes, to feed your
children, or start a

00:00:10.510 --> 00:00:13.590
business or buy a new suit,
whatever it may be.

00:00:13.590 --> 00:00:15.940
And you come to me, and you
say Sal, I need a dollar.

00:00:15.940 --> 00:00:18.150
I need to borrow it for roughly
a year, and I'm going to get a

00:00:18.150 --> 00:00:20.420
great job, or my children will
get a great job, and I'll

00:00:20.420 --> 00:00:21.110
pay you back in a year.

00:00:21.110 --> 00:00:24.480
And I say, oh, that sounds very
good, and I will lend you a

00:00:24.480 --> 00:00:27.350
dollar for the low price, or
the low interest rate, of

00:00:27.350 --> 00:00:30.750
100% annual interest.

00:00:30.750 --> 00:00:35.280
So if you borrow $1 at 100%
interest, if you borrow a

00:00:35.280 --> 00:00:39.840
dollar, in a year from now, I
want that dollar back, and

00:00:39.840 --> 00:00:41.280
I also want 100% of that.

00:00:41.280 --> 00:00:42.150
That's the interest rate.

00:00:42.150 --> 00:00:43.800
The interest rate is
essentially what percentage

00:00:43.800 --> 00:00:45.250
of the original
amount you borrowed.

00:00:45.250 --> 00:00:48.910
That's called the principal
in finance terms.

00:00:48.910 --> 00:00:50.770
That's how much I'm
essentially charging you

00:00:50.770 --> 00:00:51.760
to borrow the money.

00:00:51.760 --> 00:00:58.030
So it'll be $1 principal--
that's what you're borrowing,

00:00:58.030 --> 00:01:02.460
and of course, you have to pay
that back-- plus 100% interest.

00:01:05.180 --> 00:01:05.910
$1.

00:01:05.910 --> 00:01:09.190
That's 100%, right?

00:01:09.190 --> 00:01:09.870
100% interest.

00:01:13.550 --> 00:01:16.060
And a year from now, you are
going to pay me the principal

00:01:16.060 --> 00:01:20.340
plus the interest, so
you're going to pay me $2.

00:01:20.340 --> 00:01:24.040
Well, you're fairly desperate,
so you say, OK, Sal, that's OK.

00:01:24.040 --> 00:01:27.930
But seeing that this isn't the
lowest interest rate that

00:01:27.930 --> 00:01:30.090
you've ever seen-- I think the
federal funds rate is at

00:01:30.090 --> 00:01:35.830
something like 2.5 or 3%, so
clearly my 100% is what would

00:01:35.830 --> 00:01:37.720
make any loan shark proud.

00:01:37.720 --> 00:01:40.590
You figure, well, I want
to pay this thing off

00:01:40.590 --> 00:01:41.510
as soon as possible.

00:01:41.510 --> 00:01:44.010
So you say, Sal, what
happens if I have the

00:01:44.010 --> 00:01:46.810
money in six months?

00:01:46.810 --> 00:01:48.250
Well, I say, OK,
that's reasonable.

00:01:48.250 --> 00:01:51.130
For six months, since you're
only borrowing it for half as

00:01:51.130 --> 00:01:55.190
long, I tell you what:
You just have to pay me

00:01:55.190 --> 00:01:57.560
50% after six months.

00:01:57.560 --> 00:01:58.820
So this is after one year.

00:02:03.310 --> 00:02:11.120
After six months, I want you
to pay $1 principal plus 50%

00:02:11.120 --> 00:02:14.110
interest, plus 50 cents, right?

00:02:14.110 --> 00:02:15.170
That's 50%.

00:02:15.170 --> 00:02:19.360
And the logic being that if I'm
charging you 100%, I'm charging

00:02:19.360 --> 00:02:22.190
you $1 for you to keep the
money for the whole year, I'm

00:02:22.190 --> 00:02:23.790
only going to charge you
half as much to keep the

00:02:23.790 --> 00:02:25.360
money half the year.

00:02:25.360 --> 00:02:27.810
And so after six months,
I would expect you

00:02:27.810 --> 00:02:33.030
to pay me $1.50.

00:02:33.030 --> 00:02:34.300
This is after six months.

00:02:36.920 --> 00:02:39.280
And then you say, OK,
Sal, that sounds-- that

00:02:39.280 --> 00:02:40.340
makes sense so far.

00:02:40.340 --> 00:02:45.100
But let's just say that I want
to-- I intend to pay you back

00:02:45.100 --> 00:02:47.490
in six months, but just in case
I don't have the money in six

00:02:47.490 --> 00:02:50.830
months, will I still just
owe you $2 in a year?

00:02:50.830 --> 00:02:51.860
And I say no, no, no, no.

00:02:51.860 --> 00:02:56.100
That I can't deal with because
now I'm giving you the

00:02:56.100 --> 00:02:58.730
possibility of paying off
earlier, and if you pay this

00:02:58.730 --> 00:03:00.950
money earlier, then I have to
figure out where I'm going to--

00:03:00.950 --> 00:03:05.240
essentially who I'm going to
take advantage of next.

00:03:05.240 --> 00:03:07.290
While if I just lock in my
money with you, I can take

00:03:07.290 --> 00:03:09.870
advantage of you for
an entire year.

00:03:09.870 --> 00:03:14.440
So what I say is if you want
to-- what you're going to have

00:03:14.440 --> 00:03:17.900
to do is essentially reborrow
the money after six months

00:03:17.900 --> 00:03:19.650
for another six months.

00:03:19.650 --> 00:03:23.790
So instead of me paying you--
instead of me charging you 50

00:03:23.790 --> 00:03:28.970
cents for the next six months,
I'm going to charge you 50%

00:03:28.970 --> 00:03:29.810
for the next six months.

00:03:29.810 --> 00:03:31.890
So this is how you can view it.

00:03:31.890 --> 00:03:34.400
On day one, you
borrow $1 from me.

00:03:34.400 --> 00:03:37.460
In six months, you
pay $1.50, right?

00:03:37.460 --> 00:03:40.940
And we decided that 50 percent
was a fair interest rate

00:03:40.940 --> 00:03:43.050
for six months, right?

00:03:43.050 --> 00:03:45.980
So let's say that you really
do need the money for a year.

00:03:45.980 --> 00:03:49.490
So we will just charge
you another 50% for

00:03:49.490 --> 00:03:51.730
that next six months.

00:03:51.730 --> 00:03:53.523
Now that other 50% is
not going to be on your

00:03:53.523 --> 00:03:54.790
initial principal.

00:03:54.790 --> 00:03:57.280
Now, after six months,
you owe me $1.50.

00:03:57.280 --> 00:04:01.720
So I'm going to charge you-- so
now this is starting at the

00:04:01.720 --> 00:04:05.250
next period, you'd owe me
$1.50, and now I'm going to

00:04:05.250 --> 00:04:10.340
charge you 50% of that,
so that's 75 cents.

00:04:10.340 --> 00:04:13.550
So it's still a 50% interest
rate for the six months, but

00:04:13.550 --> 00:04:15.160
your principal has
increased, right?

00:04:15.160 --> 00:04:17.780
Because it was the old
principal plus the old

00:04:17.780 --> 00:04:19.990
interest, and that's how much
you owe me now, and now

00:04:19.990 --> 00:04:22.590
I'm going to charge the
interest rate on that.

00:04:22.590 --> 00:04:27.690
And so now that equals
$2.25 over a year.

00:04:27.690 --> 00:04:30.730
So you look at that, and you're
like, wow, you know, just to be

00:04:30.730 --> 00:04:34.730
able to essentially have this
option to pay earlier, I'm

00:04:34.730 --> 00:04:36.700
essentially on an annual rate.

00:04:36.700 --> 00:04:40.270
My annual rate looks a lot more
like 125% interest, right?

00:04:40.270 --> 00:04:43.730
Because my original principal--
your original principal was $1,

00:04:43.730 --> 00:04:46.510
and now you're paying $1.25
in interest, so you're

00:04:46.510 --> 00:04:50.420
paying 125% annual rate.

00:04:50.420 --> 00:04:55.800
So that looks pretty bad to
you, but you are, I guess, in a

00:04:55.800 --> 00:04:58.530
tough bind, so you agree to it.

00:04:58.530 --> 00:05:00.460
And I explained to you that
this is actually just

00:05:00.460 --> 00:05:01.320
a very common thing.

00:05:01.320 --> 00:05:02.720
Even though it looks suspicious
to you, it is called

00:05:02.720 --> 00:05:04.060
compounding interest.

00:05:04.060 --> 00:05:07.360
It means that after every
period-- if we say something

00:05:07.360 --> 00:05:11.420
compounds twice a year, after
every six months, we take the

00:05:11.420 --> 00:05:14.100
interest off of the new
amount that you owe me.

00:05:14.100 --> 00:05:16.200
You could pay me back what you
owe me at that point, or you

00:05:16.200 --> 00:05:18.560
could essentially reborrow it
at the same rate for

00:05:18.560 --> 00:05:20.180
another six months.

00:05:20.180 --> 00:05:23.890
So you say, OK, Sal, you're
overwhelming me a little

00:05:23.890 --> 00:05:25.890
bit, but I need the
money so I'll do it.

00:05:25.890 --> 00:05:28.600
But once again, you know,
on an annual basis,

00:05:28.600 --> 00:05:30.470
125% looks even worse.

00:05:30.470 --> 00:05:33.260
You know, 50% over six
months still isn't cheap.

00:05:33.260 --> 00:05:35.280
What if I have the
money in a month?

00:05:35.280 --> 00:05:38.680
What if I have the money in a
month, where I say, OK, here's

00:05:38.680 --> 00:05:40.750
the deal: same notion.

00:05:40.750 --> 00:05:47.350
Instead of charging you 100%
per year, I'm going to charge

00:05:47.350 --> 00:05:51.280
you-- so this is scenario
one, this is scenario two.

00:05:51.280 --> 00:05:53.400
I'm going to charge
you 1/12 of that.

00:05:53.400 --> 00:05:55.740
I'm going to charge you
100% divided by 12,

00:05:55.740 --> 00:05:56.470
and what is that?

00:05:56.470 --> 00:06:02.310
It's 12 goes into 100 eight
and a half times, right?

00:06:02.310 --> 00:06:04.630
Yeah, 8 times 12 is 96,
and then you get another

00:06:04.630 --> 00:06:06.220
half in there, right?

00:06:06.220 --> 00:06:09.840
So now I'll say, well, if you
want to pay me on any given

00:06:09.840 --> 00:06:18.110
month, I'll just charge
you 8.5% per month.

00:06:18.110 --> 00:06:20.660
And once again, though,
it's going to compound.

00:06:20.660 --> 00:06:25.280
So let's say you start with $1.

00:06:25.280 --> 00:06:30.260
After one month, you're going
to owe me that $1 plus 8.5%.

00:06:30.260 --> 00:06:31.540
So after one month,
you're going to owe

00:06:31.540 --> 00:06:34.890
me 1 plus 8.5% of 1.

00:06:34.890 --> 00:06:41.410
So plus 0.085, which
equals 1.085.

00:06:41.410 --> 00:06:43.690
And then after a month,
you're going to owe me

00:06:43.690 --> 00:06:47.990
this plus 8.5% of this.

00:06:47.990 --> 00:07:00.040
So it would be essentially
1.085 squared, and you can do

00:07:00.040 --> 00:07:01.520
the math to figure that out.

00:07:01.520 --> 00:07:06.490
And then after three months,
you'll owe me 1.085

00:07:06.490 --> 00:07:07.450
to the third.

00:07:07.450 --> 00:07:14.650
And after a full year, you'll
actually owe me 1.085 to the

00:07:14.650 --> 00:07:17.390
12th power, and let's
see what that is.

00:07:17.390 --> 00:07:19.440
I'm going to use my
little Excel here.

00:07:19.440 --> 00:07:25.680
Let's see, if I have
plus 1.085 to the 12th,

00:07:25.680 --> 00:07:30.650
you'll owe me $2.66.

00:07:30.650 --> 00:07:35.120
That equals $2.66.

00:07:35.120 --> 00:07:39.040
And you say, OK, that's
acceptable, reluctantly,

00:07:39.040 --> 00:07:39.850
because this is now what?

00:07:39.850 --> 00:07:43.290
166% effective interest rate.

00:07:43.290 --> 00:07:45.550
And just keep in mind, all
I'm doing is I'm compounding

00:07:45.550 --> 00:07:46.910
the interest, right?

00:07:46.910 --> 00:07:51.370
This was $1.085, and I think
that makes sense to you.

00:07:51.370 --> 00:07:53.080
And the reason why this is
squared is because this is

00:07:53.080 --> 00:07:59.050
going to-- this is just this
principal times 1.085 Another

00:07:59.050 --> 00:08:01.114
way to view it is this is the
same thing as-- I'm going to

00:08:01.114 --> 00:08:02.930
do it in a different color.

00:08:02.930 --> 00:08:12.390
It's equivalent to this
plus 0.085 times 1.085.

00:08:12.390 --> 00:08:18.560
So it's 1.085 plus
0.085 times 1.085.

00:08:18.560 --> 00:08:23.850
So if you think of this is 1
times 1.085 and this is 0.085

00:08:23.850 --> 00:08:27.410
times 1.085, then you can
distribute-- you can take out

00:08:27.410 --> 00:08:30.110
the 1.085, and you
would essentially get

00:08:30.110 --> 00:08:36.190
1.085 times 1.085.

00:08:36.190 --> 00:08:37.900
And it keeps going.

00:08:37.900 --> 00:08:39.630
So now, in this situation.

00:08:39.630 --> 00:08:42.450
we are compounding
the interest.

00:08:42.450 --> 00:08:45.060
We said it's essentially 100%
interest, but we're dividing

00:08:45.060 --> 00:08:49.000
it by 12 per month, but we're
compounding it 12 times.

00:08:49.000 --> 00:08:51.265
So, in general, what's
the formula if I want

00:08:51.265 --> 00:08:52.930
to compound it n times?

00:08:56.440 --> 00:08:57.770
So how much are you going
to have to pay me at

00:08:57.770 --> 00:08:59.530
the end of a year?

00:08:59.530 --> 00:09:01.490
Well, let's say you want to
compound-- let's say you

00:09:01.490 --> 00:09:03.480
want to pay every day.

00:09:03.480 --> 00:09:06.270
You want the ability to pay
every day, and I say that's OK,

00:09:06.270 --> 00:09:12.430
so each day, per day, I'll
charge you 100%, which was my

00:09:12.430 --> 00:09:17.570
original annual rate, divided
by 365 days in a year, but I'm

00:09:17.570 --> 00:09:19.770
going to compound it every day.

00:09:19.770 --> 00:09:24.770
So after every day,
you're going to owe 1.--

00:09:24.770 --> 00:09:26.270
what is this number?

00:09:26.270 --> 00:09:36.190
Let's see, that number is 100
divided by 365-- whoops, plus

00:09:36.190 --> 00:09:54.620
100 divided by 365,
so that's 0.27%.

00:09:54.620 --> 00:09:57.320
After every day, you're
going to owe me this much

00:09:57.320 --> 00:09:58.820
times the previous day.

00:09:58.820 --> 00:10:02.580
So after 365 days, you're
going to owe me this

00:10:02.580 --> 00:10:05.760
to the 365th power.

00:10:05.760 --> 00:10:07.480
So, in general-- oh, I just
realized I ran out of time

00:10:07.480 --> 00:10:08.530
so I will continue this
in the next video.

00:10:08.530 --> 00:10:10.200
See you soon.


Compound interest and e (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:01.260
Welcome back.

00:00:01.260 --> 00:00:03.360
In the last video, I was
confusing you with compound

00:00:03.360 --> 00:00:05.620
interest, and now I
will continue to do so.

00:00:05.620 --> 00:00:09.210
But the general notion is I
threw out this 100% interest

00:00:09.210 --> 00:00:13.550
rate, and that was if we just
say you pay me 100% of what

00:00:13.550 --> 00:00:14.710
you borrow after a year.

00:00:14.710 --> 00:00:18.710
And then we talked about what
happens if instead you want

00:00:18.710 --> 00:00:22.800
half the rate for six months,
but then I will compound the

00:00:22.800 --> 00:00:24.310
interest over the
next six months.

00:00:24.310 --> 00:00:27.020
Then we said what happens if
it's for every month or every

00:00:27.020 --> 00:00:28.780
day, and this was the
case for every day.

00:00:28.780 --> 00:00:33.830
And if I charged you 2.7% every
day, but I compound it for 365

00:00:33.830 --> 00:00:37.690
days, it becomes-- I'm
just using Excel here.

00:00:37.690 --> 00:00:40.520
Let me clear all of this.

00:00:40.520 --> 00:00:48.540
It becomes 1.027 to
the 365th power.

00:00:48.540 --> 00:00:53.070
Oh, I'm using my
Excel incorrectly.

00:00:53.070 --> 00:00:54.050
Oh, that's not right.

00:00:54.050 --> 00:01:09.650
Let's see, plus 1.027
to the 365th-- no, I'm

00:01:09.650 --> 00:01:10.400
making a mistake here.

00:01:10.400 --> 00:01:13.500
Let me make sure I got
this 1.027 right.

00:01:13.500 --> 00:01:19.690
So if I takes 100% and I divide
it by 365, so 100% is the same

00:01:19.690 --> 00:01:25.890
thing as 1 divided by 365,
that is-- oh, it's 0.00.

00:01:25.890 --> 00:01:26.330
Sorry.

00:01:26.330 --> 00:01:28.120
So I'm not going to
charge you 2% a day.

00:01:28.120 --> 00:01:30.820
Yeah, that seems high.

00:01:30.820 --> 00:01:40.450
I'm going to charge you
1.-- that is 0.00274.

00:01:40.450 --> 00:01:42.490
I'm charging you 0.2%, right?

00:01:42.490 --> 00:01:44.910
This would be-- this is the
kind of one percentage

00:01:44.910 --> 00:01:46.410
place, so this is 0.2%.

00:01:46.410 --> 00:01:52.860
So I'm charging you
0.274% per day.

00:01:52.860 --> 00:01:54.520
I don't know why I put
brackets around that.

00:01:54.520 --> 00:01:58.860
So if I were to compound that
over 365 days, you take it

00:01:58.860 --> 00:02:03.820
to the 365th power, and
what does that get me?

00:02:03.820 --> 00:02:11.070
Let's see, if I do plus
1.00274 to the 365th

00:02:11.070 --> 00:02:16.500
power, I get 2.7148.

00:02:16.500 --> 00:02:20.540
It equals 2-- that's how much
you're going to owe me.

00:02:20.540 --> 00:02:23.650
If you just keep the money, you
keep kind of reborrowing it

00:02:23.650 --> 00:02:29.430
every day, you'll owe me
$2.7148 after the

00:02:29.430 --> 00:02:30.970
end of one year.

00:02:30.970 --> 00:02:32.440
Now, let's say that's not
enough for you because this

00:02:32.440 --> 00:02:35.530
interest rate is so high,
you want the option to

00:02:35.530 --> 00:02:37.800
pay it by the hour.

00:02:37.800 --> 00:02:40.285
You want this thing to compound
every hour of the day.

00:02:40.285 --> 00:02:41.060
So let's see.

00:02:41.060 --> 00:02:42.760
Let's first of all figure
out how many hours

00:02:42.760 --> 00:02:46.250
there are in a year.

00:02:46.250 --> 00:02:47.390
So let's see.

00:02:47.390 --> 00:02:53.550
In a year, there's 365 days
times 24 hours per day, so

00:02:53.550 --> 00:02:57.390
there's 8,760 hours in a year.

00:02:57.390 --> 00:03:01.060
And then if we want to divide
100%, which is just 1, divided

00:03:01.060 --> 00:03:11.450
by that number, I could charge
you 0.01-- what is that number?

00:03:11.450 --> 00:03:14.200
Yeah, it's 0.0114.

00:03:14.200 --> 00:03:22.330
So hourly compounding-- I
should've been doing this

00:03:22.330 --> 00:03:25.590
in green since it's money.

00:03:25.590 --> 00:03:31.610
Hourly compounding, I will
charge you 100% divided by the

00:03:31.610 --> 00:03:42.590
number of hours in a year,
which equals 0.0114% per hour.

00:03:42.590 --> 00:03:48.850
So over a year, I would take
it to this power, right?

00:03:48.850 --> 00:03:55.640
So let's say after one hour,
you'll owe me 1.01-- sorry,

00:03:55.640 --> 00:04:02.960
it's 01% so it's 1.000114.

00:04:02.960 --> 00:04:05.560
That's how much you're going
to owe me after one hour.

00:04:05.560 --> 00:04:09.460
So $1 plus a very small
fraction of a cent.

00:04:09.460 --> 00:04:10.900
But then after another
hour, you're going to owe

00:04:10.900 --> 00:04:13.520
that times that again.

00:04:13.520 --> 00:04:16.010
Because this'll be the new
principal after an hour,

00:04:16.010 --> 00:04:19.390
so then you're going to
owe that same fraction

00:04:19.390 --> 00:04:22.010
times it again, right?

00:04:22.010 --> 00:04:23.960
Then after three hours,
you'll multiply it again.

00:04:23.960 --> 00:04:27.480
So after the total number of
hours in the year, which is

00:04:27.480 --> 00:04:36.660
1.000114, and there's 8,760
hours in a year, let's

00:04:36.660 --> 00:04:41.250
see what you'll owe me.

00:04:41.250 --> 00:04:58.860
So if I do plus 1.000114 to
the 8,760th power: 2.71443.

00:04:58.860 --> 00:05:03.430
So now, at the end of the year
after compounding roughly 8,700

00:05:03.430 --> 00:05:11.030
hours, you'll owe me $2.71 and
then some fractions of a penny.

00:05:11.030 --> 00:05:13.790
And I know you thought that
these videos were about e and

00:05:13.790 --> 00:05:16.620
you were just learning about
how to take advantage of

00:05:16.620 --> 00:05:20.120
someone in need, but there
should be something interesting

00:05:20.120 --> 00:05:22.050
here that maybe you observed.

00:05:24.640 --> 00:05:29.230
When we started compounding, at
first you owed me $2, just when

00:05:29.230 --> 00:05:31.690
I did one period, which
is the whole year.

00:05:31.690 --> 00:05:36.300
Then it got to $2.25, and then
it kept getting higher as we

00:05:36.300 --> 00:05:39.000
compounded shorter and shorter
periods, but it seems to be

00:05:39.000 --> 00:05:40.670
approaching some number.

00:05:40.670 --> 00:05:41.810
It seems to be
approaching, right?

00:05:41.810 --> 00:05:44.570
When I compounded every day, at
the end of the year, you owed

00:05:44.570 --> 00:05:47.190
me $2.71 and some change.

00:05:47.190 --> 00:05:50.520
And then if I compounded every
hour, which is 24 times as

00:05:50.520 --> 00:05:54.570
many compounds, you still owe
me a very similar number.

00:05:54.570 --> 00:05:57.370
So it seems like it's
gravitating towards this

00:05:57.370 --> 00:06:01.390
mystical number here, and
that mystical number is e.

00:06:01.390 --> 00:06:06.230
So let's kind of formalize what
I've been meandering around

00:06:06.230 --> 00:06:08.590
for a video and a half now.

00:06:08.590 --> 00:06:11.500
And I'll switch colors.

00:06:11.500 --> 00:06:16.080
So, in general, the amount
of money you owed me at the

00:06:16.080 --> 00:06:17.540
end of the year was the
amount you borrowed.

00:06:20.445 --> 00:06:22.210
Let me do that in blue.

00:06:22.210 --> 00:06:23.800
Let's call that the principal.

00:06:23.800 --> 00:06:25.340
That's not bright enough.

00:06:25.340 --> 00:06:31.640
The principal times 1 plus--
and what was the interest rate?

00:06:31.640 --> 00:06:33.840
It was 100%.

00:06:33.840 --> 00:06:38.240
100% divided by the number
of times you wanted to

00:06:38.240 --> 00:06:39.090
compound in the year.

00:06:39.090 --> 00:06:42.280
We'll call that n, right?

00:06:42.280 --> 00:06:45.770
And we raise that
to the n power.

00:06:45.770 --> 00:06:49.380
So in the case of when there's
only one compounding period,

00:06:49.380 --> 00:06:52.320
where you just borrowed it for
the year, and the principal

00:06:52.320 --> 00:06:54.740
in our example was 1, right?

00:06:54.740 --> 00:06:58.330
So this is just 1 times-- the
principal is just what you

00:06:58.330 --> 00:07:02.200
borrowed-- 1 plus-- 100% is the
same thing as 1, right, or

00:07:02.200 --> 00:07:05.850
1.00, and when there was one
compounding period, we just did

00:07:05.850 --> 00:07:11.860
that, and you owed me $2
at the end of the year.

00:07:11.860 --> 00:07:13.480
And this is exactly what
I've done in the last

00:07:13.480 --> 00:07:14.030
video and a half.

00:07:14.030 --> 00:07:16.780
I'm just formalizing it with
a couple of variables.

00:07:16.780 --> 00:07:19.630
When we compounded it every
month, it turned into this: the

00:07:19.630 --> 00:07:25.900
principal you borrowed was 1
times 1 plus 100% over 12 to

00:07:25.900 --> 00:07:31.930
the 12th power, which
equaled-- let's see.

00:07:31.930 --> 00:07:35.230
I erased the numbers
so I will redo it.

00:07:35.230 --> 00:07:45.090
So it's 1 plus 1 divided by 12
equals 1 plus-- I'm using Excel

00:07:45.090 --> 00:07:48.070
for those of you who have never
seen it before-- 12

00:07:48.070 --> 00:07:50.300
to the 12th power.

00:07:50.300 --> 00:07:58.980
That equaled 2.613.

00:07:58.980 --> 00:08:03.030
And when I compounded every
day, I got-- the principal you

00:08:03.030 --> 00:08:11.210
borrowed was that times 1 plus
1 over 365 to the 365th power,

00:08:11.210 --> 00:08:14.780
and that equaled 2.71 and
then some, some, something.

00:08:14.780 --> 00:08:17.410
So as you see, as I make n
larger and larger in this

00:08:17.410 --> 00:08:22.270
original equation, I approach
this magical number.

00:08:22.270 --> 00:08:26.410
I approach this magical number
2.71 something, something,

00:08:26.410 --> 00:08:29.610
something, and that
magical number is e.

00:08:29.610 --> 00:08:33.250
And it amazes me that this--
and it never repeats.

00:08:33.250 --> 00:08:35.610
It's one of these
transcendental numbers like pi,

00:08:35.610 --> 00:08:37.960
and later on in future videos,
we'll see that it shows up

00:08:37.960 --> 00:08:38.930
all over the place.

00:08:38.930 --> 00:08:42.270
It shows up in random
combinatorics, it shows up in

00:08:42.270 --> 00:08:45.570
complex analysis, and as we
see here, and maybe most

00:08:45.570 --> 00:08:47.810
importantly, it shows up
in compound interest.

00:08:47.810 --> 00:08:49.980
So, in general, what we
could say is the limit.

00:08:49.980 --> 00:08:52.470
And the limit is just
what happens as you

00:08:52.470 --> 00:08:53.270
approach something.

00:08:53.270 --> 00:08:56.370
The limit as n
approaches infinity.

00:08:56.370 --> 00:08:59.840
And in our example, that's as
we compound over smaller and

00:08:59.840 --> 00:09:09.540
smaller periods of time of 1
plus 1 over n to the nth power

00:09:09.540 --> 00:09:12.960
is equal to this magical
number, and I'll do it in a

00:09:12.960 --> 00:09:16.750
bold color-- well, that's not
that bold, but I'll highlight

00:09:16.750 --> 00:09:19.120
with another bold color-- is e.

00:09:19.120 --> 00:09:21.290
And that is equal to 2.71.

00:09:21.290 --> 00:09:22.150
I forget all the digits.

00:09:22.150 --> 00:09:23.600
It keeps going on.

00:09:23.600 --> 00:09:25.020
And it's really fun
to experiment.

00:09:25.020 --> 00:09:28.610
Put in a crazy, huge-- put
in like a million there.

00:09:28.610 --> 00:09:29.920
And if you put it there,
you have to put a

00:09:29.920 --> 00:09:31.130
million there, too.

00:09:31.130 --> 00:09:33.660
And you'll see that the larger
numbers you get, you just

00:09:33.660 --> 00:09:36.000
get closer and closer and
closer to this number e.

00:09:36.000 --> 00:09:38.070
And a fun project would be
to see how many digits

00:09:38.070 --> 00:09:41.450
of e you can get.

00:09:41.450 --> 00:09:44.420
But the fact that as you
compound something over smaller

00:09:44.420 --> 00:09:47.970
and smaller periods it
converges to this number, to

00:09:47.970 --> 00:09:50.000
me is pretty interesting.

00:09:50.000 --> 00:09:52.470
So with that out of the way, in
the next video I'll show you

00:09:52.470 --> 00:09:55.610
how to figure-- and so, in
the limit, as n approaches

00:09:55.610 --> 00:09:56.800
infinity, what are you doing?

00:09:56.800 --> 00:09:58.760
You're actually
compounding continuously.

00:09:58.760 --> 00:10:01.610
You're compounding every
zillionth of a second.

00:10:01.610 --> 00:10:04.770
And the fact that you can
actually calculate an interest

00:10:04.770 --> 00:10:08.950
rate compounding every
zillionth of a second to me is

00:10:08.950 --> 00:10:10.830
a fairly amazing result.

00:10:10.830 --> 00:10:12.240
But anyway, I'll see you
in the next video because

00:10:12.240 --> 00:10:13.810
I've run out of time.


Bernoulli's example problem

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:05.450
Let's say I have a horizontal
pipe that at the left end of

00:00:05.450 --> 00:00:21.000
the pipe, the cross-sectional
area, area 1, which is equal

00:00:21.000 --> 00:00:27.900
to 2 meters squared.

00:00:27.900 --> 00:00:36.380
Let's say it tapers off so that
the cross-sectional area

00:00:36.380 --> 00:00:44.350
at this end of the pipe,
area 2, is equal to

00:00:44.350 --> 00:00:45.600
half a square meter.

00:00:50.670 --> 00:00:53.890
We have some velocity at this
point in the pipe, which is

00:00:53.890 --> 00:00:57.530
v1, and the velocity exiting
the pipe is v2.

00:00:57.530 --> 00:01:00.510
The external pressure at this
point is essentially being

00:01:00.510 --> 00:01:02.920
applied rightwards
into the pipe.

00:01:02.920 --> 00:01:17.980
Let's say that pressure
1 is 10,000 pascals.

00:01:17.980 --> 00:01:22.540
The pressure at this end, the
pressure 2-- that's the

00:01:22.540 --> 00:01:25.810
external pressure at that point
in the pipe-- that is

00:01:25.810 --> 00:01:33.190
equal to 6,000 pascals.

00:01:33.190 --> 00:01:34.620
Given this information,
let's say we have

00:01:34.620 --> 00:01:36.556
water in this pipe.

00:01:41.260 --> 00:01:44.700
We're assuming that it's laminar
flow, so there's no

00:01:44.700 --> 00:01:48.420
friction within the pipe, and
there's no turbulence.

00:01:48.420 --> 00:01:54.530
Using that, what I want to do
is, I want to figure out what

00:01:54.530 --> 00:01:59.030
is the flow or the flux of the
water in this pipe-- how much

00:01:59.030 --> 00:02:03.000
volume goes either into the pipe
per second, or out of the

00:02:03.000 --> 00:02:04.210
pipe per second?

00:02:04.210 --> 00:02:06.760
We know that those are the going
to be the same numbers,

00:02:06.760 --> 00:02:10.180
because of the equation
of continuity.

00:02:10.180 --> 00:02:15.500
We know that the flow, which
is R, which is volume per

00:02:15.500 --> 00:02:20.100
amount of time, is the same
thing as the input velocity

00:02:20.100 --> 00:02:21.350
times the input area.

00:02:24.010 --> 00:02:30.860
The input area is 2, so it's
2v1, and that also equals the

00:02:30.860 --> 00:02:37.750
output area times output
velocity, so it equals 1/2 v2.

00:02:37.750 --> 00:02:44.635
We could rewrite this, that v1
is equal to 1/2 R, and that v2

00:02:44.635 --> 00:02:46.510
is equal to 2R.

00:02:46.510 --> 00:02:50.650
This immediately tells us that
v2 is coming out at a faster

00:02:50.650 --> 00:02:56.690
rate, and this is based on
the size of the openings.

00:02:56.690 --> 00:03:00.810
We know, because V2 is coming
out at a faster rate, but we

00:03:00.810 --> 00:03:03.630
also know because we have much
higher pressure at this end

00:03:03.630 --> 00:03:06.600
than at this end, that the water
is flowing to the right.

00:03:06.600 --> 00:03:08.980
The pressure differential, the
pressure gradient, is going to

00:03:08.980 --> 00:03:10.060
the right, so the water
is going to

00:03:10.060 --> 00:03:11.070
spurt out of this end.

00:03:11.070 --> 00:03:13.170
And it's coming in this end.

00:03:13.170 --> 00:03:16.380
Let's use Bernoulli's equation
to figure out what the flow

00:03:16.380 --> 00:03:20.110
through this pipe is.

00:03:20.110 --> 00:03:32.740
Let's just write it down: P1
plus rho gh1 plus 1/2 rho v1

00:03:32.740 --> 00:03:39.370
squared is equal to
P2 plus rho gh2

00:03:39.370 --> 00:03:44.590
plus 1/2 rho v2 squared.

00:03:44.590 --> 00:03:48.000
This pipe is level, and the
height at either end is the

00:03:48.000 --> 00:03:51.500
same, so h1 is going
to be equal to h2.

00:03:51.500 --> 00:03:54.270
These two terms are going to be
equal, so we can cross them

00:03:54.270 --> 00:03:57.360
out-- we can subtract that value
from both sides, and

00:03:57.360 --> 00:04:00.570
we're just left with P1.

00:04:00.570 --> 00:04:01.630
What's P1?

00:04:01.630 --> 00:04:13.280
P1 is 10,000 pascals plus 1/2
rho times v1 squared.

00:04:13.280 --> 00:04:14.720
What's v1?

00:04:14.720 --> 00:04:18.940
That's R over 2-- we figured
that out up here.

00:04:18.940 --> 00:04:26.160
v2 times R over 2 squared is
equal to P2, and that's 6,000

00:04:26.160 --> 00:04:33.520
pascals plus 1/2 rho
times v2 squared.

00:04:33.520 --> 00:04:39.360
We figured out what v2 is--
v2 is 2R squared.

00:04:39.360 --> 00:04:41.570
Let's just do some
simplification, and so let's

00:04:41.570 --> 00:04:45.130
subtract 6,000 from both sides,
and we're left with

00:04:45.130 --> 00:05:09.470
4,000 plus rho R squared over
8 is equal to 1/2 times R

00:05:09.470 --> 00:05:10.070
squared times 4.

00:05:10.070 --> 00:05:15.570
So this is 2 rho R squared.

00:05:15.570 --> 00:05:20.830
We could multiply both sides of
this equation by 8, just to

00:05:20.830 --> 00:05:23.010
get rid of this in the
denominator, so we would get

00:05:23.010 --> 00:05:33.620
32,000 plus rho R squared is
equal to 16 rho R squared.

00:05:33.620 --> 00:05:35.710
Subtract rho R squared from both
sides of this question,

00:05:35.710 --> 00:05:49.550
and we get 32,000 is equal
to 15 rho R squared.

00:05:49.550 --> 00:05:50.140
Then what's rho?

00:05:50.140 --> 00:05:51.430
What's the density of water?

00:05:51.430 --> 00:05:56.200
The density of water is 1,000
kilograms per meter cubed, so

00:05:56.200 --> 00:05:57.900
this is 1,000.

00:05:57.900 --> 00:06:03.920
Let's divide both sides
by 15 times rho.

00:06:03.920 --> 00:06:14.350
We get R squared is equal to
32,000 divided by 15 rho-- rho

00:06:14.350 --> 00:06:22.470
is 1,000, so R squared is equal
to 32,000 over 15,000,

00:06:22.470 --> 00:06:25.980
which is the same thing
is 32 over 15.

00:06:25.980 --> 00:06:30.320
R is equal to the square root
of 32 over 15, and that's

00:06:30.320 --> 00:06:31.880
going to be meters
cubed per second.

00:06:45.010 --> 00:06:52.920
I get 32 divided by 15 is equal
to 2.1, and the square

00:06:52.920 --> 00:06:56.040
root of that 1.46.

00:06:56.040 --> 00:07:06.760
So the answer is R is equal to
1.46 meters cubed per second.

00:07:06.760 --> 00:07:09.830
That is the volume of water that
is either entering the

00:07:09.830 --> 00:07:13.340
system in any given second, or
exiting the system in any

00:07:13.340 --> 00:07:14.290
given second.

00:07:14.290 --> 00:07:16.910
We can figure out the
velocities, too-- what's the

00:07:16.910 --> 00:07:18.790
velocity exiting the system?

00:07:18.790 --> 00:07:20.000
What's two times that?

00:07:20.000 --> 00:07:25.010
It's 2.8 meters per second
exiting the system, and going

00:07:25.010 --> 00:07:31.650
in it is half that, so it's
0.8 meters per second.

00:07:31.650 --> 00:07:34.902
Hopefully, that gives you--
actually, 0.7 meters per

00:07:34.902 --> 00:07:40.700
second-- a bit more intuition on
fluids, and that's all I'm

00:07:40.700 --> 00:07:41.810
going to do for today.

00:07:41.810 --> 00:07:43.370
I'll see you in the next video,
and we're going to do

00:07:43.370 --> 00:07:44.620
some stuff on thermodynamics.


Bernoulli's equation (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.300
Before we move on, I just wanted
to make sure that you

00:00:02.300 --> 00:00:04.140
understood that last point that
I made at the end of that

00:00:04.140 --> 00:00:05.440
last video.

00:00:05.440 --> 00:00:09.000
We said that the pressure
inputting into this, that we

00:00:09.000 --> 00:00:11.570
could view this cup with a hole
in it as essentially a

00:00:11.570 --> 00:00:14.990
pipe, where the opening on the
top of the cup is the input to

00:00:14.990 --> 00:00:17.200
the pipe, and this little
mini-hole is the output to the

00:00:17.200 --> 00:00:19.270
pipe, and we said that
this is a vacuum.

00:00:19.270 --> 00:00:20.380
Let's say this is vacuum
all around.

00:00:20.380 --> 00:00:23.640
I know when I drew it last time,
I closed it, but we have

00:00:23.640 --> 00:00:25.040
a vacuum everywhere.

00:00:25.040 --> 00:00:28.440
Since there's a vacuum
everywhere, the pressure at

00:00:28.440 --> 00:00:33.330
this point P1 is
equal to zero.

00:00:33.330 --> 00:00:36.390
The point I wanted to make is
because we have a hole here,

00:00:36.390 --> 00:00:41.970
the pressure at that point at
P2 is also equal to zero.

00:00:41.970 --> 00:00:44.860
You can almost view it as maybe
the atmospheric pressure

00:00:44.860 --> 00:00:46.730
at that point, but since
we're in a vacuum,

00:00:46.730 --> 00:00:48.430
that pressure is zero.

00:00:48.430 --> 00:00:50.150
That might have been a little
confusing to you, because you

00:00:50.150 --> 00:00:55.680
said, well, wait, I thought at
depth, if I had a point at

00:00:55.680 --> 00:00:58.220
that same height, that I would
actually have a pressure at

00:00:58.220 --> 00:01:03.010
that point of rho gh.

00:01:03.010 --> 00:01:04.040
That's true.

00:01:04.040 --> 00:01:04.970
That's completely true.

00:01:04.970 --> 00:01:09.000
You do have an innate pressure
in the liquid at that point of

00:01:09.000 --> 00:01:11.750
rho gh, and actually, that's
what's causing the

00:01:11.750 --> 00:01:13.120
liquid to come out.

00:01:13.120 --> 00:01:20.790
But that's actually taken care
of in the potential energy

00:01:20.790 --> 00:01:21.960
part of the equation.

00:01:21.960 --> 00:01:23.740
Let me rewrite Bernoulli's
equation.

00:01:23.740 --> 00:01:37.560
The input pressure plus rho g h1
plus rho V1 squared over 2

00:01:37.560 --> 00:01:46.560
is equal to the output pressure
plus rho g h2 plus

00:01:46.560 --> 00:01:49.360
PV2 squared over 2.

00:01:49.360 --> 00:01:52.110
I think you understand that this
term is pretty close to

00:01:52.110 --> 00:01:57.330
zero if the rate at which the
surface moves is very slow if

00:01:57.330 --> 00:02:00.290
this surface area is much
bigger than this hole.

00:02:00.290 --> 00:02:02.920
It's like if you poked a hole in
Hoover Dam, that whole lake

00:02:02.920 --> 00:02:05.900
is going to move down very, very
slowly, like 1 trillionth

00:02:05.900 --> 00:02:08.110
of the speed at which the
water's coming out at the

00:02:08.110 --> 00:02:11.190
other end, so you could
ignore this term.

00:02:11.190 --> 00:02:14.880
We also defined that the hole
was at zero, so the height of

00:02:14.880 --> 00:02:17.580
h2 is zero.

00:02:17.580 --> 00:02:21.320
It simplified down to the input
pressure, the pressure

00:02:21.320 --> 00:02:25.540
at the top of the pipe, or at
the left side of the pipe,

00:02:25.540 --> 00:02:31.070
plus rho gh1.

00:02:31.070 --> 00:02:35.790
This isn't potential energy,
but this was kind of the

00:02:35.790 --> 00:02:39.560
potential energy term when we
derived Bernoulli's equation,

00:02:39.560 --> 00:02:45.750
and that equals the output
pressure, or the pressure at

00:02:45.750 --> 00:02:50.120
the output of the hole, at the
right side of the hole, plus

00:02:50.120 --> 00:02:52.660
the kinetic energy PV2.

00:02:52.660 --> 00:02:54.830
It's the kinetic energy term,
because it doesn't actually

00:02:54.830 --> 00:02:56.840
doesn't add up completely to
kinetic energy, because we

00:02:56.840 --> 00:02:57.980
manipulated it.

00:02:57.980 --> 00:03:01.780
I just wanted to really
make the point that

00:03:01.780 --> 00:03:02.710
is definitely zero.

00:03:02.710 --> 00:03:04.950
I think that is clear
to you, because we

00:03:04.950 --> 00:03:06.450
have a vacuum up here.

00:03:06.450 --> 00:03:09.800
The pressure at that point is
zero, so we can ignore that.

00:03:09.800 --> 00:03:14.070
The question is what is
the pressure here?

00:03:14.070 --> 00:03:16.220
This pressure is zero, because
we have a vacuum here.

00:03:16.220 --> 00:03:20.520
If I were to say that the
pressure over here at this

00:03:20.520 --> 00:03:32.470
hole is equal to pgh, then I
would have the situation where

00:03:32.470 --> 00:03:39.810
pgh is equal to pgh plus
PV squared over 2.

00:03:39.810 --> 00:03:40.810
What does that mean?

00:03:40.810 --> 00:03:44.930
When I say that that pressure
at the output of the pipe is

00:03:44.930 --> 00:03:47.310
pgh, that means that
I'm applying some

00:03:47.310 --> 00:03:49.750
pressure into that hole.

00:03:49.750 --> 00:03:52.800
Essentially, that pressure I'm
applying into the hole is

00:03:52.800 --> 00:03:58.550
exactly just enough offset
to offset the

00:03:58.550 --> 00:04:00.280
pressure at this depth.

00:04:00.280 --> 00:04:02.230
Because of that, none of
the water will move.

00:04:02.230 --> 00:04:05.170
You could imagine that if this
is the hole, let's say that's

00:04:05.170 --> 00:04:09.760
the opening of the hole, and I
have some water particles, or

00:04:09.760 --> 00:04:15.320
some fluid particles, let's say
that these are the atoms,

00:04:15.320 --> 00:04:22.160
we're saying innately at any
point that there is a pressure

00:04:22.160 --> 00:04:27.360
at this point that's equal to
rho gh, but this is P2.

00:04:27.360 --> 00:04:28.830
How much pressure
am I exerting on

00:04:28.830 --> 00:04:31.680
this end of the hole?

00:04:31.680 --> 00:04:39.300
If I exert rho gh at this end,
then these molecules that were

00:04:39.300 --> 00:04:41.900
just about to exit the hole
aren't going to exit, because

00:04:41.900 --> 00:04:44.940
they're going to get the same
pressure from every direction.

00:04:44.940 --> 00:04:47.370
What we said in the last video,
and I really want to--

00:04:47.370 --> 00:04:49.880
because this is a subtle point--
is that the outside

00:04:49.880 --> 00:04:54.270
pressure, being on the outside
part of the hole, is zero, and

00:04:54.270 --> 00:04:57.550
because of that, we end up--
this term is zero, and we

00:04:57.550 --> 00:05:01.410
essentially end up with that
the change in the potential

00:05:01.410 --> 00:05:03.530
energy all becomes kinetic
energy, which is something

00:05:03.530 --> 00:05:06.190
we're familiar with from
just our kinematics

00:05:06.190 --> 00:05:08.520
and our energy equations.

00:05:08.520 --> 00:05:12.530
With that out of the way, let
me do another problem.

00:05:12.530 --> 00:05:15.260
Actually, I will do that next
problem in the next video,

00:05:15.260 --> 00:05:17.360
just so we have a clean
cut between videos.

00:05:17.360 --> 00:05:18.870
See you soon.


Bernoulli's equation (part 3)

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:04.940
Where we left off, we had this
canister, because it had a

00:00:04.940 --> 00:00:07.790
closed top and it had a vacuum
above the fluid.

00:00:07.790 --> 00:00:15.380
The fluid on top had an area
of A1, and I poked a little

00:00:15.380 --> 00:00:18.490
hole with a super-small
area A2.

00:00:18.490 --> 00:00:20.800
I said that the area of
A2 is so small, it's

00:00:20.800 --> 00:00:22.980
1/1,000 of area 1.

00:00:22.980 --> 00:00:25.330
Then we used the continuity
equation.

00:00:25.330 --> 00:00:29.040
We said the velocity, the rate
at which the surface is moving

00:00:29.040 --> 00:00:34.510
up here, V1, times area 1, the
whole surface area of the

00:00:34.510 --> 00:00:37.800
liquid, has to be equal to the
output velocity, which we're

00:00:37.800 --> 00:00:39.840
trying to figure out as a
function of everything else

00:00:39.840 --> 00:00:41.580
times this output area.

00:00:45.090 --> 00:00:46.990
I made a mistake.

00:00:46.990 --> 00:00:48.820
I don't know if I did this in
the last video, or I did this

00:00:48.820 --> 00:00:50.570
is in the mistake video.

00:00:50.570 --> 00:00:56.340
So we know that the initial top
velocity times this top

00:00:56.340 --> 00:01:01.520
area is equal to the output
velocity times-- instead of

00:01:01.520 --> 00:01:10.770
writing area 2, we could write
area 1 over 1,000.

00:01:10.770 --> 00:01:13.580
You can get rid of the area 1 on
both sides, and then you're

00:01:13.580 --> 00:01:21.475
saying that the velocity up here
is equal to the rate at

00:01:21.475 --> 00:01:24.640
which the top of the surface
moves down and is equal to

00:01:24.640 --> 00:01:29.580
1/1,000 of the velocity of the
liquid spurting out of this

00:01:29.580 --> 00:01:32.460
little hole.

00:01:32.460 --> 00:01:38.160
With that, we actually have the
three variables for the

00:01:38.160 --> 00:01:40.540
left-hand side of Bernoulli's
equation.

00:01:40.540 --> 00:01:42.920
What are the variables on
the left-hand side?

00:01:42.920 --> 00:01:49.490
What is the pressure at this
point where we have a hole?

00:01:49.490 --> 00:01:50.630
This is an important thin.

00:01:50.630 --> 00:01:52.770
When we talk about Bernoulli's--
let me rewrite

00:01:52.770 --> 00:01:53.550
Bernoulli's equation.

00:01:53.550 --> 00:02:13.010
It's P1 plus rho gh1 plus rho V1
squared over 2 is equal to

00:02:13.010 --> 00:02:23.650
P2 plus rho gh2 plus rho
V2 squared over 2.

00:02:23.650 --> 00:02:26.750
We figured out all of these
terms. Now let's figure out

00:02:26.750 --> 00:02:29.660
the things that we have
to input here.

00:02:29.660 --> 00:02:32.080
What is the pressure
at point two?

00:02:32.080 --> 00:02:32.970
This is the important thing.

00:02:32.970 --> 00:02:35.170
You might want to say, and this
was my initial reaction,

00:02:35.170 --> 00:02:37.800
too, and that why I made a
mistake, is that what 's the

00:02:37.800 --> 00:02:40.250
pressure at this depth
in the fluid?

00:02:40.250 --> 00:02:42.690
That's not what Bernoulli's
equation is telling us.

00:02:42.690 --> 00:02:45.330
Bernoulli's equation is telling
us actually what is

00:02:45.330 --> 00:02:47.743
the external pressure
at that hole.

00:02:52.670 --> 00:02:55.530
When we did the derivation, we
were saying how much work--

00:02:55.530 --> 00:02:57.440
this was kind of the work
term, although we played

00:02:57.440 --> 00:02:59.300
around with it a little bit.

00:02:59.300 --> 00:03:01.950
But if we look at the water
that's spurting out of the

00:03:01.950 --> 00:03:10.670
hole, it's not doing any work,
because it's not actually

00:03:10.670 --> 00:03:13.720
exerting force against anything
so it's not actually

00:03:13.720 --> 00:03:14.940
doing work.

00:03:14.940 --> 00:03:17.770
When we think about the
pressure, the output pressure,

00:03:17.770 --> 00:03:20.180
it's not the pressure at that
depth of the fluid.

00:03:20.180 --> 00:03:24.480
You should think of it as the
external pressure at the hole.

00:03:24.480 --> 00:03:28.340
In this case, there is no
external pressure at the hole.

00:03:28.340 --> 00:03:32.610
Let's say that if we closed
the hole, then

00:03:32.610 --> 00:03:34.120
at that point, sure.

00:03:34.120 --> 00:03:36.870
The pressure would be the
pressure that's being exerted

00:03:36.870 --> 00:03:39.750
by the outside of the canister
to contain the water, in which

00:03:39.750 --> 00:03:41.200
case, we would end up
with no velocity.

00:03:41.200 --> 00:03:42.570
The water wouldn't
spurt anywhere.

00:03:42.570 --> 00:03:44.270
But now we're seeing the
external pressure is zero.

00:03:44.270 --> 00:03:47.480
That's what the hole essentially
creates.

00:03:47.480 --> 00:03:51.990
We're going to say that P2 is
zero, so this pressure was

00:03:51.990 --> 00:03:54.850
zero, because we're
in a vacuum.

00:03:54.850 --> 00:03:58.440
P2 is also zero, so both
of these are zero.

00:03:58.440 --> 00:04:01.430
Remember, that's the
external pressure.

00:04:01.430 --> 00:04:05.180
P1 is the external pressure to
the input to the pipe, and you

00:04:05.180 --> 00:04:06.670
can view this as a pipe.

00:04:13.900 --> 00:04:23.600
I could redraw it as a pipe that
looks like it has a big

00:04:23.600 --> 00:04:25.770
hole on the top, and it goes
down to some level to a

00:04:25.770 --> 00:04:28.510
super-small hole like that.

00:04:28.510 --> 00:04:30.750
This would be a vacuum, and the
fluid is just going in and

00:04:30.750 --> 00:04:34.260
it's spurting out of this end.

00:04:34.260 --> 00:04:37.090
Anyway, the pressure going into
the pipe is zero, and we

00:04:37.090 --> 00:04:39.030
said since we put a hole, the
pressure coming out of the

00:04:39.030 --> 00:04:43.030
pipe is zero, so we're
doing no work.

00:04:43.030 --> 00:04:46.790
What is this term?

00:04:46.790 --> 00:04:50.580
This was the potential energy
term, and we said that h1 is

00:04:50.580 --> 00:04:51.980
equal to h.

00:04:51.980 --> 00:04:56.220
We're saying that this is zero
height, so now this simplifies

00:04:56.220 --> 00:05:04.620
to rho times gravity times h
plus rho times V1 squared.

00:05:04.620 --> 00:05:12.370
V1, we said, is equal to this,
so this is rho over 2 times V2

00:05:12.370 --> 00:05:16.170
over 1,000 squared.

00:05:16.170 --> 00:05:19.490
I just substituted V2
over 1,000 for V1.

00:05:19.490 --> 00:05:22.570
That equals the pressure at
the hole, the external

00:05:22.570 --> 00:05:30.390
pressure at the hole, which
is zero, plus h2.

00:05:30.390 --> 00:05:33.720
This is h2 right here, which
we said is zero.

00:05:33.720 --> 00:05:36.870
We determined that the hole was
poked at height zero, so

00:05:36.870 --> 00:05:38.940
this is also zero.

00:05:38.940 --> 00:05:42.370
That equals this kinetic
energy-like term.

00:05:42.370 --> 00:05:44.150
It's not exactly
kinetic energy.

00:05:44.150 --> 00:05:51.710
It's rho times V squared
divided by 2.

00:05:51.710 --> 00:05:54.080
One thing that we can
immediately see is that we

00:05:54.080 --> 00:05:55.860
have all these rhos on both
sides of the equation, so we

00:05:55.860 --> 00:06:02.560
can divide both sides by rho and
get rid of all of those.

00:06:02.560 --> 00:06:05.050
Then we can multiply both sides
of the equation by 2,

00:06:05.050 --> 00:06:17.500
and we get 2gh plus V2 squared
over-- what's 1,000 squared--

00:06:17.500 --> 00:06:18.750
over 1 million.

00:06:22.590 --> 00:06:28.840
That is equal to V2 squared.

00:06:28.840 --> 00:06:32.050
We could do the exact thing.

00:06:32.050 --> 00:06:36.700
We could subtract 1 over 1
million V2 squared from both

00:06:36.700 --> 00:06:41.510
sides, and we would get 0.999999
V2 squared, but let's

00:06:41.510 --> 00:06:43.970
just say for the sake of
simplicity, or let's say, if

00:06:43.970 --> 00:06:47.430
this wasn't 1,000, but 1
million, and that this surface

00:06:47.430 --> 00:06:50.780
was much bigger, we see that
this term becomes very, very,

00:06:50.780 --> 00:06:52.090
very small.

00:06:52.090 --> 00:06:54.980
If that hole is one millionth
of the surface area, then it

00:06:54.980 --> 00:07:00.250
becomes really insignificant,
so we can ignore this term

00:07:00.250 --> 00:07:02.200
because it just makes things
complicated, and we're

00:07:02.200 --> 00:07:04.290
assuming this is a really,
really large number, and that

00:07:04.290 --> 00:07:10.130
this hole is much smaller than
the surface area of the fluid.

00:07:10.130 --> 00:07:11.910
This is like poking a
hole in Hoover Dam.

00:07:11.910 --> 00:07:14.290
Hoover Dam is backing up this
huge lake, and you poke a hole

00:07:14.290 --> 00:07:16.940
in it, so that hole is going to
be this very small fraction

00:07:16.940 --> 00:07:18.160
of the surface area
of the fluid.

00:07:18.160 --> 00:07:21.490
You can only make this
assumption when that output

00:07:21.490 --> 00:07:24.090
hole is much smaller than
the input hole.

00:07:24.090 --> 00:07:26.770
With that said, what is
the output velocity?

00:07:26.770 --> 00:07:30.130
The velocity-- you just take
the square root of both

00:07:30.130 --> 00:07:36.390
sides-- is the square
root of 2gh.

00:07:36.390 --> 00:07:40.610
That is the output velocity.

00:07:40.610 --> 00:07:47.330
What is the amount of liquid
that flows out per second?

00:07:47.330 --> 00:07:48.470
We figured that out already.

00:07:48.470 --> 00:07:54.300
It's a column of fluid that
comes out, so per second, the

00:07:54.300 --> 00:07:56.930
length of the column of fluid
will be the velocity times

00:07:56.930 --> 00:08:01.600
time, and then the cross-section
of that column

00:08:01.600 --> 00:08:04.010
is equal to the output area.

00:08:04.010 --> 00:08:08.720
If I wanted to know the flow
coming out, the flow coming

00:08:08.720 --> 00:08:13.740
out or the flux coming out would
be equal to the hole's

00:08:13.740 --> 00:08:17.670
area times the hole's
output velocity.

00:08:17.670 --> 00:08:24.420
That would equal the area times
the square root of 2gh.

00:08:24.420 --> 00:08:28.430
We could use that actually solve
problems in the future

00:08:28.430 --> 00:08:30.850
if we had actual numbers.

00:08:30.850 --> 00:08:32.539
I only have a minute
and a half left.

00:08:32.539 --> 00:08:34.620
I'll see you in the
next video.


Bernoulli's equation (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:01.960
This is just a quick review
of what we were

00:00:01.960 --> 00:00:03.050
doing in the last video.

00:00:03.050 --> 00:00:08.320
We had this oddly shaped pipe
and the fluid coming in had

00:00:08.320 --> 00:00:09.780
input velocity V1.

00:00:09.780 --> 00:00:13.010
The pressure on the left-hand
side pushing to the right is

00:00:13.010 --> 00:00:19.530
P1, and the area of
this hole is A1.

00:00:19.530 --> 00:00:22.090
Everything that the same
variables with the 2 on it is

00:00:22.090 --> 00:00:23.800
coming out of the pipe.

00:00:23.800 --> 00:00:27.500
What we just set up in that last
video is we said by the

00:00:27.500 --> 00:00:33.280
law of conservation of energy,
essentially the joules, the

00:00:33.280 --> 00:00:37.080
energy at this point in the
system, or that we're putting

00:00:37.080 --> 00:00:39.570
into the system, has to be equal
to the energy coming out

00:00:39.570 --> 00:00:41.650
of the system.

00:00:41.650 --> 00:00:44.360
We used that information to set
up this big equation, but

00:00:44.360 --> 00:00:45.320
it's not too complicated.

00:00:45.320 --> 00:00:48.560
We figured out that the work
going into the system was the

00:00:48.560 --> 00:00:52.730
input pressure times the mass of
volume over some period of

00:00:52.730 --> 00:00:58.590
time divided by the density of
whatever type of liquid we

00:00:58.590 --> 00:00:59.580
had, which was the
potential energy.

00:00:59.580 --> 00:01:03.350
This is just typically mgh,
where the mass is the mass of

00:01:03.350 --> 00:01:05.040
this column of fluid.

00:01:05.040 --> 00:01:08.530
We're saying, how much work
was done over some

00:01:08.530 --> 00:01:10.550
period of time T?

00:01:10.550 --> 00:01:11.710
That's the way I would
think about it.

00:01:11.710 --> 00:01:15.510
How much energy was there over
some period of time T?

00:01:15.510 --> 00:01:17.920
The kinetic energy over that
period of time would have been

00:01:17.920 --> 00:01:23.300
the mass of this volume of
fluid times its velocity

00:01:23.300 --> 00:01:24.370
squared divided by 2.

00:01:24.370 --> 00:01:26.870
That's typical kinetic energy.

00:01:26.870 --> 00:01:29.510
Of course, that has to be
equal to essentially the

00:01:29.510 --> 00:01:35.250
output energy, and so this is
the output work, or how much

00:01:35.250 --> 00:01:40.060
work a column of water could
do on the output side.

00:01:40.060 --> 00:01:43.110
It's an equivalent volume
of water, remember that.

00:01:43.110 --> 00:01:45.840
In some period of time T,
whatever volume of water this

00:01:45.840 --> 00:01:50.240
was, an equivalent volume of
water-- maybe it'll be a

00:01:50.240 --> 00:01:53.505
longer cylinder now, because
it's going to be going faster.

00:01:56.040 --> 00:01:59.040
So on the output side, it's
this longer cylinder that

00:01:59.040 --> 00:02:00.700
we're talking about, but it's
going to be the same volume

00:02:00.700 --> 00:02:01.655
and the same mass.

00:02:01.655 --> 00:02:06.640
So what we say is that the work
that this column can do

00:02:06.640 --> 00:02:10.650
in that same amount of time
would be the output pressure

00:02:10.650 --> 00:02:13.556
times the mass of this column
divided by the density of the

00:02:13.556 --> 00:02:15.690
column-- which is the same
because the density of the

00:02:15.690 --> 00:02:19.230
liquid is the same throughout--
times the mass of

00:02:19.230 --> 00:02:22.820
this column, which is the same
as the mass of this column

00:02:22.820 --> 00:02:26.890
because the volume and the
density hasn't changed, so

00:02:26.890 --> 00:02:28.510
they're the same mass.

00:02:28.510 --> 00:02:30.960
Now, this column has more
potential energy.

00:02:30.960 --> 00:02:35.060
It's up at h2, which I'm
assuming is higher than h1.

00:02:35.060 --> 00:02:38.590
This kinetic energy is just the
mass of this cylinder of

00:02:38.590 --> 00:02:41.780
fluid times its velocity
squared, which is the output

00:02:41.780 --> 00:02:42.840
velocity divided by 2.

00:02:42.840 --> 00:02:48.770
This is potential energy
out, and this is

00:02:48.770 --> 00:02:50.920
kinetic energy out.

00:02:50.920 --> 00:02:52.930
These equal each other.

00:02:52.930 --> 00:02:57.180
This setup is Bernoulli's
equation, but let's see if we

00:02:57.180 --> 00:03:00.150
can clean it up so that we can
get rid of variables that we

00:03:00.150 --> 00:03:02.020
don't have to know about.

00:03:02.020 --> 00:03:04.280
One thing that we see is that
there's an m in every term, so

00:03:04.280 --> 00:03:04.990
let's get rid of them.

00:03:04.990 --> 00:03:06.860
Divide both sides of
this equation by m.

00:03:12.070 --> 00:03:13.320
We get that.

00:03:15.940 --> 00:03:18.640
I don't like this density in the
denominator here, so let's

00:03:18.640 --> 00:03:21.590
multiply both sides of this
equation by density, and what

00:03:21.590 --> 00:03:24.800
we're left with is-- let me
write this in a vibrant color.

00:03:27.860 --> 00:03:32.470
P, the input pressure, plus--
and we're multiplying

00:03:32.470 --> 00:03:34.320
everything by this rho,
this density.

00:03:34.320 --> 00:03:42.050
So we have input pressure plus
rho g h1, the input height,

00:03:42.050 --> 00:03:49.760
the initial height, plus
rho v squared over 2.

00:03:54.920 --> 00:03:58.290
This is rho v squared over
2, and that equals-- we

00:03:58.290 --> 00:04:05.600
multiplied both sides by rho, so
we get the input velocity,

00:04:05.600 --> 00:04:10.990
so that equals the pressure
out plus the density times

00:04:10.990 --> 00:04:13.260
gravity times the
output height.

00:04:16.019 --> 00:04:16.935
Let's make everything
consistent.

00:04:16.935 --> 00:04:20.145
I wrote 2's here, so let's just
say this is pressure 2,

00:04:20.145 --> 00:04:26.880
this is height 2, plus rho times
the velocity squared.

00:04:30.100 --> 00:04:34.180
This is Bernoulli's equation,
and it has all sorts of what I

00:04:34.180 --> 00:04:37.230
would say is fairly neat
repercussions.

00:04:37.230 --> 00:04:41.790
For example, let's assume that
the height stays constant, so

00:04:41.790 --> 00:04:44.070
we can ignore these middle
terms. If the height is

00:04:44.070 --> 00:04:50.770
constant, if I have a higher
velocity and this whole term

00:04:50.770 --> 00:04:54.810
is constant, then my pressure
is going to be lower.

00:04:54.810 --> 00:04:59.490
Think about it: If height is
constant, this doesn't change,

00:04:59.490 --> 00:05:02.520
but if this velocity increases,
but this whole

00:05:02.520 --> 00:05:05.360
thing is constant, pressure
has to decrease.

00:05:05.360 --> 00:05:09.400
Similarly, if pressure
increases, then velocity is

00:05:09.400 --> 00:05:10.430
going to decrease.

00:05:10.430 --> 00:05:12.360
That might be a little
unintuitive, but the other

00:05:12.360 --> 00:05:13.270
way, it makes a lot of sense.

00:05:13.270 --> 00:05:16.840
When velocity increases, this
pressure is going to decrease,

00:05:16.840 --> 00:05:20.050
and that's actually what makes
planes fly and all sorts of

00:05:20.050 --> 00:05:21.330
neat things happen,
but we'll get more

00:05:21.330 --> 00:05:22.550
into that in a second.

00:05:22.550 --> 00:05:25.920
Let's see if we can use
Bernoulli's equation to do

00:05:25.920 --> 00:05:26.790
something useful.

00:05:26.790 --> 00:05:28.740
You should memorize this,
and it shouldn't

00:05:28.740 --> 00:05:30.710
be too hard to memorize.

00:05:30.710 --> 00:05:33.140
It's pressure, and then you
have this potential energy

00:05:33.140 --> 00:05:35.370
term, but instead of mass,
you have density.

00:05:35.370 --> 00:05:37.160
You have this kinetic
energy term.

00:05:37.160 --> 00:05:38.410
It's not kinetic energy
anymore, because we

00:05:38.410 --> 00:05:40.180
manipulated it some,
but instead of

00:05:40.180 --> 00:05:43.290
mass, you have density.

00:05:43.290 --> 00:05:45.070
With that said, let's
do a problem.

00:05:45.070 --> 00:05:47.170
I'll keep this down here, since
you probably haven't

00:05:47.170 --> 00:05:50.680
memorized it as yet.

00:05:50.680 --> 00:05:53.076
Let me erase everything else.

00:05:53.076 --> 00:05:55.930
That's not how I wanted
to erase it.

00:05:59.070 --> 00:06:00.900
That's how I wanted
to erase it.

00:06:00.900 --> 00:06:04.960
I wanted to erase it like that
without getting rid of

00:06:04.960 --> 00:06:06.155
anything useful.

00:06:06.155 --> 00:06:07.940
OK, that's good enough.

00:06:07.940 --> 00:06:10.464
And then let me clean up.

00:06:10.464 --> 00:06:12.568
Clean up all this stuff.

00:06:17.580 --> 00:06:20.300
Let's say that I have a cup.

00:06:26.330 --> 00:06:28.100
I'll just draw a cup.

00:06:28.100 --> 00:06:31.715
It's easier to draw sometimes
then to draw straight lines

00:06:31.715 --> 00:06:32.680
and all of that.

00:06:32.680 --> 00:06:34.170
No, that's too dark.

00:06:34.170 --> 00:06:35.420
Do purple.

00:06:37.280 --> 00:06:39.120
I'm using a super-wide tool.

00:06:39.120 --> 00:06:40.440
I have to switch the length.

00:06:40.440 --> 00:06:41.410
OK, so that's my cup.

00:06:41.410 --> 00:06:42.110
It has some fluid.

00:06:42.110 --> 00:06:47.870
Actually, let's say it has a
top to it, and I have some

00:06:47.870 --> 00:06:50.090
fluid in it.

00:06:50.090 --> 00:06:51.910
Maybe it happens to be red.

00:06:51.910 --> 00:06:55.370
We haven't been dealing with
red fluids as yet,.

00:06:55.370 --> 00:06:59.450
Let me-- oh, I didn't
want to do that.

00:06:59.450 --> 00:07:01.980
So you know there's
a fluid there.

00:07:01.980 --> 00:07:04.780
And let's say that there's no
air here, so this is a vacuum.

00:07:08.660 --> 00:07:12.840
Let's say that h-- we don't know
what units are, but let's

00:07:12.840 --> 00:07:17.150
say h meters below the
surface of the fluid.

00:07:17.150 --> 00:07:19.350
This is all fluid here.

00:07:19.350 --> 00:07:26.860
I poke a hole right there, and
fluid starts spurting out.

00:07:26.860 --> 00:07:32.870
My question to you is, what is
this output velocity of the

00:07:32.870 --> 00:07:36.120
fluid as a function
of this height?

00:07:36.120 --> 00:07:37.000
Let me tell you something
else.

00:07:37.000 --> 00:07:39.930
Let's say that this hole is so
small, let's call the area of

00:07:39.930 --> 00:07:44.840
that hole A2, and let's say that
the surface area of the

00:07:44.840 --> 00:07:47.600
water is A1.

00:07:47.600 --> 00:07:51.570
Let's say that hole is so small
that the surface area

00:07:51.570 --> 00:07:59.560
the water-- let's say that A2
if equal to 1/1,000 of A1.

00:07:59.560 --> 00:08:02.270
This is a small hole relative
to the surface

00:08:02.270 --> 00:08:05.340
area of this cup.

00:08:05.340 --> 00:08:07.790
With that said, let's see what
we can do about figuring out

00:08:07.790 --> 00:08:09.430
the velocity coming out.

00:08:09.430 --> 00:08:12.030
Bernoulli's equation tells us
that the input pressure plus

00:08:12.030 --> 00:08:14.700
the input potential energy plus
the input kinetic energy

00:08:14.700 --> 00:08:16.100
is equal to the output,
et cetera.

00:08:18.960 --> 00:08:21.190
So what is the input pressure?

00:08:21.190 --> 00:08:24.690
Well, the input pressure, the
pressure at this point,

00:08:24.690 --> 00:08:27.340
there's no air or no fluid above
it, so the pressure at

00:08:27.340 --> 00:08:30.500
that point is zero.

00:08:30.500 --> 00:08:34.020
What is the input height?

00:08:34.020 --> 00:08:38.280
Let's just assume that the hole
is done at height 0, h

00:08:38.280 --> 00:08:45.060
equals 0, so the input
height h1 is just h.

00:08:45.060 --> 00:08:49.930
If this is 0, then this height
right here is h.

00:08:49.930 --> 00:08:52.440
What is the input velocity?

00:08:52.440 --> 00:08:54.950
We know from the continuity
equation, or whatever that

00:08:54.950 --> 00:09:00.580
thing was called, that the input
velocity times the input

00:09:00.580 --> 00:09:04.030
area is equal to the
output velocity

00:09:04.030 --> 00:09:05.330
times the output area.

00:09:09.660 --> 00:09:15.360
We also know that the output
area is equal to 1/1,000 of

00:09:15.360 --> 00:09:22.810
the input area-- and this is
area 2-- so we know that the

00:09:22.810 --> 00:09:28.510
input velocity times area 1 is
equal to the output velocity

00:09:28.510 --> 00:09:31.230
times 1/1,000 of area 1.

00:09:31.230 --> 00:09:36.590
We could say area 1 over
1,000 and divide both

00:09:36.590 --> 00:09:38.030
sides by area 1.

00:09:38.030 --> 00:09:42.170
We know that the input velocity
is equal to V2 over

00:09:42.170 --> 00:09:46.290
1,000, so that's good to know.

00:09:46.290 --> 00:09:51.790
These are the three inputs into
the left-hand side of

00:09:51.790 --> 00:09:53.570
Bernoulli's equation.

00:09:53.570 --> 00:09:57.150
What's on the right-hand side
of Bernoulli's equation?

00:09:57.150 --> 00:09:58.400
What's P2?

00:10:02.280 --> 00:10:04.850
What's the pressure
at this point?

00:10:04.850 --> 00:10:05.820
Oh, I just ran out of time.

00:10:05.820 --> 00:10:08.290
I'll continue this into
the next video.


Bernoulli's equation (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:05.710
Let's say we have a pipe again--
this is the opening--

00:00:05.710 --> 00:00:07.380
and we have fluid going
through it.

00:00:07.380 --> 00:00:14.860
The fluid is going with a
velocity of v1, the pressure

00:00:14.860 --> 00:00:19.560
entering the pipe is P1, and
then the area of this opening

00:00:19.560 --> 00:00:22.770
of the pipe is A1.

00:00:22.770 --> 00:00:27.030
It could even go up, and
the other end is

00:00:27.030 --> 00:00:28.280
actually even smaller.

00:00:31.790 --> 00:00:36.940
The fluid-- the liquid-- is
exiting the pipe with velocity

00:00:36.940 --> 00:00:42.950
v2, the pressure that it
exerts as it goes out.

00:00:42.950 --> 00:00:44.880
If there was a membrane on the
outside, how much pressure

00:00:44.880 --> 00:00:47.980
would it exert on it as it
pushes it out on the adjacent

00:00:47.980 --> 00:00:53.580
water is P2, and the area of
the smaller opening-- it

00:00:53.580 --> 00:00:56.300
doesn't have to be
smaller-- is A2.

00:00:56.300 --> 00:01:05.710
Let's say that this opening is
at a height, on average, of

00:01:05.710 --> 00:01:14.790
h1, and the water exiting this
opening is on average at a

00:01:14.790 --> 00:01:15.960
height of h2.

00:01:15.960 --> 00:01:18.720
We won't worry too much about
the differential between the

00:01:18.720 --> 00:01:21.260
top of the pipe and the bottom
of the pipe-- we'll assume

00:01:21.260 --> 00:01:23.610
that these h's are much
bigger relative to

00:01:23.610 --> 00:01:24.860
the size of the pipe.

00:01:28.020 --> 00:01:30.150
With that set up-- and remember,
there's fluid going

00:01:30.150 --> 00:01:38.070
through this thing-- let's go
back to what keeps showing up,

00:01:38.070 --> 00:01:40.260
which is the law of conservation
of energy, which

00:01:40.260 --> 00:01:43.830
is in any closed system, the
amount of energy that you put

00:01:43.830 --> 00:01:47.290
into something is equal to
the amount of energy

00:01:47.290 --> 00:01:48.605
that you get out.

00:01:48.605 --> 00:01:57.030
So energy in is equal
to energy out.

00:01:57.030 --> 00:02:00.390
What's the energy that you put
into a system, or that the

00:02:00.390 --> 00:02:02.970
system starts off with
at this end?

00:02:02.970 --> 00:02:11.210
It's the work that you input
plus the potential energy at

00:02:11.210 --> 00:02:17.180
that point of the system, plus
the kinetic energy at that

00:02:17.180 --> 00:02:18.430
point of the system.

00:02:20.970 --> 00:02:22.840
Then we know from the
conservation of energy that

00:02:22.840 --> 00:02:30.780
that has to equal the output
work plus the output potential

00:02:30.780 --> 00:02:35.110
energy plus the output
kinetic energy.

00:02:35.110 --> 00:02:37.110
A lot of times in the past,
we've just said that the

00:02:37.110 --> 00:02:40.710
potential energy input plus the
kinetic energy input is

00:02:40.710 --> 00:02:43.680
equal to the potential energy
output plus the kinetic energy

00:02:43.680 --> 00:02:48.260
output, but the initial energy
in the system can

00:02:48.260 --> 00:02:49.580
also be done by work.

00:02:49.580 --> 00:02:52.640
So we just added work to this
equation that says that the

00:02:52.640 --> 00:02:55.650
energy in is equal to
the energy out.

00:02:55.650 --> 00:02:57.900
With that information, let's
see if we can do anything

00:02:57.900 --> 00:03:00.800
interesting with this pipe
that I've drawn.

00:03:00.800 --> 00:03:04.960
So what's the work that's being
put into this system?

00:03:04.960 --> 00:03:10.200
Work is force times distance,
so let's just focus on this.

00:03:10.200 --> 00:03:18.410
It's the force in times the
distance in, and so over a

00:03:18.410 --> 00:03:22.050
period of time, t, what
has been done?

00:03:22.050 --> 00:03:24.740
We learned in the last video
that over a period of time, t,

00:03:24.740 --> 00:03:29.510
the fluid here might have
moved this far.

00:03:29.510 --> 00:03:31.300
What is this distance?

00:03:31.300 --> 00:03:35.740
This distance is the input
velocity times whatever amount

00:03:35.740 --> 00:03:40.880
of time we're dealing with, so
T-- so that's the distance.

00:03:40.880 --> 00:03:43.500
What's the force?

00:03:43.500 --> 00:03:47.170
The force is just pressure times
area, and we can figure

00:03:47.170 --> 00:03:52.630
that out by just dividing
force by, area and then

00:03:52.630 --> 00:03:57.890
multiply by area, so we get
input force divided by area

00:03:57.890 --> 00:04:00.820
input, times area input.

00:04:00.820 --> 00:04:03.870
It's divided and multiplied by
the same number-- that's

00:04:03.870 --> 00:04:05.060
pressure, that's area.

00:04:05.060 --> 00:04:10.010
It's equal to the input distance
over that amount of

00:04:10.010 --> 00:04:17.779
time, and that's velocity times
time, so the work input

00:04:17.779 --> 00:04:33.240
is equal to the input pressure
times the input area times

00:04:33.240 --> 00:04:37.136
input velocity times time.

00:04:39.730 --> 00:04:46.390
What is this area times velocity
times time, times

00:04:46.390 --> 00:04:48.050
this distance?

00:04:48.050 --> 00:04:51.670
That's the volume of fluid
that flowed in over that

00:04:51.670 --> 00:04:53.940
amount of time.

00:04:53.940 --> 00:04:58.400
So that equals the volume of
fluid over that period of

00:04:58.400 --> 00:05:04.660
time, so we could call that
volume in, or volume i--

00:05:04.660 --> 00:05:07.200
that's the input volume.

00:05:07.200 --> 00:05:17.110
We know that density is just
mass per volume, or that

00:05:17.110 --> 00:05:21.060
volume times density is equal
to mass, or we know that

00:05:21.060 --> 00:05:25.410
volume is equal to mass
divided by density.

00:05:25.410 --> 00:05:28.050
The work that I'm putting into
the system-- I know I'm doing

00:05:28.050 --> 00:05:33.110
a lot of crazy things, but it'll
make sense so far-- is

00:05:33.110 --> 00:05:36.880
equal to the input pressure
times the amount of volume of

00:05:36.880 --> 00:05:39.280
fluid that moved over
that period of time.

00:05:39.280 --> 00:05:43.020
That volume of fluid is equal to
the mass of the fluid that

00:05:43.020 --> 00:05:46.590
went in at that period of time,
and we'll call that the

00:05:46.590 --> 00:05:54.190
input mass, divided
by the density.

00:05:54.190 --> 00:05:57.740
Hopefully, that makes a
little bit of sense.

00:05:57.740 --> 00:06:01.140
As we know, the input volume
is going to be equal to the

00:06:01.140 --> 00:06:04.070
output volume, so the input
mass-- because the density

00:06:04.070 --> 00:06:06.250
doesn't change-- is equal to the
output mass, so we don't

00:06:06.250 --> 00:06:08.800
have to write an input and
output for the mass.

00:06:08.800 --> 00:06:11.290
The mass is going to be
constant; in any given amount

00:06:11.290 --> 00:06:13.220
of time, the mass that enters
the system will be equivalent

00:06:13.220 --> 00:06:16.460
to the mass that exits
the system.

00:06:16.460 --> 00:06:18.800
There we go: we have an
expression, an interesting

00:06:18.800 --> 00:06:21.810
expression, for the work being
put into the system.

00:06:21.810 --> 00:06:25.010
What is the potential energy
of the system on

00:06:25.010 --> 00:06:26.350
the left-hand side?

00:06:26.350 --> 00:06:36.050
The potential energy of the
system is going to be equal to

00:06:36.050 --> 00:06:44.180
that same mass of fluid that I
talked about times gravity

00:06:44.180 --> 00:06:50.500
times this input height-- the
initial height-- times h1.

00:06:50.500 --> 00:06:57.170
The initial kinetic energy of
the fluid equals the mass of

00:06:57.170 --> 00:07:00.170
the fluid-- this mass right
here, of that same cylinder

00:07:00.170 --> 00:07:04.020
volume that I keep pointing to--
times the velocity of the

00:07:04.020 --> 00:07:06.300
fluid squared.

00:07:06.300 --> 00:07:08.620
We remember this from kinetic
energy divided by 2.

00:07:08.620 --> 00:07:11.410
So what's the total energy at
this point in the system over

00:07:11.410 --> 00:07:12.280
this period of time?

00:07:12.280 --> 00:07:15.600
How much energy has gone
into the system?

00:07:15.600 --> 00:07:19.080
It's going to be the work
done, which is the input

00:07:19.080 --> 00:07:23.120
pressure-- I'm running out
of space, so let me

00:07:23.120 --> 00:07:24.330
erase all of this.

00:07:24.330 --> 00:07:27.265
I'll probably have to run out of
time, too, but that's OK--

00:07:27.265 --> 00:07:28.515
it's better than
being confused.

00:07:53.620 --> 00:07:55.270
Back to what we were doing.

00:07:55.270 --> 00:07:59.950
So, the total energy going into
the system is the work

00:07:59.950 --> 00:08:01.760
being done into the system,
and I rewrote it in this

00:08:01.760 --> 00:08:06.920
format, which is the input
pressure-- we'll call that

00:08:06.920 --> 00:08:15.000
P1-- times the mass divided by
the density of the liquid,

00:08:15.000 --> 00:08:16.130
whatever it is.

00:08:16.130 --> 00:08:21.920
This is work in plus-- and
what's the potential energy?

00:08:21.920 --> 00:08:25.720
I wrote it right here-- that's
just mgh, where m is the mass

00:08:25.720 --> 00:08:30.187
of this volume of fluid, h is
its average height, and you

00:08:30.187 --> 00:08:34.539
could almost think of how high
the center of mass above the

00:08:34.539 --> 00:08:36.850
surface of the planet.

00:08:36.850 --> 00:08:40.090
Since we have a g here, we
assume we're on Earth, so this

00:08:40.090 --> 00:08:45.440
is h1, because the height
actually changes, so this is

00:08:45.440 --> 00:08:53.270
potential energy input plus
the kinetic energy mv1

00:08:53.270 --> 00:08:56.130
squared over 2.

00:08:56.130 --> 00:08:59.920
That is the kinetic
energy input.

00:09:02.930 --> 00:09:08.250
We know that this has
to equal the energy

00:09:08.250 --> 00:09:09.500
coming out of the system.

00:09:33.110 --> 00:09:34.700
This is going to be equal
to the same thing

00:09:34.700 --> 00:09:38.070
on the output side.

00:09:38.070 --> 00:09:41.040
This is going to be equal to
the work out, so that'll be

00:09:41.040 --> 00:09:46.530
the output pressure times the
mass divided by the density

00:09:46.530 --> 00:09:52.470
plus the output potential
energy, which will just be mg

00:09:52.470 --> 00:10:00.440
h2, plus the outbound kinetic
energy, which will be mv2

00:10:00.440 --> 00:10:02.740
squared divided by 2.

00:10:02.740 --> 00:10:04.790
I just realized I'm
out of time.

00:10:04.790 --> 00:10:06.040
I will continue this
in the next video.

00:10:06.040 --> 00:10:07.660
See you soon.