Discussions of conditions for Hardy Weinberg

WEBVTT
Kind: captions
Language: en

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- [Voiceover] In the introductory video

00:00:03.130 --> 00:00:04.819
to the Hardy-Weinberg equation,

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I gave some conditions for
the Hardy-Weinberg equation

00:00:07.328 --> 00:00:09.530
to hold, and what I
wanna do in this video is

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go into a little bit more depth,

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and have a little bit more of a discussion

00:00:12.620 --> 00:00:15.185
on the conditions for the
Hardy-Weinberg equation.

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Now just to review what the Hardy-Weinberg

00:00:16.821 --> 00:00:18.171
equation is all about,

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if we have a population with the gene,

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say for eye color, and let's say that gene

00:00:22.146 --> 00:00:24.955
comes in two versions, one
is the allele that produces

00:00:24.955 --> 00:00:27.902
blue, one is the allele
that produces brown,

00:00:27.902 --> 00:00:30.084
if p is the frequency of the blue allele,

00:00:30.084 --> 00:00:31.944
q is the frequency of the brown allele.

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Well, and if they're
the only two versions,

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if you add the frequency
of p, of the blue,

00:00:36.526 --> 00:00:38.381
plus the frequency of the brown,

00:00:38.381 --> 00:00:39.659
they're gonna add up to 100%, or one.

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And if you square both sides of this,

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you would get this
expression right over here.

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And we talk about that
this is the probability,

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or you could say, the frequency,

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of being a homozygous for the blue.

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This is the probability of
having two alleles for the brown.

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And then right here in the middle,

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this is the probability
of being a heterozygote,

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and why is that?

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Well cause you can get
a blue from your mom

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and a brown from your dad,

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or a blue from your dad
and a brown from your mom.

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So there's two ways to
get that pq combination.

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Now the key idea is Hardy-Weinberg assumes

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a stable allele frequency,

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so let me write that really big.

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Because all of these other
conditions that you might see

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are really like, well, what
are all the different ways

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that you could somehow not
have stable allele frequency?

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So let me write this down,
the stable allele frequency.

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Stable allele frequency.

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So a lot of times there's
a temptation to memorize

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a bunch of this stuff,
you might wanna do that.

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But the more important thing
is to get the underlying idea.

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And the underlying idea is,
well, will something somehow

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cause the allele frequency to be unstable?

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And actually, another way to
say stable allele frequency

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is no evolution.

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No evolution.

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Evolution is a change in
the inheritable traits

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in a population, and that
will include a change

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in allele frequency.

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And if you think about the
two ways that you could have

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a population evolving, well,
you can have selection.

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So we're gonna assume no selection.

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Actually, there's more than two ways,

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you could have genetic engineering
and all sorts of things.

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So we're gonna assume the mainstream ways,

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I guess you could say, we
can assume no selection,

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we can assume no genetic drift.

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Remember, selection is
certain traits that make

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that organism more fit
for that environment,

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well those traits are more
likely to be passed on.

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Genetic drift is random chance changes

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in the allele frequency.

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It could be due to small populations,

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it could be due to
members of the population

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migrating or some type
of bottleneck effect,

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some natural disaster that really gets you

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to that small population.

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So that's the big picture.

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But given that big picture,

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I wanna dive deep into
some of the assumptions

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that you might see in your biology class,

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just so you feel comfortable with them

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and you see that we're
talking about the same thing.

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So the ones that I mentioned in that

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introductory video are no selection,

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and that's consistent with no evolution.

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I also talk about no net mutation,

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also consistent with no evolution.

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Once again, we don't want to
change the allele frequency.

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If there was net mutation, one of those,

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maybe some of those blue
versions of the gene

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got a mutation and they're
now maybe a different version,

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or they're definitely not blue anymore,

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so the allele frequency would change.

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The reason why we care
about large population

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is mainly for genetic drift.

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If you have a very small population,

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just due to random chance,

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it's more likely that
the allele frequencies

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can change appreciably.

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The other conditions
that you will often see

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are things like random, random mating.

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That whether an organism has
the blue or the brown version

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of the gene, that that
doesn't make them any more

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or less desirable to a
member of the opposite sex.

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And if you think about, you might say,

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well isn't that a form of selection?

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And you'd say well yes, it kind of is,

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but this is sometimes
broken out as another way.

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Now also, no migration,
that you don't have,

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the population isn't
growing by other organisms

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entering it or isn't
shrinking by other organisms

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leaving or there's not
a mixing of population

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between two populations.

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And once again, it's
all in, it's all because

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we care about stable allele frequencies.

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Now if we wanna go even further than that,

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and sometimes you will hear
these types of things mentioned,

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although I just mentioned
the five mainstream things,

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which all boil down to
stable allele frequency,

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no evolution, no selection,
no genetic drift.

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But sometimes, we are
assuming that we are dealing

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with diploid organisms,
that you're getting one set

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of chromosomes from your
mom, one set of chromosomes

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from your dad, one version
of an allele from your mom,

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one version of an allele from your dad.

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And you might say well, how
can you be other than diploid?

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Well you could be a, there
are tetraploid populations,

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especially this can happen in plants.

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Or you could get two sets of
chromosomes from your mom,

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and two sets of chromosomes from your dad.

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We are assuming sexual reproduction.

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That we're not dealing with cloning

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or just budding, where
you're just a copy of

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another organism from
generation to generation.

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We're assuming that whether
you are blue or brown,

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whether you have those versions,

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that that's not correlated
with what sex you are.

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So allele frequency, allele
frequency same in all sexes,

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in all sexes.

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And we're assuming sexual
reproduction, once again,

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we're assuming one where
there's only two sexes,

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so you could, you know, if
you were to think about,

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if you were to let your
imagination go wild,

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you could imagine a lot of
other constraints to put here

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or other ways that the,
where you could no longer

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apply the Hardy-Weinberg.

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Where this is with two alleles,

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we're assuming sexual reproduction,

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diploid, you're getting from
your mom, from your dad,

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and just here are all the conditions

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that help us ensure that we
have a stable allele frequency.

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Now the one thing you're saying, okay,

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I can, diploid, sexual reproduction, okay.

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But isn't there always a
chance for a little bit

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of genetic drift?

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Isn't there just, the
history of the world,

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is that we have this evolution,

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and the answer is yes.

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And so the actual reality
is that there's very few

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places where you can point to,

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very few populations, if any,

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where you can say, oh, that's a pure,

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we can purely apply Hardy-Weinberg there.

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But like a lot of things
in the applied sciences,

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it's a very good approximation
for many populations.

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And so that's why it is useful.


LC natural response example

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So in previous videos,

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we worked out an expression

00:00:03.731 --> 00:00:07.326
for the current i in an
LC circuit like this,

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and what we found

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was that i is square root of
capacitance over inductance,

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times the starting voltage, V knot,

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times sine omega knot t.

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And omega knot is the natural frequency,

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and we say that omega knot

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was equal to square root
of one over L times C.

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And V knot is the starting
voltage on the capacitor.

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And our original assumption

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was that i was zero to begin with,

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and that's the expression for the current.

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So what I wanna do now
is a specific example,

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and we'll do that down here.

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First I'm gonna install a switch

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into our circuit so that
we can add some energy

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and it won't go anywhere.

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Then for C, C is gonna be
equal to a quarter of a farad,

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and L is gonna be equal to one henry.

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All right?

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And on the capacitor, I'm
gonna put enough charge

00:01:26.460 --> 00:01:31.041
to bring this up to 10 volts
before we close the switch.

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And then at time equals
zero, we'll close the switch.

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And what we wanna find is, what is i of t?

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We have L, we have C, we
have a starting voltage,

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and so now we have everything we need

00:01:50.253 --> 00:01:52.022
to work out the current.

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Let's do that, okay?

00:01:53.302 --> 00:01:55.717
So first off we'll do omega knot.

00:01:55.717 --> 00:02:00.373
Omega knot equals one over
the square root of LC,

00:02:00.373 --> 00:02:01.540
so that equals

00:02:03.206 --> 00:02:07.373
square root of one over one
henry times one quarter farad,

00:02:11.788 --> 00:02:14.538
which equals square root of four,

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or equals two,

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and that's in units of radians per second.

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That's the natural frequency.

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We know the natural frequency right here,

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and now we can work out the rest of it.

00:02:33.388 --> 00:02:37.471
So we can just fill in i
equals square root of C,

00:02:39.567 --> 00:02:41.817
which is one quarter farad,

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divided by one henry,

00:02:46.487 --> 00:02:47.540
times V knot.

00:02:47.540 --> 00:02:49.457
V knot was was 10 volts

00:02:51.480 --> 00:02:53.751
times sine omega knot t.

00:02:53.751 --> 00:02:55.918
Sine, omega knot is two t.

00:02:59.275 --> 00:03:01.108
And finally, i equals,

00:03:03.984 --> 00:03:08.766
this is the square root of
one quarter or one half,

00:03:08.766 --> 00:03:09.881
so that's one half,

00:03:09.881 --> 00:03:13.298
times 10 is five, i equals five sine 2 t.

00:03:23.515 --> 00:03:25.318
So for this specific circuit,

00:03:25.318 --> 00:03:27.690
that's the answer for this current here.

00:03:27.690 --> 00:03:31.857
Now I wanna show you what
that actually looks like.

00:03:36.835 --> 00:03:41.002
So this is a plot of i of
t equals five sine two t.

00:03:47.678 --> 00:03:50.202
And this is what a sine
wave looks like with time,

00:03:50.202 --> 00:03:52.941
so the axes are time in seconds,

00:03:52.941 --> 00:03:55.495
and this axis goes up to five amperes

00:03:55.495 --> 00:03:58.793
and then down to minus five amperes

00:03:58.793 --> 00:04:02.960
and continues on that way, and
it basically goes on forever.

00:04:04.181 --> 00:04:05.755
Now what I wanna do next is work out

00:04:05.755 --> 00:04:07.392
the voltage in the circuit.

00:04:07.392 --> 00:04:09.736
We didn't talk about the voltage yet.

00:04:09.736 --> 00:04:12.903
So I'll sketch the circuit again here.

00:04:17.576 --> 00:04:18.733
Here's L and C.

00:04:18.733 --> 00:04:20.290
And this is the voltage right here,

00:04:20.290 --> 00:04:22.935
voltage across both guys.

00:04:22.935 --> 00:04:25.185
We've already worked out i.

00:04:26.502 --> 00:04:27.828
And now we wanna find v.

00:04:27.828 --> 00:04:29.411
What's v?

00:04:29.411 --> 00:04:31.463
That's what we're looking for.

00:04:31.463 --> 00:04:34.963
So, we know that i equals five sine two t,

00:04:41.111 --> 00:04:43.954
and to find v, one of
the easy ways to find v

00:04:43.954 --> 00:04:47.244
is to use the inductor equation.

00:04:47.244 --> 00:04:50.661
We know that v equals L di/dt.

00:04:51.671 --> 00:04:55.838
That's just the basic
inductor i v equation, right?

00:04:56.987 --> 00:04:59.798
And let's see what happens here.

00:04:59.798 --> 00:05:03.298
v equals L is one, times di/dt.

00:05:05.079 --> 00:05:08.579
So that's d/dt of 5 sine two t.

00:05:14.063 --> 00:05:15.835
Now let's take that derivative, okay?

00:05:15.835 --> 00:05:17.317
I'll go up here.

00:05:17.317 --> 00:05:20.817
v equals five comes out of the derivative.

00:05:24.044 --> 00:05:25.030
Sine two t.

00:05:25.030 --> 00:05:28.257
The derivative of sine of two t

00:05:28.257 --> 00:05:30.674
is two times cosine of two t.

00:05:37.372 --> 00:05:38.205
All right?

00:05:38.205 --> 00:05:42.288
So our voltage solution
is 10 times cosine two t.

00:05:52.288 --> 00:05:54.311
So something interesting
just happened here.

00:05:54.311 --> 00:05:55.513
Let me show you.

00:05:55.513 --> 00:05:58.548
We started with the current
being a sine function,

00:05:58.548 --> 00:06:01.268
and we eventually took the
derivative of that sine function

00:06:01.268 --> 00:06:03.202
and now we have a cosine function.

00:06:03.202 --> 00:06:06.785
So we went from sine
to cosine for voltage.

00:06:08.049 --> 00:06:09.311
That means the voltage doesn't look

00:06:09.311 --> 00:06:12.050
quite exactly like the current, okay?

00:06:12.050 --> 00:06:15.217
Let me show you a plot of the voltage.

00:06:18.747 --> 00:06:19.715
Here's the voltage here.

00:06:19.715 --> 00:06:20.965
This is v of t.

00:06:21.905 --> 00:06:25.988
And that we decided was
equal to 10 cosine two t.

00:06:28.031 --> 00:06:31.107
Starts at a value of 10,
the time equals zero,

00:06:31.107 --> 00:06:33.601
and then forms a cosine
wave that goes up and down

00:06:33.601 --> 00:06:36.268
between plus and minus 10 volts.

00:06:37.569 --> 00:06:38.863
Now let me show you what it looks like

00:06:38.863 --> 00:06:41.763
when we plot both i and
v on the same graph,

00:06:41.763 --> 00:06:46.421
and we can see the timing
relationship between them.

00:06:46.421 --> 00:06:49.421
So this is a plot of i of t in blue,

00:06:52.345 --> 00:06:54.095
and v of t in orange.

00:06:57.331 --> 00:06:59.992
One them is a sine wave,
the current is a sine wave,

00:06:59.992 --> 00:07:02.409
and voltage is a cosine wave.

00:07:04.987 --> 00:07:06.817
So this is what we were going for.

00:07:06.817 --> 00:07:09.588
This is the natural
response of an LC circuit,

00:07:09.588 --> 00:07:12.028
and we had two specific
component values in it.

00:07:12.028 --> 00:07:14.819
We saw that we came out with
both current and voltage

00:07:14.819 --> 00:07:17.319
looking like sinusoidal waves.

00:07:18.724 --> 00:07:19.662
This LC circuit,

00:07:19.662 --> 00:07:22.189
this is where sine waves
come from in electronics.

00:07:22.189 --> 00:07:23.242
It's pretty cool.


LC natural response derivation 3

WEBVTT
Kind: captions
Language: en

00:00:01.359 --> 00:00:03.689
- [Voiceover] In the last
video we took a guess at what

00:00:03.689 --> 00:00:06.759
the solution was for our
differential equation and we came

00:00:06.759 --> 00:00:11.679
up with an exponential as
our guess and as we did the

00:00:11.679 --> 00:00:16.341
analysis we developed a
characteristic equation.

00:00:16.341 --> 00:00:18.704
We ended up with a complex
answer for one of the

00:00:18.704 --> 00:00:22.871
adjustable parameters, the
natural frequency of our circuit.

00:00:25.964 --> 00:00:27.921
So here's the form of
our proposed solution,

00:00:27.921 --> 00:00:30.774
now this is getting kinda
complicated but I promise you

00:00:30.774 --> 00:00:33.371
we're gonna simplify this pretty soon.

00:00:33.371 --> 00:00:37.304
We have two solutions for s, s
one and s two that we plugged

00:00:37.304 --> 00:00:40.422
in as plus and minus j omega naught,

00:00:40.422 --> 00:00:43.472
and we have more adjustable
amplitude parameters that we

00:00:43.472 --> 00:00:45.055
have to figure out.

00:00:50.447 --> 00:00:53.439
So now in this video we're
gonna continue on solving our

00:00:53.439 --> 00:00:56.678
differential equation, this
is our proposed solution,

00:00:56.678 --> 00:00:59.424
this is a pretty complicated
looking expression and

00:00:59.424 --> 00:01:02.592
what we're gonna call on
here now is something really

00:01:02.592 --> 00:01:06.819
important in electronics and
in general and it's called

00:01:06.819 --> 00:01:08.236
Euler's Identity.

00:01:15.243 --> 00:01:18.785
And we're gonna use this
identity to figure out what to do

00:01:18.785 --> 00:01:22.387
with these complex exponential terms.

00:01:22.387 --> 00:01:25.237
So if you search on this term
on Kahn Academy you'll find

00:01:25.237 --> 00:01:28.586
an explanation of where
these identities come from

00:01:28.586 --> 00:01:30.771
but I'm just gonna state 'em right here.

00:01:30.771 --> 00:01:34.105
And what it says is that
Euler's identitiy is

00:01:34.105 --> 00:01:38.272
e to the j x anything up
there equals cosign x plus j

00:01:46.028 --> 00:01:49.528
times sine x, that's one of the identities

00:01:51.999 --> 00:01:56.166
and the other identity is
e to the minus j x equals

00:01:59.554 --> 00:02:02.054
cosign of x minus j sine x.

00:02:08.699 --> 00:02:11.337
So these are useful because
we have this exponential

00:02:11.337 --> 00:02:15.004
function with the complex
unit inside of it,

00:02:16.010 --> 00:02:18.080
when we go over to this side,

00:02:18.080 --> 00:02:20.408
this is sort of a normal complex number,

00:02:20.408 --> 00:02:23.360
cosign x is some number
between plus and minus one,

00:02:23.360 --> 00:02:26.780
sine x is some number
between plus and minus one,

00:02:26.780 --> 00:02:28.963
and it's just a normal complex number

00:02:28.963 --> 00:02:32.494
so this may help us
simplify our life here as we

00:02:32.494 --> 00:02:34.128
move forward.

00:02:34.128 --> 00:02:37.765
So that's a really important
identity that we get to use

00:02:37.765 --> 00:02:39.765
to solve our LC circuit.

00:02:43.221 --> 00:02:45.473
Now I'm gonna go back and
rewrite these two exponential's

00:02:45.473 --> 00:02:47.806
using Euler's Identity here.

00:02:48.872 --> 00:02:52.475
This is gonna get big but it'll
collapse down pretty soon.

00:02:52.475 --> 00:02:56.642
Okay, let me move over here,
i equals k one times e to the

00:02:58.586 --> 00:03:02.813
plus j omega t let's use this
one here so that would be

00:03:02.813 --> 00:03:06.980
cosign x is omega naught t
plus j sine x is omega naught t

00:03:22.837 --> 00:03:24.920
and we use the first one.

00:03:25.890 --> 00:03:30.057
Okay, now the second term is
plus k two times now we have

00:03:32.867 --> 00:03:37.587
the negative up in the
exponent so we use this one

00:03:37.587 --> 00:03:41.754
cosign omega naught t minus
j sine omega naught t.

00:03:50.525 --> 00:03:52.927
Now we have our solution spread
all out across the screen

00:03:52.927 --> 00:03:56.199
and let's see if we can
tidy things up here.

00:03:56.199 --> 00:03:58.951
So what I'm gonna do is I'm
gonna gather all the cosign

00:03:58.951 --> 00:04:02.173
terms together, this one and this one

00:04:02.173 --> 00:04:05.972
and then I'll gather
the sine terms together.

00:04:05.972 --> 00:04:10.139
Okay so i equals cosign omega
naught t, and the cosign is

00:04:16.167 --> 00:04:19.750
multiplied by k one
and plus k two alright.

00:04:29.945 --> 00:04:31.695
Now let's add to that

00:04:36.483 --> 00:04:40.650
we have j sine and j sine so
I'll write sine omega naught t

00:04:43.664 --> 00:04:47.831
over here and j times what,
j times k one and this time

00:04:53.685 --> 00:04:57.852
we have this minus sign
that makes it minus k two,

00:05:00.292 --> 00:05:04.459
alright so now, current is
some number times cosign plus

00:05:07.427 --> 00:05:10.249
j times some number times sine.

00:05:10.249 --> 00:05:12.633
Now these are two arbitrary
constants and I'm just gonna

00:05:12.633 --> 00:05:16.716
make up another one, I'm
just call this one a one

00:05:21.452 --> 00:05:25.285
we'll call this one a
two and we'll call that,

00:05:26.873 --> 00:05:29.706
I'll use the j, k one minus k two,

00:05:34.097 --> 00:05:35.930
now I can rewrite this

00:05:38.533 --> 00:05:41.450
as i equals a one cos mega naught t

00:05:50.017 --> 00:05:53.100
plus a two sine omega naught t, good.

00:06:00.377 --> 00:06:02.507
So from now on we're
gonna work with these a's

00:06:02.507 --> 00:06:04.854
and if I ever want to know
what the original k's were

00:06:04.854 --> 00:06:07.591
I would just come back
to these equations here,

00:06:07.591 --> 00:06:11.758
once I figure out a, I can
figure out both the k's.

00:06:12.619 --> 00:06:14.786
So let's keep pressing on,

00:06:15.633 --> 00:06:17.996
how do we figure out a one and a two?

00:06:17.996 --> 00:06:20.522
Well to do that we're gonna
use the initial conditions,

00:06:20.522 --> 00:06:21.355
the ic's.

00:06:24.364 --> 00:06:28.531
And if we think back we
remember that in our original

00:06:29.467 --> 00:06:34.382
schematic we had some q
here which means we had a

00:06:34.382 --> 00:06:38.549
plus or minus v zero, v
naught, and we said the current

00:06:42.673 --> 00:06:45.090
through here started at zero.

00:06:46.207 --> 00:06:48.787
So that's our initial conditions,

00:06:48.787 --> 00:06:51.870
v of time equals zero equals v naught

00:06:53.502 --> 00:06:57.169
and current of time
equals zero equals zero.

00:07:01.020 --> 00:07:02.736
Let's use these two values
to help us figure out what

00:07:02.736 --> 00:07:04.403
a one and a two are.

00:07:05.575 --> 00:07:08.041
And we'll do that in the
next video as we continue the

00:07:08.041 --> 00:07:10.470
derivation of the natural response.


Multiplying by j is rotation

WEBVTT
Kind: captions
Language: en

00:00:00.772 --> 00:00:02.464
- [Voiceover] Okay,
there's one more feature

00:00:02.464 --> 00:00:05.008
of complex numbers that
I want to share with you

00:00:05.008 --> 00:00:07.341
and we'll do that down here.

00:00:10.775 --> 00:00:15.101
So, our definition of
j is j squared equals

00:00:15.101 --> 00:00:17.424
minus one, and now what I want to do

00:00:17.424 --> 00:00:20.731
is a sequence of multiplications by j.

00:00:20.731 --> 00:00:25.405
This is a really important
property of this imaginary unit.

00:00:25.405 --> 00:00:27.617
We're gonna do powers of j

00:00:27.617 --> 00:00:30.099
and I'm going to plot them,
while we're doing that

00:00:30.099 --> 00:00:32.541
I'm gonna plot them on the imaginary,

00:00:32.541 --> 00:00:34.511
or on the complex plane.

00:00:34.511 --> 00:00:38.678
So, this is the real axis and
this is the imaginary axis.

00:00:39.551 --> 00:00:40.966
So, I'm gonna take powers of j,

00:00:40.966 --> 00:00:45.077
so first one j to the zero,
and anything to the zero

00:00:45.077 --> 00:00:47.660
is one, so j to the zero is one

00:00:48.605 --> 00:00:51.719
and if we plot that on the imaginary axis,

00:00:51.719 --> 00:00:54.802
here's one, and there's no real part.

00:00:58.628 --> 00:01:02.187
So, it's just right on the real axis.

00:01:02.187 --> 00:01:05.182
Okay, let's do j to the one,

00:01:05.182 --> 00:01:06.977
that's j times itself one time,

00:01:06.977 --> 00:01:09.130
so that is equal to j.

00:01:09.130 --> 00:01:12.708
If I plot that number, that's up here,

00:01:12.708 --> 00:01:16.875
that's up on the imaginary
axis right there, there's j.

00:01:18.941 --> 00:01:22.905
Alright, this has no real
part, it's all imaginary.

00:01:22.905 --> 00:01:24.508
Let's keep going.

00:01:24.508 --> 00:01:28.359
Let's do j squared and
what is that equal to?

00:01:28.359 --> 00:01:29.732
Well, I wrote it down up here,

00:01:29.732 --> 00:01:32.999
j squared is equal to minus one.

00:01:32.999 --> 00:01:34.907
So, j squared is equal to minus one,

00:01:34.907 --> 00:01:36.802
where's that in the complex plane?

00:01:36.802 --> 00:01:39.302
That's over here at minus one,

00:01:42.955 --> 00:01:46.657
on the real axis, no imaginary part.

00:01:46.657 --> 00:01:48.285
Let's do the next one.

00:01:48.285 --> 00:01:49.702
Let's do j cubed.

00:01:50.964 --> 00:01:52.388
What is that equal to?

00:01:52.388 --> 00:01:56.555
J to the third power is
equal to j squared times j

00:02:00.766 --> 00:02:02.892
and we have those two right here,

00:02:02.892 --> 00:02:06.642
j squared is minus one
and j to the one is j,

00:02:09.938 --> 00:02:11.855
so that equals minus j,

00:02:14.163 --> 00:02:15.364
and where do we plot that?

00:02:15.364 --> 00:02:18.131
We plot that down on the imaginary axis

00:02:18.131 --> 00:02:22.411
in the complex plane, right here, minus j.

00:02:22.411 --> 00:02:24.509
Okay, now we've got four answers.

00:02:24.509 --> 00:02:26.678
Let's go one or two more.

00:02:26.678 --> 00:02:30.156
Okay, j to the fourth is equal to what?

00:02:30.156 --> 00:02:33.489
It's equal to j squared times j squared.

00:02:34.347 --> 00:02:35.376
Let's look that up.

00:02:35.376 --> 00:02:37.959
It's minus one times minus one,

00:02:38.838 --> 00:02:41.182
and what does that equal to?

00:02:41.182 --> 00:02:42.515
That equals one.

00:02:44.192 --> 00:02:45.327
So, let's go plot that.

00:02:45.327 --> 00:02:46.810
Well, we've already plotted it,

00:02:46.810 --> 00:02:49.268
so that's this answer right here.

00:02:49.268 --> 00:02:50.858
Alright, so we already have that,

00:02:50.858 --> 00:02:52.895
let's do one more.

00:02:52.895 --> 00:02:55.286
J to the fifth, what is that equal to?

00:02:55.286 --> 00:02:58.286
That equals j to the fourth times j,

00:03:01.093 --> 00:03:04.189
and j to the fourth is
right here, it's one times j

00:03:04.189 --> 00:03:06.939
equals j, let's go plot that one.

00:03:10.076 --> 00:03:11.451
Well, we've already plotted it,

00:03:11.451 --> 00:03:13.589
it's already right here.

00:03:13.589 --> 00:03:17.166
Okay, so, you can see
there's a pattern here,

00:03:17.166 --> 00:03:19.166
one j minus one minus j.

00:03:21.541 --> 00:03:24.425
One j, it's gonna by
minus one and minus j,

00:03:24.425 --> 00:03:26.008
it keeps repeating.

00:03:27.094 --> 00:03:29.183
Here's what's interesting
about this, okay.

00:03:29.183 --> 00:03:32.605
If we draw this as
vectors, if I draw these

00:03:32.605 --> 00:03:36.112
imaginary numbers as
vectors, when I multiply

00:03:36.112 --> 00:03:39.173
by j, when I multiplied one by j,

00:03:39.173 --> 00:03:41.173
it rotated it 90 degrees

00:03:44.220 --> 00:03:46.132
and then that was the
first step right here

00:03:46.132 --> 00:03:49.830
when I multiplied one times j, I got j.

00:03:49.830 --> 00:03:51.472
Now, when we went to j squared,

00:03:51.472 --> 00:03:52.810
we ended up at minus one.

00:03:52.810 --> 00:03:54.698
So, multiplying by j again,

00:03:54.698 --> 00:03:59.033
caused the vector to
go down here, like that

00:03:59.033 --> 00:04:02.385
and that was another 90 degrees

00:04:02.385 --> 00:04:04.302
and if I take it again,

00:04:05.200 --> 00:04:07.367
the next one went this way

00:04:11.018 --> 00:04:13.685
and the final one went this way.

00:04:15.392 --> 00:04:16.665
So, this is the property of j,

00:04:16.665 --> 00:04:19.413
this is the key property
of the imaginary unit,

00:04:19.413 --> 00:04:21.544
multiplying by the imaginary unit,

00:04:21.544 --> 00:04:25.784
the nature of it is
this 90 degree rotation.

00:04:25.784 --> 00:04:28.511
There's this idea of a number that causes

00:04:28.511 --> 00:04:32.494
other numbers to rotate
and that's the feature of j

00:04:32.494 --> 00:04:34.335
that makes it super important

00:04:34.335 --> 00:04:36.176
and it's the reason we use

00:04:36.176 --> 00:04:39.834
imaginary numbers in
electrical engineering.

00:04:39.834 --> 00:04:43.167
So, the key idea here is that j rotates.

00:04:48.233 --> 00:04:49.220
That's the point.

00:04:49.220 --> 00:04:51.917
That's what we love about j.

00:04:51.917 --> 00:04:53.176
So, the last thing I want to mention

00:04:53.176 --> 00:04:55.300
is the negative powers of j.

00:04:55.300 --> 00:04:57.990
What happens if we have
j to the minus one?

00:04:57.990 --> 00:04:59.278
Let's figure out what that means.

00:04:59.278 --> 00:05:02.793
So, that, of course, is one over j

00:05:02.793 --> 00:05:05.626
and if I multiply this by j over j

00:05:08.131 --> 00:05:09.984
and anything over itself is one,

00:05:09.984 --> 00:05:12.504
so I haven't changed the value of this

00:05:12.504 --> 00:05:16.671
and that equals j on top
over j times j or j squared

00:05:21.121 --> 00:05:22.885
and what is j squared equal to?

00:05:22.885 --> 00:05:25.918
We wrote it down right here,
j squared is minus one,

00:05:25.918 --> 00:05:29.918
so this equals j over
minus 1 or equals minus j.

00:05:34.552 --> 00:05:37.148
So, whenever we see a j in a fraction,

00:05:37.148 --> 00:05:39.748
j to the minus one,
basically it introduces

00:05:39.748 --> 00:05:41.701
a minus sign and the j comes up

00:05:41.701 --> 00:05:43.965
to the top out of the fraction.

00:05:43.965 --> 00:05:47.048
So, j to the minus one equals minus j

00:05:48.997 --> 00:05:53.164
and we'll use that occasionally
to help us in our math.

00:05:54.074 --> 00:05:55.375
So, this was a really quick review

00:05:55.375 --> 00:05:58.458
of complex numbers and if any of this

00:05:59.539 --> 00:06:01.793
was new to you, I really encourage

00:06:01.793 --> 00:06:04.923
you to go back and watch Sal's
videos on complex numbers

00:06:04.923 --> 00:06:07.217
and if this is something
you've seen before,

00:06:07.217 --> 00:06:09.209
I hope it knocked off some of the rust

00:06:09.209 --> 00:06:12.212
and we're ready to go
and use these numbers.


LC natural response derivation 4

WEBVTT
Kind: captions
Language: en

00:00:01.423 --> 00:00:03.698
- [Voiceover] So now we're
gonna use the initial conditions

00:00:03.698 --> 00:00:07.971
to figure out our values, our
two constant values A1 and A2.

00:00:07.971 --> 00:00:11.601
That is in our proposed
solution for current

00:00:11.601 --> 00:00:13.184
for the LC circuit.

00:00:14.706 --> 00:00:16.271
So one thing we need to do,

00:00:16.271 --> 00:00:18.433
because this is a second order equation.

00:00:18.433 --> 00:00:20.907
We need to have two initial
conditions for the variable

00:00:20.907 --> 00:00:22.685
that we're studying here.

00:00:22.685 --> 00:00:24.049
So we're studying I.

00:00:24.049 --> 00:00:26.545
Right now we have one
initial condition for I,

00:00:26.545 --> 00:00:30.212
and because we have a
second order equation.

00:00:34.537 --> 00:00:37.370
That means we need two ic's for I.

00:00:41.360 --> 00:00:43.689
So we have one initial
condition right here,

00:00:43.689 --> 00:00:47.272
and what we'd like to
know is what is di dt

00:00:49.172 --> 00:00:50.755
at time equal zero?

00:00:52.280 --> 00:00:54.902
So the other piece of
information we have is this v.

00:00:54.902 --> 00:00:57.107
V knot at time equal zero.

00:00:57.107 --> 00:01:00.149
So let's use that, and we'll just

00:01:00.149 --> 00:01:02.429
plug that straight into
the inductor equation.

00:01:02.429 --> 00:01:05.929
so the inductor equation at t equals zero.

00:01:07.033 --> 00:01:10.899
The voltage across the inductor is v knot,

00:01:10.899 --> 00:01:13.399
and that equals L times di dt.

00:01:18.576 --> 00:01:22.743
Alright and that means the
di dt equals v knot over L.

00:01:27.799 --> 00:01:30.993
So now I have two initial conditions

00:01:30.993 --> 00:01:32.458
in terms of I.

00:01:32.458 --> 00:01:35.375
There's one, and there's one there.

00:01:36.651 --> 00:01:40.568
And we can use these now
to go after A1 and A2.

00:01:44.742 --> 00:01:48.271
First off let's plug in
I for time equals zero,

00:01:48.271 --> 00:01:51.675
and then see if we can work
our something over here.

00:01:51.675 --> 00:01:54.355
So that means that time equals zero.

00:01:54.355 --> 00:01:58.522
The current is zero and that
equals A1 cosign omega knot

00:02:03.866 --> 00:02:06.199
times, t is zero, times zero

00:02:10.265 --> 00:02:11.765
plus A2 times sine

00:02:16.551 --> 00:02:18.634
of omega knot times zero,

00:02:22.938 --> 00:02:25.263
and what does this evaluate to?

00:02:25.263 --> 00:02:29.430
Okay this is sine of zero
and sine of zero is zero,

00:02:31.424 --> 00:02:33.591
and cosine of zero is one.

00:02:40.241 --> 00:02:43.324
So that comes up with zero equals A1.

00:02:45.577 --> 00:02:49.744
Okay, and A1 equals zero
means that this entire term

00:02:51.254 --> 00:02:54.801
of our solution just dropped out.

00:02:54.801 --> 00:02:57.450
Alright let me write what we end up with.

00:02:57.450 --> 00:02:59.950
I equals A2 sine omega knot t.

00:03:08.203 --> 00:03:11.184
This whole term here just
dropped out of the solution.

00:03:11.184 --> 00:03:13.504
So here's our proposed solution down here.

00:03:13.504 --> 00:03:17.004
Now we need to go after A2, let's do that.

00:03:20.664 --> 00:03:22.433
As you might suspect, we're gonna use

00:03:22.433 --> 00:03:25.877
our second initial condition to do that.

00:03:25.877 --> 00:03:29.095
So to use our initial
condition we need di dt.

00:03:29.095 --> 00:03:30.764
So let's take di dt of this.

00:03:30.764 --> 00:03:34.764
So we're gonna take d dt
of this whole equation,

00:03:37.044 --> 00:03:40.127
and on the left side we'll get di dt,

00:03:42.448 --> 00:03:45.448
and on the other side we'll get d dt

00:03:46.405 --> 00:03:48.405
of A2 sine omega knot t.

00:03:54.195 --> 00:03:55.640
Okay so far so good.

00:03:55.640 --> 00:03:57.723
Let's roll it down again.

00:04:01.032 --> 00:04:02.596
So let's take that derivative.

00:04:02.596 --> 00:04:06.763
We get di dt equals, A2
comes out of the derivative,

00:04:10.513 --> 00:04:14.680
and the derivative of sine
omega knot t with respect to t

00:04:16.652 --> 00:04:19.985
is omega knot times cosine omega knot t,

00:04:25.659 --> 00:04:27.997
and we apply our initial condition.

00:04:27.997 --> 00:04:30.573
Let's go to t equals zero,

00:04:30.573 --> 00:04:33.823
and we know the di dt was v knot over L

00:04:36.909 --> 00:04:41.076
equals A2 omega knot cosine
of omega knot times zero,

00:04:49.813 --> 00:04:52.396
and cosine of zero goes to one,

00:04:54.749 --> 00:04:56.923
and so we can solve for A2.

00:04:56.923 --> 00:04:58.923
A2 equals V knot over L,

00:05:03.405 --> 00:05:05.905
and omega knot goes down here.

00:05:09.443 --> 00:05:12.711
So now we solve for our
second adjustable parameter

00:05:12.711 --> 00:05:14.375
and we can write I.

00:05:14.375 --> 00:05:16.625
I was A2 sine omega knot t.

00:05:22.344 --> 00:05:24.575
So let's fill it in for A2.

00:05:24.575 --> 00:05:27.825
I equals A2 is v knot over L omega knot

00:05:31.344 --> 00:05:33.344
times sine omega knot t,

00:05:35.433 --> 00:05:37.328
and I wanna go back now.

00:05:37.328 --> 00:05:39.520
I wanna write this a little bit different.

00:05:39.520 --> 00:05:44.479
I wanna go back and plug in
our value for omega knot.

00:05:44.479 --> 00:05:48.646
So if we remember, we said
omega knot equals one over LC

00:05:52.236 --> 00:05:54.665
and the square root of that whole thing.

00:05:54.665 --> 00:05:56.832
So now L omega knot equals

00:06:00.537 --> 00:06:03.454
square root of one over LC times L,

00:06:07.556 --> 00:06:11.639
and that equals square
root of L squared over LC,

00:06:17.647 --> 00:06:20.980
and that equals square root of L over C.

00:06:24.727 --> 00:06:27.977
Lastly I'll write one over L omega knot

00:06:29.934 --> 00:06:34.101
equals square root of C
over L, just the reciprocal,

00:06:36.501 --> 00:06:38.918
and now we can write I equals

00:06:41.948 --> 00:06:44.948
square root of C over L times V knot

00:06:53.091 --> 00:06:54.841
sine of omega knot t,

00:06:59.271 --> 00:07:03.334
and that is the solution
for the natural response

00:07:03.334 --> 00:07:05.100
of an LC circuit.

00:07:05.100 --> 00:07:07.426
It's in the form of a sine wave

00:07:07.426 --> 00:07:10.565
and the frequency is
determined by omega knot.

00:07:10.565 --> 00:07:13.249
Which is the two component values,

00:07:13.249 --> 00:07:15.622
and the amplitude is
determined by the energy

00:07:15.622 --> 00:07:16.803
we started with.

00:07:16.803 --> 00:07:19.073
Which is represented here by V knot,

00:07:19.073 --> 00:07:22.656
and the ratio of the two components again.

00:07:25.453 --> 00:07:27.554
So this is why I said at the beginning

00:07:27.554 --> 00:07:30.804
that this is where sine waves are born.


Origins of life

WEBVTT
Kind: captions
Language: en

00:00:00.106 --> 00:00:02.371
- [Voiceover] We have many
videos on Khan Academy

00:00:02.371 --> 00:00:05.415
on things like evolution
and natural selection.

00:00:05.415 --> 00:00:07.719
We think we have a fairly
solid understanding

00:00:07.719 --> 00:00:10.931
of how life can evolve
to give us the variety,

00:00:10.931 --> 00:00:12.964
the diversity, and the complexity

00:00:12.964 --> 00:00:14.380
that we've seen around us,

00:00:14.380 --> 00:00:17.703
but it still leaves unanswered
a very fundamental question.

00:00:17.703 --> 00:00:21.560
And this might be the
biggest question known to us,

00:00:21.560 --> 00:00:23.467
and that is the origins of life.

00:00:23.467 --> 00:00:26.260
How did life first
emerge, at least on Earth,

00:00:26.260 --> 00:00:27.777
and that even starts to
lead to other questions

00:00:27.777 --> 00:00:30.194
about is there life outside of this planet

00:00:30.194 --> 00:00:31.770
and what could it be like?

00:00:31.770 --> 00:00:34.229
And so let's start with
what we actually know

00:00:34.229 --> 00:00:36.412
and I'm gonna start with a timeline.

00:00:36.412 --> 00:00:39.260
So let's go one billion years ago,

00:00:39.260 --> 00:00:41.697
let's go two billion years ago,

00:00:41.697 --> 00:00:43.514
three billion years ago,

00:00:43.514 --> 00:00:45.618
four billion years ago.

00:00:45.618 --> 00:00:47.291
So this is now.

00:00:47.291 --> 00:00:49.010
And once again, we're talking
about a billion years ago.

00:00:49.010 --> 00:00:51.319
You'll sometimes see the abbreviation BYA,

00:00:51.319 --> 00:00:55.318
billion years ago, which
is an unfathomable amount

00:00:55.318 --> 00:00:57.733
of time going into the past.

00:00:57.733 --> 00:01:00.439
But we know that Earth along with the rest

00:01:00.439 --> 00:01:02.368
of the solar system was formed

00:01:02.368 --> 00:01:03.285
around 4.6,

00:01:04.845 --> 00:01:05.762
4.6 billion

00:01:06.914 --> 00:01:08.096
years ago,

00:01:08.096 --> 00:01:09.883
so that's when Earth was formed.

00:01:09.883 --> 00:01:13.353
And right at 4.6 or
even, you wait a casual

00:01:13.353 --> 00:01:16.594
100 million years after
that 4.5 billion years ago,

00:01:16.594 --> 00:01:18.738
we believe that Earth wasn't very suitable

00:01:18.738 --> 00:01:21.208
for even very simple life to form,

00:01:21.208 --> 00:01:24.758
and that's because the solar
system was a crazy place.

00:01:24.758 --> 00:01:28.988
You had collisions of all scales
happening all of the time.

00:01:28.988 --> 00:01:31.363
The moon itself was
formed from the collision

00:01:31.363 --> 00:01:35.408
of two planet-sized objects,
one we call it the proto-Earth

00:01:35.408 --> 00:01:38.650
and another planet-sized
object and they collided

00:01:38.650 --> 00:01:40.615
and then they started to spin around

00:01:40.615 --> 00:01:42.494
and one part became the moon.

00:01:42.494 --> 00:01:46.304
It was tidally linked with the Earth.

00:01:46.304 --> 00:01:48.321
But you can imagine,
that's not an environment

00:01:48.321 --> 00:01:50.650
where it would be easy for life to form.

00:01:50.650 --> 00:01:52.485
And even once the moon was formed,

00:01:52.485 --> 00:01:55.568
you had a heavy bombardment of things

00:01:57.030 --> 00:01:59.919
in the solar system, the solar
system was a messy place.

00:01:59.919 --> 00:02:02.277
It took a long time for the stability

00:02:02.277 --> 00:02:04.828
that we now observe out there.

00:02:04.828 --> 00:02:06.716
And so that continued, we believe,

00:02:06.716 --> 00:02:09.549
until about 3.9 billion years ago,

00:02:10.625 --> 00:02:13.124
which is the earliest
that we currently think

00:02:13.124 --> 00:02:16.486
that Earth might have
been suitable for life.

00:02:16.486 --> 00:02:17.978
Before that, there might have been pockets

00:02:17.978 --> 00:02:21.370
where the bombardment
stops and maybe some type

00:02:21.370 --> 00:02:22.859
of primitive life might have formed,

00:02:22.859 --> 00:02:24.529
but then they would have gone away

00:02:24.529 --> 00:02:25.582
with the heavy bombardment.

00:02:25.582 --> 00:02:26.415
But who knows?

00:02:26.415 --> 00:02:28.013
Maybe they could have
survived that somehow.

00:02:28.013 --> 00:02:30.529
But that's the current mainstream belief.

00:02:30.529 --> 00:02:34.528
The other thing we know is
that we see fossil evidence

00:02:34.528 --> 00:02:37.613
for life 3.5 billions years ago.

00:02:37.613 --> 00:02:39.269
And these are stromatolites.

00:02:39.269 --> 00:02:41.372
This is fossil evidence, microorganisms,

00:02:41.372 --> 00:02:44.962
they formed these structures
that actually continue

00:02:44.962 --> 00:02:48.355
to be formed today,
these types of structures

00:02:48.355 --> 00:02:50.298
continue to be formed today.

00:02:50.298 --> 00:02:52.483
And although it might not
feel like microorganisms

00:02:52.483 --> 00:02:56.221
are complex life, when you
think about what has to happen

00:02:56.221 --> 00:02:59.601
within a microorganism, they're
actually incredibly complex,

00:02:59.601 --> 00:03:00.855
and especially if you compare them

00:03:00.855 --> 00:03:04.702
to very simple, non-living organisms.

00:03:04.702 --> 00:03:07.535
So our current belief is someplace

00:03:09.088 --> 00:03:13.413
in this region life must
have arisen on Earth.

00:03:13.413 --> 00:03:15.596
But that still doesn't,
even if we were able

00:03:15.596 --> 00:03:18.129
to answer that question, oh,
it was exactly 3.7 billion

00:03:18.129 --> 00:03:20.565
years ago was the first
time that some RNA decided,

00:03:20.565 --> 00:03:24.409
or not decided, ended up getting
in the right confirmation

00:03:24.409 --> 00:03:26.982
so it could replicate itself in some way,

00:03:26.982 --> 00:03:30.813
even if we know that date,
it still leaves unanswered

00:03:30.813 --> 00:03:33.935
maybe the more interesting
question, which is the how.

00:03:33.935 --> 00:03:36.879
The how is really, at least
to me, more important,

00:03:36.879 --> 00:03:39.941
more interesting than the when.

00:03:39.941 --> 00:03:42.677
And to the how question, there's
a couple of layers on it.

00:03:42.677 --> 00:03:45.072
The first is, let's just
start with the most simple

00:03:45.072 --> 00:03:48.137
molecules that we would
have expected to find

00:03:48.137 --> 00:03:49.930
on early Earth.

00:03:49.930 --> 00:03:51.944
Here are some examples
of it right over here.

00:03:51.944 --> 00:03:55.694
This is H2O, or more
commonly-known as water.

00:03:57.284 --> 00:04:01.451
Right over here is CO2, more
commonly-known as carbon.

00:04:02.431 --> 00:04:03.453
That's a little hard to see,

00:04:03.453 --> 00:04:06.286
let me do it with a lighter color.

00:04:07.305 --> 00:04:08.138
So we have

00:04:10.854 --> 00:04:13.437
carbon dioxide right over here.

00:04:14.457 --> 00:04:17.350
Here we have molecular
nitrogen, you have some ammonia,

00:04:17.350 --> 00:04:20.366
you have some phosphate, and
many other of the elements

00:04:20.366 --> 00:04:23.384
that we see on Earth today,
they might have been available

00:04:23.384 --> 00:04:27.856
in that early Earth, but
how do they form at least,

00:04:27.856 --> 00:04:31.509
even the next step up, which
is the slightly more complex,

00:04:31.509 --> 00:04:34.945
or actually a good bit more
complex organic molecules.

00:04:34.945 --> 00:04:37.569
And when people talk about
organic molecules they might

00:04:37.569 --> 00:04:39.712
be talking about things like this.

00:04:39.712 --> 00:04:41.651
These are amino acids.

00:04:41.651 --> 00:04:46.561
These are the building blocks
of proteins, amino acids.

00:04:46.561 --> 00:04:49.145
You see over here nucleotides.

00:04:49.145 --> 00:04:53.891
These are the building blocks
of RNA, DNA, other things.

00:04:53.891 --> 00:04:55.427
And so the first question is,

00:04:55.427 --> 00:04:57.499
and these aren't the only
simple organic molecules.

00:04:57.499 --> 00:05:00.737
You could think about sugars
and all sorts of other things.

00:05:00.737 --> 00:05:04.285
But the question is, is it realistic?

00:05:04.285 --> 00:05:06.224
Do we at least understand how we can go

00:05:06.224 --> 00:05:08.345
from these very simple molecules up here

00:05:08.345 --> 00:05:12.294
to these more complex, often
called organic, molecules?

00:05:12.294 --> 00:05:14.707
And the simple answer is we
now have a lot of evidence

00:05:14.707 --> 00:05:16.158
that this is doable,

00:05:16.158 --> 00:05:19.929
that you can go from these
things to these things,

00:05:19.929 --> 00:05:24.098
abiotically, without the presence of life.

00:05:24.098 --> 00:05:26.136
You'll hear that word abiotic a lot.

00:05:26.136 --> 00:05:28.460
Think about it, antibiotic,
you're killing life,

00:05:28.460 --> 00:05:29.425
you're killing bacteria.

00:05:29.425 --> 00:05:31.738
Abiotic, that is without life.

00:05:31.738 --> 00:05:34.897
And the points of
evidence that we now have

00:05:34.897 --> 00:05:38.377
are we believe and we've seen evidence

00:05:38.377 --> 00:05:42.486
that there's amino acids and
organic molecules related

00:05:42.486 --> 00:05:46.295
to them on comets,
meteorites, on other planets,

00:05:46.295 --> 00:05:49.165
that they formed spontaneously in space,

00:05:49.165 --> 00:05:52.089
once again, without the
presence of life there.

00:05:52.089 --> 00:05:54.489
We've even been able to form amino acids

00:05:54.489 --> 00:05:56.525
and other molecules like this

00:05:56.525 --> 00:05:59.672
from these more simple
elements in the laboratory.

00:05:59.672 --> 00:06:01.255
The most famous experiment there

00:06:01.255 --> 00:06:04.391
is the Miller and Urey Experiment.

00:06:04.391 --> 00:06:07.533
This was in the 1950's,
where they were able to show

00:06:07.533 --> 00:06:09.719
with some energy, they provided a spark.

00:06:09.719 --> 00:06:11.899
You could imagine that in the early Earth

00:06:11.899 --> 00:06:13.975
it could have been from lighting.

00:06:13.975 --> 00:06:17.679
And they tried to set up a mix
of gases that they believed

00:06:17.679 --> 00:06:21.217
was similar to the atmospheric
mix in the early Earth

00:06:21.217 --> 00:06:23.750
which didn't have much oxygen
in the atmosphere then.

00:06:23.750 --> 00:06:25.845
We needed life to
actually start to produce

00:06:25.845 --> 00:06:27.153
some of that oxygen.

00:06:27.153 --> 00:06:28.333
And even though today we think

00:06:28.333 --> 00:06:30.795
that they probably didn't
have the mix of gases right,

00:06:30.795 --> 00:06:32.700
they did do something significant.

00:06:32.700 --> 00:06:35.789
They were able to show
that with that mix of gases

00:06:35.789 --> 00:06:37.755
at least they thought
were in that atmosphere,

00:06:37.755 --> 00:06:39.933
and some energy being
added to that system,

00:06:39.933 --> 00:06:42.371
that they were able to form some

00:06:42.371 --> 00:06:44.552
of these organic molecules.

00:06:44.552 --> 00:06:46.431
So we should feel pretty good

00:06:46.431 --> 00:06:48.958
that at least this first step is doable.

00:06:48.958 --> 00:06:51.545
Now the next question is
these organic molecules

00:06:51.545 --> 00:06:53.570
by themselves, that's not life.

00:06:53.570 --> 00:06:56.298
In fact, these aren't even
the most complex molecules

00:06:56.298 --> 00:06:59.112
that are, we believe, essential for life.

00:06:59.112 --> 00:07:02.313
Proteins are where things start
to get really interesting,

00:07:02.313 --> 00:07:04.188
and a protein,

00:07:04.188 --> 00:07:05.754
a protein,

00:07:05.754 --> 00:07:07.195
or proteins are one of the places,

00:07:07.195 --> 00:07:10.847
a protein might have
thousands of amino acids,

00:07:10.847 --> 00:07:12.180
thousands

00:07:12.180 --> 00:07:13.430
of amino acids.

00:07:16.207 --> 00:07:19.760
Things like DNA and RNA,
also we believe essential

00:07:19.760 --> 00:07:22.324
for life, or at least life as we know it,

00:07:22.324 --> 00:07:25.513
could be made up of tens
of millions of nucleotides

00:07:25.513 --> 00:07:27.857
for one DNA molecule.

00:07:27.857 --> 00:07:30.445
So for example, this is just a small part

00:07:30.445 --> 00:07:33.312
of a DNA molecule, but
you can already see much,

00:07:33.312 --> 00:07:37.336
much more complex than
what we see over here.

00:07:37.336 --> 00:07:40.650
And there, too, we have
evidence that you can go

00:07:40.650 --> 00:07:43.165
from the amino acids to the proteins,

00:07:43.165 --> 00:07:45.963
or you can go from the
nucleotides to the DNA

00:07:45.963 --> 00:07:47.450
without the presence of life,

00:07:47.450 --> 00:07:50.239
that these things can happen spontaneously

00:07:50.239 --> 00:07:53.869
if you have the right context,
the right energy available,

00:07:53.869 --> 00:07:57.304
some people believe,
or it's been observed,

00:07:57.304 --> 00:08:00.880
that if you have the right surfaces

00:08:00.880 --> 00:08:04.259
that these molecules can be
organized in the right way

00:08:04.259 --> 00:08:06.845
to form these more complex things.

00:08:06.845 --> 00:08:07.821
Now, I know what you're thinking.

00:08:07.821 --> 00:08:10.173
Alright, proteins are really cool,

00:08:10.173 --> 00:08:12.969
DNA, RNA is really cool, but
then how does that become life?

00:08:12.969 --> 00:08:14.642
At what point would we start going,

00:08:14.642 --> 00:08:16.623
"That was a proto-life form?"

00:08:16.623 --> 00:08:19.458
And this is where we
really get into the area

00:08:19.458 --> 00:08:22.386
of the unknown because we don't know.

00:08:22.386 --> 00:08:24.877
And there's a couple of
hypotheses out there.

00:08:24.877 --> 00:08:28.511
One of them is called
the RNA World Hypothesis.

00:08:28.511 --> 00:08:29.590
I'll write that down.

00:08:29.590 --> 00:08:31.120
RNA

00:08:31.120 --> 00:08:32.579
World

00:08:32.579 --> 00:08:34.027
Hypothesis.

00:08:34.027 --> 00:08:37.329
And this is the idea
that the first proto-life

00:08:37.329 --> 00:08:40.229
was self-replicating RNA molecules.

00:08:40.229 --> 00:08:41.915
And the reason why people tend to focus in

00:08:41.915 --> 00:08:44.731
on RNA a little bit more than DNA is

00:08:44.731 --> 00:08:46.288
that even in cells today,

00:08:46.288 --> 00:08:48.584
RNA doesn't just store information,

00:08:48.584 --> 00:08:51.668
it can actually play a role as a catalyst.

00:08:51.668 --> 00:08:54.442
And when you think about things like tRNA

00:08:54.442 --> 00:08:56.963
and you think about ribosomal RNA.

00:08:56.963 --> 00:09:00.543
And so maybe some of that
first proto-life was RNA,

00:09:00.543 --> 00:09:02.663
information that replicated itself

00:09:02.663 --> 00:09:05.321
and catalyzed the replication of itself.

00:09:05.321 --> 00:09:07.388
Maybe it somehow got organized

00:09:07.388 --> 00:09:10.507
into membrane-bound structures
so it could separate

00:09:10.507 --> 00:09:12.665
so you had environments
that were separated

00:09:12.665 --> 00:09:14.246
from the outside world.

00:09:14.246 --> 00:09:16.951
But the simple answer is we don't know.

00:09:16.951 --> 00:09:19.128
Another mainstream hypothesis

00:09:19.128 --> 00:09:21.944
is the Metabolism First Hypothesis.

00:09:21.944 --> 00:09:22.777
Metabolism

00:09:23.987 --> 00:09:26.015
Metabolism First.

00:09:26.015 --> 00:09:29.905
And this is the idea that
a lot of basic pathways

00:09:29.905 --> 00:09:32.344
that you might study
in a biochemistry book,

00:09:32.344 --> 00:09:35.661
that these were first
just happening, well,

00:09:35.661 --> 00:09:36.710
all of this could have been happening

00:09:36.710 --> 00:09:38.486
in this primordial soup where you had

00:09:38.486 --> 00:09:41.321
these organic molecules
in the right conditions,

00:09:41.321 --> 00:09:43.548
maybe around heat vents and whatever else,

00:09:43.548 --> 00:09:45.765
but the Metabolism First is
that some of these mechanisms

00:09:45.765 --> 00:09:47.923
that we now study in biochemistry,

00:09:47.923 --> 00:09:49.519
these might have happened
outside of a cell

00:09:49.519 --> 00:09:51.711
or outside of life and they
just kept creating more

00:09:51.711 --> 00:09:54.373
and more complexity, but
at some point these things

00:09:54.373 --> 00:09:57.891
started happening in self-organizing,

00:09:57.891 --> 00:10:00.058
membrane-bound structures.

00:10:01.157 --> 00:10:03.438
Maybe there's some kind
of combination of the two.

00:10:03.438 --> 00:10:05.969
The simple answer is we just don't know,

00:10:05.969 --> 00:10:07.764
but there's some fascinating clues.

00:10:07.764 --> 00:10:09.653
Even if we observe current biology,

00:10:09.653 --> 00:10:12.085
and even in fact if we
see the commonalities

00:10:12.085 --> 00:10:16.085
of things that happen,
central dogma of biology,

00:10:17.169 --> 00:10:21.716
if we see how proteins,
which structures are common

00:10:21.716 --> 00:10:23.361
to all life as we know
it, it might give us clues

00:10:23.361 --> 00:10:27.045
or hints at what some of
that very earliest life

00:10:27.045 --> 00:10:29.712
or proto-life was actually like.


Complex numbers

WEBVTT
Kind: captions
Language: en

00:00:01.318 --> 00:00:02.200
- [Voiceover] This video's going to be

00:00:02.200 --> 00:00:04.650
a quick review of complex numbers.

00:00:04.650 --> 00:00:07.480
If you studied complex numbers in the past

00:00:07.480 --> 00:00:09.079
this will knock off some of the rust

00:00:09.079 --> 00:00:12.661
and it will help explain
why we use complex numbers

00:00:12.661 --> 00:00:14.730
in electrical engineering.

00:00:14.730 --> 00:00:16.812
If complex numbers are new to you,

00:00:16.812 --> 00:00:17.985
I highly recommend you go look

00:00:17.985 --> 00:00:21.754
on the Khan Academy videos that Sal's done

00:00:21.754 --> 00:00:26.091
on complex numbers and those
are in the Algebra II section.

00:00:26.091 --> 00:00:27.612
So let's get started.

00:00:27.612 --> 00:00:30.157
The complex numbers are
based on the concept

00:00:30.157 --> 00:00:32.907
of the imaginary j, the number j,

00:00:37.311 --> 00:00:39.161
in electrical engineering
we use the number j

00:00:39.161 --> 00:00:40.910
instead of I.

00:00:40.910 --> 00:00:44.327
And j squared is defined to be minus one.

00:00:46.841 --> 00:00:49.341
So that's the definition of j.

00:00:52.173 --> 00:00:54.811
And that's referred to
as an imaginary number.

00:00:54.811 --> 00:00:56.399
I don't really like the name imaginary

00:00:56.399 --> 00:00:58.351
but that's what we call it.

00:00:58.351 --> 00:01:02.250
It's a real useful concept
in electrical engineering.

00:01:02.250 --> 00:01:04.032
So with that definition we define

00:01:04.032 --> 00:01:06.958
a complex number and the usual variable

00:01:06.958 --> 00:01:09.643
we often use for that is a z,

00:01:09.643 --> 00:01:12.367
and a complex number has a real part,

00:01:12.367 --> 00:01:15.340
we'll call that x and
it has an imaginary part

00:01:15.340 --> 00:01:17.423
that we're gonna call jy.

00:01:18.720 --> 00:01:20.170
So j is explicit out here.

00:01:20.170 --> 00:01:22.141
This is the imaginary part of the number.

00:01:22.141 --> 00:01:24.391
This is the real part of z.

00:01:25.569 --> 00:01:27.091
So based on what this number looks like,

00:01:27.091 --> 00:01:29.661
this suggests that we can maybe plot this

00:01:29.661 --> 00:01:31.321
on a two dimensional plot.

00:01:31.321 --> 00:01:33.522
And we'll call this the complex plane.

00:01:33.522 --> 00:01:36.592
And the complex plane looks like this.

00:01:36.592 --> 00:01:37.568
We can plot two parts,

00:01:37.568 --> 00:01:40.331
we'll have a real part over here

00:01:40.331 --> 00:01:42.609
on what is usually the x axis.

00:01:42.609 --> 00:01:45.276
And we'll have an imaginary part

00:01:46.571 --> 00:01:48.130
which is the vertical axis.

00:01:48.130 --> 00:01:51.140
So this is referred to
as the complex plane.

00:01:51.140 --> 00:01:52.942
And if I have a complex number z,

00:01:52.942 --> 00:01:54.782
I could represent it on this plane

00:01:54.782 --> 00:01:58.161
by basically going over x like this,

00:01:58.161 --> 00:02:00.078
going over a distance x

00:02:01.232 --> 00:02:02.899
and up a distance y.

00:02:05.690 --> 00:02:07.391
That will give me an imaginary number

00:02:07.391 --> 00:02:08.474
and that's z.

00:02:10.447 --> 00:02:13.823
So z is a location in this complex space.

00:02:13.823 --> 00:02:18.301
And that's one representation
of a complex number.

00:02:18.301 --> 00:02:21.532
So the other common way to
represent a complex number

00:02:21.532 --> 00:02:24.949
is by drawing a line from the origin here

00:02:26.160 --> 00:02:29.243
and going right through z, like that.

00:02:31.249 --> 00:02:34.352
And then we basically have some radius,

00:02:34.352 --> 00:02:38.159
r, from the origin to distance out to z

00:02:38.159 --> 00:02:41.330
and it's measured by some angle like that.

00:02:41.330 --> 00:02:43.640
That angle would be theta.

00:02:43.640 --> 00:02:45.730
So in the orange is r and theta

00:02:45.730 --> 00:02:48.272
and in the blue here we have x and y

00:02:48.272 --> 00:02:49.933
and those are two
different ways to represent

00:02:49.933 --> 00:02:52.130
exactly the same number z.

00:02:52.130 --> 00:02:54.047
So over here I can say,

00:02:55.601 --> 00:02:58.518
I can say z equals r at some angle,

00:03:01.020 --> 00:03:04.047
that's the angle symbol of theta.

00:03:04.047 --> 00:03:06.192
Now I can go over here and I can work out

00:03:06.192 --> 00:03:07.751
how we convert between the two,

00:03:07.751 --> 00:03:10.810
how do I convert from r to y and x

00:03:10.810 --> 00:03:13.022
and how do I go the other way.

00:03:13.022 --> 00:03:14.650
So one thing I notice is I just used

00:03:14.650 --> 00:03:16.710
some simple trigonometry.

00:03:16.710 --> 00:03:19.360
So this distance here if I know r,

00:03:19.360 --> 00:03:21.889
say I know r, this distance here x,

00:03:21.889 --> 00:03:24.556
is equal to the cosine of theta,

00:03:26.049 --> 00:03:27.799
times the distance r,

00:03:29.170 --> 00:03:30.591
r cosine theta.

00:03:30.591 --> 00:03:34.174
So I can say x equals r cosine of theta.

00:03:38.303 --> 00:03:40.302
So if I want to figure
out the y distance here

00:03:40.302 --> 00:03:42.052
and I know r already,

00:03:43.170 --> 00:03:46.592
let me just move, here's
the y distance right here.

00:03:46.592 --> 00:03:50.259
I can say y equals r
times the sin of theta.

00:03:52.960 --> 00:03:55.127
That's this distance here.

00:04:01.489 --> 00:04:05.772
Okay, so if I know r and theta
this is how I get x and y.

00:04:05.772 --> 00:04:06.651
Now let's go the other way.

00:04:06.651 --> 00:04:11.420
Suppose I know x and y and
I want to know r and theta.

00:04:11.420 --> 00:04:14.082
So r, this is a right triangle here,

00:04:14.082 --> 00:04:16.250
there's our right triangle,

00:04:16.250 --> 00:04:17.840
so I use the Pythagorean theorem.

00:04:17.840 --> 00:04:21.022
So to convert from x and y to r

00:04:21.022 --> 00:04:23.341
I use the Pythagorean theorem,

00:04:23.341 --> 00:04:27.091
r squared equals x squared plus y squared.

00:04:31.241 --> 00:04:33.131
And now if I want to
find theta I use another

00:04:33.131 --> 00:04:35.261
little bit of trigonometry,

00:04:35.261 --> 00:04:38.094
tangent is opposite over adjacent,

00:04:40.207 --> 00:04:43.101
opposite over adjacent is y over x,

00:04:43.101 --> 00:04:46.101
so tangent of theta equals y over x.

00:04:54.220 --> 00:04:56.110
So if you're gonna do
this on your calculator

00:04:56.110 --> 00:04:59.329
you would say that theta equals

00:04:59.329 --> 00:05:01.996
the inverse tangent of y over x.

00:05:08.161 --> 00:05:10.672
So there's two conversions

00:05:10.672 --> 00:05:13.505
between two different forms of the

00:05:14.529 --> 00:05:16.362
of the complex number.

00:05:17.612 --> 00:05:19.150
We want to be able to
use these conversions

00:05:19.150 --> 00:05:20.529
and we want to be able to use either

00:05:20.529 --> 00:05:23.181
of these two representations freely

00:05:23.181 --> 00:05:26.528
and go back and forth between them.

00:05:26.528 --> 00:05:29.340
Now there's a third representation

00:05:29.340 --> 00:05:31.451
that's also gonna be really useful to us.

00:05:31.451 --> 00:05:32.992
Now what I'm gonna do is I'm gonna take

00:05:32.992 --> 00:05:35.411
this x and y expression here

00:05:35.411 --> 00:05:38.701
and I'm gonna put it back into this way,

00:05:38.701 --> 00:05:41.811
this rectangular way of writing z.

00:05:41.811 --> 00:05:44.381
What that looks like is,

00:05:44.381 --> 00:05:47.464
z equals x is r times cosine theta

00:05:53.330 --> 00:05:57.497
and y is equal to r, I'll
put the r out front here,

00:05:58.621 --> 00:06:01.621
r sin theta with a j in front of it.

00:06:03.770 --> 00:06:06.437
So I can write plus j sin theta.

00:06:13.363 --> 00:06:16.991
Now if you look closely at
this expression right here,

00:06:16.991 --> 00:06:18.041
we recognize this.

00:06:18.041 --> 00:06:22.781
We recognize this as one
side of Euler's Formula.

00:06:22.781 --> 00:06:24.682
And the other side of Euler's Formula

00:06:24.682 --> 00:06:28.515
I could rewrite z as r
times e to the j theta.

00:06:34.853 --> 00:06:36.693
And this is called the exponential form

00:06:36.693 --> 00:06:38.520
of a complex number.

00:06:38.520 --> 00:06:40.089
All this means, what does this mean here,

00:06:40.089 --> 00:06:41.288
what is this thing?

00:06:41.288 --> 00:06:43.801
This means exactly the same thing as this

00:06:43.801 --> 00:06:45.723
and this is one of the two ways

00:06:45.723 --> 00:06:48.312
we can write complex numbers.

00:06:48.312 --> 00:06:52.395
So this r e to the j theta,
that is z right here,

00:06:53.243 --> 00:06:57.173
that means a complex
number sitting out here

00:06:57.173 --> 00:07:00.756
at radius r from the
origin at angle theta.

00:07:01.672 --> 00:07:04.121
That's what you think of when you see

00:07:04.121 --> 00:07:06.650
e to the j something written down.

00:07:06.650 --> 00:07:11.270
It's just a representation
of a complex number.

00:07:11.270 --> 00:07:13.770
And this form is going
to be particularly useful

00:07:13.770 --> 00:07:15.432
because if you remember
when we were solving

00:07:15.432 --> 00:07:17.490
all those differential equations,

00:07:17.490 --> 00:07:20.657
we always liked exponential solutions.

00:07:22.210 --> 00:07:24.790
So I want to put some
squares around these guys.

00:07:24.790 --> 00:07:28.623
These are the three ways
that we can represent

00:07:29.609 --> 00:07:33.276
a complex number and
they're all equivalent.

00:07:36.531 --> 00:07:38.507
And I'll go over here
just as a reminder note,

00:07:38.507 --> 00:07:41.174
I'll write down Euler's Formula.

00:07:47.700 --> 00:07:50.071
That's where that comes from.

00:07:50.071 --> 00:07:51.461
And let me write that down over here.

00:07:51.461 --> 00:07:54.961
Euler's Formula is e to the j theta equals

00:07:57.350 --> 00:07:59.850
cosine theta plus j sin theta.

00:08:10.569 --> 00:08:13.352
The other form has a negative exponent.

00:08:13.352 --> 00:08:17.519
E to the minus j theta equals
cosine theta minus j sin theta

00:08:25.446 --> 00:08:27.687
So that's Euler's Formula.

00:08:27.687 --> 00:08:31.542
And Sal has videos on how
to derive this equation

00:08:31.542 --> 00:08:35.433
and you would search on this
term here in Khan Academy.


LC natural response derivation 2

WEBVTT
Kind: captions
Language: en

00:00:01.164 --> 00:00:02.720
- [Voiceover] In the last video,

00:00:02.720 --> 00:00:05.372
we set up this differential equation

00:00:05.372 --> 00:00:07.566
that described an LC circuit,

00:00:07.566 --> 00:00:09.802
and now we're gonna go about solving

00:00:09.802 --> 00:00:12.052
this second order circuit.

00:00:13.630 --> 00:00:15.470
The technique that works here is the same

00:00:15.470 --> 00:00:16.910
that worked with first order

00:00:16.910 --> 00:00:19.017
ordinary differential equations.

00:00:19.017 --> 00:00:20.207
We're looking for a function of I

00:00:20.207 --> 00:00:22.575
that makes this whole equation true,

00:00:22.575 --> 00:00:25.652
and we're gonna do that
by guessing at an I,

00:00:25.652 --> 00:00:27.091
put it back into the equations,

00:00:27.091 --> 00:00:28.222
and see if it works.

00:00:28.222 --> 00:00:29.378
If it works, we win.

00:00:29.378 --> 00:00:31.615
If it doesn't work, we have
to think of something else,

00:00:31.615 --> 00:00:34.054
and we keep doing that until we solve it.

00:00:34.054 --> 00:00:37.261
So, knowing what I know about derivatives,

00:00:37.261 --> 00:00:39.084
and knowing and looking at this equation,

00:00:39.084 --> 00:00:42.084
can I guess at a plausible solution?

00:00:43.121 --> 00:00:44.607
So what we have here is two things.

00:00:44.607 --> 00:00:47.195
We have two terms that
have to add up to zero

00:00:47.195 --> 00:00:48.278
for all time,

00:00:49.634 --> 00:00:52.832
and so that means that
whatever function I pick for I,

00:00:52.832 --> 00:00:54.590
and there's some scaling factors here.

00:00:54.590 --> 00:00:57.590
Lemme write this equation like this.

00:01:09.623 --> 00:01:12.657
So what I have here is
a scaling factor times I

00:01:12.657 --> 00:01:14.584
has to somehow fully subtract

00:01:14.584 --> 00:01:17.251
from the second derivative of I.

00:01:18.830 --> 00:01:22.997
So these two terms have to
somehow look alike for all time.

00:01:23.948 --> 00:01:27.067
There's one function I
know where its derivatives

00:01:27.067 --> 00:01:28.907
sort of look like what we started with,

00:01:28.907 --> 00:01:31.305
and that's the exponential function.

00:01:31.305 --> 00:01:32.580
So I'm actually gonna make a guess.

00:01:32.580 --> 00:01:34.830
I'm gonna guess that I of T

00:01:36.329 --> 00:01:37.974
is something of the form

00:01:37.974 --> 00:01:39.557
some constant times

00:01:42.153 --> 00:01:44.070
the exponential of time

00:01:47.425 --> 00:01:49.548
with some other scale factor.

00:01:49.548 --> 00:01:52.879
Now, K is an adjustable parameter,

00:01:52.879 --> 00:01:54.879
and that's an amplitude.

00:01:57.938 --> 00:02:00.736
So it tells us how big the signal is,

00:02:00.736 --> 00:02:02.013
and what's S?

00:02:02.013 --> 00:02:04.921
S is up in the exponent along with T,

00:02:04.921 --> 00:02:05.989
and we know that by the time

00:02:05.989 --> 00:02:08.519
we take an exponent of anything

00:02:08.519 --> 00:02:10.830
that whatever's up here
has to have no units,

00:02:10.830 --> 00:02:13.455
so that means that ST, S times T,

00:02:13.455 --> 00:02:14.645
has no units,

00:02:14.645 --> 00:02:16.395
and that means that S

00:02:17.232 --> 00:02:20.001
has units of one over time

00:02:20.001 --> 00:02:22.751
so S is a frequency of some sort.

00:02:26.858 --> 00:02:28.982
And in particular, it's
gonna be a radian frequency.

00:02:28.982 --> 00:02:31.982
It's gonna be in radians per second.

00:02:37.330 --> 00:02:41.080
So, S is gonna be called
a natural frequency.

00:02:44.794 --> 00:02:46.444
So let's keep working on this.

00:02:46.444 --> 00:02:50.293
We're gonna plug I back into our equation,

00:02:50.293 --> 00:02:51.960
and see if it works.

00:02:57.183 --> 00:02:59.185
So we can plug I straight into here,

00:02:59.185 --> 00:03:01.328
and we need the second derivative of I.

00:03:01.328 --> 00:03:05.328
Let's first take the first
derivative D, DT of I

00:03:08.248 --> 00:03:09.665
equals D, DT of K

00:03:13.872 --> 00:03:14.872
E to the ST,

00:03:16.172 --> 00:03:17.580
and we can take that derivative,

00:03:17.580 --> 00:03:19.663
and that equals K times S

00:03:23.193 --> 00:03:24.693
times E to the ST.

00:03:26.626 --> 00:03:28.972
So it's also an exponential,

00:03:28.972 --> 00:03:30.718
and now we need the second derivative,

00:03:30.718 --> 00:03:33.198
so we wanna take the
derivative of this guy,

00:03:33.198 --> 00:03:37.031
so second derivative of
I with respect to time

00:03:39.937 --> 00:03:42.687
equals the first derivative of S,

00:03:48.534 --> 00:03:49.784
K, E to the ST,

00:03:52.319 --> 00:03:53.736
and that equals--

00:03:55.819 --> 00:03:56.754
Another S comes down,

00:03:56.754 --> 00:03:58.504
so it's S squared, K,

00:04:00.446 --> 00:04:01.446
E to the ST.

00:04:02.872 --> 00:04:06.639
Good, now we have our second derivative.

00:04:06.639 --> 00:04:07.929
We can plug that in here,

00:04:07.929 --> 00:04:09.552
so let's do that.

00:04:09.552 --> 00:04:12.740
So the equation becomes S squared,

00:04:12.740 --> 00:04:13.990
K, E to the ST,

00:04:15.323 --> 00:04:16.740
plus one over LC,

00:04:18.585 --> 00:04:20.335
times K, E to the ST,

00:04:23.548 --> 00:04:24.548
equals zero.

00:04:26.644 --> 00:04:27.805
Let's do a little factoring.

00:04:27.805 --> 00:04:28.925
There's a common term here.

00:04:28.925 --> 00:04:30.417
There's a common term.

00:04:30.417 --> 00:04:31.555
K, E to the ST,

00:04:31.555 --> 00:04:33.245
K to the E, to the ST,

00:04:33.245 --> 00:04:35.605
so lemme factor that out.

00:04:35.605 --> 00:04:36.772
K, E to the ST

00:04:37.819 --> 00:04:40.053
times, what's left?

00:04:40.053 --> 00:04:42.220
S squared plus one over LC

00:04:47.298 --> 00:04:48.298
equals zero.

00:04:50.485 --> 00:04:53.381
Okay, we have how many
adjustable parameters here?

00:04:53.381 --> 00:04:54.464
We have K, S.

00:04:56.636 --> 00:04:57.469
Those are the two.

00:04:57.469 --> 00:05:00.670
L and C are constants that
are values of our circuit.

00:05:00.670 --> 00:05:03.552
So, we need to find some values of K and S

00:05:03.552 --> 00:05:06.147
that make this equation zero.

00:05:06.147 --> 00:05:08.897
Okay, I can make K equal to zero,

00:05:09.797 --> 00:05:12.173
and that would mean that zero equals zero,

00:05:12.173 --> 00:05:13.816
so our amplitude would be zero.

00:05:13.816 --> 00:05:15.563
So if we put nothing in the circuit,

00:05:15.563 --> 00:05:16.706
we get nothing out.

00:05:16.706 --> 00:05:18.471
Okay, that's totally boring.

00:05:18.471 --> 00:05:21.426
So that's not so
interesting of a solution.

00:05:21.426 --> 00:05:24.461
Now, does E to the ST ever become zero?

00:05:24.461 --> 00:05:27.775
E to something, does it ever become zero?

00:05:27.775 --> 00:05:28.608
It never does.

00:05:28.608 --> 00:05:31.775
If I let T go to plus infinity,

00:05:31.775 --> 00:05:33.382
and S is negative,

00:05:33.382 --> 00:05:35.681
then E to the ST would become zero,

00:05:35.681 --> 00:05:37.715
but plus infinity time from now

00:05:37.715 --> 00:05:40.285
is pretty far in the future,

00:05:40.285 --> 00:05:42.603
and I don't wanna wait that long.

00:05:42.603 --> 00:05:45.051
So the interesting solution becomes

00:05:45.051 --> 00:05:48.218
can we make S squared plus one over LC

00:05:51.310 --> 00:05:52.477
equal to zero?

00:05:53.922 --> 00:05:55.237
This equation is referred to

00:05:55.237 --> 00:05:57.820
as the characteristic equation.

00:06:06.098 --> 00:06:10.265
So let's see what happens
when we try to solve this.

00:06:11.799 --> 00:06:13.239
Well, the first step of this is

00:06:13.239 --> 00:06:15.158
I'm gonna get S squared

00:06:15.158 --> 00:06:17.241
equals minus one over LC,

00:06:18.968 --> 00:06:20.885
or S equals square root

00:06:23.796 --> 00:06:25.546
of minus one over LC.

00:06:28.047 --> 00:06:29.392
Uh oh, look what happens here.

00:06:29.392 --> 00:06:32.256
We're taking the square
root of a negative number.

00:06:32.256 --> 00:06:33.896
So what's gonna happen here?

00:06:33.896 --> 00:06:37.968
We're gonna get an imaginary
number for our answer.

00:06:37.968 --> 00:06:39.551
I can write this as

00:06:42.681 --> 00:06:44.862
square root of minus one

00:06:44.862 --> 00:06:47.612
times square root of one over LC,

00:06:50.667 --> 00:06:53.507
and that gives me two answers,

00:06:53.507 --> 00:06:56.066
and I'm gonna call the first answer S one,

00:06:56.066 --> 00:06:57.703
and that's gonna be equal to J,

00:06:57.703 --> 00:07:00.002
which is the imaginary number.

00:07:00.002 --> 00:07:03.062
That's the square root of minus one

00:07:03.062 --> 00:07:05.408
for electrical engineers,

00:07:05.408 --> 00:07:06.241
times what?

00:07:06.241 --> 00:07:08.991
Times square root of one over LC,

00:07:11.271 --> 00:07:13.239
and the other, S two,

00:07:13.239 --> 00:07:15.913
is the negative of that,

00:07:15.913 --> 00:07:16.746
minus J.

00:07:22.270 --> 00:07:23.624
So these are two possible solutions

00:07:23.624 --> 00:07:26.601
to our differential equation.

00:07:26.601 --> 00:07:30.375
Now, I'm gonna give a nickname
to this expression here

00:07:30.375 --> 00:07:32.215
because I don't wanna write it so much.

00:07:32.215 --> 00:07:34.798
I'm gonna call this omega knot.

00:07:38.855 --> 00:07:43.261
So this is a lower case
omega Greek letter.

00:07:43.261 --> 00:07:45.967
This is the capital omega

00:07:45.967 --> 00:07:47.245
that we use for Ohms,

00:07:47.245 --> 00:07:48.748
but little omega is often used

00:07:48.748 --> 00:07:50.935
as this variable here.

00:07:50.935 --> 00:07:53.435
So I can say that S one equals

00:07:56.335 --> 00:07:57.918
plus J, omega knot,

00:08:00.689 --> 00:08:03.772
and S two equals minus J, omega knot.

00:08:07.080 --> 00:08:09.000
So we're gonna use this
notation for a little bit,

00:08:09.000 --> 00:08:12.908
but just remember I made
that simple substitution.

00:08:12.908 --> 00:08:13.843
We have two different roots

00:08:13.843 --> 00:08:16.949
that can cause our differential
equation to go to zero.

00:08:16.949 --> 00:08:21.639
So when we combine these
into a solution for I,

00:08:21.639 --> 00:08:24.320
we're gonna use a
combination of these two.

00:08:24.320 --> 00:08:26.035
We don't know which one it is.

00:08:26.035 --> 00:08:27.155
It could be both.

00:08:27.155 --> 00:08:29.879
So they could be
superimposed on each other,

00:08:29.879 --> 00:08:31.905
and this is what superposition is for.

00:08:31.905 --> 00:08:34.302
So we're gonna now have,

00:08:34.302 --> 00:08:35.481
we need to have K one.

00:08:35.481 --> 00:08:37.187
We'll have two constants,

00:08:37.187 --> 00:08:38.520
E to the S one T

00:08:42.120 --> 00:08:44.537
plus K two, E to the S two T,

00:08:50.748 --> 00:08:52.681
and I could also write this as--

00:08:52.681 --> 00:08:55.403
Let's fill in some of the numbers here,

00:08:55.403 --> 00:08:57.320
K one, E to the plus J,

00:09:00.245 --> 00:09:01.412
omega knot, T,

00:09:03.745 --> 00:09:04.662
plus K two,

00:09:07.054 --> 00:09:09.721
E to the minus, J omega knot, T.

00:09:12.876 --> 00:09:16.203
So that's my proposed solution,

00:09:16.203 --> 00:09:17.410
and what we need to do now,

00:09:17.410 --> 00:09:18.316
we found S.

00:09:18.316 --> 00:09:19.476
We found two values of S.

00:09:19.476 --> 00:09:21.224
Now we have these two constants.

00:09:21.224 --> 00:09:23.405
We gotta work those out.

00:09:23.405 --> 00:09:24.828
So, in the next video,

00:09:24.828 --> 00:09:27.277
we'll use the initial conditions

00:09:27.277 --> 00:09:29.535
to figure out what K one and K two are.


The early Temperance movement - part 2

WEBVTT
Kind: captions
Language: en

00:00:00.541 --> 00:00:01.574
- [Voiceover] Hey, it's Becca,

00:00:01.574 --> 00:00:04.720
and this is Temperance, Part Two.

00:00:04.720 --> 00:00:06.636
And in this video, I'll be talking more

00:00:06.636 --> 00:00:09.069
about how exactly over the course

00:00:09.069 --> 00:00:11.986
of the 1830s until mostly the 1860s

00:00:15.332 --> 00:00:17.856
the temperance movement
took root in America

00:00:17.856 --> 00:00:21.093
and how it became this
national phenomenon.

00:00:21.093 --> 00:00:23.641
So, a lot of it had to do
with temperance societies.

00:00:23.641 --> 00:00:24.546
So, in the last video,

00:00:24.546 --> 00:00:27.131
I mentioned that the
American Temperance Society

00:00:27.131 --> 00:00:30.131
was founded in 1826, so that's 1826.

00:00:31.496 --> 00:00:33.899
But the American Temperance
Society really was this group

00:00:33.899 --> 00:00:38.030
of upper-class Northern
white dudes saying,

00:00:38.030 --> 00:00:39.816
"Oh, well, we should probably make sure

00:00:39.816 --> 00:00:41.060
"that people don't drink as much."

00:00:41.060 --> 00:00:45.227
But this idea didn't take off
until the rise of teetotalism.

00:00:46.179 --> 00:00:49.182
So, teetotalism is very
different than temperance.

00:00:49.182 --> 00:00:50.182
Teetotalism.

00:00:51.147 --> 00:00:52.864
And so, teetotalism is the idea

00:00:52.864 --> 00:00:55.745
that people should not temper
their alcohol consumption,

00:00:55.745 --> 00:00:57.916
that they should drink no alcohol.

00:00:57.916 --> 00:01:01.126
So, the origin of this word
is debated among historians

00:01:01.126 --> 00:01:03.310
and there's kinda two funny stories.

00:01:03.310 --> 00:01:06.649
One is the idea that when
you would sign a pledge,

00:01:06.649 --> 00:01:09.333
so let's say I was going to pledge

00:01:09.333 --> 00:01:11.370
that I would drink no alcohol

00:01:11.370 --> 00:01:13.808
and join the American
Temperance Society, right here,

00:01:13.808 --> 00:01:16.609
I would have to sign my name like that

00:01:16.609 --> 00:01:18.760
and write my name, Becca.

00:01:18.760 --> 00:01:22.860
Or the other idea is that there
was this temperance activist

00:01:22.860 --> 00:01:24.457
and he was trying to convince people

00:01:24.457 --> 00:01:26.261
to stop drinking alcohol.

00:01:26.261 --> 00:01:28.563
And he said, "You don't have
to stop drinking hard alcohol,

00:01:28.563 --> 00:01:31.768
"you have to-to-to-to totally abstain,"

00:01:31.768 --> 00:01:34.919
and that's where teetotalism
came from, his stutter.

00:01:34.919 --> 00:01:38.348
Just kind of a fun little
factoid about teetotalism.

00:01:38.348 --> 00:01:42.087
But this idea of signing a
pledge to drink no alcohol

00:01:42.087 --> 00:01:45.015
was really popular among
these different societies

00:01:45.015 --> 00:01:46.353
that started popping up.

00:01:46.353 --> 00:01:48.584
So, the American Temperance Society

00:01:48.584 --> 00:01:50.617
was not quite as effective.

00:01:50.617 --> 00:01:53.028
But the Washingtonian Temperance Society

00:01:53.028 --> 00:01:56.281
started in the 1840s, in 1840, actually,

00:01:56.281 --> 00:01:59.901
the Washington Temperance Society.

00:01:59.901 --> 00:02:02.070
And the Washington Temperance Society

00:02:02.070 --> 00:02:04.390
was different than the
American Temperance Society

00:02:04.390 --> 00:02:07.153
because it kind of looked
a little bit more like

00:02:07.153 --> 00:02:10.800
the 19th century version
of Alcoholics Anonymous.

00:02:10.800 --> 00:02:13.691
People would come together
and talk about their problem.

00:02:13.691 --> 00:02:15.729
There wasn't really a treatment aspect,

00:02:15.729 --> 00:02:18.539
it wasn't super effective
in stopping people

00:02:18.539 --> 00:02:21.176
from drinking alcohol because
a pledge, people realized,

00:02:21.176 --> 00:02:24.802
wasn't actually going to stop
alcoholics from drinking.

00:02:24.802 --> 00:02:27.081
However, the Washingtonian
Temperance Society

00:02:27.081 --> 00:02:29.626
was more this group of middle class men

00:02:29.626 --> 00:02:31.160
and they would all come together

00:02:31.160 --> 00:02:34.242
and try and curb their consumption.

00:02:34.242 --> 00:02:36.563
So, during this time, there were also lots

00:02:36.563 --> 00:02:40.288
of prohibitory laws being
passed by the states.

00:02:40.288 --> 00:02:42.521
So, different states at different times

00:02:42.521 --> 00:02:45.421
during the early-1800s started to try

00:02:45.421 --> 00:02:49.209
and curb consumption by enacting laws.

00:02:49.209 --> 00:02:50.846
They realized that the pledges,

00:02:50.846 --> 00:02:52.856
you know, me signing my name like this,

00:02:52.856 --> 00:02:55.085
didn't actually help that much,

00:02:55.085 --> 00:02:57.678
and so they needed to
do something legally.

00:02:57.678 --> 00:03:01.845
The first temperance law was
passed by Maine in 1838, Maine,

00:03:02.933 --> 00:03:05.825
and this law just outlawed
the sale of hard liquor.

00:03:05.825 --> 00:03:09.060
But slowly, states across the country

00:03:09.060 --> 00:03:11.433
started banning alcohol
consumption altogether.

00:03:11.433 --> 00:03:13.397
So, this was kinda happening
all throughout here

00:03:13.397 --> 00:03:17.564
and 12-15 states had some sort
of regulatory law on alcohol.

00:03:18.974 --> 00:03:23.141
So, over this time period,
from the 1830s to the 1860s,

00:03:24.366 --> 00:03:27.157
Americans were not just taking pledges

00:03:27.157 --> 00:03:30.983
like they were up here with
the American Temperance Society

00:03:30.983 --> 00:03:32.938
and the Washingtonian Temperance Society,

00:03:32.938 --> 00:03:36.641
but they were actually enacting laws.

00:03:36.641 --> 00:03:38.588
Temperance went really mainstream.

00:03:38.588 --> 00:03:39.751
It wasn't just this idea

00:03:39.751 --> 00:03:41.079
that you were going to sign a pledge

00:03:41.079 --> 00:03:43.646
to stop drinking hard alcohol,
there were gonna be laws

00:03:43.646 --> 00:03:46.613
that would bind you to drink no alcohol.

00:03:46.613 --> 00:03:49.010
So, on top of this legal transition,

00:03:49.010 --> 00:03:51.958
there was also a big
social and media campaign

00:03:51.958 --> 00:03:54.359
about the terrors and evils of alcohol.

00:03:54.359 --> 00:03:57.590
So, right over here is
The Drunkard's Progress.

00:03:57.590 --> 00:03:59.333
So, this is really famous lithograph

00:03:59.333 --> 00:04:01.169
created by Nathaniel Currier.

00:04:01.169 --> 00:04:05.044
This was in 1846, so Drunkard's
Progress, right over here.

00:04:05.044 --> 00:04:08.038
And The Drunkard's
Progress, as you can see,

00:04:08.038 --> 00:04:11.623
shows the kind of cyclic
nature of the alcoholic.

00:04:11.623 --> 00:04:13.385
First, he's just drinking at home,

00:04:13.385 --> 00:04:15.548
then he's drinking with friends.

00:04:15.548 --> 00:04:19.256
And then, oh, what is going on there?

00:04:19.256 --> 00:04:22.660
He is not going to be going
to the factory today for work.

00:04:22.660 --> 00:04:24.659
So, the drunk started not as a drunk

00:04:24.659 --> 00:04:27.574
but as your average guy just
having a drink here or there.

00:04:27.574 --> 00:04:30.697
Then you would see them go
through each of these steps,

00:04:30.697 --> 00:04:32.666
you see Step 2, Step 3, Step 4.

00:04:32.666 --> 00:04:35.051
Over here, Step 5 seems
like he's just hanging out

00:04:35.051 --> 00:04:37.669
with his buddies, having a good time,

00:04:37.669 --> 00:04:39.883
but then it really slowly deteriorated

00:04:39.883 --> 00:04:42.954
into something that Americans didn't want.

00:04:42.954 --> 00:04:47.121
So then, in 1853, I guess
that's kind of in here, 1853,

00:04:48.033 --> 00:04:50.086
this media campaign just took off

00:04:50.086 --> 00:04:54.165
with Ten Nights in a Bar-Room,
so Ten Nights in a Bar-Room.

00:04:54.165 --> 00:04:56.660
Here it is, Ten Nights in a Bar-Room.

00:04:56.660 --> 00:04:58.920
This is one of the pictures in the book.

00:04:58.920 --> 00:05:02.561
This really had just
huge mainstream reach;

00:05:02.561 --> 00:05:04.230
almost everyone read it.

00:05:04.230 --> 00:05:06.056
And then they started
putting on plays of it,

00:05:06.056 --> 00:05:08.968
depicting just how drunk people got

00:05:08.968 --> 00:05:12.233
and how terrible that was
for everyone involved.

00:05:12.233 --> 00:05:14.980
Ten Night in a Bar-Room had
this really national reach

00:05:14.980 --> 00:05:17.829
and it was similar to that
of Harriet Beecher Stowe

00:05:17.829 --> 00:05:21.422
and Uncle Tom's Cabin in
the abolition movement.

00:05:21.422 --> 00:05:22.833
So, this was this rhetoric

00:05:22.833 --> 00:05:25.622
that made the temperance
movement take off.

00:05:25.622 --> 00:05:27.881
So, you're probably wondering, then what?

00:05:27.881 --> 00:05:29.465
What happened to temperance?

00:05:29.465 --> 00:05:33.632
Why did prohibition not happen until 1920?

00:05:33.632 --> 00:05:36.335
And so, this has a lot to do
with the abolition movement.

00:05:36.335 --> 00:05:38.066
So, the abolition movement was taking off

00:05:38.066 --> 00:05:40.435
right around here, abolition.

00:05:40.435 --> 00:05:42.405
And the abolition movement was the idea

00:05:42.405 --> 00:05:46.238
that slavery had to be
ended right now, today.

00:05:47.271 --> 00:05:50.109
Abolition was the focus
of the American people

00:05:50.109 --> 00:05:54.900
come the mid-1800s, and this
really put temperance on hold.

00:05:54.900 --> 00:05:58.827
And so, temperance would
come back after the Civil War

00:05:58.827 --> 00:06:00.952
and after slavery was abolished.

00:06:00.952 --> 00:06:03.875
So, you can learn more
about postbellum temperance,

00:06:03.875 --> 00:06:07.448
postbellum, that means
after the Civil War,

00:06:07.448 --> 00:06:10.811
postbellum temperance and prohibition

00:06:10.811 --> 00:06:15.008
in the Khan Academy
article titled Prohibition.

00:06:15.008 --> 00:06:16.672
You can check that out and
learn a little bit more

00:06:16.672 --> 00:06:18.365
about how this whole crazy story ended.


The early Temperance movement - part 1

WEBVTT
Kind: captions
Language: en

00:00:00.483 --> 00:00:02.659
- [Voiceover] Hi, this is
Becca from Khan Academy,

00:00:02.659 --> 00:00:05.273
and today I'm going to be
talking about temperance.

00:00:05.273 --> 00:00:07.356
So, what was the temperance movement?

00:00:07.356 --> 00:00:09.036
In this video, I'll talk a little bit

00:00:09.036 --> 00:00:13.005
about what temperance
was, what its causes were,

00:00:13.005 --> 00:00:17.899
and how it started to
develop in the early-1800s.

00:00:17.899 --> 00:00:20.342
Temperance was the idea
that Americans drank

00:00:20.342 --> 00:00:23.837
way too much alcohol and needed
to temper their consumption.

00:00:23.837 --> 00:00:25.887
It started as kind of this idea

00:00:25.887 --> 00:00:28.013
that people should just
drink a little bit less,

00:00:28.013 --> 00:00:31.971
they should drink less whiskey,
less rum, less hard alcohol.

00:00:31.971 --> 00:00:34.550
And then, slowly, it started to take on

00:00:34.550 --> 00:00:36.920
this kind of prohibitory character.

00:00:36.920 --> 00:00:39.522
So, again, it was the
idea that we just needed

00:00:39.522 --> 00:00:42.355
to temper our alcohol consumption.

00:00:43.356 --> 00:00:46.944
And so, how did the
temperance movement take root?

00:00:46.944 --> 00:00:49.359
The temperance movement
kinda has three main causes

00:00:49.359 --> 00:00:50.990
that I like to think about.

00:00:50.990 --> 00:00:55.157
So, the three main causes were
the Second Great Awakening,

00:00:56.211 --> 00:00:58.378
the Industrial Revolution,

00:01:02.385 --> 00:01:03.799
and growing nativism

00:01:03.799 --> 00:01:08.393
and, frankly, racism that
started as new immigrants

00:01:08.393 --> 00:01:12.467
were coming to America in the early-1800s.

00:01:12.467 --> 00:01:15.053
So, this was all kinda
happening right around here,

00:01:15.053 --> 00:01:16.264
and so I'll talk a little bit more

00:01:16.264 --> 00:01:19.227
about each of these causes
for the temperance movement

00:01:19.227 --> 00:01:21.234
and how it began.

00:01:21.234 --> 00:01:23.686
So, I'll start by talking about
the Second Great Awakening.

00:01:23.686 --> 00:01:26.953
So, the Second Great
Awakening was this time period

00:01:26.953 --> 00:01:31.455
in the early-1800s that
focused a lot of social reforms

00:01:31.455 --> 00:01:35.990
around capturing moral
good or Christian ideals

00:01:35.990 --> 00:01:39.478
so Christian ideals,
here's the little cross,

00:01:39.478 --> 00:01:41.961
within our social institutions.

00:01:41.961 --> 00:01:45.951
So, this happened in
education, in prisons,

00:01:45.951 --> 00:01:48.593
in the first women's rights movement.

00:01:48.593 --> 00:01:50.666
And so, this was all going on in the 1800s

00:01:50.666 --> 00:01:51.589
and it was about this idea

00:01:51.589 --> 00:01:53.767
that we needed to be
good and moral people,

00:01:53.767 --> 00:01:56.633
and we needed our social
institutions to reflect that.

00:01:56.633 --> 00:01:58.996
So, temperance can be seen as a part

00:01:58.996 --> 00:02:00.682
of the Second Great Awakening.

00:02:00.682 --> 00:02:02.539
And so, down here, you can kind of see

00:02:02.539 --> 00:02:04.163
the Second Great Awakening image here.

00:02:04.163 --> 00:02:05.930
This is the idea that the family

00:02:05.930 --> 00:02:10.012
was also intimately affected
by people being too drunk.

00:02:10.012 --> 00:02:12.298
Here's the father and he's really drunk

00:02:12.298 --> 00:02:14.807
and things are kind of going to mayhem.

00:02:14.807 --> 00:02:16.084
People were just too drunk,

00:02:16.084 --> 00:02:19.236
and this was tearing apart
lots of different institutions,

00:02:19.236 --> 00:02:22.871
including the family, including education,

00:02:22.871 --> 00:02:24.590
including the workplace.

00:02:24.590 --> 00:02:26.179
And so, that's a good transition

00:02:26.179 --> 00:02:28.440
to talking about the
Industrial Revolution.

00:02:28.440 --> 00:02:30.405
So, the Industrial
Revolution was also going on

00:02:30.405 --> 00:02:31.620
at this time period

00:02:31.620 --> 00:02:34.545
and people could no longer
be drunk on the job, right?

00:02:34.545 --> 00:02:36.703
So, people used to be artisans.

00:02:36.703 --> 00:02:38.443
They used to just kind
of sit in their home,

00:02:38.443 --> 00:02:42.494
make their shoes or sew something by hand,

00:02:42.494 --> 00:02:44.153
and they could be drunk while doing that.

00:02:44.153 --> 00:02:47.549
But now, if you're kind
of in a factory setting,

00:02:47.549 --> 00:02:49.676
people were getting their fingers cut off

00:02:49.676 --> 00:02:52.211
by these new machines that were promoted

00:02:52.211 --> 00:02:54.701
in the Industrial Revolution
because they were drunk

00:02:54.701 --> 00:02:56.539
while trying to operate the machinery.

00:02:56.539 --> 00:02:57.771
So, with this new industry,

00:02:57.771 --> 00:03:00.435
workers could no longer
be drunk on the job.

00:03:00.435 --> 00:03:02.524
And so, the final cause is this nativism

00:03:02.524 --> 00:03:05.850
that people were seeing with
new Catholic immigrants.

00:03:05.850 --> 00:03:09.050
So, there were Catholic immigrants
coming into the country.

00:03:09.050 --> 00:03:12.567
And lots of Protestants
were very anti-Catholic

00:03:12.567 --> 00:03:14.499
and anti-immigration.

00:03:14.499 --> 00:03:17.214
They decided that the
Catholics were drunks.

00:03:17.214 --> 00:03:18.653
They did drink a lot,

00:03:18.653 --> 00:03:21.093
but it was definitely this
kind of racist sentiment

00:03:21.093 --> 00:03:24.144
that was percolating within
the Protestant community,

00:03:24.144 --> 00:03:27.018
and this kind of aligned
itself with the Whig Party.

00:03:27.018 --> 00:03:30.101
So, the Whigs became more Protestant;

00:03:31.349 --> 00:03:34.226
they were really big temperance people.

00:03:34.226 --> 00:03:36.077
And the Catholics more aligned themselves

00:03:36.077 --> 00:03:37.660
with the Democrats.

00:03:38.973 --> 00:03:41.906
The sentiment towards
these immigrant populations

00:03:41.906 --> 00:03:44.739
had this kind of political effect.

00:03:45.727 --> 00:03:48.441
So, at this time, temperance
was starting to become

00:03:48.441 --> 00:03:50.239
more of a political movement

00:03:50.239 --> 00:03:53.588
and different social groups
were taking this more seriously.

00:03:53.588 --> 00:03:56.513
There were some state-level organizations.

00:03:56.513 --> 00:03:59.787
It was just becoming more
of a social phenomenon.

00:03:59.787 --> 00:04:04.621
In 1825, right over here,
this really famous preacher,

00:04:04.621 --> 00:04:08.788
Lyman Beecher, did his six
sermons on the sins of alcohol.

00:04:10.231 --> 00:04:12.648
And so, these sermons in 1825

00:04:14.598 --> 00:04:17.445
solidified this idea in the American mind

00:04:17.445 --> 00:04:21.391
that it was anti-Christian
to be a huge drinker,

00:04:21.391 --> 00:04:23.522
and this idea really took root.

00:04:23.522 --> 00:04:27.410
This is becoming kind of
this larger social phenomenon

00:04:27.410 --> 00:04:29.909
and there start to be
not just more state-level

00:04:29.909 --> 00:04:32.987
or community-level
societies against drinking,

00:04:32.987 --> 00:04:35.891
you see the first ever
national organization.

00:04:35.891 --> 00:04:40.615
So, the first national
temperance society was in 1826,

00:04:40.615 --> 00:04:45.545
down here, with the American
Temperance Society, so the ATS.

00:04:45.545 --> 00:04:47.152
And I'll talk more about the ATS

00:04:47.152 --> 00:04:49.279
and the kind of nationalization

00:04:49.279 --> 00:04:51.964
of the temperance movement
in the next video.