Regional attitudes about slavery, 1754-1800

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:01.480
- [Kim] This is a chart

00:00:01.480 --> 00:00:06.480
that shows the percentage
of the total population

00:00:06.650 --> 00:00:08.240
of each of these colonies,

00:00:08.240 --> 00:00:10.270
and then later, states,

00:00:10.270 --> 00:00:14.500
that was made up by enslaved Africans.

00:00:14.500 --> 00:00:17.370
Starting in the year 1754,

00:00:17.370 --> 00:00:19.000
which we'll show in purple,

00:00:19.000 --> 00:00:21.450
and comparing that to the year 1800,

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which we'll show here in green.

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Now some of these numbers are estimates,

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since the census didn't exist before 1790,

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but this can help us get an overall sense

00:00:31.760 --> 00:00:33.770
of how the institution of slavery

00:00:33.770 --> 00:00:35.310
changed in these years.

00:00:35.310 --> 00:00:37.430
Now on the left side here,

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we have the data from the Northern part

00:00:40.420 --> 00:00:41.363
of the country.

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Now in purple you can see

00:00:43.520 --> 00:00:45.010
the percentage of the population

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of each colony that was enslaved in 1754,

00:00:48.700 --> 00:00:51.790
and generally, it wasn't
a very large amount,

00:00:51.790 --> 00:00:55.360
think New York here had
the most at about 14%.

00:00:55.360 --> 00:00:58.490
By 1800, however, you can see that

00:00:58.490 --> 00:01:01.860
the percent went down considerably.

00:01:01.860 --> 00:01:03.550
Some, like Massachusetts,

00:01:03.550 --> 00:01:08.170
outlawed slavery altogether,
so they're at 0%,

00:01:08.170 --> 00:01:11.350
and others like New York or Rhode Island,

00:01:11.350 --> 00:01:15.050
began a process of gradual emancipation,

00:01:15.050 --> 00:01:17.220
phasing out slavery.

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Another thing to note is that

00:01:18.740 --> 00:01:20.770
new territories in the West

00:01:20.770 --> 00:01:22.830
that were claimed by the United States

00:01:22.830 --> 00:01:26.020
either had very small amounts of slavery,

00:01:26.020 --> 00:01:30.100
like in the Indiana
Territory, or none at all,

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like in the Northwest Territory,

00:01:32.270 --> 00:01:33.900
where slavery was outlawed.

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Now let's compare that to the same period

00:01:36.470 --> 00:01:38.070
in Southern Regions.

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In most of the South in 1754,

00:01:41.530 --> 00:01:42.640
enslaved people made up

00:01:42.640 --> 00:01:46.430
a very significant
portion of the population.

00:01:46.430 --> 00:01:49.250
In some cases, like in South Carolina,

00:01:49.250 --> 00:01:53.580
outnumbering whites at
a rate of more than 60%

00:01:53.580 --> 00:01:54.820
of the population.

00:01:54.820 --> 00:01:58.930
And unlike in the North,
as you see here in teal,

00:01:58.930 --> 00:02:02.100
in 1800, enslaved people remained

00:02:02.100 --> 00:02:04.740
a really quite large percentage

00:02:04.740 --> 00:02:06.040
of the population.

00:02:06.040 --> 00:02:09.080
In some cases, the percent of people

00:02:09.080 --> 00:02:11.210
who were enslaved grew considerably,

00:02:11.210 --> 00:02:14.360
like in Delaware or in Georgia,

00:02:14.360 --> 00:02:17.400
and in other places stayed about the same.

00:02:17.400 --> 00:02:20.650
In places like Virginia
and South Carolina,

00:02:20.650 --> 00:02:23.040
the percent of the
population that was enslaved

00:02:23.040 --> 00:02:25.130
actually dropped over this time period,

00:02:25.130 --> 00:02:27.520
but not because there were fewer slaves,

00:02:27.520 --> 00:02:29.380
but rather because the white population

00:02:29.380 --> 00:02:32.480
had grown faster than
the enslaved population

00:02:32.480 --> 00:02:33.930
in the same time period.

00:02:33.930 --> 00:02:37.510
And unlike in the Northern Territories,

00:02:37.510 --> 00:02:39.730
the percentage of enslaved people

00:02:39.730 --> 00:02:41.810
in the new Southern Territories

00:02:41.810 --> 00:02:43.470
claimed by the United States,

00:02:43.470 --> 00:02:47.460
Kentucky, Tennessee, Mississippi,

00:02:47.460 --> 00:02:49.320
was quite significant as well.

00:02:49.320 --> 00:02:51.250
So what's happening here?

00:02:51.250 --> 00:02:53.580
Why, over this 50 year period,

00:02:53.580 --> 00:02:57.320
did slavery begin to die out in the North,

00:02:57.320 --> 00:03:00.300
while surviving, and even expanding West,

00:03:00.300 --> 00:03:01.360
in the South?

00:03:01.360 --> 00:03:04.950
How much change was there
over this time period

00:03:04.950 --> 00:03:08.160
in the regional attitudes about slavery?

00:03:08.160 --> 00:03:11.610
And how would we even go
about figuring that out?

00:03:11.610 --> 00:03:13.690
One way that we can attempt

00:03:13.690 --> 00:03:16.650
to measure the extent of change

00:03:16.650 --> 00:03:20.020
is by identifying the
things that stayed the same

00:03:20.020 --> 00:03:21.750
during this time period,

00:03:21.750 --> 00:03:23.803
which we call Continuities,

00:03:25.060 --> 00:03:29.610
or aspects of society that continued on

00:03:29.610 --> 00:03:32.300
being the same as they were before,

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and Changes, aspects of society

00:03:35.750 --> 00:03:39.190
that were different than before.

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There are many different
aspects of society

00:03:41.740 --> 00:03:44.527
that historians can trace changes

00:03:44.527 --> 00:03:46.660
and continuities in over time,

00:03:46.660 --> 00:03:50.940
like changes in politics
or art or gender roles,

00:03:50.940 --> 00:03:52.010
but for this video,

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I'm gonna choose Economics and Ideas.

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Why am I choosing those two?

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Well, first and foremost,

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because slavery was an economic practice.

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It was a source of labor

00:04:03.760 --> 00:04:05.200
that supported the production

00:04:05.200 --> 00:04:06.990
of agricultural products.

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Second, I know that in this time period,

00:04:09.870 --> 00:04:11.410
the American Revolution

00:04:11.410 --> 00:04:13.470
introduced radical changes

00:04:13.470 --> 00:04:16.030
in ideas about Liberty and Equality,

00:04:16.030 --> 00:04:18.730
which stand in stark contrast

00:04:18.730 --> 00:04:20.570
to the institution of slavery.

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So let's compare the Economic systems

00:04:23.840 --> 00:04:28.840
and Ideas about slavery
in each of these Regions,

00:04:28.950 --> 00:04:33.950
in 1754 and 1800, to see what changed

00:04:35.000 --> 00:04:36.870
and what stayed the same.

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So first, let's look at
the Economic systems.

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In the North, where the
climate and the soil

00:04:42.890 --> 00:04:45.280
was not suited for large scale

00:04:45.280 --> 00:04:46.790
plantation agriculture,

00:04:46.790 --> 00:04:51.480
the economy centered around
family farms in 1754.

00:04:51.480 --> 00:04:55.060
Proximity to the coast facilitated fishing

00:04:55.060 --> 00:04:56.290
and shipping,

00:04:56.290 --> 00:04:58.670
and the people who actually did this work

00:04:58.670 --> 00:05:01.870
tended to be either
the owners of the farms

00:05:01.870 --> 00:05:05.650
or the boats, or indentured servants,

00:05:05.650 --> 00:05:08.640
and a small number of enslaved laborers

00:05:08.640 --> 00:05:09.850
who worked for them.

00:05:09.850 --> 00:05:12.670
None of these activities really required

00:05:12.670 --> 00:05:15.570
the labor of a large
number of enslaved people.

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All right, well, what about in 1800?

00:05:17.630 --> 00:05:20.440
Farming and fishing and shipping

00:05:20.440 --> 00:05:22.250
were still going strong,

00:05:22.250 --> 00:05:25.840
but as the first Industrial Revolution

00:05:25.840 --> 00:05:30.470
began to take off in the late 1700s,

00:05:30.470 --> 00:05:33.470
factories began to spring up in the North,

00:05:33.470 --> 00:05:35.720
and the cheapest labor
source for these factories

00:05:35.720 --> 00:05:37.680
was immigrant laborers,

00:05:37.680 --> 00:05:40.750
not slaves, not indentured servants.

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Now let's look at the Economic systems

00:05:42.620 --> 00:05:46.300
of the South in 1754 and 1800.

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In 1754, plantation agriculture

00:05:49.780 --> 00:05:51.700
based on enslaved labor

00:05:51.700 --> 00:05:55.600
was the central economic
system, and it was also

00:05:55.600 --> 00:05:58.360
the central economic system in 1800.

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But unlike in the North,

00:06:00.560 --> 00:06:03.360
where the technological innovations

00:06:03.360 --> 00:06:05.130
of the Industrial Revolution

00:06:05.130 --> 00:06:07.270
made indentured servants

00:06:07.270 --> 00:06:09.890
or enslaved laborers unnecessary,

00:06:09.890 --> 00:06:11.840
the major technological innovation

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of the cotton gin

00:06:13.330 --> 00:06:15.970
made cotton easier to process,

00:06:15.970 --> 00:06:19.040
and therefore, saved it
as a profitable crop,

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which made slave owners eager to expand

00:06:22.400 --> 00:06:24.190
the institution of slavery,

00:06:24.190 --> 00:06:26.510
rather than phase it out.

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Next, let's take a look at the ideas

00:06:28.690 --> 00:06:30.860
about slavery, which were commonly held

00:06:30.860 --> 00:06:32.260
in each of the regions.

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In 1754 in the North,

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I would say a few people questioned

00:06:37.260 --> 00:06:39.160
the existence of slavery.

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Northern Shipping was involved

00:06:41.250 --> 00:06:42.640
in the Atlantic Slave Trade.

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The idea that masters
deserve to rule slaves

00:06:45.880 --> 00:06:47.330
was as natural as the idea

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that kings deserve to rule their subjects.

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Even free people of color
were rarely considered

00:06:53.370 --> 00:06:55.530
full British citizens,

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the way people of English descent were.

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How did things change in 1800?

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The Revolutionary War had a strong effect

00:07:03.070 --> 00:07:05.750
on the ideas about slavery in the North.

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Northerners apply the Revolutionary ideals

00:07:08.260 --> 00:07:10.070
of Liberty and Equality

00:07:10.070 --> 00:07:13.430
to long-standing ideas about social rules,

00:07:13.430 --> 00:07:16.311
and they found that slavery

00:07:16.311 --> 00:07:17.670
was incompatible with them.

00:07:17.670 --> 00:07:19.880
So after the Revolution, Northern States

00:07:19.880 --> 00:07:21.370
either abolished slavery

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or began to phase it out

00:07:23.400 --> 00:07:25.040
with gradual emancipation.

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But I wanna caution you

00:07:26.540 --> 00:07:29.030
into thinking that this
turned the North into

00:07:29.030 --> 00:07:30.330
a racial utopia.

00:07:30.330 --> 00:07:31.670
People of African descent

00:07:31.670 --> 00:07:32.870
were still not permitted

00:07:32.870 --> 00:07:34.730
to become American citizens,

00:07:34.730 --> 00:07:36.100
even though the immigrants,

00:07:36.100 --> 00:07:39.780
who were beginning to
flood in from Europe were.

00:07:39.780 --> 00:07:42.920
So I think it would be
fair to say overall,

00:07:42.920 --> 00:07:46.200
that the North had never
really needed slavery,

00:07:46.200 --> 00:07:50.270
and therefore, when
the Revolution led many

00:07:50.270 --> 00:07:52.720
to question the institution of slavery,

00:07:52.720 --> 00:07:55.520
it was pretty easy to abolish it.

00:07:55.520 --> 00:07:57.930
How does that compare to
the South's transition

00:07:57.930 --> 00:07:59.080
in this time period?

00:07:59.080 --> 00:08:02.580
Well, by 1754, the notion that

00:08:02.580 --> 00:08:04.000
slavery was natural

00:08:04.000 --> 00:08:07.030
and enslaved Africans were property

00:08:07.030 --> 00:08:09.270
was pretty well-ingrained.

00:08:09.270 --> 00:08:11.130
But the ideas of the Revolution

00:08:11.130 --> 00:08:14.330
didn't have quite the
same effect on the South

00:08:14.330 --> 00:08:15.860
as they did in the North.

00:08:15.860 --> 00:08:19.040
If anything, by 1800, white Southerners

00:08:19.040 --> 00:08:21.490
were determined to enact

00:08:21.490 --> 00:08:23.740
harsher slave codes than ever

00:08:23.740 --> 00:08:26.000
because they feared uprisings,

00:08:26.000 --> 00:08:28.370
since the ideals of
the American Revolution

00:08:28.370 --> 00:08:30.770
had inspired the Haitians

00:08:30.770 --> 00:08:32.310
to overthrow the French.

00:08:32.310 --> 00:08:33.790
So why did these changes

00:08:33.790 --> 00:08:36.510
in continuities over time matter?

00:08:36.510 --> 00:08:39.740
Well, because by 1800, you start to see

00:08:39.740 --> 00:08:43.040
attitudes around slavery crystallizing

00:08:43.040 --> 00:08:44.300
in the United States,

00:08:44.300 --> 00:08:48.380
creating a North where slavery was rare

00:08:48.380 --> 00:08:51.720
and considered a perversion of the ideals

00:08:51.720 --> 00:08:52.980
of the Revolution,

00:08:52.980 --> 00:08:56.710
and a South where slavery was central

00:08:56.710 --> 00:08:58.280
and considered natural

00:08:58.280 --> 00:09:00.480
and perhaps even a desirable way

00:09:00.480 --> 00:09:02.120
to organize society.

00:09:02.120 --> 00:09:06.740
And as white settlers
from each of these regions

00:09:06.740 --> 00:09:09.650
moved farther and farther West,

00:09:09.650 --> 00:09:12.010
into territories that were opened up

00:09:12.010 --> 00:09:14.960
by the American victory
in the Revolutionary War

00:09:14.960 --> 00:09:16.530
and by Indian removal,

00:09:16.530 --> 00:09:18.750
both Northerners and Southerners

00:09:18.750 --> 00:09:21.700
would export their Economic systems

00:09:21.700 --> 00:09:24.000
and their ideas about slavery

00:09:24.000 --> 00:09:26.570
into these adjacent Western lands.

00:09:26.570 --> 00:09:29.430
By 1820, this gradual process

00:09:29.430 --> 00:09:32.753
would erupt into Sectional
crisis over slavery.


Interpreting direction of motion from position-time graph

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:03.100
- [Instructor] An object
is moving along a line.

00:00:03.100 --> 00:00:05.790
The following graph gives
the object's position,

00:00:05.790 --> 00:00:08.840
relative to its starting point, over time.

00:00:08.840 --> 00:00:12.260
For each point on the graph,
is the object moving forward,

00:00:12.260 --> 00:00:13.770
backward, or neither?

00:00:13.770 --> 00:00:16.840
So pause this video and
try to figure that out.

00:00:16.840 --> 00:00:20.070
All right, so we see we
have position in meters

00:00:20.070 --> 00:00:23.750
versus time, so for example,

00:00:23.750 --> 00:00:27.150
this point right over here
tells us that after one second,

00:00:27.150 --> 00:00:30.560
we are four meters ahead
of our starting point.

00:00:30.560 --> 00:00:34.400
Or for example, this point right over here

00:00:34.400 --> 00:00:38.630
says that after four seconds,
we are almost it seems,

00:00:38.630 --> 00:00:42.030
almost four meters behind
our starting point.

00:00:42.030 --> 00:00:43.880
So let's look at each of
these points and think about

00:00:43.880 --> 00:00:46.820
whether we're moving forward,
backward, or neither.

00:00:46.820 --> 00:00:49.550
So at this point right over here,

00:00:49.550 --> 00:00:53.750
at that moment, we're
about 2 1/2 meters in front

00:00:53.750 --> 00:00:54.820
of our starting point.

00:00:54.820 --> 00:00:57.920
We're at a positive
position of 2 1/2 meters.

00:00:57.920 --> 00:01:02.160
But as time goes on,
we are moving backwards

00:01:02.160 --> 00:01:04.690
closer and closer to the starting point.

00:01:04.690 --> 00:01:08.430
So this is we are moving backward.

00:01:08.430 --> 00:01:09.520
One way to think about it,

00:01:09.520 --> 00:01:11.640
at this time we're at 2 1/2 meters.

00:01:11.640 --> 00:01:14.700
If you go forward about half a second,

00:01:14.700 --> 00:01:16.630
we are then back at our starting point.

00:01:16.630 --> 00:01:18.520
So we had to go backwards.

00:01:18.520 --> 00:01:22.660
And if we look at this
point right over here,

00:01:22.660 --> 00:01:25.870
it looks like we were going
backwards this entire time

00:01:25.870 --> 00:01:29.950
while our curve is downward sloping.

00:01:29.950 --> 00:01:31.480
But at this point right over here,

00:01:31.480 --> 00:01:33.690
when we are about it
looks like five meters

00:01:33.690 --> 00:01:38.630
behind our starting point,
we start going forward again.

00:01:38.630 --> 00:01:42.080
But right at that moment,
we are going neither

00:01:42.080 --> 00:01:43.440
forward nor backwards.

00:01:43.440 --> 00:01:45.480
It's right at that moment
where we just finished

00:01:45.480 --> 00:01:48.350
going backwards and we're
about to go forward,

00:01:48.350 --> 00:01:49.810
and one way to think about it is,

00:01:49.810 --> 00:01:52.870
what would be the slope of the
tangent line at that point,

00:01:52.870 --> 00:01:55.086
and the slope of the
tangent line at that point

00:01:55.086 --> 00:01:58.600
would be horizontal.

00:01:58.600 --> 00:02:01.193
And so this is neither.

00:02:02.230 --> 00:02:05.330
So we can use that same technique
to think about this point.

00:02:05.330 --> 00:02:08.120
The slope is positive.

00:02:08.120 --> 00:02:10.880
And we see that all right,
right at that moment,

00:02:10.880 --> 00:02:12.860
it looks like we are
at the starting point,

00:02:12.860 --> 00:02:15.170
but if you fast forward even a few,

00:02:15.170 --> 00:02:18.910
even a fraction of a
second, we are now in front

00:02:18.910 --> 00:02:20.340
of our starting point.

00:02:20.340 --> 00:02:22.380
So we are moving forward.

00:02:22.380 --> 00:02:24.920
We are moving forward right over here.

00:02:24.920 --> 00:02:27.630
And at this point, we are
at our starting point,

00:02:27.630 --> 00:02:28.960
but if you think about
what's going to happen

00:02:28.960 --> 00:02:29.793
a moment later.

00:02:29.793 --> 00:02:31.710
A moment later we're gonna
be a little bit behind

00:02:31.710 --> 00:02:35.992
our starting point, and so
here we are moving backward.

00:02:35.992 --> 00:02:37.523
And we're done.


Interpreting direction of motion from velocity-time graph

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:03.940
- [Instructor] An object
is moving along a line.

00:00:03.940 --> 00:00:08.940
The following graph gives the
object's velocity over time.

00:00:09.040 --> 00:00:11.320
For each point on the graph,

00:00:11.320 --> 00:00:15.150
is the object moving forward,
backward, or neither?

00:00:15.150 --> 00:00:18.010
So pause this video and see
if you can figure that out.

00:00:18.010 --> 00:00:19.610
All right, now let's do this together.

00:00:19.610 --> 00:00:21.590
And so we can see these different points

00:00:21.590 --> 00:00:24.150
on this velocity-versus-time graph.

00:00:24.150 --> 00:00:26.800
And the important thing to realize is

00:00:26.800 --> 00:00:30.280
is if the velocity is
positive, we're moving forward.

00:00:30.280 --> 00:00:33.740
If the velocity is negative,
we're moving backward.

00:00:33.740 --> 00:00:37.120
And if the velocity is
zero, we're not moving

00:00:37.120 --> 00:00:39.550
either forward nor backwards,

00:00:39.550 --> 00:00:41.750
or neither forward nor backwards.

00:00:41.750 --> 00:00:43.620
So right over here we see

00:00:43.620 --> 00:00:45.270
that our velocity is positive.

00:00:45.270 --> 00:00:47.400
It's a positive two meters per second.

00:00:47.400 --> 00:00:51.450
So that means that we are moving forward.

00:00:51.450 --> 00:00:54.220
Now, over here our velocity
is zero meters per second.

00:00:54.220 --> 00:00:56.123
So this is neither.

00:00:57.300 --> 00:00:58.830
Now, over here our velocity

00:00:58.830 --> 00:01:00.940
is negative four meters per second.

00:01:00.940 --> 00:01:01.860
So one way to think about it

00:01:01.860 --> 00:01:04.900
is we're moving four
meters per second backward.

00:01:04.900 --> 00:01:06.563
So I'll write backward.

00:01:07.850 --> 00:01:10.200
Now, this is interesting, this last point.

00:01:10.200 --> 00:01:11.190
'Cause you might be tempted to say

00:01:11.190 --> 00:01:12.590
all right, I'm oscillating.

00:01:12.590 --> 00:01:13.640
I'm going up.

00:01:13.640 --> 00:01:15.080
Then I'm going down.

00:01:15.080 --> 00:01:16.330
Then I'm going back up.

00:01:16.330 --> 00:01:17.750
Maybe I'm moving forward here.

00:01:17.750 --> 00:01:20.170
But remember, what we're
thinking about here,

00:01:20.170 --> 00:01:21.680
this isn't position versus time.

00:01:21.680 --> 00:01:23.470
This is velocity versus time.

00:01:23.470 --> 00:01:26.350
So if our velocity is negative,
we're moving backwards.

00:01:26.350 --> 00:01:30.760
And here our velocity is still negative.

00:01:30.760 --> 00:01:33.090
It's becoming less negative,
but it's still negative.

00:01:33.090 --> 00:01:35.790
So we are still moving,

00:01:35.790 --> 00:01:37.980
we are still moving backward.

00:01:37.980 --> 00:01:41.790
If we were at this point right
over here or at this point,

00:01:41.790 --> 00:01:43.650
then we would be moving forward

00:01:43.650 --> 00:01:45.363
if our velocity were positive.


Worked example: Motion problems with derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.270
- [Instructor] A particle
moves along the x-axis.

00:00:03.270 --> 00:00:06.830
The function x of t gives
the particle's position

00:00:06.830 --> 00:00:09.810
at any time t is greater
than or equal to zero,

00:00:09.810 --> 00:00:12.240
and they give us x of t right over here.

00:00:12.240 --> 00:00:16.700
What is the particle's velocity
v of t at t is equal to two?

00:00:16.700 --> 00:00:18.870
So pause this video, see
if you can figure that out.

00:00:18.870 --> 00:00:21.970
Well, the key thing to
realize is that your velocity

00:00:21.970 --> 00:00:25.610
as a function of time is
the derivative of position.

00:00:25.610 --> 00:00:28.480
And so this is going to be equal to,

00:00:28.480 --> 00:00:31.230
we just take the derivative
with respect to t up here.

00:00:31.230 --> 00:00:33.120
So derivative of t to the third

00:00:33.120 --> 00:00:35.530
with respect to t is three t squared.

00:00:35.530 --> 00:00:36.470
If that's unfamiliar,

00:00:36.470 --> 00:00:39.050
I encourage you to review the power rule.

00:00:39.050 --> 00:00:41.910
The derivative of negative four t squared

00:00:41.910 --> 00:00:44.640
with respect to t is negative eight t.

00:00:44.640 --> 00:00:48.860
And derivative of three t with
respect to t is plus three.

00:00:48.860 --> 00:00:50.610
Derivative of a constant doesn't change

00:00:50.610 --> 00:00:52.930
with respect to time, so that's just zero.

00:00:52.930 --> 00:00:55.830
And so here we have velocity
as a function of time.

00:00:55.830 --> 00:00:59.180
And so if we want to know our
velocity at time t equals two,

00:00:59.180 --> 00:01:02.010
we just substitute two
wherever we see the t's.

00:01:02.010 --> 00:01:04.620
So it's gonna be three times four,

00:01:04.620 --> 00:01:06.070
three times two squared,

00:01:06.070 --> 00:01:10.400
so it's 12 minus eight
times two, minus 16,

00:01:10.400 --> 00:01:15.100
plus three, which is
equal to negative one.

00:01:15.100 --> 00:01:16.690
And you might say negative one

00:01:16.690 --> 00:01:19.100
by itself doesn't sound like a velocity.

00:01:19.100 --> 00:01:20.853
Well, if they gave us
units, if they told us

00:01:20.853 --> 00:01:24.200
that x was in meters and
that t was in seconds,

00:01:24.200 --> 00:01:25.490
well, then x would be,

00:01:25.490 --> 00:01:26.990
well, I already said would be in meters,

00:01:26.990 --> 00:01:30.150
and velocity would be negative
one meters per second.

00:01:30.150 --> 00:01:30.983
You might also be saying,

00:01:30.983 --> 00:01:32.480
well, what does the negative means?

00:01:32.480 --> 00:01:35.850
Well, that means that we
are moving to the left.

00:01:35.850 --> 00:01:39.040
Remember, we're moving along the x-axis.

00:01:39.040 --> 00:01:41.350
So if our velocity's negative,

00:01:41.350 --> 00:01:45.320
that means that x is decreasing
or we're moving to the left.

00:01:45.320 --> 00:01:47.660
What is the particle's acceleration a of t

00:01:47.660 --> 00:01:48.670
at t equals three?

00:01:48.670 --> 00:01:51.420
So pause this video again,
and see if you can do that.

00:01:51.420 --> 00:01:53.490
Well, here the realization is

00:01:53.490 --> 00:01:55.473
that acceleration is a function of time.

00:01:55.473 --> 00:01:58.060
It's just the derivative of velocity,

00:01:58.060 --> 00:02:01.320
which is the second
derivative of our position,

00:02:01.320 --> 00:02:03.030
which is just going to be equal to

00:02:03.030 --> 00:02:05.250
the derivative of this right over here.

00:02:05.250 --> 00:02:06.750
And so I'm just going to get

00:02:06.750 --> 00:02:10.840
derivative of three t squared
with respect to t is six t.

00:02:10.840 --> 00:02:12.610
Derivative of negative eight t

00:02:12.610 --> 00:02:14.870
with respect to t is minus eight.

00:02:14.870 --> 00:02:16.580
And derivative of a constant is zero.

00:02:16.580 --> 00:02:18.980
So it's just going to
be six t minus eight.

00:02:18.980 --> 00:02:21.640
So our acceleration at time t equals three

00:02:21.640 --> 00:02:26.640
is going to be six times three,
which is 18, minus eight,

00:02:27.530 --> 00:02:32.190
so minus eight, which is going
to be equal to positive 10.

00:02:32.190 --> 00:02:34.330
All right, now they ask
us what is the direction

00:02:34.330 --> 00:02:36.970
of the particle's motion at t equals two?

00:02:36.970 --> 00:02:39.540
Well, I already talked about
this, but pause this video and

00:02:39.540 --> 00:02:41.623
see if you can answer that yourself.

00:02:42.500 --> 00:02:44.800
Well, we've already looked
at the sign right over here.

00:02:44.800 --> 00:02:46.850
The fact that we have a
negative sign on our velocity

00:02:46.850 --> 00:02:49.200
means we are moving towards the left.

00:02:49.200 --> 00:02:51.530
So I'll fill that in right over there.

00:02:51.530 --> 00:02:54.810
At t equals three, is the particle's speed

00:02:54.810 --> 00:02:58.030
increasing, decreasing, or neither?

00:02:58.030 --> 00:03:00.670
So pause this video,
and try to answer that.

00:03:00.670 --> 00:03:03.160
All right, now we have
to be very careful here.

00:03:03.160 --> 00:03:05.420
If it says is the particle's velocity

00:03:05.420 --> 00:03:07.840
increasing, decreasing, or neither,

00:03:07.840 --> 00:03:10.460
then we would just have to
look at the acceleration.

00:03:10.460 --> 00:03:12.650
We see that the acceleration is positive,

00:03:12.650 --> 00:03:16.790
and so we know that the
velocity is increasing.

00:03:16.790 --> 00:03:19.210
But here they're not saying
velocity, they're saying speed.

00:03:19.210 --> 00:03:23.720
And just as a reminder, speed
is the magnitude of velocity.

00:03:23.720 --> 00:03:26.240
So, for example, at time t equals two,

00:03:26.240 --> 00:03:28.450
our velocity is negative one.

00:03:28.450 --> 00:03:29.960
If the units were meters and second,

00:03:29.960 --> 00:03:31.860
it would be negative
one meters per second.

00:03:31.860 --> 00:03:34.040
But our speed would just
be one meter per second.

00:03:34.040 --> 00:03:36.330
Speed, you're not talking
about the direction,

00:03:36.330 --> 00:03:38.470
so you would not have that sign there.

00:03:38.470 --> 00:03:40.290
And so in order to figure out

00:03:40.290 --> 00:03:43.410
if the speed is increasing
or decreasing or neither,

00:03:43.410 --> 00:03:46.010
if the acceleration is positive

00:03:46.010 --> 00:03:48.010
and the velocity is positive,

00:03:48.010 --> 00:03:50.550
that means the magnitude of
your velocity is increasing.

00:03:50.550 --> 00:03:52.490
So that means your speed is increasing.

00:03:52.490 --> 00:03:54.040
If your velocity is negative

00:03:54.040 --> 00:03:56.190
and your acceleration is also negative,

00:03:56.190 --> 00:03:58.700
that also means that
your speed is increasing.

00:03:58.700 --> 00:04:02.240
But if your velocity and
acceleration have different signs,

00:04:02.240 --> 00:04:05.650
well, that means that
your speed is decreasing.

00:04:05.650 --> 00:04:08.750
The magnitude of your
velocity would become less.

00:04:08.750 --> 00:04:12.370
So let's look at our velocity
at time t equals three.

00:04:12.370 --> 00:04:16.250
Our velocity at time three, we
just go back right over here,

00:04:16.250 --> 00:04:18.040
it's going to be three times nine,

00:04:18.040 --> 00:04:20.890
which is 27, three times three squared,

00:04:20.890 --> 00:04:23.180
minus 24

00:04:23.180 --> 00:04:24.400
plus three,

00:04:24.400 --> 00:04:25.233
plus three.

00:04:25.233 --> 00:04:28.930
So this is going to be equal to six.

00:04:28.930 --> 00:04:31.280
So our velocity and acceleration are both,

00:04:31.280 --> 00:04:32.900
you could say, in the same direction.

00:04:32.900 --> 00:04:34.270
They are both positive.

00:04:34.270 --> 00:04:38.630
And so our velocity's only
going to become more positive,

00:04:38.630 --> 00:04:41.920
or the magnitude of our velocity
is only going to increase.

00:04:41.920 --> 00:04:44.770
So our speed is increasing.

00:04:44.770 --> 00:04:47.190
If our velocity was

00:04:47.190 --> 00:04:50.480
negative at time t equals three,

00:04:50.480 --> 00:04:55.050
then our speed would be decreasing
because our acceleration

00:04:55.050 --> 00:04:57.650
and velocity would be going
in different directions.


Volume with cross sections: squares and rectangles (no graph)

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.150
- [Instructor] The base of a
solid is the region enclosed

00:00:03.150 --> 00:00:06.190
by the graphs of y is
equal to negative x squared

00:00:06.190 --> 00:00:10.350
plus six x minus one
and y is equal to four.

00:00:10.350 --> 00:00:15.010
Cross sections of the solid
perpendicular to the x-axis

00:00:15.010 --> 00:00:18.320
are rectangles whose height is x.

00:00:18.320 --> 00:00:22.530
Express the volume of the
solid with a definite integral.

00:00:22.530 --> 00:00:25.790
So pause this video, and see
if you can have a go at that.

00:00:25.790 --> 00:00:27.210
All right, now what's
interesting about this

00:00:27.210 --> 00:00:29.260
is they've just given us the
equations for the graphs,

00:00:29.260 --> 00:00:31.140
but we haven't visualized them yet.

00:00:31.140 --> 00:00:32.970
And we need to visualize
them, or at least I like

00:00:32.970 --> 00:00:35.480
to visualize them so I can
think about this region

00:00:35.480 --> 00:00:36.930
that they're talking about.

00:00:36.930 --> 00:00:39.210
So maybe a first thing
to do is think about,

00:00:39.210 --> 00:00:42.200
well, where do these two lines intersect?

00:00:42.200 --> 00:00:44.320
So when do we have the same y-value?

00:00:44.320 --> 00:00:45.420
Or another way to think about it is

00:00:45.420 --> 00:00:48.070
when does this thing equal four?

00:00:48.070 --> 00:00:49.680
So if we set them equal to each other,

00:00:49.680 --> 00:00:51.750
we have negative x squared

00:00:51.750 --> 00:00:55.240
plus six x minus one is equal to four.

00:00:55.240 --> 00:00:56.600
This will give us the x-values

00:00:56.600 --> 00:00:58.160
where these two lines intersect.

00:00:58.160 --> 00:01:02.740
And so we will get, if
we want to solve for x,

00:01:02.740 --> 00:01:04.980
we can subtract four from both sides.

00:01:04.980 --> 00:01:09.980
Negative x squared plus six x
minus five is equal to zero.

00:01:10.720 --> 00:01:13.110
We can multiply both
sides by negative one.

00:01:13.110 --> 00:01:16.200
We will get x squared minus six x

00:01:16.200 --> 00:01:18.660
plus five is equal to zero.

00:01:18.660 --> 00:01:21.460
And then this is pretty
straightforward to factor.

00:01:21.460 --> 00:01:23.310
One times five is five,

00:01:23.310 --> 00:01:26.360
or actually I say negative one
times negative five is five.

00:01:26.360 --> 00:01:29.690
And negative one plus
negative five is negative six.

00:01:29.690 --> 00:01:32.090
So it's going to be x minus one

00:01:32.090 --> 00:01:35.280
times x minus five is equal to zero.

00:01:35.280 --> 00:01:39.140
And so these intersect
when x is equal to one

00:01:39.140 --> 00:01:41.890
or x is equal to five.

00:01:41.890 --> 00:01:43.720
Since we have a negative out front

00:01:43.720 --> 00:01:45.590
of the second-degree term right over here,

00:01:45.590 --> 00:01:49.070
we know it's going to be a
downward-opening parabola.

00:01:49.070 --> 00:01:52.750
And we know that we
intersect y equals four

00:01:52.750 --> 00:01:55.220
when x is equal to one and x equals five.

00:01:55.220 --> 00:01:58.130
And so the vertex must
be right in between them,

00:01:58.130 --> 00:02:00.630
so the vertex is going
to be at x equals three.

00:02:00.630 --> 00:02:03.350
So let's actually visualize
this a little bit.

00:02:03.350 --> 00:02:05.980
So it's going to look something like this.

00:02:05.980 --> 00:02:07.650
I'll draw it with some perspective
'cause we're gonna have

00:02:07.650 --> 00:02:10.120
to think about a three-dimensional shape.

00:02:10.120 --> 00:02:13.220
So that's our y-axis.

00:02:13.220 --> 00:02:14.680
This is

00:02:15.720 --> 00:02:17.450
our

00:02:17.450 --> 00:02:19.120
x-axis.

00:02:19.120 --> 00:02:21.020
And let me draw some y-values.

00:02:21.020 --> 00:02:22.340
So one,

00:02:22.340 --> 00:02:23.470
two,

00:02:23.470 --> 00:02:24.600
three,

00:02:24.600 --> 00:02:26.120
four,

00:02:26.120 --> 00:02:27.400
five,

00:02:27.400 --> 00:02:28.420
six,

00:02:28.420 --> 00:02:30.300
seven, eight.

00:02:30.300 --> 00:02:31.960
This is probably sufficient.

00:02:31.960 --> 00:02:33.930
Now we have y is equal to four,

00:02:33.930 --> 00:02:36.370
which is going to look
something like this.

00:02:36.370 --> 00:02:38.310
So that is

00:02:38.310 --> 00:02:39.883
y is equal to four.

00:02:41.220 --> 00:02:44.060
And then we have y is
equal to negative x squared

00:02:44.060 --> 00:02:47.130
plus six x minus one,
which we know intersects

00:02:47.130 --> 00:02:49.650
y equals four at x equals
one or x equals five.

00:02:49.650 --> 00:02:51.670
So let's see, one,

00:02:51.670 --> 00:02:52.750
two,

00:02:52.750 --> 00:02:53.990
three,

00:02:53.990 --> 00:02:54.960
four,

00:02:54.960 --> 00:02:55.820
five.

00:02:55.820 --> 00:02:59.370
So x equals one, so we have
that point right over there.

00:02:59.370 --> 00:03:01.890
One comma four, and then
we have five comma four.

00:03:01.890 --> 00:03:05.770
And then we know the vertex
is when x is equal to three,

00:03:05.770 --> 00:03:07.100
so it might look something like this.

00:03:07.100 --> 00:03:08.840
We could substitute three back in here.

00:03:08.840 --> 00:03:13.170
So let's see, y will
equal to negative nine,

00:03:13.170 --> 00:03:14.310
three squared,

00:03:14.310 --> 00:03:16.140
plus 18

00:03:16.140 --> 00:03:17.420
minus one.

00:03:17.420 --> 00:03:18.580
And so what is that going to be?

00:03:18.580 --> 00:03:20.750
That's going to be y is
going to be equal to eight.

00:03:20.750 --> 00:03:23.060
So we have the point three comma eight.

00:03:23.060 --> 00:03:26.120
So this is five, six, seven,
eight, yep, right about there.

00:03:26.120 --> 00:03:29.200
And so we are dealing with a situation,

00:03:29.200 --> 00:03:31.390
we're dealing with a situation that looks

00:03:31.390 --> 00:03:33.650
something like this.

00:03:33.650 --> 00:03:36.750
This is the region in question.

00:03:36.750 --> 00:03:38.940
So that's going to be
the base of our solid.

00:03:38.940 --> 00:03:40.960
And they say cross sections of the solid

00:03:40.960 --> 00:03:43.120
perpendicular to the x-axis,

00:03:43.120 --> 00:03:44.770
so let me draw one of
those cross sections.

00:03:44.770 --> 00:03:49.390
So this is a cross section
perpendicular to the x-axis,

00:03:49.390 --> 00:03:52.540
are rectangles, whose height is x,

00:03:52.540 --> 00:03:55.423
so this is going to have
height x right over here,

00:03:56.270 --> 00:03:57.410
height x.

00:03:57.410 --> 00:04:01.200
Now what is this, the
width, I guess we could say,

00:04:01.200 --> 00:04:02.920
of this rectangle?

00:04:02.920 --> 00:04:03.890
Well, it's going to be the difference

00:04:03.890 --> 00:04:05.180
between these two functions.

00:04:05.180 --> 00:04:06.090
It's going to be these,

00:04:06.090 --> 00:04:09.110
this upper function minus
this lower function.

00:04:09.110 --> 00:04:11.930
So it's going to be, that
right over there is going

00:04:11.930 --> 00:04:16.920
to be negative x squared
plus six x minus one

00:04:16.920 --> 00:04:19.680
and then minus four,

00:04:19.680 --> 00:04:21.420
minus the lower function.

00:04:21.420 --> 00:04:24.700
So that could be simplified
as negative x squared

00:04:24.700 --> 00:04:27.400
plus six x minus five.

00:04:27.400 --> 00:04:29.390
And so if we wanna figure out the volume

00:04:29.390 --> 00:04:32.140
of this little section right over here,

00:04:32.140 --> 00:04:35.740
we'd multiply x times this,

00:04:35.740 --> 00:04:37.270
and then we would multiply that

00:04:37.270 --> 00:04:39.400
times an infinitesimal small,

00:04:39.400 --> 00:04:42.780
infinitesimally small depth, dx.

00:04:42.780 --> 00:04:47.030
And then we can just
integrate from x equals one

00:04:47.030 --> 00:04:48.580
to x equals five.

00:04:48.580 --> 00:04:49.500
So let's do that.

00:04:49.500 --> 00:04:52.060
The volume of just this
little slice over here

00:04:52.060 --> 00:04:53.350
is going to be the base,

00:04:53.350 --> 00:04:58.100
which is negative x squared
plus six x minus five,

00:04:58.100 --> 00:05:01.950
times the height, times x,

00:05:01.950 --> 00:05:05.020
times the depth, times dx.

00:05:05.020 --> 00:05:08.580
And then what we wanna do is
we wanna sum up all of these.

00:05:08.580 --> 00:05:10.480
So you could imagine right
over here, you would have,

00:05:10.480 --> 00:05:12.480
or like right over here,

00:05:12.480 --> 00:05:14.160
you would have a cross
section that looks like this.

00:05:14.160 --> 00:05:15.800
X is now much larger.

00:05:15.800 --> 00:05:19.450
The height is x, so now it
looks something like this.

00:05:19.450 --> 00:05:21.120
So I'm just drawing two cross sections,

00:05:21.120 --> 00:05:23.320
just so you get the idea.

00:05:23.320 --> 00:05:24.350
So these are the,

00:05:24.350 --> 00:05:26.950
this is any one cross
section for a given x,

00:05:26.950 --> 00:05:28.350
but now we want to integrate.

00:05:28.350 --> 00:05:32.490
Our x is going from x
equals one to x equals five,

00:05:32.490 --> 00:05:35.450
x equals one to x equals five.

00:05:35.450 --> 00:05:37.700
And there you have it, we
have expressed the volume

00:05:37.700 --> 00:05:40.440
of that solid as a definite integral.

00:05:40.440 --> 00:05:41.740
And it's worth noting that this is,

00:05:41.740 --> 00:05:44.640
that this definite integral,
if you distribute this x,

00:05:44.640 --> 00:05:46.840
if you multiply it by all of these terms,

00:05:46.840 --> 00:05:47.810
it's very solvable.

00:05:47.810 --> 00:05:48.643
You don't need a,

00:05:48.643 --> 00:05:50.120
or it's very solvable
without a calculator.

00:05:50.120 --> 00:05:51.970
You're just going to get
a polynomial over here

00:05:51.970 --> 00:05:53.810
that you have to take
the antiderivative of

00:05:53.810 --> 00:05:56.083
in order to evaluate
the definite integral.


Volume with cross sections perpendicular to y-axis

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.050
- [Instructor] Let R be
the region enclosed by

00:00:03.050 --> 00:00:07.220
y is equal to four times the
square root of nine minus x

00:00:07.220 --> 00:00:09.510
and the axes in the first quadrant.

00:00:09.510 --> 00:00:12.650
And we can see that region
R is gray right over here.

00:00:12.650 --> 00:00:16.120
Region R is the base of a solid.

00:00:16.120 --> 00:00:20.460
For each y-value the cross
section of the solid taken

00:00:20.460 --> 00:00:25.170
perpendicular to the y-axis
is a rectangle whose base

00:00:25.170 --> 00:00:28.250
lies in R and whose height is y.

00:00:28.250 --> 00:00:32.230
Express the volume of the
solid with a definite integral.

00:00:32.230 --> 00:00:34.603
So pause this video and
see if you can do that.

00:00:35.900 --> 00:00:37.590
Alright, now let's do this together.

00:00:37.590 --> 00:00:40.310
And first let's just try
to visualize the solid

00:00:40.310 --> 00:00:41.910
and I'll try to do it by drawing

00:00:43.090 --> 00:00:44.280
this little bit of perspective.

00:00:44.280 --> 00:00:46.250
So if that's our y-axis

00:00:46.250 --> 00:00:51.250
and then this is our
x-axis right over here.

00:00:52.130 --> 00:00:56.650
And I can redraw region R,
looks something like this.

00:00:56.650 --> 00:01:00.840
And now let's just imagine a
cross section of our solid.

00:01:00.840 --> 00:01:02.970
So it says the cross section solid

00:01:02.970 --> 00:01:05.030
taken perpendicular to the y-axis,

00:01:05.030 --> 00:01:06.590
so let's pick a y-value right over here.

00:01:06.590 --> 00:01:09.080
We're gonna go
perpendicular to the y-axis.

00:01:09.080 --> 00:01:12.370
It says whose base lies in R.

00:01:12.370 --> 00:01:14.320
So the base would look like that,

00:01:14.320 --> 00:01:16.570
it would actually be the
x-value that corresponds

00:01:16.570 --> 00:01:18.130
to that particular y-value.

00:01:18.130 --> 00:01:19.920
So I'll just write x right over here.

00:01:19.920 --> 00:01:23.190
And then the height is y.

00:01:23.190 --> 00:01:28.190
So the height is goin'a be
whatever our y-value is.

00:01:28.650 --> 00:01:31.640
And then if we wanted to
calculate the volume of

00:01:31.640 --> 00:01:36.270
just a little bit, a slice that
has an infinitesimal depth,

00:01:36.270 --> 00:01:40.810
we could think about that
infinitesimal depth in terms of y.

00:01:40.810 --> 00:01:45.610
So we could say its depth,
right over here, is d y.

00:01:45.610 --> 00:01:48.150
D y, and we could draw
other cross sections.

00:01:48.150 --> 00:01:51.150
For example, right over
here, our y is much lower,

00:01:51.150 --> 00:01:53.620
it might look some, so our
height will be like that.

00:01:53.620 --> 00:01:58.620
But then our base is the
corresponding x-value

00:01:58.710 --> 00:02:01.660
that sits on the curve
right over that x y pair,

00:02:01.660 --> 00:02:02.990
that would sit on that curve.

00:02:02.990 --> 00:02:06.770
And so this cross section
would look like this.

00:02:06.770 --> 00:02:08.860
And once again, if we wanted to put,

00:02:08.860 --> 00:02:10.120
if we wanted to calculate its volume,

00:02:10.120 --> 00:02:12.855
we could say there's
an infinitesimal volume

00:02:12.855 --> 00:02:14.480
and it would have depth d y.

00:02:14.480 --> 00:02:17.580
And so as we've learned
many times in integration,

00:02:17.580 --> 00:02:19.110
what we wanna do is
think about the volume of

00:02:19.110 --> 00:02:21.770
one of these, I guess
you could say, slices,

00:02:21.770 --> 00:02:24.260
and then integrate across all of them.

00:02:24.260 --> 00:02:25.880
Now there's a couple
of ways to approach it.

00:02:25.880 --> 00:02:28.170
You could try to integrate
with respect to x,

00:02:28.170 --> 00:02:30.630
or you could integrate with respect to y.

00:02:30.630 --> 00:02:32.840
I'm gonna argue it's much
easier to integrate with respect

00:02:32.840 --> 00:02:35.770
to y here 'cause we already
have things in terms of d y.

00:02:35.770 --> 00:02:37.590
The volume of this little slice

00:02:37.590 --> 00:02:42.500
is going to be y times x times d y.

00:02:42.500 --> 00:02:44.850
Now if wanna integrate with respect to y,

00:02:44.850 --> 00:02:46.973
we want everything in terms of y.

00:02:47.892 --> 00:02:51.140
And so what you do is
express x in terms of y.

00:02:51.140 --> 00:02:53.510
So here we just have to solve for x,

00:02:53.510 --> 00:02:55.530
so one way to do this is, let's see,

00:02:55.530 --> 00:02:58.120
we can square both sides of,

00:02:58.120 --> 00:03:00.160
oh, actually let's divide
both sides by four.

00:03:00.160 --> 00:03:03.200
So you get y over four is equal to

00:03:03.200 --> 00:03:05.100
the square root of nine minus x.

00:03:05.100 --> 00:03:06.940
Now we can square both sides.

00:03:06.940 --> 00:03:11.320
Y squared over 16 is
equal to nine minus x.

00:03:11.320 --> 00:03:13.760
And then, let's see, we could

00:03:13.760 --> 00:03:16.190
multiply both sides by negative one.

00:03:16.190 --> 00:03:18.840
So negative y squared over 16

00:03:18.840 --> 00:03:20.640
is equal to x minus nine.

00:03:20.640 --> 00:03:22.540
And now we could add nine to both sides.

00:03:22.540 --> 00:03:27.540
And we get nine minus y
squared over 16 is equal to x.

00:03:27.760 --> 00:03:30.130
And so we could substitute
that right over there.

00:03:30.130 --> 00:03:33.150
So another way to express
the volume of this little

00:03:33.150 --> 00:03:36.080
slice right over here
of infinitesimal depth,

00:03:36.080 --> 00:03:38.180
d y depth, is going to be

00:03:38.180 --> 00:03:43.130
y times nine minus y squared

00:03:43.130 --> 00:03:46.020
over 16 d y.

00:03:46.020 --> 00:03:49.380
And if we wanna find the
volume of the whole figure,

00:03:49.380 --> 00:03:51.490
it's gonna look something like,

00:03:51.490 --> 00:03:55.761
something like that, we're
just goin'a integrate

00:03:55.761 --> 00:03:59.240
from y equals zero to y is equal to 12.

00:03:59.240 --> 00:04:01.770
So integrate from y is equal to zero

00:04:01.770 --> 00:04:03.780
to y is equal to 12.

00:04:03.780 --> 00:04:05.390
And that's all they asked us to do

00:04:05.390 --> 00:04:07.810
to express the volume
as a definite integral,

00:04:07.810 --> 00:04:09.330
but this is actually a definite integral

00:04:09.330 --> 00:04:12.170
that you could solve without a calculator.

00:04:12.170 --> 00:04:16.720
If you multiply both of these terms by y,

00:04:16.720 --> 00:04:19.210
well then you're just goin'a
have a polynomial in terms of y

00:04:19.210 --> 00:04:22.360
and we know how to take
the antiderivative of that

00:04:22.360 --> 00:04:24.260
and then evaluate a definite integral.


Learnstorm Classroom Celebrations

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.833
(students shouting and cheering)


Endosymbiosis theory

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:02.690
- [Instructor] When we look
inside of eukaryotic cells,

00:00:02.690 --> 00:00:05.440
we see membrane-bound organelles,

00:00:05.440 --> 00:00:08.320
and some of these
membrane-bound organelles

00:00:08.320 --> 00:00:10.410
are particularly interesting.

00:00:10.410 --> 00:00:14.420
For example, here is a
diagram of a chloroplast

00:00:14.420 --> 00:00:17.140
that are found in plant or algal cells,

00:00:17.140 --> 00:00:18.310
and we know that this is

00:00:18.310 --> 00:00:20.640
where the photosynthesis takes place,

00:00:20.640 --> 00:00:23.740
but what's really interesting
above and beyond that

00:00:23.740 --> 00:00:26.530
is that it seems that chloroplasts have

00:00:26.530 --> 00:00:28.740
a lot of the machinery necessary

00:00:28.740 --> 00:00:33.030
for being a prokaryotic cell on its own.

00:00:33.030 --> 00:00:37.690
We don't see it acting on its
own, but it has its own DNA.

00:00:37.690 --> 00:00:40.690
It has ribosomes, which
we know are the site

00:00:40.690 --> 00:00:44.490
where we go from messenger RNA to protein.

00:00:44.490 --> 00:00:47.710
Similarly, another interesting
membrane-bound organelle

00:00:47.710 --> 00:00:49.910
that we see in eukaryotic cells,

00:00:49.910 --> 00:00:51.477
and this would include even animal cells

00:00:51.477 --> 00:00:55.250
and the cells in your and
my bodies, are mitochondria,

00:00:55.250 --> 00:00:56.940
and mitochondria are often viewed

00:00:56.940 --> 00:01:00.530
as the energy factories
of eukaryotic cells,

00:01:00.530 --> 00:01:05.320
where we can leverage oxygen
in order to produce ATP,

00:01:05.320 --> 00:01:10.110
and like chloroplasts,
mitochondria has its own DNA.

00:01:10.110 --> 00:01:13.160
It also has mitochondrial ribosomes.

00:01:13.160 --> 00:01:14.530
Here are some just diagrams

00:01:14.530 --> 00:01:16.350
of how mitochondria might look

00:01:16.350 --> 00:01:19.380
inside of a larger eukaryotic cell.

00:01:19.380 --> 00:01:21.930
And so, evolutionary biologists

00:01:21.930 --> 00:01:23.530
for many decades looked at this

00:01:23.530 --> 00:01:25.510
and said, well, why do these things exist?

00:01:25.510 --> 00:01:30.020
Why do they almost look like
prokaryotic cells on their own?

00:01:30.020 --> 00:01:33.690
And there's even examples
of prokaryotic cells,

00:01:33.690 --> 00:01:36.020
independent prokaryotic bacteria,

00:01:36.020 --> 00:01:39.730
that live in symbiosis
inside of other cells,

00:01:39.730 --> 00:01:41.270
and they look an awful lot

00:01:41.270 --> 00:01:43.880
like mitochondria and chloroplasts.

00:01:43.880 --> 00:01:46.800
And so, if we fast-forward to the 1960s,

00:01:46.800 --> 00:01:48.990
someone named Lynn Margulis comes

00:01:48.990 --> 00:01:52.600
on the scene with endosymbiosis theory,

00:01:52.600 --> 00:01:56.810
and her view is, is that these
membrane-bound organelles

00:01:56.810 --> 00:01:58.780
like mitochondria and chloroplasts,

00:01:58.780 --> 00:02:01.500
if we go deep into our evolutionary past,

00:02:01.500 --> 00:02:03.990
say, two and a half billion years ago,

00:02:03.990 --> 00:02:05.540
their ancestors were actually

00:02:05.540 --> 00:02:08.220
independent prokaryotic organisms

00:02:08.220 --> 00:02:12.110
that could produce energy
aerobically, or using oxygen,

00:02:12.110 --> 00:02:14.460
and precursors to what
we would consider today

00:02:14.460 --> 00:02:16.990
to be modern eukaryotic cells

00:02:16.990 --> 00:02:19.910
that might have already had
some membrane-bound structures,

00:02:19.910 --> 00:02:22.850
like a nucleus and
maybe some other things,

00:02:22.850 --> 00:02:26.720
that they could only metabolize
things anaerobically.

00:02:26.720 --> 00:02:28.430
They couldn't leverage oxygen,

00:02:28.430 --> 00:02:31.360
while these other characters
could leverage oxygen,

00:02:31.360 --> 00:02:34.220
and then they could have become symbionts,

00:02:34.220 --> 00:02:36.900
where the one that could leverage oxygen

00:02:36.900 --> 00:02:38.310
to produce more energy

00:02:38.310 --> 00:02:41.830
would get engulfed into the larger cell,

00:02:41.830 --> 00:02:43.230
and that larger cell is able

00:02:43.230 --> 00:02:45.170
to provide nutrients and protection,

00:02:45.170 --> 00:02:47.730
while the smaller cell
that's engulfed inside

00:02:47.730 --> 00:02:51.450
of it is able to better
metabolize the nutrients

00:02:51.450 --> 00:02:54.340
and leverage oxygen to
produce more energy,

00:02:54.340 --> 00:02:57.940
and that over time, this
symbiotic relationship

00:02:57.940 --> 00:03:00.010
became even more connected,

00:03:00.010 --> 00:03:05.010
so that the smaller organism
could not operate by itself,

00:03:05.210 --> 00:03:07.170
that it lost some of its DNA

00:03:07.170 --> 00:03:09.230
that was necessary to act independently,

00:03:09.230 --> 00:03:11.240
and some of it might
have gotten incorporated

00:03:11.240 --> 00:03:13.320
into the DNA of the larger cell.

00:03:13.320 --> 00:03:17.560
And those smaller organisms
are what eventually evolved

00:03:17.560 --> 00:03:20.640
into what we consider
today to be mitochondria.

00:03:20.640 --> 00:03:22.450
This is a fascinating theory,

00:03:22.450 --> 00:03:24.460
and it's actually been proven out.

00:03:24.460 --> 00:03:28.320
When Lynn Margulis first
published this in the late 1960s,

00:03:28.320 --> 00:03:31.770
she wasn't taken that seriously,
but in the decades since,

00:03:31.770 --> 00:03:33.780
it's been validated as we've looked

00:03:33.780 --> 00:03:36.820
at the DNA structures of
mitochondria and chloroplasts,

00:03:36.820 --> 00:03:39.270
that this actually is
the most likely theory

00:03:39.270 --> 00:03:41.610
of how they emerged in our cells.

00:03:41.610 --> 00:03:43.810
And so, it's just a fascinating glimpse

00:03:43.810 --> 00:03:45.850
of evolution in general.

00:03:45.850 --> 00:03:47.900
We talk a lot about natural selection

00:03:47.900 --> 00:03:50.780
and the role of variation and mutations,

00:03:50.780 --> 00:03:54.030
but Lynn Margulis introduces another idea

00:03:54.030 --> 00:03:55.920
that could catalyze evolution,

00:03:55.920 --> 00:03:58.240
and that's that of symbiosis,

00:03:58.240 --> 00:04:01.030
and we see symbiosis
throughout the natural world.

00:04:01.030 --> 00:04:02.320
And her argument is,

00:04:02.320 --> 00:04:05.870
sometimes those symbionts
can become so codependent

00:04:05.870 --> 00:04:08.993
on each other that they
merge into one organism.


Water potential worked example

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.120
- [Instructor] A zucchini
squash was peeled

00:00:02.120 --> 00:00:05.000
and cut into six identical cubes.

00:00:05.000 --> 00:00:07.740
After being weighed, each cubed was soaked

00:00:07.740 --> 00:00:11.260
in a different sucrose
solution for 24 hours

00:00:11.260 --> 00:00:14.480
in an open container and
at a constant temperature

00:00:14.480 --> 00:00:17.460
of 21 degrees Celsius.

00:00:17.460 --> 00:00:21.060
The cubes were then removed
from the sucrose solutions,

00:00:21.060 --> 00:00:24.820
carefully blotted on paper
towels and weighed again.

00:00:24.820 --> 00:00:27.830
The percent change in
mass due to a net gain

00:00:27.830 --> 00:00:31.210
or loss of water was
calculated for each cube

00:00:31.210 --> 00:00:33.640
and the results are
shown in the graph below.

00:00:33.640 --> 00:00:35.960
I put the graph to the
right right over here.

00:00:35.960 --> 00:00:38.420
A straight line is drawn on the graph

00:00:38.420 --> 00:00:41.070
to help in estimating results from other

00:00:41.070 --> 00:00:43.650
sucrose concentrations not tested.

00:00:43.650 --> 00:00:46.470
Using the straight line
on the graph above,

00:00:46.470 --> 00:00:49.260
calculate the water potential in bars

00:00:49.260 --> 00:00:52.630
of the zucchini squash
at 21 degrees Celsius.

00:00:52.630 --> 00:00:55.790
Give your answer to one decimal place.

00:00:55.790 --> 00:00:59.170
So pause this video and see
if you can have a go at it.

00:00:59.170 --> 00:01:01.420
Alright, now let's do this together.

00:01:01.420 --> 00:01:04.600
So we have our six
cubes of zucchini squash

00:01:04.600 --> 00:01:06.950
and we have each of these data points

00:01:06.950 --> 00:01:10.310
are the data for each of those cubes.

00:01:10.310 --> 00:01:13.780
So for example, this
data point shows a cube

00:01:13.780 --> 00:01:15.730
that was put into a solution that contains

00:01:15.730 --> 00:01:18.590
no sucrose and when it
was put into a solution

00:01:18.590 --> 00:01:20.770
that contains no sucrose,

00:01:20.770 --> 00:01:24.240
it gained 20% mass and that mass would be

00:01:24.240 --> 00:01:26.630
from the water going into the cube.

00:01:26.630 --> 00:01:28.490
So one way to think about it is

00:01:28.490 --> 00:01:30.640
it got put into a solution,

00:01:30.640 --> 00:01:34.580
and so you had a net inflow of water

00:01:34.580 --> 00:01:39.570
into that sucrose cube, enough
to increase its mass by 20%.

00:01:39.570 --> 00:01:42.450
This second data point right over here,

00:01:42.450 --> 00:01:44.795
it was put into a solution that is

00:01:44.795 --> 00:01:49.640
.20 moles per liter of sucrose

00:01:49.640 --> 00:01:52.290
and that also had an increase of mass

00:01:52.290 --> 00:01:55.400
but it was a smaller increase of mass.

00:01:55.400 --> 00:01:57.430
And then we can go all
the way to the extreme

00:01:57.430 --> 00:02:02.430
right over here where
we had a zucchini cube

00:02:03.010 --> 00:02:06.290
put into a solution that
had a lot of sucrose in it,

00:02:06.290 --> 00:02:08.440
one mole per liter, and there,

00:02:08.440 --> 00:02:10.940
you have a net outflow of water.

00:02:10.940 --> 00:02:14.380
It deceased the cube's mass by 30%.

00:02:14.380 --> 00:02:17.870
So how do we figure
out the water potential

00:02:17.870 --> 00:02:19.870
of the zucchini squash?

00:02:19.870 --> 00:02:21.560
Well one way to think about it is

00:02:21.560 --> 00:02:24.140
we could figure out the water
potential of zucchini squash

00:02:24.140 --> 00:02:26.190
is if we put it into a solution

00:02:26.190 --> 00:02:28.660
that has a certain sucrose concentration

00:02:28.660 --> 00:02:32.880
and if we have no net
inflow or outflow of water

00:02:32.880 --> 00:02:35.680
in either direction, that
means that the zucchini squash

00:02:35.680 --> 00:02:38.890
and the sucrose solution
have the same water potential

00:02:38.890 --> 00:02:40.400
and we see that happening,

00:02:40.400 --> 00:02:43.220
at least we can interpellate
it right over here

00:02:43.220 --> 00:02:45.130
because if we just follow that line,

00:02:45.130 --> 00:02:49.840
it looks like if we were
to put our zucchini cubes,

00:02:49.840 --> 00:02:53.740
our zucchini squash cubes
into a sucrose solution

00:02:53.740 --> 00:02:56.890
that is, this looks about, let's see.

00:02:56.890 --> 00:02:59.500
This would be .3, so let's just call that

00:02:59.500 --> 00:03:04.500
0.36 molar solution, 0.36 molar solution.

00:03:07.240 --> 00:03:09.380
Well, then it has the same water potential

00:03:09.380 --> 00:03:11.810
as that solution because
you have no net inflow

00:03:11.810 --> 00:03:13.400
or outflow of water.

00:03:13.400 --> 00:03:15.720
So the zucchini squash water potential

00:03:15.720 --> 00:03:17.940
is going to be the same
as the water potential

00:03:17.940 --> 00:03:22.940
of a solution of sucrose
that has 0.36 moles per liter

00:03:23.600 --> 00:03:26.470
so we just have to figure out
the water potential of that.

00:03:26.470 --> 00:03:28.590
Well, the water potential formula we know

00:03:28.590 --> 00:03:30.140
and we've seen this in other videos

00:03:30.140 --> 00:03:32.540
is equal to our solute potential

00:03:32.540 --> 00:03:34.740
plus our pressure potential.

00:03:34.740 --> 00:03:36.560
We know that these are open containers,

00:03:36.560 --> 00:03:39.200
so the pressure potential is going to be 0

00:03:39.200 --> 00:03:41.470
and so we just need to figure
out the solute potential

00:03:41.470 --> 00:03:44.780
of our sucrose solution
with this concentration.

00:03:44.780 --> 00:03:47.140
Well, we know our solute potential

00:03:47.140 --> 00:03:50.300
is going to be equal to negative I,

00:03:50.300 --> 00:03:52.430
where this is the ionization constant.

00:03:52.430 --> 00:03:54.130
For every molecule of sucrose,

00:03:54.130 --> 00:03:57.100
does it disassociate into
when it gets put into water?

00:03:57.100 --> 00:03:58.730
Well, sucrose doesn't disassociate it,

00:03:58.730 --> 00:04:03.420
so it would just be one times our molarity

00:04:03.420 --> 00:04:06.740
times what's sometimes
called a pressure constant

00:04:06.740 --> 00:04:07.620
but it's really the same thing as

00:04:07.620 --> 00:04:10.510
a universal gas constant you
might have learned in chemistry

00:04:10.510 --> 00:04:13.050
times the temperature in Kelvin.

00:04:13.050 --> 00:04:15.350
So this is going to be equal to negative,

00:04:15.350 --> 00:04:16.950
I already said that this I is one

00:04:16.950 --> 00:04:18.930
because the sucrose just stays sucrose

00:04:18.930 --> 00:04:21.260
when you put it in water, so negative one

00:04:21.260 --> 00:04:26.260
times 0.36, I'll write
the units, moles per liter

00:04:30.440 --> 00:04:33.570
times R and if you're doing
something like an AP test,

00:04:33.570 --> 00:04:34.810
they actually give you these formulas

00:04:34.810 --> 00:04:39.317
and they even give you R
as being 0.0831 liter bars

00:04:44.540 --> 00:04:49.540
per mole Kelvin and then times
the temperature in Kelvin.

00:04:49.650 --> 00:04:51.670
Well if we're dealing
with 21 degrees Celsius,

00:04:51.670 --> 00:04:56.450
we just add 273 to that,
so times 294 Kelvin

00:04:56.450 --> 00:04:58.350
and now we can just
get our calculator out.

00:04:58.350 --> 00:05:03.350
Negative one times .36
times .0831 times 294

00:05:11.520 --> 00:05:16.070
is equal to if I round
to one decimal place,

00:05:16.070 --> 00:05:17.883
it's gonna be negative 8.8.

00:05:18.780 --> 00:05:22.240
Negative 8.8 and the
units work out as well.

00:05:22.240 --> 00:05:24.640
Moles cancel with moles,
liters cancel with liters,

00:05:24.640 --> 00:05:26.538
Kelvin cancels with Kelvin,
and we're just with,

00:05:26.538 --> 00:05:30.640
left with negative 8.8 bars.

00:05:30.640 --> 00:05:31.683
And we are done.


Water potential

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:02.440
- [Instructor] So right here,
I have a container of water

00:00:02.440 --> 00:00:04.110
that is open to the atmosphere.

00:00:04.110 --> 00:00:06.090
It's standard atmospheric
pressure up here.

00:00:06.090 --> 00:00:09.120
And let's just assume that
everything in our system,

00:00:09.120 --> 00:00:10.970
the air and the water, the container,

00:00:10.970 --> 00:00:13.920
everything is 21 degrees Celsius.

00:00:13.920 --> 00:00:15.740
Now, our chamber is,

00:00:15.740 --> 00:00:19.900
or our container of water is
divided into two chambers,

00:00:19.900 --> 00:00:21.510
a left chamber and a right chamber,

00:00:21.510 --> 00:00:24.080
which is divided by a
semipermeable membrane.

00:00:24.080 --> 00:00:25.850
It's permeable only to the water,

00:00:25.850 --> 00:00:27.830
but not to the solute in the water.

00:00:27.830 --> 00:00:32.140
And on the left-hand
side, we have pure water.

00:00:32.140 --> 00:00:35.120
So it's just molecules of H2O.

00:00:35.120 --> 00:00:36.670
You could view that as distilled water.

00:00:36.670 --> 00:00:38.230
While on the right-hand side,

00:00:38.230 --> 00:00:39.690
we don't have pure water.

00:00:39.690 --> 00:00:41.310
We have some sodium chloride

00:00:41.310 --> 00:00:42.610
that has been dissolved in the water.

00:00:42.610 --> 00:00:44.950
And we want to get a little
quantitative in this video,

00:00:44.950 --> 00:00:48.720
so let's say it is 0.25

00:00:48.720 --> 00:00:53.600
moles per liter or 0.25 molar

00:00:53.600 --> 00:00:55.600
sodium chloride right over here.

00:00:55.600 --> 00:00:58.420
Well, we know intuitively,
from our study of osmosis,

00:00:58.420 --> 00:01:00.190
what is going to happen here.

00:01:00.190 --> 00:01:03.340
Since our membrane is
only permeable to water

00:01:03.340 --> 00:01:07.580
and the right side is
hypertonic relative to the left,

00:01:07.580 --> 00:01:10.100
it has a higher concentration of solute,

00:01:10.100 --> 00:01:14.710
we will have the water
going from left to right.

00:01:14.710 --> 00:01:16.240
But what we're going to do in this video

00:01:16.240 --> 00:01:19.350
is get a little bit more
quantitative in thinking about,

00:01:19.350 --> 00:01:21.540
well, what is the potential of the water

00:01:21.540 --> 00:01:24.400
to go from the left to the right?

00:01:24.400 --> 00:01:28.170
And to do that, we're gonna
talk about something known as

00:01:28.170 --> 00:01:29.120
water

00:01:30.380 --> 00:01:31.940
potential,

00:01:31.940 --> 00:01:35.213
which is denoted by the Greek letter psi.

00:01:36.410 --> 00:01:39.610
Now, by convention, the water potential

00:01:39.610 --> 00:01:43.870
of the pure distilled water
when it is in an open container

00:01:43.870 --> 00:01:46.720
exposed to standard atmospheric pressure,

00:01:46.720 --> 00:01:48.070
that is going to be,

00:01:48.070 --> 00:01:52.310
so psi over here is going
to be equal to zero bars.

00:01:52.310 --> 00:01:55.790
And as you can see
here, the units for psi,

00:01:55.790 --> 00:01:59.160
for water potential, are the
same as the units for pressure.

00:01:59.160 --> 00:02:01.880
Here, we're using bars,
but you could use PSI

00:02:01.880 --> 00:02:04.730
or atmospheres or something else.

00:02:04.730 --> 00:02:06.650
But what is going to
be the water potential

00:02:06.650 --> 00:02:07.920
on the right-hand side?

00:02:07.920 --> 00:02:10.810
Well, generally speaking,
you can break down

00:02:10.810 --> 00:02:13.690
your water potential into
subcomponents, and the what,

00:02:13.690 --> 00:02:15.600
the subcomponents that
we typically talk about,

00:02:15.600 --> 00:02:19.330
but there are more, are the
component due to the solute

00:02:19.330 --> 00:02:23.130
and the component due to pressure.

00:02:23.130 --> 00:02:25.410
This'll often be known
as the solute potential

00:02:25.410 --> 00:02:27.510
and the pressure potential.

00:02:27.510 --> 00:02:31.150
Now, because this right chamber
is open to the atmosphere

00:02:31.150 --> 00:02:33.390
and it's standard atmospheric pressure,

00:02:33.390 --> 00:02:36.430
the pressure potential in this situation

00:02:36.430 --> 00:02:38.630
is going to be equal to zero.

00:02:38.630 --> 00:02:40.940
So our total water potential
on the right-hand side

00:02:40.940 --> 00:02:44.330
is going to be the same thing
as our solute potential.

00:02:44.330 --> 00:02:45.920
Now, to figure out our solute potential,

00:02:45.920 --> 00:02:47.620
we will turn to a formula,

00:02:47.620 --> 00:02:49.640
and I'll show you a formula
sheet right over here,

00:02:49.640 --> 00:02:51.130
that's given by the College Board

00:02:51.130 --> 00:02:53.780
when folks take the AP Biology Exam.

00:02:53.780 --> 00:02:57.280
But we'll see that this formula
is actually quite intuitive

00:02:57.280 --> 00:02:59.850
and related to the ideal gas law

00:02:59.850 --> 00:03:02.260
that you first studied in chemistry.

00:03:02.260 --> 00:03:05.250
It tells us that

00:03:05.250 --> 00:03:10.070
our solute potential
is going to be equal to

00:03:10.070 --> 00:03:13.700
negative i, and I'll talk
about what that is in a second,

00:03:13.700 --> 00:03:17.720
times our molar concentration C

00:03:17.720 --> 00:03:21.000
times our pressure constant R.

00:03:21.000 --> 00:03:22.780
Now, they're calling it
the pressure constant

00:03:22.780 --> 00:03:25.510
in this context, but
it's actually the same R

00:03:25.510 --> 00:03:27.520
that we see in the ideal gas law,

00:03:27.520 --> 00:03:32.520
times the temperature
in Kelvin, so times T.

00:03:32.930 --> 00:03:35.840
So first, let's start with
i, so let me write this down.

00:03:35.840 --> 00:03:37.760
This is going to be equal to,

00:03:37.760 --> 00:03:39.440
it's all going to be negative

00:03:39.440 --> 00:03:42.440
because all of these are
going to be positive values.

00:03:42.440 --> 00:03:45.240
So i is known as the ionization constant.

00:03:45.240 --> 00:03:46.610
And one way to think about it is,

00:03:46.610 --> 00:03:48.720
for every molecule of solute,

00:03:48.720 --> 00:03:52.410
how many molecules does
it disassociate into

00:03:52.410 --> 00:03:54.080
when you put it into the water?

00:03:54.080 --> 00:03:56.110
So if I take a molecule
of sodium chloride,

00:03:56.110 --> 00:03:58.040
we know that when we put it into water,

00:03:58.040 --> 00:03:59.220
it's going to disassociate

00:03:59.220 --> 00:04:01.560
into sodium ions and chloride ions,

00:04:01.560 --> 00:04:06.560
or into two ions for every one
molecule of sodium chloride.

00:04:06.610 --> 00:04:08.460
So here, our ionization constant

00:04:08.460 --> 00:04:10.970
is going to be equal to two.

00:04:10.970 --> 00:04:13.140
Now, what's our molar concentration?

00:04:13.140 --> 00:04:15.120
Well, we already know that.

00:04:15.120 --> 00:04:18.840
Our molar concentration
is going to be 0.25 molar,

00:04:18.840 --> 00:04:21.980
or we could write it as 0.25

00:04:21.980 --> 00:04:25.500
moles per liter.

00:04:25.500 --> 00:04:27.410
And then we're gonna multiply that

00:04:27.410 --> 00:04:29.810
times our pressure constant,

00:04:29.810 --> 00:04:32.020
which is the same as
the ideal gas constant.

00:04:32.020 --> 00:04:33.760
And that is going to be,

00:04:33.760 --> 00:04:35.860
they give it to us right over here, 0.0831

00:04:38.340 --> 00:04:41.500
liter bars,

00:04:41.500 --> 00:04:44.200
liter bars per mole

00:04:45.120 --> 00:04:46.140
Kelvin.

00:04:46.140 --> 00:04:49.460
And then we are going
to multiply that times

00:04:49.460 --> 00:04:52.130
the temperature in Kelvin.

00:04:52.130 --> 00:04:53.440
Now, what's that going to be?

00:04:53.440 --> 00:04:56.390
Well, it's going to be our
temperature in Celsius plus 273.

00:04:56.390 --> 00:04:58.360
So you add 21 plus 273,

00:04:58.360 --> 00:05:02.670
you're gonna get 294 Kelvin.

00:05:02.670 --> 00:05:06.700
And so this is going to
be approximately equal to,

00:05:06.700 --> 00:05:09.280
we're going to have
this negative out front,

00:05:09.280 --> 00:05:11.500
and if we look at our units,

00:05:11.500 --> 00:05:14.090
let's see, we have moles
canceling with moles,

00:05:14.090 --> 00:05:15.630
liters canceling with liters,

00:05:15.630 --> 00:05:18.270
and then Kelvin canceling with Kelvin.

00:05:18.270 --> 00:05:21.360
So our units are going to be
in bars, which makes sense

00:05:21.360 --> 00:05:22.920
because that's what we
already talked about.

00:05:22.920 --> 00:05:25.330
That's the units that
we're dealing with here.

00:05:25.330 --> 00:05:27.280
And so let me get a calculator out.

00:05:27.280 --> 00:05:28.360
See, we already have the negative.

00:05:28.360 --> 00:05:30.610
So it's gonna be two times .25

00:05:32.170 --> 00:05:33.110
times .0831

00:05:35.720 --> 00:05:38.150
times 294

00:05:38.150 --> 00:05:41.070
is equal to, and I'll just
round to one decimal place.

00:05:41.070 --> 00:05:42.970
So 12.2, but I have that negative there,

00:05:42.970 --> 00:05:47.130
so approximately negative 12.2 bars.

00:05:47.130 --> 00:05:51.660
So our solute potential over
here is negative 12.2 bars,

00:05:51.660 --> 00:05:53.590
or it's approximately equal to that.

00:05:53.590 --> 00:05:56.500
And that's going to be equal
to our total water potential

00:05:56.500 --> 00:05:58.470
because the pressure potential is zero.

00:05:58.470 --> 00:05:59.940
So how do we interpret that?

00:05:59.940 --> 00:06:01.210
Well, one way to think about it is

00:06:01.210 --> 00:06:03.180
you can almost view it
as, since it's negative,

00:06:03.180 --> 00:06:04.930
relative to the left-hand side,

00:06:04.930 --> 00:06:07.950
that it would pull water
in, that the water,

00:06:07.950 --> 00:06:10.750
if the membrane is only
permeable to the water,

00:06:10.750 --> 00:06:14.300
is going to go from high
potential to low potential.

00:06:14.300 --> 00:06:18.380
And that's exactly what we
thought would happen intuitively.

00:06:18.380 --> 00:06:20.780
Now, another thing that
we could think about is,

00:06:20.780 --> 00:06:23.740
well, how would we keep
the water from flowing?

00:06:23.740 --> 00:06:25.370
Well, then we could add some

00:06:25.370 --> 00:06:27.280
pressure potential right over here.

00:06:27.280 --> 00:06:30.650
Maybe we could put a piston over here,

00:06:30.650 --> 00:06:33.410
and we could apply some pressure

00:06:33.410 --> 00:06:36.490
above and beyond the standard
atmospheric pressure.

00:06:36.490 --> 00:06:39.000
Well, how much pressure
would we have to apply here

00:06:39.000 --> 00:06:41.790
to kinda push down on that
water so that we don't have

00:06:41.790 --> 00:06:44.790
a net inflow from the left to the right?

00:06:44.790 --> 00:06:48.490
Well, imagine if this was 12.2,

00:06:48.490 --> 00:06:52.390
12.2 bars that we were to apply up here.

00:06:52.390 --> 00:06:53.600
Well, then all of a sudden,

00:06:53.600 --> 00:06:56.220
this pressure potential wouldn't be zero.

00:06:56.220 --> 00:06:58.370
It would be positive 12.2.

00:06:58.370 --> 00:06:59.400
And then on the right,

00:06:59.400 --> 00:07:03.400
we would have negative 12.2
bars of solute potential

00:07:03.400 --> 00:07:07.160
plus positive 12.2 bars
of pressure potential.

00:07:07.160 --> 00:07:10.730
And so that would add up to
be a zero water potential,

00:07:10.730 --> 00:07:12.400
in which case, since the left

00:07:12.400 --> 00:07:15.010
and the right-hand side would
have the same water potential,

00:07:15.010 --> 00:07:19.010
you would not have a net flow
of water in either direction.

00:07:19.010 --> 00:07:21.330
So as you can see, this
idea of water potential

00:07:21.330 --> 00:07:23.610
allows us to think a
little bit more precisely

00:07:23.610 --> 00:07:25.410
and a little bit more quantitatively

00:07:25.410 --> 00:07:28.883
about these ideas of osmosis and tonicity.


Social consequences of revolutionary ideals

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:02.470
- [Instructor] During
the American Revolution,

00:00:02.470 --> 00:00:05.730
everyone became a little
bit of a philosopher.

00:00:05.730 --> 00:00:07.480
Walking down the street in Boston

00:00:07.480 --> 00:00:10.630
past coffee houses and taverns,

00:00:10.630 --> 00:00:13.720
you might hear ordinary
people debating equality

00:00:13.720 --> 00:00:15.420
and natural rights.

00:00:15.420 --> 00:00:18.300
Before it was even a political revolution,

00:00:18.300 --> 00:00:22.250
the American Revolution
was a revolution of ideas.

00:00:22.250 --> 00:00:24.670
You see these ideas
all over the literature

00:00:24.670 --> 00:00:26.020
of the time period,

00:00:26.020 --> 00:00:29.570
perhaps most famously in
Thomas Jefferson's language

00:00:29.570 --> 00:00:33.630
in the Declaration that
all men are created equal,

00:00:33.630 --> 00:00:36.900
endowed with certain unalienable rights,

00:00:36.900 --> 00:00:39.820
the idea that everyone
has the natural right

00:00:39.820 --> 00:00:43.080
to life, liberty, and
the pursuit of happiness.

00:00:43.080 --> 00:00:45.520
You would see it in the most famous slogan

00:00:45.520 --> 00:00:49.660
of the revolution, no taxation
without representation,

00:00:49.660 --> 00:00:52.380
the idea that the people would have a say

00:00:52.380 --> 00:00:54.210
in the laws that affect them.

00:00:54.210 --> 00:00:55.740
The revolution even went so far

00:00:55.740 --> 00:00:58.450
as to criticize the idea of monarchy,

00:00:58.450 --> 00:01:01.860
which pamphleteer Thomas
Paine called absurd.

00:01:01.860 --> 00:01:06.520
These ideas criticized
long-standing social norms

00:01:06.520 --> 00:01:09.370
about who deserved to rule

00:01:09.370 --> 00:01:12.940
versus who deserved to be ruled.

00:01:12.940 --> 00:01:15.410
It's likely that most of the founders

00:01:15.410 --> 00:01:18.370
thought these ideas primarily applied

00:01:18.370 --> 00:01:19.590
to the political struggle

00:01:19.590 --> 00:01:22.090
between the colonies and Great Britain,

00:01:22.090 --> 00:01:24.780
but it wasn't just elite white men

00:01:24.780 --> 00:01:27.670
who considered these radical notions

00:01:27.670 --> 00:01:31.440
and thought about how to
apply them to their own lives.

00:01:31.440 --> 00:01:35.850
These revolutionary ideals
increased the awareness

00:01:35.850 --> 00:01:38.910
of inequalities in society more broadly

00:01:38.910 --> 00:01:41.580
and caused some people to call for changes

00:01:41.580 --> 00:01:45.570
in voting rights, in the
institution of slavery,

00:01:45.570 --> 00:01:48.040
and the status of women.

00:01:48.040 --> 00:01:52.110
One major change that the
revolutionary ideals caused

00:01:52.110 --> 00:01:56.200
was an expansion of political democracy

00:01:56.200 --> 00:01:57.790
in state governments.

00:01:57.790 --> 00:01:59.390
So after the revolution,

00:01:59.390 --> 00:02:02.720
all the states wrote new constitutions.

00:02:02.720 --> 00:02:04.300
And every one of them

00:02:04.300 --> 00:02:07.650
instituted a representative government

00:02:07.650 --> 00:02:09.830
through an elected legislature.

00:02:09.830 --> 00:02:12.910
And almost all of them also reduced

00:02:12.910 --> 00:02:16.870
the property requirements for
voting and office-holding,

00:02:16.870 --> 00:02:19.500
which previously had limited the franchise

00:02:19.500 --> 00:02:21.620
to wealthier white men.

00:02:21.620 --> 00:02:23.150
After the revolution,

00:02:23.150 --> 00:02:26.220
a majority of white men
had the right to vote.

00:02:26.220 --> 00:02:29.200
And voting rights would continue to expand

00:02:29.200 --> 00:02:33.883
until all white men had the
right to vote by the 1830s.

00:02:34.730 --> 00:02:38.740
In fact, some state
constitutions, like New Jersey's,

00:02:38.740 --> 00:02:41.530
didn't specify who could vote,

00:02:41.530 --> 00:02:45.390
so long as they met the new
lower property requirements.

00:02:45.390 --> 00:02:48.240
So between the revolution and 1807

00:02:48.240 --> 00:02:49.480
when they changed the law,

00:02:49.480 --> 00:02:53.050
property-owning women and free
people of color could vote.

00:02:53.050 --> 00:02:55.200
Another social change that was brought on

00:02:55.200 --> 00:02:58.760
by these revolutionary
ideals was the emergence

00:02:58.760 --> 00:03:03.330
of the abolition movement to
end the practice of slavery.

00:03:03.330 --> 00:03:07.150
So almost all of the
founders were slave owners,

00:03:07.150 --> 00:03:09.770
and they didn't seem to
notice any contradiction

00:03:09.770 --> 00:03:12.590
between the idea that
all men are created equal

00:03:12.590 --> 00:03:15.460
and keeping Africans in perpetual bondage.

00:03:15.460 --> 00:03:18.620
But that did not mean that
others missed the connection.

00:03:18.620 --> 00:03:20.480
During the revolution itself,

00:03:20.480 --> 00:03:23.270
many enslaved people escaped,

00:03:23.270 --> 00:03:25.890
filed petitions for freedom,

00:03:25.890 --> 00:03:29.300
or they joined the
military to gain freedom

00:03:29.300 --> 00:03:31.260
on the side of the Americans,

00:03:31.260 --> 00:03:33.870
but more frequently on
the side of the British

00:03:33.870 --> 00:03:37.840
who offered freedom in exchange
for serving in the army.

00:03:37.840 --> 00:03:39.700
After the revolution,

00:03:39.700 --> 00:03:42.960
northern states either abolished slavery,

00:03:42.960 --> 00:03:47.110
or they began a process
of gradual emancipation,

00:03:47.110 --> 00:03:50.070
so saying that enslaved people

00:03:50.070 --> 00:03:52.880
who were children might remain in slavery

00:03:52.880 --> 00:03:55.080
until they were in their mid to late 20s

00:03:56.140 --> 00:03:59.680
and then would be free, and
their children would be free,

00:03:59.680 --> 00:04:02.870
so that over the course of the late 1800s

00:04:02.870 --> 00:04:07.090
and early 1900s, in most northern states,

00:04:07.090 --> 00:04:11.750
slavery was either entirely eliminated

00:04:11.750 --> 00:04:13.750
or phased out to the point

00:04:13.750 --> 00:04:16.670
that there was a very
small enslaved population.

00:04:16.670 --> 00:04:19.990
Southern states did not abolish slavery

00:04:19.990 --> 00:04:21.910
in response to the revolution,

00:04:21.910 --> 00:04:26.610
and this would continue to
expand sectional tensions

00:04:26.610 --> 00:04:29.870
between the north and the
south until the Civil War.

00:04:29.870 --> 00:04:32.260
The last change in social values

00:04:32.260 --> 00:04:36.330
that I wanna discuss relates
to women's role in the war.

00:04:36.330 --> 00:04:38.840
Women played a pretty crucial role

00:04:38.840 --> 00:04:41.270
in supporting the independence movement,

00:04:41.270 --> 00:04:43.870
and they also drew on the rhetoric

00:04:43.870 --> 00:04:46.020
of revolutionary ideas

00:04:46.020 --> 00:04:49.700
to support their claims for
an improvement in status.

00:04:49.700 --> 00:04:54.160
Abigail Adams, who was
the wife of John Adams,

00:04:54.160 --> 00:04:57.080
wrote to him while he was
in Philadelphia working

00:04:57.080 --> 00:04:59.710
on the draft of the
Declaration of Independence

00:04:59.710 --> 00:05:02.140
with Thomas Jefferson and others.

00:05:02.140 --> 00:05:04.730
She wrote, in the new Code of Laws

00:05:04.730 --> 00:05:08.070
which I suppose it will be
necessary for you to make,

00:05:08.070 --> 00:05:10.820
I desire you would remember the ladies

00:05:10.820 --> 00:05:12.760
and be more generous and favorable

00:05:12.760 --> 00:05:14.780
to them than your ancestors.

00:05:14.780 --> 00:05:17.440
If particular care and
attention is not paid

00:05:17.440 --> 00:05:20.650
to the ladies, we are
determined to foment a rebellion

00:05:20.650 --> 00:05:22.210
and will not hold ourselves bound

00:05:22.210 --> 00:05:25.890
by any laws in which we have
no voice or representation.

00:05:25.890 --> 00:05:28.480
In general, the revolution didn't result

00:05:28.480 --> 00:05:32.250
in a huge change of status for white women

00:05:32.250 --> 00:05:34.730
and none at all for enslaved women,

00:05:34.730 --> 00:05:39.500
but one idea did emerge that
would foment social change

00:05:39.500 --> 00:05:41.070
farther down the line,

00:05:41.070 --> 00:05:44.420
and that was the idea of
republican motherhood.

00:05:44.420 --> 00:05:47.220
So republican motherhood was the notion

00:05:47.220 --> 00:05:52.220
that for this new democratic
American nation to work,

00:05:52.270 --> 00:05:55.560
the country was going to
need virtuous citizens.

00:05:55.560 --> 00:05:59.280
And who taught men how
to be virtuous citizens?

00:05:59.280 --> 00:06:00.370
Their mothers.

00:06:00.370 --> 00:06:02.500
So thinkers of the time period,

00:06:02.500 --> 00:06:04.990
particularly a Philadelphia physician

00:06:04.990 --> 00:06:07.560
and singer of the
Declaration of Independence

00:06:07.560 --> 00:06:09.290
named Benjamin Rush,

00:06:09.290 --> 00:06:14.130
argued that women should
receive more robust education

00:06:14.130 --> 00:06:16.460
in order to better educate their sons.

00:06:16.460 --> 00:06:19.960
So instead of only learning
household skills or etiquette,

00:06:19.960 --> 00:06:22.900
women should learn
philosophy and mathematics.

00:06:22.900 --> 00:06:26.840
Now, while this concept
was mainly in the service

00:06:26.840 --> 00:06:31.080
of improving the education
and virtue of men,

00:06:31.080 --> 00:06:34.330
it did result in the
expansion of women's education

00:06:34.330 --> 00:06:38.090
and the founding of new
schools and colleges for girls.

00:06:38.090 --> 00:06:41.140
And many of the girls who
attended those schools would go on

00:06:41.140 --> 00:06:43.850
to be major reformers and activists

00:06:43.850 --> 00:06:46.690
in the women's rights and
abolitionist movements

00:06:46.690 --> 00:06:48.140
in the 19th century.

00:06:48.140 --> 00:06:51.060
I wanna finish by just
briefly taking a look

00:06:51.060 --> 00:06:54.300
at John Adams' response
to his wife's letter

00:06:54.300 --> 00:06:56.700
encouraging hin to remember the ladies,

00:06:56.700 --> 00:06:59.510
which is not nearly as
famous as her letter,

00:06:59.510 --> 00:07:01.640
but I think still tells us a lot

00:07:01.640 --> 00:07:04.310
about the ideas of the time period.

00:07:04.310 --> 00:07:07.930
He said, as to your
extraordinary Code of Laws,

00:07:07.930 --> 00:07:09.610
I cannot but laugh.

00:07:09.610 --> 00:07:12.450
We have been told that our
struggle has loosened the bands

00:07:12.450 --> 00:07:13.720
of government everywhere,

00:07:13.720 --> 00:07:16.600
that children and
apprentices were disobedient,

00:07:16.600 --> 00:07:19.300
that schools and colleges
were grown turbulent,

00:07:19.300 --> 00:07:21.180
that Indians slighted their guardians

00:07:21.180 --> 00:07:24.080
and Negroes grew insolent
to their masters,

00:07:24.080 --> 00:07:26.300
but your letter was the first intimation

00:07:26.300 --> 00:07:29.040
that another tribe more
numerous and powerful

00:07:29.040 --> 00:07:31.440
than all the rest were grown discontented.

00:07:31.440 --> 00:07:33.730
And there at the end
he's referring to women.

00:07:33.730 --> 00:07:37.270
You can tell that John
kinda takes this as a joke.

00:07:37.270 --> 00:07:39.640
He can't imagine that the ideas

00:07:39.640 --> 00:07:43.350
that he and the other
founders used so persuasively

00:07:43.350 --> 00:07:45.980
in the American Revolution
would lead women

00:07:45.980 --> 00:07:47.490
and African-Americans

00:07:47.490 --> 00:07:50.680
and working class people like apprentices

00:07:50.680 --> 00:07:53.460
to apply those same notions to themselves.

00:07:53.460 --> 00:07:57.070
But perhaps the joke was
on John Adams after all,

00:07:57.070 --> 00:07:59.153
because that is exactly what they did.