Volume flow rate and equation of continuity

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:05.010
Everything we've done so far has
been stationary fluids, or

00:00:05.010 --> 00:00:08.440
static fluids, and we've been
dealing with static pressure.

00:00:08.440 --> 00:00:10.760
We were trying to figure out
what happens when everything's

00:00:10.760 --> 00:00:11.770
in a steady state.

00:00:11.770 --> 00:00:13.460
Now let's work on what
happens when the

00:00:13.460 --> 00:00:14.860
fluid is actually moving.

00:00:14.860 --> 00:00:16.260
Let's imagine a pipe.

00:00:20.530 --> 00:00:23.500
Let's say one end of the pipe
has a larger area than the

00:00:23.500 --> 00:00:25.260
other end, or at least
a different area.

00:00:28.690 --> 00:00:31.830
So this is one end of the pipe,
and this is the other

00:00:31.830 --> 00:00:33.080
end of a pipe.

00:00:36.480 --> 00:00:39.600
It's filled with some fluid,
some liquid, actually, in our

00:00:39.600 --> 00:00:46.550
example, so there's just a bunch
of liquid in this fluid.

00:00:50.910 --> 00:00:55.980
Let's say this area
at the entrance is

00:00:55.980 --> 00:00:59.620
called the area in.

00:00:59.620 --> 00:01:05.349
That's the area of the opening
into the pipe, and let's call

00:01:05.349 --> 00:01:09.540
this area out.

00:01:09.540 --> 00:01:13.100
It's the area of the opening
coming out of the pipe.

00:01:13.100 --> 00:01:16.090
Let's think about what
happens if this

00:01:16.090 --> 00:01:19.200
liquid is actually moving.

00:01:19.200 --> 00:01:24.230
Let's say it's moving into the
pipe with the velocity V in.

00:01:26.770 --> 00:01:30.920
Let's think about how much
volume moves into the pipe

00:01:30.920 --> 00:01:33.430
after T seconds.

00:01:33.430 --> 00:01:38.540
After T seconds, if you
think about it, you'd

00:01:38.540 --> 00:01:41.490
have this much area.

00:01:41.490 --> 00:01:44.210
If you think about what was
right here, it will then be

00:01:44.210 --> 00:01:46.130
moved to the right
by how much?

00:01:49.580 --> 00:01:52.820
We could just go back to our
basic kinematic formula:

00:01:52.820 --> 00:01:55.420
distance is equal to
rate times time.

00:01:55.420 --> 00:02:00.150
The distance something travels
equals velocity times time, so

00:02:00.150 --> 00:02:07.000
after T seconds, whatever fluid
was here, it would have

00:02:07.000 --> 00:02:08.479
an area of about that much.

00:02:08.479 --> 00:02:10.800
Whatever fluid was there
would have traveled

00:02:10.800 --> 00:02:12.520
how much to the right?

00:02:12.520 --> 00:02:16.660
It would have traveled-- let's
assume that the pipe doesn't

00:02:16.660 --> 00:02:21.670
change too much in diameter or
in radius from here to here.

00:02:21.670 --> 00:02:28.046
It would have traveled velocity
times time, so V in

00:02:28.046 --> 00:02:30.240
times time.

00:02:30.240 --> 00:02:33.920
It could be meters or whatever
our length units are.

00:02:33.920 --> 00:02:38.870
After T seconds, essentially
this much water has traveled

00:02:38.870 --> 00:02:40.690
into the pipe.

00:02:40.690 --> 00:02:43.870
You could imagine a cylinder
of water here.

00:02:43.870 --> 00:02:46.650
Once again, I know I made it
look like it's getting wider

00:02:46.650 --> 00:02:48.610
the whole time, but let's assume
that its width doesn't

00:02:48.610 --> 00:02:51.800
change that much over the T
seconds or whatever units of

00:02:51.800 --> 00:02:53.840
time we're looking at.

00:02:53.840 --> 00:02:56.640
What is the volume of this
cylinder of water?

00:02:56.640 --> 00:03:08.280
The volume-in over the T seconds
is equal to the area,

00:03:08.280 --> 00:03:11.610
or the left-hand side
of the cylinder.

00:03:11.610 --> 00:03:14.990
Let me draw the cylinder in a
more vibrant color so you can

00:03:14.990 --> 00:03:16.240
figure out the volume.

00:03:21.526 --> 00:03:26.740
So it equals this side, the left
side of the cylinder, the

00:03:26.740 --> 00:03:31.040
input area times the length
of the cylinder.

00:03:31.040 --> 00:03:33.510
That's the velocity of the
fluid times the time that

00:03:33.510 --> 00:03:40.690
we're measuring, times the input
velocity times time.

00:03:40.690 --> 00:03:44.120
That's the amount of volume
that came in.

00:03:44.120 --> 00:03:48.270
If that volume came into the
pipe-- once again, we learned

00:03:48.270 --> 00:03:51.050
several videos ago that the
definition of a liquid is a

00:03:51.050 --> 00:03:53.640
fluid that's incompressible.

00:03:53.640 --> 00:03:56.900
It's not like no fluid could
come out of the pipe and all

00:03:56.900 --> 00:03:58.020
of the fluid just
gets squeezed.

00:03:58.020 --> 00:04:03.560
The same volume of fluid would
have to come out of the pipe,

00:04:03.560 --> 00:04:05.680
so that must equal
the volume out.

00:04:11.230 --> 00:04:14.950
Whatever comes into the pipe has
to equal the volume coming

00:04:14.950 --> 00:04:15.810
out of the pipe.

00:04:15.810 --> 00:04:20.290
One assumption we're assuming in
this fraction of time that

00:04:20.290 --> 00:04:23.160
we're dealing with is also that
there's no friction in

00:04:23.160 --> 00:04:27.770
this liquid or in this fluid,
that it actually is not

00:04:27.770 --> 00:04:30.480
turbulent and it's
not viscous.

00:04:30.480 --> 00:04:32.870
A viscous fluid is really just
something that has a lot of

00:04:32.870 --> 00:04:36.340
friction with itself and that
it won't just naturally move

00:04:36.340 --> 00:04:37.970
without any resistance.

00:04:37.970 --> 00:04:40.390
When something is not viscous
and has no resistance with

00:04:40.390 --> 00:04:43.670
itself and moves really without
any turbulence, that's

00:04:43.670 --> 00:04:44.950
called laminar flow.

00:04:44.950 --> 00:04:49.800
That's just a good word to
know about and it's the

00:04:49.800 --> 00:04:52.210
opposite of viscous flow.

00:04:52.210 --> 00:04:53.990
Different things have different
viscosities, and

00:04:53.990 --> 00:04:55.260
we'll probably do
more on that.

00:04:55.260 --> 00:04:57.870
Like syrup or peanut
butter has a

00:04:57.870 --> 00:04:59.350
very, very high viscosity.

00:04:59.350 --> 00:05:01.290
Even glass actually
is a fluid with a

00:05:01.290 --> 00:05:02.725
very, very high viscosity.

00:05:05.450 --> 00:05:09.470
I think there's some kinds of
compounds and magnetic fields

00:05:09.470 --> 00:05:11.300
that you could create that have
perfect laminar flow, but

00:05:11.300 --> 00:05:14.970
this is kind of a perfect
situation.

00:05:14.970 --> 00:05:18.330
In these circumstances, the
volume in, because the fluid

00:05:18.330 --> 00:05:21.040
can't be compressed, it's
incompressible, has to equal

00:05:21.040 --> 00:05:22.920
the volume out.

00:05:22.920 --> 00:05:27.160
What's the volume out over
that period of time?

00:05:27.160 --> 00:05:31.040
Similarly, we could draw this
bigger cylinder-- that's the

00:05:31.040 --> 00:05:37.930
area out-- and after
T seconds, how much

00:05:37.930 --> 00:05:39.390
water has come out?

00:05:39.390 --> 00:05:42.700
Whatever water was here at the
beginning of our time period

00:05:42.700 --> 00:05:50.570
will have come out and we can
imagine the cylinder here.

00:05:50.570 --> 00:05:56.130
What is the width
of the cylinder?

00:05:56.130 --> 00:05:58.840
What's going to be the velocity
that the liquid is

00:05:58.840 --> 00:06:00.570
coming out on the
right-hand side?

00:06:04.260 --> 00:06:07.590
Capital V is volume, and
lowercase v is for velocity,

00:06:07.590 --> 00:06:10.240
so it's going to be the output
velocity-- that's a lowercase

00:06:10.240 --> 00:06:12.640
v-- times the same time.

00:06:16.440 --> 00:06:20.580
So what is the volume that has
come out in our time T?

00:06:20.580 --> 00:06:25.160
It's just going to be this area
times this width, so the

00:06:25.160 --> 00:06:29.030
output volume over that same
period of time is equal to the

00:06:29.030 --> 00:06:32.570
output area of this pipe
times the output

00:06:32.570 --> 00:06:36.410
velocity times time.

00:06:36.410 --> 00:06:38.650
Once again, I know I keep saying
this, but this is kind

00:06:38.650 --> 00:06:43.020
of the big ah-hah moment, is
in that amount of time, the

00:06:43.020 --> 00:06:45.160
volume in this cylinder
has to equal the

00:06:45.160 --> 00:06:46.190
volume in this cylinder.

00:06:46.190 --> 00:06:49.490
Maybe it's not as wide, or
something like that, but their

00:06:49.490 --> 00:06:50.220
volumes are the same.

00:06:50.220 --> 00:06:54.610
You can't get more water here
all of a sudden than what's

00:06:54.610 --> 00:06:57.510
going in, and likewise, you
can't put more water into the

00:06:57.510 --> 00:06:59.370
left side than what's coming out
of the right side, because

00:06:59.370 --> 00:07:00.860
it's incompressible

00:07:00.860 --> 00:07:07.190
These two volumes equal each
other, so we know the area of

00:07:07.190 --> 00:07:11.030
the opening onto to the left
hand of the pipe times the

00:07:11.030 --> 00:07:16.160
input velocity times the
duration of time we're talking

00:07:16.160 --> 00:07:22.450
about is equal to the output
area times the output velocity

00:07:22.450 --> 00:07:24.830
times the duration of time
we're talking about.

00:07:28.680 --> 00:07:32.000
It's the same time on both sides
of this equation, so we

00:07:32.000 --> 00:07:37.250
could say that the input area
times the input velocity is

00:07:37.250 --> 00:07:42.570
equal to the output area times
the output velocity.

00:07:42.570 --> 00:07:48.040
This is actually called in fluid
motion the equation of

00:07:48.040 --> 00:07:51.370
continuity, and it leads to
some interesting things.

00:07:51.370 --> 00:07:53.300
We'll do some problems
with it in a second.

00:07:53.300 --> 00:07:54.560
One thing that I want to
introduce at this point as

00:07:54.560 --> 00:07:57.670
well is what is the
volume per second?

00:07:57.670 --> 00:08:00.000
Because this is also something
we're going to deal with in a

00:08:00.000 --> 00:08:02.390
second, probably in the next
video, because I'm about to

00:08:02.390 --> 00:08:03.590
run out of time.

00:08:03.590 --> 00:08:07.630
We said that in T seconds we
have this amount of volume

00:08:07.630 --> 00:08:10.420
coming in and it's the same
coming in as coming out.

00:08:10.420 --> 00:08:12.040
So what is the volume
per second?

00:08:12.040 --> 00:08:17.060
It's this big capital Vi
per amount of time, and

00:08:17.060 --> 00:08:18.220
we call that flux.

00:08:18.220 --> 00:08:20.430
We'll learn a lot about flux,
especially when we start doing

00:08:20.430 --> 00:08:23.870
vector calculus, but flux is
just how much of something

00:08:23.870 --> 00:08:26.900
crosses a surface in
an amount of time.

00:08:26.900 --> 00:08:28.930
It's how much a volume
crosses a surface in

00:08:28.930 --> 00:08:29.640
an amount of time.

00:08:29.640 --> 00:08:32.890
So in this case, the
surface is the

00:08:32.890 --> 00:08:33.840
left-hand side of cylinder.

00:08:33.840 --> 00:08:36.020
And we're saying how much
crosses in amount of time?

00:08:36.020 --> 00:08:39.830
We figured out it's that input
volume, which crosses in every

00:08:39.830 --> 00:08:41.820
T seconds, and this
is called flux.

00:08:45.930 --> 00:08:48.650
You've probably heard of the
flux capacitor in Back To The

00:08:48.650 --> 00:08:51.040
Future, and maybe we can think
about what they were

00:08:51.040 --> 00:08:52.320
trying to hint at.

00:08:52.320 --> 00:08:59.970
Let's see if we can use flux
and these ideas to come up

00:08:59.970 --> 00:09:01.965
with some other interesting
equations.

00:09:08.360 --> 00:09:11.290
We know that the volume per
time is equal to flux.

00:09:11.290 --> 00:09:12.180
This is a big V.

00:09:12.180 --> 00:09:16.700
V is equal to flux, and actually
the variable people

00:09:16.700 --> 00:09:18.760
generally use for flux is R.

00:09:18.760 --> 00:09:23.640
Of course, it's in meters
cubed per second.

00:09:23.640 --> 00:09:24.640
That's its unit.

00:09:24.640 --> 00:09:33.480
We also know that the input area
times input velocity--

00:09:33.480 --> 00:09:37.820
that's a lowercase v-- is equal
to the output area times

00:09:37.820 --> 00:09:42.810
output velocity, and this is
called the equation of

00:09:42.810 --> 00:09:43.810
continuity.

00:09:43.810 --> 00:09:47.150
It holds true whenever
we have laminar flow.

00:09:47.150 --> 00:09:48.230
Actually, I'm about to
run out of time.

00:09:48.230 --> 00:09:50.330
In the next video, I'm actually
going to use some of

00:09:50.330 --> 00:09:54.580
this information to figure out
how much power is there in a

00:09:54.580 --> 00:09:57.150
system where we have fluid
going through a pipe.

00:09:57.150 --> 00:09:58.600
See you soon.


Buoyant force example problems

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.240
Let's say that I have some
object, and when it's outside

00:00:04.240 --> 00:00:12.970
of water, its weight
is 10 newtons.

00:00:12.970 --> 00:00:15.850
When I submerge it in water-- I
put it on a weighing machine

00:00:15.850 --> 00:00:26.690
in water-- its weight
is 2 newtons.

00:00:26.690 --> 00:00:28.440
What must be going on here?

00:00:28.440 --> 00:00:31.670
The water must be exerting some
type of upward force to

00:00:31.670 --> 00:00:35.640
counteract at least
8 newtons of the

00:00:35.640 --> 00:00:36.890
object's original weight.

00:00:43.460 --> 00:00:45.450
That difference is the
buoyant force.

00:00:45.450 --> 00:00:48.600
So the way to think about is
that once you put the object

00:00:48.600 --> 00:00:52.630
in the water-- it could be a
cube, or it could be anything.

00:01:00.170 --> 00:01:08.540
We know that we have a downward
weight that is 10

00:01:08.540 --> 00:01:12.230
newtons, but we know that once
it's in the water, the net

00:01:12.230 --> 00:01:16.070
weight is 2 newtons, so there
must be some force acting

00:01:16.070 --> 00:01:20.460
upwards on the object
of 8 newtons.

00:01:20.460 --> 00:01:22.620
That's the buoyant force that
we learned about in the

00:01:22.620 --> 00:01:25.250
previous video, in the video
about Archimedes' principle.

00:01:25.250 --> 00:01:27.380
This is the buoyant force.

00:01:30.040 --> 00:01:38.320
So the buoyant force is equal
to 10 minus 2 is equal to 8.

00:01:38.320 --> 00:01:40.140
That's how much the water's
pushing up.

00:01:40.140 --> 00:01:42.350
And what does that
also equal to?

00:01:42.350 --> 00:01:49.520
That equals the weight of the
water displaced, so 8 newtons

00:01:49.520 --> 00:01:52.780
is equal to weight of
water displaced.

00:01:52.780 --> 00:01:54.990
What is the weight of
the water displaced?

00:01:54.990 --> 00:01:59.510
That's the volume of the water
displaced times the density of

00:01:59.510 --> 00:02:04.460
water times gravity.

00:02:04.460 --> 00:02:06.350
What is the volume of
water displaced?

00:02:06.350 --> 00:02:11.910
It's just the volume of water,
divide 8 newtons by the

00:02:11.910 --> 00:02:15.130
density of water,
which is 1,000

00:02:15.130 --> 00:02:19.790
kilograms per meter cubed.

00:02:19.790 --> 00:02:28.930
A newton is 1 kilogram meter
per second squared.

00:02:31.770 --> 00:02:32.450
Then, what's gravity?

00:02:32.450 --> 00:02:39.730
It's 9.8 meters per
second squared.

00:02:39.730 --> 00:02:42.820
If we look at all the units,
they actually do turn out with

00:02:42.820 --> 00:02:45.430
you just ending up having
just meters cubed, but

00:02:45.430 --> 00:02:46.470
let's do the math.

00:02:46.470 --> 00:03:03.870
We get 8 divided by 1,000
divided by 9.8 is equal to 8.2

00:03:03.870 --> 00:03:06.510
times 10 to the negative 4.

00:03:06.510 --> 00:03:16.650
V equals 8.2 times 10 to the
minus 4 cubic meters.

00:03:16.650 --> 00:03:21.420
Just knowing the difference in
the weight of an object-- the

00:03:21.420 --> 00:03:22.760
difference when I put
it in water-- I can

00:03:22.760 --> 00:03:23.710
figure out the volume.

00:03:23.710 --> 00:03:26.340
This could be a fun game
to do next time your

00:03:26.340 --> 00:03:27.670
friends come over.

00:03:27.670 --> 00:03:31.970
Weigh yourself outside of water,
then get some type of

00:03:31.970 --> 00:03:36.700
spring or waterproof weighing
machine, put it at the bottom

00:03:36.700 --> 00:03:38.965
of your pool, stand on it, and
figure out what your weight

00:03:38.965 --> 00:03:41.490
is, assuming that you're dense
enough to go all the

00:03:41.490 --> 00:03:42.150
way into the water.

00:03:42.150 --> 00:03:44.330
You could figure out somehow
your weight in water, and then

00:03:44.330 --> 00:03:46.390
you would know your volume.

00:03:46.390 --> 00:03:46.940
There's other ways.

00:03:46.940 --> 00:03:50.170
You could just figure out how
much the surface of the water

00:03:50.170 --> 00:03:52.850
increases, and take
that water away.

00:03:52.850 --> 00:03:53.630
This was interesting.

00:03:53.630 --> 00:03:56.760
Just knowing how much the
buoyant force of the water was

00:03:56.760 --> 00:03:59.610
or how much lighter we are when
the object goes into the

00:03:59.610 --> 00:04:02.800
water, we can figure out the
volume of the object.

00:04:02.800 --> 00:04:05.880
This might seem like a very
small volume, but just keep in

00:04:05.880 --> 00:04:17.100
mind in a meter cubed, you
have 27 square feet.

00:04:17.100 --> 00:04:28.020
If we multiply that number
times 27, it equals 0.02

00:04:28.020 --> 00:04:31.650
square feet roughly.

00:04:31.650 --> 00:04:35.280
In 0.02 square feet, how many--
in a square foot,

00:04:35.280 --> 00:04:41.210
there's actually-- 12 to the
third power times 12 times 12

00:04:41.210 --> 00:04:47.630
is equal to 1,728 times 0.02.

00:04:47.630 --> 00:04:52.060
So this is actually
34 square inches.

00:04:52.060 --> 00:04:55.560
The object isn't as small as you
may have thought it to be.

00:04:55.560 --> 00:04:59.280
It's actually maybe a little bit
bigger than 3 inches by 3

00:04:59.280 --> 00:05:02.500
inches by 3 inches, so it's
a reasonably sized object.

00:05:02.500 --> 00:05:03.840
Anyway, let's do another
problem.

00:05:08.280 --> 00:05:11.840
Let's say I have some balsa
wood, and I know that the

00:05:11.840 --> 00:05:18.780
density of balsa wood is 130
kilograms per meter cubed.

00:05:24.330 --> 00:05:28.440
I have some big cube of balsa
wood, and what I want to know

00:05:28.440 --> 00:05:33.040
is if I put that-- let
me draw the water.

00:05:33.040 --> 00:05:38.400
I have some big cube of balsa
wood, which I'll do in brown.

00:05:38.400 --> 00:05:40.570
So I have a big cube of balsa
wood and the water should go

00:05:40.570 --> 00:05:41.910
on top of it, just so
that you see it's

00:05:41.910 --> 00:05:45.410
submerged in the water.

00:05:45.410 --> 00:05:48.640
I want to know what percentage
of the cube goes below the

00:05:48.640 --> 00:05:50.590
surface of the water?

00:05:50.590 --> 00:05:52.160
Interesting question.

00:05:52.160 --> 00:05:54.790
So how do we do that?

00:05:54.790 --> 00:05:57.625
For the object to be at rest,
for this big cube to be at

00:05:57.625 --> 00:06:02.080
rest, there must be zero net
forces on this object.

00:06:02.080 --> 00:06:09.140
In that situation, the buoyant
force must completely equal

00:06:09.140 --> 00:06:13.510
the weight or the force
of gravity.

00:06:13.510 --> 00:06:16.550
What's the force of gravity
going to be?

00:06:16.550 --> 00:06:19.130
The force of gravity is just the
weight of the object, and

00:06:19.130 --> 00:06:27.110
that's the volume of the balsa
wood times the density of the

00:06:27.110 --> 00:06:31.950
balsa wood times gravity.

00:06:31.950 --> 00:06:34.130
What's the buoyant force?

00:06:34.130 --> 00:06:39.960
The buoyant force is equal to
the volume of the displaced

00:06:39.960 --> 00:06:46.090
water, but that's also the
volume of the displaced water

00:06:46.090 --> 00:06:50.200
and it's the volume of the cube
that's been submerged.

00:06:50.200 --> 00:06:52.170
The part of the cube that's
submerged, that's volume.

00:06:52.170 --> 00:06:53.400
That's also equal
to the amount of

00:06:53.400 --> 00:06:55.690
volume of water displaced.

00:06:55.690 --> 00:06:57.970
We could say that's the volume
of the block submerged, which

00:06:57.970 --> 00:07:00.330
is the same thing, remember,
as the volume of the water

00:07:00.330 --> 00:07:05.300
displaced times the density
of water times gravity.

00:07:05.300 --> 00:07:06.820
Remember, this is density
of water.

00:07:06.820 --> 00:07:09.880
So remember, the buoyant force
is just equal to the weight of

00:07:09.880 --> 00:07:13.000
the water displaced and that's
just the volume of the water

00:07:13.000 --> 00:07:16.230
displaced times the density
of water times gravity.

00:07:16.230 --> 00:07:18.560
Of course, the volume of the
water displaced is the exact

00:07:18.560 --> 00:07:20.710
same thing as the volume
of the block

00:07:20.710 --> 00:07:22.850
that's actually submerged.

00:07:22.850 --> 00:07:25.710
Since the block is stationary,
it's not accelerating upwards

00:07:25.710 --> 00:07:28.490
or downwards, we know that
these two quantities must

00:07:28.490 --> 00:07:29.900
equal each other.

00:07:29.900 --> 00:07:34.340
So V, the volume of the wood,
the entire volume, not just

00:07:34.340 --> 00:07:38.210
the amount that's submerged,
times the density of the wood

00:07:38.210 --> 00:07:43.450
times gravity must equal the
volume of the wood submerged,

00:07:43.450 --> 00:07:47.610
which is equal to the volume of
the water displaced times

00:07:47.610 --> 00:07:52.580
the density of water
times gravity.

00:07:52.580 --> 00:07:53.890
We have the acceleration
of gravity.

00:07:53.890 --> 00:07:56.140
We have that on both sides,
so we can cross it out.

00:07:56.140 --> 00:07:58.400
Let me switch colors to
ease the monotony.

00:08:01.130 --> 00:08:07.360
What happens if we divide both
sides by the volume of the

00:08:07.360 --> 00:08:08.610
balsa wood?

00:08:13.130 --> 00:08:14.050
I'm just rearranging
this equation.

00:08:14.050 --> 00:08:15.850
I think you'll figure it out.

00:08:15.850 --> 00:08:19.300
We divide both sides by that,
and you get the volume

00:08:19.300 --> 00:08:23.080
submerged divided by the volume
of the balsa wood-- I

00:08:23.080 --> 00:08:27.450
just divided both sides by VB
and switched sides-- is equal

00:08:27.450 --> 00:08:35.590
to the density of the balsa
wood divided by

00:08:35.590 --> 00:08:37.970
the density of water.

00:08:37.970 --> 00:08:38.549
Does that make sense?

00:08:38.549 --> 00:08:43.140
I just did a couple of quick
algebraic operations, but

00:08:43.140 --> 00:08:45.100
hopefully that got rid of
the g, and that should

00:08:45.100 --> 00:08:47.200
make sense to you.

00:08:47.200 --> 00:08:49.270
Now we're ready to solve
our problem.

00:08:49.270 --> 00:08:52.520
My original question is
what percentage of

00:08:52.520 --> 00:08:54.450
the object is submerged?

00:08:54.450 --> 00:08:56.160
That's exactly this number.

00:08:56.160 --> 00:08:59.530
If we say this is the volume
submerged over the total

00:08:59.530 --> 00:09:02.450
volume, this is the
percent submerged.

00:09:02.450 --> 00:09:07.030
That equals the density of
balsa wood, which is 130

00:09:07.030 --> 00:09:10.550
kilograms per meter cubed,
divided by the density of

00:09:10.550 --> 00:09:15.940
water, which is 1,000 kilograms
per meter cubed, so

00:09:15.940 --> 00:09:20.180
130 divided by 1,000 is 0.13.

00:09:20.180 --> 00:09:24.370
Vs over VB is equal to
0.13, which is the

00:09:24.370 --> 00:09:26.370
same thing as 13%.

00:09:26.370 --> 00:09:31.730
So, exactly 13% percent of this
balsa wood block will be

00:09:31.730 --> 00:09:33.140
submerged in the water.

00:09:33.140 --> 00:09:34.660
That's pretty neat to me.

00:09:34.660 --> 00:09:36.470
It actually didn't have
to be a block.

00:09:36.470 --> 00:09:38.490
It could have been shaped
like a horse.

00:09:38.490 --> 00:09:40.450
I'll see you in the
next video.


Archimedes principle and buoyant force

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.080
Let's say we have
a cup of water.

00:00:03.080 --> 00:00:04.330
Let me draw the cup.

00:00:06.800 --> 00:00:11.530
This is one side of the cup,
this is the bottom of the cup,

00:00:11.530 --> 00:00:15.130
and this is the other
side of the cup.

00:00:15.130 --> 00:00:16.325
Let me say that it's
some liquid.

00:00:16.325 --> 00:00:19.920
It doesn't have to be water,
but some arbitrary liquid.

00:00:19.920 --> 00:00:21.260
It could be water.

00:00:21.260 --> 00:00:23.850
That's the surface of it.

00:00:23.850 --> 00:00:26.570
We've already learned that the
pressure at any point within

00:00:26.570 --> 00:00:30.970
this liquid is dependent
on how deep

00:00:30.970 --> 00:00:32.580
we go into the liquid.

00:00:32.580 --> 00:00:35.450
One point I want to make before
we move on, and I

00:00:35.450 --> 00:00:39.730
touched on this a little bit
before, is that the pressure

00:00:39.730 --> 00:00:43.460
at some point isn't just acting
downwards, or it isn't

00:00:43.460 --> 00:00:44.480
just acting in one direction.

00:00:44.480 --> 00:00:46.975
It's acting in all directions
on that point.

00:00:46.975 --> 00:00:50.620
So although how far we go down
determines how much pressure

00:00:50.620 --> 00:00:53.000
there is, the pressure is
actually acting in all

00:00:53.000 --> 00:00:54.850
directions, including up.

00:00:54.850 --> 00:00:58.330
The reason why that makes sense
is because I'm assuming

00:00:58.330 --> 00:01:02.950
that this is a static system,
or that the fluids in this

00:01:02.950 --> 00:01:05.500
liquid are stationary, or you
even could imagine an object

00:01:05.500 --> 00:01:07.360
down here, and it's
stationary.

00:01:07.360 --> 00:01:09.800
The fact that it's stationary
tells us that the pressure in

00:01:09.800 --> 00:01:12.000
every direction must be equal.

00:01:12.000 --> 00:01:13.970
Let's think about a
molecule of water.

00:01:13.970 --> 00:01:15.870
A molecule of water, let's say
it's roughly a sphere.

00:01:20.920 --> 00:01:24.440
If the pressure were different
in one direction or if the

00:01:24.440 --> 00:01:28.070
pressure down were greater than
the pressure up, then the

00:01:28.070 --> 00:01:30.310
object would start accelerating
downwards,

00:01:30.310 --> 00:01:32.950
because its surface area
pointing upwards is the same

00:01:32.950 --> 00:01:36.240
as the surface area pointing
downwards, so the force

00:01:36.240 --> 00:01:37.410
upwards would be more.

00:01:37.410 --> 00:01:39.550
It would start accelerating
downwards.

00:01:39.550 --> 00:01:43.520
Even though the pressure is a
function of how far down we

00:01:43.520 --> 00:01:45.900
go, at that point,
the pressure is

00:01:45.900 --> 00:01:48.150
acting in every direction.

00:01:48.150 --> 00:01:51.790
Let's remember that, and now
let's keep that in mind to

00:01:51.790 --> 00:01:54.530
learn a little bit about
Archimedes' principle.

00:01:54.530 --> 00:02:01.840
Let's say I submerge a cube into
this liquid, and let's

00:02:01.840 --> 00:02:13.370
say this cube has dimensions
d, so every side is d.

00:02:17.180 --> 00:02:19.560
What I want to do is I want to
figure out if there's any

00:02:19.560 --> 00:02:22.870
force or what is the net
force acting on this

00:02:22.870 --> 00:02:25.070
cube due to the water?

00:02:25.070 --> 00:02:27.770
Let's think about what the
pressure on this cube is at

00:02:27.770 --> 00:02:29.980
different points.

00:02:29.980 --> 00:02:32.760
At the depths along the side of
the cube, we know that the

00:02:32.760 --> 00:02:35.310
pressures are equal, because
we know at this depth right

00:02:35.310 --> 00:02:38.350
here, the pressure is going to
be the same as at that depth,

00:02:38.350 --> 00:02:40.110
and they're going to offset each
other, and so these are

00:02:40.110 --> 00:02:41.900
going to be the same.

00:02:41.900 --> 00:02:44.080
But one thing we do know, just
based on the fact that

00:02:44.080 --> 00:02:47.480
pressure is a function of depth,
is that at this point

00:02:47.480 --> 00:02:50.510
the pressure is going to be
higher-- I don't know how much

00:02:50.510 --> 00:02:54.010
higher-- than at this point,
because this point is deeper

00:02:54.010 --> 00:02:55.330
into the water.

00:02:55.330 --> 00:02:59.275
Let's call this P1.

00:02:59.275 --> 00:03:03.210
Let's call that pressure on top,
PT, and let's call this

00:03:03.210 --> 00:03:06.260
point down here PD.

00:03:06.260 --> 00:03:07.910
No, pressure on the
bottom, PB.

00:03:11.330 --> 00:03:17.220
What's going to be the net
force on this cube?

00:03:17.220 --> 00:03:22.330
The net force-- let's call that
F sub N-- is going to be

00:03:22.330 --> 00:03:27.420
equal to the force acting
upwards on this object.

00:03:27.420 --> 00:03:29.280
What's the force acting
upwards on the object?

00:03:29.280 --> 00:03:35.830
It's going to be this pressure
at the bottom of the object

00:03:35.830 --> 00:03:38.730
times the surface area at the
bottom of the object.

00:03:38.730 --> 00:03:41.530
What's the surface area at
the bottom of the object?

00:03:41.530 --> 00:03:42.930
That's just d squared.

00:03:42.930 --> 00:03:45.770
Any surface of a cube is d
squared, so the bottom is

00:03:45.770 --> 00:03:54.440
going to be d squared minus--
I'm doing this because I

00:03:54.440 --> 00:03:56.850
actually know that the pressure
down here is higher

00:03:56.850 --> 00:03:58.630
than the pressure here, so this
is going to be a larger

00:03:58.630 --> 00:04:01.090
quantity, and that the net force
is actually going to be

00:04:01.090 --> 00:04:04.430
upwards, so that's why I can
do the minus confidently up

00:04:04.430 --> 00:04:07.570
here-- the pressure
at the top.

00:04:07.570 --> 00:04:09.590
What's the force at the top?

00:04:09.590 --> 00:04:14.620
The force at the top is going to
be the pressure on the top

00:04:14.620 --> 00:04:16.750
times the surface area of
the top of the cube,

00:04:16.750 --> 00:04:20.700
right, times d squared.

00:04:20.700 --> 00:04:24.540
We can even separate out the d
squared already at that point,

00:04:24.540 --> 00:04:30.980
so the net force is equal to
the pressure of the bottom

00:04:30.980 --> 00:04:33.920
minus the pressure of the top,
or the difference in pressure

00:04:33.920 --> 00:04:37.150
times the surface area of either
the top or the bottom

00:04:37.150 --> 00:04:39.840
or really any of the
sides of the cube.

00:04:39.840 --> 00:04:41.460
Let's see if we can figure
what these are.

00:04:41.460 --> 00:04:45.770
Let's say the cube is submerged
h units or h meters

00:04:45.770 --> 00:04:48.560
into the water.

00:04:48.560 --> 00:04:51.780
So what's the pressure
at the top?

00:04:51.780 --> 00:04:55.600
The pressure at the top is
going to be equal to the

00:04:55.600 --> 00:04:58.240
density of the liquid-- I keep
saying water, but it could be

00:04:58.240 --> 00:05:02.610
any liquid-- times how
far down we are.

00:05:02.610 --> 00:05:08.080
So we're h units down, or maybe
h meters, times gravity.

00:05:08.080 --> 00:05:10.700
And what's the pressure
the bottom?

00:05:10.700 --> 00:05:15.060
The pressure at the bottom
similarly would be the density

00:05:15.060 --> 00:05:18.600
of the liquid times the depth,
so what's the depth?

00:05:18.600 --> 00:05:21.270
It would be this h and then
we're another d down.

00:05:24.226 --> 00:05:29.480
It's h plus d-- that's our total
depth-- times gravity.

00:05:29.480 --> 00:05:31.946
Let's just substitute both of
those back into our net force.

00:05:31.946 --> 00:05:35.750
Let me switch colors to keep
from getting monotonous.

00:05:35.750 --> 00:05:40.360
I get the net force is equal to
the pressure at the bottom,

00:05:40.360 --> 00:05:42.120
which is this.

00:05:42.120 --> 00:05:52.730
Let's just multiply it out, so
we get p times h times g plus

00:05:52.730 --> 00:05:54.480
d times p times g.

00:05:58.580 --> 00:06:00.910
I just distributed this out,
multiplied this out.

00:06:00.910 --> 00:06:06.230
That's the pressure at the
bottom, then minus the

00:06:06.230 --> 00:06:13.950
pressure at the top, minus phg,
and then we learned it's

00:06:13.950 --> 00:06:17.860
all of that times d squared.

00:06:17.860 --> 00:06:20.350
Immediately, we see something
cancels out.

00:06:20.350 --> 00:06:23.905
phg, phg subtract.

00:06:23.905 --> 00:06:25.950
It cancels out, so we're
just left with--

00:06:25.950 --> 00:06:27.100
what's the net force?

00:06:27.100 --> 00:06:36.000
The net force is equal to dpg
times d squared, or that

00:06:36.000 --> 00:06:42.490
equals d cubed times
the density of the

00:06:42.490 --> 00:06:45.740
liquid times gravity.

00:06:45.740 --> 00:06:50.000
Let me ask you a question:
What is d cubed?

00:06:50.000 --> 00:06:51.830
d cubed is the volume
of this cube.

00:06:51.830 --> 00:06:54.780
And what else is it?

00:06:54.780 --> 00:06:56.800
It's also the volume of
the water displaced.

00:06:56.800 --> 00:06:59.680
If I stick this cube into the
water, and the cube isn't

00:06:59.680 --> 00:07:03.050
shrinking or anything-- you
can even imagine it being

00:07:03.050 --> 00:07:05.860
empty, but it doesn't have to be
empty-- but that amount of

00:07:05.860 --> 00:07:08.640
water has to be moved out
of the way in order for

00:07:08.640 --> 00:07:11.040
that cube to go in.

00:07:11.040 --> 00:07:15.360
This is the volume of
the water displaced.

00:07:15.360 --> 00:07:16.680
It's also the volume
of the cube.

00:07:24.200 --> 00:07:28.010
This is the density-- I keep
saying water, but it could be

00:07:28.010 --> 00:07:30.770
any liquid-- of the liquid.

00:07:30.770 --> 00:07:32.560
This is the gravity.

00:07:32.560 --> 00:07:33.190
So what is this?

00:07:33.190 --> 00:07:38.800
Volume times density is the mass
of the liquid displaced,

00:07:38.800 --> 00:07:42.240
so the net force is
also equal to the

00:07:42.240 --> 00:07:47.120
mass of liquid displaced.

00:07:47.120 --> 00:07:52.260
Let's just say mass times
gravity, or we could say that

00:07:52.260 --> 00:07:55.660
the net force acting on this
object is-- what's the mass of

00:07:55.660 --> 00:07:57.120
the liquid displaced
times gravity?

00:07:57.120 --> 00:08:05.750
That's just the weight
of liquid displaced.

00:08:05.750 --> 00:08:08.030
That's a pretty interesting
thing.

00:08:08.030 --> 00:08:12.990
If I submerge anything, the net
force acting upwards on

00:08:12.990 --> 00:08:15.630
it, or the amount that I'm
lighter by, is equal to the

00:08:15.630 --> 00:08:18.370
weight of the water
being displaced.

00:08:18.370 --> 00:08:20.960
That's actually called
Archimedes' principle.

00:08:20.960 --> 00:08:24.260
That net upward force due to
the fact that there's more

00:08:24.260 --> 00:08:27.090
pressure on the bottom than
there is on the top, that's

00:08:27.090 --> 00:08:28.280
called the buoyant force.

00:08:28.280 --> 00:08:30.950
That's what makes
things float.

00:08:30.950 --> 00:08:35.280
I'll leave you there to just to
ponder that, and we'll use

00:08:35.280 --> 00:08:37.919
this concept in the next couple
of videos to actually

00:08:37.919 --> 00:08:40.789
solve some problems.
I'll see you soon.


Pressure at a depth in a fluid

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.740
In the last video, we showed
that any external pressure on

00:00:04.740 --> 00:00:08.490
a liquid in a container
is distributed

00:00:08.490 --> 00:00:10.000
evenly through the liquid.

00:00:10.000 --> 00:00:13.340
But that only applied to-- and
that was called Pascal's

00:00:13.340 --> 00:00:16.950
principle-- external pressure.

00:00:16.950 --> 00:00:19.670
Let's think a little bit about
what the internal pressure is

00:00:19.670 --> 00:00:20.620
within a liquid.

00:00:20.620 --> 00:00:23.760
We're all familiar, I think,
with the notion of the deeper

00:00:23.760 --> 00:00:26.930
you go into a fluid or the
deeper you dive into the

00:00:26.930 --> 00:00:29.350
ocean, the higher the
pressure is on you.

00:00:29.350 --> 00:00:32.619
Let's see if we can think about
that a little bit more

00:00:32.619 --> 00:00:35.420
analytically, and get a
framework for what the

00:00:35.420 --> 00:00:38.270
pressure is at any depth
under the water, or

00:00:38.270 --> 00:00:39.730
really in any fluid.

00:00:39.730 --> 00:00:44.680
Here I've drawn a cylinder, and
in that cylinder I have

00:00:44.680 --> 00:00:47.620
some fluid-- let's not assume
that it's water, but some

00:00:47.620 --> 00:00:50.160
fluid, and that's
the blue stuff.

00:00:50.160 --> 00:00:52.250
I'm also assuming that I'm doing
this on a planet that

00:00:52.250 --> 00:00:55.670
has the same mass as Earth, but
it has no atmosphere, so

00:00:55.670 --> 00:00:57.630
there's a vacuum up here--
there's no air.

00:00:57.630 --> 00:00:59.590
We'll see later that the
atmosphere actually adds

00:00:59.590 --> 00:01:00.940
pressure on top of this.

00:01:00.940 --> 00:01:04.800
Let's assume that there's no
air, but it's on a planet of

00:01:04.800 --> 00:01:07.770
the same mass, so the
gravity is the same.

00:01:07.770 --> 00:01:10.530
There is gravity, so the
liquid will fill this

00:01:10.530 --> 00:01:12.810
container on the bottom
part of it.

00:01:12.810 --> 00:01:15.840
Also, the gravitational constant
would be the same as

00:01:15.840 --> 00:01:19.040
Earth, so we can imagine this is
a horrible situation where

00:01:19.040 --> 00:01:22.020
Earth has lost its magnetic
field and the solar winds have

00:01:22.020 --> 00:01:23.820
gotten rid of Earth's
atmosphere.

00:01:23.820 --> 00:01:26.300
That's very negative, so we
won't think about that, but

00:01:26.300 --> 00:01:28.000
anyway-- let's go back
to the problem.

00:01:28.000 --> 00:01:35.640
Let's say within this cylinder,
I have a thin piece

00:01:35.640 --> 00:01:40.340
of foil or something that
takes up the entire

00:01:40.340 --> 00:01:43.000
cross-sectional area
of the cylinder.

00:01:43.000 --> 00:01:46.290
I did that just because I want
that to be an indicator of

00:01:46.290 --> 00:01:49.150
whether the fluid is moving
up or down or not.

00:01:49.150 --> 00:01:52.760
Let's say I have that in the
fluid at some depth, h, and

00:01:52.760 --> 00:01:56.650
since the fluid is completely
static-- nothing's moving--

00:01:56.650 --> 00:02:00.335
that object that's floating
right at that level, at a

00:02:00.335 --> 00:02:02.440
depth of h, will
also be static.

00:02:02.440 --> 00:02:05.060
In order for something to be
static, where it's not

00:02:05.060 --> 00:02:07.490
moving-- what do we
know about it?

00:02:07.490 --> 00:02:11.060
We know that the net forces on
it must be zero-- in fact,

00:02:11.060 --> 00:02:13.050
that tells that it's
not accelerating.

00:02:13.050 --> 00:02:15.470
Obviously, if something's not
moving, it has a velocity of

00:02:15.470 --> 00:02:18.170
zero, and that's a constant
velocity-- it's not

00:02:18.170 --> 00:02:20.000
accelerating in any direction,
and so its net

00:02:20.000 --> 00:02:22.830
forces must be zero.

00:02:22.830 --> 00:02:32.810
This force down must be
equal to the force up.

00:02:32.810 --> 00:02:38.050
So what is the force down
acting on this cylinder?

00:02:38.050 --> 00:02:41.860
It's going to be the weight of
the water above it, because

00:02:41.860 --> 00:02:45.780
we're in a gravitational
environment, and so this water

00:02:45.780 --> 00:02:47.030
has some mass.

00:02:49.310 --> 00:02:53.470
Whatever that mass is, times
the gravitational constant,

00:02:53.470 --> 00:02:56.170
will equal the force down.

00:02:56.170 --> 00:02:57.240
Let's figure out what that is.

00:02:57.240 --> 00:03:00.960
The force down, which is the
same thing is the force up, is

00:03:00.960 --> 00:03:12.860
going to equal the mass of
this water, times the

00:03:12.860 --> 00:03:15.490
gravitational constant.

00:03:15.490 --> 00:03:19.020
Actually, I shouldn't say
water-- let me change this,

00:03:19.020 --> 00:03:20.760
because I said that this is
going to be some random

00:03:20.760 --> 00:03:23.140
liquid, and the mass
is a liquid.

00:03:23.140 --> 00:03:27.360
The force down is going to be
equal to the mass of the

00:03:27.360 --> 00:03:31.500
liquid times gravity.

00:03:31.500 --> 00:03:33.220
What is that mass
of the liquid?

00:03:33.220 --> 00:03:35.565
Well, now I'll introduce you to
a concept called density,

00:03:35.565 --> 00:03:38.460
and I think you understand what
density is-- it's how

00:03:38.460 --> 00:03:41.150
much there is of something in
a given amount of volume, or

00:03:41.150 --> 00:03:42.290
how much mass per volume.

00:03:42.290 --> 00:03:44.560
That's the definition
of density.

00:03:44.560 --> 00:03:47.670
The letter people use for
density is rho-- let me do

00:03:47.670 --> 00:03:50.440
that in a different
color down here.

00:03:50.440 --> 00:04:01.476
rho, which looks like a p to
me, equals mass per volume,

00:04:01.476 --> 00:04:03.940
and that's the density.

00:04:03.940 --> 00:04:12.380
The units are kilograms per
meter cubed-- that is density.

00:04:12.380 --> 00:04:15.170
I think you might have an
intuition that if I have a

00:04:15.170 --> 00:04:24.930
cubic meter of lead-- lead is
more dense than marshmallows.

00:04:24.930 --> 00:04:28.690
Because of that, if I have a
cubic meter of lead, it will

00:04:28.690 --> 00:04:32.320
have a lot more mass, and in a
gravitational field, weigh a

00:04:32.320 --> 00:04:36.100
lot more than a cubic meter
of marshmallows.

00:04:36.100 --> 00:04:39.050
Of course, there's always that
trick people say, what weighs

00:04:39.050 --> 00:04:42.740
more-- a pound of feathers,
or a pound of lead?

00:04:42.740 --> 00:04:46.600
Those, obviously, weigh the
same-- the key is the volume.

00:04:46.600 --> 00:04:50.010
A cubic meter of lead is going
to weigh a lot more than a

00:04:50.010 --> 00:04:52.180
cubic meter of feathers.

00:04:52.180 --> 00:04:54.890
Making sure that we now know
what the density is, let's go

00:04:54.890 --> 00:04:57.360
back to what we were
doing before.

00:04:57.360 --> 00:05:00.330
We said that the downward force
is equal to the mass of

00:05:00.330 --> 00:05:04.300
the liquid times the
gravitational force, and so

00:05:04.300 --> 00:05:06.390
what is the mass
of the liquid?

00:05:06.390 --> 00:05:09.250
We could use this formula right
here-- density is equal

00:05:09.250 --> 00:05:12.490
to mass times volume, so we
could also say that mass is

00:05:12.490 --> 00:05:15.660
equal to density times volume.

00:05:15.660 --> 00:05:18.130
I just multiply both sides of
this equation times volume.

00:05:21.240 --> 00:05:26.570
In this situation, force down is
equal to-- let's substitute

00:05:26.570 --> 00:05:28.430
this with this.

00:05:28.430 --> 00:05:31.300
The mass of the liquid is equal
to the density of the

00:05:31.300 --> 00:05:35.910
liquid times the volume of the
liquid-- I could get rid of

00:05:35.910 --> 00:05:39.110
these l's-- times gravity.

00:05:39.110 --> 00:05:43.110
What's the volume
of the liquid?

00:05:43.110 --> 00:05:45.150
The volume of the liquid
is going to be the

00:05:45.150 --> 00:05:49.685
cross-sectional area of the
cylinder times the height.

00:05:49.685 --> 00:05:53.110
So let's call this
cross-sectional area A.

00:05:53.110 --> 00:05:58.290
A for area-- that's the area
of the cylinder or the foil

00:05:58.290 --> 00:06:01.290
that's floating within
the water.

00:06:01.290 --> 00:06:05.820
We could write down that the
downward force is equal to the

00:06:05.820 --> 00:06:09.460
density of the fluid-- I'll stop
writing the l or f, or

00:06:09.460 --> 00:06:13.800
whatever I was doing
there-- times the

00:06:13.800 --> 00:06:15.310
volume of the liquid.

00:06:15.310 --> 00:06:19.870
The volume of the liquid is just
the height times the area

00:06:19.870 --> 00:06:21.390
of the liquid.

00:06:21.390 --> 00:06:27.110
So that is just times the height
times the area and then

00:06:27.110 --> 00:06:28.360
times gravity.

00:06:35.920 --> 00:06:39.560
We've now figured out if we knew
the density, this height,

00:06:39.560 --> 00:06:42.840
the cross-sectional area, and
the gravitational constant, we

00:06:42.840 --> 00:06:44.100
would know the force
coming down.

00:06:44.100 --> 00:06:46.800
That's kind of vaguely
interesting, but let's try to

00:06:46.800 --> 00:06:48.610
figure out what the pressure
is, because that's what

00:06:48.610 --> 00:06:50.180
started this whole discussion.

00:06:50.180 --> 00:06:54.720
What is the pressure when you go
to deep parts of the ocean?

00:06:54.720 --> 00:06:59.903
This is the force-- what is the
pressure on this foil that

00:06:59.903 --> 00:07:01.190
I have floating?

00:07:01.190 --> 00:07:05.060
It's the force divided by the
area of pressure on this foil.

00:07:05.060 --> 00:07:08.400
So I would take the force and
divide it by the area, which

00:07:08.400 --> 00:07:11.780
is the same thing as A,
so let's do that.

00:07:11.780 --> 00:07:15.470
Let's divide both sides of this
equation by area, so the

00:07:15.470 --> 00:07:19.850
pressure coming down--
so that's P sub d.

00:07:25.610 --> 00:07:28.900
The downward pressure at that
point is going to be equal

00:07:28.900 --> 00:07:31.460
to-- keep in mind, that's going
to be the same thing as

00:07:31.460 --> 00:07:33.950
the upward pressure, because the
upward force is the same.

00:07:33.950 --> 00:07:36.680
The area of whether you're going
upwards or downwards is

00:07:36.680 --> 00:07:37.380
going to be the same thing.

00:07:37.380 --> 00:07:40.130
The downward pressure is going
to be equal to the downward

00:07:40.130 --> 00:07:44.430
force divided by area, which is
going to be equal to this

00:07:44.430 --> 00:07:46.260
expression divided by area.

00:07:46.260 --> 00:07:49.630
Essentially, we can just get
rid of the area here, so it

00:07:49.630 --> 00:07:57.550
equals PhAg divided by A-- we
get rid of the A's in both

00:07:57.550 --> 00:08:02.580
situations-- so the downward
pressure is equal to the

00:08:02.580 --> 00:08:07.480
density of the fluid, times the
depth of the fluid, or the

00:08:07.480 --> 00:08:11.040
height of the fluid above it,
times the gravitational

00:08:11.040 --> 00:08:13.400
constant Phg.

00:08:13.400 --> 00:08:15.430
As I said, the downward pressure
is equal to the

00:08:15.430 --> 00:08:16.910
upward pressure-- how
do we know that?

00:08:16.910 --> 00:08:19.360
Because we knew that the upward
force is the same as

00:08:19.360 --> 00:08:20.230
the downward force.

00:08:20.230 --> 00:08:26.170
If the upward force were less,
this little piece of foil

00:08:26.170 --> 00:08:28.480
would actually accelerate
downwards.

00:08:28.480 --> 00:08:31.050
The fact that it's static-- it's
in one place-- lets us

00:08:31.050 --> 00:08:34.250
know that the upward force is
equal to the downward force,

00:08:34.250 --> 00:08:35.390
so the upward pressure
is equal to

00:08:35.390 --> 00:08:37.520
the downward pressure.

00:08:37.520 --> 00:08:41.640
Let's use that in an example.

00:08:41.640 --> 00:08:46.650
If I were on the same planet,
and this is water, and so the

00:08:46.650 --> 00:08:53.330
density of water-- and this is
something good to memorize--

00:08:53.330 --> 00:08:58.805
is 1,000 kilograms
per meter cubed.

00:09:02.710 --> 00:09:06.400
Let's say that we have no
atmosphere, but I were to go

00:09:06.400 --> 00:09:09.230
10 meters under the
water-- roughly 30

00:09:09.230 --> 00:09:10.000
feet under the water.

00:09:10.000 --> 00:09:11.980
What would be the
pressure on me?

00:09:11.980 --> 00:09:16.840
My pressure would be the density
of water, which is

00:09:16.840 --> 00:09:19.760
1,000 kilograms per meter
cubed-- make sure your units

00:09:19.760 --> 00:09:21.750
are right, and I'm running out
of space, so I don't have the

00:09:21.750 --> 00:09:25.110
units-- times the height,
10 meters, times the

00:09:25.110 --> 00:09:28.970
gravitational acceleration, 9.8
meters per second squared.

00:09:28.970 --> 00:09:30.440
It's a good exercise for
you to make sure

00:09:30.440 --> 00:09:32.210
the units work out.

00:09:32.210 --> 00:09:36.000
It's 10,000 times 9.8, so the
pressure is going to be equal

00:09:36.000 --> 00:09:41.200
to 98,000 pascals.

00:09:41.200 --> 00:09:42.500
This actually isn't that
much-- it just

00:09:42.500 --> 00:09:43.960
sounds like a lot.

00:09:43.960 --> 00:09:47.100
We'll actually see that this
is almost one atmosphere,

00:09:47.100 --> 00:09:50.450
which is the pressure at sea
level in France, I think.

00:09:50.450 --> 00:09:53.380
Anyway, I'll see you
in the next video.


Finding height of fluid in a barometer

WEBVTT
Kind: captions
Language: en

00:00:01.220 --> 00:00:04.070
In the last video, we learned
that the pressure at some

00:00:04.070 --> 00:00:10.120
depth in a fluid is equal to the
density of the fluid times

00:00:10.120 --> 00:00:13.670
how deep we are in the fluid,
or how high is the column of

00:00:13.670 --> 00:00:16.990
fluid above us times gravity.

00:00:16.990 --> 00:00:19.870
Let's see if we can use that
to solve a fairly typical

00:00:19.870 --> 00:00:22.750
problem that you'll see in your
physics class, or even on

00:00:22.750 --> 00:00:24.660
an AP physics test.

00:00:24.660 --> 00:00:26.000
Let's say that I have a bowl.

00:00:37.900 --> 00:00:41.740
And in that bowl, I have
mercury, and then I also have

00:00:41.740 --> 00:00:45.000
this kind of inverted test
tube that I stick in the

00:00:45.000 --> 00:00:47.480
middle of-- this is the side
view of the bowl, and I'll

00:00:47.480 --> 00:00:48.950
draw everything shortly.

00:00:48.950 --> 00:00:50.960
Let's say my test tube looks
something like this.

00:00:58.320 --> 00:01:00.090
Let's say I have no air in this
test tube-- there's a

00:01:00.090 --> 00:01:03.120
vacuum here-- but the outside
of the bowl, this whole area

00:01:03.120 --> 00:01:04.950
out here, this is exposed
to the air.

00:01:04.950 --> 00:01:08.530
We are actually on Earth, or
actually in Paris, France, at

00:01:08.530 --> 00:01:14.550
sea level, because that's what
an atmosphere is defined as--

00:01:14.550 --> 00:01:15.940
the atmospheric pressure.

00:01:15.940 --> 00:01:17.970
Essentially, the way you could
think about it-- the weight of

00:01:17.970 --> 00:01:21.830
all of the air above us is
pushing down on the surface of

00:01:21.830 --> 00:01:24.270
this bowl at one atmosphere.

00:01:24.270 --> 00:01:27.530
An atmosphere is just the
pressure of all of the air

00:01:27.530 --> 00:01:31.000
above you at sea level
in Paris, France.

00:01:31.000 --> 00:01:32.650
And in the bowl,
I have mercury.

00:02:05.380 --> 00:02:08.536
Let's say that that mercury--
there's no air in here, and it

00:02:08.536 --> 00:02:11.000
is actually going to go up
this column a little bit.

00:02:11.000 --> 00:02:14.200
We're going to do the math as
far as-- one, we'll see why

00:02:14.200 --> 00:02:16.830
it's going up, and then we'll do
the math to figure out how

00:02:16.830 --> 00:02:19.110
high up does it go.

00:02:19.110 --> 00:02:24.030
Say the mercury goes up
some distance-- this

00:02:24.030 --> 00:02:25.280
is all still mercury.

00:02:29.498 --> 00:02:31.820
And this is actually how a
barometer works; this is

00:02:31.820 --> 00:02:34.860
something that measures
pressure.

00:02:34.860 --> 00:02:39.730
Over here at this part, above
the mercury, but still within

00:02:39.730 --> 00:02:45.040
our little test tube, we have
a vacuum-- there is no air.

00:02:45.040 --> 00:02:47.680
Vacuum is one of my favorite
words, because it has

00:02:47.680 --> 00:02:48.930
two u's in a row.

00:02:52.400 --> 00:02:55.740
We have this set up, and so my
question to you is-- how high

00:02:55.740 --> 00:02:58.430
is this column of mercury
going to go?

00:03:02.210 --> 00:03:05.270
First of all, let's just have
the intuition as to why this

00:03:05.270 --> 00:03:07.660
thing is going up
to begin with.

00:03:07.660 --> 00:03:09.780
We have all this pressure from
all of the air above us-- I

00:03:09.780 --> 00:03:12.320
know it's a little un-intuitive
for us, because

00:03:12.320 --> 00:03:15.140
we're used to all of that
pressure on our shoulders all

00:03:15.140 --> 00:03:18.520
of the time, so we don't really
imagine it, but there

00:03:18.520 --> 00:03:21.340
is literally the weight of
the atmosphere above us.

00:03:21.340 --> 00:03:25.830
That's going to be pushing down
on the surface of the

00:03:25.830 --> 00:03:28.420
mercury on the outside
of the test tube.

00:03:28.420 --> 00:03:31.630
Since there's no pressure here,
the mercury is going to

00:03:31.630 --> 00:03:34.220
go upwards here.

00:03:34.220 --> 00:03:36.810
This state that I've drawn is
a static state-- we have

00:03:36.810 --> 00:03:40.260
assumed that all the
motion has stopped.

00:03:40.260 --> 00:03:41.267
So let's try to solve
this problem.

00:03:41.267 --> 00:03:44.300
Oh, and there are a couple of
things we have to know before

00:03:44.300 --> 00:03:46.140
we do this problem.

00:03:46.140 --> 00:03:50.710
It's mercury, and we know the
specific gravity-- I'm using

00:03:50.710 --> 00:03:53.000
terminology, because a lot of
these problems, the hardest

00:03:53.000 --> 00:04:09.350
part is the terminology--
of mercury is 13.6.

00:04:09.350 --> 00:04:12.230
That's often a daunting
statement on a test-- you know

00:04:12.230 --> 00:04:13.860
how to do all the math, and all
of a sudden you go, what

00:04:13.860 --> 00:04:15.060
is specific gravity?

00:04:15.060 --> 00:04:19.589
All specific gravity is, is the
ratio of how dense that

00:04:19.589 --> 00:04:21.300
substance is to water.

00:04:21.300 --> 00:04:29.460
All that means is that
mercury is 13.6

00:04:29.460 --> 00:04:37.365
times as dense as water.

00:04:40.260 --> 00:04:42.070
Hopefully, after the last
video-- because I told you

00:04:42.070 --> 00:04:44.240
to-- you should have memorized
the density of water.

00:04:44.240 --> 00:04:48.870
It's 1,000 kilograms per meter
cubed, so the density of

00:04:48.870 --> 00:04:53.210
mercury-- let's write that down,
and that's the rho, or

00:04:53.210 --> 00:04:56.400
little p, depending on how you
want to do it-- is going to be

00:04:56.400 --> 00:05:01.240
equal to 13.6 times the density
of water, or times

00:05:01.240 --> 00:05:11.960
1,000 kilograms per
meter cubed.

00:05:11.960 --> 00:05:14.830
Let's go back to the problem.

00:05:14.830 --> 00:05:17.060
What we want to know
is how high this

00:05:17.060 --> 00:05:19.350
column of mercury is.

00:05:19.350 --> 00:05:22.290
We know that the pressure--
let's consider this point

00:05:22.290 --> 00:05:25.700
right here, which is essentially
the base of this

00:05:25.700 --> 00:05:27.270
column of mercury.

00:05:27.270 --> 00:05:30.030
What we're saying is the
pressure on the base of this

00:05:30.030 --> 00:05:33.630
column of mercury right here, or
the pressure at this point

00:05:33.630 --> 00:05:39.800
down, has to be the same thing
as the pressure up, because

00:05:39.800 --> 00:05:42.180
the mercury isn't moving--
we're in a static state.

00:05:42.180 --> 00:05:45.180
We learned several videos ago
that the pressure in is equal

00:05:45.180 --> 00:05:50.350
to the pressure out on
a liquid system.

00:05:50.350 --> 00:05:53.740
Essentially, I have one
atmosphere pushing down here

00:05:53.740 --> 00:05:56.380
on the outside of the surface,
so I must have one atmosphere

00:05:56.380 --> 00:05:57.630
pushing up here.

00:05:59.820 --> 00:06:03.590
The pressure pushing up at this
point right here-- we

00:06:03.590 --> 00:06:05.520
could imagine that we have
that aluminum foil there

00:06:05.520 --> 00:06:09.850
again, and just imagine where
the pressure is hitting-- is

00:06:09.850 --> 00:06:14.450
one atmosphere, so the pressure
down right here must

00:06:14.450 --> 00:06:18.090
be one atmosphere.

00:06:18.090 --> 00:06:20.840
What's creating the pressure
down right there?

00:06:20.840 --> 00:06:25.330
It's essentially this column
of water, or it's this

00:06:25.330 --> 00:06:28.380
formula, which we learned
in the last video.

00:06:28.380 --> 00:06:31.000
What we now know is that the
density of the mercury, times

00:06:31.000 --> 00:06:33.880
the height of the column of
water, times the acceleration

00:06:33.880 --> 00:06:37.300
of gravity on Earth-- which is
where we are-- has to equal

00:06:37.300 --> 00:06:40.810
one atmosphere, because it has
to offset the atmosphere

00:06:40.810 --> 00:06:43.820
that's pushing on the outside
and pushing up here.

00:06:43.820 --> 00:06:52.880
The density of mercury is this:
13.6 thousand, so 13,600

00:06:52.880 --> 00:06:58.550
kilogram meters per
meter cubed.

00:06:58.550 --> 00:07:01.060
That's the density times the
height-- we don't know what

00:07:01.060 --> 00:07:03.695
the height is, that's going to
be in meters-- times the

00:07:03.695 --> 00:07:06.460
acceleration of gravity,
which is 9.8

00:07:06.460 --> 00:07:09.340
meters per second squared.

00:07:09.340 --> 00:07:12.350
It's going to be equal
to one atmosphere.

00:07:12.350 --> 00:07:14.020
Now you're saying-- Sal,
this is strange.

00:07:14.020 --> 00:07:15.850
I've never seen this atmosphere
before-- we've

00:07:15.850 --> 00:07:18.600
talked a lot about it, but how
does an atmosphere relate to

00:07:18.600 --> 00:07:21.330
pascals or newtons?

00:07:21.330 --> 00:07:23.320
This is something else you
should memorize: one

00:07:23.320 --> 00:07:32.120
atmosphere is equal to 103,000
pascals, and that also equals

00:07:32.120 --> 00:07:38.660
103,000 newtons per
meter squared.

00:07:38.660 --> 00:07:41.350
One atmosphere is how much we're
pushing down out here.

00:07:41.350 --> 00:07:43.230
So it's how much we're pushing
up here, and that's going to

00:07:43.230 --> 00:07:45.560
be equal to the amount of
pressure at this point from

00:07:45.560 --> 00:07:47.650
this column of mercury.

00:07:47.650 --> 00:07:54.380
One atmosphere is exactly this
much, which equals 103,000

00:07:54.380 --> 00:07:56.800
newtons per meters squared.

00:07:59.430 --> 00:08:07.660
If we divide both sides by
13,609.8, we get that the

00:08:07.660 --> 00:08:21.130
height is equal to 103,000
newtons per meter cubed, over

00:08:21.130 --> 00:08:36.659
13,600 kilograms per meter cubed
times 9.8 meters per

00:08:36.659 --> 00:08:37.909
second squared.

00:08:40.480 --> 00:08:41.940
Make sure you always have the
units right-- that's the

00:08:41.940 --> 00:08:44.600
hardest thing about these
problems, just to know that an

00:08:44.600 --> 00:08:49.210
atmosphere is 103,000 pascals,
which is also the same as

00:08:49.210 --> 00:08:52.410
newtons per meter squared.

00:08:52.410 --> 00:09:01.810
Let's just do the math, so let
me type this in-- 103,000

00:09:01.810 --> 00:09:17.000
divided by 13,600 divided
by 9.8 equals 0.77.

00:09:17.000 --> 00:09:19.250
We were dealing with newtons,
so height is

00:09:19.250 --> 00:09:22.870
equal to 0.77 meters.

00:09:22.870 --> 00:09:24.720
And you should see that the
units actually work, because

00:09:24.720 --> 00:09:27.030
we have a meters cubed in the
denominator up here, we have a

00:09:27.030 --> 00:09:29.060
meters cubed in the denominator
down here, and

00:09:29.060 --> 00:09:32.260
then we have kilogram meters
per second squared here.

00:09:32.260 --> 00:09:35.780
We have newtons up here,
but what's a newton?

00:09:35.780 --> 00:09:39.720
A newton is a kilogram meter
squared per second, so when

00:09:39.720 --> 00:09:42.700
you divide you have kilogram
meters squared per second

00:09:42.700 --> 00:09:44.060
squared, and here you
have kilogram

00:09:44.060 --> 00:09:45.390
meter per second squared.

00:09:45.390 --> 00:09:47.080
When you do all the division of
the units, all you're left

00:09:47.080 --> 00:09:51.450
with is meters, so we have 0.77
meters, or roughly 77

00:09:51.450 --> 00:09:54.850
centimeters-- is how high this
column of mercury is.

00:09:54.850 --> 00:09:57.255
And you can make a barometer out
of it-- you can say, let

00:09:57.255 --> 00:09:59.760
me make a little notch on this
test tube, and that represents

00:09:59.760 --> 00:10:01.550
one atmosphere.

00:10:01.550 --> 00:10:04.540
You can go around and figure
out how many atmospheres

00:10:04.540 --> 00:10:05.780
different parts of
the globe are.

00:10:05.780 --> 00:10:07.200
Anyway, I've run out of time.

00:10:07.200 --> 00:10:08.940
See you in the next video.


Pressure and Pascal's principle (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:01.250
Welcome back.

00:00:01.250 --> 00:00:02.990
To just review what I was doing
on the last video before

00:00:02.990 --> 00:00:07.050
I ran out of time, I said that
conservation of energy tells

00:00:07.050 --> 00:00:09.980
us that the work I've put into
the system or the energy that

00:00:09.980 --> 00:00:11.300
I've put into the system--
because they're really the

00:00:11.300 --> 00:00:14.740
same thing-- is equal to the
work that I get out of the

00:00:14.740 --> 00:00:17.440
system, or the energy that
I get out of the system.

00:00:17.440 --> 00:00:20.480
That means that the input work
is equal to the output work,

00:00:20.480 --> 00:00:23.540
or that the input force times
the input distance is equal to

00:00:23.540 --> 00:00:25.610
the output force times the
output distance-- that's just

00:00:25.610 --> 00:00:27.020
the definition of work.

00:00:27.020 --> 00:00:30.040
Let me just rewrite this
equation here.

00:00:30.040 --> 00:00:33.730
If I could just rewrite this
exact equation, I could say--

00:00:33.730 --> 00:00:42.780
the input force, and let me just
divide it by this area.

00:00:42.780 --> 00:00:44.900
The input here-- I'm pressing
down this piston that's

00:00:44.900 --> 00:00:47.580
pressing down on this
area of water.

00:00:47.580 --> 00:00:54.560
So this input force-- times
the input area.

00:00:54.560 --> 00:00:58.380
Let's call the input 1, and
call the output 2 for

00:00:58.380 --> 00:00:59.630
simplicity.

00:01:01.590 --> 00:01:04.069
Let's say I have a piston
on the top here.

00:01:04.069 --> 00:01:09.320
Let me do this in a good color--
brown is good color.

00:01:09.320 --> 00:01:15.060
I have another piston here, and
there's going to be some

00:01:15.060 --> 00:01:18.140
outward force F2.

00:01:18.140 --> 00:01:20.500
The general notion is that I'm
pushing on this water, the

00:01:20.500 --> 00:01:22.740
water can't be compressed, so
the water's going to push up

00:01:22.740 --> 00:01:25.470
on this end.

00:01:25.470 --> 00:01:28.880
The input force times the input
distance is going to be

00:01:28.880 --> 00:01:32.540
equal to the output force times
the output distance

00:01:32.540 --> 00:01:34.560
right-- this is just the law of
conservation of energy and

00:01:34.560 --> 00:01:37.590
everything we did with
work, et cetera.

00:01:37.590 --> 00:01:40.060
I'm rewriting this equation, so
if I take the input force

00:01:40.060 --> 00:01:46.610
and divide by the input area--
let me switch back to green--

00:01:46.610 --> 00:01:50.660
then I multiply by the
area, and then I just

00:01:50.660 --> 00:01:53.222
multiply times D1.

00:01:53.222 --> 00:01:55.730
You see what I did here-- I just
multiplied and divided by

00:01:55.730 --> 00:01:56.710
A1, which you can do.

00:01:56.710 --> 00:01:59.230
You can multiply and divide by
any number, and these two

00:01:59.230 --> 00:02:00.260
cancel out.

00:02:00.260 --> 00:02:03.160
It's equal to the same thing
on the other side, which is

00:02:03.160 --> 00:02:07.420
F2-- I'm not good at managing
my space on my whiteboard--

00:02:07.420 --> 00:02:13.790
over A2 times A2 times D2.

00:02:13.790 --> 00:02:15.270
Hopefully that makes sense.

00:02:15.270 --> 00:02:20.890
What's this quantity right here,
this F1 divided by A1?

00:02:20.890 --> 00:02:24.640
Force divided by area, if you
haven't been familiar with it

00:02:24.640 --> 00:02:26.780
already, and if you're just
watching my videos there's no

00:02:26.780 --> 00:02:29.450
reason for you to be, is
defined as pressure.

00:02:29.450 --> 00:02:34.260
Pressure is force in a given
area, so this is pressure--

00:02:34.260 --> 00:02:35.620
we'll call this the
pressure that I'm

00:02:35.620 --> 00:02:38.165
inputting into the system.

00:02:40.700 --> 00:02:43.360
What's area 1 times
distance 1?

00:02:43.360 --> 00:02:46.950
That's the area of the tube
at this point, the

00:02:46.950 --> 00:02:48.840
cross-sectional area,
times this distance.

00:02:48.840 --> 00:02:51.750
That's equal to this volume
that I calculated in the

00:02:51.750 --> 00:02:53.520
previous video-- we could
say that's the

00:02:53.520 --> 00:02:56.410
input volume, or V1.

00:02:56.410 --> 00:03:02.420
Pressure times V1 is equal to
the output pressure-- force 2

00:03:02.420 --> 00:03:06.115
divided by area 2 is the output
pressure that the water

00:03:06.115 --> 00:03:07.670
is exerting on this piston.

00:03:07.670 --> 00:03:11.560
So that's the output
pressure, P2.

00:03:11.560 --> 00:03:14.700
And what's area 2 times D2?

00:03:14.700 --> 00:03:18.110
The cross sectional area, times
the height at which how

00:03:18.110 --> 00:03:20.490
much the water's being displaced
upward, that is

00:03:20.490 --> 00:03:21.740
equal to volume 2.

00:03:24.460 --> 00:03:26.680
But what do we know about
these two volumes?

00:03:26.680 --> 00:03:29.420
I went over it probably
redundantly in the previous

00:03:29.420 --> 00:03:33.690
video-- those two volumes are
equal, V1 is equal to V2, so

00:03:33.690 --> 00:03:36.170
we could just divide both
sides by that equation.

00:03:36.170 --> 00:03:42.040
You get the pressure input is
equal to the pressure output,

00:03:42.040 --> 00:03:43.380
so P1 is equal to P2.

00:03:52.480 --> 00:03:54.410
I did all of that just to show
you that this isn't a new

00:03:54.410 --> 00:03:57.190
concept: this is just the
conservation of energy.

00:03:57.190 --> 00:04:00.120
The only new thing I did is I
divided-- we have this notion

00:04:00.120 --> 00:04:03.200
of the cross-sectional area,
and we have this notion of

00:04:03.200 --> 00:04:06.190
pressure-- so where
does that help us?

00:04:06.190 --> 00:04:10.320
This actually tells us-- and
you can do this example in

00:04:10.320 --> 00:04:13.660
multiple situations, but I like
to think of if we didn't

00:04:13.660 --> 00:04:16.170
have gravity first, because
gravity tends to confuse

00:04:16.170 --> 00:04:19.070
things, but we'll introduce
gravity in a video or two-- is

00:04:19.070 --> 00:04:25.490
that when you have any external
pressure onto a

00:04:25.490 --> 00:04:29.490
liquid, onto an incompressible
fluid, that pressure is

00:04:29.490 --> 00:04:32.960
distributed evenly throughout
the fluid.

00:04:32.960 --> 00:04:36.770
That's what we essentially just
proved just using the law

00:04:36.770 --> 00:04:40.030
of conservation of energy, and
everything we know about work.

00:04:40.030 --> 00:04:43.370
What I just said is called
Pascal's principle: if any

00:04:43.370 --> 00:04:46.390
external pressure is applied to
a fluid, that pressure is

00:04:46.390 --> 00:04:49.660
distributed throughout
the fluid equally.

00:04:49.660 --> 00:04:51.640
Another way to think about it--
we proved it with this

00:04:51.640 --> 00:04:59.710
little drawing here-- is, let's
say that I have a tube,

00:04:59.710 --> 00:05:01.520
and at the end of the
tube is a balloon.

00:05:01.520 --> 00:05:04.720
Let's say I'm doing this
on the Space Shuttle.

00:05:04.720 --> 00:05:09.320
It's saying that if I increase--
say I have some

00:05:09.320 --> 00:05:10.570
piston here.

00:05:14.350 --> 00:05:17.560
This is stable, and
I have water

00:05:17.560 --> 00:05:18.810
throughout this whole thing.

00:05:22.480 --> 00:05:25.950
Let me see if I can use that
field function again-- oh no,

00:05:25.950 --> 00:05:29.760
there must have been a
hole in my drawing.

00:05:29.760 --> 00:05:31.120
Let me just draw the water.

00:05:31.120 --> 00:05:36.970
I have water throughout this
whole thing, and all Pascal's

00:05:36.970 --> 00:05:39.410
principle is telling us that
if I were to apply some

00:05:39.410 --> 00:05:51.360
pressure here, that that net
pressure, that extra pressure

00:05:51.360 --> 00:05:55.040
I'm applying, is going to
compress this little bit.

00:05:55.040 --> 00:05:56.770
That extra compression is
going to be distributed

00:05:56.770 --> 00:05:58.130
through the whole balloon.

00:05:58.130 --> 00:06:00.330
Let's say that this right here
is rigid-- it's some kind of

00:06:00.330 --> 00:06:01.450
middle structure.

00:06:01.450 --> 00:06:06.150
The rest of the balloon is going
to expand uniformly, so

00:06:06.150 --> 00:06:08.830
that increased pressure I'm
doing is going through the

00:06:08.830 --> 00:06:09.100
whole thing.

00:06:09.100 --> 00:06:12.980
It's not like the balloon will
get longer, or that the

00:06:12.980 --> 00:06:16.010
pressure is just translated
down here, or that just up

00:06:16.010 --> 00:06:17.615
here the balloon's going to get
wider and it's just going

00:06:17.615 --> 00:06:19.170
to stay the same length there.

00:06:19.170 --> 00:06:22.800
Hopefully, that gives you a
little bit of intuition.

00:06:22.800 --> 00:06:25.050
Going back to what I had drawn
before, that's actually

00:06:25.050 --> 00:06:28.430
interesting, because that's
actually another simple or

00:06:28.430 --> 00:06:32.220
maybe not so simple machine
that we've constructed.

00:06:32.220 --> 00:06:36.190
I almost defined it as a
simple machine when I

00:06:36.190 --> 00:06:37.100
initially drew it.

00:06:37.100 --> 00:06:40.520
Let's draw that weird thing
again, where it looks like

00:06:40.520 --> 00:06:44.440
this, where I have
water in it.

00:06:54.390 --> 00:06:56.678
Let's make sure I fill it, so
that when I do the fill, it

00:06:56.678 --> 00:06:59.720
will completely fill, and
doesn't fill other things.

00:07:02.450 --> 00:07:05.820
This is cool, because this is
now another simple machine.

00:07:05.820 --> 00:07:15.320
We know that the pressure in is
equal to the pressure out.

00:07:21.330 --> 00:07:26.990
And pressure is force divided
by area, so the force in,

00:07:26.990 --> 00:07:32.080
divided by the area in, is equal
to the force out divided

00:07:32.080 --> 00:07:34.200
by the area out.

00:07:37.350 --> 00:07:40.200
Let me give you an example:
let's say that I were to apply

00:07:40.200 --> 00:07:49.450
with a pressure in equal
to 10 pascals.

00:07:49.450 --> 00:07:51.590
That's a new word, and it's
named after Pascal's

00:07:51.590 --> 00:07:55.090
principle, for Blaise Pascal.

00:07:55.090 --> 00:07:56.070
What is a pascal?

00:07:56.070 --> 00:08:02.310
That is just equal to 10 newtons
per meter squared.

00:08:02.310 --> 00:08:06.460
That's all a pascal is-- it's
a newton per meter squared,

00:08:06.460 --> 00:08:08.770
it's a very natural unit.

00:08:08.770 --> 00:08:12.930
Let's say my pressure in is 10
pascals, and let's say that my

00:08:12.930 --> 00:08:20.930
input area is 2 square meters.

00:08:20.930 --> 00:08:22.560
If I looked the surface of the
water there it would be 2

00:08:22.560 --> 00:08:31.500
square meters, and let's say
that my output area is equal

00:08:31.500 --> 00:08:38.640
to 4 meters squared.

00:08:41.780 --> 00:08:45.550
What I'm saying is that I can
push on a piston here, and

00:08:45.550 --> 00:08:50.470
that the water's going to push
up with some piston here.

00:08:50.470 --> 00:08:53.220
First of all, I told you what my
input pressure is-- what's

00:08:53.220 --> 00:08:55.840
my input force?

00:08:55.840 --> 00:09:00.900
Input pressure is equal to input
force divided by input

00:09:00.900 --> 00:09:06.260
area, so 10 pascals is equal to
my input force divided by

00:09:06.260 --> 00:09:09.470
my area, so I multiply
both sides by 2.

00:09:09.470 --> 00:09:13.640
I get input force is equal
to 20 newtons.

00:09:13.640 --> 00:09:15.650
My question to you is what
is the output force?

00:09:15.650 --> 00:09:17.810
How much force is the
system going to push

00:09:17.810 --> 00:09:20.180
upwards at this end?

00:09:20.180 --> 00:09:24.880
We know that must if my input
pressure was 10 pascals, my

00:09:24.880 --> 00:09:28.270
output pressure would
also be 10 pascals.

00:09:28.270 --> 00:09:34.390
So I also have 10 pascals is
equal to my out force over my

00:09:34.390 --> 00:09:37.760
out cross-sectional area.

00:09:37.760 --> 00:09:40.880
So I'll have a piston here,
and it goes up like that.

00:09:40.880 --> 00:09:46.710
That's 4 meters, so I do 4
times 10, and so I get 40

00:09:46.710 --> 00:09:49.450
newtons is equal to
my output force.

00:09:49.450 --> 00:09:50.740
So what just happened here?

00:09:50.740 --> 00:09:55.920
I inputted-- so my input force
is equal to 20 newtons, and my

00:09:55.920 --> 00:10:00.550
output force is equal to 40
newtons, so I just doubled my

00:10:00.550 --> 00:10:03.840
force, or essentially I had a
mechanical advantage of 2.

00:10:03.840 --> 00:10:07.530
This is an example of a
simple machine, and

00:10:07.530 --> 00:10:09.040
it's a hydraulic machine.

00:10:09.040 --> 00:10:10.310
Anyway, I've just
run out of time.

00:10:10.310 --> 00:10:11.560
I'll see you in the next video.


Pressure and Pascal's principle (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.100
Let's learn a little
bit about fluids.

00:00:04.100 --> 00:00:07.940
You probably have some notion of
what a fluid is, but let's

00:00:07.940 --> 00:00:10.110
talk about it in the physics
sense, or maybe even the

00:00:10.110 --> 00:00:12.280
chemistry sense, depending
on in what context you're

00:00:12.280 --> 00:00:13.470
watching this video.

00:00:13.470 --> 00:00:14.680
So a fluid is anything
that takes the

00:00:14.680 --> 00:00:16.030
shape of its container.

00:00:16.030 --> 00:00:28.550
For example, if I had a glass
sphere, and let's say that I

00:00:28.550 --> 00:00:31.110
completely filled this glass
sphere with water.

00:00:31.110 --> 00:00:32.640
I was going to say that we're in
a zero gravity environment,

00:00:32.640 --> 00:00:34.130
but you really don't
even need that.

00:00:34.130 --> 00:00:39.300
Let's say that every cubic
centimeter or cubic meter of

00:00:39.300 --> 00:00:41.010
this glass sphere is
filled with water.

00:00:44.380 --> 00:00:46.810
Let's say that it's not a glass,
but a rubber sphere.

00:00:46.810 --> 00:00:49.700
If I were to change the shape of
the sphere, but not really

00:00:49.700 --> 00:00:53.520
change the volume-- if I were
to change the shape of the

00:00:53.520 --> 00:00:56.730
sphere where it looks like this
now-- the water would

00:00:56.730 --> 00:01:01.220
just change its shape
with the container.

00:01:01.220 --> 00:01:03.620
The water would just change in
the shape of the container,

00:01:03.620 --> 00:01:07.410
and in this case, I
have green water.

00:01:07.410 --> 00:01:11.580
The same is also true if that
was oxygen, or if that was

00:01:11.580 --> 00:01:13.110
just some gas.

00:01:13.110 --> 00:01:16.450
It would fill the container,
and in this situation, it

00:01:16.450 --> 00:01:20.130
would also fill the newly
shaped container.

00:01:20.130 --> 00:01:26.070
A fluid, in general, takes the
shape of its container.

00:01:31.420 --> 00:01:34.510
And I just gave you two examples
of fluids-- you have

00:01:34.510 --> 00:01:41.370
liquids, and you have gases.

00:01:41.370 --> 00:01:43.590
Those are two types of fluid:
both of those things take the

00:01:43.590 --> 00:01:45.180
shape of the container.

00:01:45.180 --> 00:01:48.180
What's the difference between
a liquid and a gas, then?

00:01:48.180 --> 00:01:55.760
A gas is compressible, which
means that I could actually

00:01:55.760 --> 00:02:00.110
decrease the volume of this
container and the gas will

00:02:00.110 --> 00:02:02.500
just become denser within
the container.

00:02:02.500 --> 00:02:05.750
You can think of it as if I blew
air into a balloon-- you

00:02:05.750 --> 00:02:07.240
could squeeze that balloon
a little bit.

00:02:07.240 --> 00:02:09.800
There's air in there, and at
some point the pressure might

00:02:09.800 --> 00:02:11.200
get high enough to pop
the balloon, but

00:02:11.200 --> 00:02:12.400
you can squeeze it.

00:02:12.400 --> 00:02:13.780
A liquid is incompressible.

00:02:21.080 --> 00:02:23.020
How do I know that a liquid
is incompressible?

00:02:23.020 --> 00:02:25.750
Imagine the same balloon filled
with water-- completely

00:02:25.750 --> 00:02:26.660
filled with water.

00:02:26.660 --> 00:02:30.760
If you squeezed on that balloon
from every side-- let

00:02:30.760 --> 00:02:33.550
me pick a different color-- I
have this balloon, and it was

00:02:33.550 --> 00:02:34.660
filled with water.

00:02:34.660 --> 00:02:37.370
If you squeezed on this balloon
from every side, you

00:02:37.370 --> 00:02:39.900
would not be able to change the
volume of this balloon.

00:02:39.900 --> 00:02:42.360
No matter what you do, you would
not be able to change

00:02:42.360 --> 00:02:44.870
the volume of this balloon, no
matter how much force or

00:02:44.870 --> 00:02:48.190
pressure you put from any side
on it, while if this was

00:02:48.190 --> 00:02:53.650
filled with gas-- and magenta,
blue in for gas-- you actually

00:02:53.650 --> 00:02:56.120
could decrease the volume by
just increasing the pressure

00:02:56.120 --> 00:02:59.520
on all sides of the balloon.

00:02:59.520 --> 00:03:00.780
You can actually squeeze
it, and make the

00:03:00.780 --> 00:03:02.060
entire volume smaller.

00:03:02.060 --> 00:03:03.930
That's the difference between
a liquid and a gas-- gas is

00:03:03.930 --> 00:03:06.870
compressible, liquid isn't, and
we'll learn later that you

00:03:06.870 --> 00:03:09.710
can turn a liquid into a gas,
gas into a liquid, and turn

00:03:09.710 --> 00:03:11.850
liquids into solids, but we'll
learn all about that later.

00:03:11.850 --> 00:03:15.620
This is a pretty good working
definition of that.

00:03:15.620 --> 00:03:17.920
Let's use that, and now we're
going to actually just focus

00:03:17.920 --> 00:03:20.740
on the liquids to see if we
could learn a little bit about

00:03:20.740 --> 00:03:25.410
liquid motion, or maybe even
fluid motion in general.

00:03:25.410 --> 00:03:34.470
Let me draw something else--
let's say I had a situation

00:03:34.470 --> 00:03:40.160
where I have this weird shaped
object which tends to show up

00:03:40.160 --> 00:03:43.050
in a lot of physics books, which
I'll draw in yellow.

00:03:43.050 --> 00:03:45.700
This weird shaped container
where it's relatively narrow

00:03:45.700 --> 00:03:51.250
there, and then it goes
and U-turns into

00:03:51.250 --> 00:03:54.010
a much larger opening.

00:03:58.480 --> 00:04:04.780
Let's say that the area of this
opening is A1, and the

00:04:04.780 --> 00:04:09.110
area of this opening is A2--
this one is bigger.

00:04:09.110 --> 00:04:15.550
Now let's fill this thing with
some liquid, which will be

00:04:15.550 --> 00:04:18.930
blue-- so that's my liquid.

00:04:23.890 --> 00:04:26.740
Let me see if they have
this tool-- there

00:04:26.740 --> 00:04:27.500
you go, look at that.

00:04:27.500 --> 00:04:28.880
I filled it with liquid
so quickly.

00:04:32.430 --> 00:04:35.190
This was liquid-- it's not just
a fluid, and so what's

00:04:35.190 --> 00:04:36.250
the important thing
about liquid?

00:04:36.250 --> 00:04:38.950
It's incompressible.

00:04:38.950 --> 00:04:44.610
Let's take what we know about
force-- actually about work--

00:04:44.610 --> 00:04:47.990
and see if we can come up with
any rules about force and

00:04:47.990 --> 00:04:49.250
pressure with liquids.

00:04:49.250 --> 00:04:50.680
So what do we know about work?

00:04:50.680 --> 00:04:54.490
Work is force times distance, or
you can also view it as the

00:04:54.490 --> 00:04:57.910
energy put into the system--
I'll write it down here.

00:04:57.910 --> 00:05:03.480
Work is equal to force
times distance.

00:05:03.480 --> 00:05:08.790
We learned in mechanical
advantage that the work in--

00:05:08.790 --> 00:05:13.470
I'll do it with that I--
is equal to work out.

00:05:13.470 --> 00:05:15.500
The force times the distance
that you've put into a system

00:05:15.500 --> 00:05:16.940
is equal to the force
times the distance

00:05:16.940 --> 00:05:17.490
you put out of it.

00:05:17.490 --> 00:05:19.820
And you might want to review
the work chapters on that.

00:05:19.820 --> 00:05:21.700
That's just the little law of
conservation of energy,

00:05:21.700 --> 00:05:24.030
because work in is just the
energy that you're putting

00:05:24.030 --> 00:05:26.160
into a system-- it's measured
in joules-- and the work out

00:05:26.160 --> 00:05:28.010
is the energy that comes
out of the system.

00:05:28.010 --> 00:05:30.910
And that's just saying that
no energy is destroyed or

00:05:30.910 --> 00:05:34.270
created, it just turns into
different forms. Let's just

00:05:34.270 --> 00:05:36.940
use this definition: the force
times distance in is equal to

00:05:36.940 --> 00:05:38.260
force times distance out.

00:05:52.570 --> 00:05:56.080
Let's say that I pressed
with some force

00:05:56.080 --> 00:05:57.510
on this entire surface.

00:05:57.510 --> 00:06:02.360
Let's say I had a piston-- let
me see if I can draw a piston,

00:06:02.360 --> 00:06:04.320
and what's a good color for
a piston-- so let's add a

00:06:04.320 --> 00:06:05.690
magenta piston right here.

00:06:08.230 --> 00:06:14.350
I push down on this magenta
piston, and so I pushed down

00:06:14.350 --> 00:06:20.380
on this with a force of F1.

00:06:20.380 --> 00:06:25.790
Let's say I push it
a distance of D1--

00:06:25.790 --> 00:06:26.970
that's its initial position.

00:06:26.970 --> 00:06:30.460
Its final position-- let's see
what color, and the hardest

00:06:30.460 --> 00:06:32.980
part of these videos is picking
the color-- after I

00:06:32.980 --> 00:06:36.500
pushed, the piston
goes this far.

00:06:36.500 --> 00:06:41.120
This is the distance that I
pushed it-- this is D1.

00:06:41.120 --> 00:06:46.130
The water is here and I push
the water down D1 meters.

00:06:46.130 --> 00:06:50.890
In this situation, my work
in is F1 times D1.

00:06:50.890 --> 00:06:55.540
Let me ask you a question: how
much water did I displace?

00:06:55.540 --> 00:06:57.650
How much total water
did I displace?

00:06:57.650 --> 00:06:59.370
Well, it's this volume?

00:06:59.370 --> 00:07:02.370
I took this entire volume and
pushed it down, so what's the

00:07:02.370 --> 00:07:05.530
volume right there
that I displaced?

00:07:05.530 --> 00:07:09.360
The volume there is going to
be-- the initial volume that

00:07:09.360 --> 00:07:14.510
I'm displacing, or the
volume displaced, has

00:07:14.510 --> 00:07:16.640
to equal this distance.

00:07:16.640 --> 00:07:21.250
This is a cylinder of liquid,
so this distance times the

00:07:21.250 --> 00:07:24.040
area of the container
at that point.

00:07:24.040 --> 00:07:25.565
I'm assuming that it's constant
at that point, and

00:07:25.565 --> 00:07:32.620
then it changes after that,
so it equals area 1 times

00:07:32.620 --> 00:07:36.680
distance 1.

00:07:36.680 --> 00:07:41.510
We also know that that liquid
has to go someplace, because

00:07:41.510 --> 00:07:42.990
what do we know about
a liquid?

00:07:42.990 --> 00:07:47.510
We can't compress it, you can't
change its total volume,

00:07:47.510 --> 00:07:50.830
so all of that volume is going
to have to go someplace else.

00:07:50.830 --> 00:07:53.060
This is where the liquid was,
and the liquid is going to

00:07:53.060 --> 00:07:57.400
rise some level-- let's say that
it gets to this level,

00:07:57.400 --> 00:08:00.640
and this is its new level.

00:08:00.640 --> 00:08:05.260
It's going to change some
distance here, it's going to

00:08:05.260 --> 00:08:07.930
change some distance there,
and how do we know what

00:08:07.930 --> 00:08:09.400
distance that's going to be?

00:08:09.400 --> 00:08:12.960
The volume that it changes
here has to go someplace.

00:08:12.960 --> 00:08:14.910
You can say, that's going to
push on that, that's all going

00:08:14.910 --> 00:08:17.230
to push, and that liquid
has to go someplace.

00:08:17.230 --> 00:08:19.470
Essentially it's going to end
up-- it might not be the exact

00:08:19.470 --> 00:08:21.630
same molecules, but that might
displace some liquid here,

00:08:21.630 --> 00:08:23.530
that's going to displace some
liquid here and here and here

00:08:23.530 --> 00:08:25.700
and here and all the way until
the liquid up here gets

00:08:25.700 --> 00:08:27.880
displaced and gets
pushed upward.

00:08:27.880 --> 00:08:30.550
The volume that you're pushing
down here is the same volume

00:08:30.550 --> 00:08:32.470
that goes up right here.

00:08:32.470 --> 00:08:37.830
So what's the volume-- what's
the change in volume, or how

00:08:37.830 --> 00:08:40.610
much volume did you
push up here?

00:08:40.610 --> 00:08:44.580
This volume here is going to be
the distance 2 times this

00:08:44.580 --> 00:08:49.990
larger area, so we could say
volume 2 is going to be equal

00:08:49.990 --> 00:08:55.080
to the distance 2 times
this larger area.

00:08:55.080 --> 00:08:58.170
We know that this liquid is
incompressible, so this volume

00:08:58.170 --> 00:09:01.750
has to be the same
as this volume.

00:09:01.750 --> 00:09:05.470
We know that these two
quantities are equal to each

00:09:05.470 --> 00:09:12.410
other, so area 1 times distance
1 is going to be

00:09:12.410 --> 00:09:16.120
equal to this area times
this distance.

00:09:21.600 --> 00:09:22.580
Let's see what we can do.

00:09:22.580 --> 00:09:25.540
We know this, that the force
in times the distance in is

00:09:25.540 --> 00:09:28.550
equal to the force out times
the distance out.

00:09:28.550 --> 00:09:31.010
Let's take this equation-- I'm
going to switch back to green

00:09:31.010 --> 00:09:33.990
just so we don't lose
track of things--

00:09:33.990 --> 00:09:37.510
and divide both sides.

00:09:37.510 --> 00:09:39.230
Let's rewrite it--
so let's say I

00:09:39.230 --> 00:09:41.730
rewrote each input force.

00:09:41.730 --> 00:09:43.530
Actually, I'm about to run out
of time, so I'll continue this

00:09:43.530 --> 00:09:44.400
into the next video.

00:09:44.400 --> 00:09:45.980
See you soon.