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# Finding height of fluid in a barometer

Video transcript

In the last video, we learned
that the pressure at some depth in a fluid is equal to the
density of the fluid times how deep we are in the fluid,
or how high is the column of fluid above us times gravity. Let's see if we can use that
to solve a fairly typical problem that you'll see in your
physics class, or even on an AP physics test. Let's say that I have a bowl. And in that bowl, I have
mercury, and then I also have this kind of inverted test
tube that I stick in the middle of-- this is the side
view of the bowl, and I'll draw everything shortly. Let's say my test tube looks
something like this. Let's say I have no air in this
test tube-- there's a vacuum here-- but the outside
of the bowl, this whole area out here, this is exposed
to the air. We are actually on Earth, or
actually in Paris, France, at sea level, because that's what
an atmosphere is defined as-- the atmospheric pressure. Essentially, the way you could
think about it-- the weight of all of the air above us is
pushing down on the surface of this bowl at one atmosphere. An atmosphere is just the
pressure of all of the air above you at sea level
in Paris, France. And in the bowl,
I have mercury. Let's say that that mercury--
there's no air in here, and it is actually going to go up
this column a little bit. We're going to do the math as
far as-- one, we'll see why it's going up, and then we'll do
the math to figure out how high up does it go. Say the mercury goes up
some distance-- this is all still mercury. And this is actually how a
barometer works; this is something that measures
pressure. Over here at this part, above
the mercury, but still within our little test tube, we have
a vacuum-- there is no air. Vacuum is one of my favorite
words, because it has two u's in a row. We have this set up, and so my
question to you is-- how high is this column of mercury
going to go? First of all, let's just have
the intuition as to why this thing is going up
to begin with. We have all this pressure from
all of the air above us-- I know it's a little un-intuitive
for us, because we're used to all of that
pressure on our shoulders all of the time, so we don't really
imagine it, but there is literally the weight of
the atmosphere above us. That's going to be pushing down
on the surface of the mercury on the outside
of the test tube. Since there's no pressure here,
the mercury is going to go upwards here. This state that I've drawn is
a static state-- we have assumed that all the
motion has stopped. So let's try to solve
this problem. Oh, and there are a couple of
things we have to know before we do this problem. It's mercury, and we know the
specific gravity-- I'm using terminology, because a lot of
these problems, the hardest part is the terminology--
of mercury is 13.6. That's often a daunting
statement on a test-- you know how to do all the math, and all
of a sudden you go, what is specific gravity? All specific gravity is, is the
ratio of how dense that substance is to water. All that means is that
mercury is 13.6 times as dense as water. Hopefully, after the last
video-- because I told you to-- you should have memorized
the density of water. It's 1,000 kilograms per meter
cubed, so the density of mercury-- let's write that down,
and that's the rho, or little p, depending on how you
want to do it-- is going to be equal to 13.6 times the density
of water, or times 1,000 kilograms per
meter cubed. Let's go back to the problem. What we want to know
is how high this column of mercury is. We know that the pressure--
let's consider this point right here, which is essentially
the base of this column of mercury. What we're saying is the
pressure on the base of this column of mercury right here, or
the pressure at this point down, has to be the same thing
as the pressure up, because the mercury isn't moving--
we're in a static state. We learned several videos ago
that the pressure in is equal to the pressure out on
a liquid system. Essentially, I have one
atmosphere pushing down here on the outside of the surface,
so I must have one atmosphere pushing up here. The pressure pushing up at this
point right here-- we could imagine that we have
that aluminum foil there again, and just imagine where
the pressure is hitting-- is one atmosphere, so the pressure
down right here must be one atmosphere. What's creating the pressure
down right there? It's essentially this column
of water, or it's this formula, which we learned
in the last video. What we now know is that the
density of the mercury, times the height of the column of
water, times the acceleration of gravity on Earth-- which is
where we are-- has to equal one atmosphere, because it has
to offset the atmosphere that's pushing on the outside
and pushing up here. The density of mercury is this:
13.6 thousand, so 13,600 kilogram meters per
meter cubed. That's the density times the
height-- we don't know what the height is, that's going to
be in meters-- times the acceleration of gravity,
which is 9.8 meters per second squared. It's going to be equal
to one atmosphere. Now you're saying-- Sal,
this is strange. I've never seen this atmosphere
before-- we've talked a lot about it, but how
does an atmosphere relate to pascals or newtons? This is something else you
should memorize: one atmosphere is equal to 103,000
pascals, and that also equals 103,000 newtons per
meter squared. One atmosphere is how much we're
pushing down out here. So it's how much we're pushing
up here, and that's going to be equal to the amount of
pressure at this point from this column of mercury. One atmosphere is exactly this
much, which equals 103,000 newtons per meters squared. If we divide both sides by
13,609.8, we get that the height is equal to 103,000
newtons per meter cubed, over 13,600 kilograms per meter cubed
times 9.8 meters per second squared. Make sure you always have the
units right-- that's the hardest thing about these
problems, just to know that an atmosphere is 103,000 pascals,
which is also the same as newtons per meter squared. Let's just do the math, so let
me type this in-- 103,000 divided by 13,600 divided
by 9.8 equals 0.77. We were dealing with newtons,
so height is equal to 0.77 meters. And you should see that the
units actually work, because we have a meters cubed in the
denominator up here, we have a meters cubed in the denominator
down here, and then we have kilogram meters
per second squared here. We have newtons up here,
but what's a newton? A newton is a kilogram meter
squared per second, so when you divide you have kilogram
meters squared per second squared, and here you
have kilogram meter per second squared. When you do all the division of
the units, all you're left with is meters, so we have 0.77
meters, or roughly 77 centimeters-- is how high this
column of mercury is. And you can make a barometer out
of it-- you can say, let me make a little notch on this
test tube, and that represents one atmosphere. You can go around and figure
out how many atmospheres different parts of
the globe are. Anyway, I've run out of time. See you in the next video.