Fluids (part 4) Using our understanding of fluid pressure to figure out the height of a column of mercury.
Fluids (part 4)
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- 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.
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