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0:00
Bernoulli's equation. You might have heard of this before. A lot of science fair
0:06
projects, including one that I helped my grandson on a couple years ago, are on
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Bernoulli's equation, Bernoulli's law, Bernoulli's principle, got a lot of different
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names.
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So, we're going to add another adjective to our list of adjectives that we described.
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We're going to add another adjective to our list of adjectives that we describe flows
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with. Steady we've talked about a couple of times. Incompressible
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means you can't squash it. The density is always the same. And now we're going
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to add another one, inviscid, meaning that it's not viscous.
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You can ignore the viscosity of the flow. The
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viscosity is the resistance to shear in the
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flow. If you think about honey,
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if you put honey on the table and you put a book on top of
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that layer of honey, it's hard to push that book along the layer. And the
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reason is viscosity resists emotion. But we're going to ignore that in
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this particular concept. V, elevation Y,
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and density rho at two points are related by
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the following equation.
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Here's point one. Here's point two.
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Pressure at point one. Pressure
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at point two. The pressure
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at point one plus one -half rho v1 squared.
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1 half mv squared. And this is one way that helps me remember this equation.
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This looks like mgh.
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This term here, if you replace the roll with an m, looks like a kinetic
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energy. This term here, if you replace the roll with an m, This term here,
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if you replace the rho with an m and the y with an h,
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looks like potential energy.
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In fact, this is the kinetic energy per unit volume,
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because if I take this and I divide by a volume, mass per unit volume
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is a density. So that is a kinetic energy per unit volume, and this is
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a potential energy per unit volume. and this is a potential energy per unit. And
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this is a pressure. So pressure plus kinetic energy per unit volume plus
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potential energy per unit volume equals the same thing
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evaluated at some other location in the fluid. That's the Bernoulli
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equation. It's very powerful. It follows from the work
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energy theorem and it is a statement of conservation of energy for fluids. It follows
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from the work energy theorem, and it is a statement.
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So what does it say? If the height, so here's y1
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and here's y2.
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So they're measured from the same spot, measured as zero at some
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common location. If both of those are the same, y1 is the same. If both
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of those are the same, y1 equals y2, then this y1
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equals this y2, and these two terms cancel each other. So in this case, where
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they're both high, both at the same height, both points are at the same height,
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you've got just two terms on each side of the equation, the pressure and the
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one -half rho v squared. What that says
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is that... What it says is that if the velocity is high,
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if you increase the velocity, then you have to decrease the pressure because
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this quantity, the sum of the two, has to be a constant in the fluid.
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So increasing the velocity means decreased pressure. And that's what we've said here.
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And vice versa. A low velocity means high pressure because you have to add up
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these two terms and always get the same thing out of those two terms. So
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for example, this case here. We expect
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a high velocity here at point one where the pressure is P1.
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That means, this high velocity means that the pressure is going to be low there.
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So here the pressure will be low and this velocity will be high, but over
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here at 0 .2, the pressure is high and the velocity is
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low. We already know that the velocity is low by conservation
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of the volume flow rate has to be conserved. We already talked about
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that and did an example. The velocity has to be low. Bernoulli's equation says that
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the velocity has to be low. pressure has to be high. You say, well, just
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a second here, Dr. Edwards, because that seems counterintuitive to me. To me,
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the pressure should be high on this nozzle, because it's creating a high
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velocity. And here's a case where your intuition fails utterly. We'll do a
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demonstration to show that your intuition does fail you. This actually has lower pressure.
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than this. Here's where the high pressure is. One
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example of that is varicose veins. Where a
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vein gets enlarged, the pressure actually is high there
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and it continues to enlarge more and that's a cause for varicose
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veins. It doesn't balloon out where it's narrow, it balloons out where it's big.
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So also there's a dependence on the elevation. If you get up higher, you'll be
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at a lower pressure. And that's just the whole gravitational potential energy type of idea.
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It's actually seen in the
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equation that we derived for the
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static pressure. the higher you go, the lower the pressure is until you reach the
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surface where it's atmospheric pressure. So that's the same idea that comes in here. In
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fact, if you just look at this term, and this term, and this term, and
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this term, if you say that y2 minus y1 is h, then,
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I'm sorry, y1 minus y2 equals h,
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let's see, p2 equals p1
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plus rho g
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y1 minus y2 and replace this by
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h, then we've recovered the equation that we had before, p2 equals
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p1 plus rho g h. Here h and y serve
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a similar function, but H is the difference between two, two heights.
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Okay, if you still don't believe me, here's a demonstration. This is a
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Venturi tube. I've got a source of air that'll come in to this
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side of the tube. It'll pass through this region where the tube is wide, and
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then here where the tube is quite narrow, and then back to wide again.
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Since the same amount of air has to pass and then back to wide again.
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through this wide cross -section as passes through here,
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namely the volumetric flow rate has to be the same. The number of cubic
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centimeters per second passing this spot has to be the same as the
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volumetric flow rate that passes this spot,
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and because this spot has a smaller cross -sectional area,
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the air has to flow faster here. So we've got a high velocity region here,
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a low velocity region here. But Bernoulli's principle says that the higher the velocity, the
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lower the pressure. So in this spot here, contrary to what your intuition might say,
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instead of the pressure being high here, the pressure is actually low. And we can
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demonstrate that using this colored water. I filled the bottom of this
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U -shaped glass tubing with water that has a little bit of food
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coloring added to it. What you'll see is when we
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turn on the air, this column of water in the middle will rise
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and essentially acts as a suction for the water. So this one will
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rise higher than either of the other two.
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So that's a Venturi tube, a demonstration of Bernoulli's principle.
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and very important in a lot of areas of fluid dynamics.
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All right, conceptual example. When a truck is stationary, the tarp on the
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truck is flat. So if it's just sitting here, but it bulges upward and the
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truck moves down the highway. Why is that? Well, as it's moving
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down the highway, the truck's velocity is this way, it's moving through the fluid. But
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from the truck's perspective, the air is rushing over the top of the truck, as
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well as below the truck and along the sides of the truck, at a high
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velocity. So above the tarp, the velocity is high,
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and the pressure must therefore be low. Below the tarp,
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whatever's between the... well you've got air between the tarp and whatever the contents are
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in the trailer, the air is not moving at all, or at least not as
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much as it is on top of the tarp. You'll get a, you'll have a
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low velocity and a high pressure. So you get high pressure below
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and a low pressure above,
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because it's high velocity. so high V
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and a low P on top.
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And here we have a low V and a high P.
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So the pressure is higher underneath, that causes the tarp to come up.
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Another demo showing how Bernoulli's principle
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works. High velocity is low pressure. It raises the paper up.
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This is another demonstration of Bernoulli's principle. I'm going to create a
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stream of air using this hose and send it across this piece of
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paper that's been taped to the tabletop and is dangling off of the edge of
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the tabletop. What you'll be looking for is to... or what you'll be thinking
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about hopefully, is that the velocity will be high on the top side of this
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paper. That high velocity means the pressure will be low.
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And the pressure on the bottom side of the paper is atmospheric pressure. And so
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it will be greater than that low pressure on top and the atmospheric pressure will
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push the piece of paper up and overcome. the small amount of weight
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that this paper has and bring it into a vertical position
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just below the stream of air. This is how a
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clarinet reed works. When the stream of air from the player's mouth goes over the
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reed, it creates a low -pressure region which causes the
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reed to move, and then when the air goes down, the air goes down. It
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causes the reed to move and then when the air goes the other direction, the
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reed moves in the other direction. So that causes the reed to vibrate and causes
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the instrument then to sound.
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Yes, we've now got the paper up. We've got high velocity and low pressure here.
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I can disrupt it and the paper will go down.
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And this is actually a principle upon which a lot of musical instruments work. The
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lips of the trumpet, there's a Bernoulli effect that is relevant there.
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The Tacoma Narrows bridge collapse, same thing. the
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air stream over the top of the bridge will cause a lower pressure and the
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bridge to rise and that's what caused that Tacoma Narrows Bridge to come down.
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This is the way airfoils work. What happens in an airfoil
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is that two air molecules
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that, one that ends up going above the wing and the other
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one that goes below the wing, the one that ends up going over the wing
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travels a farther distance than the one that's below the wing. And if they meet
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up after they both made their trek across the airfoil,
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if they meet up at the same time, so they started out together and they
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end up together, that would be a streamlined flow, a steady flow, then that would
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mean in order for them to meet up after they're done, Then that would mean,
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in order for them to meet up after their journey is over, that would mean
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that the guy on the top had to be traveling faster. If he's
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traveling faster, that's a high velocity
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and the pressure must therefore be lower by Bernoulli's principle. And down here
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we have a low velocity
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So what that says is that there's a higher pressure pushing up on the wing
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than there is pushing down on the wing. This is a low pressure
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pushing down and a high pressure pushing up. Which one wins? Well, the one that's
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pushing up. That's what provides the lift for the wing. That's only usually 30 or
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40 percent of the lift provided for wings. The other way that you can provide
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lift is by putting these ailerons down or actually just tilting the wing up The
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other way that you can provide lift is by putting...
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so that the air hits it and reflects downward. That's called dynamic pressure. But
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this Bernoulli principle is an important piece of the puzzle.
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Another demo of Bernoulli's principle using a spool and a paper. This is a simple
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demonstration, one that you can do at home, a demonstration of Bernoulli's principle.
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I have one of my wife's spools for thread.
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I have a pin, and it's stuck through a piece of
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just construction paper, all there is to it. You can amaze your friends and neighbors
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with this one. Hollow out the spool, put the pin
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through the spool, and the... the part that's not so really
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amazing is if you suck up through this hole the
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paper will of course because of the suction I'm evacuating the hole here and the
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paper is sucked up because of that because the pressure below is atmospheric and I've
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created a small vacuum in here the more A more interesting one, the one that
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demonstrates Bernoulli's principle is when you blow vertically down through this hole. interesting one that
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the one that demonstrates Bernoulli's principle is
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So it seems unexpected that you would have thought that
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the air coming down through that hole would blow the piece
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of construction paper away from the spool. but it's not what happens.
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What happens instead, if this is the spool and this is the piece of paper,
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the airflow goes down through the center hole, hits the piece of paper and
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exits to either side. Well, it exits
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radially in all directions, so it's coming down through here and going out in all
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directions. velocity is high
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here, the pressure
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must be low by Bernoulli's principle. So you have a low
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pressure up in this region between the paper and the bottom of the spool,
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but you have high pressure, well, or normal pressure, atmospheric pressure
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down here. so the pressure pushing up on the paper
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is greater than the pressure
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in this in this region and
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and that difference in pressure the high minus the low is what holds the paper
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in place and that actually counteracts the effect of the air
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hitting the surface the air will hit the surface and will cause an impulse
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like a rocket when it's shooting out its fuel, hitting the surface, actually
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pushes it down. But this pressure effect, which is due to the Bernoulli principle, is
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the greater of the two effects and holds it in place.
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Yet another demo. This is a demonstration of Bernoulli's principle.
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I taught for 24 years in West Virginia and our home in West Virginia had
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a heat vent that was on the floor and so the heat, the hot air
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came up through that vent into the room. This is a demonstration. And one day
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the kids had a balloon and the balloon was balancing in mid -air above that
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heat vent. And I thought, how in the world does that work? And it's actually
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quite interesting. So this is the phenomenon. We've
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got a stream of air that comes up vertically
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in the case of West Virginia through the heat vent and then you've got a
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rather light object in that case a balloon, in this case a foam ball that's
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perched and being balanced and levitated by that stream of air. The
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principle is that the uprushing air and levitated by that stream of air. holds the
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ball up. There's some momentum transfer when that the air hits the bottom of the
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ball and it holds it up. So that part is not so mysterious. The part
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that is mysterious and was to me was what happens if that ball gets
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off -centered. So, let's say it moves to the left
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and this shows the uncentered situation with a ball So let's say it moves to
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the left and this
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to the left and the stream of air going past it on the right. Well,
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why does that ball want to move back into that rushing air? You'd think that
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it wouldn't want to move back into the air and it would just fall off
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to the side. But what happens is that Bernoulli's principle comes into play.
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Over here, over here there's a lot of air on the right side that's rushing
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past the ball. And Bernoulli's principle says that if you have a high
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velocity, a high airspeed, then you have a low pressure. So this
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air on this side of the, the right side of the ball exerts a smaller
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amount of pressure on this ball than the atmospheric pressure does on the other
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side so these this arrow represents the atmospheric pressure pushing the ball to the right
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and this represents the less than atmospheric pressure that pushes
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tries to push the ball back to the left the atmospheric pressure from the left
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wins and pushes the ball back into the centered configuration where now
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you've got equal pressures on each side and the ball is happy. so
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you can take this up to any scale that you want to,
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including this rather large scale demonstration.
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So we're building a house now in North Logan, and the builder, Sierra Homes,
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puts, near each sink, puts a pipe, a vent
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to the outside. What's the purpose for that? Well, our current house in Logan,
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sometimes there's a sewer smell that comes up in the kitchen sink. And why is
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that? If there's another appliance, like a dishwasher, nearby, what
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happens is that... if the washing machine or the
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dishwasher or whatever has a high velocity flow through
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this point, high velocity
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and low pressure. Well that low pressure region can,
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well actually let's look at it over here, Without a vent, you get a high
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velocity and a low pressure.
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I don't think you can read that, but that says high velocity, low pressure. What
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happens, you get a high velocity passing through here, you get this trap, and what
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that can do is it can, it can empty out the trap. It can suck
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that fluid from the trap here, allowing the sewer gases to come up into the
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sink. So it's disgusting. So what you do instead is you vent it to the
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outside so that when you've got a high velocity here, the pressure is
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forced to be the atmospheric pressure outside and it doesn't empty this trap.
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Wise practices based on physics. Wise construction practices.
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Physics of baseball. If the ball isn't spinning, then it just
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moves through the air and, well, with a knuckleball, if it's a knuckleball,
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then you get some cool patterns across the stitching, which can cause the ball to
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move erratically. But if you do a curveball and the ball is spinning
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in this direction, what it does is the ball kind of carries the air and
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encourages the flow across the top side of the ball, whereas... the
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ball moving this way is opposite to the direction of the flow of air crosses
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it, and so it slows the air here. So we get a low
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velocity and a high pressure.
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And up here we get a high velocity
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and a low
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pressure. So on this side of the ball we have a
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low pressure, on this side of the ball we have a high pressure, so there's
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a net deflection force that causes the ball to curve. That's how
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curve balls and how they work.
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So here's a demo of this effect.
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So, on that last one, it's interesting,
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the batter thought the ball was coming toward him and he ducked. and it went
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clear to the other side of the home plate. And it went clear to the
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other side of the... Fascinating.
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Alright, another example. The tank is open to the atmosphere at the top. Find an
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expression for the speed of the liquid leaving the pipe at the bottom.
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So in this particular case, we're looking at atmospheric pressure. up here
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on the top.
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Atmospheric pressure on the top, but we're also allowing this to exit.
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the liquid at the bottom.
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So there's going to be atmospheric pressure at the bottom. Now, you say, well hang
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on, what does that happen? Doesn't the pressure increase with depth? And it does. So
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there'll be increasing pressure as we get deeper and deeper, and then as we go
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between here and here, the pressure's going to be atmospheric again Because it's open to
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the atmosphere. And anything that's open to the atmosphere is at atmospheric pressure.
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So how fast is this going to be going? So
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here's how to do this problem. Pressure 1 and pressure 2 at the two different
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points. So here's point 2 at the top. Here's point 1 at the bottom. Both
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of those pressures are going to equal atmospheric pressure. So P1 equals P2.
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So we can cancel them out. They're both equal to the same thing. And
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the difference between the heights, y2 minus y1, we're just going to call
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h. So I'm going to subtract this term from both sides of the equation,
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and we'll get rho g times y2 minus y1, but that's h. So that's this
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term right here. And then we're going to ignore...
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I mean certainly the velocity at 1 here is huge.
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But the velocity up here is essentially 0. It's moving down a little bit, but
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compared to this velocity, it's negligible. So we're going to ignore the velocity of
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e2 and set it to 0. And that leaves
29:37
us with one -half rho v1 squared equals rho gh. And that allows... us to
29:42
calculate the speed of the water where it exits
29:48
down here.
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