Hi. It’s Mr. Andersen. And this is AP Physics essentials video 88. It is on Bernoulli’s
Equation. And a great application of this is the reason why docks are not solid, why
they are made of pylons so water can flow underneath it. If you were to have a big boat
like this, we are looking at it from above, come into a dock, if it was solid it would
force all of the water out of the way. And that fluid as it is moving faster would have
a lower pressure. There would be a higher pressure on the outside of the boat which
would slam it up against the dock. And so Bernoulli’s Equation is really conservation
of energy inside a fluid. And to solve most of the problems you will first have to understand
how the continuity equation works. Continuity equation is equal to A1 times V1 equals A2
time V2. Where A1 is the cross-sectional area and V1 is going to be the velocity. So if
we look at a pipe like this, and so fluid is flowing through it from left to right,
in two different points. A1 is going to be the cross-sectional area, so the area of this
pipe here. And A2 is going to be the area over here. And so if we look at the velocity,
that is what V1 stands for, since we have a large cross-sectional area here we are going
to have a relatively low velocity. And then as we move to the right, since we are decreasing
that area, we are going to have a high velocity. And so if you know the velocity at any point
in the pipe, since that whole fluid is moving, if you know the cross-sectional area anywhere
else you are going to be able to figure out the velocity there. An application of this,
if you have ever had a hose, as water comes out of it, it has a large cross-sectional
area. If you put your thumb in front of it, what are we doing? We are decreasing the cross-sectional
area. What happens to the velocity? It increases. Now if we look at Bernoulli’s Equation it
is somewhat scary when you see it the first time. It looks like this. And so this is going
to be Bernoulli’s Equation. And it is frightening to look at. But what we have really added
here are simply two things. We have added y, which is going to be the height of the
pipe. Because that is the potential energy. And then the other thing we have added is
rho, which is going to be density of the fluid. Because those can affect the amount of energy
that we have. And so if we break apart this equation, we have also got one on the left
side and two on the right side. If we look at this first one, P1 stands for the pressure.
So inside the fluid itself, how much is that fluid pushing in on a point inside the fluid
itself. So that is going to be the pressure energy. We then have this, rho g y 1. Now
that seems confusing. But let me kind of write it out a different way. If we were to instead
of write rho or density, if I were to say the mass, this is g, gravitational field strength
and then if we were to look at y if I wrote h there or the height, what is m g h? You
know that. That is just the potential energy of the fluid. And so that second bit of this
equation is going to be the potential energy of the fluid. And then if we write this next
one, what is 1/2mv squared? That would be kinetic energy. But we are writing density
since we are dealing with a fluid. So that is going to be the kinetic energy. And so
those are the three ways we can get energy on the left side of that pipe. It is the pressure
of the fluid. It is the potential energy, how far it is in relation to gravity. And
then the last is going to be the speed of that fluid. And since this is the conservation
of energy, in these two points of the pipe, since we know the energy over here, and we
know it is going to be an equal amount of energy over here, we can solve some pretty
complex problems. So for example if we were to look at this pipe right here, on both sides
they both has the same height or the same y value. So I have taken those out of this
equation. And so where is the velocity going to be faster? It is going to be faster on
the right side of the pipe. So we are going to have a faster velocity on the right side.
Slower velocity on the left side. So what has to be our pressure on the right side?
Well to make it equal on either side, conservation of energy, we are going to have to have a
lower pressure. So just like in my example of the ship, if the fluid is moving fast,
then it is going to have a lower pressure. And so let’s start with a continuity equation,
which is simple. We have a phet simulation. You can see as the fluid is moving, it is
moving faster on the right. And so if we use a flux meter, on the left side the cross-sectional
area is 10 meters squared. And the speed is going to be 0.5 meters per second. So that
would be my A1 times my V1. If I move over the the right side, now my cross-sectional
area is 1, what is my velocity? 5 meters per second. And so I can just solve for the one
I do not know and I can figure that out. If we were to just change the pipe, now the cross-sectional
area is 5 meters squared. What is going to be my speed? It is simply going to be 1 meter
per second. And so continuity equation is very, very simple. If we were to apply Bernoulli’s
Equation, what we are really adding to it is the density of the fluid and then the height
of the fluid. And so let me give you a thought experiment. Let’s say we were to take a
half gallon of milk. And I were to pop three holes in it and quickly put tape over it like
that. So the whole thing is filled up with a fluid. And then I were to simply pull the
tape off the side. So what are are going to get? Streams of milk coming out of the carton.
But do you think those streams would look like this, like this or like that? If you
were to look at its from the side? Which is going to be the correct way the streams are
going to come out? Well the correct answer is C. Why is that? It is because if we look
down here at the the bottom, this is like the second part of the pipe. So on the left
side of the pipe we have way more potential energy inside the milk. And so that is going
to be converted to more kinetic energy and the stream is going to go out farther. Where
as if we go to the top it is not going to have as much potential energy above it. And
so it is not going to be able to have as much velocity. So it is not going to go out as
far. And so understanding both sides of the pipe, where is the potential energy greater?
It really helps you understand how Bernoulli’s Equation works. And so it is written like
this. Left side remember is going to be the pressure energy, the potential energy and
then the kinetic energy. And they are going to balance. So it is the conservation of energy.
If we know what is on the left side we can solve for what is on the right side. So let’s
try to solve a problem using Bernoulli’s Equation. So I have a pipe. I have a left
side and a right side. On the left side you can see the velocity is lower. I am giving
you the pressure on the left side. But what we are going to solve for is the pressure
on the right side. So let’s go through this equation. On the left side I am giving you
the pressure on the right side we do not know what that pressure is. On the left side do
we have less or more potential energy in the pipe on the left side then we would on the
right side. Well the density of the fluid, since it is water is going to be the same
on both sides. And since the pipe on the right side is going to have a higher height it is
going to have a higher potential energy on the right side. What about this? Are we going
to have higher kinetic energy on the right side or the left side? Since it is going faster
on the right side, our velocity is faster on the right side. It is also going to have
a faster kinetic energy or higher kinetic energy on the right side. And since I am giving
you the pressure on the left side as 128, we would expect to have a pressure on the
right side that has to be lower, since we have higher potential and kinetic energy on
the right side of the pipe. So let me show you how I would solve this. I would write
it out with the things I know. So the density of the water, since it is the same fluid,
is going to be 1000 kilograms per meter second. And we know the gravitational field strength
is 9.8 meters per second squared. On the left side of the pipe we know the pressure, 129
kilopascals. We know the velocity and then we also know the height. And on the right
side what we do not know is going to be the pressure. That is the unknown. But on the
right side again we have a higher height, higher velocity so we have higher energy right
there. It should be a lower pressure. So if I were to write it out I would write it out
like this. Now I am not going to include the units because it simply would not fit on the
screen. Watch one thing that could screw you up since it is 129 kilopascals, I am writing
that out as 129,000 pascals because that is the unit that we are going to use. I then
solve it. And so this right here is going to be my potential energy left side, kinetic
energy left side. If we compare that to the potential and kinetic on the right side you
could see on the right side both of those values are greater. But we are simply solving
for pressure 2. And so using significant digits I am getting around 90 kilopascals on the
right side. And that is because I am limited by this. It is not a very precise measurement
in the velocity. So let’s test that and see if that comes out. On the right side it
is around 87 kilopascals. So it is around 90 kilopascals. And so we can use Bernoulli’s
Equation not only to calculate but we can use it to analyze a confusing situations when
we have a fluid. So how does a curveball work? When you throw a curveball it should drop.
So as the pitcher tries, or rather as the batter tries to hit it, it is going to drop
right at the end. And so how does that work? How does a curveball work? Using the fluid
of air. So as you through a ball, let’s say it is not spinning, as it is moving through
the air the fluid of the air is going to move around it. But since it is not spinning, the
velocity on the top and the bottom is going to be exactly the same. But when you throw
a curveball you start spinning it so it is spinning like this and it generates a wind
on the top which counteracts the wind that is already coming in, and so it decreases
the speed of the wind on the top. But on the bottom, since the wind over the ball and now
the wind generated by the spin of the ball are both going in the same direction, the
fluid on the bottom is going to be going faster. So Bernoulli’s Equation, what happens to
our pressure if the fluid is going faster, pressure is going to decrease. Since we have
lower pressure on the bottom there is going to be a force pushing it down. So that is
how a curveball curves. And so did you learn to construct an explanation of Bernoulli’s
Equation using the conservation of energy? Remember on the equation left side equals
right side. Could you use it to make some calculations and solve some problems in a
moving fluid? And then finally could you use it to show where the pressure changes? So
in a dock, in a curveball or in any fluid? I hope so. And I hope that was helpful.