PROFESSOR: Here's what I wanted to show you today.
This is an apparatus to help us see the Bernoulli equation.
It tells us a whole lot.
It tells us how changes in height can affect the velocity of a fluid,
and how the velocity of that fluid can change the pressure.
A whole bunch of things operate because of this.
Airplane wings work because of the Bernoulli equation.
Hydroelectric dams work because of the Bernoulli equation.
A whole lot of stuff needs to be calculated using the Bernoulli
equation in order to make an irrigation system work.
All right, so the first thing I wanted to show you is how changes in height
affect the velocity of a fluid.
All right.
What we have is a great big column filled up with red water
just to make it a little bit easier for you to see the fluid.
Now, if I hold these two next to each other,
you can see that the height is the same in both tubes.
If I move it down real fast, it will adjust itself.
Comes right back to exactly the same height.
If I move it up real fast, it'll drop right down to exactly the same height.
It's neat, you can watch it go back and forth, and then it levels off right
there.
Now what we're going to see, is if I lower this hose,
there'll come a point where the water in the tube
is right here, the exact same height as it is over here.
And if I lower it just a little bit, we start to make a mess.
Water comes out very slowly.
If I lift it back up, the flow stops.
If I lower it a little bit more, it comes out slowly.
Now, the change in height between these two points is very small.
If I increase that change in height, the water comes up a whole lot faster.
If I bring it all the way down here, the water comes out a whole lot faster.
If I hold it way down here, the water comes out really fast.
As I slowly bring this back, you can see how
the velocity of the water leaving that tube slows down until it stops again.
Let's go find out how this works it's called the Bernoulli equation.
All right, hi let's start talking about the Bernoulli equation.
We're going to set this up, talk about a few different variables,
see what happens to this.
If we change that, what happens to that if we change this?
And then we're going to see a few more demonstrations
about some cool things with the Bernoulli equation.
All right, first, let's take a look at the types
of things we could solve now we know the Bernoulli equation.
It looks like one of the more confusing formulas I've ever
seen in my whole life, but it turns out a lot of things cancel out,
and a lot of things simplify.
So the whole equation looks like this the pressure on one side,
and rho gy on one side, and one half rho v squared on one side,
will always be equal to the pressure at some other spot,
and rho gy at some other height, and one half rho v
squared for some other velocity.
Now, an easier way to look at this is just say that everything on this side
will be constant.
Doesn't mean that these won't change, it just means that everything over here
still has to add up to everything that's on the side.
All right, we should be able to use this equation
to figure out what's the velocity at this height, one,
and what's the velocity at this height, two,
and we should be able to figure out why it's moving faster.
To make this equation much easier to look at,
we can just simply say that the pressure on top of the water, p1,
is so close to the pressure at other points, cancel that.
Now, the reason is, there's an atmospheric pressure here,
and there's an atmospheric pressure here, that cancels out of the equation.
The next thing that actually cancels out of this equation
is the velocity that this top level is dropping.
Since the area up here is much, much larger
than the area of the opening over here, this velocity is much, much smaller.
It's essentially zero, so we don't need this in our equation either.
All right, I'm going to rewrite this, and it should look a whole lot easier.
I'm going to put rho g times my change in y,
is going to be equal to 1/2 rho v squared.
Rho is the density of water, it cancels out on both sides.
And when I rearrange this to solve for the velocity,
I get the square root of 2 times g times change in y.
Coincidentally, that's the same formula for how fast
something is if you drop it vertically.
All right, let's see what else we can do with the Bernoulli equation.
OK, that last one was a little bit too easy.
Let's kick it up a notch, try something a little bit
more exciting with the Bernoulli equation.
Let's imagine that we have a water pump down here
and it forces water up a hill.
So let's start down here at point 1, come up here to point B. I'm sorry,
let's call this one point A, and this one point B.
So our water pump moves water from point A
to point B. Point B is 2 meters higher than point A.
And the area where the water comes out at the top,
is 1/5 as much as it is at A. The other thing that we know
is that the velocity is 5 meters per second down here.
Here's what we need to find, the gauge pressure at point A.
This is going to tie together all three formulas
that we've seen so far in fluids.
We're going to have to use the continuity equation,
a1 v1 is equal to a2 v2.
We are going to have to use our definition of gauge pressure.
That was total pressure minus the atmospheric pressure
is equal to the gauge pressure.
And of course, we're going to need to use the entire Bernoulli
equation this time.
OK, let's start out.
Let's use this continuity equation to find the velocity at B.
They didn't tell us what the area was at A,
all it told us is that at B, it's 1/5 as much.
So if the area at A is 1, and the area at B is 1/5 as much,
I should be able to solve it from this.
We were told the velocity at point A was 5,
so that tells me the velocity up here at B is 25 meters per second.
OK, there's one more list.
One more thing to add to my list of given, 25
meters per second, velocity at B.
OK, now let's set up the big Bernoulli equation and see what we know.
We've got total pressure at 1, And rho gy at point A
write those A's so I don't get confused.
And 1/2 rho v A squared, has to be equal to the pressure at B. What
is the pressure at B?
When that water comes out of that hose, it's in the atmosphere now.
That means the pressure B, is the atmospheric pressure.
So we know that atmospheric pressure is the pressure at point B,
and rho g times the height at point B, and 1/2 rho v squared.
All right, I think now it's just a great big giant substitution.
Let's see what we can plug-in here.
We don't know the pressure at A, that's what we're trying to find.
We can conveniently say that the height at A is at zero.
That's our lowest point, so that term cancels out.
We don't need to worry about that.
We know the velocity at A was 5.
We know the pressure at B was the atmospheric pressure.
And the density of water times gravity times the height at point B,
and 1/2 rho times the velocity at B, we found that that was 25.
And we're going to have to square that.
All right, let's plug-in everything we know.
Pressure at A. All right, let me grab my calculator,
and set this up to find gauge pressure.
All right, let me double check this real quick.
And my calculator is telling me that the gauge pressure at point A
is 320,000 pascals.
All right, well we've seen some easy stuff with the Bernoulli equation,
we've seen some complicated stuff with the Bernoulli equation.
Let's take a look conceptually at what we can just figure out
without using numbers for the Bernoulli equation.