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AP Physics 2: Exam Prep26 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Sarah Friedl*

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Bernoulli's equation describes an important relationship between pressure, speed, and height of an ideal fluid. In this lesson you will learn Bernoulli's equation, as well as see through an example how in an ideal fluid, the dynamics of that fluid remain constant.

If you're like me, you really dig recycling. It's amazing that we can take a material that has been used for one purpose and then either change its form or function so that it can be used again for another purpose. The universe likes recycling, too!

In fact, all of the energy on Earth, other planets, and the rest of the universe is constant - it is not created or destroyed, only transferred from one object to another, or transformed from one form to another. This is concept is so vital to how things work in the universe that it's called the **law of conservation of energy**.

We can apply the same idea of conservation to fluids. This is because as a fluid moves through a pipe or tube, the relationship between the pressure, velocity, and height remains constant. When one of these variables changes, that change is 'recycled' as a respective change in another variable.

For example, if the pressure in the fluid increases, the speed of the fluid decreases to compensate. Likewise, if the area that the fluid travels through becomes smaller, the speed increases because the same amount of volume has to travel through that smaller area.

However, there are some caveats attached to these fluids that make them 'ideal.' Fluid dynamics is a very complex subject, and we don't even fully understand some of the ways that fluids move. So we have to make a few assumptions to create an 'ideal' fluid that allows us to understand its movement.

First we assume that the fluid is **incompressible**, meaning that its density doesn't change. Second, we assume that the fluid is **non-viscous**, meaning that there is no resistance to the fluid's movement. Finally, we assume that the flow is **laminar**, which means that it is steady and constant. It would be quite difficult if we tried to work with a fluid that was moving around all willy-nilly on us.

We can neatly package the concept of fluid conservation in **Bernoulli's equation**, which relates pressure, speed, and height at any two points within an ideal fluid.

This relationship can be written as an equation:

*P1 + ½ ρv1^2 + ρgh1 = P2 + ½ ρv2^2 + ρgh2*

where *P* is the pressure in the fluid, *ρ* is the density of the fluid, *g* is the acceleration due to gravity (9.80 m/s^2), *h* is the height of the fluid off the ground, and *v* is the velocity of the fluid.

Can you see how if one variable changes at point 1, then something else must also change to keep the equation, well, equal?

You may still be having some difficulty grasping this concept and relating it to the conservation of energy, so let's work through an actual example.

Say that some water flows through an S-shaped pipe. At one end, the water in the pipe has a pressure of 150,000 Pascal (Pa), a speed of 5.0 m/s, and a height of 0.0 m. At the other end, the speed of the water is 10 m/s, and the height is now 2.0 m. Since the density of water is 1000 kg/m^3, all you're missing is the pressure at the second point, and this can be determined by rearranging Bernoulli's equation to get it alone on one side.

So to start, our equation looks like this:

*P1 + ½ ρv1^2 + ρgh1 = P2 + ½ ρv2^2 + ρgh2*

To make things easier, let's rearrange our equation first, and then plug in our values. To get P2 alone, we rearrange things so our equation looks like this:

*P2 = P1 + ½ ρv1^2 - ½ ρv2^2 + ρgh1 - ρgh2*

What's great is that we can simplify this even further:

*P2 = P1 + ½ρ(v1^2 - v2^2) + ρg(h1 - h2)*

Once we fill in our known values, our equation reads:

*P2* = 150,000 Pa + ½ * 1000 kg/m^3 * ((5 m/s)^2 - (10 m/s)^2) + 1000 kg/m^3 * 9.80 m/s^2 * (0 m - 2 m)

Once we do the math, we find our pressure at the second point of the pipe to be 92,900 Pa.

In this case, the pressure decreased because it gained elevation. But also note that the speed was greater at this point (*v2* = 10 m/s), which makes sense. And, if we arrange our equation so that it reads as it did originally, both sides would still be equal. Can you see how as one variable changes, the others change to accommodate and conserve the relationship within the fluid?

If you wanted to, you could examine this relationship further by solving for any one missing variable. As long as you know the others, you can shuffle the equation around to see how the variables keep a balance not only in Bernoulli's equation but also in our ideal fluid.

**The law of conservation of energy** is a helpful guide to understanding conservation in ideal fluids. This law states that energy can't be created or destroyed, it only changes form or transfers between objects.

In a flowing fluid, we can see this same concept of conservation through **Bernoulli's equation**, expressed as *P1 + ½ ρv1^2 + ρgh1 = P2 + ½ ρv2^2 + ρgh2*. This equation relates pressure, speed, and height at any two points within an ideal fluid.

Because it is an equation, both sides must be equal. Even if the individual components of pressure, speed, and height are different at one point in a tube, the relationship between them will be the same as the relationship between those variables at another point. Knowing this, we can see how even when we rearrange the equation to find a missing value, both sides will always be the same because the relationship within an ideal fluid is the same across the entire fluid.

Once you are done with this lesson you should be able to:

- Comprehend the concept of conservation in ideal fluids
- Express Bernoulli's equation
- Solve an equation for a fluid's pressure, speed or height using Bernoulli's equation

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AP Physics 2: Exam Prep26 chapters | 140 lessons

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