Fluid Dynamics - AP Physics 2

To begin with, it will be necessary to make use of Bernoulli's equation:

For the situation described in the question stem, we'll designate the top of the container as point 1, and the hole where water is flowing out as point 2.

To begin simplifying things, it's important to realize a few things. First, both points are open to the atmosphere. Therefore, the term on each side of the above equation is equal to 1atm and thus can cancel out. Secondly, since the size of the hole on the side of the tank is so small compared to the rest of the tank, the velocity of the water at point 1 is nearly equal to 0. Hence, we can cancel out the term on the left side of the equation. Thus far, we have:

Dividing everything by , we obtain:

And rearranging:

The equation relating fluid pressure, kinetic energy, and potential energy from state to state is known as the Bernoulli equation, and is as follows:

Our potential energy is the same, so we can remove that part from the equation.

is the density, which in our case is equal to 1, so it doesn't change anything here.

We have the values for the initial pressure, initial velocity, and final velocity, so we can rearrange our equation to equal final pressure.

Now, we can plug in our values.

Therefore, the final pressure is equal to .

This is solved using Bernoulli's equation and the definition of pressure. First choose the "Bernoulli points", one just inside the roof where the air is still (Point A) and one just outside where the air is moving (Point B). This will allow us to eliminate many of the terms:

Since the air is still inside,

Since our points are at the same height, the terms cancel. Rearrange:

Since we only care about the difference in pressure inside and out.

Plug in known values and solve for the difference in pressure.

Force is pressure times area:

Use Bernoulli's equation to find the pressure difference on the two sides of the kite. Call point A the inner surface (where the air is still) and point B the outer surface (where the wind is at full speed).

Since the points are at the same height, the terms cancel. Since the air is still at point A, the term is zero. Since we only care about the difference in pressure on the two sides of the kite, solve for :

Net force is pressure difference times area:

We will use Bernoulli's equation to solve this. We must do this twice: once for the air and once for the water. The central principle here is that the moving stream of air has a lower pressure than still air. In this problem we will ignore the atmospheric pressure since it is applied at the tube ends and at the surface of the water outside the vertical tube. For the air, choose our two "Bernoulli points": point A is just outside the horizontal tube and point B is just inside. Write the equation:

The heights are the same, so they cancel out of the equation. The air is still at point A, so the velocity term is zero for the left side. Finally, as mentioned, we care only about the difference in pressure:

This is saying that the pressure inside the tube is below the pressure outside.

Now we solve for the water. Our points will be on the surface outside the vertical tube (point A) where the pressure is one atmosphere, and inside the vertical tube at the surface of the risen column (point B). Write the equation:

Since the water is no longer moving, the terms containing are equal to zero. Rearrange:

Put in numbers and solve for the height difference:

The moving water on one side of the metal plate has a lower pressure than the still water on the other side, resulting in a force. We start by writing the Bernoulli equation:

We choose our two "Bernoulli points" to make the problem as easy as possible. Take point A to be on the side where the water is still, and point B on the side where the water is moving. If we make them at the same height, the terms can be subtracted from both sides. Since the water is still at point A, the velocity term on the right-hand side of the equation is zero. Rearrange to find the difference in pressure from side A to side B:

This is saying that the pressure at point B is less than the pressure at point A. Using the definition of pressure, find the resulting force:

Since Bernoulli's principle is derived from the work-energy theorem, it is a requirement that the flow be incompressible, steady, and without internal friction. Otherwise, energy would be lost to these outlets and the equation would no longer be applicable.

In the question stem, we're told that a fluid with a density of moves through a pipe at a speed of and has a pressure of . We're then told that the same fluid begins to move through the pipe at a new speed of , and we're asked to determine what the new pressure will be.

In order to answer this question, we'll need to make use of Bernoulli's equation. This equation essentially tells us that the pressure, kinetic energy, and potential energy of a moving fluid is constant. Or, put another way:

Alternatively, since we know the sum of these values is constant, we can relate the sum of these values at one instant to the sum of these values at another instant.

Furthermore, since we're told that the fluid remains flowing in a horizontal pipe, the height of the fluid does not change. Therefore, we can cancel out the potential energy terms on both sides.

Next, if we define the initial conditions as instant 1, and the final conditions as instant 2, then the term we are trying to solve for is .

Then, plugging in the values we have allows us to obtain the answer:

To solve this problem, we will use Bernoulli's equation, a simplified form of the law of conservation of energy. It applies to fluids that are incompressible (constant density) and non-viscous.

Bernoulli's equation is:

Where is pressure, is density, is the gravitational constant, is velocity, and is the height.

In our question, state 1 is at the tap and state 2 is at the nozzle. Input the variables from the question into Bernoulli's equation:

To solve this problem, we will use Bernoulli's equation, a simplified form of the law of conservation of energy. It applies to fluids that are incompressible (constant density) and non-viscous.

Bernoulli's equation is:

Where is pressure, is density, is the gravitational constant, is velocity, and is the height.

In our question, state 1 is at the tap and state 2 is at the nozzle. Input the variables from the question into Bernoulli's equation:

Simplify and solve for the final velocity.

To solve this problem, we will use Bernoulli's equation, a simplified form of the law of conservation of energy. It applies to fluids that are incompressible (constant density) and non-viscous.

Bernoulli's equation is:

Where is pressure, is density, is the gravitational constant, is velocity, and is the height. In our question, state 1 is at the entrance of the pipe and state 2 is at the exit of the pipe. Both states of the pipe are at the same height because the pipe is horizontal. Input the variables from the question into Bernoulli's equation and solve for the final pressure:

To solve this problem, we will use Bernoulli's equation, a simplified form of the law of conservation of energy. It applies to fluids that are incompressible (constant density) and non-viscous.

Bernoulli's equation is:

Where is pressure, is density, is the gravitational constant, is velocity, and is the height.

In our question, state 1 is at the entrance of the pipe and state 2 is at the exit of the pipe. Both states of the pipe are at the same height because the pipe is horizontal. Input the variables from the question into Bernoulli's equation and solve for the final velocity:

To solve this problem, we will use Bernoulli's equation, a simplified form of the law of conservation of energy. It applies to fluids that are incompressible (constant density) and non-viscous.

Bernoulli's equation is:

Where is pressure, is density, is the gravitational constant, is velocity, and is the height. Since the height of the pipe is constant, we can eliminate the height term leaving us with:

The velocity is increasing, making the velocity term negative. We are subtracting some amount from , therefore , meaning the pressure is decreasing.

In this question, we're presented with a scenario in which blood is flowing through a blood vessel with a clot. We're asked to determine how the partial pressure of oxygen changes as the blood flows pass the clot. To answer this, we'll need to understand the concepts of the continuity equation, Bernoulli's equation, and partial pressures.

Since the blood is flowing past an obstruction, and because the volume flow rate is the same everywhere, we can use the continuity equation to show the relationship between velocity and area.

What this equation shows is that, when the volume flow rate is constant, the velocity of the fluid is inversely proportional to the area in which it is flowing. When the blood passes through the narrow space caused by the obstruction, it temporarily speeds up and thus has a higher velocity.

Next, we can use the Bernoulli equation to show how the velocity is related to pressure.

In this case, we can assume that the height does not change, and thus we can take the potential energy component out of the equation. What we are left with is an inverse relationship between velocity and pressure. Thus, as the velocity increases, the pressure decreases.

The next thing to do is relate total pressure with partial pressure. Remember that partial pressure refers to the proportion of pressure that a certain gas contributes to the overall pressure when there is a mixture of gasses. For instance, the partial pressure of oxygen in the atmosphere is roughly , meaning that oxygen is responsible for contributing of the overall pressure of the atmosphere. The expression for partial pressure is as follows.

Again, what this means is that the partial pressure of a certain gas is dependent on both the fraction of that gas out of all other gasses present, and also the total pressure. Even though the proportion of oxygen will remain the same as blood flows pass the clot, the total pressure in that region decreases. Thus, the partial pressure of oxygen in this region will decrease as well.

All in all, let's summarize what happened by bringing it all together. As the blood flows through a narrower space due to the obstruction, its velocity increases. As a result of the increase in velocity, the pressure lessens. And as the pressure decreases, so too does the partial pressure of any of the gasses within that region, including that of oxygen.

The velocity of the water can be determined from the following formula:

We need to calculate the volumetric flow rate and the cross-sectional area. For the flow rate:

Rearrange to solve for volumetric flow rate:

Next, calculate cross-sectional area:

Now we can solve for the velocity:

To find the answer to this question, we'll need to use the continuity equation to determine the flow rate, which will be the same in both pipes.

We'll also need to calculate the area of the pipe using the equation:

Solve the combined equation for and plug in known values to find the velocity of the water through the second pipe.

Use the continuity equation for incompressible fluids.

The cross sectional area of the garden hose at both ends are circular regions. Rewrite the equation replacing areas with the formula for an area of a circle and solve for the velocity at the second point.

This is a volume flow rate problem. Because the water is an incompressible fluid, we can apply the flow rate equation:

Find the surface area of the pond:

Substitute into the flow rate equation:

When the area halves, the velocity of the fluid will double. However, the volume flow rate (the product of these two quantities) will remain the same. In other words, the volume of water flowing through location 1 per second is the same as the volume of water flowing through location 2 per second.