Question

A water pipe narrows from a radius of \(r_{1}=5.00 \mathrm{~cm}\) to a radius of \(r_{2}=2.00 \mathrm{~cm} .\) If the speed of the water in the wider part of the pipe is \(2.00 \mathrm{~m} / \mathrm{s}\), what is the speed of the water in the narrower part?

Short answer

Answer: The speed of the water in the narrower part of the pipe is 25.00 m/s.

Step-by-step solution

Write down the continuity equation for fluids

The continuity equation for fluids is given by: \(A_1v_1 = A_2v_2\) Where \(A_1\), \(A_2\) are the cross-sectional areas of the wider and narrower parts of the pipe respectively, and \(v_1\), \(v_2\) are the speeds of the water in the wider and narrower parts of the pipe.

Calculate the cross-sectional areas of the respective parts of the pipe

The cross-sectional areas of the pipe can be calculated as the area of a circle, using the radii given: \(A_1 = \pi r_1^2\) \(A_2 = \pi r_2^2\) Plug in the given values for \(r_1\) and \(r_2\): \(A_1 = \pi (5.00 \mathrm{~cm})^2\) \(A_2 = \pi (2.00 \mathrm{~cm})^2\) Convert the area to \(\mathrm{m^2}\): \(A_1 = \pi (0.050 \mathrm{~m})^2\) \(A_2 = \pi (0.020 \mathrm{~m})^2\)

Rearrange the continuity equation and solve for \(v_2\)

We can rearrange the continuity equation to solve for the speed of the water in the narrower part of the pipe, \(v_2\): \(v_2 = \frac{A_1v_1}{A_2}\) Plug in the values for \(A_1\), \(A_2\), and \(v_1\): \(v_2 = \frac{\pi(0.050 \mathrm{~m})^2 \cdot 2.00 \mathrm{~m}/\mathrm{s}}{\pi(0.020 \mathrm{~m})^2}\) The \(\pi\) terms cancel out: \(v_2 = \frac{(0.050 \mathrm{~m})^2 \cdot 2.00 \mathrm{~m}/\mathrm{s}}{(0.020 \mathrm{~m})^2}\)

Calculate the speed of the water in the narrower part of the pipe

Perform the calculation: \(v_2 = \frac{(0.050 \mathrm{~m})^2 \cdot 2.00 \mathrm{~m}/\mathrm{s}}{(0.020 \mathrm{~m})^2} = 25.00 \mathrm{~m}/\mathrm{s}\) The speed of the water in the narrower part of the pipe is \(25.00 \mathrm{~m}/\mathrm{s}\).

Key concepts

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics that describes how fluids behave when flowing through different areas of a pipe or channel. It's based on the principle of conservation of mass, which states that mass cannot be created or destroyed in a closed system. This equation relates the cross-sectional areas and the velocities of the fluid at two points along the flow.

When a fluid flows from a larger area to a smaller one, the equation shows us that if the cross-sectional area decreases, the velocity of the flow has to increase, and vice versa. The continuity equation is mathematically expressed as:
\[A_1v_1 = A_2v_2\]
This formula tells us that the product of the cross-sectional area (\(A\)) and the flow velocity (\(v\)) at one point (\(1\)) is equal to the product of the cross-sectional area and flow velocity at another point (\(2\)). This equation becomes very useful when we want to determine the flow characteristics of a fluid when it encounters changes in pipe diameter, such as in the exercise problem provided.

Cross-sectional Area

In the context of fluid flow, the cross-sectional area is the area through which a fluid passes at a given point in a pipe or a conduit. It's an important factor because, together with fluid velocity, it determines the rate of flow of a fluid. To imagine this, think of a garden hose: when you put your thumb over the end, narrowing the opening, the water sprays out faster and further because you've reduced the cross-sectional area.

The cross-sectional area is typically calculated using geometric formulas depending on the shape of the cross-section. For pipes and most water channels, which are round, we use the area formula for a circle:
\[A = \$\pi r^2\$\]
where \(A\) is the area and \(r\) is the radius of the pipe. Understanding how to calculate cross-sectional area is essential for solving problems involving the continuity equation, as demonstrated in the step-by-step solution.

Water Flow Velocity

Water flow velocity is a measure of how quickly the water is moving at a given point in a flow system, usually measured in meters per second (\(\mathrm{\frac{m}{s}}\)). It's a dynamic parameter that changes based on the geometry of the system and can vary within different sections of a pipe or riverbed.

For example, in the solution above, the water's initial velocity in the wider portion of the pipe is given as 2 m/s. Due to the narrowing of the pipe, this velocity increases in the smaller section to maintain the continuity of flow. Faster flow velocities can lead to different fluid mechanics phenomena like turbulence or increased pressure drops, which are important considerations in engineering and environmental applications.

Conservation of Mass in Fluids

In fluid dynamics, the concept of the conservation of mass tells us that the mass of fluids remains constant as they flow from one place to another. This is also known as the principle of mass continuity. For a non-compressible liquid like water, this principle means that the amount of water flowing into a section of pipe in a given time is equal to the amount of water flowing out.

Using the principle of conservation of mass alongside the continuity equation, engineers can predict and control flows in pipelines, design efficient systems, and avoid problems like backflow or bursting due to pressure build-up. The application of this principle in real-life scenarios ensures that, in the absence of external forces or phase changes, we can reliably design systems that transport fluids effectively.