Problem 37 - University Physics with Modern Physics

Question

Water is flowing in a pipe with a circular cross section but with varying cross-sectional area, and at all points the water completely fills the pipe. (a) At one point in the pipe the radius is 0.150 \(\mathrm{m}\) . What is the speed of the water at this point if water is flowing into this pipe at a steady rate of 1.20 \(\mathrm{m}^{3} / \mathrm{s} ?\) (b) At a second point in the pipe the water speed is 3.80 \(\mathrm{m} / \mathrm{s} .\) What is the radius of the pipe at this point?

Step-by-Step Solution

Answer

(a) 16.97 m/s; (b) 0.125 m.

Step 1: Find the Area of the First Cross-section

Given that at one point the radius of the pipe is \( r = 0.150 \) m, we can calculate the cross-sectional area \( A_1 \) using the formula for the area of a circle, \( A = \pi r^2 \). Thus, \( A_1 = \pi (0.150)^2 \).

Step 2: Calculate the Speed of Water at the First Point

We know the volume flow rate \( Q = 1.20 \ \mathrm{m}^3/\mathrm{s} \). The flow rate is defined as \( Q = A_1 v_1 \), where \( A_1 \) is the cross-sectional area and \( v_1 \) is the velocity we need to find. Solving for \( v_1 \), we get \( v_1 = \frac{Q}{A_1} \). Substitute \( A_1 = \pi (0.150)^2 \) into the equation and calculate \( v_1 \).

Step 3: Understand the Continuity Equation

According to the principle of conservation of mass, the mass flow rate must be constant throughout the pipe. This implies that \( A_1 v_1 = A_2 v_2 \), where \( A_2 \) and \( v_2 \) are the cross-sectional area and velocity at a second point in the pipe.

Step 4: Calculate the Area of the Second Cross-section Using Velocity

Given \( v_2 = 3.80 \ \mathrm{m}/\mathrm{s} \), use the continuity equation \( A_1 v_1 = A_2 v_2 \) to solve for \( A_2 \). We already have \( v_1 \) from Step 2. Substitute \( v_1 \) and \( v_2 \) and calculate \( A_2 = \frac{A_1 v_1}{v_2} \).

Step 5: Calculate the Radius at the Second Point

The area of the second cross-section, \( A_2 \), is \( A_2 = \pi r_2^2 \). Solve for \( r_2 \) using \( r_2 = \sqrt{ \frac{A_2}{\pi} } \). Use the \( A_2 \) from Step 4 to find \( r_2 \).

Key Concepts

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics. It states that the product of the cross-sectional area of a pipe and the fluid velocity must remain constant along the length of the pipe. This concept is rooted in the conservation of mass, which means that the same amount of fluid must pass through each section of the pipe in a steady flow.

The continuity equation is expressed mathematically as:

• \( A_1 v_1 = A_2 v_2 \)

Where:

• \( A_1 \) and \( v_1 \) are the cross-sectional area and velocity at the first point.
• \( A_2 \) and \( v_2 \) are the cross-sectional area and velocity at a second point.

This equation helps to determine how changes in the pipe's size impact the fluid flow. If the cross-sectional area increases, the velocity decreases, and vice versa. Understanding this equation is crucial for solving and analyzing fluid flow problems in a variety of applications.

Volume Flow Rate

Volume flow rate is a measure of the volume of fluid that passes through a given cross-sectional area of a pipe per unit of time. It is usually denoted by the symbol \( Q \), and it indicates the flow capacity of a system.

The volume flow rate can be calculated using the equation:

• \( Q = A v \)

Where:

• \( Q \) is the volume flow rate.
• \( A \) is the cross-sectional area through which the fluid flows.
• \( v \) is the velocity of the fluid.

In the context of the exercise, the volume flow rate remains constant at 1.20 m³/s, indicating a steady flow. This consistency allows us to apply the continuity equation and calculate variables such as the velocity of the fluid and the size of the cross-sectional area at different points in the pipe.

Cross-Sectional Area

The cross-sectional area of a pipe is pivotal in determining the fluid dynamics within the pipe. It refers to the area of the slice or section of the pipe through which the fluid flows. For a circular pipe, this area can be calculated using the formula for the area of a circle:

• \( A = \pi r^2 \)

Where:

• \( A \) is the cross-sectional area.
• \( r \) is the radius of the pipe.

In the given exercise, the radius at one section of the pipe is 0.150 m, allowing us to find the cross-sectional area by squaring the radius and multiplying by \( \pi \). Calculating this area is essential for solving problems related to fluid flow rate and velocity, as it's a critical component in both the continuity equation and flow rate calculations.

Velocity Calculation

Velocity calculation is a fundamental aspect when analyzing fluid flow in pipes. The velocity of fluid can be derived using the knowledge of volume flow rate and cross-sectional area, making use of the equation:

• \( v = \frac{Q}{A} \)

Where:

• \( v \) is the velocity of the fluid.
• \( Q \) is the volume flow rate.
• \( A \) is the cross-sectional area.

For the section of the pipe in question, once the area has been calculated from the given radius of 0.150 m, the velocity can be found by dividing the constant flow rate, 1.20 m³/s, by the calculated area. Understanding how to calculate velocity using these parameters is crucial for resolving real-world engineering problems and ensuring efficient fluid transport systems.