Problem 44 - College Physics

Question

Water is flowing in a cylindrical pipe of varying circular 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 the volume flow rate in the pipe is 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) 0.317 m.

Step 1: Understand Volume Flow Rate Formula

The volume flow rate \(Q\) in a cylindrical pipe is given by the formula \(Q = A \cdot v\), where \(A\) is the cross-sectional area, and \(v\) is the velocity of the fluid.

Step 2: Calculate the Cross-Sectional Area at First Point

At the first point, the radius \(r_1 = 0.150\, \mathrm{m}\). The cross-sectional area \(A_1\) is given by \(A_1 = \pi r_1^2\). Calculate \(A_1\) as follows:\[A_1 = \pi (0.150)^2 = 0.0707\, \mathrm{m}^2.\]

Step 3: Solve for Water Speed at First Point

We know the volume flow rate \(Q = 1.20\, \mathrm{m}^3/\mathrm{s}\). Using the formula \(Q = A_1 \cdot v_1\), solve for the velocity \(v_1\):\[v_1 = \frac{Q}{A_1} = \frac{1.20}{0.0707} = 16.97\, \mathrm{m/s}.\]

Step 4: Find the Area of the Second Point using Water Speed

At the second point, the water speed \(v_2 = 3.80\, \mathrm{m/s}\). Using the flow rate formula, \(Q = A_2 \cdot v_2\), solve for \(A_2\):\[A_2 = \frac{Q}{v_2} = \frac{1.20}{3.80} = 0.3158\, \mathrm{m}^2.\]

Step 5: Calculate the Radius at the Second Point

Using the cross-sectional area formula \(A_2 = \pi r_2^2\), solve for \(r_2\):\[r_2 = \sqrt{\frac{A_2}{\pi}} = \sqrt{\frac{0.3158}{\pi}} = 0.317\, \mathrm{m}.\]

Key Concepts

Volume Flow Rate

Volume flow rate is a fundamental concept in fluid dynamics that helps us measure how much volume of fluid flows through a particular section of a pipe over a given period. It's important because it tells us the efficiency and speed of fluid movement. For a pipe, the volume flow rate (\(Q\)) is expressed as:\[Q = A \cdot v\]where:

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

This equation illustrates that the flow rate is directly proportional to both the area and the flow speed. Thus, changes in either the area of the pipe or the speed of the fluid will result in changes to the flow rate. Understanding volume flow rate helps us solve problems related to fluid movement, especially when designing and analyzing pipe systems.

Cylindrical Pipe

A cylindrical pipe is often used in fluid flow problems. Its geometry is simple, making calculations straightforward. Such pipes have a constant, circular cross-section, letting fluids move efficiently. In fluid dynamics, the behavior of fluid in a cylindrical pipe is often analyzed using the concepts of flow rate, velocity, and cross-sectional area.

The main feature of a cylindrical pipe is its radius. The radius impacts the pipe's cross-sectional area and thus influences the flow characteristics. For a set flow rate, a wider pipe (larger radius) allows slower fluid velocity, and conversely, a narrower pipe increases fluid velocity. This relationship is key when designing systems to control or predict fluid behavior.

Cross-Sectional Area

The cross-sectional area (\(A\)) of a pipe is crucial because it's one of the primary determinants of how fluid flows within that pipe. For a cylindrical pipe, this area can be calculated using the formula:\[A = \pi r^2\]where \(r\) is the radius of the pipe. This formula tells us that the area increases with the square of the radius, meaning even small changes in the radius can significantly alter the area.

The cross-sectional area is directly linked to the volume flow rate as it, along with fluid velocity, determines how much fluid passes through a section per second. In our exercise, knowing the area at different points allows us to calculate relevant velocities and radii, making it a vital step in understanding fluid behavior through pipes.

Velocity of Fluid

The velocity (\(v\)) of a fluid in a pipe indicates how quickly the fluid is moving through the system. It's a critical factor in fluid dynamics because it affects the force and energy within flowing fluids. In the context of a cylindrical pipe, velocity is derived from the flow rate and the cross-sectional area using the formula:\[v = \frac{Q}{A}\]Here, \(Q\) is the volume flow rate and \(A\) is the cross-sectional area. This equation shows that velocity is inversely proportional to the cross-sectional area for a constant flow rate. Thus, when the pipe narrows (smaller \(A\)), the velocity increases, and when the pipe widens, the velocity decreases.

Understanding the velocity of fluid provides insight into kinetic energy within the pipe system and helps predict how other factors, like pressure changes, might occur as fluid travels.