Problem 52 - College Physics

Question

A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7200 \(\mathrm{cm}^{3} / \mathrm{s}\) . At one point in the pipe, where the radius is \(4.00 \mathrm{cm},\) the water's absolute pressure is \(2.40 \times 10^{5}\) Pa. At a second point in the pipe, the water passes through a constriction where the radius is 2.00 \(\mathrm{cm} .\) What is the water's absolute pressure as it flows through this constriction?

Step-by-Step Solution

Answer

The absolute pressure at the constriction is approximately 224627 Pa.

Step 1: Understand the Problem

We have a pipe with two different radii, and we need to find the absolute pressure at the constriction (second point) given the discharge rate, radius, and pressure at the first point.

Step 2: Apply the Continuity Equation

The continuity equation relates the flow velocities at the two points:\[ A_1 \cdot v_1 = A_2 \cdot v_2 \] where \(A_1\) and \(A_2\) are the cross-sectional areas at the first and second point respectively. The cross-sectional area \(A\) for a circular pipe is given by \(A = \pi r^2\). The flow rate is given as 7200 \(\text{cm}^3/\text{s}\), which allows us to find \(v_1\) and \(v_2\).

Step 3: Calculate Velocities at Both Points

First, calculate the areas at both points:\[ A_1 = \pi \times (4.00 \text{ cm})^2 = 50.27 \text{ cm}^2 \]\[ A_2 = \pi \times (2.00 \text{ cm})^2 = 12.57 \text{ cm}^2 \]Now, use the flow rate to find velocities:\[ v_1 = \frac{7200 \text{ cm}^3/\text{s}}{50.27 \text{ cm}^2} = 143.21 \text{ cm/s} \]\[ v_2 = \frac{7200 \text{ cm}^3/\text{s}}{12.57 \text{ cm}^2} = 572.82 \text{ cm/s} \]

Step 4: Apply Bernoulli's Equation

Bernoulli's equation relates the pressures and velocities at two points along the streamline:\[ P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \]Given that the pressure at point 1 \(P_1\) is \(2.40 \times 10^5\) Pa, density of water \(\rho\) is approximately \(1000 \text{ kg/m}^3\), we can solve for \(P_2\).

Step 5: Solve for the Absolute Pressure at the Constriction

First, convert velocities to m/s:\[ v_1 = 1.4321 \text{ m/s} \] \[ v_2 = 5.7282 \text{ m/s} \]Now plug the values into Bernoulli's equation:\[ 2.40 \times 10^5 \text{ Pa} + \frac{1}{2} \times 1000 \times (1.4321)^2 = P_2 + \frac{1}{2} \times 1000 \times (5.7282)^2 \]Simplify and solve for \(P_2\):\[ 240000 + 1026.17 = P_2 + 16398.99 \]\[ P_2 = 240000 + 1026.17 - 16398.99 = 224627.18 \text{ Pa} \]

Step 6: Final Answer

The absolute pressure as the water flows through the constriction is approximately \(224627 \text{ Pa}\).

Key Concepts

Continuity Equation

The continuity equation is a fundamental concept in fluid dynamics that helps us understand how fluids behave when moving through different sections of a pipe. It states that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another, as long as no fluid is added or taken away.

• The equation is written as: \[ A_1 \cdot v_1 = A_2 \cdot v_2 \]
• \(A_1\) and \(A_2\) represent the cross-sectional areas, while \(v_1\) and \(v_2\) are the flow velocities at two different points.

The application of this equation in our problem allows us to understand how velocities are inversely affected by changes in cross-sectional area. If the area decreases, the velocity must increase to maintain the same flow rate. This continuity is essential in analyzing and predicting fluid behavior in various engineering applications.

Fluid Flow

Fluid flow describes the motion of liquid molecules as they move through a system or medium. Here, we examine how water travels through a pipe in a sprinkler system. Key characteristics of fluid flow include:

• Flow Rate: The given flow rate of water is 7200 \(\text{cm}^3/\text{s}\). This tells us the volume of water moving through the pipe per second.
• Flow Velocity: This is the speed at which fluid moves. By using the continuity equation, we calculate the velocities at different pipe sections, showing that smaller pipes mean faster flow.

The fluid flow is central to understanding how changes in pipe diameter affect the behavior of the moving water. Engineers consider fluid flow when designing systems to ensure efficient movement and distribution of liquids.

Absolute Pressure

Absolute pressure is a measure of the total pressure exerted on a fluid, which includes atmospheric pressure. In the context of the sprinkler system, it's crucial to know the water pressure at different pipe sections. Absolute pressure helps us determine the energy available to move the fluid.
In our problem, the absolute pressure at the first point is given as \(2.40 \times 10^5\) Pa. Bernoulli's equation is then used to calculate the pressure at the constriction:
\[ P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \]
Understanding absolute pressure is vital for ensuring that sufficient force is available to propel the fluid through the system. It impacts how systems are built to prevent inefficiencies or damage.

Cross-Sectional Area

The cross-sectional area of a pipe is directly related to the ease with which fluid flows through it. It indicates how much of the pipe is open for fluid to pass through. Calculating the cross-sectional area involves the formula for the area of a circle:
\[ A = \pi r^2 \]

• At the first point in the pipe, we calculated \(A_1\) with a radius of 4.00 cm, giving an area of 50.27 \(\text{cm}^2\).
• At the constriction, the second point has a radius of 2.00 cm, resulting in an area of 12.57 \(\text{cm}^2\).

Knowing the cross-sectional area is essential for using the continuity and Bernoulli’s equations effectively. Smaller areas lead to faster fluid speeds, as seen in the problem solution. Understanding how the cross-sectional area affects fluid dynamics aids in designing systems that optimize flow efficiency.