Problem 47 - University Physics with Modern 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 in the constricted section is 1.92×10⁵ Pa.

Step 1: Understand the Given Information

We have a pipe with two different sections. The first section has a radius of 4.00 cm and a water flow rate of 7200 cm³/s while the absolute pressure is 2.40×10⁵ Pa. The second section has a radius of 2.00 cm, and we need to find the pressure at this point.

Step 2: Calculate the Flow Velocity in the First Section

Using the formula for volumetric flow rate, \(Q = A \cdot v\), where \(A\) is the cross-sectional area and \(v\) is the velocity, we can calculate the velocity in the first section. The area \(A_1 = \pi \cdot (4.00)^2 \). Therefore, \(v_1 = \frac{Q}{A_1} = \frac{7200}{\pi \cdot (4.00)^2} \). Calculate \(v_1\).

Step 3: Calculate the Flow Velocity in the Second Section

The flow rate \( Q \) remains constant, so use the same formula \(Q = A \cdot v\). The area \(A_2 = \pi \cdot (2.00)^2\). Therefore, \(v_2 = \frac{7200}{\pi \cdot (2.00)^2} \). Calculate \(v_2\).

Step 4: Apply Bernoulli's Equation

Now, apply Bernoulli’s equation, which is useful for incompressible fluids: \(P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2\), where \(P_1\) is the pressure in the first section, and \(P_2\) is the pressure in the second section. Solve for \(P_2\) using the known \(P_1\), \(v_1\), and \(v_2\). Assume water density, \(\rho = 1000\, \mathrm{kg/m^3}\).

Step 5: Solve the Equation for P_2

Substitute the computed velocities and the known pressures into Bernoulli's equation to solve for \(P_2\). Since the pipe is horizontal, the height terms cancel. Rearrange to find \(P_2:\)\[P_2 = P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2)\].Compute \(P_2\) using the given pressure \(P_1\), and calculated \(v_1\) and \(v_2\).

Key Concepts

Fluid Dynamics

Fluid dynamics is a branch of physics that focuses on the movement of liquids and gases. Understanding how fluids behave when in motion helps solve practical problems in many fields, such as engineering and meteorology. It involves various principles and equations, with Bernoulli's Principle being a key player. Bernoulli's Principle is especially important for understanding how pressure changes in moving fluids. This principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or potential energy. This is vital when analyzing problems like the one we're solving for the sprinkler system. Fluid dynamics helps us calculate the changes in pressure as water flows from one section of the pipe to another.

Volumetric Flow Rate

Volumetric flow rate, often denoted as \( Q \), is the quantity of three-dimensional space that a fluid occupies as it moves through a given area per unit time. In other words, it tells us how much fluid is passing through a section of a pipe or channel at any given moment. In the case of the sprinkler system, the given volumetric flow rate of 7200 \( \mathrm{cm}^3/\mathrm{s} \) lets us calculate the speed of the water in different sections of the pipe. The flow rate remains constant for incompressible fluids, which ensures calculations are consistent across the varying pipe diameters. Calculating the velocity involves using the cross-sectional area of the pipe and applying the formula \( Q = A \cdot v \), where \( A \) represents the area and \( v \) the velocity.

Pressure Calculation

Pressure calculation within fluid systems helps determine the force exerted by the fluid per unit area. In the sprinkler problem, we need to calculate the pressure difference between two points inside the pipe. This involves using Bernoulli's equation, \[P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2\], where \( P_1 \) and \( P_2 \) are the pressures at the first and second points respectively, and \( v_1 \) and \( v_2 \) are the velocities. By knowing the pressure at the first point and calculating the velocities, we can solve for \( P_2 \), the pressure at the narrowest part of the pipe. This approach is essential in designing systems like sprinklers, where consistent water pressure ensures effective operation.

Sprinkler System

A sprinkler system is an effective way to distribute water across a wide area, ideal for uses like golf courses and agriculture. These systems depend crucially on understanding fluid flow and pressure changes. In our problem, water is transported through pipes with varying diameters, requiring precise calculations to ensure uniform spraying. Different segments of the pipe will influence the speed and pressure of the water, which ultimately affects how it is dispersed to the surrounding area. By applying Bernoulli’s Principle, we can design sprinkler systems that balance pressure and speed, ensuring even coverage across the landscape.

Incompressible Fluid Flow

Incompressible fluid flow refers to the behavior of a fluid whose density remains constant throughout its motion. Most liquids, like water, are considered incompressible under normal conditions. This characteristic simplifies calculations in physics and engineering since the mass flow rate is conserved. In the context of our sprinkler problem, this assumption means that the volumetric flow rate stays the same across different sections of the pipe. By treating the fluid as incompressible, we straightforwardly apply Bernoulli’s equation, facilitating the calculation of pressures and velocities through the system. This allows for a precise understanding of how the fluid will behave as it travels from one point to another in the pipe, which is useful for ensuring an efficient and effective sprinkler system.