Problem 49 - College Physics

Question

What gauge pressure is required in the city water mains for a stream from a fire hose connected to the mains to reach a vertical height of 15.0 \(\mathrm{m}\) ? (Assume that the mains have a much larger diameter than the fire hose.)

Step-by-Step Solution

Answer

The required gauge pressure is 147,150 pascals.

Step 1: Understand the Problem

We need to determine the gauge pressure needed in the city's water mains for water to reach a height of 15.0 meters using the fire hose. We are given the height and need to find the pressure.

Step 2: Identify the Relevant Formula

We use Bernoulli's equation for fluid flow, which states: \[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \] Here, \(P\) is the fluid pressure, \(\rho\) is the fluid density, \(v\) is the fluid velocity, \(g\) is the acceleration due to gravity, and \(h\) is the height.

Step 3: Consider Assumptions

Assume the point of water exit from hose has negligible velocity and height, setting these terms to zero for simplifying Bernoulli's equation. This implies the pressure difference needs to offset the gravitational height as the main factor.

Step 4: Substitute Known Values

Since velocity at the top of the stream can be zero (just reaches the height), and neglecting other minor components, the equation simplifies to balancing pressure with gravitational height:\[ P_g = \rho gh \]where \(\rho\) is the density of water (\(1000 \, \text{kg/m}^3\)), \(g = 9.81 \, \text{m/s}^2\), and \(h = 15 \, \text{m}\).

Step 5: Calculate Gauge Pressure

Enter the values:\[ P_g = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 15 \, \text{m} = 147150 \, \text{Pa} \]Therefore, the required gauge pressure is 147150 pascals.

Key Concepts

Bernoulli's equation

Bernoulli's equation is a fundamental principle in fluid dynamics that describes the conservation of energy in a fluid flow. It relates the pressure, velocity, and height of a fluid at different points in a streamline. This equation can be expressed as: \[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]Here:

• \(P\) represents the fluid pressure.
• \(\rho\) is the density of the fluid.
• \(v\) is the velocity of the fluid.
• \(g\) is the acceleration due to gravity, approximately \(9.81 \, \text{m/s}^2\) on Earth.
• \(h\) is the height above a reference point.

Bernoulli's equation is invaluable for understanding how changes in a fluid's speed or elevation will affect its pressure, helping us to solve problems involving flow from fires hoses or even aircraft lift. Since energy along a streamline is constant, increasing a fluid's velocity will lower its pressure, and vice versa.

fluid dynamics

Fluid dynamics is the branch of physics concerned with the study of fluids (liquids and gases) in motion. It deals with how forces affect the movement and behavior of fluids. The principles of fluid dynamics are critical in understanding how fluids flow and interact with their surroundings. Common applications include:

• Designing and analyzing systems involving water supply, like city water mains.
• Predicting weather patterns and designing HVAC systems.
• Understanding natural phenomena such as ocean currents and atmospheric circulation.

By examining the forces and energy within a fluid system, fluid dynamics allows engineers and scientists to predict how fluids behave under various conditions, optimizing systems for efficiency and safety. Bernoulli's equation is one of many powerful tools used in the field.

fluid pressure

Fluid pressure is the perpendicular force exerted by a fluid per unit area. It is an essential concept in fluid mechanics, helping to determine how fluids move through pipes, how they apply force on surfaces, and how they interact with other systems.Key factors include:

• The density of the fluid \(\rho\), which affects pressure distribution.
• The depth of the fluid, as pressure increases with depth due to the weight above.
• The state of motion of the fluid, particularly its velocity, as described by Bernoulli's principle.

Fluid pressure is often measured in pascals (Pa) and is crucial in determining the performance of hydraulic systems, the lift on aircraft, and even atmospheric pressure. Understanding how pressure varies with changes in motion, depth, or fluid type is foundational in many engineering fields.

gauge pressure calculation

Gauge pressure refers to the pressure measured relative to the ambient atmospheric pressure. Unlike absolute pressure, which includes atmospheric pressure, gauge pressure is used in everyday applications where only the internal pressure difference matters.To calculate gauge pressure using Bernoulli's principle in the context of a fire hose, we use the simplified form:\[ P_g = \rho gh \]where

• \(P_g\) is the gauge pressure needed to achieve the desired vertical height.
• \(\rho\) is the density of water (\(1000 \, \text{kg/m}^3\)).
• \(g\) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)).
• \(h\) is the height the water needs to reach (in this exercise, 15 meters).

The calculated pressure gives us the additional pressure beyond the atmospheric pressure needed in the water mains to ensure the water reaches the specified height. Understanding gauge pressure is crucial for ensuring systems like fire hoses operate effectively and safely.