Problem 14 - Applied Physics

Question

Find the water pressure at ground level to supply water to the third floor of a building \(8.00 \mathrm{~m}\) high with a pressure of \(325 \mathrm{kPa}\) at the third-floor level.

Step-by-Step Solution

Answer

The pressure at ground level should be 403.48 kPa to supply the third floor with 325 kPa.

Step 1: Understand the Problem

We're given a building that is 8 meters tall, and we need to provide water to the third floor with a pressure of 325 kPa. We must calculate the pressure that needs to be applied at the ground level to achieve this.

Step 2: Identify Known Values

We know the height of the building is 8.00 meters and the pressure at the third floor needs to be 325 kPa. We'll use gravitational acceleration \( g = 9.81 \text{ m/s}^2 \) and the density of water \( \rho = 1000 \text{ kg/m}^3 \) in our calculations.

Step 3: Apply Bernoulli's Principle

Bernoulli's equation in simpler terms for a fluid at rest is: \( P_1 + \rho gh_1 = P_2 + \rho gh_2 \). Since the supply line is vertical, \( h_1 = 0 \) (ground level) and \( h_2 = 8 \text{ m} \) (third floor level). We want to find the pressure \( P_1 \) at ground level.

Step 4: Substitute Known Values Into Bernoulli's Equation

Using the simplified Bernoulli's equation: \[ P_1 = P_2 + \rho g h \]. \( P_2 = 325 \text{ kPa} = 325000 \text{ Pa} \), \( \rho = 1000 \text{ kg/m}^3 \), \( g = 9.81 \text{ m/s}^2 \), and \( h = 8 \text{ m} \).So, \[ P_1 = 325000 + 1000 \times 9.81 \times 8 \].

Step 5: Calculate the Pressure at Ground Level

Compute \( P_1 \) using the substituted values:\[ P_1 = 325000 + 1000 \times 9.81 \times 8 \]First, calculate \( 1000 \times 9.81 \times 8 = 78480 \).So, \[ P_1 = 325000 + 78480 = 403480 \text{ Pa} \].

Key Concepts

Bernoulli's Principle

Let's start by understanding Bernoulli's Principle, a cornerstone of fluid mechanics. This principle describes how the pressure within a moving fluid varies with its speed. Essentially, it tells us that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or potential energy of that fluid. Although our exercise involves a fluid at rest, Bernoulli's equation is often used in scenarios where analyzing fluid behavior is essential, like calculating pressures at different heights in a static fluid system.
In a simpler form relevant to our problem, Bernoulli's Equation states:

• The pressure plus potential energy per unit volume at one point equals the pressure plus potential energy per unit volume at another point.
• This equation takes the form: \( P_1 + \rho gh_1 = P_2 + \rho gh_2 \), where \( P \) is the pressure, \( \rho \) is the fluid density, \( g \) is gravitational acceleration, and \( h \) is the height.

For solving our problem, we aim to determine the ground level pressure needed at a specific height to ensure desired pressure at the third floor.

Fluid Mechanics

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) whether at rest or in motion. It helps us analyze fluid behavior in various conditions, involving aspects like pressure, force, and energy.
In the context of our problem, we focus on the following aspects:

• Static fluids, which are not in motion, and so only pressure differences and gravitational forces are involved.
• The density of the fluid, which in our case is water with a density of \( 1000 \text{ kg/m}^3 \).
• Gravitational acceleration \( g \), a constant \( 9.81 \text{ m/s}^2 \), which influences potential energy changes due to height differences in fluids.

Understanding these basic elements allows us to apply the principles of fluid mechanics to solve our water pressure problem effectively.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational pull. It is influenced by factors like fluid density, gravitational force, and the height of the fluid column. In the exercise, this concept helps us understand how pressure varies with depth in a fluid system.
The equation for calculating hydrostatic pressure is \( P = \rho gh \), where:

• \( P \) is the hydrostatic pressure.
• \( \rho \) is the fluid density.
• \( g \) is the gravitational acceleration.
• \( h \) is the height from the reference point.

In our exercise, as water moves vertically across an 8-meter height to the third floor, the pressure decreases by \( \rho gh \). It is crucial to calculate this change to determine the required ground-level pressure to ensure the desired pressure at higher floors.

Physics Problem Solving

Physics problem solving involves a structured approach to break down complex problems into manageable steps. Let's see how it applies to our water pressure problem.
Key steps include:

• Understanding the problem: Identifying the desired pressure requirements at the third-floor level and the height of the building.
• Identifying known and unknown values: Recognizing given data, like height \( 8\, \text{m} \) and third-floor pressure \( 325\, \text{kPa} \).
• Applying relevant equations: Using Bernoulli's equation, considering fluid mechanics principles.
• Performing calculations: Substituting known values into equations to determine unknowns like the ground-level pressure.
• Evaluating the results: Ensuring calculated pressures meet the requirements and logical interpretations fit within physical realities.

By following such a methodical process, students can confidently solve both simple and complex physics problems, ensuring a clearer understanding of fluid dynamics and hydrostatics.