Problem 7 - Applied Physics

Question

Find the water pressure (in \(\mathrm{kPa}\) ) at the \(25.0\) -m level of a water tower containing water \(50.0 \mathrm{~m}\) deep.

Step-by-Step Solution

Answer

The water pressure at the 25.0-m level is 245.25 kPa.

Step 1: Identify the Known Values

We know the height of the water column is 25.0 meters.The density of water \( \rho \) is \( 1000 \ \mathrm{kg/m^3} \).The gravitational acceleration \( g \) is \( 9.81 \ \mathrm{m/s^2} \).The formula for pressure due to a column of fluid is \( P = \rho g h \).

Step 2: Substitute Known Values into Formula

Substitute the given values into the formula \( P = \rho g h \), resulting in\[ P = (1000 \, \mathrm{kg/m^3}) \times (9.81 \, \mathrm{m/s^2}) \times (25.0 \, \mathrm{m}) \].

Step 3: Calculate the Pressure

Calculate the product:\[ P = 1000 \times 9.81 \times 25.0 = 245,250 \, \mathrm{Pa} \].This result is in pascals, where \( 1 \, \mathrm{Pa} = 1 \, \mathrm{N/m^2} \).

Step 4: Convert Pressure to kPa

Since 1 kPa is equal to 1000 Pa, we convert pascals to kilopascals:\[ 245,250 \, \mathrm{Pa} = 245.25 \, \mathrm{kPa} \].

Key Concepts

Pressure Calculation

Pressure calculation is a fundamental concept in fluid mechanics that expresses the force exerted by a fluid per unit area. For a fluid at rest, like water, this pressure is often determined based on the height of the fluid column above the point where the pressure is being calculated. This is due to the weight of the fluid exerting a force over the area beneath it. In our exercise, we use the formula:\[ P = \rho g h \]where:

• \( P \) is the pressure.
• \( \rho \) is the fluid density, which is the mass per unit volume.
• \( g \) is the gravitational acceleration.
• \( h \) is the height of the fluid.

This formula is essential as it allows us to calculate the pressure at any depth within a fluid, based on the weight of the fluid column above it. It's a direct application of the principles of physics, relating mass, force, and surface area.

Density of Water

The density of water is a critical factor in fluid mechanics calculations. It is the measure of how much mass is contained within a given volume. In our exercise, we take the density of water as a standard value of \( 1000 \, \mathrm{kg/m^3} \). This standard value is often used for calculations under normal conditions, implying pure water at about 4°C.

• The consistency and reliability of this figure make it a fundamental constant in various scientific calculations.
• It simplifies the use of formulas and allows for more straightforward process analysis.

Understanding the density of water helps us predict how it will behave under different conditions, such as changes in pressure or temperature.

Gravitational Acceleration

Gravitational acceleration \( g \) on Earth is approx. \( 9.81 \, \mathrm{m/s^2} \). This value represents the acceleration of an object due in free fall towards the Earth, unaffected by air resistance. This concept is critical in pressure calculations, because it determines how strongly the force of gravity acts on the water mass.

• Gravitational acceleration imparts weight to the fluid, contributing directly to the pressure it produces.
• When calculating pressure in fluids, \( g \) remains a constant value, simplifying computations by providing consistency across various scenarios.

Knowledge of gravitational acceleration helps ensure accurate predictions of pressure in steady-state systems like water towers.

Water Tower Pressure

Water tower pressure is a practical application of the principles of fluid mechanics, where pressures are needed at various points in a distribution system. The height of water in a tower is vital since it determines the potential energy given to the water, translating to pressure at any point in its system.

• This pressure ensures that water can flow throughout distribution networks effectively.
• Higher water levels create more pressure, explaining why taller towers provide water to broader areas.

In our exercise, we calculated the pressure at \( 25 \, \mathrm{m} \) below the surface, revealing how control over a simple variable - height - can directly influence water system efficiency and operation.