Problem 10 - Applied Physics

Question

Find the gauge pressure of a tire with an absolute pressure of \(425 \mathrm{kPa}\).

Step-by-Step Solution

Answer

The gauge pressure of the tire is 323.7 kPa.

Step 1: Understand the Problem

We need to find the gauge pressure of a tire. Gauge pressure is the pressure relative to atmospheric pressure, while absolute pressure is the total pressure including atmospheric pressure.

Step 2: Identify Known Values

We are given that the absolute pressure in the tire is \(425 \mathrm{kPa}\). We also know that typical atmospheric pressure is approximately \(101.3 \mathrm{kPa}\).

Step 3: Apply the Formula for Gauge Pressure

Gauge Pressure \(P_g\) is given by the formula: \[ P_g = P_{abs} - P_{atm} \] where \(P_{abs}\) is the absolute pressure, and \(P_{atm}\) is the atmospheric pressure.

Step 4: Calculate Gauge Pressure

Substituting the known values into the formula, we have: \[ P_g = 425 \mathrm{kPa} - 101.3 \mathrm{kPa} \]Calculate the difference to find the gauge pressure.

Step 5: Perform the Calculation

Calculate: \[ P_g = 425 \mathrm{kPa} - 101.3 \mathrm{kPa} = 323.7 \mathrm{kPa} \]

Step 6: Verify the Calculation

Ensure the subtraction is correct: The difference of \(425 \mathrm{kPa}\) and \(101.3 \mathrm{kPa}\) is indeed \(323.7 \mathrm{kPa}\).

Key Concepts

Absolute Pressure

Absolute pressure is a crucial concept in physics and engineering. It is the total pressure exerted by a fluid, including the atmospheric pressure. This is important when you measure the pressure within a system because it accounts for all influences on that system.

To understand absolute pressure, consider it as the complete picture of pressure in an environment:

• It combines the pressure from the medium (like air or liquid) with the atmospheric pressure.
• This measurement gives a full understanding of the conditions affecting your system or object.

Absolute pressure is typically expressed in pascals (Pa) or kilopascals (kPa), which are standard units of pressure. Knowing the absolute pressure helps in accurately assessing forces acting upon a surface or volume, especially in closed systems.

When dealing with problems that involve pressures like in tires, understanding absolute pressure ensures you are working with all pressure aspects, including both internal and external forces.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere above a given point. This pressure affects many day-to-day processes and is a significant factor in weather systems.

Typically, atmospheric pressure is measured in kilopascals (kPa), with standard atmospheric pressure being around 101.3 kPa at sea level. This measurement, although variable based on location and weather, serves as a reference point for many calculations involving gases and liquids.

Consider some key points about atmospheric pressure:

• It varies with altitude; higher altitudes have lower atmospheric pressures.
• Atmospheric pressure impacts how substances behave, including boiling points and the ability of living organisms to breathe.
• In tire pressure calculations, it acts as the baseline against which other pressures are measured.

Understanding atmospheric pressure aids in calculating gauge pressure and interpreting various aspects of the environment or systems you are studying.

Pressure Calculation

Pressure calculation is essential to determining the conditions within a variety of systems, from tires to weather balloons. The objective is to understand the pressures at play and how they influence the system's performance.

One common calculation is finding the gauge pressure, which is the difference between absolute pressure and atmospheric pressure. Gauge pressure is a practical measure as it indicates how much more pressure there is inside compared to outside in the surrounding environment.

The calculation steps are straightforward:

• Identify the absolute pressure given.
• Know the standard atmospheric pressure (about 101.3 kPa).
• Use the formula: \[ P_{gauge} = P_{abs} - P_{atm} \]

For example, in calculating the pressure of a tire, subtracting the atmospheric pressure from the absolute pressure gives the gauge pressure, informing you how the tire pressure compares to the surrounding air pressure. Having the ability to calculate this accurately helps in maintaining safe and efficient operating conditions across various applications.