Problem 15 - Chemistry: The Central Science in SI Units, Global Edition

Question

A person weighing \(75 \mathrm{~kg}\) is standing on a threelegged stool. The stool momentarily tilts so that the entire weight is on one foot. If the contact area of each foot is \(5.0 \mathrm{~cm}^{2}\), calculate the pressure exerted on the underlying surface in (a) bars, \((\mathbf{b})\) atmospheres, and \((\mathbf{c})\) pounds per square inch.

Step-by-Step Solution

Answer

The pressure exerted on the surface by the person weighing 75 kg standing on one foot of a three-legged stool is approximately \(14.715\) bars, \(14.52\) atmospheres, and \(213.37\) pounds per square inch.

Step 1: (Step 1: Calculate the weight of the person in Newtons)

The weight of the person is given in kg, so we will first need to convert it into a force in Newtons using the gravitational acceleration, g = 9.81 m/s². Weight (force) = mass × gravitational acceleration Weight (force) = 75 kg × 9.81 m/s² = 735.75 N

Step 2: (Step 2: Calculate the contact area in square meters)

The contact area is given in square centimeters. To calculate pressure, we need this area in square meters: Area = 5.0 cm² × 1 m² / 10000 cm² = 5.0 × 10⁻⁴ m²

Step 3: (Step 3: Calculate the pressure exerted on the surface)

We can now calculate the pressure exerted on the surface using the formula: Pressure = Force / Area Pressure = 735.75 N / 5.0 × 10⁻⁴ m² = 1.4715 × 10⁶ N/m² or Pa

Step 4: (Step 4: Convert the pressure into bars)

To convert the pressure in Pascals (Pa) to bars, we can use the following conversion factor: 1 bar = 10⁵ Pa Pressure (bars) = Pressure (Pa) / 10⁵ Pressure (bars) = 1.4715 × 10⁶ Pa / 10⁵ = 14.715 bars

Step 5: (Step 5: Convert the pressure into atmospheres)

To convert the pressure in Pascals (Pa) to atmospheres (atm), we can use the following conversion factor: 1 atm = 101325 Pa Pressure (atm) = Pressure (Pa) / 101325 Pressure (atm) = 1.4715 × 10⁶ Pa / 101325 = 14.52 atm

Step 6: (Step 6: Convert the pressure into pounds per square inch (psi))

To convert the pressure in Pascals (Pa) to pounds per square inch (psi), we can use the following conversion factor: 1 psi = 6895 Pa Pressure (psi) = Pressure (Pa) / 6895 Pressure (psi) = 1.4715 × 10⁶ Pa / 6895 = 213.37 psi As a result, the pressure exerted on the surface by the person weighing 75 kg standing on one foot of a three-legged stool is (a) approximately 14.715 bars, (b) approximately 14.52 atmospheres, and (c) approximately 213.37 pounds per square inch.

Key Concepts

Converting Newtons to Pascals

When calculating pressure, it is essential to understand how to convert units from force to pressure. Force is commonly measured in Newtons (N), whereas pressure is measured in Pascals (Pa). The relationship between these two is defined by the formula for pressure:

• Pressure (P) is defined as the Force (F) distributed over an Area (A).
• The formula for pressure is given by: \( P = \frac{F}{A} \)
• This means that pressure is the force exerted perpendicularly over a unit area.

To convert Newtons to Pascals, you will need to consider the area over which the force is spread. If a force of 735.75 N acts on an area of \(5.0 \times 10^{-4}\) square meters, the pressure in Pascals can be calculated as:\[ P = \frac{735.75 \text{ N}}{5.0 \times 10^{-4} \text{ m}^2} = 1.4715 \times 10^6 \text{ Pa} \]Understanding this conversion is critical for any situation where precise measurements of force and surface area are needed, such as in engineering or physics problems.

Pressure Calculation

Pressure is a fundamental concept in science and engineering that describes how a force is distributed over a particular area. Calculating pressure involves assessing both the magnitude of the force and the area upon which it acts:

• For example, when a person stands on a stool, the person’s weight exerts a force on the stool’s legs.
• The force from the person’s weight is directed into a smaller area – the contact point of the stool's leg with the ground.

The calculation involves these steps:

• First, find the person's weight in Newtons by multiplying their mass by gravitational acceleration: \( \text{Weight (N)} = \text{Mass (kg)} \times 9.81 \) m/s².
• Second, convert the contact area from \(\text{cm}^2\) to \(\text{m}^2\) to match the SI units: \( 1 \text{ cm}^2 = 1 \times 10^{-4} \text{ m}^2\).
• Finally, use the formula \( P = \frac{F}{A} \) to calculate pressure.

By understanding how to perform these calculations, you can determine how forces will affect various surfaces, which is incredibly important in designing safe and effective structures.

Unit Conversion in Pressure Measurement

In real-world applications, pressure needs to be understood across different units depending on the context. This requires converting pressure from one unit to another, such as converting Pascals to bars, atmospheres, or pounds per square inch (psi). Here's how you can do it:

• To convert Pascals to bars, remember that 1 bar = 10⁵ Pa.
• Divide the pressure in Pascals by 10⁵ to get the pressure in bars.
• When converting to atmospheres (atm), use the conversion factor 1 atm = 101325 Pa.
• For psi, note that 1 psi = 6895 Pa.

Performing a unit conversion is straightforward once you know the appropriate conversion factors. For instance, converting a pressure of \(1.4715 \times 10^6\) Pa results in:

• Approximately 14.715 bars
• Approximately 14.52 atm
• Approximately 213.37 psi

Understanding these conversions is crucial for comparing pressure measurements in different scientific fields and for communicating with international partners where different units may be used.