Problem 15 - Chemistry The Central Science In Si Units

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 (c) pounds per square inch.

Step-by-Step Solution

Answer

Pressure on the surface: (a) 14.715 bars, (b) 14.52 atm, (c) 213.50 psi.

Step 1: Calculate the Force Exerted by the Person

The force exerted by the person is equal to their weight, which is the mass times the acceleration due to gravity. Given that the mass is \(75 \text{ kg}\), we use the formula \( F = m \cdot g \) where \( g = 9.81 \text{ m/s}^2 \). Thus, \( F = 75 \text{ kg} \times 9.81 \text{ m/s}^2 = 735.75 \text{ N} \).

Step 2: Convert Contact Area from cm² to m²

The contact area is given as \(5.0 \text{ cm}^2\). To convert this to square meters, use the conversion factor \(1 \text{ cm}^2 = 0.0001 \text{ m}^2\). Therefore, the area is \(5.0 \text{ cm}^2 \times 0.0001 \text{ m}^2/\text{cm}^2 = 5.0 \times 10^{-4} \text{ m}^2\).

Step 3: Calculate Pressure in Pascals

Pressure is force per unit area, calculated using \( P = \frac{F}{A} \). Substituting the values, \( P = \frac{735.75 \text{ N}}{5.0 \times 10^{-4} \text{ m}^2} = 1,471,500 \text{ Pa} \), which is equal to 14715 hectopascals (hPa).

Step 4: Convert Pressure from Pascals to Bars

1 bar is equal to \(100,000\) Pascals. To find the pressure in bars, divide by \(100,000\). Thus, \(1,471,500 \text{ Pa} = \frac{1,471,500}{100,000} = 14.715 \text{ bars}\).

Step 5: Convert Pressure from Pascals to Atmospheres

1 atmosphere is approximately equal to \(101,325\) Pascals. To convert, divide the pressure in Pascals by the number of Pascals per atmosphere: \(1,471,500 \text{ Pa} = \frac{1,471,500}{101,325} \approx 14.52 \text{ atm}\).

Step 6: Convert Pressure from Pascals to Pounds per Square Inch (psi)

1 psi is approximately equal to 6895 Pascals. Convert by dividing the pressure in Pascals by this value: \(1,471,500 \text{ Pa} = \frac{1,471,500}{6895} \approx 213.50 \text{ psi}\).

Key Concepts

Force Exerted

Understanding the force exerted is crucial in calculating pressure. When we talk about force exerted by an object, we are referring to the force due to its weight, which is the product of mass and gravitational acceleration. In our exercise, a person weighing 75 kg exerts a force determined by the formula \[ F = m \cdot g \]where \( F \) is force, \( m \) is mass, and \( g \) is gravitational acceleration (approximately 9.81 m/s² on Earth). This results in an exerted force of 735.75 Newtons.
This force is what presses down on the surface beneath the stool, causing pressure. Understanding this concept ensures that we correctly determine how forces contribute to pressures in various contexts.

Unit Conversion

Unit conversion is a key step in solving physics problems, especially when dealing with dimensions like area or pressure. In our exercise, we needed to convert the contact area from square centimeters to square meters.
The conversion factor is:

• 1 cm² = 0.0001 m²

The contact area given as 5 cm² can be converted to m² by multiplying with 0.0001 m²/cm², resulting in 5.0 × 10^-4 m².
This conversion is essential because it allows us to keep consistent units for further calculations which often require International System of Units (SI units). It's crucial to perform these conversions carefully to avoid errors in later steps.

Pressure Units

When calculating pressure, it's about understanding the relationship between force and area, expressed as pressure = force/area. Pressure in physics is typically measured in Pascals (Pa). The formula used in the exercise is \[ P = \frac{F}{A} \]where \( F \) is force and \( A \) is area. After calculation, the pressure was found to be 1,471,500 Pascals.
Pascals are the metric unit for pressure, but other units like bars, atmospheres, and psi are also commonly used, particularly in specific industries. Thus, knowing how to compute in Pascals is the first step in understanding pressure measurement.

Pressure in Different Units

Different contexts require pressure to be expressed in various units. Converting Pascals to other pressure units widens comprehension across fields.

• Bars: 1 bar equals 100,000 Pascals. By dividing 1,471,500 Pa by 100,000, we get approximately 14.715 bars.
• Atmospheres (atm): An atmosphere is roughly 101,325 Pascals. For conversion, divide the pressure by 101,325, resulting in around 14.52 atm.
• Pounds per Square Inch (psi): With 1 psi about 6895 Pascals, dividing the Pascal value yields about 213.50 psi.

Each unit serves specific applications: bars in engineering, atmospheres in meteorology, and psi commonly in the United States. This variety of conversions ensures precise communication and understanding across diverse domains.