Question

At times, a pressure is stated in units of mass per unit area rather than force per unit area. Express \(P=1 \mathrm{atm}\) in the unit \(\mathrm{kg} / \mathrm{cm}^{2}\) [Hint: How is a mass in kilograms related to a force?]

Short answer

1 atm is approximately equivalent to \(1.03 \, \mathrm{kg/cm^{2}}\).

Step-by-step solution

Understand the relationship between force and mass

Force is defined as mass times acceleration. In this case, the mass is in kilograms and the acceleration is the gravitational acceleration, which is approximately \(9.8 \, \mathrm{m/s^{2}}\) on the Earth's surface. Therefore, a mass \(m\) in kilograms exerts a force of \(m \times 9.8 \, \mathrm{N}\).

Convert 1 atm to Newton per square meter (Pascal)

1 atm is equivalent to \(101325 \, \mathrm{Pa}\) or \(101325 \, \mathrm{N/m^{2}}\). This means that 1 atm exerts a force of \(101325 \, \mathrm{N}\) on each square meter.

Convert Newtons to kilograms-force

Now we can use the relationship between mass and force to convert from Newtons to kg-force. We use the approximate value for the acceleration due to gravity, so \(1 \, \mathrm{N} = 1/9.8 \, \mathrm{kgf}\). Thus, \(101325 \, \mathrm{N} = 101325/9.8 \, \mathrm{kgf} \approx 10332.65 \, \mathrm{kgf/m^{2}}\).

Convert from square meter to square centimeter

We need to convert the area from square meters to square centimeters. We know that \(1 \, \mathrm{m^{2}} = 10000 \, \mathrm{cm^{2}}\), so \(10332.65 \, \mathrm{kgf/m^{2}} = 10332.65 / 10000 \, \mathrm{kgf/cm^{2}} = 1.033 \, \mathrm{kgf/cm^{2}}\) which is approximately equal to \(1.03 \, \mathrm{kg/cm^{2}}\).

Final answer

So to convert 1 atm into the unit of kg/cm², we get 1 atm = \(1.03 \, \mathrm{kg/cm^{2}}\).

Key concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air surrounding the Earth. It plays a crucial role in weather patterns and our daily lives. At sea level, this pressure is approximately 101,325 Pascals, which equals 1 atmosphere (atm). This value provides a standard reference point for measuring other pressures. You might wonder why it's important. Well, atmospheric pressure affects everything from boiling water to inflating tires, demonstrating its widespread impact. The standard measurement of 1 atm can be converted into various units, such as kg/cm², for different applications. Understanding these conversions is essential in scientific, industrial, and environmental contexts.

Mass and Force Relationship

The connection between mass and force helps us comprehend how objects interact with the world around them. In physics, force is calculated using Newton's Second Law, where force equals mass times acceleration \[ F = m imes a \]. On Earth, the acceleration due to gravity is about 9.8 m/s². When we talk about an object's weight, we're actually referring to the force it exerts due to gravity. For example, a 1 kilogram mass exerts a force of 9.8 Newtons (N). This principle explains why heavier objects require more force to move or stop. Moreover, understanding the mass-force relationship is crucial for converting units of pressure, as pressure often combines these two aspects.

Unit Conversion

Unit conversion is a fundamental skill in scientific calculations, allowing us to translate values from one set of units to another. Proper unit conversion ensures accuracy in measurements and helps scientists and engineers communicate effectively. When converting units, it’s essential to understand the relationships between them. For example, 1 atm equals 101,325 Pascal, or N/m². By using conversion factors, you can switch from one unit to another. A conversion factor is a fraction that represents a relationship between units, such as 1 m² = 10,000 cm². Accurate conversions are vital in fields like physics, chemistry, and engineering, ensuring precise and consistent results.

Pascal to Kilogram-Force per Square Centimeter

Converting from Pascals to kg-force/cm² involves several steps that reflect the relationships between pressure, force, and units of area. First, you need to understand the conversion from Newtons to kg-force. Using gravitational acceleration, 1 Newton is approximately equal to 0.102 kg-force. Then, consider the conversion from square meters to square centimeters, since 1 m² equals 10,000 cm². Now, when you convert 1 atm, or 101,325 Pa, you first change it to Newtons and then to kg-force. Finally, adjust the area units from m² to cm². This multi-step process results in the final conversion: 1 atm approximates to 1.033 kg/cm², providing a practical measurement for various scientific and engineering applications. Through these conversions, we bridge the gap between different unit systems, making scientific data amenable for diverse contexts.