Problem 31 - College Physics

Question

A person weighs 170 lb. (a) What is his mass in kilograms? (b) Assuming the density of the average human body is about that of water (which is true), estimate his body's volume in both cubic meters and liters. Explain why the smaller unit of the liter is more appropriate (convenient) for describing a volume of this size.

Step-by-Step Solution

Answer

Mass: 77.11 kg; Volume: 0.077 m³ and 77 L; Liters are more convenient.

Step 1: Convert Weight to Mass

We need to convert the person's weight from pounds to kilograms. To do this, we use the conversion factor: 1 pound equals 0.453592 kilograms. Therefore, the mass in kilograms is calculated as:\[\text{Mass (kg)} = 170 \text{ lb} \times 0.453592 \text{ kg/lb} = 77.11064 \text{ kg}\]

Key Concepts

Weight to Mass Conversion

When dealing with physics problems, converting weight to mass is a common task. The difference between weight and mass might seem small at first, but they hold different meanings in physics. Weight is the force that gravity exerts on an object, and it's typically measured in newtons or pounds. Mass, however, is the amount of matter in an object, measured in kilograms or grams.

To convert weight to mass, especially from pounds to kilograms, we use a set conversion factor because 1 pound is equivalent to 0.453592 kilograms. This means if you weigh 170 pounds, your mass in kilograms can be calculated by multiplying your weight by this conversion factor:

• 170 lb × 0.453592 kg/lb = 77.11064 kg.

Understanding this conversion is crucial, as it allows us to express weight in terms of a mass which is more universally used in scientific calculations, such as those involving force or energy.

Density of Human Body

The density of an object tells us how much mass is contained within a given volume. For the human body, the density is typically compared to that of water. This means we assume that the body's density is approximately 1 gram per cubic centimeter (since the density of water is 1 g/cm³).

Why is this approximation useful? Because it allows us to estimate the body's volume directly once we know the mass. Knowing the mass in kilograms and assuming the density is equivalent to that of water helps simplify calculations. For a person weighing 170 pounds and converted to 77.11064 kilograms, it follows that the volume is:

• Mass/ Density = 77.11064 kg / 1000 kg/m³ = 0.07711064 m³.

This approximation is handy for estimations and offers a straightforward method to relate mass to volume when examining the human body's properties.

Volumetric Measurement

Volumetric measurement refers to assessing the space that an object occupies. In the context of the human body, it's practical to use units like cubic meters or liters for these measurements.

Given our earlier calculation, we can express the body volume in liters. Since 1 cubic meter equals 1000 liters, we convert this volume:

• 0.07711064 m³ × 1000 L/m³ = 77.11064 L.

This conversion shows why using liters can be more convenient for smaller volumes, like those of a human body. Liters simplify visualization and are easier to relate to everyday substances and containers, making them a practical unit for human body volume estimation.

Unit Conversion

Unit conversion is essential in physics for ensuring consistent measurements across different systems of units. Whether we're dealing with lengths, weights, or volumes, converting one unit to another allows for clear and consistent comparison.

In this exercise, we've converted from pounds to kilograms and from cubic meters to liters. Mastering unit conversion requires understanding conversion factors, such as:

• 1 lb = 0.453592 kg
• 1 m³ = 1000 L

By using these conversion factors, you ensure calculations remain accurate and easily interpreted, helping you solve complex physics problems while maintaining clarity in representation and comparison.