Problem 4 - College Physics

Question

(a) The density (mass divided by volume) of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\). What is this value in kilograms per cubic meter? (b) The density of blood is \(1050 \mathrm{~kg} / \mathrm{m}^{3} .\) What is this density in \(\mathrm{g} / \mathrm{cm}^{3} ?\) (c) How many kilograms are there in a \(1.00 \mathrm{~L}\) bottle of drinking water? How many pounds?

Step-by-Step Solution

Answer

(a) 1000 kg/m³; (b) 1.05 g/cm³; (c) 1.00 kg and 2.20 lb.

Step 1: Convert g/cm³ to kg/m³ for Water

To convert the density of water from \(1.00 \, \text{g/cm}^3\) to \(\text{kg/m}^3\), multiply by 1000 because there are 1000 g in 1 kg and \(1 \text{m}^3 = 10^6 \text{cm}^3\). Thus, \(1.00 \, \text{g/cm}^3 = 1.00 \times 1000 = 1000 \, \text{kg/m}^3\).

Step 2: Convert kg/m³ to g/cm³ for Blood

Given the density of blood is \(1050 \, \text{kg/m}^3\). To convert to \(\text{g/cm}^3\), divide by 1000 as \(1 \text{kg/m}^3 = 0.001 \text{g/cm}^3\). Therefore, \(1050 \, \text{kg/m}^3 = 1050 \times 0.001 = 1.05 \, \text{g/cm}^3\).

Step 3: Liters to Cubic Meters for Water Calculation

Convert 1.00 \(\text{L}\) of water to cubic meters. Since \(1 \text{L} = 0.001 \text{m}^3\), a 1.00 \(\text{L}\) bottle corresponds to 0.001 \(\text{m}^3\).

Step 4: Calculate Mass in Kilograms for Water

Using the density of water \(1000 \, \text{kg/m}^3\) from step 1 and volume from step 3, calculate the mass: \( \text{mass} = \text{density} \times \text{volume} = 1000 \, \text{kg/m}^3 \times 0.001 \, \text{m}^3 = 1.00 \text{kg}\).

Step 5: Convert Kilograms to Pounds

To convert from kilograms to pounds, use the conversion factor \(1 \text{kg} = 2.20462 \text{lb}\). Thus, \(1.00 \text{kg} \approx 1.00 \times 2.20462 = 2.20462 \text{lb}\).

Key Concepts

Unit Conversion

Unit conversion is a fundamental process in science and mathematics that allows us to translate measurements from one unit to another. In this exercise, we need to convert densities between units like grams per cubic centimeter (\( ext{g/cm}^3 \)) to kilograms per cubic meter (\( ext{kg/m}^3 \)), and vice versa. Each type of unit serves a specific context or scale, helping us measure various quantities in a way suited to the problem at hand.
Here's a simple way to approach unit conversions:

• Identify the conversion factors between the units. For instance, 1 kg = 1000 g and 1 m³ = 1,000,000 cm³.
• Determine if you need to multiply or divide by these factors to change one unit into another.
• Apply the conversion factor carefully to transform the original measurement, ensuring the units cancel correctly.

Grasping unit conversion is essential for understanding and solving problems across different scientific fields.

Metric System

The metric system is an internationally recognized system of measurement based on powers of ten, making calculations simple and harmonized.
It is important to understand how the metric system integrates with all types of measurements such as mass, length, and volume.
In this exercise, we see this through lengths (centimeters and meters), masses (grams and kilograms), and their associated volumes.
The metric system is built on basic units such as:

• Length measured in meters (m).
• Mass measured in grams (g) or kilograms (kg).
• Volume measured in liters (L) or cubic meters (m³).

Conversions within the metric system often involve shifting the decimal point, thanks to its base-10 design, making it easy to understand and work with.

SI Units

The International System of Units (SI) is the modern form of the metric system, established to provide a coherent and standardized measurement system worldwide.
In science and engineering, SI units serve as the foundation for clear communication and consistency in measurements.
This problem highlights some critical SI base units:

• The kilogram (kg) for mass.
• The meter (m) for length.
• The cubic meter (\( ext{m}^3 \) ) or liter (L) for volume.

Where precise measurements are crucial, as in studying densities or calculating physical quantities, the use of SI units ensures we avoid misunderstandings and maintains a uniform approach.

Mass and Volume Calculation

Calculating mass and volume involves using formulas that connect these properties through density.In this exercise, density serves as a bridge between mass and volume, given by the relation:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]We used this relation to convert and calculate various quantities:

• From density and volume, one can calculate mass as *mass = density × volume*.
• Converting between units of mass and volume enables us to perform computations that are foundational for many scientific applications.

Understanding these calculations equips one to tackle problems from simple everyday calculations to complex scientific investigations, ensuring one handles differences in scale with ease.