Problem 5 - College Physics

Question

\(\bullet(\) 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 \(\mathrm{g} / \mathrm{cm}^{3} ?(\mathrm{c})\) 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 grams per cubic centimeter to kilograms per cubic meter

The density of water is given as \(1.00 \ \mathrm{g/cm^3}\). To convert this to kilograms per cubic meter, recall that \(1 \ \mathrm{g} = 0.001 \ \mathrm{kg}\) and \(1 \ \mathrm{cm^3} = 10^{-6} \ \mathrm{m^3}\). Therefore, \(1.00 \ \mathrm{g/cm^3} = 1000 \ \mathrm{kg/m^3}\).

Step 2: Convert kilograms per cubic meter to grams per cubic centimeter

The density of blood is given as \(1050 \ \mathrm{kg/m^3}\). To convert this to grams per cubic centimeter, recognize that \(1 \ \mathrm{kg} = 1000 \ \mathrm{g}\) and \(1 \ \mathrm{m^3} = 10^6 \ \mathrm{cm^3}\). Hence, \(1050 \ \mathrm{kg/m^3} = 1.05 \ \mathrm{g/cm^3}\).

Step 3: Find mass of 1 liter of water in kilograms

1 liter is equivalent to \(1000 \ \mathrm{cm^3}\), and since the density of water is \(1.00 \ \mathrm{g/cm^3}\), the mass is \(1000 \ \mathrm{g}\). Converting grams to kilograms gives \(1.00 \ \mathrm{kg}\).

Step 4: Convert kilograms to pounds

Knowing that 1 kilogram is approximately 2.20462 pounds, \(1.00 \ \mathrm{kg}\) of water is \(2.20 \ \mathrm{lb}\).

Key Concepts

Water Density

Understanding the density of water is fundamental in many scientific calculations. Water is often used as a standard due to its uniform density. At 4°C, the density of water is given as precisely \(1.00\ \mathrm{g/cm^3}\). This makes that measurement widely used because it's easy to remember and work with in the metric system.

When you need to convert this to another unit, it's vital to know the relationship between the units. Since there are 1000 grams in a kilogram and 1,000,000 cubic centimeters in a cubic meter, the conversion allows us to express the density in \(1000\ \mathrm{kg/m^3}\) for easier comparisons across different substances.

Blood Density

Blood density is slightly more than that of water, largely due to its composition of various cells and proteins. In terms of metrics, blood density is often given as \(1050\ \mathrm{kg/m^3}\), which points to its greater mass per unit volume compared to water.

Translating this into grams per cubic centimeter turns it to \(1.05\ \mathrm{g/cm^3}\), a close number to that of water but slightly higher due to its more complex composition. Understanding these conversions is crucial in medical sciences where exact densities are necessary for diagnoses and treatments.

Unit Conversion

Unit conversion is a key skill in sciences to ensure measurements are uniform and comprehensible. When converting units of density, remember:

• Mass is converted by recognizing 1 gram equals 0.001 kilograms.
• Volume conversion from cubic centimeters to cubic meters involves multiplying by \(10^{-6}\).
• Conversely, converting from cubic meters to cubic centimeters entails a factor of \(10^6\).

These conversions make switching between the metric units much simpler and are a critical foundation for many scientific calculations.

Metric System

The metric system is the most widely used global standard for measurement, making conversions and comparisons convenient. It operates on a base-10 system, simplifying complex calculations. For example, converting mass from grams to kilograms simply involves moving the decimal place three spots.

The metric system's logical organization encourages uniformity and precision which are essential in fields like science and engineering. Practicing unit conversions within the metric system ensures that you can smoothly transition between different scales of measurement.

Mass Conversion

Converting mass from one unit to another is a frequent task in science and daily life. To convert mass into different units such as from kilograms to pounds, it's crucial to know that 1 kilogram equals approximately 2.20462 pounds.

Grasping this concept helps in varying contexts, from understanding your weight in another country to scientific measurements. Calculations can be performed simply by multiplying the known mass by the conversion factor. Tools like conversion charts or calculators can assist, but understanding the basic relationships ensures accurate manual conversions.