Question

Using your knowledge of metric units, English units, and the information on the back inside cover, write down the conversion factors needed to convert (a) \(\mathrm{mm}\) to \(\mathrm{nm}\), (b) \(\mathrm{mg}\) to \(\mathrm{kg}\), (c) \(\mathrm{km}\) to \(\mathrm{ft}\), \((\mathrm{d}) \mathrm{in},{ }^{3}\) to \(\mathrm{cm}^{3}\).

Short answer

(a) Conversion factor: \(1 \mathrm{mm} = 1,000,000 \mathrm{nm}\) (b) Conversion factor: \(1 \mathrm{mg} = 0.000001 \mathrm{kg}\) (c) Conversion factor: \(1 \mathrm{km} = 3280.84 \mathrm{ft}\) (d) Conversion factor: \(1 \mathrm{in}^3 = 16.3871 \mathrm{cm}^3\)

Step-by-step solution

(a) Converting millimeters (mm) to nanometers (nm)

1 millimeter (mm) is equal to 1,000,000 nanometers (nm), so to convert millimeters to nanometers, we simply multiply by 1,000,000. Conversion factor: \[1 \mathrm{mm} = 1,000,000 \mathrm{nm}\]

(b) Converting milligrams (mg) to kilograms (kg)

1 kilogram (kg) is equal to 1,000,000 milligrams (mg), so to convert milligrams to kilograms, we divide by 1,000,000. Conversion factor: \[1 \mathrm{mg} = 0.000001 \mathrm{kg}\]

(c) Converting kilometers (km) to feet (ft)

First, we need to convert kilometers (km) to meters (m), then meters to feet (ft). Conversion factors: 1. 1 kilometer (km) is equal to 1000 meters (m) \[1 \mathrm{km} = 1000 \mathrm{m}\] 2. 1 meter (m) is equal to 3.28084 feet (ft) \[1 \mathrm{m} = 3.28084 \mathrm{ft}\] Combine both conversion factors to get the conversion factor for kilometers to feet: \[1 \mathrm{km} = 1000 \mathrm{m} \times 3.28084 \mathrm{ft/m} = 3280.84 \mathrm{ft}\]

(d) Converting cubic inches (in^3) to cubic centimeters (cm^3)

First, we need to convert cubic inches (in^3) to cubic feet (ft^3), then cubic feet to cubic meters (m^3), and finally, cubic meters to cubic centimeters (cm^3). Conversion factors: 1. 1 cubic inch (in^3) is equal to 0.0005787 cubic feet (ft^3) \[1 \mathrm{in}^3 = 0.0005787 \mathrm{ft}^3\] 2. 1 cubic foot (ft^3) is equal to 0.0283168 cubic meters (m^3) \[1 \mathrm{ft}^3 = 0.0283168 \mathrm{m}^3\] 3. 1 cubic meter (m^3) is equal to 1,000,000 cubic centimeters (cm^3) \[1 \mathrm{m}^3 = 1,000,000 \mathrm{cm}^3\] Combine all three conversion factors to determine the conversion factor for cubic inches to cubic centimeters: \[1 \mathrm{in}^3 = 0.0005787 \mathrm{ft}^3 \times 0.0283168 \mathrm{m}^3/\mathrm{ft}^3 \times 1,000,000 \mathrm{cm}^3/\mathrm{m}^3 = 16.3871 \mathrm{cm}^3\]

Key concepts

Metric Units

Metric units are an integral part of the International System of Units (SI), which is used globally for scientific and everyday measurements. This system is based on multiples of ten, making it simple to understand and convert between units. For example, common metric units include meters for distance, grams for weight, and liters for volume. These units are easily scalable using prefixes such as milli (one-thousandth), centi (one-hundredth), and kilo (one thousand).

To convert between different metric units, you can use these prefixes. For instance, to convert from millimeters to nanometers, you multiply by 1,000,000 because a nanometer is one billionth of a meter, while a millimeter is one-thousandth. Similarly, when converting milligrams to kilograms, you divide by 1,000,000 because a kilogram is 1,000 grams, and each gram is 1,000 milligrams.

English Units

English units, also known as Imperial units, are traditionally used in some countries like the United States and include measurements such as inches, feet, and pounds. These units can sometimes be more complex to convert due to the lack of a consistent base, unlike metric units that rely on multiples of ten.

For example, converting inches to feet involves dividing by 12 since there are 12 inches in a foot. Similarly, to convert from cubic inches to cubic feet, you must consider that volume is three-dimensional, adding a layer of complexity. This involves using factors unique to each conversion, such as recognizing that 1 cubic foot is 1,728 cubic inches.

Conversion Factors

Conversion factors are numbers used to change one unit of measurement to another. They act as a bridge between different units. To use a conversion factor, multiply or divide the original figure by the conversion factor to switch from one unit to another.

When dealing with metric units, use the power of ten inherent in the prefixes. For example, to convert from kilometers to meters, use the factor of 1,000. Meanwhile, converting between English and metric units involves more specific factors due to the units' differing bases. An example is converting kilometers to feet, which requires utilizing the intermediate step of converting kilometers to meters first, then meters to feet.

• Conversion factors allow for consistent and accurate calculations across various units.
• Remember to always keep track of unit cancellations to ensure the result is in the desired unit.

Measurement Systems

A measurement system is a standardized system of units used to quantify physical quantities. The two primary systems in use today are the Metric system and the Imperial system.

The Metric system is preferred worldwide, recognized for its simplicity and ease of use, as it is based on the decimal system. This feature allows for straightforward scaling and unit conversion using powers of ten.

On the other hand, the Imperial system is more traditionally used in the United States and consists of units like inches, feet, and pounds. These units often require complex conversion due to their non-decimal base.

• Understanding the connections and differences between these systems is crucial for fields such as science, engineering, and international commerce.
• Cross-system conversions are important for ensuring consistency and accuracy in global measurement tasks.