Question

The unit of nanometers \((\mathrm{nm})\) is commonly used for the wavelength of visible light. What does \(1.00\) \(\mathrm{nm}\) equal in meters? What does it equal in inches? \([1\) inch \(=2.54 \mathrm{~cm}\) exactly \(]\)

Short answer

\(1.00 \,\mathrm{nm}\) is equal to \(10^{-9} \,\mathrm{m}\) and approximately equal to \(3.94 \times 10^{-9} \,\mathrm{in}\).

Step-by-step solution

Convert nanometers to meters

To convert a length given in nanometers to meters, use the fact that there are \(10^9\) nanometers in one meter. Divide the given length in nanometers by \(10^9\). \[1.00 \,\mathrm{nm} \cdot \frac{1\,\mathrm{m}}{10^9\,\mathrm{nm}} \]

Calculate the meters value

Perform the division: \[\frac{1.00}{10^9} \,\mathrm{m} = 10^{-9} \,\mathrm{m} \] So, \(1.00 \,\mathrm{nm}\) is equal to \(10^{-9} \,\mathrm{m}\).

Convert meters to centimeters

Next, we need to convert our meters value to centimeters using the fact that there are \(100\) centimeters in one meter: \(10^{-9} \,\mathrm{m} \cdot \frac{100\,\mathrm{cm}}{1\,\mathrm{m}} \)

Calculate the centimeters value

Perform the multiplication: \[(10^{-9} \cdot 100)\,\mathrm{cm} \] This simplifies to: \(10^{-7} \,\mathrm{cm} \) So, \(1.00 \,\mathrm{nm}\) is equal to \(10^{-7} \,\mathrm{cm}\).

Convert centimeters to inches

Now, we'll use the given conversion factor between inches and centimeters: \(10^{-7} \,\mathrm{cm} \cdot \frac{1\,\mathrm{in}}{2.54\,\mathrm{cm}} \)

Calculate the inches value

Perform the division: \[\frac{10^{-7}}{2.54}\,\mathrm{in}\] This results in approximately \(3.94 \times 10^{-9} \,\mathrm{in}\). So, \(1.00 \,\mathrm{nm}\) is approximately equal to \(3.94 \times 10^{-9} \,\mathrm{in}\). In conclusion, \(1.00 \,\mathrm{nm}\) is equal to \(10^{-9} \,\mathrm{m}\) and approximately equal to \(3.94 \times 10^{-9} \,\mathrm{in}\).

Key concepts

Wavelength of Visible Light

The term 'wavelength' refers to the distance between consecutive peaks or troughs in a wave. It's an essential concept in physics, particularly in the study of light waves. Visible light is the portion of the electromagnetic spectrum that can be perceived by the human eye, and it spans from roughly 380 nanometers (nm) to about 750 nm. Each color within this range has its own characteristic wavelength. For example, violet light has the shortest wavelengths that the human eye can see, around 380 to 450 nm, while red light has the longest, approximately 620 to 750 nm.

Understanding the measurement of light's wavelength is vital in various fields, from astronomy to telecommunications. The nanometer, which is one billionth of a meter, is a preferred unit for these small scales because it is well-suited to the atomic and molecular levels where light interactions typically occur.

When we discuss converting nanometers to larger units such as meters, we are scaling up so we can relate these tiny distances to something more perceivable by humans. This type of conversion is a fundamental skill in scientific computations where different scales of measurement are often encountered.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another. It is a fundamental aspect of scientific study as different systems and scales are used across various fields. To perform a unit conversion, you need a conversion factor, which is a ratio expressing how many of one unit are equivalent to another unit.

For example, in the case of converting nanometers to meters, the conversion factor is based on the definition that one meter equals one billion nanometers. Therefore, to convert from nanometers to meters, you need to divide the number of nanometers by one billion, as shown in the provided exercise. By internalizing these standard conversion factors, such as those for lengths, masses, and volumes, students can tackle a wide variety of problems more effectively.

It's important when converting units to always ensure that the units you are converting from and to are of the same dimension (such as length, mass, or time) to make a meaningful comparison.

Metric to Imperial Conversion

The metric system and the imperial system are two major types of measurement systems. The metric system is based on multiples of ten, which simplifies calculations and conversions. In contrast, the imperial system, used predominantly in the United States, often involves more complex conversion factors that are not as easily divisible.

The exercise provided focuses on converting metric units to imperial units, specifically from centimeters to inches. This conversion uses an exact conversion factor where 1 inch is defined as equal to exactly 2.54 centimeters. To perform this conversion, divide the length value in centimeters by 2.54 to obtain the equivalent length in inches. Understanding this type of conversion is essential not only within scientific disciplines but also in everyday life, such as in construction, manufacturing, and many other industries where specifications might require imperial units.