Question

If a nanotube measures \(10 \mathrm{~nm}\) in diameter, how many, laid side by side, would fit within the width of a human hair? A human hair is \(20 \times 10^{-6} \mathrm{~m}\) wide.

Short answer

2,000 nanotubes can fit side by side across the width of a human hair.

Step-by-step solution

Convert the width of the human hair from meters to nanometers

First, we need to make sure both measurements are in the same units. Since the diameter of the nanotube is given in nanometers, we should also convert the width of the human hair to nanometers. There are 1,000,000 nanometers in a meter, so we multiply the width of the hair (in meters) by 1,000,000 to get nanometers.

Calculate the number of nanotubes

To find out how many nanotubes can fit within the width of a human hair, we divide the width of the human hair in nanometers by the diameter of a single nanotube. This will give us the number of nanotubes that can be laid side by side across the width of the hair.

Perform the division

Divide the width of the human hair in nanometers by the diameter of one nanotube to find out how many nanotubes can be laid side by side across the width of a hair.

Key concepts

Unit Conversion

Understanding how to compare different sizes requires a mastery of unit conversion, which is the process of turning one type of measurement unit into another.

Consider the exercise that asks us to convert the width of a hair from meters to nanometers—to do this, we need to know that one meter is equivalent to 1,000,000 nanometers. By performing simple multiplication, we can find that a width of 20 micrometers (\(20 \times 10^{-6}\text{m}\)) translates to 20,000 nanometers (20,000 nm). It's crucial to handle these conversions accurately, as they can significantly impact the final results of calculations involving dimensions or measurements.

Nanometers

A nanometer, abbreviated as 'nm', is a unit of measurement that equals one billionth of a meter (\(1 nm = 1 \times 10^{-9} \text{m}\)).

Nanometers are commonly used to measure things at the atomic or molecular scale, such as the diameter of a nanotube in our exercise. The tiny scale of nanometers makes them perfect for representing measurements that are far too small to be seen with the naked eye and are often used in scientific fields like nanotechnology and materials science. In the context of comparing the size of a nanotube to human hair, using nanometers allows for a direct and precise comparison because both can be expressed in terms relevant to their sizes.

Scientific Measurement

Scientific measurement is a crucial aspect of understanding and describing the physical world. These measurements are based on metric units like meters, grams, and liters and can be scaled down to nanometers or up to kilometers depending on the scope of what's being measured.

Proper scientific measurement involves not only the accurate determination of quantities but also the correct application of units for those measurements. As seen in the exercise, scientific measurements can bridge the gap between the incredibly small, such as nanotubes, and the relatively large, such as human hair, thereby allowing us to make meaningful comparisons and analyses.

Dimensional Analysis

Dimensional analysis, often known as the factor-label method or the unit-factor method, is a way to convert one type of unit to another to solve problems like the one presented in our exercise.

This method relies on the fact that multiplication by one does not alter the value of a number. Conversion factors, which are ratios equal to one, can be used to keep the magnitude the same while changing the units. For instance, the ratio of 1 meter to 1,000,000 nanometers (\(1 \text{m} = 1,000,000 \text{nm}\)) serves as a conversion factor. By using dimensional analysis, we can systematically perform unit conversions and ensure that the units we end up with are the correct ones for answering the question at hand.