## Unit Conversion and Dimensional Analysis

Read this text for examples of how to calculate physical quantities and units of measurement.

### Unit Conversion and Dimensional Analysis

It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles.

Let's consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km).

The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers.

Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer.

Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:

$80 \not x \times \frac{1 \mathrm{~km}}{1000 \not \mathrm{m}}=0.080 \mathrm{~km}$

Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit.

Lengths in meters Masses in kilograms (more precise values in parentheses) Times in seconds (more precise values in parentheses)
$10^{-18}$ Present experimental limit to smallest observable detail
$10^{-30}$
Mass of an electron $\left(9.11 \times 10^{-31} \mathrm{~kg}\right)$
$10^{-23}$
Time for light to cross a proton
$10^{-15}$ Diameter of a proton
$10^{-27}$
Mass of a hydrogen atom $\left(1.67 \times 10^{-27} \mathrm{~kg}\right)$
$10^{-22}$
Mean life of an extremely unstable nucleus
$10^{-14}$ Diameter of a uranium nucleus
$10^{-15}$
Mass of a bacterium
$10^{-15}$
Time for one oscillation of visible light
$10^{-10}$ Diameter of a hydrogen atom
$10^{-5}$
Mass of a mosquito
$10^{-13}$
Time for one vibration of an atom in a solid
$10^{-8}$ Thickness of membranes in cells of living organisms
$10^{-2}$
Mass of a hummingbird
$10^{-8}$
Time for one oscillation of an FM radio wave
$10^{-6}$ Wavelength of visible light
1
Mass of a liter of water (about a quart)
$10^{-3}$
Duration of a nerve impulse
$10^{-3}$ Size of a grain of sand $10^{2}$

Mass of a person
$\quad 1$
Time for one heartbeat
$1$ Height of a 4-year-old child
$10^{3}$
Mass of a car
$10^{5}$
One day $\left(8.64 \times 10^{4} \mathrm{~s}\right)$
$10^{2}$ Length of a football field
$10^{8}$
Mass of a large ship
$10^{7}$
One year (y) $\left(3.16 \times 10^{7} \mathrm{~s}\right)$
$10^{4}$ Greatest ocean depth
$10^{12}$
Mass of a large iceberg
$10^{9}$
About half the life expectancy of a human
$10^{7}$ Diameter of the Earth
$10^{15}$
Mass of the nucleus of a comet
$10^{11}$
Recorded history
$10^{11}$ Distance from the Earth to the Sun
$10^{23}$
Mass of the Moon $\left(7.35 \times 10^{22} \mathrm{~kg}\right)$
$10^{17}$
Age of the Earth
$10^{16}$ Distance traveled by light in 1 year (a light year)
$10^{25}$
Mass of the Earth $\left(5.97 \times 10^{24} \mathrm{~kg}\right)$
$10^{18}$
Age of the universe
$10^{21}$ Diameter of the Milky Way galaxy
$10^{30}$
Mass of the Sun $\left(1.99 \times 10^{30} \mathrm{~kg}\right)$
$10^{22}$ Distance from the Earth to the nearest large galaxy (Andromeda)
$10^{42}$
Mass of the Milky Way galaxy (current upper limit)
$10^{26}$ Distance from the Earth to the edges of the known universe
$10^{53}$
Mass of the known universe (current upper limit)

Table 1.3 Approximate Values of Length, Mass, and Time

#### Nonstandard Units

While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters.

To learn more about nonstandard units, use a dictionary or encyclopedia to research different "weights and measures". Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units.

Source: Rice University, https://openstax.org/books/college-physics/pages/1-2-physical-quantities-and-units