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
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