Physics » The Nature of Physics » Physical Quantities and Units

# Unit Conversion

## Known Ranges of Length, Mass, and Time Tiny phytoplankton swims among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. Image credit: Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections

The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in the table. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See images above and below.) Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. Image credit: NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.

## 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 us 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:

$$\require{cancel} 80 \cancel{\mathrm{m}} × \cfrac{1 \; \mathrm{km}}{1000 \cancel{\mathrm{m}}} = 0.080 \; \mathrm{km}.$$

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.

### Approximate Values of Length, Mass, and Time

Lengths in metersMasses 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
$$(9.11 × 10^{-31} \; \mathrm{kg}$$
$$10^{-23}$$Time for light to cross a proton
$$10^{-18}$$Diameter of a proton$$10^{-27}$$Mass of a hydrogen atom
$$(1.67 × 10^{-27} \; \mathrm{kg}$$
$$10^{-22}$$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 person1Time for one heartbeat
$$1$$Height of a 4-year-old child$$10^{3}$$Mass of a car$$10^{5}$$One day
$$(8.64 × 10^{4} \; \mathrm{s}$$
$$10^{2}$$Length of a football field$$10^{8}$$Mass of a large ship$$10^{7}$$

One year (y)
$$(3.16 × 10^{7} \; \mathrm{s}$$

$$10^{4}$$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
$$(7.35 × 10^{22} \; \mathrm{kg}$$
$$10^{17}$$Age of the Earth
$$10^{16}$$Distance traveled by light in 1 year (a light year)$$10^{25}$$Mass of the Earth
$$(5.97 × 10^{24} \; \mathrm{kg}$$
$$10^{18}$$Age of the universe
$$10^{21}$$Diameter of the Milky Way galaxy$$10^{30}$$Mass of the Sun
$$(1.99 × 10^{30} \; \mathrm{kg}$$

$$10^{22}$$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)