Units of Measure

© 2013 by  Fernando Caracena


Measuring things

We measure something by comparing it to some kind of standard. The units used in physics were developed from those used in commerce, which are maintained internationally by government laboratories. They are known as SI units after the International System of Units. In the United States, the standards of weights and measures are maintained by the National Institute of Standards and Technology (NIST). The science  of measurement, metrology, concerns how to reconcile a number of measurements of different things. Since there are no absolute measurements, all are approximations, probable error produced by uncertainties in measurement is a major concern in computing physical results. Prof. Walter Lewin at MIT discusses the units of measure and uncertainties in measurement in this lecture.

Two aspects characterize measurements: precision and accuracy. The notion of precision concerns how tightly measurements of the same thing cluster. For example, you want to measure your own height with a scale provided for that purpose. You are in your stocking feet, already that boosts your height by a tiny bit; but wait, that tiny bit may be negligible compare to how closely your height measurements cluster in repeated trails. the thickness of your socks are below the level of precision of your height measurement.

The three fundamental units of measure in physics are time (T), length (L)  and mass (M). All other quantities are expressed in terms of products and fractions of these units. The way that these units enter into a derived quantity (X) defines what are called the dimensions ("[X]") of that quantity expressed as combinations of the fundamental units. For example, the dimensions of speed ([v]) are


where we use unbracketed symbols (M,  L, T) for the fundamental unites:

[L]=L, [T]=T and [M]=M. The units of acceleration (a) are expressed as



[a]=L T-2.

The challenge of scientifically establishing fundamental units

Suppose that in the future we make contact with a distant, extraterrestrial civilization. After we are able to determine each other's numbering system, how can we communicate to each other our fundamental units of measure? First of all, we do not know if we both have considered the same set of units to be fundamental.For example, an important dimensionless number in quantum electrodynamics is the fine structure constant, which is defined as follows:

α = e2/(4πe0ħc)



The reciprocal of α is also a dimensionless number,

α-1 = 137.035999074(44)

where the bracket error notation (44) above= +/- 0.000000044


α-1 = 137.035999074+/- 0.000000044.

High energy physics use the following natural units

c = ħ =e0 =1,

in which case

α = e2/(4π).

If the aliens are a very advanced civilization, we will have to learn each other's  physics very well before we can be able to translate each other's units of measure.

What researchers are trying to do in defining our system of units, is a way of constructing the standards of M, L, and T in any good, terrestrial, physics laboratory. If we ever contact an advanced extra terrestrial then they will really have their hands full. The SI system of units is defined in terms of three standards of measure: of time, of length and of mass.

Time (s, seconds)

In the business world we are very time oriented: time is money. We need good clocks, which all run very close to the same standard of time and can be synchronized to a universal time standard. The U. S. Federal Government maintains a websiteat NIST where you can find the current time to within a few tenths of a second as determined by an an on-site atomic clock. Your computer clock may be regulated by an online utility that maintains it to the same degree of tolerance and accuracy as the government site. In physics, the units of time used are seconds; but there are derived units of time, such as the millennium, century, year, day, hour and microsecond—all of which can be expressed in seconds. According to a Wikipedia article,  "Since 1967, the second has been defined to be: the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." We and extraterrestrial scientist have to understand what all these terms mean before their being able to reconstruct the second as a unit of time in their laboratories.

Length (m, meter)

Originally defined as one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), the definition of a meter has changed to a more universal one as the distance that light travels in a vacuum in 1/299,792,458 of a second. An intermediate definition adopted in 1960 was in terms of atomic physics:  1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum.

Given the speed of light in a vacuum in the extraterrestrial 's units, the definition of the second given above would allow them to reconstruct the meter in their laboratory.

Mass (Kg, kilogram)

The mass of one kilogram was initially defined as the mass of one liter of water, at some standard pressure and temperature. A liter is defined as 1,000 cubic centimeters (cm), where

1 cm= 10-2 meters.

However, the kilogram is currently defined as the mass of an artifact that is stored in a special place: a platinum cylinder kept in the custody of the International Bureau for Weights and Measures . That particular definition could not be conveyed to an extraterrestrial civilization unless we develop a long range transporter which could send a copy of this standard several light years.

Considering the importance of the kilogram in physics, metrologists should develop a more fundamental definition of it as soon as possible. If we make contact with extraterrestrials before that is done, we could send them the old definition in terms of a volume of water, which they should have on hand if they are also hydrocarbon based organisms. However, if they are silicon based organisms in an environment in which liquid water is a rarity then we might have trouble conveying some definition of the kilogram to them. Or, being good scientists, they could create earthlike conditions in their laboratory long enough to extract a stable kilogram standard of mass.



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