3.3 SI Units

In the making of scientific syntheses, our mutual needs for clear thinking and precision of meaning–as well as the need for clarity in communications between us have an obvious dependence on the exactness of our reference standards and the precision we choose to employ in the characterizing of those natural quantities that ultimately enter our formulations as symbols. Our usages here shall conform, with little revision, to the conventions and standards known as the International System of Units (abbreviated SI for Systeme International), supplemented where necessary by the practices of the National Bureau of Standards (NBS). In the SI system, the fundamental metric units associated with dimensions \(M\), \(L\), \(T\) are the kilogram, the meter, and the second (customarily called the “mks system”). A variety of auxiliary metric units are employed in practice (such as the kilometer, the angstrom, the gamma, and so on), but the auxiliary or supplementary set of metric units most commonly associated with dimensions \(M\), \(L\), \(T\) are the gram, the centimeter, and the second (customarily called the “cgs system”).

The following definitions for mechanical quantities have been abstracted from publications of NBS (1964,1968) and Cray (1972). For comprehensive tabulations of the quantities appropriate to optics, acoustics, and electricity, see the Handbook of the American Institute of Physics, the Handbook of Chemistry and Physics (Chemical Rubber Co.), or the Handbook of the National Bureau of Standards, which is edited and published annually.

3.3.1 Fundamental Mechanical Units

Meter. Unit of length mks system. Symbol m, dimension L. By international agreement (1960) defined to be 1,650,763.73 wavelengths of the orange-red line of krypton 86, which replaces the length standard based on the platinum-iridium meter bar in Paris.

Kilogram. Unit of mass mks system. Symbol kg, dimension M. Defined to be the mass of a certain cylinder of platinum-iridium alloy, called the International Prototype Kilogram, preserved at the International Bureau of Weights and Measures in Paris.

Second. Unit of time mks system. Symbol sec, dimension T. By international agreement (1960) defined to be 1/31,556,925.9747 of the tropical year 1900. (A “tropical year” is the interval of time between two successive passages of the sun through the vernal equinox.) A more recent international conference adopted provisionally a new definition of the second as the time corresponding to 9,192,631,770 oscillations of the cesium atom in the so-called atomic clock.

3.3.2 Temperature Scales

In 1968 the International Committee on Weights and Measures officially adopted the “International Practical Temperature Scale”, which is based on the concept of temperature (variable symbol 0, dimension 0) as being that of thermodynamic temperature. The unit magnitude of the International Scale is the degree Kelvin (symbol K). The degree Kelvin is the fraction 1/273.16 of the triple point of water. The Celsius temperature scale is defined in terms of the Kelvin scale as \[\theta_c=\frac{1^{\circ}C}{1K}\Big(\theta_k-\theta_0\Big)\] where \(\theta_c\) is the temperature magnitude on the Celsius scale, \(\theta_k\) the temperature magnitude on the Kelvin scale, and \(\theta_0\) = 273.15K (the ice point of water). The scale unit of the Celsius temperature is the degree Celsius (symbol \(^{\circ}\)C), equal in magnitude to the degree Kelvin. Reference point relationships between Kelvin, Celsius, and Fahrenheit scales are:

0K = -273.15 \(^{\circ}\)C = -459.67\(^{\circ}\)F (“absolute” zero),
273.15K = 0\(^{\circ}\)C = 32\(^{\circ}\)F (the ice point),
273.16K = 0.01\(^{\circ}\)C = 32.018\(^{\circ}\)F (equilibrium between solid, liquid, and vapor phases of water; the triple point),
373.15K = 100\(^{\circ}\)C = 212\(^{\circ}\)F (the boiling point of water).

The relationships between the Kelvin, Celsius, and Fahrenheit scales are linear and appear as straight lines when graphed.

3.3.3 Supplementary Mechanical Units

Centimeter. Unit of length cgs system. Symbol cm, dimension L. Defined to be 1/100 meter.

Gram. Unit of mass cgs system. Symbol g, dimension M. Defined to be 1/1000 kilogram.

International Yard. Unit of length. Symbol yd, dimension L. Defined by agreement between the United States and the British Commonwealth (1959) to be 0.9144 meter.

International Pound. Unit of mass. Symbol lb, dimension M. Defined by agreement between the United States and the British Commonwealth (1959) to be 0.45359237 kilogram.

Angstrom. Unit of length. Symbol \(\mathring{A}\), dimension L. Defined to be 10\(^{-8}\) cm. Gamma. See microgram.

Microgram. Unit of mass. Symbol \(\mu\)g, dimension M. Defined to be 10\(^{-6}\) g.

Micrometer. Unit of length. Symbol \(\mu\)m, dimension L. Defined to be 10\(^{-6}\) m (1/1000 cm). Equivalent to, but replaces, micron (symbol \(\mu\)).

Micron. See micrometer.

3.3.4 Angular Units

Degree. Unit of angular measure. Symbol deg or \(^{\circ}\) (superscript), dimensionless, Defined as the angle subtended at the center by a circular arc 1/360 the circumference.

Radian. Unit of angular measure. Symbol radian, dimensionless. Defined as the angle subtended at the center by a circular arc equal in length to the radius of the circle. 1 radian = 360/2\(\pi\) degrees angular measure.

3.3.5 Derived Units

Atmosphere. Unit of pressure. Symbol atm, dimension FL\(^{-2}\) or ML\(^{-1}\)T\(^{-2}\). Defined to be the pressure exerted by dry atmosphere, 0\(^{\circ}\)C, at mean sea level. Equivalent to \(1.013250 \times 10^6 dyn\cdot cm^{-2}\) in cgs units.

Bar. Unit of pressure. Symbol bar, dimension FL\(^{-2}\) or ML\(^{-1}\)T\(^{-2}\). Equal to \(10^5 nt\cdot m^{-2}\) in mks units.

British Thermal Unit (Mean). Unit of energy or quantity of heat. Symbol Btu, mechanical dimension ML\(^2\)T\(^{-2}\), thermal dimension H. Originally defined to be the quantity of heat energy required to raise the temperature of 1 lb mass of water 1\(^{\circ}\)F (averaged from 32\(^{\circ}\)F to 212\(^{\circ}\)F). Equivalent to 252.16038 gram calories.

Calorie (Mean). Unit of energy or quantity of heat. Symbol cal, mechanical dimension ML\(^2\)T\(^{-2}\), thermal dimension H. Originally defined to be the quantity of heat energy required to raise the temperature of 1 g mass of water 1\(^{\circ}\)C (averaged from 0\(^{\circ}\)C to 100\(^{\circ}\)C). Equivalent to 4.1840 joules.

Dyne. Unit of force cgs system. Symbol dyn, dimension F or MLT\(^{-2}\), unit g\(\cdot\)cm\(\cdot\)sec\(^{-2}\). Force required to give 1 g mass an acceleration of 1 cm\(\cdot\)sec\(^{-2}\).

Erg. Unit of work or energy cgs system. Symbol erg, dimension FL or ML\(^2\)T\(^{-2}\), unit dyn\(\cdot\)cm or g\(\cdot\)cm\(^2\cdot\)sec\(^{-2}\). Work done by a force of 1 dyne acting through a distance of 1 cm (10\(^7\) erg = 1 J).

Hertz. Unit of frequency. Symbol Hz, dimension T\(^{-1}\), unit sec\(^{-1}\). Equivalent to, but replaces, unit cps (cycle per second).

Joule. Unit of work or energy mks system. Symbol J, dimension FL or ML\(^2\)T\(^{-2}\), unit ntm or kg\(\cdot\)m\(^2\cdot\)sec\(^{-2}\). The work done by a force of 1 newton acting through a distance of 1 meter.

Liter. Unit of volume (liquids and gases). Symbol 1, dimension L\(^3\). Originally defined to be the volume of 1 kg airfree \(H_2O\) at 4\(^{\circ}\)C (the maximum density temperature of water). Redefined by international agreement (1964) to be 1/1000 cubic meter (10\(^{-3}\)m\(^3\)) exactly.

Newton. Unit of force mks system. Symbol N (alternate symbol nt), dimension F or MLT\(^{-2}\). Force required to give 1 kg mass an acceleration of 1 m\(\cdot\)sec\(^{-2}\).

Poise. Unit of viscosity cgs system. Symbol P, dimension FL\(^{-2}\)T or ML\(^{-1}\)T\(^{-1}\), unit dyn\(\cdot\)cm\(^{-2}\cdot\)sec or g\(\cdot\)cm\(^{-1}\cdot\)sec\(^{-1}\). Shear viscosity, or resistance to flow, is defined as the ratio of shearing stress (tangential force per unit area) in a moving fluid and its associated rate of area deformation (\(dA/Adt\)). Shear viscosity is sometimes called the dynamic viscosity.

Poundal. Unit of force. Symbol lbf (pound force), dimension F or MLT\(^{-2}\). Force required to give 1 lb mass an acceleration of 1 ft\(\cdot\)sec\(^{-2}\).

Stokes. Unit of kinematic viscosity cgs system. Symbol St, dimension L\(^2\)T\(^{-1}\), unit cm\(^2\cdot\)sec\(^{-1}\). Kinematic viscosity in units of Stokes is defined to be the ratio of the dynamic viscosity (in poise) of a fluid to its cgs density (g\(\cdot\)cm\(^{-3}\)).

Torr. Unit of pressure. Symbol torr, dimension FL\(^{-2}\) or ML\(^{-1}\)T\(^{-2}\). The pressure exerted by a column of mercury 1 mm in height, 0\(^{\circ}\)C. Equivalent to 13.3322 dyn\(\cdot\)cm\(^{-2}\) in cgs units.

Watt. Unit of power or rate of work mks system. Symbol W, dimension FLT\(^{-1}\) or ML\(^2\)T\(^{-3}\), unit J\(\cdot\)sec\(^{-1}\) or nt\(\cdot\)m\(\cdot\)sec\(^{-1}\) or kg\(\cdot\)m\(^2\cdot\)sec\(^{-3}\). Work done or energy expended at the rate of l J\(\cdot\)sec\(^{-1}\).

An appendix summarizing the SI units and dimensions is attached.

3.3.6 Auxiliary Prefixs of the Metric System

Multiples and sub-multiples	Prefix
10^-18	atto
10^-15	femto
10^-12	pico
10^-9	nano
10^-6	micro
10^-3	milli
10^-2	centi
10^-1	deci
10	deka
10²	hecto
10³	kilo
10⁶	mega
10⁹	giga
10¹²	tera