establishing simple relations between each of them and the
fundamental mechanical units. Measurements based on such
relations are called absolute measurements. The science of
dynamics, as far as that part of it is concerned which deals
with the motion and energy of material substances, starts
from certain primary definitions concerning the measurable
quantities involved. In constructing a system of physical
units, the first thing to consider is the manner in which we
shall connect the various items. What, for instance, shall be
the unit of force, and how shall it be determined by simple
reference to the units of mass, length and time?
The modern absolute system of physical measurement is founded upon dynamical notions, and originated with C. F. Gauss. We are for the most part concerned in studying motions in nature; and even when we find bodies at rest in equilibrium it is because the causes of motion are balanced rather than absent. Moreover, the postulate which lies at the base of all present-day study of physics is that in the ultimate issue we must seek for a mechanical explanation of the facts of nature if we are to reach any explanation intelligible to the human mind. Accordingly the root of all science is the knowledge of the laws of motion, and the enunciation of these laws by Newton laid the foundation of a more exact knowledge of nature than had been possible before. Our fundamental scientific notions are those of length, time, and mass. No metaphysical discussion has been able to resolve these ideas into anything simpler or to derive them from each other. Hence in selecting units for physical measurements we have first to choose units for the above three quantities.
Fundamental Units.—Two systems of fundamental units are in common use: the British system, having the yard and pound as the standard units of length and mass, frequently termed the “foot-pound-second” (F.P.S.) system; and the “centimetre-gramme-second” system (C.G.S.), having the centimetre and gramme as standard units of length and mass, termed the “metric” system. The fundamental unit of time is the same in both systems, namely, the “mean solar second,” 86,400 of which make 1 solar day (see Time). Since these systems and the corresponding standards, together with their factors of conversion, are treated in detail in the article Weights and Measures, we need only deal here with such units as receive special scientific use, i.e. other than in ordinary commercial practice. The choice of a unit in which to express any quantity is determined by the magnitude and proportional error of the measurement. In astronomy, where immense distances have to be very frequently expressed, a common unit is the mean radius of the earth's orbit, the “astronomical unit” of length, i.e. 92,900,000 miles. But while this unit serves well for the region of our solar system, its use involves unwieldy numerical coefficients when stellar distances are to be expressed. Astronomers have therefore adopted a unit of length termed the “light year,” which is the distance traversed by light in a year; this unit is 63,000 times the mean radius of the earth's orbit. The relative merits of these units as terms in which astronomical distances may be expressed is exhibited by the values of the distance of the star α. Centauri from our earth, namely, 25,000,000,000,000 miles = 275,000 astronomical units = 4·35 light years.
As another example of a physical unit chosen as a matter of convenience, we may refer to the magnitudes of the wave-lengths of light. These quantities are extremely small, and admit of correct determination to about one part in ten-thousand, and range, in the visible spectrum, from about 6 to 4 ten-millionths of a metre. Since their values are determined to four significant figures, it is desirable to choose a unit which represents the value as an integer number; the unit is therefore a ten-thousand millionth of a metre, termed a “tenth metre,” since it is 10−10 metres. Sometimes the, thousand-millionth of a metre, the “micro millimetre,” denoted by μμ, serves as a unit for wave lengths. Another relatively minute unit is the “micron,” denoted by μ, and equal to one-millionth of a metre; it is especially used by bacteriologists.
Units in Mechanics.—The quantities to be measured in mechanics (q.v.) are velocity and acceleration, dependent on the units of length and time only, momentum, force, energy or work and power, dependent on the three fundamental' units. The unit of velocity in the British system is 1 foot, 1 yard, or 1 mile per second; or the time to which the distance is referred may be expressed in hours, days, &c., the choice depending upon the actual magnitude of the velocity or on custom. Thus the muzzle velocity of a rifle or cannon shot is expressed in feet per second, whereas the speed of a train is usually expressed in miles per hour. Similarly, the unit on the metric system isr metre, or any decimal multiple thereof, per second, per hour, &c. Since acceleration is the rate of increase of velocity per unit time, it is obvious that the unit of acceleration depends solely upon the units chosen to express unit velocity; thus if the unit of velocity be one foot per second, the unit of acceleration is one foot per second per second, if one metre per second the unit is one metre per second per second, and similarly for other units of velocity. Momentum is defined as the product of mass into velocity; unit momentum is therefore the momentum of unit mass into unit velocity; in the British system the unit of mass may be the pound, ton, &c., and the unit of velocity any of those mentioned above; and in the metric system, the gramme, kilogramme, &c., may be the unit of mass, while the metre per second, or any other metric unit of velocity, is the remaining term of the product.
Force, being measured .by the change of momentum in unit time, is expressed in terms of the same units in which unit momentum is defined. The common British unit is the “poundal,” the force which in one second retards or accelerates the velocity of a mass of one pound by one foot per second. The metric (and scientific) unit, named the “dyne,” is derived from the centimetre, gramme, and second. The poundal and dyne are related as follows:—1 poundal= 13,825·5 dynes.
A common unit of force, especially among engineers, is the “weight of one pound,” by which is meant the force equivalent to the gravitational attraction of the earth on a mass of one pound. This unit obviously depends on gravity; and since this varies with the latitude and height of the place of observation (see Earth, Figure of), the “force of one pound” of the engineer is not constant. Roughly, it equals 32·17 poundals or 980 dynes. The most frequent uses of this engineer's unit are to be found in the expressions for pressure, especially in the boilers and cylinders of steam engines, and in structures, such as bridges, foundations of buildings, &c. The expression takes the form: pounds per square foot or inch, meaning a force equivalent to so many pounds' weight distributed over a square foot or inch, as the case may be. Other units of pressure (and therefore special units of force) are the “atmosphere” (abbreviated “atmo”), the force exerted on unit area by the column of air vertically above it; the “millimetre or centimetre of mercury,” the usual scientific units, the force exerted on unit area by a column of mercury one millimetre or centimetre high; and the “foot of water,” the column being one foot of Water. All these units admit of ready conversion:—1 atmo= 760 mm. mercury= 32 feet of water= 1,013,600 dynes.
Energy of work is measured by force acting over a distance. The scientific unit is the “erg,” which is the energy expended when a force of one dyne acts over one centimetre. This unit is too small for measuring the quantity of energy associated, for instance, with engines; for such purposes a unit ten-million times as great, termed the “joule,” is used. The British absolute unit is the “poundal-foot.” As we noticed in the case of units of force, common-life experience has led to the introduction of units dependent on gravitation, and therefore not invariable: the common British practical unit of this class is the “foot pound”; in the metric system its congener is the “kilogramme metre.”
Power is the rate at which force does work; it is therefore expressed by “units of energy per second.” The metric unit in use is the “watt,” being the rate equal to one joule per second. Larger units in practical use are: “kilowatt, equal to 1000 watts; the corresponding energy unit being the
