Dimensions of Units

UNITS, DIMENSIONS OF (including the principles of DYNAMIC SIMILARITY.) In the article on PHYSICAL UNITS it is shown how certain units, or standards, of physical quantities have been chosen, enabling other amounts to be specified as multiples of the respective standards. It is further described how, in accordance with the lines along which physical science has developed, it has been found to be necessary to select three only as fundamental, namely, the standards of mass, length and time; other physical quantities are not independent but can be ex pressed in terms of these three by an application of the laws of mechanics. For example, a velocity is a length divided by a time, an acceleration is a velocity divided by a time, a force is a mass multiplied by an acceleration, and so on. The particular way in which the fundamental quantities enter into the speci fication of the secondary or derived quantities can be indicated by a type of algebra based upon the definition of the quantity in question. The most explicit representation is obtained by in cluding the name of the unit as well as the numerical magnitude. Thus when a length of x feet is traversed in t seconds the velocity is x not merely – ; the area of a rectangle is t sec. t [x feet] X [y where x and y are the numerical lengths of the two sides and the symbol merely indicates the number of lengths (expressed in feet) that are multiplied together in calculating the area In the same way if we write the velocity feet we are again indicating, by usual algebraic symbols, the operation that is carried out in calculating the velocity. Metaphysical arguments as to what the possible meaning of a reciprocal second may be are completely out of place. The fundamental quantities of mass, length and time are denoted by M, L, T and the indices for the respective quantities are indicated in square brackets, thus [M]"=, [Lr2, ar.; that is to say the first term indicates that the numerical values of ni masses were multiplied together; the second, that lengths were multiplied and the third that n3 times were multiplied. Such equations indicate what are called the dimensions of the quan tity to which they refer.

Principle of Homogeneity.

The basic fact which makes a knowledge of the dimensions important is that we can add, subtract or equate together only things of like kind. We can add any number lengths, for example, or of times, or of velocities; but to add the numerical values of a length and a time, or a length and a volume, is a meaningless act so far as rational physics is concerned. This can be stated, as a positive general principle in the following words:— In any physical equation every term must have the same dimensions.

It is perhaps unfortunate that the same word—dimensions- is used in popular conversation for size (as when we speak of a hall of large dimensions) ; if we do not use it in this sense in con nection with scientific problems no harm is done.

Every term may consist of one or more factors or elements; there is nothing constraining the separate elements to have the same dimensions. Thus, by definition Hence work done and energy have the same dimensions. This could have been foreseen because energy is defined as that which diminishes, when work is done, by an amount equal to the work so done.

The greater part of this article will be concerned with appli cations of this principle of homogeneity.

Change-ratios.

A second application is concerned with the change of the numerical value of a quantity when the unit is changed. This arises from the fact that we are not yet content with having only one unit for each kind of quantity. Thus in ordinary life, money is measured in pounds, shillings, pence, francs, marks, dollars, kopecs, pesetas, etc. These do not even preserve invariable relations to one another. Keeping to the British system, if we estimate the value of an estate in shillings the numerical value is 20 times its value expressed in pounds sterling since the shilling is 41- of a pound; or, in general, the numerical value of a physical quantity varies inversely as the be observed that a is the ratio of two lengths and is therefore of zero dimensions. We could have inferred that it could only depend upon the ratio of the lengths of the arc and of the pendulum and not upon their absolute magnitudes because the "constant" must be of zero dimensions.