The dependence of the time-period upon the square-root of the determining length is characteristic of cases in which forces proportional to the volume are the sole motive power. "As examples coming under this head may be mentioned the common pendulum, sea-waves whose velocity varies as the square-root of the wave-length, and the whole theory of the comparison of ships and their models by which Froude predicted the behaviour of ships from experiments made on models of small dimensions." (Rayleigh, Theory of Sound, Art. 381, 2nd edition.) We return to this more general problem later.
In geometry, plane figures are said to be similar when a single length is sufficient to specify for each its distinction from the rest. Thus, for equilateral triangles it is sufficient to specify for each the length of one side, and for circles, the radius. For example, the areas of circles are given by
those of equilateral triangles are given by r1/4. In each
case the area is proportional to the square of the length x but the factors of proportionality are different when the figures are different. The constancy of the numerical factor is, therefore, a characteristic of similar figures.
Now, in the problem of the pendulum we can write, in general, allowing for the effect of the arc of swing If, however, we restrict attention to those cases only for which the ratio arc/length is constant we can consider them as con stituting a family (analogous to the family of similar triangles). All members of such a family are said to be dynamically similar and for these the right hand and therefore the left side also, is a constant. The similarity consists in the circumstance that dif ferent members of the family are distinguished from one another by the single quantity L. Take two pendulums in the same locality (g= const.) and having the same value of arc/length, then P cc L. If we observe one pendulum at intervals of time T, and find it is always passing through its central position at each observation; and then take a second pendulum of twice the length (and twice the maximum arc of swing) then the intervals of time must be four times as great in order that the observations shall take place at passages through its central position.
For the experimental determination of the constant, on the right, use may be made of either shorter or longer pendulums according to convenience. This is the principle underlying the
use of models in dynamical problems, to which Lord Rayleigh referred.
As a more complicated case, which brings out the weakness as well as the power of the method we may consider planetary motion under the influence of universal gravitation. Putting F for the attraction between the sun and planet the law of gravitation gives F =-y LS
and S are the masses of planet and sun respectively, L the distance apart and 7 the gravitation constant; and consequently traceable to the formation of eddies when the speed is great enough. The problem of eddying flow has not yet been solved by means of mechanical theory and this is exactly the type of prob lem which it is useful to investigate as far as possible by means of the method of dimensions. The data upon which the velocity of outflow depends are (i.) the diameter D of the tube which is assumed in the first place to be cylindrical; (ii.) the density p of the fluid; (iii.) the viscosity, ,u; (iv.) the resistance per unit area R. The temperature will be regarded as fixed; in any case it only enters in a secondary way owing to the above quantities depending upon it. The dimensions of D and p are [L] and
while for which is defined as a force per unit area
per unit gradient of velocity the dimensions are
and for R, they are
The dimensional equation is F= P X acceleration it is seen that P enters in the same way on both sides and it can be cancelled; in other words, we can define 7 in terms of the acceleration instead of in terms of the force. If this is done, the mass P does not come into the equations at all and therefore n must be zero. This subsidiary principle may be stated thus: the dimensions of any one of the physical ele ments of the equation must be defined by the simplest mechani cal equation by which its definition is possible.
The factor (P/S)n therefore disappears from the equation. The other factor can be compared with that determined by applying the laws of mechanics in the simple case of circular orbits. The centripetal force on the planet is
where w is the angular velocity and this must equal
whence
Since co= 2'Ir/T this is of the same form as that obtained by considering the dimensions.