Dimensions of Units

Oscillations.

The oscillations of a simple pendulum were taken as the introductory example. In that case the force was proportional to the volume of the moving body. A tuning fork presents a case where the force is proportional to an area, viz., the area of cross section of the prongs. According to what law will the time-period of geometrically similar tuning-forks depend? The elements of the dimensional equation are, the density, p, Young's modulus, Y, Poisson's ratio, P and a length L. Poisson's ratio is a mere number and may at first be left out of account. Hence putting [1] = [YxpYLz] and recalling that Y has the dimensions of a force per unit area it is found that x= —y and z = I; .*. Time-period = constant LAI (p/ Y). Since with any selected material, p and Y and P are constant the time period is proportional to the characteristic length. Strictly for complete similarity the amplitude should be varied in the same proportion in order that this conclusion may apply. The time period however is so nearly independent of the amplitude that we may disregard this restriction.

"If the tuning forks are not similar to one another, the prob lem is more complicated and can only be solved by reference to the mechanical equations." (Rayleigh.) Miscellaneous Vibrations.—The following results can also easily be obtained by this method; the controlling elements are in each case specified. The velocity of propagation of periodic waves controlled by gravity on the surface of deep water is as the square root of the wave-length. f(g, X).

For short ripples, which are governed principally by surface tension, a-, the velocity is proportional to the inverse square-root of the wave length. V2= f(cr, p, X). Waves of intermediate wave length are controlled both by gravity and by surface tension so that we should expect the velocity to be a function of and V2. Mechanical theory gives = gX 2 ro 2 '7r pn The time period of a Helmholtz resonator is directly as the linear dimension.

The frequency of vibration of a globe of liquid vibrating in any of its modes under its own gravitation, is independent of the diameter and directly as the square root of the density. The frequency of vibration of a drop of liquid, vibrating under "capillary" force is directly as the square root of the surface tension and inversely as the square root of the density and as the -I power of the diameter.

Weight of Drops.

In the article on SURFACE TENSION the weight, W, of drops from tubes of various diameters is discussed. If the diameter of the delivery tube is D, surface tension a, the acceleration due to gravity, g, and if these data determine W completely, Rayleigh has shown that TV I (aD) = f o- . Experiment shows that for wide changes in the variables on the right the term on the left is nearly a constant but has a mini mum value. Approximately, therefore, we expect the weight for a given liquid to be proportional to the diameter of the tube.

The deviation from strict dynamic similarity is no doubt due to omitted quantities which have some control over the situation. The most important one is viscosity and since (paD) is a non-dimensional quantity the solution is made more general by writing where the first term on the right depends only upon the nature of the fluid while the second involves Experiments are lack ing by which algebraic laws based upon this dimensional exam ination can be tested. The terms on the right seem to have comparatively little influence for the size of an oil drop from a given tube is much the same as that of a water drop although the viscosity is nearly ioo times as great and enters as the second power. This is what might be expected if the rate of formation of the drop is sufficiently slow.

This problem is a very instructive one. If we take Rayleigh's formula as the standard it might be expected that a/ must be constant for the cases to be dynamically similar, whereas wide variation of this term has scarcely any influence. There is, however, a variety of possible independent variables. Since is a numeric so also is [a/ (gpD2)][117/(o-D)].

Now, assuming the diameters of the drops to be proportional to the diameters of the tubes, W varies as whence this new "variable" is an absolute constant. This is nearly the case in practice. We may now attribute the departure from precise similarity to the diameters of the drops and tubes not being strictly proportional to one another. Let D2 be the characteristic linear dimension of the drops. If their weight be expressed in terms of the above substitution gives finally W/(o-D) = The approximate constancy of each side of this equation is now easier to understand. It must be added that large and small drops are not geometrically similar before falling; this fact also needs to be taken into account.

Temperature.

In all the cases that have been considered, up to this point, temperature has only entered in a secondary way. For example, the density, p, depends upon the tempera ture, but if we write p= the term at must be a mere numeric and provided that we know the law of variation the value p can be calculated from the standard value and the temperature does not further concern us. The same remark applies to other variables such as o- and kt. But there are many cases in which the question cannot be dealt with so simply and we are obliged to ask what are the dimensions of temperature. At first sight it appears to be a new fundamental quantity; and it can be so treated in many problems, so that we then expect it to turn up to the same power on both sides of the equation. In other cases we must seek to find a relation by which it can be expressed in mechanical units.