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April, 1930] DIMENSIONAL ANALYSIS 379
and
/ = LI' and / = TV, so that
LI' I' S5-pi.I'-t
Applying this result to our problem we find that
T = -\/L = \/OXJ5 = 0.224.
This is the relative size of an interval of time in the miniature system, and is the ratio of projector to camera speed. The reciprocal, or 4.5, is the speed at which the camera should be run with respect to the projector.
Now let us study the illusion that we have created. It is important to note that length and time mutually control one another, that is, had we run the camera just a trifle slower we would have created the illusion that the building was only 160 feet tall instead of 200. Had we run it at 5 times normal, the building would have grown in the imagination of the audience to a height of 250 feet. The illusion, then, is made perfect by two facts: first, that the building is made to represent something with which the audience is familiar here on earth, and secondly, because the audience is unthinkingly conscious of the fact that bodies on the earth fall naturally with a very definite acceleration which is practically constant for all bodies.
Having gone this far without saying very much about the forces involved, it will be well to attack the forces in our model at this point. In the case of falling bodies, which we have been considering up to this point, weight has been the active force. If we may assume that the densities of the materials in our rnodel are the same as the densities of the real objects which we are depicting, it is clear that our mass reduction factor, M, will be equal to L3 or the volume dimension. The force causing a body to fall is then m'g, and will be mg in the model, or L3 times as great. In order to be consistent then, it will be necessary to cause all forces to be reduced in proportion to L3. In the case of gravitational force, we are not greatly concerned with the exact value of mass, for it cancels on both sides of our equation of motion, and the motion is independent of mass as first pointed out.
We may now consider some special types of forces. A very interesting illustration is that of resilience forces, such as are produced by springs, bending beams (with certain limitations), and torsional