International projectionist (Oct 1931-Sept 1933)

MATHEMATICS FOR THE PROJECTIONIST Siegfried S. Meyers INSTRUCTOR IN PHYSICS, STUYVESANT HIGH SCHOOL, NEW YORK f ■"! HE motion of the film in its passI age through the projector is the -^ result of careful design of gears and shafts. Gears vary in size, but in a gear train intended to perform a certain work, each gear bears a definite relationship to the others. Let us consider a hypothetical case of gear mechanism: Problem : Suppose we desired to drive a certain disc at a speed of 60 revolutions a minute by means of a direct-current motor the normal speed of which under a load is 1,800 revolutions per minute. Solution: This speed reduction may be accomplished by any one of several methods: 1 . Couple the disc directly to the motor shaft, and reduce the speed with a series rheostat; or 2. Couple the disc directly, and reduce the speed by applying mechanical opposition to its rotation ; or 3. Couple the motor to a small gear which meshes with a larger gear. Of these three possible methods, the latter is the most economical as well as most practicable; the others having inherent power losses. Since the motor makes 1,800 revolutions in one minute, and we desire to reduce this number to ■60, the ratio is 30-to-l. Thus, the small gear must make 30 times as many revolutions as the larger gear which carries the disc. To accomplish this work two gears are so machined that the larger one will contain 30 times as many teeth (not 30 teeth more), as the smaller one. If the small gear has, say, 10 teeth, the larger gear must have 300 teeth to produce the Tequired speed of rotation (Fig. 1). The Worm Gear Where precision adjustment is required, a worm gear is most satisfactory. A worm gear consists of a threaded shaft which meshes with a cog-wheel. This ar Figiire 2 rangement is used where large ratio reductions are required. Where 14-to-l reduction is required, it is quite easy to machine two teeth on the worm and 28 teeth on the cog-wheel. In such a case it would require one complete revolution of the worm to make the shaft move 1/14 of a revolution. The rack and pinion, which is commonly employed to focus lenses, uses the worm gear principle (Fig. 2). The Graph In optics the formula is given: candle-power Intensity = distance' Reducing this to symbols : c.p. 1= d' where : d' = distance X distance c.p. = candle-power Let us consider the meaning of d'. Assume a 32-candle-power lamp is mounted one foot removed from a screen. The brilliancy or intensity of illumination of this screen is 32 foot-candles. If we should move the lamp twice as far away, or 2 feet, the illumination on the screen would naturally decrease. The distance now being twice as great, we might say offhand that the illumination would be reduced by one-half. Such is not the case, for it has been proven that the intensity does not vary inversely as the distance but rather varies inversely as the square of the distance (distance multiplied by distance). Using numerical values, let us make a table of illumination as compared with distance : Intensity (foot-candles) Distance (feet) D= Figure 1 32 1 1 8 2 4 2 4 16 'A 8 64 We may make a graph showing these values, so that we may ascertain the intensity at any distance within the limits of the graph (Fig. 3). In reading this graph it may be seen that the intensity falls off quite rapidly as the distance from the screen becomes greater. If we were to continue to move further away, the intensity would be so small as to be negligible. The equation used to arrive at this conclusion is said to be a "second degree equation," which means that one of its terms varies as some other term whose numerical value is squared or multiplied by itself. This equation is frequently referred to as the "law of inverse squares." In projecting an image upon a screen it is common experience to observe that the image may be made larger or smaller by simply adjusting the distance of the object from the lens. The size of the image is equal to that of the object when both the object and the screen are equidistant from the lens. This relationship is expressed by the following equation: Lo Li Do Di Where: Lo = length of the object Li = length of the image Do = distance of object from lens Di = distance of image from lens Problem : Substituting numerical values in the equation, let us consider the placing of a piece of film 1 inch in length before a lens at a distance of 4 inches. The screen is, say, 50 feet away. What is the length of the image? Solution : Lo Do Li Di Li 50X12 fa 1 1 — 1 1 — 1 1 1 r 1 [ 1 1 — 1 — 1 1 — 1 ^.o ^^ Kg ."^ ^ > !2 T '^1 Nb iH r-~. ' — ^ a _ ^ I— ^ , ' 1 ■■~ ' — •< O i f A . a. Figure 3 [23]