From our rule which says the product of the means equals the product of the extremes: 13×137=1781, product of “means.” This must equal the product of the extremes which would be 19 X Answer. In using proportion keep the following things in mind: (1) Make the number which is the same kind of thing as the required answer the third term. Make the answer the fourth term. (2) See whether the answer will be greater or less than the third term; if less, place the less of the other two numbers for the second term; if greater, place the greater of the other two numbers for the second term. (3) Solve by knowing that the product of the means equals the product of the extremes, or by this rule: Multiply the means together and divide by the given extreme; the result will be the other extreme or answer. Let us see how these rules would be applied to a practical example. Example: A countershaft for a grinder is to be driven at 450 R. P. M. by a lineshaft that runs 200 R. P. M. If the pulley on the countershaft is 8 in. in diameter, what size pulley should be put on the lineshaft? A proportion can be formed of the pulley diameters and their revolutions per minute. Applying the rules of proportion, we get the following analysis and solution to the problem. (1) The diameter of the lineshaft pulley is the unknown answer. The other number of the same kind is the diameter of the countershaft pulley (8 in.). So we have the ratio. 8:Answer (2) If the countershaft pulley is to run faster, its diameter must be smaller than the other one. Therefore, the answer is greater than 8. Hence, the greater revolutions (450) will be placed as the second term and the other R. P. M. (200) will be the first term. Therefore, we have the completed proportion: (3) Solving this we get: 200:450-8:Answer 450×8=3600, product of means. 3600÷200=18, Answer. Hence, an 18-in. pulley should be put on the lineshaft to give the desired Speed to the countershaft. Sometimes the letter X is used to represent the unknown number whose value is sought. The following is an example of such a case. Hence, 6:40 5:X. Find what number X stands for. = 58. Speeds and Diameters of Pulleys.—As shown in an example previously worked, if two pulleys are belted together, their diameters and revolutions per minute can be written in a proportion having diameters in one ratio and R. P. M. in the other ratio of the proportion. It will be noticed from the example which was worked, that the numbers which form the means apply to the same pulley, while the extremes both refer to the other pulley. Then, since the product of the means equals the product of the extremes, we obtain the following simple relation for pulleys belted together: The product of the diameter and revolutions of one pulley equals the product of the diameter and revolutions of the other. This gives us the following simple rule for working pulley problems. Rule for Finding the Speeds or Diameters of Pulleys.-Take the pulley of which we know both the diameter and the R. P. M., and multiply these two numbers together. Then divide this product by the number that is known of the other pulley. The result is the desired number. Examples: 1. A 36-in. pulley running 240 R. P. M. is belted to a 15-in. pulley. Find the R. P. M. of the 15-in. pulley 36×240=8640, the product of the known diameter and revolutions. 8640 15 567, the R. P. M. of the 15-in. pulley, Answer. 2. A 36-in. grindstone is to be driven at a speed of 800 Ft. P. M. from a 6-in. pulley on the lineshaft which is running 225 R. P. M. What size pulley must be put on the grindstone arbor? 59. Gear Ratios.-The same principles as are applied to pulleys can be applied to gears. If we have two gears running together as shown in Fig. 9, the product of the diameter and R. P. M. of one gear will be equal to the product of the diameter and R. P. M. of the other. In studying gearing, we do not deal with the diameters so much as we do with the numbers of teeth. We find that gears are generally designated by the numbers of teeth. For example, we talk of 16 tooth gears and 24 tooth gears, etc., but we seldom talk about gears of certain diameters. In making these calculations for gears, we can use the numbers of teeth instead of the diameters. When a gear is revolving, the number of teeth that pass a certain point in one minute will be the product of the number of teeth times the R. P. M. of the gear. FIG. 9. If this gear is driving another one, as in Fig. 9, each tooth on the one gear will shove along one tooth on the other one. Consequently, the product of the number of teeth times R. P. M. of the second gear will be the same as for the first gear. This gives us our rule for the relation of the speeds and numbers of teeth of gears. Rule for Finding the Speeds or Numbers of Teeth of Gears.Take the gear of which we know both the R. P. M. and the number of teeth and multiply these two numbers together. Divide their product by the number that is known about the other gear. The quotient will be the unknown number. Example: A 38 tooth gear running 360 R. P. M. is to drive another gear at 190 R. P. M. What must be the number of teeth on the other gear? 38×360=13,680, the product of the number of teeth and revolutions of one gear. 190X Answer = 13,680 111. (a) If you draw $33.00 on pay day and another man draws $22.00, what is the ratio of your pay to his? (b) What is the ratio of his pay to yours? 112. The speeds of two pulleys are in the ratio of 1:4. If the faster one goes 260 R. P. M., how fast does the slower one go? 113. Two castings are weighed and the ratio of their weights is 5:2. If the lighter one weighs 80 lb., what does the heavier one weigh? 114. Find the unknown number in each of the following proportions: 115. If it takes 72 lb. of metal to make 14 castings, how many pounds are required to make 9 castings? If the 14 tooth gear 116. A 14 tooth gear is driving a 26 tooth gear. runs 225 revolutions per minute, what is the speed of the 26 tooth gear? 117. A 12 in. lineshaft pulley runs 280 revolutions and is belted to a machine running 70 revolutions. What must be the size of the pulley on the machine? 118. A lineshaft runs 250 R. P. M. A grinder with a 6 in. pulley is to run 1550 R. P. M. Determine size of pulley to put on the lineshaft to run the grinder at the desired speed. 119. An apprentice was given 100 bolts to thread. He completed threefifths of this number in 45 minutes and then the order was increased so that it took him 2 hours for the entire lot. How many bolts did he thread? 120. A 42 in. planer has a cutting speed of 30 ft. per minute and the ratio of cutting speed to return speed of the table is 1:2.8. What is the return speed in feet per minute? CHAPTER IX PULLEY AND GEAR TRAINS-CHANGE GEARS 60. Direct and Inverse Proportions.-A proportion formed of numbers of castings and the weights of metal required to make them is a direct proportion, because the amount of metal required increases directly as the number of castings increases. When two pulleys (or gears) are running together, one driving the other, the larger of the two is the one that runs the slower. The proportion formed from their diameters and revolutions is, therefore, called an Inverse Proportion, because the larger pulley runs at the slower speed. The number of revolutions of one pulley is said to vary inversely as its diameter, since the greater the diameter, the less the number of revolutions it will make. In every pair of gears one of them is driving the other, so the one can be called the driving gear, or the driver, and the other the driven gear, or the follower. These names are in quite general use to designate the gears and to assist in keeping the proportions in the right order. Accordingly, we have the proportion: No. of teeth of): (R. R. P. M. of driven driver = on driver on driven This is an inverse proportion because the driver and the driven are in the reverse order in the second ratio from what they are in the first ratio. Perhaps this can be seen better if the ratios are written as fractions. Here the reason for the name "inverse proportion" is easily seen. The second fraction has the driver and the driven inverted from what they are in the first fraction. This method of writing proportions as fractions is much used in solving problems in gears or pulleys. 61. Gear Trains.-A gear train consists of any number of gears used to transmit motion from one point to another. Fig. |