This number 3.1416 is, without doubt, the most used in practical work of any figure in mathematics. In writing formulas, it is quite common to represent this decimal by the Greek letter π (pronounced "pi"), instead of writing out the whole number. For this reason, the number 3.1416 is given the name "pi." Where it is more convenient and extreme accuracy is not It, therefore, gives values of the circumference slightly too large, but in many cases it is sufficiently accurate and saves time. Examples: 1. What length of steel sheet would be needed to roll into a drum 32 in. in diameter? When rolled up, the length of the sheet will become the circumference of a 32-in. circle. The circumference must be π times 32. 3.1416X32=100.5+ in. The length of the sheet must, therefore, be 100 in. and, if it is to be lapped and riveted, we would have to add a suitable allowance of 1 in. or so for making the joint. 2. A circular steel tank measures 37 ft. 8 in. in circumference. What is its diameter? If the circumference of a circle is 3.1416 times the diameter, then the diameter can be obtained by dividing the circumference by 3.1416. 51. Formulas.-A formula, in mathematics, is a rule in which mathematical signs and letters have been used to take the place of words. We say that "the circumference of a circle equals 3.1416 times the diameter." This is a rule. merely write C=πXD But suppose we This is the same rule expressed as a formula. We have used C instead of the words "the circumference of a circle;" the sign= replaces the word "equals;" the symbol 7 is used instead of the number 3.1416; X stands for "times;" and D stands for "the diameter." We found in the second example under Article 50 that, when the circumference is given, we can obtain the diameter by dividing the circumference by T. As a formula this would be written This arrangement is useful when we want to get the diameters of trees, chimneys, tanks, and other large objects. We can easily measure their circumferences and, by dividing by 3.1416, we get the diameters. Formulas do not save much, if any space, because it is necessary usually to explain what the letters stand for. They have, however, the great advantage that intricate mathematical operations can be shown much more clearly than if they were written out in a long sentence or statement. One can usually see in one glance at a formula just what is to be done, with the numbers that are given in the problem, to find the quantity that is unknown. Example: What is the circumference in feet of a 16-in. emery wheel? C=3.1416 X 16=50.2656 in. 50.2656 in.÷12=4.1888 ft., Answer. Explanation: We have the diameter given and want to get the circumference. We, therefore, use the formula which says that С=XD. πis always 3.1416 and D in this case is 16 in. Then C comes out 50.2656 in. But the problem calls for the circumference in feet. This is of the number of inches, or it is the number of inches divided by 12. In the work of this chapter, the circumferences of circles are always used in feet, and, consequently, should always be calculated in feet. If we use D in feet, we will get C in feet, while, if D is in inches, C will come out in inches. If the diameter can be reduced to exact feet, it is easiest to use the diameter in feet when multiplying by π, rather than to reduce to feet after multiplying. Example: What is the circumference of a 48-in. fly wheel? 48 in. +124 ft., the diameter. C=3.1416X4-12.5664 ft., Answer. This is much shorter than it would be to multiply 3.1416 by 48 and then divide the product by 12. 52. Circumferential Speeds.-When a fly wheel or emery wheel or any circular object makes one complete revolution, each point on the circumference travels once around the circumference and returns to its starting-point. When the wheel turns ten times, the point will have travelled a distance of ten times the circumference. In one minute, it will travel a distance equal to the product of the circumference times the number of revolutions per minute. The distance, in feet per minute, travelled by a point on the circumference of a wheel is called its Circumjerential Speed, Rim Speed, or Surface Speed. It is also sometimes called Peripheral Speed, because the circumference is sometimes given the name of periphery. It is the surface speed by which we determine how to run our fly wheels, belts, emery wheels, and grindstones, and what speeds to use in cutting materials in a machine. Written as a formula: S=CXN where: S is the surface speed C is the circumference N is the number of revolutions per minute (R. P. M.). Expressed in words this formula states that the surface speed of any wheel is equal to the circumference of the wheel multiplied by the number of revolutions per minute. Example: What would be the rim speed of a 7 ft. fly wheel when running at 210 revolutions per minute? 22 7 C=X7-22 ft. S=CXN S=22×210=4620 4620 ft. per min., Answer. Explanation: First we find the circumference of the wheel, by multiplying the diameter by л. Here is a case where it is much easier to use for than to use the decimal 3.1416, and the result is sufficiently accurate for our purposes. We get 22 ft. for the circumference. We can now get the rim speed, which is equal to the product of the circumference times the number of revolutions per minute; or S-CXN. C being 22 ft. and N being 210 revolutions per minute, we find that S is 4620 ft. per min. Hence, the rim of this fly wheel travels at a speed of 4620 ft. per minute. If we have given a certain speed which is wanted and have the circumference of the wheel, then the R. P. M. (revolutions per minute) will be obtained by dividing the desired speed by the circumference. In the example just worked, if we want to give the fly wheel a rim speed of 5280 ft. per minute, it requires no argument to show that the wheel will have to run at 5280÷22=240 revolutions per minute. In such a case, we would use our formula in the form This formula expresses the same relation as S=NXC, but now it is rearranged to enable us to find the R. P. M. when the rim speed and the circumference are given. Sometimes, especially with emery wheels, we know the proper surface speed and we have an arbor belted to run a certain number of R. P. M. The problem then is to find the proper size of stone to order. The desired speed divided by the number of R. P. M. will give the circumference, and from this we can figure the diameter of the stone. Here again we have merely rearranged the formula S=CXN so as to be in more suitable form for finding the circumference when the surface speed and the R. P. M. are given. 53. Grindstones and Emery Wheels.-Makers of emery wheels and grindstones usually give the proper speed for the stones in feet per minute. This refers to the distance that a point on the circumference of the stone should travel in 1 minute and is called the "surface speed" or the "grinding speed." The proper speed at which to run grindstones depends on the kind of grinding to be done and the strength of the stones. For heavy grinding they can be run quite fast. For grinding edge tools they must be run much slower to get smooth surfaces and to prevent heating the fine edges of the tools. The following surface speeds may be taken as representing good practice: Grindstones: For machinists' tools, 800 to 1000 ft. per minute. Grindstones for very rapid grinding: Coarse Ohio stones, 2500 ft. per minute. Fine Huron stones, 3000 to 3400 ft. per minute. Sometimes the rule is given for grindstones as follows: "Run. at such a speed that the water just begins to fly." This is a speed of about 800 ft. per minute and would be a good average speed for sharpening all kinds of tools. Examples: 1. A 36-in. grindstone, used for sharpening carpenters' and patternmakers' tools, is run at 60 R. P. M. Is this speed correct? We must first find the circumference and then the surface speed to see if it falls between the allowable limits. 36 in.÷12-3 ft., the diameter C=3.1416×3=9.4248 ft. S=CXN S=9.4248X60=565.488 Explanation: First we find the circumference, which comes out 9.4248 ft. Using this and the R. P. M., we find S to be 565 F. P. M. (feet per minute). As this lies between the allowed limits (550 to 600 F. P. M.) the speed of the stone is correct. 2. At what R. P. M. should a 50-in. Huron stone be run if it is to be used for rough grinding? Emery wheels are usually run at a speed of about 5500 ft. per minute. A good, ready rule, easy to remember, is a speed of a mile a minute. Most emery wheel arbors are fitted with two pulleys of different diameters. When the wheel is new, the larger pulley on the arbor should be used and, when the wheel becomes worn down sufficiently, the belt should be shifted to the smaller pulley. Never shift the belt on an emery wheel, however, without first calculating the effect on the surface speed of the wheel. Many serious accidents have been caused by emery wheels bursting as a result of being driven at too great a speed. Before cutting a new wheel on an arbor the resultant surface speed |