CHAPTER VII CIRCUMFERENCES OF CIRCLES; GRINDING SPEEDS CUTTING AND 52. Shop Uses. In the operation of almost any machine, care must be used to determine the speed which will give the best results. Lathes, milling machines, boring mills, etc. are provided with means for changing the speed, according to the nature of the work being done. Emery wheels and grindstones, however, are often set up and run at any speed which the pulleys happen to have, regardless of the diameter. If an emery wheel of large size is put on a spindle that has been belted to drive a smaller wheel, the speed may be too great for the larger wheel and, if the difference is considerable, the large wheel may fly to pieces. Every mechanic should know how to calculate the proper sizes of pulleys to use for emery wheels or grindstones, the correct speed at which to run the work in his lathe, or the most economical speeds to use for belts and pulleys. A few data on this subject may be useful and will afford applications for arithmetical principles. 53. Circles. To understand what has just been mentioned, it is necessary to obtain a knowledge of circles and their properties. The distance across a circle, measured straight through the center, is called the Diameter. Circles are generally designated by their diameters. Thus a 6-in. circle means a circle 6 in. in diameter. Sometimes the radius is used. The Radius is the distance from the center to the edge or circumference and is, therefore, just half the diameter. If a circle is designated by the radius, we should be careful to say so. Thus, there would be no misunderstanding if we said "a circle of 5-in. radius;" but unless the word "radius" is used, we always understand that the measurement given is the diameter. The Circumference is the name given to the distance around the circle, as indicated in Fig. 14. The circumference of any circle is always 3.1416 times the diameter. In other words, if we measure the diameter with a string and lay this off around the circle, it will take a little over three times the length of the string to go once around the circle. 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 (pronounced "pi"), instead of writing out the whole number. For this reason, the number 3.1416 is given the name "pi." 22 Where it is more convenient and extreme accuracy is not required, the fraction may be used for instead of the more exact value 3.1416. 22 = 3 = 3.1429 It, therefore, gives values of the circumference slightly too large, but in many cases it is sufficiently accurate and saves time. Example: What length of steel sheet would be needed to roll into a drum 32 in. in diameter? When the sheet is rolled up, its length will become the circumference of a 32-in. circle. The circumference must be π times 32. The length of the sheet must, therefore, be 1001⁄2 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. 54. Formulas.-A formula, in mathematics, is a rule in which mathematical signs and letters are 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. But suppose we merely write C = TX D = 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 is used instead of the number 3.1416; X stands for "times"; and D stands for "the diameter." 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, in order to find the quantity that is unknown. Example: What is the circumference in feet of a 16-in. emery wheel? Explanation: We have the diameter given and want to get the circumference. We, therefore, use the formula which says that C = TX D. T is always 3.1416, and D in this case is 16 in. Then C, But the problem calls for the circumference in feet. This is of inches, or it is the number of inches divided by 12. is 50.2656 in. of the number In the work of this chapter, the circumferences of circles are always used in feet, and, consequently, should always be calculated in feet. If D is in feet, we will get C in feet, while, if D is in inches, C will be in inches. If the diameter can be reduced to exact feet, it is easier to use the diameter in feet when multiplying by π, rather than to reduce to feet after multiplying. This is much shorter than it would be to multiply 3.1416 by 48 and then divide the product by 12. Frequently it is desired to find the diameter of some object, such as a tree, tank, or chimney, where it is an easy matter to measure the circumference but quite difficult to measure the diameter. From the preceding formula we note that Circumference 3.1416 X diameter = Since we multiply the diameter by 3.1416 to get the circumference, we can reverse the operation and divide the circumference by 3.1416 to get the diameter. In other words we "work the problem backwards," dividing instead of multiplying. Consider a circle having a diameter of 2 in. 3.1416 X Diameter = Circumference = 6.2832 in. circumference If we work backwards from the circumference, we get A circular steel tank measures 37 ft. 8 in. in circumference. What is its diameter? 55. Circumferential Speeds. When a flywheel or emery wheel or any circular object makes one complete revolution, each point on the rim or circumference travels once around the circumference and returns to its starting point. Thus any point on the rim travels a distance equal to the circumference when the wheel turns around once. When the wheel turns ten times, the point will have traveled a distance of ten times the circumference. In 1 min. 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, traveled by a point on the circum |