63. Screw Cutting.-Most lathes are equipped with a small plate giving the necessary gears to use for cutting different threads, but every good machinist should know how to calculate the proper gear setting for such work. This is a simple problem in gear trains and should cause no difficulty for the man who understands the principles of gear trains.

The lathe carriage and tool are moved by a "lead screw" having usually 2, 4, 6, or 8 threads per inch. If a lathe has a lead screw having 6 threads per inch, each revolution of the lead screw will move the carriage & in.; a 4 pitch screw would move the carriagein. for each revolution of the screw. Then, if the spindle of the lathe and the lead screw turn at the same speed, the lathe will cut a thread of the same pitch as that on the lead screw. If a finer thread is wanted than that on the lead screw, the spindle should make more turns than does the lead screw. Suppose we want to cut 24 threads per inch and have a 6 thread per inch lead screw. It will require 6 turns of the lead screw to move the carriage 1 inch. Meanwhile, the work should revolve 24 times. Then the ratio of spindle speed to lead screw speed should be 4:1

Speed of spindle

Speed of lead screw

=

Threads per inch to be cut
Threads per inch on lead screw

The first driving gear is that on the spindle, while the last driven

gear is that on the end of the lead screw.

Threads per inch to be cut Threads per inch on lead screw

Hence,

Product of Nos. of teeth on driven gears
Product of Nos. of teeth on driving gears

PROBLEMS

121. In Fig. 10, if we removed the 48 tooth gear and put a 64 tooth gear in its place, what would be the speed ratio of A to B?

122. In Fig. 11, if B makes 6 revolutions, how many turns will C make and how many will A make?

123. What would be the speed ratio of the train of Fig. 12 if we put a 32T gear on at C and a 48T gear at D?

Determine the size of

124. The lineshaft in Fig. 15 runs 250 R. P. M. lineshaft pulley to run the grinder at 1550 R. P. M. using the countershaft as shown in the figure.

125. A machinist wishes to thread a pipe on a lathe having 2 threads per inch on the lead screw. There are to be 11 threads per inch on the pipe. What is the ratio of the speeds of the spindle and the lead screw?

126. Two gears are to have a speed ratio of 4.6 to 1. If the smaller gear has 15 teeth, what must be the number of teeth on the larger gear?

127. It has been decided to equip the punch in Fig. 14 with a motor drive by replacing the fly wheel with a large gear to be driven by a small pinion on the motor. If the motor runs 800 R. P. M., and has a 16 tooth pinion, what must be the number of teeth on the other gear? Speed of the punch to be 20 strokes per minute.

128. A street car is driven through a single pair of gears; a large gear on the axle being driven by a smaller one on the motor shaft. If a car has 33-in. wheels and a gear ratio of 1:4, how fast would the car go when the motor is running 1200 R. P. M.?

129. Fig. 16 shows the head stock for a lathe. The cone pulley carries with it the cone pinion A, which drives the back gear B. B is connected solidly with the back pinion C which drives the face gear D. If the gears have the following numbers of teeth, determine the back gear ratio (speed of A: speed of D):

130. If you were to cut a 20 pitch thread on a lathe having a 4 pitch lead screw, what would be the ratio of the speeds of the spindle and the lead screw?

CHAPTER X

AREAS AND VOLUMES OF SIMPLE FIGURES

64. Squares. In taking up the calculation of areas of surfaces and the volumes and weights of objects, the expressions "square" and "square root" will be met and must be understood. To one unfamiliar with these names and the corresponding operations the signs and operations themselves seem difficult. They are in reality very simple. The square of a number is simply the product of the number multiplied by itself; the square of 2 is 2×2=4; the square of 5 is 5X5-25; the square of 12.5 is 12.5X12.5= 156.25. Instead of writing 2×2 or 5×5, it is customary to write 22 and 52. These are read "2 squared" and "5 squared." 12.52 12.5 squared, and so on. The little 2 at the upper right hand corner is called the Exponent.

=

65. Square Root.-The square root of a given number is simply another number which, when multiplied by itself (or squared), produces the given number. Thus, the square root of 4 is 2, since 2 multiplied by itself (2×2) gives 4. The square root of 9 is 3, since 3×3=32=9. Square root is the reverse of square, so if the square of 5 is 25 the square root of 25 is 5. The mathematical sign of square root, called the radical sign, is √. Then √9-3; 25-5. These expressions are read "the square root of 9=3"; "the square root of 25=5". Square roots of larger numbers can usually be found in handbooks and the actual process of calculating them, which is somewhat complicated, will be taken up later on.

=

66. Cubes and Higher Powers.-In the same way that 22 (2 squared) =2×2=4, 23 (2 cubed) = 2×2×2=8. The exponent simply indicates how many times the number is used as a factor, or how many times it is multiplied together. 48=4X4X4-64. 38=3X3X3=27.

Just as square root is the reverse of square, so cube root is the reverse of cube. The sign for cube root is V. So if 33 = 3X3X3=27, then 27-3. Sometimes a factor is repeated more than 3 times, in which case, the exponent indicates the

number of times. 24 means 2×2×2×2 and is read "2 to the fourth power." 25=2X2X2X2X2 and is read "2 to the fifth power," and so on. The roots are indicated in the same way. 16-fourth root of 16-2; 5=5X5X55-625, etc.

67. Square Measure.-Before going further, it will be well to get clearly in mind just what the term "Square" means in terms of the things we see. Areas of figures are measured in terms of the "square" unit. For instance, if the dimensions of the base of a milling machine are 3 ft. by 5 ft., the floor space covered by this base is 15 square feet. In this case the area is

FIG. 17.

measured by the unit known as the square foot. A Square Foot is a surface bounded by a square having each side 1 ft. in length. In case of the milling machine base represented in Fig. 17, there are by actual count 15 sq. ft. in this surface and this is readily seen to be the product of the length and the breadth of the base, since 3×5=15.

The Square Inch is another common unit of area. This is much smaller than the square foot, being only one-twelfth as great each way. If a square foot is divided into square inches it will be seen to contain 12×12 or 144 sq. in. (see Fig. 18). It will be readily seen that the area of any square is equal to the product of the side of the square by itself. In other words, the area of a square equals the side "squared" (referring to the process explained in Article 64). Looking at it the other way around, the square of any number can be represented by the area of a square figure, one side of which represents the number itself. The actual things which the number represents makes no difference whatever. If the side of a square is 5 in., the area is