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 12.5×12.5= 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.

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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 3x3=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.

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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. 43=4X4×4=64. 33=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=5×5×5×5=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

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.

25 sq. in.; if the side is 5 ft., the area is 25 sq. ft. If we simply have the number 5, its square is 25, no matter what kind of things the 5 may refer to.

As mentioned before, 1 sq. ft. is the area of a square 1 ft. on each side and, if divided into square inches, will be found to contain 122 or 144 sq. in. Likewise, a square yard is 3 ft. on each side and, therefore, contains 329 sq. ft. The following table gives the relation between the units ordinarily used in measuring areas:

FIG. 18.

MEASURES OF AREA (SQUARE MEASURE)

144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet-1 square yard (sq. yd.)

30 square yards = 1 square rod (sq. rd.)
160 square rods = 1 acre (A)

640 acres 1 square mile (sq. mi.)

68. Area of a Circle.-If a circle is drawn in a square as shown in Fig. 19, it is easily seen that it has a smaller area than the square because the corners are cut off. The area of the circle is always a definite part of the area of the square drawn on its diameter, the area of the circle being always .7854 times the area of the square. This number .7854 happens to be just onefourth of the number 3.1416 given in Chapter VII. Just why this is so will be shown later on. If the diameter of the circle = 10 in., as in Fig. 19, the area of the square is 100 sq. in. and the area of the circle is .7854 × 100=78.54 sq. in. You can prove this to your own satisfaction in the following manner. Cut a

square of cardboard of any size, and from the center describe a circle as shown just touching on all four sides. Weigh the square, and then cut out the circle and weigh it. The circle will weigh .7854 Xweight of the square. A pair of balances such as are found in a drug store are the best for this experiment.

AREA 78.54

-10".

FIG. 19.

Rule for Area of Circle.-The area of any circle is obtained by squaring the diameter and then multiplying this result by .7854. If written as a formula this rule would read

If you think a little you will see that, if the diameter is doubled, the area is increased four times. This can also be seen from Fig. 20. The diameter of the large circle is twice that of one of the small circles, but its area is four times that of one of the small circles. This is a very important and useful law and may be stated as follows: "The areas of similar figures are to each other as the squares of their like dimensions." A 2 in. circle contains 22.7854 3.1416 sq. in., while a 6 in. circle contains 62X .7854 28.2744 sq. in., or nine times as much. This we can find

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