58. Pulleys and Belts.-If the rim of a pulley is run at too great a speed, the pulley may burst. The rim speeds of pulleys are calculated in the same manner as are grinding and cutting speeds. A general rule for cast-iron pulleys is that they should not have a rim speed of over a mile a minute (5280 f.p.m.). This speed may be exceeded somewhat if care is taken that the pulley is well balanced and is sound and of good design.

The proper speeds for belts is taken up fully in a later chapter under the general subject of belting. It is well, however, to point out now that the speed at which any belt is traveling through the air is practically the same as that of the rim of either of the pulleys over which the belt runs; and if we neglect the small amount of slipping which usually occurs between a belt and its pulleys, we can say that the speed of a belt is the same as the rim speed of the pulleys. It will be seen from this that if two pulleys are connected by a belt, their rim speeds are practically the same.

PROBLEMS

106. Steel hoops are used to reinforce a wooden tank whose outside diameter is 9 ft. 6 in. If each hoop is made from a flat strip of steel, bent into a circle, and riveted together at the ends, how long a strip is required for each hoop, allowing 2 in. extra for lapping the ends?

107. A broken emery wheel in a toolroom is replaced temporarily by a larger wheel until a new one can be obtained. The larger wheel is 7 in. in diameter and runs at 3600 r.p.m. What is its surface speed?

108. The Bridgeport Safety Emery Wheel Co., Bridgeport, Conn., makes an emery wheel 36 in. in diameter and recommends a speed of 425 to 450 r.p.m. Calculate the surface speeds of this wheel at 425 and 450 r.p.m.

109. The belt conveyor of Fig. 15 is driven by the pulley in the foreground, which is 2 ft. in diameter and runs at 54 r.p.m. What is the speed of the belt in f.p.m.? If approximately 249 lb. of coal are carried by every 10 ft. of belt, how many tons of coal does it convey per hour?

110. A line shaft rotates at 450 r.p.m. What diameter pulley is required for driving a cupola fan if a belt speed of 4250 f.p.m. is required? 111. A grindstone 3 ft. in diameter is used for grinding carpenters' tools. How many r.p.m. should it run?

112. At how many r.p.m. should a 14-in. high-speed drill be run to give a cutting speed of 80 f.p.m.

113. At how many r.p.m. should an 8-in. shaft be driven in a lathe to give a cutting speed of 60 f.p.m.?

114. A contractor purchases a 10 hp. motor having a speed of 1800 r.p.m. to drive a centrifugal pump. He knows that at belt speeds

greater than 5000 f.p.m. the belt will slip badly. What is the diameter of the largest pulley that can be used on the motor without causing too much slippage?

115. In the appraisal of a certain gas company's property, it was necessary to obtain the diameter of a large gas holder. The diameter could not be measured directly, so the engineer measured the circumference, which he found to be 550 ft. What was the diameter of the holder?

CHAPTER VIII

RATIO AND PROPORTION

59. Ratios. In comparing the relative sizes of two quantities, we refer to one as being a multiple or a fraction of the other. If one casting weighs 600 lb., and another weighs 200 lb., we say that the first one is three times as heavy as the second, or that the second is one-third as heavy as the first. This relation between two quantities of the same kind is called a ratio.

In comparing the speeds of two pulleys, one of which runs 40 r.p.m. and the other one 160 r.p.m., we say that their speeds are "as 40 is to 160," or "as 1 is to 4." In this sentence, "40 is to 160" is a ratio, and so also is "1 is to 4" a ratio.

Ratios may be written in three ways. For example, the ratio of (or relation between) the diameters of two pulleys which are 12 in. and 16 in. in diameter may be written as a fraction, 1; or, since a fraction means division, it may be written 12 ÷ 16; or, again, the line in the division sign is sometimes left out and it becomes 12:16. The last method, 12:16, is the one most used and will be followed here. It is read "twelve is to sixteen."

A ratio may be reduced to lower terms the same as a fraction, without changing the value of the ratio. If one bin in the stock room contains 1000 washers, while another bin contains 3000, then the ratio of the contents of the first bin to the contents of the second is "as 1000 is to 3000." The ratio of 1000 to 3000 can be reduced by dividing both by 1000. This leaves the ratio 1 to 3.

Hence, the ratio between the contents of the bins is also 1 to 3. Likewise, the ratio 24:60 can be reduced to 2:5 by dividing both terms by 12. If we write it as a fraction we can easily see that

The ratio of the 1000 washers to the 3000 washers is 1000:3000

or 1:3.

The ratio of 8 in. to 12 in. is 8:12 or 2:3.

The ratio of $1 to $1.50 is 1:1 or 2:3.

The ratio of 30 castings to 24 castings is 30:24 or 5:4.

60. Proportion.-When two ratios are equal, the four numbers (or terms) are said to be in proportion. The two ratios 2:4 and 8:16 are clearly equal, because we can reduce 8:16 to 2:4 and we can, therefore, write 2:4 8:16. When written thus, these four numbers form a proportion.

=

Likewise, we can say that the numbers 6, 8, 15, and 20 form a proportion because the ratio 6:8 is equal to the ratio of 15:20.

6:8 = 15:20

Now, it will be noticed that if the first and fourth terms of this proportion be multiplied together, their product will be equal to the product of the second and third terms:

This is true of any proportion and forms the basis for an easy way of working practical examples, where we do not know one term of the proportion but know the other three. The first and fourth terms are called the extremes, and the second and third are called the means. Then we have the rule: "The product of the means is equal to the product of the extremes."

This relation can be simply expressed as a formula.

Let a, b, c, and d represent the four terms of any proportion so that

Let us now see of what practical use this is. We will take this example:

If it requires 137 lb. of metal to make 19 castings, how many pounds will it take to make 13 castings from the same pattern?