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 travelling 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

101. A stack is measured with a tape line and its circumference found to be 88 in. What is the diameter of the stack?

102. An emery wheel 16 in. in diameter runs 1300 R. P. M. Find the surface speed.

103. The Bridgeport Safety Emery Wheel Co., Bridgeport, Conn., build an emery wheel 36 in. in diameter and recommend a speed of 425-450 revolutions. Calculate the surface speeds at 425 and at 450 revolutions.

104. An emery wheel runs 1000 R. P. M. What should be its diameter to give a surface speed of 5500 ft.?

105. A grindstone 3 ft. in diameter is to be used for grinding carpenters' tools; how many R. P. M. should it run?

106. Calculate the belt speed on a high-speed automatic engine carrying a 48 in. pulley and running at 250 R. P. M.

107. How many revolutions will a locomotive driving wheel, 72 in. in diameter, make in going 1 mile?

108. What would be the rim speed in feet per minute of a fly wheel 14 ft. in diameter running 80 R. P. M.?

109. At how many R. P. M. should an 8 in. shaft be driven in a lathe to give a cutting speed of 60 ft. per minute?

110. At what R. P. M. should a 1 in. high speed drill be run to give a cutting speed of 80 ft. per minute? If the drill is fed .01 in. per revolution, how long will it take to drill through 2 in. of metal?

CHAPTER VIII

RATIO AND PROPORTION

56. 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 revolutions per minute and the other one 160 revolutions per minute, 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 can be written as a fraction, 1; or, since a fraction means division, it can 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 relation 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 as 1 is 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. 57. Proportion.-When two ratios are equal, the four 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 very nicely and 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?

Now very clearly the ratio between the number of castings 19:13 is the same as the ratio of the weights, but one of the weights we do not know. Writing the proportion out and putting the word answer" for the number which we are to find, we have

19:13=137:Answer

From our rule which says the product of the means equals the product of the extremes:

13X137-1781, product of "means."

This must equal the product of the extremes which would be 19 X Answer.

In using proportion keep the following things in mind:

(1) Make the number which is the same kind of thing as the required answer the third term. Make the answer the fourth term.

(2) See whether the answer will be greater or less than the third term; if less, place the less of the other two numbers for the second term; if greater, place the greater of the other two numbers for the second term.

(3) Solve by knowing that the product of the means equals the product of the extremes, or by this rule: Multiply the means together and divide by the given extreme; the result will be the other extreme or answer.

Let us see how these rules would be applied to a practical example.

Example:

A countershaft for a grinder is to be driven at 450 R. P. M. by a lineshaft that runs 200 R. P. M. If the pulley on the countershaft is 8 in. in diameter, what size pulley should be put on the lineshaft? A proportion can be formed of the pulley diameters and their revolutions per minute. Applying the rules of proportion, we get the following analysis and solution to the problem.

The

(1) The diameter of the lineshaft pulley is the unknown answer. other number of the same kind is the diameter of the countershaft pulley (8 in.). So we have the ratio.

8:Answer

(2) If the countershaft pulley is to run faster, its diameter must be smaller than the other one. Therefore, the answer is greater than 8. Hence, the greater revolutions (450) will be placed as the second term and the other R. P. M. (200) will be the first term. Therefore, we have the completed proportion:

(3) Solving this we get:

200:450-8:Answer

450×8-3600, product of means. 3600÷200=18, Answer.

Hence, an 18-in. pulley should be put on the lineshaft to give the desired speed to the countershaft.

Sometimes the letter X is used to represent the unknown number whose value is sought. The following is an example of such a case.

Hence,

=

6:40 5:X. Find what number X stands for. 40X 5 200, product of the means.

58. Speeds and Diameters of Pulleys.-As shown in an example previously worked, if two pulleys are belted together, their diameters and revolutions per minute can be written in a proportion having diameters in one ratio and R. P. M. in the other ratio of the proportion. It will be noticed from the example which was worked, that the numbers which form the means apply to the same pulley, while the extremes both refer to the other pulley. Then, since the product of the means equals the product of the extremes, we obtain the following simple relation for pulleys belted together: The product of the diameter and revolutions of one pulley equals the product of the diameter and revolutions of the other. This gives us the following simple rule for working pulley problems.

Rule for Finding the Speeds or Diameters of Pulleys.—Take the pulley of which we know both the diameter and the R. P. M., and multiply these two numbers together. Then divide this product by the number that is known of the other pulley. The result is the desired number.

Examples:

1. A 36-in. pulley running 240 R. P. M. is belted to a 15-in. pulley. Find the R. P. M. of the 15-in. pulley.

36×240-8640, the product of the known diameter and revolutions. 8640÷15=576, the R. P. M. of the 15-in. pulley, Answer.

2. A 36-in. grindstone is to be driven at a speed of 800 R. P. M. from a 6-in. pulley on the lineshaft which is running 225 R. P. M. What size pulley must be put on the grindstone arbor?

=

Explanation: First we must find the R. P. M. for the grindstone as explained in Chapter VII. To get

=85 R. P. M., nearly. the required surface speed we find 85 R. P. M. necessary.

6X225=1350

1350÷85-16 in., nearly.

Use a 16-in. pulley on the arbor.

Now we have the R. P. M. and the size of the lineshaft pulley. The product of these two numbers is 1350. Dividing this by the R.

P. M. of the grindstone arbor gives 16 in. as the nearest even size of pulley,

so we will use that size.