12 



THE POPULAR EDUCATOR. 



It is evident that, if we are to gain any power, the nut must 

 not be allowed to turn together with the screw ; and hence we 

 have different modes of using the screw, according as the screw 

 itself or the nut is fixed. When used to fasten the beams of a 

 house together, or to strain the wire of a fence, the screw is 

 prevented from rotating, and the nut turned by a wrench ; the 

 screw is thus drawn forward, and the required strain applied. 

 In a carpenter's vice, on the other hand, the nut is fixed, and 

 the pressure applied by turning the screw. The gain is in each 

 case just the same, the difference being merely one of conve- 

 nience in applying it. 



Now we shall easily be able to see the amount of power 

 gained. If a particle be placed at the point of a screw and 

 prevented from turning with it, it will, after one revolution of 

 the screw, have been raised through a distance equal to that 

 between two threads of the screw, while any point in the cir- 

 cumference of the screw will have passed through a space equal 

 to that circumference. If, then, the power be applied at the 

 surface of the screw, it will bear the same proportion to the 

 resistance that the distance between two threads of the screw 

 does to its circumference. 



In practice, however, the power is nearly always applied at 

 the extremity of a lever, as at d in Fig. 79 a, so that it becomes a 

 combination of the lever and inclined plane. In a thumb-screw 

 the flattened part acts as a lever, and when a screw is driven by 

 a screwdriver we usually grasp it at the broadest part, and 

 thus gain a leverage. More commonly, however, a long lever is 

 put through the head of the screw. 



In all such cases we can easily ascertain the gain from the 

 fundamental principle of virtual velocities. Hence, we have the 

 following rule : Measure the circumference of the circle de- 

 scribed by the power, and divide this 

 by the distance between two threads 

 of the screw ; the result will be the 

 mechanical gain. 



Thus, if the power describe a circle 

 whose circumference is 10 feet, and 

 the distance between two threads be 

 ^ inch, we have a gain of 10 feet 

 divided by J inch, or 480. There is, 

 however, a difficulty here. We can- 

 not easily measure the actual space 

 through which the power passes, nor 

 can we calculate it with absolute 

 accuracy. It is, however, usually 

 near enough if wo take the circum- 

 ference as 3} times the diameter. 

 The fraction is more exactly 3'14159, 

 but you may always use 3f without 

 being far wrong. Thus, if the radius of a circle be 2 feet 6 

 inches, its diameter is 5 feet, and its circumference 3^ times 5 

 feet, or about 15 feet 8i inches. We see then, now, how to 

 work a question like the following : In the screw of a book- 

 binder's press there are 3 thretids to an inch, and a force of 10 

 pounds is applied to a lever 14 inches long. What force are 

 the books pressed with ? The gain is 14 X 2 X 3 divided 

 by |, which equals 264; and as the power is 10 pounds, the 

 pressure is 264 X 10, or 2,640 pounds. The real pressure is, 

 however, less than this, as a portion of the power (sometimes 

 set down at a third) is employed in overcoming friction. 

 Still, this is not altogether lost, for it prevents the screw 

 turning back when the pressure is removed. We have clearly 

 two ways of increasing our gain in the screw, we can either 

 lengthen our lever or make our threads closer ; but we soon 

 reach a practical limit to either of these, as the lever becomes 

 inconveniently long, or else the threads so narrow that they 

 are stripped off by the pressure. 



To obviate this difficulty, an arrangement known, after the 

 inventor, as Hunter's screw was planned. Fig. 80 represents 

 this. A hollow screw, A, of rather large diameter, is cut and made 

 to work through a strong fixed nut ; another screw, B, of smaller 

 diameter is fixed to the upper board of the press, a female screw 

 being cut in the interior of the first, into which this may work. 

 Supposing now that both screws have the same number of 

 threads in a foot, the board will not move at all when the upper 

 screw is turned, for the fixed screw will enter the hollow of it 

 exactly the same distance as it is depressed. But if the upper 

 one has, say 24 threads in a foot, and the other 25, the one will 



Fig. 80. 



Fig. 81. 



have moved downwards JL of a foot while the other will have 

 risen i only, and the board will be depressed by the difference 

 between the two, which is ^ of a foot. It is obvious that we 

 may diminish as much as we like the difference between the two 

 threads, without at all decreasing their strength, and the more 

 nearly they are alike, the greater power we gain. 



There is a modification of the screw, or rather a combination 



of it with the wheel and axle, 

 which is frequently used. It 

 is known as the endless 

 screw, and is represented in 

 Fig. 81. A thread is cut 

 upon an axle, which is turned 

 by a winch, and the teeth of 

 the wheel catch in the thread 

 of the screw and are thus 

 pressed forward as the winch 

 is turned, each revolution ad- 

 vancing the wheel one tooth. 

 Hence the winch must be 

 turned as many times as there 

 are teeth in the wheel in 

 order to raise the weight a. 

 distance equal to the circum- 

 ference of the axle ; and since, in the ordinary wheel and axle, 

 the power is to the weight as the radius of the wheel is to that 

 of the axle, so here, the gain is expressed by the length of the 

 arm to which the power is applied, multiplied by the number 

 of teeth in the wheel, and divided by the radius of the axle. 



In all these cases it has been supposed that the screw has 

 only one thread. Occasionally it has two, and then the gain is 

 only one-half. 



We must now give a few more examples for practice, and also 

 the answers to those in our last lesson. 

 EXAMPLES. 



1. An ascent is 120 yards long, and rises in this length 10 feet : what 

 power is required to sustain a weight of 7,236 pounds on it ? 



2. A road rises 1 foot in 25 : what strain is required to sustain a 

 wagon, weighing 1 ton, on the incline ? 



3. A wedge is 11 inches long and 2 inches thick : what resistance 

 will a pressure of 112 pounds on its head overcome ? 



4. A screw has four threads in the inch : what force must be applied 

 to a lever 1 foot long to press with a force of 3,000 pounds ? 



5. The lever of a screw is 2 feet 6 inches long, and is moved with a 

 force of 6 pounds. Required the pressure, there being three threads 

 to the inch. 



6. In Hunter's screw, if one have 10 and the other 11 threads in a 

 foot, and the lever is 1 foot 9 inches long, what is the gain ? 



7. An endless screw is driven by a 12-inch crank. The axle is 2. 

 inches in radius, and the wheel has 45 teeth. What weight will a 

 power of 8 ounces sustain ? 



ANSWEES TO QUESTIONS IN LESSON XIV. 



1. He must press with a force of 74 pounds. 



2. Six feet from the heavier boy, as there the moments about the 

 fulcrum will be equal, for 6x72 = 8 x54. 



3. 2,700 pounds. 18x60x54 



4. He must have a force of 16 -J-? pounds. The gain is -g = 

 270; and two tons divided by this give 161 pound. 



5. A little over 69 pounds. 



6. I must pull with a force of 98 pounds through a space of 4 feet. 



7. 155 pounds. The middle rope sustains 20 pounds of the weight. 



8. The front man will bear $ of the weight, or 93J pounds, the other 

 56i pounds. 



LESSON'S IN GERMAN. XLV. 



SECTION XCV. IDIOMATIC PHEASES (continued). 



SBerfl) (worth), like its equivalent in our language, is used in 

 designating the value of things ; as : DiefcS $fcrfc tft treityunbett 

 iu~rcu ujcvtlj, this horse is worth three hundred florins. When, 

 however, the amount of one's wealth is referred to, some phrase 

 like the following is employed :S'r fat etn aSermogcn sen je^n 

 Saufcnb u(t>cn ; or, @r fat jefa Saufenb ulbcn im SScnnogen, he is 

 worth ten thousand florins. 



1. 2luSfommen (a coming or getting out), with fatten, forms the 

 phrase, Gin SluSfemmcn fa6en, " to have a competency or subsis- 

 tence ; " as : 3n biefcm Sante fat bet 2lr6eit cin guteS 3lufommen. 

 ttftfjrenb cr in ten mcifien Sanbern europa'8 nut em nctfaurftigeS fat, m 

 this country the labourer has a good subsistence, while in (the) 

 most countries of Europe he has only a scanty (one). 



