CHAPTER XI.

THE MECHANICAL POWERS.

222. =Machines not Sources of Power.=--The Mechanical Powers, as they are termed, are six in number--viz., the Lever, the Wheel and Axle, the Inclined Plane, the Screw, and the Wedge. They are not, strictly speaking, powers; for, as you will see in the course of our investigation, they are merely means of applying power to advantage, and are not in reality sources of power. The true sources of power are the causes of motion treated of in § 181. The instrument or machine can not create power, and the only use of all the variety of tools and machinery is to enable us to _apply_ power in such a manner, with such a velocity, and in such a direction, as to effect the objects which we have in view. The term Mechanical Power, then, is not strictly proper as applied to those contrivances which commonly have this name; but the term is in so general use that it would not be well to alter it.

Every instrument, however simple and insignificant, and every machine, however large or complicated, is an example of some one of the six Mechanical Powers, or of a combination of them. I will proceed to consider each of these separately. In doing this certain terms will be used which I will first explain. _Power_ is the force by which a machine or instrument is moved. _Weight_ is the resistance to be overcome. If the resistance be in some other form than that of weight it is called technically by this name. So what is called Power may be in the form of weight. The _fulcrum_ is the point on which the instrument or machine is supported while it is in motion.

223. =The Lever.=--The Lever is the most simple of all the Mechanical Powers, and is therefore in universal use. Though the savage makes use of but few tools in comparison with the civilized man, he uses the lever almost constantly in some form or other. The wedge is the only one of the other Mechanical Powers that he uses to any great extent. Levers are of three kinds, which I will notice separately.

224. =Lever of the First Kind.=--In the lever of the first kind the fulcrum or prop is between the weight and the power. The common crow-bar or hand-spike is a familiar example, as seen in Fig. 157--the stone, S, or other heavy body to be moved being the weight, the stone or block of wood, F, on which the bar rests being the fulcrum, and the pressure of the hand, H, the power. The nearer the fulcrum is to the weight, or the farther is the power from the fulcrum, the greater is the force of the lever. This may be illustrated on Fig. 158. Here the short arm of the lever, as it is called, C W, is one eighth of the length of the long arm, A C. If the weight hanging at the end of the short arm be 72 pounds, a weight of 9 pounds, or the force of a hand amounting to this, will balance it at the end of the long arm. But if the power should be applied at only four times the distance from the fulcrum at which the weight is, then it would require a force of 18 pounds to balance the 72 pounds on the short arm. Similar variations can be made by altering the length of the short arm. The power and the weight will balance each other if the weight multiplied by the length of the short arm, and the power multiplied by the length of the long arm, give equal products.

225. =Scales and Steelyards.=--In the common scale-beam we have a lever, the two arms of which are equal, and therefore equal weights suspended at the ends balance. If they be not exactly equal, a heavier weight will be necessary on the shorter arm than on the longer. The inequality will injure the buyer if the prop be too near the scale in which the weights are placed, and the seller if it be too near that which holds the article to be sold. Any difference can be easily detected by changing the places of the article and the weights. Whenever cheating is practiced by the "false balance," it is of course done in a small way, to avoid any observation by the eye of the inequality of the two arms of the scale-beam, and the weight of the scale hanging from the shorter arm is made a little greater than that of the other, so that they may balance. Scales may be rendered very accurate by making the fulcrum or pivot of hardened steel, and of a wedge shape, with a sharp edge, in order to avoid friction as much as possible. The steelyard differs from the scale-beam in having the arms of different lengths. The principles on which this instrument is constructed were developed in what I said of Fig. 158. When either with the balance or the steelyard two weights balance each other the centre of the weights and the apparatus taken together is just over the fulcrum, § 195. We see in this the reason that it is necessary to have the prop near the large weight when we wish to balance it by a small one.

226. =Other Examples.=--Scissors are double levers of the first kind. The fulcrum is the rivet, the weight or the resistance to be overcome is the article to be cut, and the power is applied to the long arms of the levers by the fingers. With large shears hard substances can be cut. Even plates of iron are cut like paper by shears which are worked by a steam-engine.--Pincers are double levers. The hinge, or rivet, is the fulcrum.--The common hammer, as used in drawing nails, is a good example of the power of this kind of lever. Though crooked, it acts in the same way with a straight lever. The fulcrum is the point on the board where the hammer rests, and this is between the resistance to be moved, the nail, and the power, that is, the hand which grasps the handle.

227. =No Gain of Power in this Lever.=--I will now illustrate the truth that there is no gain or saving of power in this lever, though at first thought it would seem that there is. Let _a b_, Fig. 159, represent a lever, and _e_ its fulcrum. Let the arm _a e_ be twice as long as _e b_. A pound, therefore, suspended from _a_ will balance two pounds at _b_. If, now, when the weights are suspended, the long arm be raised so that the lever shall be in the position represented by the line _c d_, and then let go, the one pound at _c_, balancing the two pounds at _d_, will bring the lever again to the position _a b_. It will be perceived that the end of the long arm of the lever moves through the space _a c_, which is larger than _b d_, through which the end of the short arm moves, in the same time. The one-pound weight, in fact, falls two feet in raising the two-pound weight one foot, and it moves twice as far as a one-pound weight suspended at _i_ would. If a one-pound weight could raise a two-pound weight without thus moving through twice as much space we might then say that there is an actual gain of power in the lever. But it evidently makes no difference whether one pound moves through two feet or two pounds through one foot; the force is the same in both cases. For the momentum or force of a moving body is in proportion to its weight and velocity, § 201; and therefore the pound weight, moving through two feet, has as much momentum as the two-pound weight moving through one foot in the same time. The small weight does the same amount of work that the larger one would by moving twice as far in the same time as the larger, just as a boy, who carries a load half as large as a man, will do as much work as the man if he carry it twice as fast.

228. =The See-Saw.=--We see the same thing illustrated in the see-saw, Fig. 160. The man, being much heavier than the boy, is nearer the prop, and as they move up and down the boy passes over a much larger space than the man, describing an arc in a much larger circle.

229. =Archimedes's Lever.=--Archimedes said that if he could have a lever long enough and a prop strong enough he could move the world by his own weight. But he did not think how far he would have to move to do this, from the vast difference between his weight and the weight of the earth. "He would have required," says Dr. Arnot, "to move with the velocity of a cannon-ball for millions of years to alter the position of the earth by a small part of an inch."

230. =An Analogy.=--You will remember that in the case of the Hydrostatic Paradox, the Hydrostatic Bellows, and Bramah's Press (§ 131, § 132, and § 133), great effects are produced by a small power. But this small power has to execute an extensive motion in order to produce these effects. Thus, as stated in § 132, if the area of the top of the Hydrostatic Bellows be one thousand times the area of the tube, though the water poured into the tube will raise a very great weight on the bellows, the water in the tube must fall ten inches in raising the weight the hundredth part of an inch. So when the pressure of the hand on the long arm of a lever moves a great weight, as a heavy stone, the weight is moved but a little, while the extent of the hand's motion is comparatively very great.

231. =Lever of the Second Kind.=--In the second kind of lever the weight is between the fulcrum and the power, as you see in Fig. 161. The same rule of equilibrium applies here as in the case of the lever of the first kind. The 72 pounds of weight can be sustained by 8 pounds of power, because the power acts on the lever at 9 times the distance from the fulcrum that the weight does, for 1×72 = 9×8. The common wheel-barrow, Fig. 162 (p. 180), is an example of this kind of lever. The point at which the wheel presses on the ground is the fulcrum, and the weight is the load, its downward pressure from its centre of gravity being indicated at M. Of course the nearer the load is to the fulcrum the easier it is, on starting, to raise the handles. The crow-bar can be used as a lever of this kind when its point is placed beyond the weight to be raised. The chipping-knife, Fig. 163, is another example. The end, F, attached to the board, is the fulcrum, the hand pressing at P the power, and the resistance of the substance R, which is to be cut, is the weight. Nut-crackers have a similar arrangement. In shutting a door by pushing it near its edge we move a lever of this kind. The hinge is the fulcrum, and the weight is between this and the hand.

We see, then, the reason that the slight push of a hand shutting the door may even crush a finger when caught in it at the side where the hinges are. The finger is a resistance so near the fulcrum that the power moving through a great space acts upon it with immense force. The same explanation applies to the severe bite of the finger when it is caught in the hinge of a pair of tongs. The oar of a boat is a lever of this kind, the weight to be moved being the boat, which is between the power, the hand of the rower, and the fulcrum, the resisting water.

232. =Lever of the Third Kind.=--In the third kind of lever the power is between the fulcrum and the weight, as seen in Fig. 164. In the first two kinds of lever the power may be less than the weight, but in this the power must always be greater than the weight. This lever has, then, no mechanical advantage, as that expression is commonly used. Applying the same rule here as in the other levers, see what is the result. If the weight, as in Fig. 164, be 9 times as far from the fulcrum as the power is, it will require a power equal to a weight of 648 pounds to sustain a weight of 72 pounds, for 9×72=1×648.

233. =Examples.=--When a man puts his foot against the end of a ladder, and raises it by taking hold of one of the rounds, the ladder is a lever of this kind. It is evident that he spends his force upon it at a great mechanical disadvantage, for the power is applied much nearer to the fulcrum than the weight of the ladder, taken as a whole, is. If you push a door to by placing your hand very near the hinges, you do not shut it as easily as when you take hold of it at its edge. In the first case it is a lever of the third kind, and the hand moves through a small space, and therefore must exert a considerable force; while in the latter case the door is a lever of the second kind, and the hand, moving through a greater space, puts forth less force. When we use a pair of tongs we use a pair of levers of the third kind. They are an instrument in which convenience rather than power is needed. We can not grasp any thing very firmly with them because the power is so much nearer to the fulcrum than the weight to be lifted. For this reason a pinch with the ends of the tongs is nothing compared with one in the hinge. The most beautiful example of this lever we have in the moving apparatus of animals. Take, for example, the principal muscle which bends the elbow, as represented in Fig. 165 (p. 182). This comes down from the shoulder in front of the bone of the arm, and is inserted just below the elbow-joint into one of the bones of the forearm. It pulls upon the forearm very near the fulcrum, which is the elbow-joint, and so acts at a great mechanical disadvantage. The object of this arrangement is to secure quickness of movement, which is here, as in almost all muscular motions, of more importance than great strength. When great weights are lifted the fact that the muscles act at such mechanical disadvantage makes the exhibition of power wonderful.

234. =Compound Levers.=--When several levers are connected together we call the whole apparatus a compound lever. Let each of the levers in Fig. 166 be 3 inches long, the long arms being 2 inches, and the short ones 1 inch. One pound at A will, according to the rule, balance 2 at B, and 2 at B will balance 4 at C, and 4 at C will balance 8 at D. Therefore 1 pound at A will balance 8 pounds at D. And you see that an equilibrium is effected when the power is to the weight as the product of all the short arms is to the product of all the long arms. The compound lever is used in weighing heavy loads--as hay, coal, etc. You have a representation of the arrangement in Fig. 167. The load, W, stands on a platform, A B, which rests upon two levers, E D and E C. The long arms of these levers are E G and E F, and the short arms are G D and F C. The ends of the long arms press upon the fulcrum of the lever, H I. The pressure is now transmitted from the end of the long arm by the rod, I K, to a small lever, K L, where a small weight or power, P, balances the weight of the heavy load, W. The two objects secured by this arrangement are accuracy and the occupation of a small space.

235. =Wheel and Axle.=--The mechanical power next in simplicity to the lever is the Wheel and Axle. The most familiar applications of this power we see in drawing water and in raising heavy articles in stores. The principle of this power is the same as that of the lever, as may be shown in Fig. 168, which represents a section of the wheel and axle. The power, P, hangs by a cord which goes round the wheel, and the weight, W, by a cord around the axle. We may consider the power as pulling on a lever represented by A B, the long arm of which is A C, and the short arm B C. You see that the wheel and axle, then, may be viewed as a constant succession of levers, and it is therefore sometimes called the perpetual lever. And the same rule of equilibrium applies here as in the simple lever.

236. =Windlass.=--In the common windlass the power is applied to a winch or crank, D C B, Fig. 169, instead of a wheel. In estimating the power of this arrangement B C must be considered the long arm of the lever, and half of the diameter of the axle, B A, as its short arm.

237. =Capstan.=--In the capstan, represented in Figs. 170 and 171, the axle is in a vertical position. The top of it is pierced with holes, into which levers are introduced. In Fig. 170 you see the instrument as it is commonly used in moving buildings. Sometimes horse-power is applied at the ends of the levers. Great power is exerted by this instrument; but we have the same fact here as in all cases where a small force produces a great effect--the effect is slow, and the force passes over a great space in producing it. The moving of a building a foot requires many circuits of the horse around the axle. Fig. 171 gives us the capstan as it is commonly on board ship. The head of it is circular, with many holes for levers, so that many men can work together in raising a heavy anchor.

238. =Fusee of a Watch.=--In the fusee of a watch we have a wheel and axle of a peculiar construction. When we wind up a watch the chain is wound around the spiral path-way on the fusee, B, Fig. 172, and at the same time the spring is coiled up tightly in the round box, A. The spring, in gradually uncoiling itself, turns this round box around, and thus pulls upon the chain, _c_, making the fusee to revolve, and so give motion to other parts of the machinery. Now the spring, in its effort to uncoil, acts strongest at first; and therefore if the fusee were of uniform size the watch would go fastest when first wound up, and go gradually slower as it run down. This difficulty is obviated by giving the power a small wheel to pull on at first, and gradually enlarging the wheel as the spring uncoils. This is because, in order to produce a certain effect on a given weight by a power, the less the power is the longer must be the arm of the lever on which the power acts.

239. =The Pulley.=--The third mechanical power is the Pulley. Pulleys are _fixed_ or _movable_. In Fig. 173 you have a fixed pulley. There is no mechanical advantage in this pulley, for its action may be conceived of as the action of successive levers of equal arms, B F and A F, and therefore equilibrium requires an equality of the power and weight. But this pulley is often a great convenience. For example, a man can raise himself or some weight to any desired elevation, as seen in Fig. 174. It is used also in effecting descents. With two fixed pulleys a horizontal force may be used in raising a weight vertically, as seen in Fig. 175. In using a fixed pulley either one or the other of two objects is attained--applying force where we could not otherwise apply it, and changing the direction of its application.

240. =Movable Pulley.=--You have a representation of a movable pulley in Fig. 176 (p. 186). It is evident here that the force of the weight is equally divided between the cords, A B, so that the cord B, extending over the fixed pulley, needs to have a weight, P, but half of the weight W to balance it. A movable pulley is sometimes called a "runner," and a fixed pulley is commonly connected with it, in order to give the desired direction to the force. Many pulleys are often connected together in various ways, as seen in Fig. 177. It is easy to estimate in such cases the relation of the power to the weight on the principles developed in relation to the lever. If, for example, in the system of pulleys on the left, the weight be 36 pounds, the two cords of the first pulley will each sustain a weight of 18 pounds, those of the next pulley each 9 pounds, and those of the next each 4½ pounds. The weight W then will be balanced by the weight P if it weigh 4½ pounds.

241. =Inclined Plane.=--The fourth mechanical power is the Inclined Plane. This being a very simple contrivance is much used, especially when heavy bodies are to be raised only a small height, as in getting large boxes and hogsheads into stores. The mechanical advantage of the inclined plane may be illustrated on Fig. 178. The line A _c_ represents an inclined plane. If a weight be drawn up this plane it is raised only the height B _c_. A smaller power is requisite to draw the weight up the plane than to raise it perpendicularly; and the power necessary will be the less the longer the plane. A power which would balance a weight on an inclined plane would be to the weight as the height of the plane to its length. Thus if A _c_ be twice as long as B _c_, a weight of four pounds on the plane may be balanced by a two-pound weight suspended by a cord passing from the weight over the summit of the plane. A flight of stairs is an inclined plane in regard to the principle on which the ascent is effected, the projections in it being for the purpose of affording a sure footing in making the ascent or the descent. So likewise hogsheads are let down the steps of a cellar-way by ropes, and it makes no difference in the principle of the operation whether the steps have or have not planks laid along them. It is supposed that the immense stones in the pyramids and other massive Egyptian structures were put into their position by means of the inclined plane. Roads, when they are not level, are inclined planes, and the steeper the inclination the more power is required to draw a load up the road. Great mistakes were formerly made in carrying roads too frequently over high hills. Besides failing to take advantage of the principles of the inclined plane, in many cases the horse in going over a hill passes over quite as much space as he would if the road were made to go round the base of the hill, and sometimes even more. If the hill were a perfect hemisphere, a road over it would be just equal in length to a road around its base to the opposite point.

242. =The Wedge.=--This is the fifth of the mechanical powers. It may be considered as two inclined planes placed with their bases together, as seen in Fig. 179. Indeed, sometimes the wedge has one side only inclined, it being only half of the ordinary wedge. The difference between the inclined plane and the wedge in operation is, that in the first the inclined plane is fixed, and the weight is made to move up along its surface, while in the latter the weight, that is, the resistance, is stationary, and the surface of the plane is made to move along upon it. The power of the wedge is estimated just as the power of the inclined plane is, that is, by comparing the thickness of the wedge with the length of its side. The less the thickness of the wedge compared with its length, obviously the more powerful is the wedge as a penetrating instrument. The wedge is used for splitting blocks of wood and stone, for producing great pressures, for raising heavy bodies, etc. All cutting and piercing instruments, knives, razors, axes, needles, pins, nails, etc., act on the principle of the wedge.

243. =The Screw.=--This is the sixth mechanical power. The principle of it is essentially that of the inclined plane. The "thread" running around the screw is an inclined plane which is spiral instead of straight, and so is also the corresponding part in the nut an inclined plane running in the opposite direction. In the common screw the nut is fixed, and the screw is made to play up and down in it; but sometimes the screw is fixed, and the nut is made to play around it. The screw acts like a wedge, and has the same relation to a straight wedge that a road winding up a hill has to a straight road of the same length and rise. Especially does the comparison hold when the screw is forced into wood; the wedge goes straight into the wood, but the edge of the screw's thread enters the wood spirally.

To estimate the force of the screw we compare the length of one turn of the thread around it with the height to which the thread rises in going round. Let _a b_, Fig. 180, represent one turn of the thread, and _b c_ the height to which it goes. It is clear from the figure that the principle which applies to the inclined plane and to the wedge applies here also. As the less is the height of the plane the easier it is for a weight to be drawn up it; and as the less is the depth of the wedge the less is it resisted; so, also, the less the height of the turn of the screw's thread the easier is it to move the screw, and the greater is the force which it exerts. Hence the prodigious power of a screw with a thread which rises very slowly in its spiral turns. Screws are much used when great pressure is required, as in pressing oils and juices from vegetable substances, in compressing cotton into bales, in bringing together with firm grasp the jaws of the vice, etc. In turning the screw a bar is used, so that we have in this instrument the combined advantages of the screw and the lever. That you may have some idea of the power of these two instruments acting together I will suppose a case. Let the weight to be raised by a screw be 10,000 pounds. Let a turn of the screw be 10 inches long, and the rise be but one inch. Then, so far as the screw is concerned, the power requisite to raise the 10,000 pounds will be 1000--the ratio of the height of the thread's turn to its length. But the power of the lever is yet to be estimated. Let the length of the lever, passed through the head of the screw so that it is equal on each side, be 30 inches. The diameter of the screw is about three inches, or one-tenth of the diameter of the circle described by the end of the lever. It will now take but a power of 100 pounds to raise the weight, the ratio of the radius of the screw to half the length of the lever.

244. =Truly but Three Mechanical Powers.=--The Wheel and Axle, you have seen, is merely a modification of the Lever, and the Wedge and the Screw are modifications of the Inclined Plane. The Mechanical Powers are, then, in reality but three--the Lever, Pulley, and Inclined Plane. And these are the elements of all machinery, from the simplest tool that is used for the most common purposes to the most complicated and powerful engine which the ingenuity of man ever designed. The principle upon which a pin is shaped is identical with that of the wedge, by which large masses are cleft in two; and the instrument by which the finest textures are cut by delicate fingers is arranged on the same principle with those varied contrivances by which immense weights are raised by a comparatively small power, viz., the principle of the lever.

245. =Friction in Machinery.=--You have seen, as we have proceeded, that the Mechanical Powers, though thus named, do not generate power. So far from this, there is really a loss of power in their use, chiefly from friction. In raising a weight, for example, directly by the hand, there is no loss from this cause; but if you use a pulley you have the friction of the cord upon it, and a loss of power in proportion to the amount of friction. In some cases the loss of power from this cause is so great as to call for a considerable variation from such calculations as we have made in this chapter in regard to the relations of power and weight in machinery. In the operations of the screw friction has a great influence in diminishing the power of the instrument.

246. =The Real Advantages of the Mechanical Powers.=--If there is then no saving, but a loss of power in tools and machinery, what, let us inquire, are their advantages?

If one man can do alone by the aid of some instrument what would otherwise require the exertion of many men, though he be slow in doing it, yet it is a great advantage. Thus one man can with a lever move a stone which perhaps it would require thirty men to move without it, and though it take him thirty times as long, it saves him the trouble of getting a company of men to help him. So if a man can raise his goods by a wheel and axle to the upper loft of his store, though he raise them slower than several men would lift them directly by ropes, it is an advantage to him, as it saves the hiring of a company of laborers. A few men by a capstan can raise an anchor which could be raised without it only by a large company of men.

Another advantage often is that there may be intervals of rest in applying the force without any loss. This is obvious in the case of the pulley, but still more so in the case of the screw. It is friction in both these cases which enables the workman to rest. It saves to him all that he has gained by opposing any tendency to slip back. We see the same thing in the wedge. When this is driven into wood, it remains because it is prevented from returning by the friction of the wood against its sides. It is the same cause which holds a nail in its place, and opposes any effort to draw it out. In driving the wedge the workman can have as long intervals as he pleases between his blows, because friction saves all that is gained. This effect is very well exemplified in the capstan, Fig. 170. It requires but little exertion of the man who sits there to hold the rope, because the few turns of it around the axle prevent its slipping easily.

A third advantage which often attends the use of tools and machines is that force may be made to produce motion at various distances, in various directions, and in various degrees of velocity. Thus as to distance, a man standing on the ground can raise a weight to the top of a house by a pulley. So, also, a water-wheel may by the connections of machinery produce motion at considerable distances from it. Then as to direction, horizontal motion may be converted into vertical, rotary into straight, etc. The velocity of motion is generally varied by cog-wheels. Thus a wheel of 60 cogs revolving once in a minute, playing on a wheel of 10 cogs, will make it revolve once in 6 seconds.

Another advantage of tools and machines is that they secure a better mode of applying power than we otherwise could have. Thus when several men are pulling on a rope much power is lost by their pulling irregularly, a difficulty which is removed by the pulley. The same can be said of applying pressure by the screw. One man presses more steadily, and therefore more effectually, than fifty men would without the screw. The arrangements of tools and machines are so made as to provide convenient ways of applying our strength. An instrument, for example, for moving a weight by hand is so shaped as to hold the weight well, and also to afford a good handle for the hand to grasp. The common claw hammer is a very good illustration. We grasp the nail by an iron claw, with the handle we can apply not merely the force of the hand, but that of the whole arm, and then we have the immense lever power of the instrument. We have a good illustration of convenience in an instrument, in what is called a Lewis, represented in Fig. 181. It is used for raising blocks of stone in building. It has three parts, A B C. It is used in this way: A hole is made in the upper part of the block of stone to be raised in shape like the instrument; then A and C are inserted, and B is pushed in between them. With the ring, D, bolted through the instrument the stone is raised to its place by the ordinary machinery. The principle of the instrument, you see, is that of the wedge.

247. =Man a Tool-Making Animal.=--Though there is no actual saving of power in the tools and machines which man uses, yet so great are the advantages which he reaps from them, that more than two thousand years ago a philosopher thought that man could not be better distinguished from brutes than by calling him a tool-making animal. If the distinction was so striking in the time of Aristotle, when tools and machines were so few in number and so rudely contrived, and so few of the sources of power were appropriated by man to his use, how much more striking is it now, with all the variety and perfection of instruments and machinery, and with the ever-extending appropriation of the sources of power furnished by the elements. The power which air and water and gravitation give is applied constantly with more and more variety and effect; and the appropriation of that mighty source of power, steam, is wholly a modern invention.