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If several levers be combined together balance the weighit at the point S of the in such a manner, as that a weight being lever A. This method of combining levers appended to the first lever, may be sup- is frequently used in machines and instruported by a power applied to the last, as ments, and is of great service, either in in fig. 5, which consists of three levers of obtaining a greater power, or in applying the first kind, and is so contrived, that a it with more convenience. power applied at the point L of the lever The balance, an instrument of very exC, may sustain a weight at the point $ of tensive use in comparing the weights of the lever A, the power must here be to the bodies, is a lever of the first kind, whose weight, in a ratio, or proportion, compound- arms are of equal length. The points from ed of the several ratios, which those powers which the weights are suspended being that can sustain the weight by the help of equally distant from the centre of motion, each lever, when used singly and apart from will move with equal velocity ; consequently the rest, have to the weight. For instance, if equal weights be applied, their momenta if the power which can sustain the weight will be equal, and the balance will remain W by the help of the lever A, be to the in equilibrio. In order to have a balance weight as 1 to 6; and if the power which as perfect as possible, it is necessary to can sustain the same weight, by the lever attend to the following circumstances : 1. B alone, be to the weight as 1 to 4; and The arms of the beam ought to be exactly if the power wbich could sustain the same equal, both as to weight and length. 2. weight by the lever C, be to the weight as The points from which the scales are sus. 1 to 5; then the power which will sustain pended, should be in a right line, passing the weight by help of the three levers through the centre of gravity of the beam; joined together, will be to the weight in a for by this, the weights will act directly proportion consisting of the several propor against each other, and no part of either tions multiplied together, of 1 to 5, 1 to 4, will be lost, on account of any oblique and 1 to 5; that is as 1:5 X 4 X 5, or direction. 3. If the fulcrum be placed in of 1 : 100. For since, in the lever A, a the centre of gravity of the beam, and if power eqnal to one-fifth of the weight W the fulcrum and the points of suspension pressing down the lever at L, is sufficient be in the same right line, the balance will to balance the weight, and since it is the have no tendency to one position more same thing whether that power be applied than another, but will rest in any position to the lever A at L, or the lever B at S, it may be placed in, whether the scales be the point S bearing on the point L, a power on or off, empty or loaded. If the centre equal to one-fifth of the weight P, being of gravity of the beam, when level, be applied to the point S of the lever B, will immediately above the fulcrum, it will support the weight; but one-fourth of the
overset by the smallest action; that is, the saine power being applied to the point L end which is lowest will descend; and it of the lever B, and pushing the same np. will do this with more swiftness, the higher ward, will as effectually depress the point the centre of gravity be, and the less the S of the same lever, as if the whole power points of suspension be loaded. But if were applied at S; conseqnently' a power the centre of gravity of the beam be imequal to one-fourth of one-fifth, that is, one mediately below tlie fulcrum, the beam twentieth of the weight P, being applied to will not rest in any position but when level; the point L of the lever B, and pushing up and if disturbed from that position, and the same, will support the weight : in like then left at liberty, it will vibrate, and at mavner, it matters not whether that force last come to rest on the level. In a babe applied to the point L of the lever B, or lance, therefore, the fulcrum ought always to the point S of the lever C, since, if to be placed a little above the centre of s be raised, L, which rests on it, must be gravity. Its vibrations will be quicker, raised also; but one-fifth of the power ap- and its horizontal tendency stronger, the plied at the point L of the lever C, and lower the centre of gravity, and the less pressing it downwards, will as effectually the weight upon the points of suspension. raise the point S of the same lever, as it the . 4. The friction of the beam upon the axis whole power were applied at S, and pushed ought to be as little as possible ; because, up the same; consequently a power equal should the friction be great, it will require to one-fifth of one-twentieth, that is, one- a considerable force to overcome it; upon hundredth part of the weight P, being ap- ' which account, though one weight should a plied to the point L of the lever C, will little exceed the other, it will not prepon
derate, the excess not being sufficient to DX. If this arm be divided into as many overcome the friction, and bear down equal parts as it will contain, each equal the beam. 5. The pivots, which form the to G D, the single weighi P (which we may axis or fulcrum, should be in a straight line, suppose to be one pound) will serve for and at right angles to the beam. 6. The weighing any thing as heavy as itself, or as arms should be as long as possible, relative many times heavier as there are divisions in ly to their thickness, and the purposes for the arm D X, or any quantity between its which they are intended, as the longer they own weight and that quantity. As for are the more sensible is the balance. They example, if P be one pound, and placed should also be made as stiff and inflexible at the first division 1 in the arm D X, it as possible ; for if the beam be too weak, will balance one pound in the scale at W; it will bend, and become untrue. 7. The if it be removed to the second division at rings, or the piece on which the axis bears, 2, it will balance two poinds in the scale ; should be bard and well polished, parallel if to the third, three pounds; and so on to to each other, and of an oval form, that the the end of the arm D X. If any of these axis may always keep its proper bearing, integral divisions be subdivided into as or remain always at the lowest point. 8. many equal parts as a pound contains If the arms of a balance be unequal, the ounces, and the weight P be placed at any weights in equipoise will be unequal in the of these subdivisions, so as to counte:poise same proportion. The equality of the arms what is in the scale, the pounds and odd is of use, in scientific pursuits, chiefly in ounces therein will by that means be as. the making of weights by bisection. A certained. In the Danish and Swedish balance with unequal arms will weigh as steel-yard, the body to be weighed, and the accurately as another of the same work- constant weight, are fixed at the extremities manship with equal arms, provided the of the steel-yard, but the point of suspenstandard weight itself be first counter- sion or centre of motion moves along the poised, then taken out of the scale, and the lever till the equilibrium takes place. The thing to be weighed be put into the scale, centre of motion therefore shews the weight and adjusted against the counterpoise. Or, of the body. when proportional quantities only are con- The wheel and axle, or axis in peritro. sidered, the bodies under examination may chio, is a machine much used, and is made be weighed against the weights, taking care in a variety of forins. It consists of a wheel always to put the weights in the same with an axle fixed to it, so as to turn round scale; for then, though the bodies may not with it; the power being applied at the cir. be really equal to the weights, yet their cumference of the wheel, the weight to be proportions amongst each other will be the raised is fastened to a rope which coils same as if they had been accurately so. 9. round the axle. Very delicate balauces are not only useful A B (fig. 7.) is a whcel, and C D an axle in nice experiments, but are likewise much fixed to it, and which moves round with it. more expeditious than others in conjmon If the rope which goes round the wheel be weighing. If a pair of scales, with a cer- pulled, and the wheel turned once round, it tain load, be barely sensible to one-tenth is evident that as much rope will be drawn of a grain, it will require a considerable off as the circumference of the wheel; but time to ascertain the weight to that degree while the wheel turns once round, the axle of accuracy, because the turn must be ob-. turns once round; and consequently the served several times over, and is very small. rope by which the weight is suspended will But if no greater accuracy were required, wind once round the axis, and the weight and scales were used, which would turn will be raised tlmrough a space equal to the with one-hundredth of a grain, a tenth of a circumference of the axis. The velocity of grain more or less would inake so great a
the power, therefore, will be to that of the difference in the turn, that it would be seen weight, as the circumference of thic wheel immediately.
to that of the axis. In order, therefore, The statera, or Roman steel-yard, is a that the power and the weight may be in lever of the first kind, and is used for find- equilibrio, the power must be to the weight ing the weights of different bodies, by one as the circumference of the wheel to that single weight placed at different distances of the axis. Circles being to each other as from the prop or centre of motion D, fig. 6. their respective diameters, the power is to For, the shorter arm D G is of such a weight the weight, as the diameter also of the axis as exactly to counterpoise the lovger arm to that of the wheel. Thus, suppose the
diameter of the wheel to be eight inches, ference of the axle of the first, and as the and the diameter of the axis to be one inch; former is much greater than the latter, it then one ounce acting as the power P, will is evident that the first wheel must go balance eight ounces as a weight W; and round as many times more than the second, a small additional force will canse the wheel as the circumference of the second wheel to turn with its axis, and raise the weight ; exceeds that of the first asle. In order to and for every inch which the weight rises, a balance here, the power must be to the the power will fall eight inches.
weight, as the product of the circum. The wheel and axis may be considered as ferences, or diameters of the two axles mul. a kind of perpetual lever, (fig. 8.) of which tiplied together, is to the circumferences or the fulcrum is the centre of the axis, and diameters of the two wheels. This will the long and short arms the diameter of the become sufficiently clear, if it be considered wheel and the diameter of the axis. From as a compound lever, which was explained this it is evident, that the longer the wheel, above. Instead of a combination of two and the smaller the axis, the stronger is the wheels, three or four wheels may work in power of this machine; but then the weight each other, or any number; and by thus in must rise slower in proportion. A capstan creasing the number of wheels, or by prois a cylinder of wood, with boles in it, into portioning the wheels to the axis, any dewhich are put bars, vr levers, to turn it gree of power may be acquired. To this round ; these are like the spokes of a wheel sort of engine belong all cranes for raising withont the rim. Sometimes the axis is great weights ; and in this case the wheel turned by a winch fastened to it, which, in may have cogs all round it, instead of this respect, serves for a wheel, and is more handles ; and a small lanthorn, or trundle, powerful, in proportion to the largeness of may be made to work in the cogs, and be the circle it describes, compared with the turned by a winch; which will make the diameter of the axle. When the parts of power of the engine to exceed the power of the axis differ in thickness, and weights are the man who works it, as much as the num. suspended at the different parts, they may ber of revolutions of the winch exceeds be snstained by one and the same power ap- those of the axle, when multiplied by the plied to the circumference of the wheel, excess of the length of the winch above the provided the product arising from the mul- length of the semi-diameter of the axle, tiplication of the power into the diameter added to the semi-diameter or half thick of the wheel, be equal to the sum of the ness of the rope, by which the weight products arising from the multiplication of is drawn up. See CRANE. the several weights into the diameters of The construction of the main-spring-box those parts of the axis from which they are of the fusee of a watch round which the suspended. In considering the theory of chain is coiled will illustrate the principle the wheel and axle, we have supposed the of the wheel and axis. The box may be rope that goes round the axle-to have no considered as the wheel, and the fusee the sensible thickness ; but as in practice this axle or pinion to which the chain communi. cannot be the case, if it is a thick rope, or if cates the motion of the box. The power there be several folds of it round the axis, resides in the spring wound round an axis you must measure to the middle of the in the centre of the box, and the weight is outside rope to obtain the diameter of the applied to the lower circumference of the axis, for the distance of the weight from the fusee. As the force of the spring is greatest centre is increased by the coiling up of the when newly wound up, and gradually derope.
creases as it unwinds itself, it is necessary If teeth are cut in the circumference that the fusee should bave different radii, so of a wheel, and if they work in the teeth of that the chain may act upon the smallest another wheel of the same size as fig. 9. it is part of the fusee when its force is greatest, evident that both the wheels will revolve in and upon the largest part of the fusee when the same time; and the weight appended the force is least, for the equable motion of to the axle of the wheel B, will be raised in the watch requires that the inequality in the the same time as if the axle had been fixed action of the spring should be counteracted to the wheel A. But if the teeth of the se- so as to produce an uniform effect. cond wheel be made to work in teeth made The pulley is a small wheel turning on an in the axle of the first, as at fig. 10. as every axis, with a drawing rope passing over it; part of the circumference of the second the small wheel is usually called a sheeve, wheel is applied successively to the circum- and is so fixed in a box, or block, as to be
moveable round a pin passing through its vided by the number of lower pullies; centre. Pullies are of two kinds ; fixed, that is, as twice the number of lower pollies which do not move ont of their places ; and is to 1, so is the weight suspended to the moveable, which rise and fall with the power. But if the extremity C (fig. 14.) weight.
be tixed to the lower block, it will sustain When a pulley is fixed, as Plate II. Me. half as much as a pulley; consequently here chanics, fig. 11. two equal weights suspended the rule will be, as twice the number of to the ends of a rope passing over it will pullies adding unity is to 1, so is the weight balance each other, for they stretch the to the power. These rules hold good, whatrope equally, and if either of them be ever may be the namber of pullies in the pulled down through any given space, the blocks. If, instead of one rope going round other will rise through an equal space in the all the pullies, the rope belonging to each same time; and consequently, as the velo- pulley be made fast at top, as in fig. 15, a cities of both are equal, they must balance different proportion between the power and each other. This kind of pulley, therefore, the weight will take place. Here it is evi. gives no mechanical advantage ; but its use dent, that each pulley doubles the power ; consists in changing the direction of the thus, if there are two pullies, the power power, and sometimes enabling it to be ap. will sustain four times the weight; if three plied with more convenience. By it, a man pullies, eight times the weight ; if four pulmay raise a weight to any point, as the top lies, sixteen times ; and so on: that is, the of a building, without moving from the power P of 11b. will sustain a weight W of place he is in ; whereas, otherwise, he 161b. would have been obliged to ascend with the When pullies in blocks are placed perweight; it also enables several men toge. pendicularly under each other, on separate ther to apply their strength to the weight by pins, they occupy considerable space, and means of the rope. The moveable pulley would not in general answer ; it is, there. represented at A (fig. 12.) is fixed to the fore, common to place all the pullies in weight W, and rises and falls with it. In each block on the same pin, by the side of comparing this to a lever, the fulcrum must each other, as in fig. 16. but the advantage be considered as at A, the weight acts upon and rule for the power, are the same here as the centre c, and the power is applied at the in fig. 13 and 14. A pair of blocks with the extremity of the lever D. The power, rope fastened round it, is commonly called therefore, being twice as far from the ful- a tackle. crum as the weight is, the proportion be- To avoid, in a great measure, the friction tween the power and weight, in order to of several pullies running on different pivots, balance each other, must be ás 1 to 2. Mr. James White, a very able mechanic, Whence it appears, that the use of this pulley invented the concentric pulley, (fig. 17.) for doubles the power, and that a man may which he obtained a patent. O and R are raise twice as muel by it as by his strength two brass pullies in which grooves are cut; alone. Again, every moveable polley hangs round these a cord is passed, by which by two ropes equally stretched, and which means the two answer the same purpose of must, consequently, bear equal parts of the so many distinct pullies as there are grooves; weight ; but the rope A B being made fast and the advantage gained is found by at B, half the weight is sustained by it, and doubling the number of grooves in the the other part of the rope, to which the lower block. In this case the advantage power is applied, has bnt half the weight to gained is 12, that is, a power of 121b. will support; consequently the advantage gained balance a weight of 144.
The concen by this pulley is as 2 to 1. When the upper tric pulley removes very considerably the and tixed block contains two pullies, which shaking motion of the common pulley as only turn npon their axis, and the lower well as the friction. moveable block contains also two, which The inclined plane is of very great 'nse in not only turn on their axis, but rise with the rolling up heavy bodies, such as casks, weight F (fig. 13.) the advantage gained is wheel-barrows, &c. It is formed by placing as 4 to 1. For each lower pulley will be boards, or earth, in a sloping direction. The acted upon by an equal part of the weight; force with which a body descends upon an and because in each pulley that moves with inclined plane, is to the force of its absolute the weight a double increase of power is gravity, by which it would descend perpengained, the force by which F may be sus- dicularly in free space, as the height of the tained will be equal to balf the weight di. plane is to its length. For suppose tbe
plane A B (fig. 18,) to be parallel to the ho- B F the length of one of its sides ; and OF rizon, the cylinder C will keep at rest on is its sharp edge, which is entered into the any part of the plane where it is laid. "If wood intended to be split, by the force of a the plane be placed perpendicularly, as hammer or mallet striking perpendicularly A B, (fig. 19,) the cylinder C will descend on its back. Thus, A B (fig. 22.) is a wedge with its whole force of gravity, because the driven into the cleft CED of the wood plane contributes nothing to its support or
FG. When the wood does not cleave at hindrance; and therefore it would require any distance before the wedge, there will a power equal to its whole weight to keep be an equilibrium between the power im. it from descending. Let A B (fig. 10.) be a pelling the wedge downward and the replane parallel to the horizon, and A Da sistance of the wood acting against the two plane inclined to it; and suppose the whole sides of the wedge, when the power is to length A D to be four times as great as the the resistance as half the thickness of the perpendicular D B. In this case, the cy wedge at its back is to the length of either linder E will be supported upon the plane of its sides ; because the resistance then D A, and kept from rolling, by a power acts perpendicularly to the sides of the equal to a fourth part of the weight of the wedge. But when the resistance on each cylinder ; therefore a weight may be rolled side acts parallel to the back, the power up this inclined plane, hy a third part of the that balances the resistances on both sides power which would be snfficient to draw it will be, as the length of the whole back of up by the side of an upright wall. It must the wedge is to double its perpendicular also be evident, that the less the angle of height. elevation, or the gentler the ascent is, the When the wood cleaves at any distance greater will be the weight which a given before the wedge (as it generally does) the power can draw up; for the steeper the in- power impelling the wedge will not be to clined plaue is, the less does it support of the resistance of the wood as the length on the weight ; and the greater the tendency the back of the wedge is to the length of which the weight has to roll, consequently both its sides, but as half the length of the the more difficult for the power to support back is to the length of either side of the it: the advantage gained by this mechanical cleft, estimated from the top or acting part power, therefore, is as great as its length of the wedge. For, if we suppose the exceeds its perpendicular height. To the wedge to be lengthened down from the top inclined plane may be reduced all hatchets, CE, to the bottom of the cleft at D, the chisels, and other edge-tools.
same proportion will hold ; namely, that The inclined plane, when combined with the power will be to the resistance as half other machinery, is often of great use in the the length of the back of the wedge is to elevation of weights : it has been likewise the length of either of its sides : or, which made use of in the late Duke of Bridge amounts to the same thing, as the whole water's canal. After this canal has extended length of the back is to the length of both about 40 miles on the same level, it is joined the sides. The wedge is a very great meto a subterraneous navigation about 12 miles chanical power, since not only wood, but long, by means of an inclined plane, and even rocks, can be split by it; which it this subterraneous portion is again connect- would be impossible to effect by the lever, ed by an inclined plane with another por. wheel, and axle, or pulley ; for the force of tion 100 feet above it. This plane is a the blow, or stroke, shakes the cohering stratum of stone which slopes one foot in parts, and thereby makes them separate four, and is about 450 feet long. The more easily. boats are conveyed from one level to another The sixth and last mechanical power is by means of a windlass, so that a loaded the screw; which cannot properly be called boat descending along the plane turns the a simple machine, because it is never used axis of the windlass, and raises an empty without the application of a lever or winch boat.
to assist in turning it ; and then it becomes The fifth mechanical power or machine is a compound engine of a very great force, the wedge ; which may be considered as either in pressing the parts of bodies closer two equally inclined planes, joined together together, or in raising great weights. It at their bases ; then D G (fig. 21.) is the may be conceived to be made by cutting a whole thickness of the wedge at its back piece of paper, A B C (fig. 23.) into the A B GD, where the power is applied; form of an inclined plane, or half wedge ; E F is the depth or height of the wedge; and then wrapping it ronnd a cylinder (fig. VOL. IV.