« НазадПродовжити »
Leibnitz and others call these mechani- there is in every combination of bodies, cal curves transcendental, and dissent from and in every single body which may be Des Cartes in excluding them out of geome- considered as made up of a number of try. Leibnitz found a new kind of trans. lesser bodies, a centre of pressure or gracendental equations, whereby these curves vity. This discovery Archimedes applied are defined ; but they do not continue con- to particular cases, and pointed out the mestantly the same in all points of the curve, thod of finding the centre of gravity of as algebraic ones do,
plane surfaces, whether bounded by a paMECHANICS, is the science which rallellogram, a triangle, a trapesium, or a treats of the laws of the equilibrium and parabola. See Centre of gravity. motion of solid bodies; of the forces by Galileo, towards the close of the sixteenth which bodies, whether animate or inani- century, made many important discoveries mate, may be made to act upon one ano. on this subject. In a small treatise on statics, ther; and of the means by which these may he proved that it required an equal power to be increased, so as to overcome such as are raise two different bodies to altitudes, in most powerful. As this science is closely the inverse ratio of their weights, or that connected with the arts of life, and particu- the same power is requisite to raise ten larly with those which existed even in the pounds to the height of one hundred feet, rudest ages of society, the construction of and twenty pounds fifty feet. It is imposmachines must have been practised long be- sible for us to follow this great inan in all fore the theory upon which their principles his discoveries. In his works, which were depend conld have been understood. Hence published early in the seventeenth century, we find in use among the ancients, the le he discusses the doctrine of equable mover, the pulley, the crane, the capstan, and tions in various theorems, containing the many other simple machines, at a period different relations between the velocity of when mechanics, as a science, were un- the moving body, the space which it de. known. In the remains of Egyptian archi.' scribes, and the time employed in its detecture are beheld the most surprising scription. He treats also of accelerated marks of mecbanical genius. The eleva. motion, considers all bodies as heavy, and tion of immense and ponderous masses of composed of heavy parts, and infers that stone to the tops of their stupendous fa- the total weight of the body is proportional brics, must have required an accumulation to the number of the particles of which it of mechanical power, which is not in the is composed. On this subject he reasons possession of modern architects. We are in the following manner: “As the weight jodebted to Archimedes for the foundation of a body is a power always the same in of this science: he demonstrated, that when quantity, and as it constantly acts without a balance with unequal arms is in equilibrio, interruption, the body must be continually by means of two weights in its opposite receiving from it equal impulses in equal scales, these weights must be reciprocally and successive instants of time. When the proportional to the arms of the balance. body is prevented from falling, by being From this general principle the mathemati- placed on a table, its weight is incessantly cian might have deduced all the other pro. impelling it downwards; but these impulses perties of the lever, but he did not follow are destroyed by the resistance of the ta. the discovery through all its consequences. ble, which prevents it from yielding to In demonstrating the leading property of them. But where the body falls freely, the lever, he lays it down as an axiom, that the impulses which it perpetually receives if the two arms of the balance are equal, are perpetually accumulating, and remain the weights must be equal, to give an equi- in the body unchanged in every respect, librinm. Reflecting on the construction of except the diminution which they expe. the balance, which moved upon a fulcrum, rience from the resistance of the air : bence he perceived that the two weights exerted it follows, that a body falling freely is unithe same pressure on the fulcrum as if they formly accelerated, or receives equal increhad both rested on it. He then advanced ments of velocity in equal times. He then another step, and considered the sum of demonstrated that the time in which any these two weights as combined with a third, space is described by a motion uniformly and then the sum of the three, with a fourth, accelerated from rest, is equal to the time and so on, and perceived that in every such in which the same space would be describ. combination the fulcrum must supported by an uniform equable motion, with half their united weight; and, therefore, that the final velocity of the accelerated motion, and that in every motion uniformly accele- sustained. The points of suspension are rated from rest, the spaces described are those points where the weights really are, in the duplicate ratio of the times of de- or from which they hang freely. The scription: after this he applied the doctrine power and the weight are always supposed to the ascent and descent of bodies on in- to act at right angles to the lever, except clined planes. For a more particular ac- it be otherwise expressed. The lever is count we may refer to Dr. Keil's “ Phy- distinguished into three sorts, according to sics." Under the articles Centre of gra. the different situations of the fulcrum, or vity, DYNAMICS, ELASTICITY, Force, prop, and the power, with respect to each GRAVITATION, Motion, &c. will be other. 1. When the prop is placed be. found much relating to the doctrine of tween the power and the weight, as in steelmechanics; we shall therefore in this place . yards, scissars, pincers, &c. 2. When the chiefly treat of the mechanical powers, prop is at one end of the lever, the power which are usually reckoned six in number: at the other, and the weight between them, viz. the lever ; the wheel and axis, or, as it as in cutting knives fastened at, or near is frequently called," the axis in peritro- the point of the blade ; also in oars moving chio ;" the pulley; the inclined plane; the a boat, the water being the fulcrum. 3. wedge; and the screw. Some writers on When the prop is at one end, the weight this subject reduce the six to two, riz. the at the other, and the power applied belever, and the inclined plane; the polley, tween them, as in tongs, sheers, &c.; and wheel and axis being, in their estima. The lever of the first kind is principally tion, assemblages of the lever; and the used for loosening large stones ; or to raise wedge and the screw being modifications great weights to small heights, in order to of the inclined plane.
get ropes under them, or other means of When two forces act against each other, raising them to still greater heights: it is by the intervention of a machine, the one the most common species of lever. ABC is denominated the power, and the other (Plate I, Mechanics, fig. 1.) is a lever of this the weight. The weight is the resistance kind, in which F is the fulcrum, A the end to be overcome, or the effect to be pro. at which the power is applied, and C the duced. The power is the force, whether end where the weight acts. To find when animate or inanimate, which is employed an equilibrium will take place between the to overcome that resistance, or to produce power and the weight, in this as well as in the required effect.
every other species of lever, we must ob. The power and weight are said to ba. serve that when the momenta, or quantities lance each other, or to be in equilibrio, of force, in two bodies are equal, they will when the effort of the one to produce mo. balance each other. Now, let us consider tion in one direction, is equal to the effort when this will take place in the lever. of the other to produce it in the opposite Suppose the lever AB, fig. 2, to be turned directio; or when the weight opposes that on its axis, or fulcrum, so as to come into degree of resistance which is precisely re- the situation DC; as the end D is farthest quired to destroy the action of the power. from the centre of motion, and as it has The power of a machine is calculated when moved through the arch AD in the same it is in a state of equilibrium. Having dis- time as the end B moved through the arch covered what quantity of power will be re- BC, it is evident that the velocity of AB quisite for this purpose, it will then be ne. must have been greater than that of B. cessary to add so much more, viz. one. But the momenta being the products of the fourth, or, perhaps, one-third, to overcome quantities of matter multiplied into the velothe friction of the machine, and give it mo- cities, the greater the velocity, the less the tion.
quantity of matter to obtain the same proThe lever is the simplest of all machines, duct. Therefore, as the velocity of A is and is a straight bar of iron, wood, or other the greatest, it will require less matter to material, supported on, and moveable about produce an equilibrium than B. a prop called the fulcrum. In the lever, Let us now examine how much more there are three circumstances to be prin- weight B will require than A, to balance. cipally attended to: 1. The fulcrum, or As the radii of circles are in proportion to prop, by which it is supported, or on which their circumferences, they are also proporit turns as a centre of motion : 2. The tionate to similar parts of them; therefore, power to raise and support the weight: 3. as the arches, AD, CB, are similar, the The resistance or weight to be raised or radius, or arm, DE, bears the same proportion to E C that the arch A D bears to CB. weight to change places, so that the power But the arcbes A D and C B represent the may be betweeu the weight and the prop, velocities of the ends of the lever, because it will become a lever of the third kind; they are the spaces which they moved over in which, that there may be a balance bein the same time; therefore the arms D E tween the power and the weight, the in. and EC may also represent these velocities. tensity of the power must exceed the inHence, an equilibrium will take place, when tensity of the weight just as much as the the length of the arm A E, multiplied into distance of the weight from the prop exthe power A, shall equal EB, multiplied ceeds the distance of the power. Thus, into the weight B ; and consequently, that let E, fig. 4, be the prop of the lever E F, the shorter E B is, the greater must be the and W, a weight of one pound, placed three weight B; that is, the power and the times as far from the prop as the power P weight must be to each other inversely, as acts at F, by the cord going over the fixed their distances from the fulcrum. Thus, pulley D: in this case, the power must be suppose A E, the distance of the power equal to three pounds, in order to support from the prop, to be twenty inches, and the weight of one pound. To this sort of EB, the distance of the weight from the lever are generally referred the bones of a prop, to be eight inches, also the weight man's arm; for when lie lifts a weight by to be raised at B to be five pounds; then the hand, the muscle that exerts its force the power to be applied at A, must be two to raise that weight, is fixed to the bone pounds; because the distance of the weight about one tenth part as far below the from the fulcrum eight, multiplied into the elbow as the land is. And the elbow being weight tive, makes forty; therefore twenty, the centre round which the lower part of the distance of the power from the prop, the arm turns, the muscle must therefore must be multiplied by two, to get an equal exert a force ten times as great as the product; which will produce an equilie weight that is raised. As this kind of lever brium.
is a disadvantage to the moving power, it The second kind of lever, when the is used as little as possible; but in some weight is between the fulcrum and the power, cases it cannot be avoided; such as that is represented by fig. 3, in which A is the of a ladder, which being fixed at one end, fulcrom, B the weight, and C the power. is by the strength of a man's arms reared The advantage gained by this lever, as in against a wall. the first, is as great as the distance of the What is called the hammer-lever, differs power from the prop exceeds the distance in nothing but its form from a lever of the of the weight from it. Thus, if the point tirst kind. Its name is derived from its use, a, on which the power acts, be seven times that of drawing a nail out of wood by a as far from A as the point b, on which the hammer. Suppose the shaft of a hammer weight acts, then one pound applied at C to be five times as long as the iron part will raise seyen pounds at B. This lever which draws the nail, the lower part resting shews the reason why two men carrying a on the board, as a fulcrum; then, by pullburden upon a stick between them, bearing backwards the end of the shaft, a man shares of the hurden which are to one ano- will draw a pail with one-fifth part of the ther in the inverse proportion of their dis- power that he must tise to pull it out with tances from it.
a pair of pincers; in which case, the pail It is likewise applicable to the case of would move as fast as his hand; but with two horses of unequal strength to be so the hammer, the hand moves five times as yoked, as that each horse may draw a part much as the nail, by the time that the nail proportionable to his strength; which is is drawn out. Hence it is evident, that done by so dividing the beam they pull, in every species of lever there will be an that the point of traction may be as much equilibrium, when the power is to the weight nearer to the stronger horse than to the as the distance of the weight from the fulweaker, as the strength of the former ex. crum is to the distance of the power from ceeds that of the latter. To this kind of the fulcrum, In experiments with the lever may be reduced rudders of ships, lever we take care that the parts are per. doors turning upon hinges, &c. The hinges fectly balanced before the weights and being the centre of motion, the hand ap- powers are applied. The bar, therefore, plied to the lock is the power, while the has the short end so much thicker than the door is the weight to be moved.
long arm, as will be sufficient to balance If in this lever we suppose the power and it on the prop,
If several levers be combined together balance the weiglit 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 instru. ported 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 ex. C, 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, componnd- 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 eqnal 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, shonld 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 equal to one-fifth of the weight w the fulcrum and the points of snspension 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 Lend 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; conseqnenty a power the centre of gravity of the beam beinnequal to one-fourth of one-fifth, that is, one mediately below the 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 manner, 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 onght 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 if 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; npon 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 DX, 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 anequal in the these sub isions, 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 asthe 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 peritrosidered, 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 forms. 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- palled, 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 enake 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 tlic 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 longer ar to that of the wheel. Thus, suppose the