and the velocity must be calculated by the space passed through in a given time. Now, if I examine the lever, (Fig. 18. 20.) and consider it as an inflexible bar turning on a centre of motion, it is evident, that the same time is used for the motion both of the weight and the power, but the spaces passed over are very different; that which the power passes through, being as much greater as that passed by the weight, as the length of the distance of the power from the prop, is greater than the distance of the weight from the prop; and the velocities being as the spaces passed in the same time, must be greater in the same proportion. Consequently, the velocity of P, the power, multiplied into its weight, will be equal to the smaller velocity of w, multiplied into its weight, and thus their momenta being equal, they will balance one another. Father. This applies to the first and second kind of lever; what do you say to the third? Charles. In the third, the velocity of the power P, (Fig. 21.) being less than that of the weight w, it is evident, in order that their momenta may be equal, that the weight acting at P, must be as much greater than that of w, as A c is less than B C, and then they will be in equilibrio. Father. The second mechanical power is the Wheel and Axis, which gains power in proportion as the circumference of the wheel is greater. nearest to him, and then the weaker must be considered as the power. Again, two horses may be so yoked to a carriage that each shall draw a part proportional to his strength, by dividing the beam in such a manner, that the point of traction, or drawing, may be as much nearer to the stronger horse than to the weaker, as the strength of the former exceeds that of the latter. We will now describe the third kind of lever. In this the prop or fulcrum c (Fig. 21.) is at one end, the weight w at the other, and the power P is applied at в somewhere between the prop and weight. Charles. In this case, the weight being further from the centre of motion than the power, must pass through more space than it. Father. And what is the consequence of that ? Charles. That the power must be greater than the weight and as much greater as the distance of the weight from the prop exceeds the distance of the power from it, that is, to balance a weight of three pounds at A, there will require the exertion of a power P, acting at B, equal to five pounds. Father. Since then a lever of this kind is a disadvantage to the moving power, it is but seldom used, and only in cases of necessity; such as in that of a ladder, which being fixed at onc end against a wall or other obstacle, is by the strength of a man's arm raised into a perpendicular situation. But the most important application of this third kind of lever, is manifest in the structure of the limbs of animals, particularly in those of man; to take the arm as an instance; when we lift a weight by the hand, it is effected by means of muscles coming from the shoulder blade, and terminating about one-tenth as far below the elbow as the hand is now the elbow being the centre of motion round which the lower part of the arm turns, according to the principle just laid down, the muscles must exert a force ten times as great as the weight that is raised. At first view this may appear a disadvantage, but what is lost in power is gained in velocity, and thus the human figure is better adapted to the various functions it has to perform. CONVERSATION XVII. Of the Wheel and Axis. Father. Well, Emma, do you understand the principle of the lever, which we discussed so much at large yesterday? than that of the axis; this machine may be referred to the principle of the lever; A B (Plate 111. Fig. 23.) is the wheel, Q D its axis, and if the circumference of the wheel be eight times as great as that of the axis, then a single pound r, will balance a weight w, of eight pounds. Charles. Is it by an instrument of this kind that water is drawn from those deep wells so common in many parts of the country? Father. It is; but as in most cases of this kind only a single bucket is raised at once, there requires but little power in the operation, and therefore, instead of a large wheel as A B, an iron handle fixed at в is made use of, which, you know, by its circular motion, answers the purpose of a wheel. Charles. I once raised some water by a machine of this kind, and I found, that as the bucket ascended nearer the top the difficulty increased. f Father. That must always be the case, where the wells are so deep as to cause, in the ascent, the rope to coil more than once the length of the axis, because the advantage gained is in proportion as the circumference of the wheel is greater than that of the axis; so that if the circumference of the wheel be 12 times greater than that of the axis, 1 pound applied at the former will balance 12 hanging at the latter; but by the coiling of the rope round the axis, the difference between the circumference of the wheel, and that of the axis, continually diminishes, consequently the advantage gained is less every time a new coil of rope is wound on the whole length of the axis; this explains why the difficulty of drawing the water, or any other weight, increases as it ascends nearer the top. Charles. Then by diminishing the axis, or by increasing the length of the handle, advantage is gained? Father. Yes, by either of those methods you may gain power, but it is very evident, that the axis cannot be diminished beyond a certain limit, without rendering it too weak to sustain the weight; nor can the handle be managed, if it be constructed on a scale much larger than what is commonly used. Charles. We must, then, have recourse to the wheel with spikes standing out of it at certain distances from each other to serve as levers. Father. You may by this means increase your power according to your wish, but it must be at the expense of time, for you know that a simple handle may be turned several times, while you are pulling the wheel round once. To the principle of the wheel and axis, may be referred the capstan windlass, and all those numerous kinds of cranes, which are to be seen at the different wharves on the banks of the Thames. Rous'd from repose, aloft the sailors swarm, |