Fig. 63. Fig. 63 represents a simple pulley, with a single fixed wheel. In other forms of the machine, the wheel moves up and down, with the weight. The pulley is arranged among the simple mechanical powers; but when several are connected, the machine is called a system of pulleys, or a compound pulley. One of the most obvious advantages of the pulley is, its enabling men to exert their own power, in places where they cannot go themselves. Thus, by means of a rope and wheel, a man can stand on the deck of a ship, and hoist a weight to the topmast. By means of two fixed pulleys, a weight may be raised upward, while the power moves in a horizontal direction. The weight will also rise vertically through the same space that the rope is drawn horizontally. พ Fig. 64. a Fig. 64 represents two fixed pulleys, as they are arranged for such a purpose.` In the erection of a lofty edifice, suppose the upper pulley to be suspended to some part of the building; then a horse, pulling at the a, would raise the weight w vertically, as far as he went horizontally. rope In the use of the wheel of the pulley, there is no mechanical advantage, except that which arises from removing the friction, and diminishing the imperfect flexibility of the rope. In the mechanical effects of this machine, the result would be the same, did it slide on a smooth surface with the same ease that its motion makes the wheel revolve. The action of the pulley is on a different principle from that of the wheel and axle. A system of wheels, as already explained, acts on the same principle as the compound lever. What is a simple pulley? What is a system of pulleys, or a compound pulley? What is the most obvious advantage of the pulley? How must two fixed pulleys be placed, to raise a weight vertically, as far as the power gors horizontally? What is the advantage of the wheel of the pulley ? But the mechanical efficacy of a system of pulleys, is derived entirely from the division of the weight among the strings em ployed in suspending it. In the use of the single fixed pulley, there can be no mechanical advantage, since the weight rises as fast as the power descends. This is obvious by fig. 63; where it is also apparent that the power and weight must be exactly equal, to balance each other. In the single moveable pulley, fig. 65, the same rope passes from the fixed point a, tc the power p. It is evident, here, that the weight is supported equally by the two parts of the string between which it hangs. Therefore, if we call the weight w ten pounds, five pounds will be supported by one string, and five by the other. The pow er, then, will support twice its own weight, so that a person pulling with a force of five pounds at p, will raise ten pounds at w. The mechanical force, therefore, in respect to the power, is as two to one. In this example, it is supposed there are only two ropes, each of which bears an equal part of the weight. If the number of ropes be increased, the . weight may be increased with the same power; or the power may be diminished in proportion as the number of ropes is increased. In fig. 66, the number of ropes sustaining the weight is four, and therefore, the weight may be four times as great as the power. This principle must be evident, since it is plain that each rope sustains an equal part of the weight. The weight may therefore be considered as divided into four parts, and each part sustained by one rope. In fig. 67, there is a system of pulleys represented, in which the weight is sixteen times the power. How does the action of the pulley differ from that of the wheel and axle ? Is there any mechanical advantage in the fixed pulley? What weight at p, fig. 65, will balance ten pounds at w? Suppose the number of ropes to be increased, and the weight increased, must the power be increased also? 2 Fig. 67. 1 The tension of the rope d, e, is evidently equal to the power, p, because it sustains it: d, being a moveable pulley, must sustain a weight equal to twice the power; but the weight which it sustains, is the tension of the second rope, d, c. Hence the tension of the second rope is twice that of the first, and, in like manner, the tension of the third rope is twice that of the second, and so on, the weight being equal to twice the tension of the last rope. Suppose the weight w, to be six teen pounds, then the two ropes, 8 and 8, would sustain just 8 pounds each, this being the whole weight divided equally between them. The next two ropes, 4 and 4, would evidently sustain but half this whole weight, because the other half is already sustained by a rope, fixed at its upper end. The next two ropes sustain but half of 4, for the same reason; and the next pair, 1 and 1, for the same reason, will sustain only half of 2. Lastly, the power p, will balance two pounds, because it sustains but half this weight, the other half being sustained by the same rope, fixed at its upper end. It is evident, that in this system, each rope and pulley which is added, will double the effect of the whole. Thus, by adding another rope and pulley beyond 8, the weight w might be 32 pounds, instead of 16, and still be balanced by the same power. In our calculations of the effects of pullies, we have allowed nothing for the weight of the pullies themselves, or for the friction of the ropes. In practice, however, it will be found, Suppose the weight, fig. 66, to be 32 pounds, what will each rope bear? Explain fig. 67, and show what part of the weight each rope sustains, and why 1 pound at p, will balance 16 pounds at w. Explain the reason why each additional rope and pulley will double the effect of the whole, or why its weight may be double by that of all the others, with the same power. that nearly one third must be allowed for friction, and that the power, therefore, to actually raise the weight, must be about one third greater than has been allowed. The pulley, like other machines, obeys the law of virtual velocities, already applied to the lever and wheel. Thus, "in a system of pullies, the ascent of the weight, or resistance, is as much less than the descent of the power, as the weight is greater than the power." If, as in the last example, the weight is 16 pounds, and the power 1 pound, the weight will rise only one foot, while the power descends 16 feet. In the single fixed pulley, the weight and power are equal, and consequently, the weight rises as fast as the power descends. With such a pulley, a man may raise himself up to the mast head by his own weight. Suppose a rope is thrown over a pulley, and a man ties one end of it round his body, and takes the other end in his hands. He may raise himself up, because, by pulling with his hands, he has the power of throwing more of his weight on that side than on the other, and when he does this, his body will rise. Thus, although the power and the weight are the same individual, still the man can change his centre of gravity, so as to make the power greater than the weight, or the weight greater than the power, and thus can elevate one half his weight in succession. The fourth simple The Inclined Plane. This power consists of a plain, smooth surface, which is inclined towards, or from the earth. It is represented by fig. 68, where from a to b is the inclined plane; the line from d to a, is its height, and that from b to d, its base. A board, with one end on the ground, and the other end resting on a block, becomes an inclined plane. This machine, being both useful and easily constructed, is in very general use, especially where heavy bodies are to be raised only to a small height. Thus a man, by means of an In compound machines, how much of the power must be allowed for the friction? How may a man raise himself up by means of a rope and single fixed pulley? What is an inclined plane? inclined plane, which he can readily construct with a board, or couple of bars, can raise a load into his wagon, which ten men could not lift with their hands. The power required to force a given weight up an inclined plane, is in a certain proportion to its height, and the length of its base, or, in other words, the force must be in proportion to the rapidity of its inclination. Fig. 69. р The power p, fig. 69, pulling a weight up the inclined plane, from c to d, only raises it in a perpendicular direction from e to d, by acting along the whole length of the plane. If the plane be twice as long as it is high, that is, if the line from c to d be double the length of that from e to d, then one pound at p will balance two pounds any where between d and c. It is evident, by a glance at this fig., that were the base, that is, the line from e to c, lengthened, the height from e to d being the same, that a less power at p, would balance an equal weight any where on the inclined plane; and so, on the contrary, were the base made shorter, that is, the plane more steep, the power must be increased in proportion. W Fig. 70. Suppose two inclined planes, fig. 70, of the same height, with bases of different lengths; then the weight and power will be to each other as the length of the planes. If the length from a to b, is two feet, and that from b to c, one foot, then two pounds at d will balance four pounds at w, and so in this proportion, whether the planes be longer or shorter. a The same principle, with respect to the vertical velocities of the weight and power, applies to the inclined plane, in common with the other mechanical powers. On what occasions is this power chiefly used? Suppose a man wants to load a barrel of cider into his wagon, how does he make an inclined plane for this purpose? To roll a given weight up an inclined plane, to what must the force be proportioned? Explain fig. 69. If the length of the long plane, fig. 70, be double that of the short one, what must be the proportion between the power and the weight? |