Science for the School and Family, Part I. Natural Philosophy

by Worthington Hooker

Fig. 158.

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.

Fig. 159.

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.

Fig. 160.

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.