Examples of the First Kind of Lever.

125

The common fire-poker is a simple lever. It rests on the bar of the grate as its prop, and displaces or breaks the caked coal behind as the resistance.

The mast of a ship, with sails set upon it, acts sometimes as a long lever, having the wind filling the sails as the power, turning upon the centre of buoyancy of the vessel as the fulcrum, and lifting the weight at the centre of gravity as the resistance. For this reason lofty sails make a ship heel or lean over greatly, and if used in open boats, are dangerous. In some of the islands in the Eastern and Pacific Oceans, for the sake of sailing swiftly, boats are used so extremely narrow and sharp, that to counteract the overturning tendency of their large sails, they have an outrigger or projecting plank to windward, on the extremity of which one or more of the crew may sit as a balance.

No instance of the lever that has the prop between the forces is more interesting than the weighing-beam or balance: whether with equal arms, forming the common scale-beam, or with unequal arms, forming the steelyard.

The Common Balance or Weighing Beam.

K

Fig. 5.

233. We have seen (223) why quantities of matter, attached at equal distances from the prop, must be equal to each other in order to balance. A lever, therefore, which enables us to place masses thus exactly in opposition to each other, and which turns easily on its axis, becomes a weighing-beam. Of this the annexed figure shows a common form. The axis or pivot at c is sharpened below, wedge-like, that the beam may turn easily, and that its centre of motion may be nicely determined;-in a delicate balance for philosophical purposes, the axis is almost sharp as a knife edge, and rests on some hard smooth surface of support, so as to turn with even the thousandth part of a grain. The scales also of a weighing beam are suspended on sharp edges, to facilitate motion, and to determine nicely the points cf suspension. If the two arms of a beam be not of exactly equal length, a smaller weight at the end of the longer will balance a greater weight at the end of the shorter. An excess of a tenth or other proportion in the length of a beam-arm, to which merchandize is attached, would cheat the buyer of exactly that proportion of his purchase. This case may be detected instantly, by changing the places of the two things balanced; for so, the lighter would be at

126

The Common Balance-the Steelyard.

the short arm, and would then appear doubly light. A beam intended for delicate purposes, and required, therefore, to turn easily, must have its centre of gravity very near the axis on which the beam turns; for if otherwise, the beam will be in the predicament of a ship with the ballast either too high or too low in the former case, when once inclined, it would fall over, and could not recover itself; in the latter, it would tend too strongly to remain horizontal, and therefore would be less free to move. The proper situation of the centre of gravity is a little below the axis or line of support, so that the beam may return with sufficient readiness from any state of inclination to its horizontal position of rest.

234. It is possible but troublesome to weigh very accurately even with a weighing-beam which is not itself accurately made, provided it has very free motion. First balance very nicely in one scale the substance to be weighed by sand or other matter put into the other; then remove the substance, and put weights into the same scale, until a perfect balance is again produced. Such weights will be the exact equivalent or weight of the substance, however unequal the arms of the balance may be. A projecting rod or plank or branch of a tree, may thus be made to answer the purpose of a weighing-beam, by first attaching the substance to be weighed to its extremity, and observing minutely how far it bends, and then trying what weights will bend it as much.

235. The steelyard is a lever of the first kind with unequal arms, and any weight, as b, on the long arm (fig. 36), will balance a weight,

that the steelyard when bare hangs horizontally, from having a greater mass of matter on the short arm to counterbalance the long slender arm from which the shifting weight hangs. When this is not the case, a corresponding allowance has to be made.

The Chinese, who are so remarkable for the simplicity to which they have reduced all their common implements, weigh any small objects by a delicate pocket steelyard. It is a rod of wood or ivory, about six inches long, with a silk cord passing through it at a par.

Examples of the Second Kind of Lever.

127

ticular part, to serve as a fulcrum, and with a sliding weight on the long arm, and a small scale attached to the short one.

236. The following are examples of levers of the second kind with both forces on the same side of the prop or fulcrum, and where the more distant force acts as the power.

A common wheelbarrow is a lever of this sort, in using which a man bears as much less than the whole weight of the load, as the centre of gravity of the load is nearer to the axle of the wheel than to his hands.

a d

с

b

When two porters carry, on a pole, a b (fig. 37), a load placed midway between them, as at c, each bears a half, for the pole becomes a lever, of which each porter is a fulcrum, as regards the other; but if the load be at d, the man at a bears three-quarters of its weight, and the man at ʼn only one-quarter, the latter

Fig. 37.

being three times as far from the weight as the former.

Two horses drawing a plough act from the ends of a cross bar, of which the middle usually is hooked to the plough. The horses must thus pull equally, to keep the bar directly across. When, on heavy land, three horses are yoked, and two of them are made to draw from one end of the bar, it must be attached to the plough by a hook, not at its middle, but half as far from one end of it as from the other.

The oar of a boat is a lever of this kind, where singularly the purpose of fulcrum is momentarily served by the unstable water.

The common nut-crackers may cited be as another instance, by the lever-power of which a person can break a shell many times stronger than he could break with the bare fingers.

The consideration of this kind of lever explains why a finger caught near the hinge of a shutting door is so severely crushed. The momentum of the door acts by a comparatively long lever, upon a feeble resistance placed very near the fulcrum.

The circumstance of the branch of a tree giving way, when in autumn overloaded with fruit, or in winter with snow, also exhibits the action of this kind of lever. The resistance is the cohesion of the upper side of the branch, and the fulcrum is the part below which is last broken.

237. The following are examples of the lever of the third

kind, where the two forces are on the same side of the pivot,

128

Examples of the Third Kind of Lever.

but where that nearest to the pivot acts as the power. In

this kind, the power is greater than the resistance.

The hand of a man, who pushes open a gate while standing near the hinges, moves through much less space than the end of the gate, and hence must act with great force.

When a man uses the common fire-tongs, the ends move much farther than his fingers, and therefore with less strength. No one fears a pinch thus given with the ends of the fire-tongs.

Beautiful instances of this modification of lever are exhibited in the limbs of animals. In these the distant extremities, as the hand or foot, require to have great range and freedom of motion, without clumsiness of the limb; and the object has been attained most perfectly by the tendons or ropes which move the limbs being attached near to the joints, which constitute the pivots or fulcra of the bone-levers.

In the human arm, the deltoid muscle, which forms the cushion of the shoulder, by contracting its fibres less than an inch, can raise the hand a yard or more; and of course, if it overcome a force of many pounds placed there, it must itself be acting with a force very intense. What extraordinary strength of muscle, then, is displayed by a man who lifts another man at the end of an extended arm !

How powerful, again, must be the wing-muscles of birds, which, by this kind of action, sustain themselves in the sky for many hours together! The great albatross, with wings extended fourteen feet, is seen in the stormy solitudes of the Southern Ocean, accompanying ships for whole days, without ever resting on the water.

A contraction of about one inch of the glutei muscles of the hip gives to the human step a length of four feet.

While the erroneous opinion prevailed, that machines increase power, instead of, as they do, merely accommodating forces to purposes, this last kind of lever, where a great force acting through a short distance is made to give great extent of motion and other benefits, was viewed by many as an unprofitable contrivance, and was called the losing lever.

238. It is almost unnecessary to say, that the same rule of comparative velocities ascertains the relations required between power and resistance, where a combination of levers is used, as where there is only one. If a lever which makes one balance four, be applied to work a second lever which does the same, one pound at the long arm of the first will balance sixteen pounds at the short arm of the second, and would balance sixty-four at the short arm of a third, and so on.

The Bent Lever.- Wheel and Axle.

d

129

239. The general rule for the lever, that a force may be less intense the farther it is from the pivot, supposes always that the force acts at right angles, or directly across the lever; for if there be any obliquity, there is a corresponding diminution of effect, as explained under the head of resolution of forces, at page 57. For instance, one pound at b on the end of the long arm of the bent lever, b d a, has influence only as if it were acting directly at the end of a shorter horizontal arm, d f, because its weight does not act directly across b d; and the two-pound weight at a acts only as if it were on a horizontal arm, de; now e being only half as far from the centre as ƒ, two pounds at a, in the position of the lever here shown, would just balance the one pound at b. In every case, the exact influence of weights is known by referring them to places directly above or below them, on a supposed horizontal lever, ef.

Fig. 38.

The bent-lever balance, in common use for letter-weighing, is made on the principle here explained. It has on one side a heavy weight as at a, and on the other side a scale attached at b; and the weight of anything put into the scale is indicated by the position then assumed by the lever, marked by the point at which it cuts an arc of divisions placed behind it. In any common weigh-beam, the point of suspension of the scales being a little below the axis of motion of the beam, there is to a certain degree the property of the bent-lever balance, enough to require notice in very nice experiments.

"The Wheel and Axle."

240. The next of the simple machines is the wheel and axle (fig. 39); d marks a wheel, and e an axle affixed to it. In turning together, the wheel would take up, or throw off, as much more rope than the axle, as its circumference or diameter were greater than that of the axle. If the proportions were as four to one, a pound at b, hanging from the circumference of the wheel, would balance four pounds at a, hanging from the opposite side of the axle. The proportions are equally in

a

Fig. 39.