Philosophy in Sport Made Science in Earnest / Being an Attempt to Illustrate the First Principles of Natural Philosophy by the Aid of Popular Toys and Sports

by John Ayrton Paris

The Lever being the simplest of all the mechanic powers, is in general considered the first. It is an inflexible rod or bar of any kind, so disposed as to turn on a pivot or prop, which is always called its fulcrum. It has the weight or resistance to be overcome attached to some one part of its length, and the power which is to overcome that resistance applied to another; and, since the power, resistance, and fulcrum admit of various positions with regard to each other, so is the lever divided into three kinds or modifications, distinguished as the first, second, and third kinds of lever. That portion of it which is contained between the fulcrum and the power, is called the acting part or arm of the lever; and that part which is between the fulcrum and resistance, its resisting part or arm.

In the lever of the first kind, the fulcrum is placed between the power and the resistance. A poker, in the act of stirring the fire, well illustrates this subject; the bar is the fulcrum, the hand the power, and the coals the resistance to be overcome. Another common application of this kind of lever is the crow-bar, or hand-spike, used for raising a large stone or weight. In all these cases power is gained in proportion as the distance from the fulcrum to the power, or part where the men apply their strength, is greater than the distance from the fulcrum to that end under the stone or weight. A moment’s reflection will show the rationale of this fact; for it is evident that if both the arms of the lever be equal, that is to say, if the fulcrum be midway between the power and weight, no advantage can be gained by it, because they pass over equal spaces in the same time; and, according to the fundamental principle already laid down, as advantage or power is gained, time must be lost; but, since no time is lost under such circumstances, there cannot be any power gained. If, now, we suppose the fulcrum to be so removed towards the weight, as to make the acting arm of the lever three times the length of the resisting arm, we shall obtain a lever which gains power in the proportion of three to one, that is, a single pound weight applied at the upper end will balance three pounds suspended at the other. A pair of scissors consists of two levers of this kind, united in one common fulcrum; thus the point at which the two levers are screwed together is the fulcrum; the handles to which the power of the fingers is applied, are the extremities of the acting part of the levers, and the cutting part of the scissors are the resisting parts of the levers; the longer, therefore, the handles, and the shorter the points of the scissors, the more easily you cut with them. A person who has any hard substance to cut, without any knowledge of the theory, diminishes as much as possible the length of the resisting arms, or cutting part of the scissors, by making use of that part of the instrument nearest the screw or rivet. Snuffers are levers of a similar description; so are most kind of pincers, the power of which consists in the resisting arm being very short in comparison with the acting one.

In the lever of the second kind, the resistance or weight is between the fulcrum and the power. Numberless instances of its application daily present themselves to our notice; amongst which may be enumerated the common cutting knife, used by last and patten makers, one end of which is fixed to the work-bench by a swivel-hook. Two men carrying a load between them, by one or more poles, as a sedan chair, or as brewers carrying a cask of beer, in which case either the back or front man may be considered as the fulcrum, and the other as the power. Every door which turns upon its hinges is a lever of this kind; the hinges may be considered as the fulcrum, or centre of motion; the whole door is the weight to be moved, and the power is applied to that side on which the handle is usually fixed. Nut-crackers, oars, rudders of ships, likewise fall under the same division. The boat is the weight to be moved, the water is the fulcrum, and the waterman at the oar is the power. The masts of ships are also levers of the second kind, for the bottom of the vessel is the fulcrum, the ship the weight, and the wind acting against the sail is the moving power. In this kind of lever the power or advantage is gained in proportion as the distance of the power is greater than the distance of the weight from the fulcrum; if, for instance, the weight hang at one inch from the fulcrum, and the power acts at five inches from it, the power gained is five to one; because, in such a case, the power passes over five times as great a space as the weight. It is thus evident why there is considerable difficulty in pushing open a heavy door, if the hand is applied to the part next the hinges, although it may be opened with the greatest ease in the usual method. In the third kind of lever, the fulcrum is again at one of the extremities, the weight or resistance at the other; and it is now the power which is applied between the fulcrum and resistance. As in this case the weight is farther from the centre of motion than the power, such a lever is never used, except in cases of absolute necessity, as in the case of lifting up a ladder perpendicularly, in order to place it against a wall. The man who raises it cannot place his hands on the upper part of the ladder; the power, therefore, is necessarily placed much nearer the fulcrum than the weight; for the hands are the power, the ground the fulcrum, and the upper part of the ladder the weight. The use of the common fire-tongs is another example, but the circumstance that principally gives this lever importance is, that the limbs of men and animals are actuated by it; for the bones are the levers, while the joints are the fulcra, and the muscles which give motion to the limbs, or produce the power, are inserted and act close to the joints, while the action is produced at the extremities; the consequence of such an arrangement is, that although the muscles must necessarily exert an enormous contractile force to produce great action at the extremities, yet a celerity of motion ensues which could not be equally well provided for in any other manner. We may adduce one example in illustration of this fact. In lifting a weight with the hand, the lower part of the arm becomes a lever of the third kind; the elbow is the fulcrum; the muscles of the fleshy part of the arm the power; and as these are nearer to the elbow than the hand, it is necessary that their power should exceed the weight to be raised. The disadvantage, however, with respect to power, is more than compensated by the convenience resulting from this structure of the arm; and it is no doubt that which is best adapted to enable it to perform its various functions. From these observations it must appear, that although this arrangement must be mentioned as a modification of the lever, it cannot, in strictness, be called a mechanical power; since its resisting arm is in all cases, except one, longer than the acting arm, and in that one case is equal to it, on which account it never can gain power, but in most instances must lose it.

The Wheel and Axle is the next mechanical power to be considered; it must be well known to every reader who has seen a village well; for it is by this power that the bucket is drawn up, although in such cases, instead of a wheel attached to the axle, there is generally only a crooked handle, which answers the purpose of winding the rope round the axle, and thus raising the bucket, as may be seen in the engraving at the head of our third chapter. It is evident, however, that this crooked handle is equivalent to a wheel; for the handle describes a circle as it revolves, while the straight piece which is united to the axle corresponds with the spoke of a wheel. This power may be resolved into a lever; in fact, what is it but a lever moving round an axle? and always retaining the effect gained during every part of the motion, by means of a rope wound round the butt end of the axle; the spoke of the wheel being the long arm of the lever, and the half diameter of the axle its short arm. The axle is not in itself a mechanical power, for it is as impotent as a lever whose fulcrum is in the centre; but add to it the wheel, and we have a power which will increase in proportion as the circumference of the wheel exceeds that of the axle. This arises from the velocity of the circumference being so much greater than that of the axle, as it is farther from the centre of motion; for the wheel describes a great circle in the same space of time that the axle describes a small one; therefore the power is increased in the same proportion as the circumference of the wheel is greater than that of the axle. Those who have ever drawn a bucket from a well by this machine, must have observed, that as the bucket ascended nearer the top the difficulty increased: such an effect must necessarily follow from the views we have just offered; for whenever the rope coils more than once the length of the axle, the difference between its circumference and that of the wheel is necessarily diminished. To the principle of the wheel and axle may be referred the capstan, windlass, and all those numerous kinds of cranes which are to be seen at the different wharfs on the banks of the river Thames. It is scarcely necessary to add that the force of the windmill depends upon a similar power. The treadmill furnishes another striking example. The wheel and axle is sometimes used to multiply motion, instead of to gain power, as in the multiplying wheel of the common jack, to which it is applied when the weight cannot conveniently have a long line of descent; a heavy weight is in this case made to act upon the axle, while the wheel, by its greatest circumference, winds up a much longer quantity of line than the simple descent of the weight could require, and thus the machine is made to go much longer without winding than it otherwise would do.

The Pulley is a power of very extensive application. Every one must have seen a pulley; it is a circular and flat piece of wood or metal, with a string which runs in a groove round it. Where, however, this is fixed, it cannot afford any power to raise a weight; for it is evident that, in order to raise it, the power must be greater than the weight, and that if the rope be pulled down one inch, the weight will only ascend the same space; consequently, there cannot be any mechanical advantage from the arrangement. This, however, is not the case where the pulley is not fixed. Suppose one end of the rope be fastened to a hook in the ceiling, and that to the moveable pulley on the rope a cask be attached, is it not evident that the hand applied to the other extremity of the rope will sustain it more easily than if it held the cask suspended to a cord without a pulley? Experience shows that this is the fact, and theory explains it by suggesting that the fixed hook sustains half the weight, and that the hand, therefore, has only the other half to sustain. The hook will also afford the same assistance in raising the weight as in sustaining it; if the hand has but one half the weight to sustain, it will also have only one half the weight to raise; but observe, says Mrs. Marcet, that in raising the weight, the velocity of the hand must be double that of the cask; for, in order to raise the weight one inch, the hand must draw each of the strings one inch; the whole string is therefore shortened two inches, while the weight is raised only one. Pulleys then act on the same principle as the lever, the deficiency of strength of the power being compensated by its superior velocity. It will follow, from these premises, that the greater the number of pulleys connected by a string, the more easily the weight is raised, as the difficulty is divided amongst the number of strings, or rather of parts into which the string is divided by the pulleys. Several pulleys, thus connected, form what is called a system, or tackle of pulleys. They may have been seen suspended from cranes, to raise goods into warehouses, and in ships to draw up the sails.

The Inclined Plane is a mechanic power which is seldom used in the construction of machinery, but applies more particularly to the moving or raising of loads upon slopes or hills, as in rolling a cask up or down a sloping plank into or out of a cart or cellar, or drawing a carriage up a sloping road or hill, all which operations are performed with less exertion than would be required if the same load were lifted perpendicularly. It is a power which cannot be resolved into that of the lever: it is a distinct principle, and those writers who have attempted to simplify the mechanical powers, have been obliged to acknowledge the inclined plane is elementary. The method of estimating the advantage gained by this mechanical power is very easy; for just as much as the length of the plane exceeds its perpendicular height, so much is the advantage gained; if, for instance, its length be three times greater than its height, a weight could be drawn to its summit with a third part of the strength required for lifting it up at the end; but, in accordance with the principle so frequently alluded to, such a power will be at the expense of time, for there will be three times more space to pass over. The reason why horses are eased by taking a zig-zag direction, in ascending or descending a steep hill, will appear from the preceding account of the action of the inclined plane, because in this way the effective length of the inclining surface is increased while its height remains the same.

The Wedge is rather a compound, than a distinct mechanical power; since it is composed of two inclined planes, and in action frequently performs the functions of a lever. It is sometimes employed in raising bodies; thus the largest ship may be raised to a small height by driving a wedge below it; but its more common application is that of dividing and cleaving bodies. As an elevator, it resembles exactly the inclined plane; for the action is obviously the very same, whether the wedge be pushed under the load, or the load be drawn under [sic] the wedge. But when the wedge is drawn forward, the percussive tremor excited destroys, for an instant, the adhesion or friction at its sides, and augments prodigiously the effect. From this principle chiefly is derived the power of the wedge in rending wood and other substances. It then acts besides as a lever, insinuating itself into the cleft as fast as the parts are opened by the vibrating concussion. To bring the action of the wedge, therefore, under a strict calculation, would be extremely difficult, if not impossible. Its effects are chiefly discovered by experience. All the various kinds of cutting tools, such as axes, knives, chisels, saws, planes, and files, are only different modifications of the wedge.

The Screw is a most efficient mechanic power, and is of great force and general application. It is in reality nothing more than an inclined plane formed round a cylinder, instead of being a continued straight line. Its power is, therefore, estimated by taking its circumference, and dividing this by the distance between any two of its threads; for what is taking the circumference of a screw, but another mode of measuring the length of the inclined plane which wraps round it? and taking the distance between one thread and the next to it, is but measuring the rise of that inclined plane in such length; and from the properties of the inclined plane, it follows, that the closer the threads of a screw are together in proportion to its diameter, the greater will be the power gained by it.

Note 24, p. [165].--The cycloid.

A cycloid is a peculiar curve line; and is described by any one point of a circle as it rolls along a plane, and turns round its centre; thus, for instance, the nail on the felly of a cart-wheel traces a cycloid in the air as the wheel proceeds. This curve is distinguished by some remarkable properties, the most important of which is that mentioned in the text, viz. that any body moving in such a curve, by its own weight, or swing, will pass through all distances of it in exactly the same time; and it is for such a reason that pendulums are made to swing in cycloids, in order that they may move in equal times, whether they go through a long or a short part of the same curve. Where the arc described is small, a portion of the circle will be sufficiently accurate, because it will be seen that such an arc will not deviate much from an equal portion of a cycloidal curve.