Robert S. Ball

“ Many of our illustrations will be designedly drawn from very commonplace sources: by this means I would try to impress upon you that mechanics is not a science that exists in books merely, but that it is a study of those principles which are constantly in action about us. Our own bodies, our houses, our vehicles, all the implements and tools which are in daily use—in fact all objects, natural and artificial, contain illustrations of mechanical principles. ”

“ Here we learn the important truth that two forces not in the same direction can be replaced by a single force. The process is called the composition of forces, and the single force is called the resultant of the two forces. ”

“ We are now to consider the important subject of the application of mechanical principles to structures. These are fixtures, while machines are adapted for motion; a roof or a bridge is a structure, but a crane or a screw-jack is a machine. Structures are employed for supporting weights, and the mechanical powers give the means of raising them.
344. A structure has to support both its own weight and also any load that is to be placed upon it. Thus a railway bridge must at all times sustain what is called the permanent load, and frequently, of course, the weight of one or more trains. ”

“ If we try to conceive a body free in space, and not acted upon by any force, it is more natural to suppose that such a body, when once started, should go on moving uniformly for ever, than that its velocity should be altered. ”

“ When the weight driving the clock is wound up, a store of energy is communicated which is doled out to the pendulum in a very small impulse, at every vibration. The clock-weight is just large enough to be able to counterbalance the retarding forces when the pendulum has a proper amplitude of vibration. In all machines there is some energy lost in maintaining the parts in motion in opposition to friction and other resistances; in clocks this represents the whole amount of the force, as there is no external work to be performed. ”

“ Gravity is a force which attracts every particle of matter; it acts not merely on those parts of a body which lie on the surface, but it equally affects those in the interior. This is proved by observing that [Pg 53] a body has the same weight, however its shape be altered: for example, suppose I take a ball of putty which weighs 1 lb., I shall find that its weight remains unchanged when the ball is flattened into a thin plate, though in the latter case the surface, and therefore the number of superficial particles, is larger than it was in the former. ”

“ We have, then, a class of mechanical powers, of which a hammer may be taken as the type, which depend upon the storage of energy; the power of the arm is accumulated in the hammer throughout its descent, to be instantly transferred to the nail in the blow. Inertia is the property of matter which qualifies it for this purpose. Energy is developed by the explosion of gunpowder in a cannon. This energy is transferred to the ball, from which it is again in large part passed on to do work against the object which is struck. ”

“ It is a rule sometimes useful to practical men that a cast iron bar one foot long by one inch square would break with about a ton weight. If the iron be of the same quality as that which we have used, this result is too small, but the error is on the safe side; the real strength will then be generally a little greater than the strength calculated from this rule. What we have said (Art. 403) with reference to the precaution for safety in bars of wood applies also to cast iron. The load which the beam has to bear in ordinary practice should only be a small fraction of that which would break it. ”

“ A sleigh is an admirable vehicle on a smooth surface such as ice, but it is totally unadapted for use on common roads; the reason being that the amount of friction between the sleigh and the road is so great that to move the sleigh the horse would have to exert a force which would be very great compared with the load he was drawing. But a vehicle properly mounted on wheels moves with the greatest ease along the road, for the circumference of the wheel does not slide, and consequently there is no friction between the wheel and the road ”

“ The laws governing the lever may be deduced from the principle of work; the load, if nearer than the power to the fulcrum, is moved through a smaller distance than the power. Thus, for example, in the lever of the first order: if the load be 12 times as far as the power from the fulcrum, then for every inch the load moves it can be demonstrated that the power must move 12 inches. The number of units of work applied at one end of a machine is equal to the number yielded at the other, always excepting the loss due to friction, which is, however, so small in the lever that we may neglect it. ”

“ Of course, if g were not projected horizontally, we should not have arrived at this result; all we assert is that whatever be the motion of a body, it will (when possible) be at the end of a second, sixteen feet nearer the earth than it would have been if gravity had not acted. If the body be projected horizontally, its descent is due to gravity alone, and is neither accelerated nor retarded by the horizontal velocity. What this experiment proves is, that the mere fact of a body having velocity does not affect the action of gravity thereon. ”

“ Hence each stone in falling moves towards the centre of the earth, and this is consequently the direction of the force. We therefore assert that the earth has an attraction for the stone, in consequence of which it tries to get as near the earth’s centre as possible, and this attraction is called the force of gravitation. ”

“ Let us finally try a simple pendulum two-thirds of the length of the bar. I make the experiment, and find that the ball and the bar return to my hand precisely at the same instant. Therefore two-thirds of the length of the bar is the length of the isochronous simple pendulum.
We may state generally that the time of vibration of a uniform bar about one end equals that of a simple pendulum whose length is two-thirds of the bar; no doubt the bar we have used is not strictly uniform, because of the edges; but in the positions they occupy, their influence on the time of vibrations is imperceptible. ”

“ It is an universal law of nature that action and reaction are equal and opposite. Hence when any agent acts to set a body in motion, or to modify its motion in any way, the body reacts on the agent, and this force has been called the Kinetic reaction. ”

“ The most common rotating pieces in machines are wheels of various kinds, and in these the centre of gravity is evidently identical with the centre of rotation; but if from any cause a wheel, which is to turn [Pg 64] rapidly, has an extra weight attached to one part, this weight must be counterpoised by one or more on other portions of the wheel, in order to keep the centre of gravity of the whole in its proper place. Thus it is that the driving wheels of a locomotive are always weighted so as to counteract the effect of the crank and restore the centre of gravity to the axis of rotation. ”

“ An engine which turns a number of machines in a factory should work uniformly. Irregularities of motion may be productive of loss and various inconveniences. An engine would work irregularly either from variation in the production of steam, or from the demands upon the power being lessened or increased. Even if the first of these sources of irregularity could be avoided by care, it is clear that the second could not. Some machines in the mill are occasionally stopped, others occasionally set in motion, and the engine generally tends to go faster the less it has to do. ”

“ Let x be the force usefully employed in raising, and y the force of friction, which acts equally in either direction against the production of motion; then to raise the load the power applied must be sufficient to overcome both x and y, and therefore we have x + y = 9·0. When the weight is to be lowered the force x of course aids in the lowering, but x alone is not sufficient to overcome the friction; it requires the addition of 3·4 lbs., and we have therefore x + 3·4 = y, and hence x = 2·8, y = 6·2. ”

“ Gravity produces different effects upon different substances. This is commonly expressed by saying that some substances are heavier than others; for example, I have here a piece of wood and a piece of lead of equal bulk. The lead is drawn to the earth with a greater force than the wood. Substances are usually termed heavy when they sink in water, and light when they float upon it. But a body sinks in water if it weighs more than an equal bulk of water, and floats if it weigh less. ”

“ We often look with attention at the attraction of a piece of iron by a magnet, and justly so, for the phenomenon is very interesting, and yet the falling of a stone is produced by a far grander and more important force than the force of magnetism.
86. It is gravity which causes the weight of bodies. I hold a piece of lead in my hand: gravity tends to pull it downwards, thus producing a pressure on my hand which I call weight. Gravity acts with slightly varying intensity at various parts of the earth’s surface. ”

“ It is not always practicable to employ numerous hands for the purpose, nor is a steam-engine or other great source of power at all times available. But what are called the mechanical powers enable the forces at our disposal to be greatly increased. One man, by their aid, can exert as much force [Pg 100] as several could without such assistance; and when they are employed to augment the power of several men or of a steam-engine, gigantic weights, amounting sometimes to hundreds of tons, can be managed with facility. ”

“ We shall next consider the mechanical efficiency of the differential pulley-block. The block (Fig. 37) which we shall use is intended to be worked by one man, and will raise any weight not exceeding a quarter of a ton.
We have already learned that with this block the power must act through sixteen feet for the load to be raised one foot. Hence, were it not for friction, the power need only be the sixteenth part of the load. A few trials will show us that the real efficiency is not so large, and that in fact more than half the work exerted is merely expended upon overcoming friction. ”

“ Bodies lying on the earth are whirled around in a great circle every day. Hence, if there were not some force drawing them to the centre, they would fly off at a tangent. A part of the earth’s attraction goes for this purpose, and the remainder, which is the apparent weight, is thus diminished by a quantity increasing from the pole to the equator (Art. 86) . ”

“ In the first lecture I showed how a large weight could be raised by a smaller weight, and I stated that this subject would again occupy our attention. I now fulfil this promise. The questions to be discussed involve the most advantageous methods of employing a small force to overcome a greater. Here is a subject of practical importance. A man of average strength cannot raise more than a hundredweight without great exertion, yet the weights which it is necessary to lift and move about often weigh many hundredweights, or even many tons. ”

“ We often recognize it as a destroyer or impeder of motion, as a waster of our energy, and as a source of loss or inconvenience. But, on the other hand, friction is often indirectly the means of producing motion, and of this we have a splendid example in the locomotive engine. The engine being very heavy, the wheels are pressed closely to the rails; there is friction enough to prevent the wheels slipping, consequently when the engines force the wheels to turn round they must roll onwards. ”

“ Where great resistances have to be overcome, but where the distance through which the resistance must be urged is short, the lever or the screw is generally found to be the most appropriate means of [Pg 150] increasing power. When, however, the resistance has to be moved a considerable distance, the aid of the pulley, or the wheel and axle, or sometimes of both combined, is called in. The wheel and axle is the form of mechanical power which is generally used when the distance is considerable through which a weight must be raised, or through which some resistance must be overcome. ”

“ When w is 270 lbs. the compression on a b is 300 lbs., and the tension on b c is 405 lbs., so that both tie and strut attain their breaking loads together. The principle of duly apportioning the strength of each piece to the load it has to carry, involves the essence of sound engineering. In that greatest of mechanical feats, the construction of a mighty railway bridge across a wide span, attention to this principle is of vital importance. Such a bridge has to bear the occasional load of a passing train, but it has always to support the far greater load of the bridge materials. ”

“ If the magnitude of b exceeds a certain limit, then a is pulled along the table and b descends; but if b be smaller than this limit, both a and b remain at rest. When b is not heavy enough to produce motion it is supported by the tension of the cord, which is itself neutralized by the friction produced by a certain coherence between a and the table. Friction is by this experiment proved to be a force, because it prevents the motion of b. Indeed friction is generally manifested as a force by destroying motion, though sometimes indirectly producing it. ”

“ We can readily calculate from the formula how much power will be required to raise a given weight; for example, suppose 200 lbs. be attached to the moveable pulley, we find that 111 lbs. must be applied as the power. But in order to raise 200 lbs. one foot, the power exerted must act over two feet; hence the number of foot-pounds required is 2 × 111 = 222. The quantity of energy that is lost is 22 foot-pounds. Out of every 222 foot-pounds applied, 200 are usefully employed; that is to say, about 90 per cent. of the applied energy is utilized, while the remaining 10 per cent. ”