Chapter XIV: Introduction (1)

483. Kinetics is that branch of mechanics which treats of the action of forces in the production of motion. We shall find it rather more difficult than the subjects with which we have been hitherto occupied; the difficulties in kinetics arise from the introduction of the element of _time_, into our calculations. The principles of kinetics were unknown to the ancients. Galileo discovered some of its truths in the seventeenth century; and, since his time, the science has grown rapidly. The motion of a falling body was first correctly apprehended by Galileo; and with this subject we can appropriately commence.

THE FIRST LAW OF MOTION.

484. Velocity, in ordinary language, is supposed to convey a notion of rapid motion. Such is not precisely the meaning of the word in mechanics. By velocity is merely meant the _rate_ at which a body moves, whether the rate be fast or be slow. This rate is most conveniently measured by the number of feet moved over in one second. Hence when it is said the velocity of a body is 25, it is meant that if the body continued to move for one second with its velocity unaltered, it would in that time have moved over 25 feet.

485. The first law of motion may be stated thus. _If no force act upon a body, it will, if at rest, remain for ever at rest; or if in motion, it will continue for ever to move with a uniform velocity._ We know this law to be true, and yet no one has ever seen it to be true for the simple reason that we cannot realise the condition which it requires. We cannot place a body in the condition of being unacted upon by any forces. But we may convince ourselves of the truth of the law by some such reasoning as the following. If a stone be thrown along the road, it soon comes to rest. The body leaves the hand with a certain initial velocity and is not further acted upon by it. Hence, if no other force acted on the stone, we should expect, if the first law be true, that it would continue to run on for ever with the original velocity at the moment of leaving the hand. But other forces do act upon the stone; the attraction of the earth pulls it down; and, when it begins to bound and roll upon the ground, friction comes into operation, deprives the stone of its velocity, and brings it to rest. But let the stone be thrown upon a surface of smooth ice; when it begins to slide, the force of gravity is counteracted by the reaction of the ice: there is no other force acting upon the stone except friction, which is small. Hence we find that the stone will run on for a considerable distance. It requires but little effort of the imagination to suppose a lake whose surface is an infinite plane, perfectly smooth, and that the stone is perfectly smooth also. In such a case as this the first law of motion amounts to the assertion that the stone would never stop.

486. We may, in the lecture room, see the truth of this law verified to a certain extent by Atwood's machine (Fig, 66). This machine has been devised for the purpose of investigating the laws of motion by actual experiment. It consists principally of a pulley C, mounted so that its axle rests upon two pairs of wheels, as shown in the figure; it being the object of this contrivance to enable the wheel to revolve with the utmost freedom. A pair of equal weights A, B, are attached by a silken thread, which passes over the pulley; each of the weights is counterbalanced by the other: so that when the two are in motion, we may consider either as a body not acted upon by any forces, and it will be found that it moves uniformly, as far as the size of the apparatus will permit.

487. 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. The true proof of the first law of motion is, that all consequences properly deduced from it, in combination with other principles, are found to be verified. Astronomy presents us with the best examples. The calculation of the time of an eclipse is based upon laws which in themselves assume the first law of motion; hence, when we invariably find that an eclipse occurs precisely at the moment for which it has been predicted, we have a splendid proof of the sublime truth which the first law of motion expresses.

THE EXPERIMENT OF GALILEO FROM THE TOWER OF PISA.

488. The contrast between heavy bodies and light bodies is so marked that without trial we hardly believe that a heavy body and a light body will fall from the same height in the same time. That they do so Galileo proved by dropping a heavy ball and a light ball together from the top of the Leaning Tower at Pisa. They were found to reach the ground simultaneously. We shall repeat this experiment on a scale sufficiently reduced to correspond with the dimensions of the lecture room.

489. The apparatus used is shown in Fig. 67. It consists of a stout framework supporting a pulley H at a height of about 20 feet above the ground. This pulley carries a rope; one end of the rope is attached to a triangular piece of wood, to which two electro-magnets G are fastened. The electro-magnet is a piece of iron in the form of a horse-shoe, around which is coiled a long wire. The horse-shoe becomes a magnet immediately an electric current passes through the wire; it remains a magnet as long as the current passes, and returns to its original condition the moment the current ceases. Hence, if I have the means of controlling the current, I have complete control of the magnet; you see this ball of iron remains attached to the magnet as long as the current passes, but drops the instant I break the current. The same electric circuit includes both the magnets; each of them will hold up an iron ball F when the current passes, but the moment the current is broken both balls will be released. Electricity travels along a wire with prodigious velocity. It would pass over many thousands of miles in a second; hence the time that it takes to pass through the wires we are employing is quite inappreciable. A piece of thin paper interposed between the magnets and the balls will ensure that they are dropped simultaneously; when this precaution is not taken one or both balls may hesitate a little before commencing to descend. A long pair of wires E, B, must be attached to the magnets, the other ends of the wires communicating with the battery D; the triangle and its load is hoisted up by means of the rope and pulley and the magnets thus carry the balls to a height of 20 feet: the balls we are using weigh about 0·25 lb. and 1 lb. respectively.

490. We are now ready to perform the experiment. I break the circuit; the two balls are disengaged simultaneously; they fall side by side the whole way, and reach the ground together, where it is well to place a cushion to receive them. Thus you see the heavy ball and the light one each require the same amount of time to fall from the same height.

491. But these balls are both of iron; let us compare together balls made of different substances, iron and wood for example. A flat-headed nail is driven into a wooden ball of about 2"·5 in diameter, and by means of the iron in the nail I can suspend this ball from one of the magnets; while either of the iron balls we have already used hangs from the other. I repeat the experiment in the same manner, and you see they also fall together. Finally, when an iron ball and a cork ball are dropped, the latter is within two or three inches of its weighty companion when the cushion is reached: this small difference is due to the greater effect of the resistance of the air on the lighter of the two bodies. There can be no doubt that in a vacuum all bodies of whatever size or material would fall in precisely the same time.

492. How is the fact that all bodies fall in the same time to be explained? Let us first consider two iron balls. Take two equal particles of iron; it is evident that these fall in the same time; they would do so if they were very close together, even if they were touching, but then they might as well be in one piece: and thus we should find that a body consisting of two or more iron particles takes the same time to fall as one (omitting of course the resistance of the air). Thus it appears most reasonable that two balls of iron, even though unequal in size, should fall in the same time.

493. The case of the wooden ball and the iron ball will require more consideration before we realise thoroughly how much Galileo’s experiment proves. We must first explain the meaning of the word _mass_ in mechanics.

494. It is not correct to define _mass_ by the introduction of the idea of weight, because the _mass_ of a body is something independent of the existence of the earth, whereas weight is produced by the attraction of the earth. It is true that weight is a convenient means of measuring _mass_, but this is only a consequence of the property of gravity which the experiment proves, namely, that the attraction of gravity for a body is proportional to its _mass_.

495. Let us select as the unit of mass the mass of a piece of platinum which weighs 1 lb. at London; it is then evident that the mass of any other piece of platinum should be expressed by the number of pounds it contains: but how are we to determine the mass of some other substance, such as iron? A piece of iron is defined to have the same mass as a piece of platinum, if the same force acting on either of the bodies for the same time produces the same velocity. This is the proper test of the equality of masses. The mass of any other piece of iron will be represented by the number of times it contains a piece equal to that which we have just compared with the platinum; similarly of course for other substances.

496. The magnitude of a force acting for the time unit is measured by the product of the mass set in motion and the velocity which it has acquired. This is a truth established, like the first law of motion, by indirect evidence.

497. Let us apply these principles to explain the experiment which demonstrated that a ball of wood and a ball of iron fall in the same time. Forces act upon the two bodies for the same time, but the magnitudes of the forces are proportional to the mass of each body multiplied into its velocity, and, since the bodies fall simultaneously, their velocities are equal. The forces acting upon the bodies are therefore proportional to their masses; but the force acting on each body is the attraction of the earth, therefore, the gravitation to the earth of different bodies is proportional to their masses.

498. We may here note the contrast between the attraction of gravitation and that of a magnet. A magnet attracts iron powerfully and wood not at all; but the earth draws all bodies with forces depending on their masses and their distances, and not on their chemical composition.

THE SPACE DESCRIBED BY A FALLING BODY IS PROPORTIONAL TO THE SQUARE OF THE TIME.

499. We have next to discover the law by which we ascertain the distance a body falling from rest will move in a given time; it is not possible to experiment directly upon this subject, as in two seconds a body will drop 64 feet and acquire an inconveniently large velocity; we can, however, resort to Atwood’s machine (Fig. 66) as a means of diminishing the motion. For this purpose we require a clock with a seconds pendulum.

500. On one of the equal cylinders A I place a slight brass rod, whose weight gives a preponderance to A, which will consequently descend. I hold the loaded weight in my hand, and release it simultaneously with the tick of the pendulum. I observe that it descends 5" before the next tick. Returning the weight to the place from whence it started, I release it again, and I find that at the second tick of the pendulum it has travelled 20". Similarly we find that in three seconds it descends 45". It greatly facilitates these experiments to use a little stage which is capable of being slipped up and down the scale, and which can be clamped to the scale in any position. By actually placing the stage at the distance of 5", 20", or 45" below the point from which the weight starts, the coincidence of the tick of the pendulum with the tap of the weight on its arrival at the stage is very marked.

501. These three distances are in the proportion of 1, 4, 9; that is, as the squares of the numbers of seconds 1, 2, 3. Hence we may infer that _the distance traversed by a body falling from rest is proportional to the square of the time_.

502. The motion of the bodies in Atwood’s machine is much slower than the motion of a body falling freely, but the law just stated is equally true in both cases so that in a free fall the distance traversed is proportional to the square of the time. Atwood’s machine cannot directly tell us the distance through which a body falls in one second. If we can find this by other means, we shall easily be able to calculate the distance through which a body will fall in any number of seconds.

A BODY FALLS 16' IN THE FIRST SECOND.

503. The apparatus by which this important truth maybe demonstrated is shown in Fig. 67. A part of it has been already employed in performing the experiment of Galileo, but two other parts must now be used which will be briefly explained.

504. At A a pendulum is shown which vibrates once every second; it need not be connected with any clockwork to sustain the motion, for when once set vibrating it will continue to swing some hundreds of times. When this pendulum is at the middle of its swing, the bob just touches a slender spring, and presses it slightly downwards. The electric current which circulates about the magnets G (Art. 489) passes through this spring when in its natural position; but when the spring is pressed down by the pendulum, the current is interrupted. The consequence is that, as the pendulum swings backwards and forwards, the current is broken once every second. There is also in the circuit an electric alarm bell C, which is so arranged that, when the current passes, the hammer is drawn from the bell; but, when the current ceases, a spring forces the hammer to strike the bell. When the circuit is closed, the hammer is again drawn back. The pendulum and the bell are in the same circuit, and thus every vibration of the pendulum produces a stroke of the bell. We may regard the strokes from the bell as the ticks of the pendulum rendered audible to the whole room.

505. You will now understand the mode of experimenting. I draw the pendulum aside so that the current passes uninterruptedly. An iron ball is attached to one of the electro-magnets, and it is then gently hoisted up until the height of the ball from the ground is about 16'. A cushion is placed on the floor in order to receive the falling body. You are to look steadily at the cushion while you listen for the bell. All being ready, the pendulum, which has been held at a slight inclination, is released. The moment the pendulum reaches the middle of its swing it touches the spring, rings the bell, breaks the current which circulated around the magnet, and as there is now nothing to sustain the ball, it drops down to the cushion; but just as it arrives there, the pendulum has a second time broken the electric circuit, and you observe the falling of the ball upon the cushion to be identical with the second stroke of the bell. As these strokes are repeated at intervals of a second, it follows that the ball has fallen 16' in one second. If the magnet be raised a few feet higher, the ball may be seen to reach the cushion after the bell is heard. If the magnet be lowered a few feet, the ball reaches the cushion before the bell is heard.

506. We have previously shown that the space is proportional to the square of the time. We now see that when the time is one second, the space is 16 feet. Hence if the time were two seconds, the space would be 4 × 16 = 64 feet; and in general the space in feet is equal to 16 multiplied by the square of the time in seconds.

507. By the help of this rule we are sometimes enabled to ascertain the height of a perpendicular cliff, or the depth of a well. For this purpose it is convenient to use a stop-watch, which will enable us to measure a short interval of time accurately. But an ordinary watch will do nearly as well, for with a little practice it is easy to count the beats, which are usually at the rate of five a second. By observing the number of beats from the moment the stone is released till we see or hear its arrival at the bottom, we determine the time occupied in the act of falling. The square of the number of seconds (taking account of fractional parts) multiplied by 16 gives the depth of the well or the height of the cliff in feet, provided it be not high.

THE ACTION OF GRAVITY IS INDEPENDENT OF THE MOTION OF THE BODY.

508. We have already learned that the effect of gravity does not depend upon the actual chemical composition of the body. We have now to learn that its effect is uninfluenced by any motion which the body may possess. Gravity pulls a body down 16' per second, if the body starts from rest. But suppose a stone be thrown upwards with a velocity of 20 feet, where will it be at the end of a second? Did gravity not act upon the stone, it would be at a height of 20 feet. The principle we have stated tells us that gravity will draw this stone towards the earth through a distance of 16', just as it would have done if the stone had started from rest. Since the stone ascends 20' in consequence of its own velocity, and is pulled back 16' by gravity, it will, at the end of a second, be found at the height of 4'. If, instead of being shot up vertically, the body had been projected in any other direction, the result would have been the same; gravity would have brought the body at the end of one second 16' nearer the earth than it would have been had gravity not acted. For example, if a body had been shot vertically downwards with a velocity of 20', it would in one second have moved through a space of 36'.

509. We shall illustrate this remarkable property by an experiment. The principle of doing so is as follows:—Suppose we take two bodies, A and B. If these be held at the same height, and released together, of course they reach the ground at the same instant; but if A, instead of being merely dropped, be projected with a horizontal velocity at the same moment that B is released, it is still found that a and B strike the floor simultaneously.

510. You may very simply try this without special apparatus. In your left hand hold a marble, and drop it at the same instant that your right hand throws another marble horizontally. It will be seen that the two marbles reach the ground together.

511. A more accurate mode of making the experiment is shown by the contrivance of Fig. 68.

In this we have an arrangement by which we ensure that one ball shall be released just as the other is projected. At A B is shown a piece of wood about 2" thick; the circular portion (2' radius) on which the ball rests is grooved, so that the ball only touches the two edges and not the bottom of the groove. Each edge of the groove is covered with tinfoil C, but the pieces of tinfoil on the two sides must not communicate. One edge is connected with one pole of the battery K, and the other edge with the other pole, but the current is unable to pass until a communication by a conductor is opened between the two edges. The ball G supplies the bridge; it is covered with tinfoil, and therefore, as long as it rests upon the edges, the circuit is complete; the groove is so placed that the tangent to it at the lowest point B is horizontal, and therefore, when the ball rolls down the curve, it is projected from the bottom in a horizontal direction. An india-rubber spring is used to propel the ball; and by drawing it back when embraced by the spring, I can communicate to the missile a velocity which can be varied at pleasure. At H we have an electro-magnet, the wire around which forms part of the circuit we have been considering. This magnet is so placed that a ball suspended from it is precisely at the same height above the floor as the tinned ball is at the moment when it leaves the groove.

512. We now understand the mode of experimenting. So long as the tinned ball G remains on the curve the bridge is complete, the current passes, and the electro-magnet will sustain H, but the moment G leaves the curve, H is allowed to fall. We invariably find that whatever be the velocity with which G is projected, it reaches the ground at the same instant as H arrives there. Various dotted lines in the figure show the different paths which G may traverse; but whether it fall at D, at E, or at F, the time of descent is the same as that taken by H. 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.

513. Though we have only shown that a horizontal velocity does not affect the action of gravity, yet neither does a velocity in any direction. This is verified, like the first law of motion, by the accordance between the consequences deduced from it and the facts of observation.

514. We may summarize these results by saying that no matter what be the material of which a particle is composed, whether it be heavy or light, moving or at rest, if no force but gravity act upon the particle for _t_ seconds, it will then be 16_t_² feet nearer the earth than it would have been had gravity not acted.

515. A proposition which is of some importance may be introduced here. Let us suppose a certain velocity and a certain force. Let the velocity be such that a point starting from A, Fig. 69, would in one second move uniformly to B. Let the force be such that if it acted on a particle originally at rest at A, it would in one second draw the particle to D; if then the force act on a particle having this velocity where will it be at the end of the second? Complete the parallelogram A B C D, and the particle will be found at C. By what we have stated the force will equally discharge its duty whatever be the initial velocity. The force will therefore make the particle move to a distance equal and parallel to A D from whatever position the particle would have assumed, had the force not acted; but had the force not acted, the particle would have been found at B: hence, when the force does act, the particle must be found at C, since B C is equal and parallel to A D.

HOW THE FORCE OF GRAVITY IS DEFINED.

516. From the formula

Distance = 16_t_²,

we learn that a body falls through 64' in 2 seconds; and as we know that it falls 16' in the first second, it must fall 48' in the next second. Let us examine this. After falling for one second, the body acquires a certain velocity, and with that velocity it commences the next second. Now, according to what we have just seen, gravity will act during the next second quite independently of whatever velocity the body may have previously had. Hence in the second second gravity pulls the body down 16', but the body moves altogether through 48'; therefore it must move through 32' in consequence of the velocity which has been impressed upon it by gravity during the first second. We learn by this that when gravity acts for a second, it produces a velocity such that, if the body be conceived to move uniformly with the velocity acquired, the body would in one second move over 32'.

517. In three seconds the body falls 144', therefore in the third second it must have fallen

144' - 64' = 80';

but of this 80' only 16' could be due to the action of gravity impressed during that second; the rest,

80' - 16' = 64',

is due to the velocity with which the body commenced the third second.

518. We see therefore that after the lapse of two seconds gravity has communicated to the body a velocity of 64' per second; we should similarly find, that at the end of the third second, the body has a velocity of 96', and in general at the end of _t_ seconds a velocity of 32_t_. Thus we illustrate the remarkable law that _the velocity developed by gravity is proportional to the time_.

519. This law points out that the most suitable way of measuring gravity is by the velocity acquired by a falling body at the end of one second. Hence we are accustomed to say that _g_ (as gravity is generally designated) is 32. We shall afterwards show in the lecture on the pendulum (XVIII.) how the value of _g_ can be obtained accurately. From the two equations, _v_ = 32_t_ and _s_ = 16_t_² it is easy to infer another very well known formula, namely, _v_² = 64_s_.

THE PATH OF A PROJECTILE IS A PARABOLA.

520. We have already seen, in the experiments of Fig. 68, that a body projected horizontally describes a curved path on its way to the ground, and we have to determine the geometrical nature of the curve. As the movement is rapid, it is impossible to follow the projectile with the eye so as to ascertain the shape of its path with accuracy; we must therefore adopt a special contrivance, such as that represented in Fig. 70.

B C is a quadrant of wood 2" thick; it contains a groove, along which the ball B will run when released. A series of cardboard hoops are properly placed on a black board, and the ball, when it leaves the quadrant, will pass through all these hoops without touching any, and finally fall into a basket placed to receive it. The quadrant must be secured firmly, and the ball must always start from precisely the same place. The hoops are easily adjusted by trial. Letting the ball run down the quadrant two or three times, we can see how to place the first hoop in its right position, and secure it by drawing pins; then by a few more trials the next hoop is to be adjusted, and so on for the whole eight.

521. The curved line from the bottom of the quadrant, which passes through the centres of the hoops, is the path in which the ball moves; this curve is a parabola, of which F is the focus and the line A A the directrix.

It is a property of the parabola that the distance of any point on the curve from the focus is equal to its perpendicular distance from the directrix. This is shown in the figure. For example, the dotted line F D, drawn from F to the centre of the lowest hoop D, is equal in length to the perpendicular D P let fall from D on the directrix A A.

522. The direction in which the ball is projected is in this case horizontal, but, whatever be the direction of projection, the path is a parabola. This can be proved mathematically as a deduction from the theorem of Art. 515.

LECTURE XVI.
_INERTIA._
Inertia.—The Hammer.—The Storing of Energy.—The
Fly-wheel.—The Punching Machine.

INERTIA.

523. A body unacted upon by force will continue for ever at rest, or for ever moving uniformly in a straight line. This is asserted by the first law of motion (Art. 485). It is usual to say that _Inertia_ is a property of all matter, by which it is meant that matter cannot of itself change its state of rest or of motion. Force is accordingly required for this purpose. In the present chapter we shall discuss some important mechanical considerations connected with the application of force in changing the state of a body from rest or in altering its velocity when in motion. In the next chapter we shall study the application of force in compelling a body to swerve from its motion in a straight line.

524. We have in earlier lectures been concerned with the application of force either to raise a weight or to overcome friction. We have now to consider the application of force to a body, not for the purpose of raising it, nor for pushing it along against a frictional resistance, but merely for the purpose of generating a velocity. Unfortunately there is a practical difficulty in the way of making the experiments precisely in the manner we should wish. We want to get a mass isolated both from gravitation and from friction, but this is just what we cannot do—that is, we cannot do it perfectly. We have, however, a simple appliance which will be sufficiently isolated for our present purpose. Here is a heavy weight of iron, about 25 lbs., suspended by an iron wire from the ceiling about 32 feet above the floor (see Fig. 82). This weight may be moved to and fro by the hand. It is quite free from friction, for we need not at present remember the small resistance which the air offers. We may also regard the gravity of the weight as neutralized by the sustaining force of the wire, and accordingly as the body now hangs at rest it may for our purposes be regarded as a body unacted upon by any force.

525. To give this ball a horizontal velocity I feel that I must apply force to it. This will be manifest to you all when I apply the force through the medium of an india-rubber spring. If I pull the spring sharply you notice how much it stretches; you see therefore that the body will not move quickly unless a considerable force is applied to it. It thus follows that merely to generate motion in this mass force has been required.

526. So, too, when the body is in motion as it now is I cannot stop it without the exertion of force. See how the spring is stretched and how strong a pull has thus been exerted to deprive the body of motion. Notice also that while a small force applied sufficiently long will always restore the body to rest, yet that to produce rest quickly a large force will be required.

527. 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_.

528. For example. When a railway train starts, the locomotive applies force to the carriages, and the speed generated during one second is added to that produced during the next, and the pace improves. The kinetic reaction of the train retards the engine from attaining the speed it would acquire if free from the train.

THE HAMMER.

529. The hammer and other tools which give a blow depend for their action upon inertia. A gigantic hammer might force in a nail by the mere weight of the head resting on the nail, but with the help of inertia we drive the nail by blows from a small hammer. We have here inertia aiding in the production of a mechanical power to overcome the considerable resistance which the wood opposes to the entrance of the nail. To drive in the nail usually requires a direct force of some hundreds of pounds, and this we are able conveniently to produce by suddenly checking the velocity of a small moving body.

530. The theory of the hammer is illustrated by the apparatus in Fig. 71. It is a tripod, at the top of which, about 9' from the ground, is a stout pulley C; the rope is about 15' long, and to each end of it A and B are weights attached. These weights are at first each 14 lbs. I raise A up to the pulley, leaving B upon the ground; I then let go the rope, and down falls A: it first pulls the slack rope through, and then, when A is about 3' from the ground, the rope becomes tight, B gets a violent chuck and is lifted into the air. What has raised B? It cannot be the mere weight of A, because that being equal to B, could only just balance B, and is insufficient to raise it. It must have been a force which raised B; that force must have been something more than the weight of A, which was produced when the motion was checked. A was not stopped completely; it only lost some of its velocity, but it could not lose any velocity without being acted upon by a force. This force must have been applied by the rope by which A was held back, and the tension thus arising was sufficient to pull up B.

531. Let us remove the 14 lb. weight from B, and attach there a weight of 28 lbs., A remaining the same as before (14 lbs.). I raise A to the pulley; I allow it to fall. You observe that B, though double the weight of A, is again chucked up after the rope has become tight. We can only explain this by the supposition that the tension in the rope exerted in checking the motion of A is at least 28 lbs.

532. Finally, let us remove the 28 lbs. from B, put on 56 lbs., and perform the experiment again; you see that even the 56 lbs. is raised up several inches. Here a tension in the rope has been generated sufficient to overcome a weight four times as heavy as A. We have then, by the help of inertia, been able to produce a mechanical power, for a small force has overcome a greater.

533. After B is raised by the chuck to a certain height it descends again, if heavier than A, and raises A. The height to which B is raised is of course the same as the height through which A descends. You noticed that the height through which 28 lbs. was raised was considerably greater than that through which the 56 lbs. was raised. Hence we may draw the inference, that when A was deprived of its velocity while passing through a short space, it required to be opposed by a greater force than when it was gradually deprived of its velocity through a longer space. This is a most important point. Supposing I were to put a hundredweight at B, I have little doubt, if the rope were strong enough to bear the strain, that though A only weighs 14 lbs., B would yet be raised a little: here A would be deprived of its motion in a very short space, but the force required to arrest it would be very great.

534. It is clear that matters would not be much altered if A were to be stopped by some force, exerted from below rather than above; in fact, we may conceive the rope omitted, and suppose A to be a hammer-head falling upon a nail in a piece of wood. The blow would force the nail to penetrate a small distance, and the entire velocity of A would have to be destroyed while moving through that small distance: consequently the force between the head of the nail and the hammer would be a very large one. This explains the effect of a blow.

535. In the case that we have supposed, the weight merely drops upon the nail: this is actually the principle of the hammer used in pile-driving machines. A pile is a large piece of timber, pointed and shod with iron at one end: this end is driven down into the ground. Piles are required for various purposes in engineering operations. They are often intended to support the foundations of buildings; they are therefore driven until the resistance with which the ground opposes their further entrance affords a guarantee that they shall be able to bear what is required.

536. The machine for driving piles consists essentially of a heavy mass of iron, which is raised to a height, and allowed to fall upon the pile. The resistance to be overcome depends upon the depth and nature of the soil: a pile may be driven two or three inches with each blow, but the less the distance the pile enters each time, the greater is the actual pressure with which the blow forces it downwards. In the ordinary hammer, the power of the arm imparts velocity to the hammer-head, in addition to that which is due to the fall; the effect produced is merely the same as if the hammer had fallen from a greater height.

537. Another point may be mentioned here. A nail will only enter a piece of wood when the nail and the wood are pressed together with sufficient force. The nail is urged by the hammer. If the wood be lying on the ground, the reaction of the ground prevents the wood from getting away and the nail will enter. In other cases the element of _time_ is all-important. If the wood be massive less force will make the nail penetrate than would suffice to move the wood quickly enough. If the wood be thin and unsupported, less force may be required to make it yield than to make the nail penetrate. The usual remedy is obvious. Hold a heavy mass close at the back of the wood. The nail will then enter because the augmented mass cannot now escape as rapidly as before.

THE STORING OF ENERGY.

538. Our study of the subject will be facilitated by some considerations founded on the principles of energy. In the experiment of Fig. 71 let A be 14 lbs., and B, on the ground, be 56 lbs. Since the rope is 15' long, A is 3' from the ground, and therefore 6' from the pulley. I raise A to the pulley, and, in doing so, expend 6 × 14 = 84 units of energy. Energy is never lost, and therefore I shall expect to recover this amount. I allow A to fall; when it has fallen 6', it is then precisely in the same condition as it was before being raised, except that it has a considerable velocity of descent. In fact, the 84 units of energy have been expended in giving velocity to A. B is then lifted to a maximum height X, in which 56 × X units of energy have been consumed. At the instant when B is at the summit X, A must be at a distance of 6 + X feet from the pulley; hence the quantity of work performed by A is 14 × (6 + X). But the work done by A must be equal to that done upon B, and therefore

14(6 + X) = 56 X,

whence X = 2. If there were no loss by friction, B would therefore be raised 2'; but owing to friction, and doubtless also to the imperfect flexibility of the rope, the effect is not so great. We may regard the work done in raising A as so much energy stored up, and when A is allowed to fall, the energy is reproduced in a modified form.

539. Let us apply the principle of energy to the pile-driving engine to which we have referred (Art. 536); we shall then be able to find the magnitude of the force developed in producing the blow. Suppose the “monkey,” that is the heavy hammer-head, weighs 560 lbs. (a quarter of a ton). A couple of men raise this by means of a small winch to a height of 15'. It takes them a few minutes to do so; their energy is then saved up, and they have accumulated a store of 560 × 15 = 8,400 units. When the monkey falls upon the top of the pile it transfers thereto nearly the whole of the 8,400 units of energy, and this is expended in forcing the pile into the ground. Suppose the pile to enter one inch, the reaction of the pile upon the monkey must be so great that the number of units of energy consumed in one inch is 8,400. Hence this reaction must be 8,400 × 12 = 100,800 lbs. If the reaction did not reach this amount, the monkey could not be brought to rest in so short a distance. The reaction of the pile upon the monkey, and therefore the action of the monkey upon the pile, is about 45 tons. This is the actual pressure exerted.

540. If the soil which the pile is penetrating be more resisting than that which we have supposed,—for example, if the pile require a direct pressure of 100 tons to force it in,—the same monkey with the same fall would still be sufficient, but the pile would not be driven so far with each blow. The pressure required is 224,000 lbs.: this exerted over a space of 0"·45 would be 8,400 units of energy; hence the pile would be driven 0"·45. The more the resistance, the less the penetration produced by each blow. A pile intended to bear a very heavy load permanently must be driven until it enters but little with each blow.

541. We may compare the pile-driver with the mechanical powers in one respect, and contrast it in another. In each, we have machines which receive energy and restore it modified into a greater power exerted through a smaller distance; but while the mechanical powers restore the energy at one end of the machine, simultaneously with their reception of it at the other, the pile-driver is a reservoir for keeping energy which will restore it in the form wanted.

542. 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. Here we see energy stored in a rapidly moving body, a case to which we shall presently return.

543. But energy can be stored in many ways; we might almost say that gunpowder is itself energy in a compact and storable form. The efforts which we make in forcing air into an air-cane are preserved as energy there stored to be reproduced in the discharge of a number of bullets. During the few seconds occupied in winding a watch, a small charge of energy is given to the spring which it expends economically over the next twenty-four hours. In using a bow my energy is stored up from the moment I begin to pull the string until I release the arrow.

544. Many machines in extensive use depend upon these principles. In the clock or watch the demand for energy to sustain the motion is constant, while the supply is only occasional; in other cases the supply is constant, while the demand is only intermittent. We may mention an illustration of the latter. Suppose it be required _occasionally_ to hoist heavy weights rapidly up to a height. If an engine sufficiently powerful to raise the weights be employed, the engine will be idle except when the weights are being raised; and if the machinery were to have much idle time, the waste of fuel in keeping up the fire during the intervals would often make the arrangement uneconomical. It would be a far better plan to have a smaller engine; and even though this were not able to raise the weight directly with sufficient speed, yet by keeping the engine continually working and storing up its energy, we might produce enough in the twenty-four hours to raise all the weights which it would be necessary to lift in the same time.

545. Let us suppose we want to raise slates from the bottom of a quarry to the surface. A large pulley is mounted at the top of the quarry, and over this a rope is passed: to each end of the rope a bucket is attached, so that when one of these is at the bottom the other is at the top, and their sizes and that of the pulley are so arranged that they pass each other with safety. A reservoir is established at the top of the quarry on a level with the pulley, and an engine is set to work constantly pumping up water from the bottom of the quarry into the reservoir. Each of the buckets is partly composed of a large tank, which can be quickly filled or emptied. The lower bucket is loaded with slates, and when ready for work, the man at the top fills the tank of the upper bucket with water: this accordingly becomes so heavy that it descends and raises the slates. When the heavier one reaches the bottom, the water from its tank is let out into the lower reservoir, from which the engine pumps, and the slates are removed from the bucket which has been raised. All is then ready for a repetition of the same operation. If the slates be raised at intervals of ten minutes, the energy of the engine will be sufficient when in ten minutes’ work it can pump up enough water to fill one tank; by the aid of this contrivance we are therefore able to accumulate for one effort the whole power of the engine for ten minutes.

THE FLY-WHEEL.

546. One of the best means of storing energy is by setting a heavy body in rapid motion. This has already been referred to in the case of the cannon-ball. In order to render this method practically available for the purposes of machinery, the heavy body we use is a fly-wheel, and the rapid motion imparted to it is that of rotation about its axis. A very large amount of energy can by this means be stored in a manageable form.

547. We shall illustrate the principle by the apparatus of Fig. 72. This represents an iron fly-wheel B: its diameter is 18", and its weight is 26 lbs.; the fly is carried upon a shaft (A) of wrought iron ¾" in diameter. We shall store up a quantity of energy in this wheel, by setting it in rapid motion, and then we shall see how we can recover from it the energy we have imparted.

548. A rope is coiled around the shaft; by pulling this rope the wheel is made to turn round: thus the rope is the medium by which my energy shall be imparted to the wheel. To measure the operation accurately, I attach the rope to the hook of the spring balance (Fig. 9); and by taking the ring of the balance in my hand, I learn from the index the amount of the force I am exerting. I find that when I walk backwards as quickly as is convenient, pulling the rope all the time, the scale shows a tension of about 50 lbs. To set the wheel rapidly in motion, I pull about 20' of rope from the axle, so that I have imparted to the wheel somewhere about 50 × 20 = 1,000 units of energy. The rope is fastened to the shaft, so that, after it has been all unwound, the wheel now rapidly rotating winds it in. By measuring the time in which the wheel made a certain number of coils of the rope around the shaft, I find that it makes about 600 revolutions per minute.

549. Let us see how the stored-up energy can be exhibited. A piece of pine 24" × 1" × 1" of which both ends are supported, requires a force of 300 lbs. applied to its centre to produce fracture (See p. 190). I arrange such a piece of pine near the wheel. As the shaft is winding in the rope, a tremendous chuck would be given to anything which tried to stop the motion. If I tied the end of the rope to the piece of pine, the chuck would break the rope; therefore I have fastened one end of a 10' length of chain to the rope, and the other has been tied round the middle of the wooden bar. The wheel first winds in the rope, then the chain takes a few turns before it tightens, and crack goes the rod of pine. The wheel had no choice; it must either stop or break the rod: but nature forbids it to be stopped, unless by a great force, which the rod was not strong enough to apply. Here I never exerted a force greater than 50 lbs. in setting the wheel in motion. The wheel stored up and modified my energy so as to produce a force of 300 lbs., which had, however, only to be exerted over a very small distance.

550. But we may show the experiment in another way, which is that represented in the figure (72). We see the chain is there attached to two 56 lb. weights. The mode of proceeding is that already described. The rope is first wound round the shaft, then by pulling the rope the wheel is made to revolve; the wheel then begins to wind in the rope again, and when the chain tightens the two 56 lbs. are raised up to a height of 3 or 4 feet. Here, again, the energy has been stored and recovered. But though the fly-wheel will thus preserve energy, it does so at some cost: the store is continually being frittered away by friction and the resistance of the air; in fact, the energy would altogether disappear in a little time, and the wheel would come to rest; it is therefore economical to make the wheel yield up what it has received as soon as possible.