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)=56x,

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.

551. These principles are illustrated by the function of the fly-wheel in a steam-engine. The pressure of the steam upon the piston varies according to the different parts of the stroke: and the fly-wheel obviates the inconvenience which would arise from such irregularity. Its great inertia makes its velocity but little augmented by the exuberant action of the piston when the pressure is greatest, while it also sustains the motion when the piston is giving no assistance. The fly-wheel is a vast reservoir into which the engine pours its energy, sudden floods alternating with droughts; but these succeed each other so rapidly, and the area of the reservoir is so vast, that its level remains sensibly uniform, and the supplies sent out to the consumers are regular and unvaried. The consumers of the energy stored in the fly-wheel of an engine are the machines in the mill; they are supplied by shafts which traverse the building, conveying, by their rotation, the energy originally condensed within the coal from which combustion has set it free.

THE PUNCHING MACHINE.

552. When energy has been stored in a fly-wheel, it can be withdrawn either as a small force acting over a great distance, or as a large force over a small distance. In the latter case the fly-wheel acts as a mechanical power, and in this form it is used in the very important machine to be next described. A model of the punching machine is shown in Fig. 73.

The punching machine is usually worked by a steam-engine, a handle will move our small model. The handle turns a shaft on which the fly-wheel f is mounted. On the shaft is a small pinion d of 40 teeth: this works into a large wheel e of 200 teeth, so that, when the fly and the pinion have turned round 5 times, e will have turned round once, c is a circular piece of wood called a cam, which has a hole bored through it, between the centre and circumference; by means of this hole, the cam is mounted on the same axle as e, to which it is rigidly fastened, so that the two must revolve together. a is a lever of the first order, whose fulcrum is at a: the remote end of this lever rests upon the cam c; the other end b contains the punch. As the wheel e revolves it carries with it the cam: this raises the lever and forces the punch down a hole in a die, into which it fits exactly. The metal to be operated on is placed under the punch before it is depressed by the cam, and the pressure drives the punch through, cutting out a cylindrical piece of metal from the plate: this model will, as you see, punch through ordinary tin.

553. Let us examine the mode of action. The machine being made to rotate rapidly, the punch is depressed once for every 5 revolutions of the fly; the resistance which the metal opposes to being punched is no doubt very great, but the lever acts at a twelve-fold advantage. When the punch comes down on the surface of the metal, one of three things must happen: either the motion must stop suddenly, or the machine must be strained and injured, or the metal must be punched. But the motion cannot be stopped suddenly, because, before this could happen, an infinite force would be developed, which must make something yield. If therefore we make the structure sufficiently massive to prevent yielding, the metal must be punched. Such machines are necessarily built strong enough to make the punching of the metal easier than breaking the machine.

554. We shall be able to calculate, from what we have already seen in Art. 248, the magnitude of the force required for punching. We there learned that about 22·5 tons of pressure was necessary to shear a bar of iron one square inch in section. Punching is so far analogous to shearing that in each case a certain area of surface has to be cut; the area in punching is measured by the cylinder of iron which is removed.

555. Suppose it be required to punch a hole 0"·5 in diameter through a plate 0"·8 thick, the area of iron that has to be cut across is ²²/7×½×4/5=1·26 square inches: and as 22·5 tons per square inch are required for shearing, this hole will require 22·5×1·26=28·4 tons. A force of this amount must therefore be exerted upon the punch: which will require from the cam a force of more than 2 tons upon its end of the lever. Though the iron must be pierced to a depth of 0"·8, yet it is obvious that almost immediately after the punch has penetrated the surface of the iron, the cylinder must be entirely cut and begin to emerge from the other side of the plate. We shall certainly be correct in supposing that the punching is completed before the punch has penetrated to a depth of 0"·2, and that for not more than this distance has the great pressure of 28 tons been exerted; for a small pressure is afterwards sufficient to overcome the friction which opposes the motion of the cylinder of iron. Hence, though so great a pressure has been required, yet the number of units of energy consumed is not very large; it is ¹/60×2240×28·4=1062.

The energy actually required to punch a hole of half an inch diameter through a plate eight-tenths of an inch thick is therefore less than that which would be expended in raising 1 cwt. to a height of ten feet.

556. The fly-wheel may be likened to the reservoir in Art. 545. The time that is actually occupied in the punching is extremely small, and the sudden expenditure of 1062 units is gradually reimbursed by the engine. If the rotating fly-wheel contain 50,000 units of energy, the abstraction of 1362 units will not perceptibly affect its velocity. There is therefore an advantage in having a heavy fly sustained at a high speed for the working of a punching machine.