Part 8

And now comes a difference between the lead and the water, which depends on the fluidity of the latter. The lead exerts no pressure on the sides of the tube, but the water does. If a small hole is cut in the side of the tube close to the bottom, and stopped with a cork, the lead will not press upon the cork. But if the column of water is high enough the cork will be driven out with as much force as before, so that the water presses just as much sideways as downwards. It is easy to satisfy oneself of this by inserting a long glass tube, with its lower end bent at right angles and fitted with a cork, into the side of the wooden pipe. The water will at once rise in the tube to the same height as it has in the pipe. Whence it is obvious that the pressure of the water on any point of the side is exactly equal to the vertical pressure at that point; for the pressure outwards is exactly balanced by that of the vertical column in the tube inwards. The water in a watering-pot always stands at the same level in the can and in the spout.

If a glass tube is bent into the shape of a =U=, and water is poured into it, the water will always stand at the same level in the two legs of the tube, whatever the shape of the bend may be, or the relative capacities of the two legs, or the inclination of the tube.

And this must needs be so, for the force with which the water tends to flow out of the one half of the arrangement depends on the vertical height of the surface of the water above the aperture of exit; so that any column of equal vertical height must balance it.

Footnote 2:

Vertical height is the height measured along a line drawn from the surface of the water perpendicularly to the surface of the earth. A plumb-line is a string to one end of which a weight is attached and thus hangs suspended. If the other end of the line is brought opposite the surface of the water the direction of the string answers to the line of vertical height.

That a column of water will stand at exactly the same level as any other with which it communicates, may be seen still more simply by placing a glass tube, open at each end, in a basin of water. However the tube may be inclined or bent, whether its lower end is wide or narrow, the column of water inside it will be at exactly the same level as the water outside it. Yet, of course, the rigid glass walls of the tube cut off all communication between the column of water inside it and the rest, except at the bottom.

In a well-ordered town, water is supplied to every house and can be drawn from taps placed in the highest stories. These are fed by pipes which lead from a cistern at the top of the house. This water is brought from a large pipe, or =main=, in the street, by a smaller house-pipe, which is often made to twist about in various directions before it reaches the cistern at the top of the house into which it delivers the water. If you followed the main, you would find that it took a long course up and down, beneath the pavement of the streets, until at last it reached the water-works. Here you would find that the main was connected with a reservoir; and either this reservoir is at a greater height than any of the cisterns into which the water is delivered, or there is some means of pumping the water from it to that height on its way to the main. Thus the reservoir, the main, and the house-pipe form one immense =U=-tube, and the water in the house-pipe tends to rise to the same level as that of the water in the reservoir, and hence flows into the cistern when the supply-pipe is open.

28. =The Transference of Motion by Moving Water: the Momentum of Moving Water.=

Suppose a wooden vat with a horizontal tap, the sectional area of the tube of which is one square inch, inserted close to the bottom, to be filled with water up to 100 inches above the tap. Then supposing the tap to be shut, the pressure upon its sectional area will be 25,250 grains, or rather more than three pounds and a half—and there is the same pressure on every square inch of the bottom of the vat.

Footnote 3:

The sectional area of a tube is the surface occupied by its cavity when it is cut across. It would be represented by the surface of a piece of cardboard, like the wad of a gun, just large enough to go into the tube.

If the tap is now turned, the water nearest to it being unsupported on its outer side, the pressure on the inner side sets it in motion, and it flows out in a stream. At first the stream shoots out violently and the water is carried to a long distance. That is to say, the weight of the column of water 100 inches high acts as a force, or cause of motion, upon the water nearest the tap, and this water is forced out with a velocity depending on that force in a horizontal direction. Now suppose that you take a common toy cup-and-ball and bring the ball into the way of the stream of water. The stream will at once strike the ball and drive it in the same direction as that which it is itself taking. The power which the moving water has of transferring or communicating motion to a body which is at rest, but free to move as the ball is, is due to its =momentum=. The greater the mass of the stream and the more rapidly it moves, the more motion will it communicate to the ball, or the heavier the ball it will move. Close to the mouth of the tap the direction of the stream is horizontal; but it very soon begins to bend downwards, and describing a rapid curve, comes to the ground. It does this for just the same reasons that a stone thrown horizontally describes a curve; and at length strikes the ground; and, in fact, the stream may be regarded as so much water thrown horizontally.

These reasons are two: firstly, as soon as the water has left the tap it is an unsupported heavy body; and, as such, it begins to fall to the ground. Secondly, the momentum of the water is continually being diminished by the resistance of the air through which it passes. For, although the air which surrounds us is so thin and movable a body that we ordinarily take no notice of it—the fact that it offers resistance to bodies which move through it is easily observed; as, for example, in using a fan. The water has to overcome this resistance, and its momentum is proportionally diminished.

If, when the water leaves the tap, the air and gravitation were alike abolished, the water, keeping its momentum, would travel for ever in the same direction.

As the water runs out, it will be observed that the velocity of the stream becomes less and the curve which it describes sharper, so that it comes to the ground sooner; and finally, when the vat is nearly empty, the stream falls nearly vertically downwards. The reason of this is that the level of the top of the water is gradually lowered; consequently, the height of the column which presses on the water close to the tap is gradually lessened, and therefore its weight is diminished. But this weight or pressure is the cause of the motion of the water, and as the cause diminishes the effect of that cause must diminish. Therefore the momentum of the water is gradually lessened and it is carried less and less far horizontally in the time which it takes to fall to the ground: until finally, it acquires no appreciable horizontal motion at all, and so falls vertically downwards from the mouth of the tap.

29. =The Energy of Moving Water.=

If a short pipe bent at right angles like the letter =L= is fitted by one arm on to the end of the tap, while the other is turned vertically upwards, and the vat is full as before; when the tap is turned, the water will shoot up into the air, and after rising for a certain distance will stop, and then fall. In fact we shall have a fountain.

Observe the difference between the vertical jet of water and the horizontal jet. If we leave the resistance of the air out of consideration, the water in the horizontal jet has no obstacle to overcome; and it might go on for ever, if its weight did not gradually cause its path to become more and more bent towards the earth, against which it eventually strikes.

When the jet is vertical the case is altered. The water thrown up vertically constantly tends to fall down vertically, as any other heavy body would do, and its momentum has to overcome the obstacle of its gravity. Any given portion of the water is, in fact, acted upon by two opposite tendencies, momentum urging it up, and gravity pulling it down. Now if two equal tendencies exactly oppose one another, the body upon which they act does not move at all; while, if one is stronger than the other, the body moves in the direction of the stronger.

Thus a portion of water which has just left the spout shoots up, because the velocity with which it is impelled upwards is sufficient to carry it through a greater space in a given time, say a second, than that through which its gravity would, in the same time, impel it downwards.

But the distance which the water will travel during this second will be the difference between the distance which it would have ascended if there had been no gravity forcing it down, and the distance which it would have descended if there had been no momentum driving it up; and, at the end of the second, the rate of its motion upwards, or its velocity, will be proportionally slower. Thus, at the end of the first second, the water has spent a certain portion of its momentum in overcoming its gravity. And as there is nothing to make good the loss, it would, if left to itself, travel more slowly, or over a less distance, in the second second than it tended to do in the first. But though the momentum of the water is diminished, its gravity, weight, or tendency to fall downwards, for a given distance in a second, remains exactly what it was, and operates in the course of the second second to exactly the same extent as in the first. Hence, at the end of the second second, the distance through which the water travels upwards is still smaller, and its velocity is still more diminished. It is obvious that, however great the disproportion between momentum and gravity to start with, gravity must gain the day in the long run under these circumstances. The store of momentum will be used up; and, after a momentary rest, the water, reduced to the condition of a body without support, will begin to be carried downwards by the unopposed action of gravity.

The case is similar to that of a boy sculling a boat, the bows of which are suddenly seized and the boat thrust violently backwards by a strong man. The boat will go stern-foremost rapidly, at first, but every stroke of the boy’s oar at the stern will retard its backward motion; until, at length, the stock of momentum conferred upon it by the man’s thrust will be completely exhausted in working against the boy, and the boat, after a momentary rest, will resume its onward course. The distance to which the boat will be propelled backwards will evidently depend upon the amount of muscular power which the man, as it were, suddenly capitalizes in the boat, and which the boat then slowly pays out.

We call people who possess much muscular or other power energetic; and we estimate their =energy= by the obstacles they overcome, or, in other words, by the =work= they do. In the present illustration the man’s energy would be measured by the distance to which the boat was propelled before it stopped.

It is easy to transfer this conception of energy, as the power of doing work, to inanimate things; and thus when a body in motion overcomes any kind of obstacles in its way, parting with its momentum and more or less coming to rest in the process, we say that it has =energy= and that it does =work=.

The energy of moving water is thus measured by the intensity of the opposing forces which it can overcome multiplied into the distance which it can travel before that energy is exhausted; that is to say, by the work it does before it is itself reduced to a state of rest. In the case under consideration, the energy by which gravity is overcome, for a greater or less time, depends upon the velocity of the stream; and this again depends upon the height of the water in the vat above the tap. Just as the energy of the horizontal stream diminished as the level of the water became lower, so does the energy of the vertical stream diminish. Hence, as the vat empties, the jet becomes shorter and shorter, until at last it sinks down to nothing.

The energy of moving water makes it, under some circumstances, one of the most destructive of natural agents; and, under others, one of the most useful servants of man. A stream is water falling down hill with a velocity depending upon the inclination of its bed. As it falls it acquires momentum and, hence, energy; and thus a mountain stream, suddenly swollen by rain or melting snow, will tear away masses of rock and sweep everything before it. Nothing can look softer or more harmless than a calm sea, but if the wind sweeping over its surface puts the water in motion, it strikes upon the shore with terrific force; and its energy is expended in throwing up great waves, which lift vast blocks or drive masses of shingle up the beach.

In all kinds of watermills it is the energy of more or less rapidly falling water which is turned to account. The water is made to flow against buckets or floats attached to the circumference of a wheel. Each bucket or float is therefore an obstacle to which the water transfers some of its own motion; it moves away and thus makes the wheel to which it is fastened turn. But the turning of the wheel brings a new obstacle in the way of the stream. This is treated in the same fashion, and the wheel turns still further, thus introducing another obstacle in the way of the stream upon which the same effect is produced. Thus each float, or bucket, is a means by which some of the momentum of the stream is, as it were, caught and transferred to the water-wheel, which consequently turns round with a certain velocity.

But this water-wheel is now a mass of matter in motion, and therefore itself contains a store of energy or power doing work. If a cord with a weight at the end of it were fastened to the axle of the wheel, the cord would be wound upon the axle, and the weight could be raised, or, in other words, so much work would be done by the turning of the wheel and we should thus have a rough measure of the amount of energy which had been given up by the stream to the wheel.