Part 3
Such an instrument is the _balance_ or _scales_, which you may see in every grocer’s shop. It is composed of a beam which moves easily on a pivot fixed to its middle, and which has a scale-pan attached to each end. So long as both scale-pans are empty the beam is horizontal; but if you put anything which has weight into one, that one goes down and the other rises. If now you either pull or push the empty scale downwards, the beam may be brought into the horizontal position again, and the effort required to bring it into the horizontal position will be the greater, the greater the weight of the body in the opposite scale. An ounce in the one scale is easily raised by the pressure of a finger in the other. A pound requires more effort; ten pounds needs putting out the strength of the arm; to raise fifty pounds involves still more exertion; while a couple of hundredweight will not be stirred by the strongest push or pull upon the empty scale.
Suppose that, instead of pressing down the empty scale, you put something that has weight into it; then, as soon as this weight is equal to that in the other scale, the beam will become horizontal. In fact, one scale has just as much tendency to move towards the centre of the earth as the other has, and as neither can go down without pulling the other up, they neutralise one another. It comes to the same thing, as if two boys of equal strength were pulling one against the other; so long as the pulls in opposite directions are equal, of course neither boy can stir; while the smallest addition of strength to one enables him to pull the other over.
22. =The Weight of the same Bulk or Volume of Water is Constant under the same conditions. Mass. Density.=
Now let two graduated thin glass measures be put into the two scales, and made to counterpoise one another exactly. Then, if even a single drop of water is put into the one measure the scale will descend, if the balance is a good one; showing that the drop has weight. If the measures are graduated accurately, then whatever volume of water is put into one, an exactly similar volume of the same water must be put into the other to make the beam level. This obviously means that =the same volume of water under the same circumstances always has the same weight=.
In § 18 it was said that bodies tend to move towards one another with a relative velocity[1] which is inversely proportional to the quantity of matter which they contain. But how are we to measure quantity of matter? Is it to be estimated by the space which it occupies; that is, by its volume? or are we to estimate the quantity of matter in a body by its weight? You will soon learn that the volume of all bodies is constantly changing in correspondence with the changes in the pressure exerted by other bodies, but more especially in correspondence with the changes of temperature to which they are subjected; while the weight of the same body, at the same point on the earth’s surface, never alters. Hence we may take the weight of a body as a measure of the quantity of matter which it contains; and it follows that, for the same weight, the larger the volume of a body the less matter it contains proportionally to its volume, and the less the volume, the more matter it contains. The proportion of its weight to its volume gives us the =density= of a body.
Footnote 1:
Velocity, or swiftness, is measured by the distance over which a body travels, in a given time. Of two bodies, one of which travels through one foot in a second, while the other travels through two feet, the latter has the greater relative velocity.
Now what is true of water is true of all other bodies or material substances. Suppose that one of the measures is emptied and replaced, the beam may be brought to the horizontal position again by means of a piece of lead cut to exactly the right size. The piece of lead will thenceforth furnish an exactly corresponding or equivalent weight for so much water; and pieces of iron or brass, which counterpoise the lead, will also be equivalents of the weight of the water, or of the lead, or of one another. But the pieces of lead, iron, or brass will obviously be of much less volume or bulk than the water which they counterpoise. Here it follows that the densities of these metals, or the quantity of matter contained in the same volume, must be much greater than in the case of water.
What are called =weights= in commerce are pieces of lead, or iron, or brass exactly equivalent in weight to a certain bulk of water under certain conditions. =An imperial gallon of water thus weighs ten pounds, and therefore an imperial pint weighs a pound and a quarter.=
23. =Equal Volumes of Different Things under the same circumstances, have Different Weights: the Density of Different Bodies is Different.=
The important fact which has just been alluded to must be considered more fully. We have seen that an imperial pint measure gives us the space which is taken up by as much water as weighs a pound and a quarter; and this space is the bulk or =volume= of that weight of water. But if you take an ordinary pound weight and a quarter-pound weight, and put them into an imperial pint measure, you will find that instead of filling it, they take up only a very small portion of the space in its interior, or in other words, of its capacity. Thus the volume of a pound and a quarter of lead, or of iron, or of brass, is very much less than the volume of the same weight of water; that is to say, the metals are =denser= than water; the same volume has greater mass or more gravity. Or, to put the case in another way, fill the tumbler with which we began half full of water, making a mark on the side exactly at the level of the top of the water. Then place it in one scale of a balance, and counterpoise it with weights in the other. Next, pour out the water, and after drying the tumbler, fill it with fine sand carefully up to the mark. The volume of sand will be equal to the volume of water. But now the same weights will no longer counterpoise it, and you will have to put more weights in the opposite scale. Volume for volume, therefore, sand is heavier than water. Throw out the sand, and put in sawdust in the same way, and you will find that a less weight than was necessary to counterpoise the water counterpoises the sawdust. Volume for volume, therefore, sawdust is lighter than water. Experiment in the same way with spirit and oil, and they will be found to be lighter than water, while treacle will be heavier, and quicksilver very much heavier than water.
24. =The Meaning of Heavy and Light—Specific Gravity.=
We are in the habit of using the words =heavy= and =light= rather carelessly. We call things that are easily lifted light, and things that are hard to lift heavy. We say that sand, which is blown about by the wind, is light, and that a block of wood is heavy, and yet we have just seen that sand is heavier, bulk for bulk, than wood. In order to get rid of this double meaning, the weight of a volume of any liquid or solid, in proportion to the weight of the same volume of water at a known temperature and pressure, is called its =specific gravity=. Water being taken as 1, anything a volume of which is twice as heavy as the same volume of water is said to have the specific gravity 2; if three times, 3; if four and a half times, 4·5, and so on. Thus the specific gravity of any liquid or solid expresses its density in proportion to that of water under the same conditions. Sawdust, oil, and spirit have a less specific gravity than water, while treacle, sand, and quicksilver have a greater specific gravity. In this sense, the former three substances are =light=, while the latter three are =heavy=.
25. =Things of greater Specific Gravity than Water sink in Water; Things of less Specific Gravity float.=
Here are two tumblers of water. Throw some sand into one and some sawdust into the other. What happens? The sand sinks to the bottom, the sawdust floats at the top. We may stir them up as we like, but the sand will tumble to the bottom and the sawdust, as obstinately, rise to the top. Thus that which is lighter than the water floats, and that which is heavier (bulk for bulk) sinks. So, if we pour some oil into the water, it floats, and if we pour some coloured spirit in carefully, it also floats; while treacle and quicksilver sink to the bottom, just as the iron-filings do.
We saw that the iron-filings sank, because iron is heavier than water. Here is a piece of the thin tinned sheet-iron that they make tin boxes of. What will happen if we drop it into the water? It is heavier than water, bulk for bulk, and therefore it will sink as you see it does.
But now here is a “tin” canister made of this very same tinned sheet-iron. We drop that into the water, and you see it does not sink at all, but floats at the top as if it were made of cork. Here is a perplexity. We were sure just now that iron is heavier than water, and here is an iron box floating! Is this an exception to the law? Not at all; for what we said was that a thing would float if it were lighter, bulk for bulk, than water. Now let us weigh the tin box, and having weighed it let us next try to find out how much the same bulk of water weighs. This may be done very simply, for the walls of the box are very thin, so that the inside of the box is very nearly as large as the whole box. Consequently, if we fill the box with water, and then weigh the water, we shall find out, very nearly, what is the weight of a bulk of water as great as that of the box. But if we do this, we shall find that the water which was contained in the box, weighs very much more than the box does. So that, bulk for bulk, the box, although it is made of iron, is really lighter than water, and that is why it floats.
You will all have heard of the iron ships which are now so common, and you may have wondered how it is, that ships made of thick plates of iron riveted together, and weighing many thousand tons, do not go to the bottom. But they are nothing but our tin canisters on a great scale, and they float because each ship weighs less than a quantity of water of the same bulk does.
It is because of this property of water to bear up things lighter than itself, and because of that other property of being easily moved which the particles of water have, that the sea, and rivers, and canals, are such great highways for mankind.
For there is nothing so heavy that it may not be made to float in water, if the box which holds it is large enough to make the weight of the whole less than the weight of the same bulk of water. And then, having once got the weight to float, the particles of water are so easily moved, that the force of the winds, or of oars, or of paddles, readily causes it to slip through the water from one place to another.
26. =A Body which Floats in Water always occupies as much Space beneath the level of the Surface of the Water as is equal to the Volume of Water which weighs as much as that Body; in other words, it displaces its own Weight of Water.=
A cubic inch of water weighs about 252 grains and a half. Suppose that the tin box in the previous experiment was square, and had the bulk of 100 cubic inches, then the weight of a corresponding volume of water would be 25,250 grains. If the box weighed 8,416 grains, just a third of its bulk would be immersed; if 12,625 grains, half; if 16,832 grains, it would sink two-thirds of its volume, and so on. Or, if, when the box is floating, you make a mark upon its side at the exact level of the surface of the water, the bulk of that portion of the box which lies below the water-level can be ascertained. Suppose it to be thirty cubic inches, then the weight of the box will be 30 × 252·5 or 7575 grains. Hence it may be said that the immersed part of a floating body takes the place of the water which it displaces, and, as it were, represents it. If you press downwards upon the floating box, there is a feeling of resistance as it descends, and when the pressure is taken off, the body immediately rises again. Hence the water presses upwards against the bottom of the floating body. But it also presses against the sides, for if the sides of the box are very thin they will be driven in. If a thin empty bottle is tightly corked and lowered into deep water the cork will be driven in, or else the bottle will be crushed.
27. =Water presses in all Directions.=
Thus water presses in all directions upon things which are immersed in it.
If a long wooden or metal pipe, placed vertically, has its lower end stopped with a cork which does not fit very tightly, and water is poured into the top of the tube, the water will at first fill the part of the tube above the cork, and its weight will exert a certain =pressure= on the cork. In fact, if the end of the tube is stopped by applying the palm of the hand closely against it, the downward pressure of the water will have to be overcome by a certain amount of effort. As the water accumulates, this downward pressure will become greater and greater until the hand is driven away, or the cork is forced out, and the water falls to the ground. The pressure in this case is the same as the weight of the water, and the cork would have been driven out equally well by a rod of lead of the same weight.
Suppose the tube to be square, and that the inside of the square measures exactly one inch each way. Then an inch of height of the tube will hold exactly one cubic inch of water. Since one cubic inch of water weighs 252 grains and a half, as much water as will fill the tube about two feet three inches and a half high, will weigh a pound (7,000 grains), and fifteen pounds of water will fill such a tube between thirty-three and thirty-four feet high. And these respective weights measure the pressure of two columns of water, one twenty-seven and a half inches high, and the other nearly thirty-four feet high, on a square inch of the surface on which they rest.
The specific gravity (§ 24) of lead is 11·45; in other words it is about eleven and a half times denser than water. Therefore if a bar of lead cut square and one inch in the side, and rather less than 1/11th of the height of a column of water, is slipped into the tube in place of the water, it will exert the same pressure on the bottom.
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[2] 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[3] 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.