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.
[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.