equal to the weight of a column of water of which the base is one square inch and the height is 28 inches. Such a column would contain 28 cubic inches of water. cubic foot of water weighs about 1000 ounces Avoirdupois, Now a so that a cubic inch weighs ounces, that is about a pound. It will be con weigh venient to remember that a column of water of which the area of the base is a square inch and the height is 28 inches weighs about a pound Avoirdupois. 360. But let us advert to the evidence for the truth of the preceding statement. We might contrive some experimental test. For instance the vessel might be placed high on supports at its corners, so as to allow of easy access to the base; then a tube might be inserted at P in which a piston should work; and the force necessary to sustain this piston in its place could be found by trial. Or we might adopt some methods of reasoning. For instance the sides of the vessel being vertical it seems obvious first that the whole pressure on the base must be equal to the whole weight of the liquid, and next that the pressure on any assigned part of the base will be proportional to the area of the part; and from these two natural suppositions the result will follow. There is also a method of reasoning which may appear somewhat artificial to the reader at first, but which well deserves attention as it is very useful in the theoretical investigations of the subject. Consider a vertical column of the liquid which has for its base an area of a square inch at P, and reaches up to the surface of the liquid. Conceive this to become solid; then we may take it as obvious that the pressure on the square inch is not altered. The weight of this solid column must be supported by the resistance of the base, which is equal and opposite to the pressure the liquid exerted on the base. For the liquid around the column will exert pressures on it only in horizontal directions, and so will in no degree counteract the weight of the column. Thus finally the pressure on the square inch of area at P is equal to the weight of the column of liquid standing on this square inch of area as base. 361. Next suppose a plane area of one square inch to be placed at any point between P and Q, in a horizontal position. The pressure on one side of it, say the upper side, will be equal to the weight of the column of liquid above it. This will appear obvious on reflection. We might suppose all the liquid below the plane area to become solid, and allow that the pressure on the plane area would remain unchanged: then this case reduces to the former. 362. Next suppose that at any point of the liquid we put an area of one square inch inclined to the horizon. The pressure on one side will be the same in amount as if the area were horizontal and at the same depth. The words at the same depth are used for brevity; they require a little explanation in order to bring out their strict sense. The depth of the inclined plane must be understood to mean the average depth, that is the depth of the centre of gravity. It would not be easy to obtain a very simple direct verification of this statement; but we may give an experimental illustration which will serve to render the meaning clear. Let there be a flat piston moving in a tube closed at the bottom and quite water-tight; and in the tube let there be a spring which resists the motion of the piston, so that a certain pressure must be exerted on the piston to maintain it at a certain position in the tube. Put the whole under the surface of the liquid; then the pressure exerted by the liquid pushes the piston in until there is equilibrium between the pressure and the resistance of the spring. Then for all positions of the piston so long as the centre of gravity of its area remains at the same depth the piston will remain in equilibrium. 363. All the results we have given in this Chapter are obtained on the supposition that the upper surface of the liquid is left free. If a lid is put on the upper surface, and pushed down, this gives rise to an additional pressure which is transmitted to every point of the liquid. It will appear hereafter that the atmosphere produces a pressure of about fifteen pounds on every square inch of surface exposed to it; and this pressure is transmitted through the liquid to a square inch of area placed in any position within the liquid. 364. But at present we leave out of consideration the action of any other force except the weight of the liquid itself; and the results at which we have arrived may be summed up briefly thus: the pressure at any point of a liquid is proportional to the depth of the point below the surface, and is the same in every direction. Like many other brief statements this would be scarcely intelligible without previous explanations. We measure pressure at any point by the pressure on a certain small area, say a square inch, so placed as to have its centre of gravity at the point; and when we say that the pressure is the same in every direction we mean that this area may be placed at any inclination to the horizon. 365. The fact that the pressure is the same in all directions round any assigned point, to which we have just drawn attention, is quite distinct from the fact that liquids transmit pressure from one point to another: both are very important properties of liquids. 366. The reader will observe that we speak of the pressure of a liquid at a point and not of the pressure on a point; in order to form a notion of the pressure of a liquid we must suppose that it is exerted on some definite area; this area may be very small, but it is not what is called a point in geometry. XXVIII. VESSELS OF ANY FORM. 367. In the preceding Chapter we supposed liquid to be contained in a vessel with vertical sides; but we must now proceed to some other cases. Let us suppose liquid to be contained in vessels which have sides that are not vertical; these sides may slope outwards as in the left-hand side diagram, or inwards D A B A B as in the right-hand side diagram. The two results which were obtained in the preceding Chapter, and summed up in Art. 364 are still true, and thus we shall be led to some curious and important consequences. 368. Consider the case represented by the left-hand side diagram. The pressure on the base of the vessel is equal to the weight of such a column of the liquid as would stand vertically over the base; thus it is less than the weight of all the liquid contained in the vessel. The weight of the liquid contained in the vessel is equal to the vertical component of the pressure on the vessel; but this does not fall entirely on the base; part falls on the inclined sides. Next consider the case represented by the right-hand side diagram. The pressure on the base of the vessel is equal to the weight of such a column of the liquid as would stand vertically over the base; thus it is greater than the weight of all the liquid contained in the vessel. In this case, as in the former, there is pressure by the liquid on all the vessel in contact with it, and therefore resistance from the vessel on the liquid. But in this case the vertical component of the resistance from the inclined sides tends downwards; and the difference between this and the resistance of the base upwards is equal to the weight of the liquid in the vessel. 369. The following is the general result. Let there be a series of vessels all having flat bases of the same area, all open at the top, and filled with the same liquid up to the same height; then the pressure on the base of any vessel will be the same, namely the weight of such a column of the liquid as would stand vertically over the base. The vessels may have any shape whatever; they may be like cups, or jugs, or decanters, or pails; and the opening at the top may be as small as we please. It is plain that we have thus a fact of the same nature as that involved in the Hydrostatic Paradox. Suppose that the vessel is in the form suggested by the diagram, large and shallow, with a tall slender neck. Pour in liquid until it fills all the shallow part and the neck up to CD. Then the pressure A B on the base of the vessel, represented by AB, is equal to the weight of such a column of the liquid as would stand on this base and reach up to CD: and it is obvious that this may be many times as large as the weight of all the liquid contained in the vessel. 370. The reader will observe that the pressure about which we are speaking is the pressure of the liquid on that side of the base with which it is in contact, and not the pressure between the other side of the base and the table or ground on which the vessel may be supposed to stand. The latter is equal to the sum of the weights of the vessel and the liquid which it contains, by the ordinary principles of mechanics. 371. Experimental evidence can be furnished of the truth of the general statement of Art. 369. Vessels are constructed of various shapes, as suggested in that Article, and having bases of the same area. These bases are not fixed to the sides of the vessels, but are kept in contact with them by forces which can be exerted by means of a lever. Then it is found that when the vessels are filled up to the same height the same force must be exerted in every case in order to keep the moveable base in its place. |