LEVELLING.
If a mass of fluid contained in a vessel is in equilihrio, any one particle of the fluid is pressed in every direction, with a force equal to a weight of the column of the fluid, whose base is equal to that particle, and whose height is the depth of the particle below the surface.
Let Fig. 5, be the particle of fluid whose depth in the vessel of fluid ABCD is ep. We may suppose, as for merly, that a portion of the water is frozen, so as to form a tube of ice cp, whose diameter is equal to that of the particle p, without any change taking place in the pres sure sustained by p. In this case, the particle p is ob viously pressed downwards with the weight of the column e fz; and, consequently, the measure of this pressure is the absolute weight of the column e12. But as the particle is in equilibrio, it must be pressed with this force in every direction.
The proposition is also true of a particle situated at ni, for draw ing the horizontal line mg; and supposing a syphon of ice fg ism to be formed, it is obvious that the column of fluid in the branch m h is in equilibrio with, or balanced by, the column in g II; consequently the particle of water at m is pressed with the same force as the par ticle at g, that is, with a column of water whose height is fg.
Cor. Hence it follows, that every particle of a vessel containing fluid is pressed with a force equal to a column of fluid, whose base is the particle, and whose height is the depth of the particle below the surface ; for, since the particle of fluid adjacent to this particle of the vessel is pressed in every direction with this force, it must exert , the same force against that particle of the vessel.
The pressure exerted by a fluid upon any given portion of the vessel which contains it, is equal to a column of the fluid whose base is the area of the given portion, and whose altitude is the depth of the centre of gravity of the portion below the fluid surface.
Let 771 n be the given portion of the vessel ABCD, Fig.
6. filled with fluid, and let us conceive this portion to be occupied by any number of particles m, 0,p, n, &c. then the pressure sustained by each of these particles, by Prop.
III. will be mxmu-l-oxox+p xpy+nxzzz,bcc.; but, by the property of the centre of gravity or inertia, (See MEertaxics,) the sum of these products is equal to the distance EF of the centre of gravity E, from the sur face at F, multiplied into the number of pai.ticles m, n, o, p; that is, nrxmu+ ox ox-1-12XPy+71 xhz=EFX n, 0, p ; consequently, since m, n, o, p, represents the area, or the number of particles in the given portion m n, the pressure upon m n = EF x m n.
Con 1. It follows from this proposition, that the pres sure sustained by the bottom of the vessel is not the same as the weight of the fluid contained in the vessel. In the cylindrical vessel shewn in Fig. 7, or in any vessel, what ever be its shape, in which the sides are perpendicular to its bottom, the pressure upon the bottom is ac curately measured by the weight of the water which it contains ; but in vessels of all other shapes, such as Fig. 8,9, the pressure on the bottom is measured by mnxm x, which in Fig. 8 is much less than the weight of water in the vessel, and in Fig. 9 much greater.
Cor. 2. The truth of what is called the Hydrostatic Paradox is easily deduced from the preceding proposi tion. Let ABCDEFGH, Fig. 10, be a vessel filled with water, then, by the proposition, the pressure upon GF = GF x GI, however narrow be the column ABCD ; that is, zhe pressure exerted upon the Bottoms of -vessels filled with fluid does not depend upon the quantity of the fluid which they contain, but solely upon its altitude. In like manner, it is obvious from Prop. II. that the water will stand at the same level a b AB, Fig. 11, in the two communicating ves sels a b c d, ABCD, consequently, any portion of fluid abed, however small, will balance any portion of fluid ABCD, however great.