Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/457

441 HYDROSTATICS,] HYDROMECHANICS 441 H is called the pressure height (Everett, Units and Physical Constants). For dry air at C. , H = 7 988 x 10 5 centimetres. Prop. III. To find the pressure on the bottom of a vessel of any form containing liquid, the bottom of the vessel being supposed a horizontal plane. The pressure at any point of the base being the same and due to the depth below the free surface, whatever be the shape of the sides of the vessel, as in fig. 5, it follows that the pressure on the base AB is equal to the weight of the liquid contained in an imagin ary cylinder, traced out by vertical lines round the base AB reaching to the surface. In order to illustrate this, let there be four vessels A, B, C, D (fig. 6), having bottoms all of the same area, and closed by plates E, F, G, H, of the same weight. Let the plates also be kept in their places by means of strings passing over pulleys and supporting equal weights u ,iv ,iv&quot;,w.&quot; The weights will measure the vertical forces acting on the plates, i.e., the bases of the vessels. It will be found that water must be poured into each vessel to the same height to cause the plates to descend. The same will be the case whatever be the shapes of the vessels, and the extreme cases of a very large base acted upon by the pressure of water in a very con- A B Fig. 5. tracted vessel, and of a very small area kept closed by a very small weight when the vessel is very much enlarged above, constitute what used to be calkj the hydrostatic paradox. Another kind of hydro- ii Fig. 6. static paradox is the hydraulic press, where water, pumped in with small exertion by a forcing pump of which the plunger is of small diameter, causes a second plunger of very much greater diameter to rise and produce a very great pressure. Prop. IV. (Archimedes s Principle). The resultant pressure of a fluid on a body immersed in it acts vertically upwards through the centre of gravity, and is equal to the weight of the displaced liquid. For, suppose the body removed and its place filled up with fluid as it would be at rest, and imagine for clearness this iluid solidified. The fluid pressures which act upon this solidified fluid are the same as before, and since the fluid is in equilibrium under its own weight and the fluid pressures, the resultant of the fluid pressures must be a single vertical force equal to the weight of the displaced liquid, and acting upwards through the centre of gravity of the displaced liquid. Corollary. If a body float at rest in fluid, the weight of the body must be equal to the weight of the displaced fluid, and the C.G.s of the body and displaced fluid must be in the same vertical line. For instance, the weight of water displaced by a ship is equal to the weight of the whole ship, masts, rigging and all When a balloon is at rest in the air, the weight of the balloon is equal to the weight of air displaced. Archimedfs s principle is employed to determine the densities of bodies ; for, if w be the weight of a body weighed in a balance in air (strictly speaking in vacua), and if w be its apparent weight when immersed in water, then w-w, the resultant upward pres- e of the water, is equal to the weight of water displaced, and the density relative to water is therefore w-ttf In very accurate weighings the weight of air displaced must be taken into account, the real weight of a body being its apparent weight in air plus the weight of air displaced. Thus, if in air a pound of lead and a pound of feathers balance in a pair of scales, if placed under a receiver and the air exhausted the feathers would preponderate. The densities of liquids and solids are also determined by hydro meters, for a description of which consult the article HYDROMETER. The numerical measure of the density of a substance being the quotient of the number of units of mass (or weight) by the number ol units of volume, it follows that in a mixture of fluids the density of the mixture is the quotient of the number of units of mass in the component fluids by the volume of the mixture. The volume of the mixture will in general be the sum of the volumes of the component fluids, except in some cases where the fluids combine chemically with each other. If equal volumes of the component fluids be taken, the density of the mixture will therefore be the arithmetical mean of the densities of the component fluids ; but if equal weights be taken, the density of the mixture will be the harmonica] mean of the densities of the component fluids, no change of volume being supposed to take place. Prop. V. To find the resultant vertical and horizontal pressures on one side of a portion of a surface immersed in fluid at rest under gravity. The resultant vertical pressure is the weight of the superincum bent fluid contained by vertical lines drawn round the bounding curve of the surface to the free surface. If, however, the free sur face does not extend over the surface, we must suppose it made to do so by filling up the empty space with fluid. The line of action of the resultant vertical pressure passes through the centre of gravity of this superincumbent fluid. The resultant horizontal pressure in any direction is equal to the resultant pressure on the plane area traced out on a plane perpendi cular to the given horizontal direction by horizontal lines drawn through the bounding line of the surface in the given direction ; this plane area is called the orthogonal projection of the surface on a plane perpendicular to the given direction. For, resolving in the given direction the weight of the enclosed liquid and the pressures on the cylindrical surface traced out by the horizontal lines acting in a direction at right angles, the horizontal components of the pressures on the ends balance, which proves the proposition. The line of action of this horizontal pressure passes through the centre of pressure of the plane area (vide &quot; Centre of Pressure &quot;). If a plane area be immersed in homogeneous liquid at rest under gravity, the resultant force acting on one side of the area will be the product of the area and the pressure at the centre of gravity of the area For, dA denotipgr iv element of the area, su&amp;lt;J. it drpth,, th resulfan* pressura which proves the proposition. General Equations of Equilibrium of any Fluid at rest under any forces. If we take .any arbitrary origin 0, and three rectangular axes of reference Ox, Oy, Oz, then, if p be the pressure, p the density, and X, Y, Z the components of the impressed force per unit of mass at the point xyz, the equilibrium of the fluid in any closed surface S requires, resolving parallel to the axis of x, the integrations extending respectively over the surface and through the volume of the space S, and I, m, n denoting the direction-cosines of the outward drawn normal at the surface element dS. But by Green s transformation and therefore leading to the differential relation Similarly, dp dy dx dp The three equations of equilibrium obtained by taking moments about the axis will be found to be satisfied identically. Hence the space variation of the pressure in any direction is equal to the resolved force per unit of volume in that direction. The resultant force is therefore in the direction of the greatest space variation of the pressure, that is, normal to the surface of equal pressure ; and the lines of force must therefore be capable of being- cut orthogonally by a system of surfaces, which will be the surface? of equal pressure.