ELEMENTS OF HYDROSTATICS. SECTION I. GENERAL PROPERTIES OF FLUIDS. ART. 1. A FLUID is a body which can be divided in any direction, and whose parts can be moved among one another by any assignable force. Elastic fluids are those whose dimensions are increased or diminished when the pressure upon them is diminished or increased. Non-elastic fluids are those whose dimensions are independent of the pressure. Water, mercury, and probably all other liquids, are in a small degree compressible. Their resistance however to compression is so great, that the conclusions obtained on the supposition of their being incompressible, are in most cases free from any sensible error. 2. Let DEF (fig. 1.) be a piston without weight exactly fitting an orifice in the plane ABC, which forms the side of a vessel containing fluid. It is manifest that the fluid can make no effort to move the piston in any other direction than that of a normal to its surface, the piston may therefore be kept at rest by a force applied at some point G in it, and acting in a direction HG perpendicular to DEF. A force equal and opposite to this is called the pressure of the fluid on DEF. 3. The pressure of a fluid at a given point is measured by the quantity p, pr being the pressure of the fluid on an indefinitely small area è contiguous to the given point. When the pressure of a fluid on a given surface is the same, wherever that surface is placed, p is the pressure on an unit of surface. When the pressure on a given surface, varies with the situation of the surface, p is the pressure which would be exerted on an unit of surface, if the pressure at each part of the unit of surface were equal to the pressure at the given point. A 4. AXIOM. When a fluid is at rest, any portion of it may become solid without disturbing either its own equilibrium, or that of the surrounding fluid. For as long as the fluid remains at rest, it makes no difference whether the parts of which it is composed, are moveable among one another, and capable of being divided in any direction, or not. 5. Fluids press equally in all directions. Let Abc (fig. 2.) be a very small prism of fluid in the interior of a fluid at rest; then (Art. 4.) the equilibrium of A b c will not be disturbed, if we suppose it to become solid. Now if R be the accelerating force at A, Abc is kept at rest by the pressure of the surrounding fluid on its ends and sides, together with R. (mass prism) acting in the direction of the force at A. But if the prism remain similar to itself while its magnitude is diminished indefinitely, R. (mass prism) vanishes compared with the pressure on either of its sides; (for the former is proportional to A a3, the latter to A ao;) and we may consider the prism to be kept at rest solely by the pressures on its ends and sides: and these pressures are respectively perpendicular and parallel to ABC, therefore they must be separately in equilibrium. And since the pressures on Ab, Ac, Cb are in equilibrium, and perpendicular to the sides AB, AC, CB of the triangle ABC, they are proportional to those sides; hence if p. Ab, q. Ac be the pressures on Ab, Ac respectively, p. Ab: q. Ac = AB: AC, therefore pq. A perpendicular to Ab, Ac respectively, and Ab, Ac may be taken perpendicular to any two given lines, therefore fluids press equally in all directions: But p, q measure the pressures of the fluid at COR. 1. Suppose the sides of the base of the prism to be indefinitely small compared with its length; then if the pressure on ABC be increased or diminished in any degree without disturbing the equilibrium of Abc, the pressure on abc must be equally increased or diminished. Hence if F, G, H...... M, N, P (fig. 3.) be any series of points in a fluid at rest, so taken that the straight lines FG, GH......MN, NP may be wholly within the fluid, and the pressure at F be increased or diminished without disturbing the equilibrium of the fluid, the pressures at G, H......M, N, P will be equally increased or diminished. COR. 2. If the fluid be acted on by no accelerating force, the pressures on ABC, abc must be equal; therefore pressure at F= pressure at G= ... = pressure at N=pressure at P: or, the pressure is the same at all points in a fluid at rest acted on by no accelerating force. 6. Let the forces P, Q, R, &c. be in equilibrium when applied to pistons A, B, C, &c. fitting cylindrical apertures in the sides of a vessel filled with fluid. Let a, b, c &c. be the areas of the pistons, and suppose the fluid to be acted on by no accelerating force. Then since the fluid is at rest, the pressures on an unit of the surface of each of the pistons must be equal, 7. Let the fluid be incompressible; p, q, r, &c. the distances of the pistons A, B, C, &c. from fixed points in the axes of the cylinders in which they play; p+dp, q+dq, r+dr, &c. their distances from the same points after they have been moved in any manner. Then since the volume of the fluid in the vessel remains the same, Sp, Sq, dr, &c. are the virtual velocities of the pistons A, B, C, &c. to which the forces P, Q, R, &c. are applied. SECTION II. ON THE EQUILIBRIUM OF NON-ELASTIC FLUIDS ACTED ON BY GRAVITY. ART. 8. THE specific gravity of a body is the weight of an unit of its volume. 9. The density of a body is the quantity of matter in an unit of its volume. 10. Let W, M, V be the number of units of weight, mass, and volume contained in the weight, mass, and volume of a given body, S its specific gravity, D its density, g the force of gravity; then S=weight of one unit=gD M=mass of V units=DV W = weight of V units = SV=gDV. 11. When a fluid acted on by gravity is at rest, the pressures are equal at all points in the same horizontal plane. Let P, Q (fig. 4.) be any two points in the same horizontal plane in the interior of a mass of fluid at rest. The equilibrium of the fluid will not be disturbed if we enclose a part of it in a tube PMQ of uniform bore, having its branches MP, MQ symmetrical with respect to a vertical line. Then since the columns MP, MQ are symmetrical, and similarly situated with respect to the direction of gravity, they will balance when the pressures at P and Q are equal. But if the pressures at P and Q be unequal, the fluid will begin to move towards that end at which the pressure is the least, and the equilibrium will be destroyed; therefore in order that the fluid may be at rest, the pressures at P and Q must be equal. 12. To find the pressure at any point in a mass of fluid at rest. Let the vertical prism AEF (fig. 5.) be a portion of a fluid at rest; then (Art. 4.) the equilibrium of AEF will not be disturbed if we suppose it to become solid. Now AEF is kept at rest by its own weight, and the pressures on its sides and ends. The pressures upon its sides act in a horizontal plane; its weight, and the pressures upon its ends, act in a vertical line; therefore the latter must be capable of maintaining equilibrium separately; therefore pressure on DEF = weight of prism of fluid AEF+ pressure on ABC. Let m, p be the pressures at A, D respectively, p the density of the fluid, ABC =k, AD=≈; then the pressure on ABC=mк, the pressure on DEF=рк, and the weight of AEF=gp≈k; therefore PK gpxk+mk, therefore p=gpx+m. COR. 1. Let A be a point in the open surface of the fluid, then m=0, and p=gpx. COR. 2. Since fluids press equally in all directions, and the pressure is the same at all points in the same horizontal plane, the pressure on a small area of any plane is ultimately equal to the pressure on an equal area of the horizontal plane that intersects it. 13. The surface of a fluid at rest is a horizontal plane. Let A, P (fig. 6.) be any two points in the surface of a fluid at rest, AB, PQ vertical straight lines intersected by a horizontal plane in B, Q; p the density of the fluid. Then, (Arts. 11, 12.) gp. PQ = pressure at Q = pressure at B=gp. AB; therefore PQ=AB, therefore A and P are in the same horizontal plane. 14. The common surface of two fluids that do not mix is a horizontal plane. Let A, P (fig. 7.) be any two points in the common surface of two fluids that do not mix; BAC, QPR vertical straight lines intersected by horizontal planes in B, Q, and in C, R; p, σ the densities of the upper and under fluids respectively. Then, (Art. 12.) pressure at A-pressure at B=gp. AB, also pressure at C- pressure at Ago. AC, pressure at C-pressure at B=g. (p. AB+o. AC) in like manner |