So we conclude that in a heavy fluid of uniform density (2) The pressure will be the same in all directions at any particular point. 34. We have next to consider in what way the pressure varies from point to point in the interior of a fluid of uniform density when it is in equilibrium, and first we shall shew that the pressure is the same at all points in the same horizontal plane. Let A and B be two points in the same horizontal plane in the interior of a fluid of uniform density. Imagine all the fluid contained in a small horizontal cylinder, of which AB is the axis, to become solid. Then the forces acting on the cylinder are to the axis. (1) The fluid pressures on its curved surface) perpendicular (2) The weight of the cylinder (3) The fluid pressure on the end 4 parallel to the axis. (4) The fluid pressure on the end B Of these (1) and (2) have no tendency to produce motion in the direction of the axis (Art. 17). Therefore, since there is no horizontal motion, fluid pressure on end A-fluid pressure on end B. And since, the ends being very small, the pressure at every point in each end may be regarded as the same, pressure at point A = pressure at point B. 35. The pressure at any point within a heavy inelastic fluid, not exposed to external pressure, is proportional to the depth of that point below the surface of the fluid. B Let P and Q be two points at different depths below the surface of the fluid. Suppose two small equal and horizontal circles to be described round P and Q as centres. Then suppose the fluid in the two small vertical cylinders PA, QB, extending from the bases P and Q to the surface, to become solid. Now the forces acting on the cylinder PA are (1) The fluid pressures on its curved surface, all of which are perpendicular to the axis. (2) The weight of the cylinder (3) The fluid pressure on the base P parallel to the axis. Of these (1) has no tendency to produce motion in the direction of the axis (Art. 17). Hence since there is no vertical motion, fluid pressure on base P= weight of cylinder PA. So also, fluid pressure on base Q=weight of cylinder QB. Hence pressure at point P: pressure at point Q :: pressure on base P: pressure on base Q, (Art. 24.) COR. If pressure at P=pressure at Q depth of Pdepth of Q. The pressure of the atmosphere on the surface of the fluid is not taken into account, but we shall shew hereafter how it affects the pressure at a point in the interior of a fluid. 36. The surface of a heavy inelastic fluid at rest is horizontal. D Let A and B be two points in the same horizontal plane in the interior of a heavy fluid at rest. Suppose the fluid contained in a small horizontal cylinder of fluid, of which AB is the axis, to become solid. Then, fluid pressure on end A = fluid pressure on end B (Art. 34), and, since the ends are equal, fluid pressure at point A = fluid pressure at point B. Hence A and B are at the same depth below the surface of the fluid (Cor. Art. 35), and if we draw AC, BD vertically to meet the surface in C, D, AC=BD, also, AC is parallel to BD; .. CD is parallel to AB (Eucl. 1. 33); .. CD is horizontal. Similarly any other point in the surface may be proved to be in the same horizontal plane with C or D; .. the surface is horizontal. 37. The proposition that the surface of a fluid at rest is horizontal is only true when a very moderate extent of surface is taken. Large surfaces of water assume, in consequence of the attraction exercised by the earth, a spherical form. The following practical results are worthy of notice: (1) All fluids find their level. If tubes of various shapes, some large and some small, some straight and others bent, be placed in a closed vessel full of water, and water be then poured into one of the tubes, the fluid will rise to a uniform height in it and all the other tubes. (2) If pipes be laid down from a reservoir to any distance, the fluid will mount to the same height as that to which it is raised in the reservoir. (3) The surface of a fluid at rest furnishes a means of observing objects at a distance in the same horizontal plane with a mark at the place of observation. 38. We have seen that in an inelastic fluid at rest the pressure at any point depends on the depth of that point below the surface of the fluid, that is, on the length of the vertical line drawn from the point to meet a horizontal line drawn through the highest point in the fluid. Thus if ABC be a conical vessel with a horizontal base, standing on a table, and filled with fluid, the pressure at any point P is determined in the following manner. B A B From A, the highest point of the fluid, draw a vertical line meeting the horizontal plane passing through P in the point Q. Then the pressure at P=prossure at Q, because P and Q are in the same horizontal plane. But pressure at Q depends on the length of AQ: therefore pressure at P depends on the length of PR, a line drawn vertically to meet the horizontal line AR. 39. If a vessel, of which the bottom is horizontal and the sides vertical, be filled with fluid, the pressure on the bottom will be equal to the weight of the fluid. Let ACDB (fig. I.) be a vessel whose bottom, CD, is horizontal, and its sides vertical. We may consider the fluid in this vessel to be made up of vertical columns of fluid. Each of these columns will press vertically downwards with its weight, and the sum of these pressures will be the weight of the fluid. Now the base of the vessel, being horizontal, will sustain all these vertical pressures; .. pressure on the base of the vessel = weight of the fluid. If the sides of the vessel be not vertical, as in figs. II. and III., the pressure on the base will be equal to the weight of a column of fluid ECDF, EC and FD being perpendicular to CD, and EF being the surface of the fluid. Hence if in the three vessels the bases are equal and on the same horizontal plane, and the fluid stands at the same height in the vessels, the pressure on the base in each case will be the same. The fluid in vessel I. produces a pressure on the base equal to its own weight. The fluid in vessel II. produces a pressure on the base less than its own weight. The fluid in vessel III. produces a pressure on the base greater than its own weight. S. H. 2 |