EXAMPLES.-I. (1) IN the experiment described in Art. 23, if the horizontal section of the small cylinder be 13 inches, and that of the larger cylinder 64 inches, find the weight supported under a pressure of 1 ton exerted on the piston of the small cylinder. (2) If the horizontal section of the small cylinder be 13 inches, and that of the large cylinder 240 inches, find the weight supported by a pressure of 3 cwt. applied to the piston of the small cylinder. (3) If the pistons are circular, the diameters being 13 inch and 50 inches, find the weight supported by a pressure of 15 lbs. applied to the smaller piston. (N.B. The areas of circles are as the squares of their diameters.) (4) A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface not capable of bearing a pressure of more than 4 pounds on the square foot. If a piston, the area of which is 2 square inches, be fitted into an aperture in the upper surface, what pressure applied to it will burst the vessel? (5) A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface not capable of bearing a pressure of more than 9 lbs. upon the square foot. If a piston, the area of which is one square inch, be fitted into an aperture in the upper surface, what pressure applied to it will burst the vessel? (6) If the horizontal section of the small cylinder be 11 inch and the diameter of the large piston 20 inches, find the lifting power of the machine under a pressure of 1 ton exerted on the piston of the small tube. (N. B. The area of a circle is times the square of the radius nearly.) 22 7 24. The pressure at any point in any direction in a fluid is a conventional expression used to denote the pressure on a unit of area imagined as containing the point, and perpendicular to the direction in question. For example, if the whole pressure of a fluid on the bottom of a vessel is 2000 lbs., and the pressure is uniform throughout, then if we take a square inch as the unit of area, and the area of the bottom of the vessel is 40 square inches, 2000 the pressure at a point in the base is lbs. or 50 lbs. 25. The student must carefully observe the distinction between the expressions "pressure on a point" and "pressure at a point": the former is zero, because a point has no magnitude. 26. If a mass of fluid is at rest, any portion of it may be supposed to become rigid without affecting the conditions of equilibrium. Thus if we consider any portion A of the fluid in a closed vessel, we may suppose the fluid in A to become solid, while the rest of the fluid remains in a fluid state, or we may suppose the fluid round A to become solid, while the fluid in A remains in a fluid state. 27. The importance of the principle laid down in the preceding article may be seen from the following considerations. The laws of Statics are proved only in the case of forces acting on rigid bodies. Now since the supposition of any part of a fluid becoming solid does not affect the action of the forces acting upon it, and since we can in that case obtain the effect of those forces by the laws of Statics, we shall know their effect on the fluid. 28. If a body of fluid, supposed to be without weight, be confined in a closed vessel, so as to exactly fill the vessel, an equal pressure will be exerted on the fluid by every equal area in the sides of the vessel (Art. 22), and we proceed to shew that the pressure is the same in all directions at every point of the fluid. For let O be any point in the fluid, and AB, CD two plane surfaces, each representing a unit of arca, passing through and parallel to two sides of the vessel EF, GH. Then drawing straight lines at right angles to AB, CD from the extremities of AB, CD to the sides of the vessel, we may imagine all the fluid except that contained in the prism ABNM to become solid. Then the pressure exerted on the fluid by the area MN will be transmitted to AB. Again, if we suppose all the fluid except that contained in the prism CDSR to become solid, the pressure exerted on the fluid by the area RS will be transmitted to CD. Now the pressures exerted on the fluid by the areas MN, RS are equal, and consequently the pressures on AB, CD will be equal, that is, the pressure at the point O is the same in all directions. Also since the distance of the point O from the sides of the vessel is not involved in the preceding considerations, it follows that the pressure is the same at every point. CHAPTER II. On the Pressure of a Fluid acted on by Gravity. 29. IN the preceding chapter we considered the consequences that result from the peculiar property, essential to all fluids, of transmitting equally in all directions the pressures applied to their surfaces. We have now to consider the effects produced by the action of gravity upon the substance of a fluid. 30. The student must mark carefully the distinction between force applied to a surface and force applied to each of the particles composing a body. As an example of these distinct forces consider the case of a book resting on a table. Force is applied to the surface of the book by the table, and thus is counterbalanced the force of gravity which acts upon each particle of which the book is composed. 31. All fluids are subject to the action of gravity in the same way as solid bodies. Each particle of a fluid has a tendency to fall to the surface of the earth, and in a mass of fluid at rest there is a particular point, called the centre of gravity, at which the resultant of all the forces exercised by the attraction of the Earth on the particles composing the fluid may be supposed to act. 32. The term density is applied to fluids, as it is to solid bodies, to denote the degree of closeness with which the parti cles are packed. When we speak of a fluid of uniform density, we mean that if from any part of the body of fluid a portion be taken, and if from any other part of the body of fluid a portion like in form and equal in volume to the former portion be taken, the weights of the two portions will be equal. 33. If a vessel be filled with a heavy fluid of uniform density the pressure at every point in the interior of the fluid will not be the same, because the pressure which results from the action of gravity will vary in magnitude according to the position of the point in the containing vessel. Consider a closed surface of small dimensions containing the point A, and suppose the fluid outside the closed surface to become solid. The fluid within the closed surface will exercise pressure against the surface at every point, and these pressures will be unequal, because the fluid is acted on by gravity. But we may conceive that, if the quantity of fluid within the surface be very small, the difference between the pressures at different points of the surface will be very small, and when the surface is indefinitely diminished the pressures exercised by the fluid at each point of the surface may be regarded as equal, and the weight of the fluid may be neglected. Thus we can consider it as the case of a weightless fluid and apply the conclusions of Art. 28. Hence all the planes of equal area which can be drawn, passing through the point A and not extending beyond the small surface, may be considered to be subject to equal pressures. |