11. A Piston is a short cylinder of wood or metal, which fits exactly the cavity of another cylinder, and works up and down alternately. 12. A Valve is a closed lid affixed to the end of a tube or hole in a piston, opening into or out of a vessel, by means of a hinge or other sort of moveable joint, in such a manner that it can be opened only in one direction. 13. A Prism is a solid figure, the ends of which are parallel equal and similar plane figures, and the sides which connect the ends are parallelograms. The figure represents a rectangular prism, in which each of the lines bounding the surfaces of the prism is at right angles to each of the four lines which it meets. 14. We shall often have to use the expression Horizontal Section of a tube or hollow cylinder, and we may explain the meaning of the expression by the following example: Suppose a gun-barrel to be placed in a vertical position: suppose a wad to be part of the way down the barrel with its upper surface exactly parallel to the top of the barrel: then suppose the barrel to be cut away so as just to leave the upper surface of the wad exposed: the area of this surface of the wad is called the horizontal section of the barrel. 15. The mathematical theory of Hydrostatics is founded on two laws, which we shall now explain. 16. LAW I. The force exerted by a fluid on any surface, with which it is in contact, is perpendicular to that surface. 17. This law is merely a repetition of the definition of a fluid given in Art. 5, and we can best explain its meaning and application by an example. If AB be a cylinder immersed in a fluid the pressures of the fluid on the curved surface are all perpendicular to the axis of the cylinder, and the pressures of the fluid on the flat ends are all parallel to the axis. Now it is a law of Statics that a force has no tendency to produce motion in a direction perpendicular to its own direction. Hence the pressures on the curved surface have no tendency to produce motion in the direction of the axis, and the pressures on the flat ends have no tendency to produce motion in a direction perpendicular to the axis. 18. LAW II. Any pressure communicated to the surface of a fluid is equally transmitted through the whole fluid in every direction. 19. A characteristic property of fluids which distinguishes them from solid bodies is this faculty which they possess of transmitting equally in all directions the pressures applied to their surfaces. It is of great importance to form a correct notion of the principle of "the equal transmission of pressure," a principle which is applicable to all fluids, inasmuch as it depends upon a property which is essential to all fluids and is not an accidental property, as weight, colour, and others. 20. Suppose then we take a vessel ABCD, in the form of a hollow rectangular prism, and place it on a horizontal table. Place a block of wood, cut to fit the vessel, so that it rests on the base BC and reaches up to the level EF. Then if we place a weight P on the top of the block an additional pressure P will be imposed on the base of the prism. Now suppose the block to be removed and the vessel filled with an incompressible fluid up to the level of EF. Suppose a piston exactly fitting the vessel to be inserted and a pressure P applied by means of it to the surface of the fluid at EF. In this case the pressure P is transmitted by means of the fluid not only to the base BC, but also to the sides of the vessel, and if we take a unit of area, as a square inch, in the side FO, and a unit of area in the base BC, the same additional pressure will be conveyed to each. 21. That fluids transmit pressure equally in all directions may be shewn experimentally in the following manner: A B ABC is a vessel of any shape filled with fluid. Close the openings by pistons, kept at rest by such a force as may be required in each case. Then it will be found that if any additional force P be applied to the piston at A, the same force P must be applied to each of the pistons at B and C to prevent them from being thrust out. If the area of the base of one of the pistons, as B, be larger than the area of the base of the piston A, it is found that the pressure which must be applied to B to keep it at rest bears the same relation to the pressure applied to A that the area of the base of B bears to the area of the base of A. 22. From the preceding article it is clear that if a body of fluid, supposed to be without weight, be confined in a closed vessel, the pressure communicated to the fluid by any area in any part of the vessel will be transmitted equally to every equal area in any other part of the vessel. It is owing to this fact that the use of a Safety Valve can be depended on. Thus, if the vessel A be full of steam and the pressure of the steam be required to be kept down to 200 lbs. on the square inch, if a valve B, whose area is a square inch, be placed at any part of the vessel, and be so loaded that it will require a force of 200 lbs. to raise it, then if the steam acquire an increase of pressure above 200 lbs. on the square inch, the valve will open, and will remain open till the pressure of the steam is just equal to 200 lbs. on the square inch. 23. Any force, however small, may by the transmission of its pressure through a fluid, be made to support any weight, however large. Suppose DE and FH to be two vertical cylinders, connected by a pipe EH, and suppose FH to have a horizontal section much larger than the horizontal section of DE: for instance, let the area of a horizontal section of FH be 400 square inches, and the area of a horizontal section of DE be 1 square inch. Now if water be poured into the cylinders, and pistons A and B be applied to the surface at D and F, whatever force we apply to A will be transmitted to each portion of the base of the piston B which is equal in area to the base of the piston A. Hence a pressure of 1lb. applied to the piston A will produce a pressure of 400 lbs. on the base of the piston B, and will therefore support a weight of 400 lbs. placed on the piston B. This effect of pressure by the medium of a fluid is often called The Hydrostatic Paradox. |