13. Rule for Gauging Cisterns or Tubs.-To find the capacity of a cistern or tub, take the dimensions, in inches and tenths of an inch. Add together the square of the head Head Diameter Midway Bottom Height D+d R Diameter FIG. 15 diameter, Fig. 15, the square of the bottom diameter, and four times the square of the midway diameter (ascertained by adding the top and bottom diameters together and dividing by 2), and divide the sum by 6, which gives the square of the true mean diameter; multiply this by the height of the cistern, and the product will be the capacity, in cylindrical inches. As there are 294 cylindrical inches in a gallon, divide this last product by 294, and the quotient will be the number of gallons contained in the cistern. A cylindrical inch is equal to the capacity of a cylinder 1 inch in diameter and 1 inch high; it is less than a cubic inch, 294 cylindrical inches being equal to 231 cubic inches. EXAMPLE.-What is the capacity, in gallons, of a tub whose bottom diameter is 10 feet, top diameter 9 feet, and height 8 feet 4 inches? SOLUTION. Top diameter, 108 in. Divided by 2)228 in. Midway diameter, 114 in. 13,008 × 100 ÷ 294 = 4,424.5 gal. Ans. 14. Should the cistern be full, warped so as not to be a perfect circle, or otherwise in such condition that the diameter of the bottom cannot be taken, the following rule, though not mathematically correct, is for all practical purposes sufficiently so, the difference being shown only in large cisterns, where the difference between the top and bottom diameters is considerable. Rule.-Divide the outside circumference of the cistern half way between the bottom and top, in inches, by 3.1416 (or multiply by 7 and divide by 22), and the result will be the mean outside diameter, in inches. Deduct from this twice the thickness of the staves, and the result will be the mean inside diameter. Multiply this sum by itself and by the height, in inches, and the product by .0034; the final product will be the capacity of the cistern, in gallons. EXAMPLE.-What is the capacity, in gallons, of a cistern having the dimensions of the tub given in the example in Art. 13 and whose outside circumference is 367.5 inches? SOLUTION.- 367.53.1416 = 117 3 in. 114 in.; 114 X 114 X 100 = 1,299,600 X .0034 = 4,418 gal. Ans. ELEMENTARY PHYSICS INTRODUCTORY DEFINITIONS 15. Natural and Physical Science. Science, which may be defined as a classified knowledge of nature, is divided into natural and physical science. Natural science concerns itself with the external form and internal structure of bodies. Physical science considers only the matter of which these bodies are composed. Geology, mineralogy, botany, and zoology, which investigate the form and structure of the earth, of minerals, of plants, and of animals, respectively, are natural sciences. Physics and chemistry, which consider the properties of matter itself, whether it is light or heavy, hard or soft, combustible or incombustible, are physical sciences. 16. Matter is anything that possesses weight; that is, is acted on by gravitation. In studying matter, physical science considers: (1) The division of which matter is capable; (2) the attractions by which these particles are held together; (3) the motions that these particles may have. Science recognizes three divisions of matter-masses, molecules, and atoms. A mass, or body, of matter is any portion of matter appreciable by the senses. A molecule is the smallest particle of matter that a body can be divided into without losing its identity. An atom is an indivisible portion of matter. Atoms unite to form molecules; a collection of molecules forms a mass, or a body. The sun and the grain of sand are masses of matter; the smallest particles of sugar or of salt that still show the properties of these substances, respectively, are molecules of sugar or of salt. The still more minute particles of carbon or hydrogen, and of oxygen, that make up the molecule of sugar, or those of chlorine and of sodium that compose the molecule of salt, are atoms. 17. Bodies, which are collections of molecules, exist in three forms or conditions: solid, liquid, and gaseous. A solid body is one whose molecules change their relative positions with great difficulty; as iron, wood, stone, etc. A liquid body is one whose molecules tend to change their relative positions easily. Liquids readily adapt themselves to the vessel that contains them, and their upper surface always tends to become level. Water, mercury, etc. belong to this class. A gaseous body, or a gas, is one whose molecules tend to separate from one another; as air, oxygen, etc. Gaseous bodies are sometimes called aeriform (air-like) bodies and are divided into two classes: permanent gases and vapors. A permanent gas is one that remains a gas at ordinary temperatures and pressures. A vapor is a body that at ordinary temperature is a liquid or a solid, but when heat is applied becomes a gas. By means of heat, nearly all bodies may be finally vaporized, MECHANICS OF FLUIDS HYDROSTATICS 18. Hydrostatics treats of liquids at rest under the action of forces. 19. Liquids are very nearly incompressible. A pressure of 15 pounds per square inch compresses water less than 20000 of its volume. 20. Hydrostatic Pressure. Fig. 16 represents two cylindrical vessels of exactly the same size. The vessel (a) is fitted with a wooden block of the same size as, and free to move in, the cylinder; the vessel (b) is filled with water, whose depth is the same as the length of the wooden block in (a). Both vessels are fitted with air-tight pistons P, each of whose areas are 10 square inches. (a) FIG. 16 (b) Suppose, for convenience, that the weights of the pistons, block, and water are neglected, and that a force of 100 pounds is applied to both pistons. The pressure per square inch will be 100 ÷ 10 = 10 pounds. In the vessel (a), this pressure will be transmitted to the bottom of the vessel, and will be 10 pounds per square inch; it is easy to see that there will be no pressure on the sides. In the vessel (b), an entirely different result is obtained. The pressure on the bottom will be the same as in the other case, that is, 10 pounds per square inch, but, owing to the fact that the molecules of the water are perfectly free to move, this pressure of 10 pounds per square inch is transmitted in every direction with the same intensity; that is to say, the pressure at any point c, d, e, f, g, h, etc., due to the force of 100 pounds, is exactly the same, and equals 10 pounds per square inch. In the foregoing explanation, it was assumed that the water possessed no weight. As a matter of fact, however, the weight of water must be taken into consideration in practice, and the pressure due to the weight of a column of water must be added to the applied pressure. Under such conditions, the pressure will be found greatest at the bottom of the cylinder and will decrease in intensity toward the top of the cylinder. 21. The fact that water transmits pressure in every direction with equal intensity may be easily proved, experimentally, by means of an apparatus like that shown in Fig. 17. Let the area of the piston a be 20 square inches; of b, 7 square FIG. 17 inches; of c, 1 square inch; of d, 6 square inches; of e, 8 square inches; and of f 4 square inches. If the pressure due to the weight of the water is neglected, and a force of 5 pounds is applied at c (whose area is 1 square inch), a pressure of 5 pounds per square inch will be transmitted in all directions, and in order that there shall be no movement, a force of 6 X 5 30 pounds must be ap = plied at d, 40 pounds at e, 20 pounds at f, 100 pounds at a, and 35 pounds at b. If a force of 99 pounds were applied to a, instead of 100 pounds, the piston a would rise, and the other pistons b, c, d, e, and f would move inwards; but, if the force applied to a were 100 pounds, they would all be in equilibrium. If 101 pounds were applied at a, the pressure per square inch |