(3) The specific gravity of mercury being 135, find the weight of one cubic inch of it, having given that a cubic foot of water weighs 1000 oz. (4) If two cubic feet of a substance weigh 100 lbs., what is its specific gravity? (5) Find the weight of 36 cubic inches of cork, whose specific gravity is 0.24. (6) A cubic foot of water weighs 1000 oz., what will bo the weight of a cubic inch of a substance whose specific gravity is 3? (7) What is the specific gravity of a body of which m cubic feet weigh n lbs. ? (8) Five cubic inches of iron weigh 22 oz., what is the specific gravity of iron? (9) Twelve cubic feet of dried oak weigh 875 lbs., what is the specific gravity of the wood? (10) Twenty-six cubic feet of ash weigh 13711⁄2 lbs., what is its specific gravity? (11) A metal, whose specific gravity is 15, is mixed with half the volume of an alloy whose specific gravity is 12, find the specific gravity of the compound. (12) Two metals are combined into a lump the volume of which is 2 cubic inches; 13 cubic inches of one metal weigh as much as the lump, and 2 cubic inches of the other metal weigh the same. What volume of each of the two metals is there in the lump? (13) Two substances whose specific gravities are 1.5 and 30 are mixed together, and form a compound whose specific gravity is 25; compare the volumes and also the weights or the two substances. (14) The specific gravity of sea-water being 1'027, what proportion of fresh water must be added to a quantity of sea-water that the specific gravity of the compound may be 1'009 ? (15) Equal weights of two substances whose densities are 3·25 and 2.75 are mixed together; find the density of the compound. (16) Equal volumes of two substances whose specific gravities are 25 and 15 are mixed together; what is the specific gravity of the compound? (17) Five cubic inches of lead, specific gravity 11·35, are mixed with the same volume of tin, specific gravity 73; what is the specific gravity of the compound? (18) A mixture is formed of equal volumes of three fluids; the densities of two are given and also the density of the mixture. What is the density of the third fluid? (19) Ten cubic inches of copper, specific gravity 8-9, are mixed with seven cubic inches of tin, specific gravity 7-3; find the specific gravity of the compound. (20) Three fluids, whose specific gravities are '7, 8 and 9 respectively, are mixed in the proportion of 5 lbs., 6 lbs., and 7 lbs. What is the specific gravity of the mixture? (21) The specific gravity of pure gold is 19′3 and of copper 8.62; required the specific gravity of standard gold, which is a mixture of eleven parts of gold and one of copper. (22) When 63 pints of sulphuric acid, specific gravity 1·82, are mixed with 24 pints of water, the mixture contains only 86 pints. What is its specific gravity ? (23) If three fluids the volumes of which are 4, 5, 6 and the specific gravities 2, 3, 4 are mixed together, determine the specific gravity of the compound. (24) The specific gravity of quartz is 2·62, and that of gold 19:35; a nugget of quartz and gold weighs 115 oz., and its specific gravity is 7-43; find the weight of gold in it. (25) An iron spoon is gilded, and the mean specific gravity of the gilded spoon is 8; those of iron and gold are 78 and 194: find the ratio of the volumes and weights of the metals employed. CHAPTER IV. On the Conditions of Equilibrium of Bodies under the Action of Fluids. 56. WHEN a body is wholly or partially immersed in a fluid, it is a general principle of Hydrostatics that the resultant pressure of the fluid on the surface of the body is equal to the weight of the fluid displaced. This principle we shall prove for two cases in Articles 57 and 61. (1) When the body is wholly immersed in the fluid: 57. To find the resultant Pressure of a Fluid on a body wholly immersed and floating in a fluid. Let A be a body floating in a fluid and wholly immersed in it. Imagine the body removed and the vacant space filled with fluid of the same kind as that in which the body floated. Then suppose this substituted fluid to become solid. The pressure at each point of its surface will still be the same as it was at the same point of the surface of A The solidified fluid is kept at rest by (1) The attractions exercised by the earth on every particle of its mass : (2) The pressures exercised by the fluid at the different points of its surface. Hence the resultants of these two sets of forces must be equal in magnitude and opposite in their lines of action. Now the resultant of set (1) is called the weight of the solidified fluid and acts vertically downwards through its centre of gravity. Hence the resultant of set (2) is equal in magnitude to the weight of the solidified fluid and acts vertically upwards through its centre of gravity. Now since the pressures on the solidified fluid are the same as on the body A, we see that the resultant pressure of the fluid on A is equal to the weight of the fluid displaced by A and acts vertically upwards through the centre of gravity of this displaced fluid. This principle we shall now apply to the following Examples in Statics. 58. Ex. (1) Find the conditions of equilibrium of a body floating in a fluid and wholly immersed in it. The body A (see diagram in Art. 57) is kept at rest by (1) Its weight, acting vertically downwards through its centre of gravity: (2) The pressures of the fluid on its surface, the resultant of which is equal to the weight of the fluid displaced by A and acts vertically upwards through the centre of gravity of the fluid displaced. Hence (1) Weight of A= weight of fluid displaced by A: (2) The centres of gravity of A and of the fluid displaced are in the same vertical line. These are the conditions of equilibrium. Note. A difficulty often occurs with beginners in conceiving how a solid body can be in equilibrium in the midst of a fluid, neither rising to the surface nor sinking to the bottom. It may however be proved by experiment that a hollow ball of copper, such as is used for a ball-tap, may be constructed of such a weight relatively to its size that when placed in water it will remain where it is placed, just as the body A is represented in the diagram. 59. Ex. (2) Find the conditions of equilibrium for a body of uniform density wholly immersed in a fluid and in part supported by a string. Let a body the measure of whose volume is V be suspended by a string from the fixed point A so as to float below the surface of a fluid. The body is kept at rest by (1) its weight, (2) the pressures of the fluid on its surface, |