CHAPTER III. ARCHIMEDES' PRINCIPLE AND BUOYANCY. EXPERIMENTAL DETERMINATION OF SPECIFIC GRAVITY, BY THE HYDROSTATIC BALANCE AND HYDROMETER. 48. The principle of Archimedes which was established as a Corollary in § 45 of the last Chapter is so important in Hydrostatics that it is advisable to restate it in a more general form, and to give an independent proof. Archimedes' Principle. "A body wholly or partially immersed in a Fluid or Fluids (not necessarily a single liquid), at rest under gravity, is buoyed up by a force equal to the weight of the displaced fluid, acting vertically upwards through the centre of gravity of the displaced fluid." To prove this principle, in the manner employed by Archimedes, we suppose the body removed, and its place filled up with fluid, arranged exactly as the fluid would be when at rest; and to fix the ideas we suppose this fluid solidified or frozen; this will not alter the thrust of the surrounding fluid, which will be exactly the same as that which acted on the body. But now this solidified fluid will remain in equilibrium of itself under two forces, the attraction of gravity on its weight and the thrust of the surrounding liquid; this thrust must therefore balance the weight of the solidified fluid and act vertically upwards through the C.G. of the solidified fluid. Therefore also on the original body, the thrust of the surrounding liquid will be equal to the weight of the displaced fluid and act vertically upwards through its C.G. COR. For a body to float freely at rest in a fluid or fluids, the weight of the body and of the fluid it displaces must be equal, and their c.G.'s in the same vertical line. The principle of Archimedes is therefore true not only of a body partly immersed in liquid, like a ship, but also of a body completely submerged in liquid, like a fish, diving bell, or submarine boat; or of a body floating in air, like a balloon; and generally of a body immersed in two or more fluids, as a ship is, strictly speaking, partly in air and partly in water. A mere increase of atmospheric pressure, due to increased temperature, will produce a uniform increase of pressure; this will not alter the draft of a ship unless accompanied by an increase of density of the air, when less water and more air would be displaced in equilibrium, and the draft of water would be diminished. Again, a body floating in a liquid, placed in the receiver of an air-pump, sinks slightly as the air is exhausted. The thrust of the surrounding fluid on the body is called the buoyancy (French poussée, German auftrieb); and Archimedes' Principle asserts that the buoyancy is equal to the weight of the displaced fluid, and acts vertically upwards through its c.G. ARCHIMEDES PRINCIPLE. 95 A similar method will show that in the general case of any applied forces (not merely gravity in parallel vertical lines), the buoyancy of a body in a fluid is equal and opposite to the resultant of all the applied forces on the fluid which would occupy in equilibrium the space vacated by the body. Thus, for instance, in the case of a sponge suspended by a thread, so as to dip partially into water, capillary attraction will cause the water to rise above the level surface, and the tension of the thread will be increased by the weight of water drawn up into the sponge. So also a feebly magnetic body, if immersed in oxygen or a liquid more magnetic than itself, will under external magnetic force appear diamagnetic, like bismuth. According to tradition, this Principle was discovered by Archimedes in his bath from observations on the buoyancy of his own body, while pondering over a method of discovering to what extent the crown of King Hiero of Syracuse (B.C. 250) had been adulterated with baser metal. (Vitruvius, de Architectura, lib. IX. c. iii. Palæmon, de Ponderibus et Mensuris, 124-163. Histoire du principe d'Archimède. Berthelot, Comptes Rendus, Dec. 1890). The Principle of Archimedes is employed not only in the determination of the density of solid and liquid substances, but also for the condition of equilibrium of floating bodies, such as ships, fishes and balloons; it may be called the fundamental Principle of Hydrostatics, as its discovery was the first step to placing the science on a sound basis. 49. Density and Specific Gravity. The density of a homogeneous body has already been defined as the weight of the unit of volume (§ 21). With British units the density is the weight in pounds per cubic foot; and with Metric Units the density is the weight in grammes per cubic centimetre, or kilogrammes per litre (cubic decimetre), or tonnes (of 1000 kg) per cubic metre (§ 8). DEFINITION. "The specific gravity of a body is the ratio of its density to the density of water"; or "is the ratio of the weight of the body to the weight of an equal volume of water." In the Metric System a litre of water, at or near its maximum density, was taken as the unit of weight (poids), and called a kilogramme; so that in this system the density and the specific gravity are the same. Now if s denotes the metric density or specific gravity of a substance, the equation gives the weight W in grammes of a volume V cm3; or the weight Win tonnes of a volume V m3 (metres cube); but the equation W = 1000s V gives the weight W in kg of a volume V m3. .(2) With British units the specific gravity s and density w (in lb/ft3) are connected by the relation where D denotes the weight in pounds of a cubic foot of water; so that the equation gives the weight W in lb of Vft3 of a substance whose S.G. (specific gravity) is denoted by 8. DENSITY OF WATER. 97 50. In rough numerical calculations it is usual to take a cubic foot of water as weighing 1000 oz or 62.5 lb; so that the equation (2) gives the weight W in oz of V ft3 of a substance of S.G. 8; but in equation (4) we must put D=62.5. A better average value is D=624, which is the density of water, in lb/ft3, at a temperature of about 53° F.; while D=62.425 is the maximum density of water, in lb/ft3, at a temperature of about 4° C. or 39.2° F. A Table, due to Mendeleeff, is printed in an Appendix, giving the density of water at different temperatures; s denoting the density in g/cm3 at the temperature t° C, from which the column of D, the density in lb/ft3, was deduced, by multiplying by (30.4794)÷453.593=log-11·7953528; while v denotes the specific volume, in cm3/g; so that s and are reciprocal. s, 51. These values of 8, D, and v refer to pure distilled water, at standard atmospheric pressure. But water may contain solid matter in suspension (as mud), or in solution (as salt), by which its density is increased. Thus in muddy water D may rise to about 75, so that a gallon of this water will weigh about 12 lb, as against the gallon of 10 lb of pure distilled water. In ordinary sea water, we generally take s=1·025, D=64, and therefore weighing 10-25 lb to the gallon; but in the |