GENERAL EXERCISES. 91 (4) A vessel in the form of a regular tetrahedron rests with one face on a horizontal table. The other faces are uniform plates, each of weight W, which can turn freely about their lowest edges, and when shut fit closely. Through a hole at the top water is poured in, and the sides are pressed out when the depth of the water is m times the height of the vessel. Show that, if the weight of water poured in be p W, then 9p(2m2 — m3) = 2(m2 — 3m+3). (5) ABCD is a quadrilateral having the sides AB, BC in the ratio 2:1, the angles B, D right angles, and the angle BAD=a. If a thin vessel of the form generated by the revolution of ABC round AD be placed with its circular opening upon a horizontal plane, and be filled with water through a small hole at A, prove that the water will be on the point of escaping by lifting the vessel if 3+5 tan2a=11 tan a. (6) A regular tetrahedron ABCD is immersed with the face ABC vertical, the side AB being horizontal and in the surface of the liquid. CE is drawn perpendicular to AB, meeting it in E. Show that the line of action of the resultant thrust on the remaining faces of the tetrahedron divides CE in F so that EF: FC 5:13. (7) Show that the limiting position of the centre of pres sure of a crescent, formed by two circles in a plane touching at a point in the surface of a liquid, when the two circles are infinitely nearly equal, is distant from the centre two-thirds of the radius. 92 GENERAL EXERCISES. (8) If a square is immersed wholly in a liquid, find the C.P.; and prove that the point is not changed if the square is turned round in its own plane. Hence prove that the C.P. is the same for all parallelograms wholly immersed, described about an ellipse at the ends of a pair of conjugate diameters. Prove also that the C.P. of all parallelograms formed by the points of contact is the same. (9) If a convex polygon of n sides is completely immersed, no side being parallel to the surface; and if x1, x2,... xn be the depths of the vertices A1, A 2, An, of which let A, and A, be the highest and lowest; and if a, a, ... an be the inclinations to the horizon of the sides A1Ä  ̧, prove that the depth of the C.P. is 1 ... 2) AnA1; xsin A,coseca,coseca,-xsin A,coseca,coseca ... ... 2 asin A coseca,coseca,-x3sin A,coseca cosecα2 +xsin A,cosecar-¡cosecar ... (10) A plane rectangular lamina is bent into the form of a cylindrical surface, of which the transverse section is a rectangular hyperbola. If it is now immersed in water so that, first, the transverse, secondly, the conjugate axes of the hyperbolic sections is in the surface, prove that the horizontal thrust on any the same immersed surface will be the same in the two cases. 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. 94 ARCHIMEDES PRINCIPLE. 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. |