DENSITY AND SPECIFIC GRAVITY. 133 (16) If the liquid employed with Nicholson's Hydrometer be water, the substance a mixture of two metals whose S.G.'s are 14 and 16, and the weights used are 16 oz, 1 oz, 2 oz; find the quantity of each metal in the mixture. (17) Show that the units may be chosen so that the specific gravity and the density of a substance are identical. A nugget of gold mixed with quartz weighs 12 (10) ounces, and has a specific gravity 6·4 (8⋅6); given that the specific gravity of gold is 19.35, and of quartz is 215, find the quantity of gold in the nugget. (18) Air is composed of oxygen and nitrogen mixed together in volumes which are as 21 to 79, or by weights which are as 23 to 77; compare the densities of the gases. (19) How many gallons of water must be mixed with 10 gallons of milk to reduce its s.G. from 1.03 to 1.02 ? (20) Bronze contains 91 per cent. by weight of copper, 6 of zinc, and 3 of tin. A mass of bell-metal (consisting of copper and tin only) and bronze fused together is found to contain 88 per cent. of copper, 4.875 of zinc, and 7·125 of tin. Find the proportion of copper and tin in bell-metal. (21) Two fluids are mixed together: first, by weights in the proportion of their volumes of equal weights; secondly, by volumes in the proportion of their weights of equal volumes; compare the specific gravities of the two mixtures. 134 GENERAL EXERCISES ON r (22) A mixture of gold with n different metals contains r per cent. of gold and r1, 2, 3, ..., în per cent. of the other metals. After repeated processes, by which portions of the other metals are taken away, the amount of gold remaining unaltered, the mixture contains 8 per cent. of gold and S1, S2, S3, ....., Sʼn per cent. of the other metals. Find what percentage of each metal remains. (23) A quart vessel is filled with a saturated solution of salt. A quart of water is poured drop by drop into the vessel, causing the solution to overflow, but is poured in so slowly that it may be supposed to diffuse quickly through the solution. Show that after the operation the amount of salt left in the solution in the vessel will be 1/e of the original amount, where e is the base of the Naperian logarithms. (24) From a vessel full of liquid of density p is removed one-nth of the contents, and it is filled up with liquid of density σ. If this operation is repeated m times, find the resulting density in the vessel. Deduce the density in a vessel of volume V, originally filled with liquid of density p, after a volume U of liquid of density has dripped into it by infinitesimal drops. (25) The mixture of a gallon of A with S.G. 81, with W2 lb of B a S.G. 82, with W1⁄2 lb of B a 2 S.G. 8; find the S.G.'s of A and B. (26) Find the chance that a solid composed of three substances whose densities are P1, P2, P2, will float in a liquid of density p DENSITY AND SPECIFIC GRAVITY. 135 (27) A vessel is filled with three liquids whose densities in descending order of magnitude are P1, P2, P3 All volumes of the liquids being equally likely prove that the chance of the density of the mixture being greater than p is (28) Describe some method of determining the absolute expansion of a liquid. A piece of copper is weighed in water at 16° and at 80°, the weights of water displaced being 50 g and 48-809 g; find the mean coefficient of cubical expansion of copper between those temperatures ; given the S.G. of water at 16° and 80° as 0.999 and 0.972. (29) The hydrometer is used to determine the S.G. of a liquid which is at a temperature higher than that of water. When the hydrometer is transferred from water to the liquid the S.G. appears at first to be s, but afterwards to be 8. Show that, neglecting the density of the air, the true S.G. at the temperature of the water is a' where a and a' are the coefficients of expansion of the hydrometer and the liquid respectively. 136 GENERAL EXERCISES. (30) Show that the coefficient of expansion of a body may be found as follows: Lets be the S.G. of the body at zero temperature compared with water at its greatest density; 1+e1, 1+e2 the volumes at temperatures t1, t2 of a unit volume at zero temperature; 1+E1, 1+E2 the volumes at t1, t2 of a unit volume of water at its greatest density; w the weight of the body in a vacuum; w1, w2 its apparent weights in water at temperatures t1, to; then ¤ ̧ — е2 — E1 — Е2—s(w1—w2)/w very nearly. = (31) Prove that, if a hydrometer of weight W sinks to certain marks on the stem in a liquid at temperatures t1 and t2, and to the same marks in the liquid at zero temperature, when weights w1 and w2 are fixed at the top of the hydrometer, the coefficients of cubical expression of the hydrometer and of the liquid are respectively (32) Determine the s.v. in cubic feet to the ton, and the density in lb per cubic foot of lead shot, cast iron spherical shot, and cast iron spherical shells with internal radius three-quarters the outside radius, given the S.G. of lead as 11·4, and of cast iron 7.2. Determine also the s.v. or roomage of earthenware pipes, and cylindrical barrels, of apparent density p. CHAPTER IV. THE EQUILIBRIUM AND STABILITY OF A SHIP OR FLOATING BODY. 84. Simple Buoyancy. The Principle of Archimedes leads immediately, as in § 48, to the Conditions of Equilibrium of a body supported freely in fluid, like a fish in water, or a balloon in air, or like a ship floating partly immersed in water (fig. 38, p. 148). The body is in equilibrium under two forces; (i.) its weight W acting vertically downwards through G, the C.G. of the body; and (ii) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upwards through B, the C.G. of the displaced fluid; and for equilibrium these two forces must be equal and directly opposed. The Conditions of Equilibrium of a body, floating like a ship on the surface of a liquid, are therefore (i.) the weight of the body must be less than the weight of the total volume of liquid it can displace, or else the body will sink to the bottom of the liquid; (ii) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body; (iii.) the c.G. B of the displaced liquid and G of the body must lie in the same vertical line GB. |