GENERAL EXERCISES. GENERAL EXERCISES ON CHAPTER III. 131 (1) The diameters of two globes are as 2:3, and their weights as 1:5; compare their specific gravities. (2) The weight of a vessel when empty is 3 oz; when filled with water, it is 9 oz; and when filled with olive oil, 8-49 oz; required the S.G. of the oil. (3) A vessel filled with water weighs 5 oz, and when a piece of platinum weighing 294 oz is placed in it, and it is filled up with water, it weighs 33 oz; prove that the s.G. of the platinum is 19.5. (4) The weight of a piece of cork in air is oz, the weight of a piece of lead in water is 64 oz, and the weight of the cork and lead together in water is 4.07 oz. Prove that the S.G. of the cork is 0.24. (5) A piece of metal weighing 36 lb in air, and 32 lb in water, is attached to a piece of wood whose weight is 30 lb, and then the compound body is found to weigh 12 lb in water. Prove that the S.G. of the wood is 0.6. (6) The s.G.'s of platinum, standard gold, and silver being respectively 21, 17.5, and 10.5, and the values of an ounce of each 30s, 80s, and 5s respectively; prove that the value of a coin composed of platinum and silver, which is equal in weight and magnitude to a sovereign, is 6s 3d. (7) A solid, whose weight is 250 grains, weighs 147 in water, and 130 in another fluid. Prove that the S.G. of the latter is 1.262. (8) A solid, whose weight is 60 grains, weighs 40 grains in water, and 30 grains in sulphuric acid; required the S.G. of the acid. 132 GENERAL EXERCISES ON (9) The S.G. of gold being 19.25, and of copper 89, what are the weights of copper and gold respectively in a compound of these metals which weighs 800 grains in air, and 750 in water? (10) A piece of gun-metal was found to weigh 1057.9 grains in air, and 934-8 grains in water; find the proportions of copper and of tin in 100 lb of the metal, the S.G. of the copper being 8788 and of tin 7291. (11) A body immersed in a liquid is balanced by a weight P, to which it is attached by a thread passing over a fixed pulley; and when half immersed, is balanced in the same manner by a weight 2P. Prove that the densities of the body and liquid are as 3 to 2. (12) It is found on mixing 63 pints of sulphuric acid, whose S.G. is 1.82, with 24 pints of water, that 1 pint is lost by their mutual penetration; find the S.G. of the compound. (13) A piece of gold immersed in a cylinder of water causes it to rise a inches; a piece of silver of the same weight causes it to rise b inches; and a mixture of gold and silver of the same weight c inches; prove that the gold and silver in the compound are by weight as b-c: c-a. (14) The S.G. of lead is 11-324; of cork is 0.24; of fir is 0.45; determine how much cork must be added to 60 lb of lead that the united bodies may weigh as much as an equal volume of fir. (15) The S.G.'s of pure gold and copper are 19.3 and 8·62; required the S.G. of standard gold, which is an alloy of 11 parts pure gold and one part copper. 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 2·15, 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 (22) A mixture of gold with n different metals contains r per cent. of gold and r1, 2, 3, ..., în per cent. 72, 73, 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 s 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. ρ (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 o has dripped into it by infinitesimal drops. (25) The mixture of a gallon of A with W1 lb of B has a with W2 lb of B a S.G. 82, with W, lb of B a S.G. 8; find the S.G.'s of A and B. S.G. 81, 3 (26) Find the chance that a solid composed of three substances whose densities are P1, P2, P3, 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 is ρ (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 8, 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. |