Woods. Alloys. Solid Metals. we deduce 1 m V of A= M, 1 nV of BM, n/m V of AV of B. This expresses the relative bulkiness which is the reciprocal of the relative density. SPECIFIC MASS AND DENSITY.—Continued. SUBSTANCE. M OF SUBSTANCE = M OF WATER OR G.M. PER C.C. LB. OF SUBSTANCE Gases at 32° Fahr. and under one atmosphere Liquids at 32° Fahr., but of pressure. water at 39°1 Fahr Ex. 1. A cubic foot of fresh water weighs 62.4 lbs., find in cubic feet the space occupied by one ton of sea water. 62.4 lbs. of fresh water cubic foot, 1.026 lbs. of salt water = = lb. of fresh water, 112 × 20 lbs. of sea water; Ex. 2. Sodium has to alcohol the relative density 1.23, and alcohol has to water the relative density 79; what is the density of sodium relatively to water? Observation-In order that two relative equivalences may be combined in this manner, they must of course be equivalences in the same respect; and the conclusion is an equivalence in the same respect. Ex. 3. Eleven cubic inches of iron weigh as much as seven cubic inches of lead, and the price per ton of lead is £15, of iron £4. The value of a certain block of lead is 367. 17s. 11d., what would be the value of a block of iron of the same size? Ex. 4. The specific gravity of gold is 19.3, that of silver is 10.4. What is the composition of an alloy of gold and silver whose specific gravity is 17·6, no change of volume being supposed to accompany the combination of the metals? Let the composition by volume of the alloy be a V gold +b V silver = a + b V alloy, 19.3 M per V, 10.4 M per V; therefore the composition by mass is 19.3 a M gold + 10·4 6 M silver = 19.3 a + 10·4 b M alloy. Hence the density of the alloy is 1. A gallon of fresh water measures 277 271 cubic inches, and contains 10 lbs. avoirdupois. A ton of sea water measures 35 cubic feet. What is the mass of a gallon of sea water in pounds and decimals? 2. If 100 cubic inches of oxygen, under certain circumstances of pressure and temperature, contains 35 grains, and a cubic inch of mercury contains 0:49 lbs., how many cubic inches of the oxygen would contain the same quantity of matter as a cubic inch of mercury? 3. Of two bodies one has a volume of 5 cubic inches, the other of one-fifth of a cubic foot; the mass of the former is 15 oz., and of the latter 12.8 lb. What is the ratio of the density of the first to that of the second? 4. A flask holds 27 oz. of water. mass is 0.95? What mass will it hold of an oil whose specific 5. A cubic foot of water contains 1,000 ounces. 502.5 ounces of lead of specific gravity 11.5, and 440 ounces of iron of specific gravity 8, are placed in a cistern of the capacity of one cubic foot. Find the quantity of water necessary to fill the cistern. 6. From the relative densities to water of zinc, iron, tin, copper, lead, find the relative densities to iron of each of the other four metals. 7. The line-density of iron wire of No. 10 Birmingham wire gauge is 4'96 lb. per 100 lineal feet; what is the line-density of copper wire and of brass wire of the same gauge? 8. Sodium has to alcohol the relative density 1.23, and water has to alcohol the relative density 1.26; what is the density of sodium relatively to water? 9. If the specific gravity of a specimen of milk be m, and that of pure milk s; calculate the proportion of water added. 10. What must be the volume of a mass of wood of relative density 0.5, in order that when it is attached to 500 gms. of iron of relative density 7, the mean density of the whole may be equal to that of water? 11. If the price of whisky, the specific gravity of which is 75, be 16s. a gallon, find the price when it is mixed with water so as to have the specific gravity '8. 12. A Prussian dollar, made of an alloy of silver and copper, has the specific gravity 10.05. Determine the relative amount of silver and of copper in it, the specific gravities of these metals being 105 and 8.7 respectively. 13. A nugget of gold mixed with quartz weighs 10 oz. gold is 19.35, of the quartz 2·15, and of the nugget 6:45. The specific gravity of gold and of the quartz contained in the nugget; find also the ratio of their volumes. 14. A mixture is made of 7 cubic centimetres of sulphuric acid (specific gravity, 1.843) and 3 cubic centimetres of distilled water; and its specific gravity when cold is found to be 1'615. Determine the contraction which has taken place. 15. The density of a mixture of two liquids being supposed to be an arithmetical mean between those of the components; determine the ratio of the volumes of the components contained in the mixture. 16. Several liquids which do not alter their volume when mixed are shaken together; determine the specific gravity of the mixture from their specific gravities. 17. Half a pint of a liquid which is half as dense again as water is mixed with a pint of water; what is the density of the mixture? 18. A rod of uniform cross section 18 in. long weighs 3 oz. ; its specific gravity is 88; what fraction of a square inch is the area of its cross section? 19. What is the mass of a cast-iron ball having a diameter of 6 inches; and of a cast-iron cylinder having the same diameter and 4 feet long? 20. A ditch 3 feet deep is dug round a square garden containing one tenth of an acre; find its width in order that the removed earth may raise the garden one foot. 21. Two liquids are mixed first by volume in the proportion of 1 to 4, and second by mass in the proportion of 4 to 1; the resulting specific masses are 2 and 3 respectively. Find the specific masses of the liquids. SECTION XXV.-MASS-VECTOR. ART. 135.-Idea of Mass-Vector. The ideas of dynamics differ from those of geometry and kinematics by the introduction of the idea of mass. From the idea of a vector we derive that of a massvector, which is proportional to a vector and to a mass. This term was introduced by Clerk-Maxwell,* and it is expressed in terms of M by L. A mass-vector can be resolved and compounded in the same manner as a simple vector. ART. 136.-Centre of Mass. The centre of mass (commonly Matter and Motion, p. 49. |