proportional to the pressure of the portion, provided that the temperature be kept constant. Hence, or k M per V = F per S, M=F per S by V. The letter k is used to denote some constant number. EXAMPLES. Ex. 1. A litre of air at 0° C. and 760 mm. pressure contains Find the mass of 73 litres at the same temperature, 1.293 gms. Ex. 2. A certain quantity of air forms a small spherical bubble of a given radius, when 5 feet below the surface of water; at what depth would the same quantity of air form a bubble of half the given radius, the change of temperature being neglected. Take the quantity of air as unit of mass M, and the first sphere as unit of volume V. The pressure of the atmosphere is equivalent to 30 feet of water. 1 M = V by (30+5) feet of water, Ex. 3. The content of the receiver of an air pump is 6 times that of the barrel. Find the pressure of the air in the receiver at the end of the 8th stroke of the piston, the atmospheric pressure being 15 lbs. to the square inch. Take the original mass of air for unit of mass, and the volume of the receiver for unit of volume; then 1 M=V by 15 lb. wt. per sq. inch. At the end of the operations we have the same volume; hence, 1 M = 15 lb. wt. per sq. inch. The volume of the barrel is one seventh that of the receiver and barrel conjointly. As the air will always distribute itself with uniform density, one seventh of the mass will be removed by the first double stroke, one seventh of the remainder by the second double stroke, and so on; hence after 8 strokes (1 − 1)8 M will be left. But 1 M=15 lb. wt. per sq. inch, ... (1 - 4)8 × 15 lb. wt. per sq. inch. 1. What is the height of the homogeneous atmosphere, when the mercurial barometer is at 30 inches? The specific gravity of air at that pressure is '00125. 2. In a tube of uniform bore a quantity of air is enclosed. What will be the length of this column of air under a pressure of three atmospheres, and what under a pressure of a third of an atmosphere; its length under the pressure of a single atmosphere being 12 inches? 3. When the height of the mercurial barometer changes from 29.55 inches to 30:33 inches, what is the change in the mass of 1,000 cubic inches of air, assuming that 100 cubic inches of air weigh 31 grains at the former pressure, and that the temperature remains constantly at 0° C.? 4. A cylindrical bell 4 feet deep, whose content is 20 cubic feet, is lowered into water until its top is 14 feet below the surface of the water, and the air is forced into it until it is three quarters full. What volume would the air occupy under the atmospheric pressure, the water-barometer being at 34 feet? 5. If the water-barometer stand at 33 feet, to what depth must a diving bell be sunk to reduce the contained air to one-third of its original volume, the height of the bell itself being neglected? 6. A diving-bell is lowered into water at a uniform rate, and air is supplied by a force-pump so as to keep the bell full, without allowing any to escape. How must the rate at which the air is supplied be varied as the bell descends? 7. An air-bubble at the bottom of a pond 10 feet deep, has a volume of 0.00006 of a cubic inch. Find what its volume becomes when it just reaches the surface, the barometer standing at 30 inches. 8. A closed indiarubber ball containing air has a volume of 4 cubic inches at a depth of 100 feet below the surface of water, whose density is unity. If the height of the water-barometer be 30 feet, determine the volume of the ball at the surface of the water, assuming the temperature to remain constant? 9. A Mariotte's tube has a uniform section of 1 square inch, and is graduated in inches; 6 cubic inches are inclosed in the shorter (closed) limb, when the mercury is at the same level in both. What volume of mercury must be poured into the longer limb in order to compress the air into two inches? The barometer stands at 30 inches. 10. Ten cubic centimetres of air are measured off at atmospheric pressure. When introduced into the vacuum of a barometer they depress the mercury which previously stood at 76 centimetres, and occupy a volume of 15 cc. By how much has the mercurial column been depressed? 11. A cylindrical tube, 2 feet long, closed at one end, is lowered down into the sea 200 feet, open end downward like a diving-bell. The atmospheric pressure at the surface being 30 inches of mercury, find how high the water rises in the tube. A column of about 32 feet of sea-water is equal in weight to a similar column of mercury of 30 inches. 12. If the pressure inside the receiver of an air-pump were reduced to of the atmospheric pressure in 4 strokes, to what would it be reduced in 6 strokes? 13. The cylinder of a single-barrelled air-pump has a sectional area of one square inch, and the length of the stroke is 4 inches. The pump is attached to a receiver whose capacity is 36 cubic inches. Compare the pressure of the air inside the cylinder, after 8 complete strokes of the pump, with the pressure before commencing the operation. 14. If the volume of the cylinder of an air-pump be that of the receiver, find the density of the air in the latter at the end of the fifth stroke. 15. A receiver attached to an air-pump has the volume of 100 cubic inches, while the cylinder has the volume of 10 cubic inches. What proportion of the original air will be left in the receiver after the completion of the fourth double stroke? 16. If the barrel of the common water-pump be 3 feet long, and the tube, supposed of the same cross-section, be 16 feet long; find how high the water will rise after the first stroke, the water-barometer being at 34 feet. SECTION XXXIII.-WORK. ART. 156.--Absolute Units of Work. Work is done when resistance is overcome; the quantity of work done is proportional to the resisting force and to the distance through which it is overcome. Let the unit of work be denoted by W, then 1 W = F resistance by L displacement. By "displacement" is meant the displacement in the direction of the force. The British absolute unit is denominated the foot-poundal, and is defined by 1 foot-poundal = poundal resistance by foot displacement. The C.G.S. absolute unit is denominated the erg, and is defined by 1 erg dyne resistance by cm. displacement. ART. 157.-Gravitation Units. Work is also measured in terms of gravitation units, by taking the corresponding gravitation unit of force instead of the absolute unit of force. The principal British unit is the foot-pound, defined by 1 foot-pound = lb. by weight by foot displacement. A metric unit is the kilogrammetre, defined similarly, 1 kilogrammetre = kilogramme by weight by metre. The principal C.G.S. unit is the gramme-centimetre, defined siinilarly. ART. 158.-Work done by a Pressure. Suppose that the resistance is a pressure uniformly distributed over the surface of application. Then and 1 F resistance (F per S) by S surface, 1 W = (F per S) by S surface by L displacement. If 1 SL2, then, since the displacement is normal to the surface of application, 1 W = (F per L) by L3 volume displacement. EXAMPLES. Ex. 1. Reduce 1 foot-pound to ergs, taking the acceleration due to gravity at 981 cm. per second per second. 1 foot-pound = lb. by weight by foot, 1 lb. 453.6 gm., weight 981 cm. per sec. per sec., 1 foot = 30.48 cm., 1 foot-pound = 981 × 453·6 × 30·48 gm. by cm. per sec. per sec. Ex. 2. A train of 120 tons runs on a level road, and the resistances to be overcome are 8 lbs. per ton. How many units of work must be expended in making a run of 40 miles, when there is no useless expenditure of steam. 8 lb. by weight, resistance = ton of mass, 1 foot-pound = lb. by weight by foot advance, 120 × 8 foot-pound = foot advance, X Ex. 3. How many units of work must be expended in raising from the ground the materials for building a uniform column 66 feet 8 inches high and 21 feet square, a cubic foot of brickwork weighing one hundredweight. As the column is uniform there is the same amount of matter in the different courses, but the height through which the matter of a course is raised increases uniformly from the bottom to the top. Hence the correct result will be got by taking the average height. |