1012 fully the 400 ope. late ons, d is each am kis d 3 1. I of has nce ine tes can та nd ler gy of 180. A railway train of 300 tons is running at 45 miles an hour. Find its energy of motion in foot-tons. Also, if the engine, while getting up the speed of the train, does work on it at an average rate of 100 horsepower, show how long it will take to get up the speed of 45 miles an hour. [Take g=32 (feet, seconds); I horse-power = 33,000 foot-pounds per minute.] Int. Sc. 1884. 181. What is the horse-power of an engine which can project 10,000 lbs. of water per minute, with a velocity of 80 feet per second, 20 per cent of the whole work done being wasted by friction, etc.? [N.B.-An agent of one horse-power can do 33,000 foot-pounds of work per minute.] Matric. 1884. 182. If a bicyclist always works at the rate of onetenth of a horse-power, and goes 12 miles an hour on the level, prove that the resistance of the road is 3.125 pounds. Int. Sc. 1885. 183. A railway train of mass 100 tons is moving at 20 miles per hour. What horse-power would be required to impart to it this velocity in 5 minutes from starting, in addition to overcoming the resistances supposed uniform and equal to 12 lbs. per ton? (One horse-power = 33,000 foot-pounds per minute.) Camb. Schol. 1884. cal of is 10 it le le IN solving the examples in this chapter, the followi facts may be assumed : The mass of a cubic foot of water is 1000 ounces 62 pounds. The specific gravity of mercury is 13.6. In examples on fluid pressure, the pressure of t atmosphere may be neglected, unless the contrary is e pressly stated. The normal atmospheric pressure is that due to column of mercury 76 centimetres in height. When t inch is taken as the unit of length, the normal barometr height may be assumed as 30 inches. Geometrical Relations.-The ratio of the circumfe ence of a circle to its diameter is 3.1416 (or approx mately 22), and is denoted by the Greek letter π. The circumference of a circle of radius r is 2πr, an its area is πr2. The area of the surface of a sphere is 4π2. 1. Define a fluid, and explain what is meant by th pressure of a fluid at any point within it. Prove that th pressure at any point of a fluid at rest is the same i every direction. 2. State and illustrate the law of transmission pressures in liquids, and explain how it is applied in th construction of hydraulic presses. 3. Distinguish between pressure and intensity of pressure; and find the dimensions of each of these quantities in terms of the fundamental units of length, time, and mass. 4. Define density and specific gravity, pointing out the essential distinction between them. Explain how it is that the density of a substance has, in the C.G.S. system, the same numerical value as its specific gravity, or density relative to water. 5. A block of mahogany 2 inches long, 1 inch broad, and inch thick is found to weigh1 461 grains : express its density in grains per cubic inch. The volume of the block is 2xx cubic inches, and its mass is 461 grains. The density, in grains per cubic inch, is 6. A ton of clay is found to occupy a volume of 18 cubic feet what is its density in pounds per cubic foot? 7. The density of water is 62 pounds per cubic foot: express this in ounces per cubic inch. 8. Basalt is three times as heavy as water what is its density in ounces per cubic inch? 9. A ton of chalk occupies a volume of 15 cubic feet what is its specific gravity referred to water as the standard substance? The mass of a cubic foot of the chalk is (2240÷31) pounds, and the mass of a cubic foot of water is 62 pounds. Hence the required specific gravity (or ratio between the masses of equal volumes) is (2240x2)-125-2240 x 2x2 = 2.312. 31 x 125 10. A cubic inch of a substance weighs half a pound: what is its specific gravity? In this and the succeeding chapters the term weight is occasionally used instead of mass, when there is no danger of confusion between the two. When a body is said to be weighed in air, the weight of the displaced air may be neglected unless the contrary is evnrecely stated 11. The specific gravity of lead is 11.4: what is t weight of a cube of lead 3 inches in the side? 12. A mahogany block of the same dimensions that in Example 5 is found to weigh 30.35 gramme express its density in grammes per cubic centimetre. 13. The specific gravity of iron is 7.7: what is t weight of an iron rod 2 inches broad, 2 inches thic and 18 feet long? 14. A body weighs 1000 pounds, and its density is fi times that of water: what is its volume? 15. How many grammes of mercury will be require to fill a cylindrical glass tube, the length of which is 7 centimetres, and internal diameter o.8 centimetre? The cross-section of the tube is Tr2=22x (0.4)2=22 x 0.1 square centimetre, and its internal volume is 70 x 22 x 0.1 35.2 cubic centimetres. The specific gravity of mercu is 13.6 approximately, and this number also represents i density in the C.G.S. system: i.e. I cubic centimetre mercury=13.6 grammes. Thus 35.2 x 13.6=478.72 grammes are required to fill the tub 16. A tube 120 centimetres long holds 600 gramme of mercury: find its cross-section and internal diameter 17. Find the mass of a piece of copper wire 5 metre in length and 1.8 millimetre in diameter, given that th density of copper is 8.8. 18. A cylindrical tube 1 metre in length and I centi metre in internal diameter weighs 100 grammes whe empty, and 210 grammes when filled with a liquid: find the specific gravity of the liquid. 19. Find the diameter of a cylindrical kilogramm weight made of brass (density 8.4), its height being 7.5 centimetres. 20. The radii of two spheres are 2 centimetres and 3 centimetres, and their masses are 200 grammes and 250 grammes respectively: compare their densities. 21. A litre of hydrogen gas weighs 0.0896 gramme, and the density of carbon-dioxide is twenty-two times s the Is as nes: the lick, five ired 70 D.16 0.16 ury its of Se. es er. es e i n that of hydrogen: how much carbon-dioxide is required to fill a gas-bag which holds 10 litres? 22. One litre of a liquid of specific gravity 1.4 is mixed with 2 litres of a liquid of specific gravity 0.96, and the mixture occupies nine-tenths of the volume of its components: what is its specific gravity? If A denote its density, then its mass in grammes is VA (1000 x 1.4)+(2000 x 0.96) = 1400+ 1920=3320. 23. If the specific gravities of two liquids be 4 and 5 respectively, find the specific gravity of a mixture containing 3 parts by weight of the former to 4 parts of the latter. 24. What would have been the specific gravity of the mixture in the last example if the proportions had been taken by volume instead of by weight? 25. Equal volumes of two liquids whose specific gravities are and 2s are mixed together, and the mixture occupies four-fifths of the sum of the volumes of its components: what is its specific gravity? 26. The density of fire-damp is one-half that of air: what is the density of foul air containing 15 per cent of fire-damp? 27. Equal weights of two liquids whose specific gravities are 0.9 and 0.7 are mixed together, and a contraction of 10 per cent occurs in the volume: what is the specific gravity of the mixture? 28. Equal volumes of three fluids are mixed. The specific gravity of the first is 1.55, that of the second 1.75, and that of the mixture is 1.6: find the specific oravity of the third. |