through the space ho (1 − s)/(1 - σ), where σ is the ratio 185. Glycerine rises in a barometer tube to a height of 26 feet when mercury stands at 30 inches. The specific gravity of mercury is 13.6: find that of glycerine. Matric. 1886. 186. A cylindrical tube a metre in length and having one end sealed contains dry air at the ordinary pressure and temperature. The tube is dipped vertically with its open end downwards into a tank of mercury, till the air within it is compressed to four-fifths of its former volume: find the distance of the top of the tube from the free surface of the mercury in the tank, the height of the barometer being 750 millimetres. Matric. 1879. 187. State the laws of compression for air and water respectively. Assuming them to hold for great ranges of pressure, show that a bubble of air will sink in water if the pressure be sufficiently increased; and calculate roughly at what pressure this will take place. (At a pressure of one atmosphere the density of air is 1/773 of the density of water.) Edinb. M.A. 1884. 188. If the height of the barometer at the sea-level be 760 millimetres, the specific gravity of mercury 13.6, and the specific gravity of sea-water 1.02, at what depth below the surface is the pressure per square centimetre equivalent to the weight of 10 kilogrammes ? Camb. Schol. 1883. 189. A barometer stands at 30 inches, and the space occupied by the Toricellian vacuum is then 2 inches; if now a bubble of air which would at atmosnhari. occupy half an inch of the tube be introduced into t tube, prove that the surface of the mercury in the tu will be lowered 3 inches. Show also that the height a correct barometer when this false one stands at inches is x+15/(32-x). Camb. Schol. 1885. 190. A cylindrical diving-bell of length h/4 is su in water till its lowest part is nh below the surface; the water fill of the bell, show that the bell conta air whose volume at atmospheric pressure would (n+18)V, V being the volume of the bell and h height of the water barometer. Camb. B.A. 1880. 191. The height of the water barometer is 333 fe a bubble of gas has a volume of 1 cubic inch at a dep of 100 feet below the surface of pure water: what w be its volume on reaching the surface? Matric. 1884. 192. A cubic foot of water weighs 1000 ounces. cylindrical test-tube is held in a vertical position a immersed mouth downwards in water. When t middle of the tube is at a depth of 32.75 feet it is fou that the water has risen half-way up the tube: find t atmospheric pressure in pounds weight per square inc Matric. 1882. 193. Having given that the density of the air 00129, the density of mercury 13.596, and the heig of the barometer 75.9 centimetres, prove that if the u of force be taken to be the weight of 800 kilogramm the numerical value of the pressure of the air will almost exactly equal to that of its density, it bei assumed that a centimetre cube of water weighs gramme. M. Tripos. 1880. 194. The height of the water barometer being feet 9 inches, and the specific gravity of mercury 13 find at what height a common barometer will stand in cylindrical diving-bell when lowered until the water fi one-tenth of the bell. How far will the surface of t water within the bell be below the external surface? Camb. Schol. 1886. 195. Suppose that a cubic foot of air weighs 1.2 ounce, and a cubic foot of water 1000 ounces. A balloon so thin that the volume of its substance may be neglected contains 1.5 cubic feet of coal-gas, and the envelope, together with the car and appendages, weighs The balloon just floats in the middle of the room, without ascending or descending: find the specific gravity of coal gas (1) compared with air, (2) compared with water. I ounce. Matric. 1882. 196. A cylindrical diving-bell weighs 2 tons, and has an internal capacity of 200 cubic feet, while the volume of the material composing it is 20 cubic feet. The bell is made to sink by weights attached to it. At what depth may the weights be removed, and the bell just not ascend, it being given that the mass of a cubic foot of water is 1000 ounces, and the height of the water barometer 33 feet? Matric. 1883. 197. What are the conditions of equilibrium of a floating body? A Cartesian diver consists of a hollow ball with a weight attached to it. There is an opening at the bottom of the ball, and it contains a quantity of air. If the diver be made of glass (specific gravity= 3.33), and weigh 7.05 grammes, find the volume of air in it in order that it may just float in water at its maximum density, and show that if it contain 5.2 cubic centimetres of air and be sunk to a depth of 500 millimetres in the water it will not rise again, the pressure of the air above the water being that due to 10 metres of water. weight of the air in the diver may be neglected.) (The Camb. Schol. 1881. 198. Atmospheric air having a volume of 50 litres at ordinary pressure (1000 grammes weight per square centimetre) is admitted into a vessel having a volume of I cubic metre, and containing only aqueous vapour at a pressure of 4 grammes weight per square centimetre. 199. Explain the action of a syphon. A syphon with vertical arms filled with mercury (p) and clos both ends is inserted into a basin of water (o). the stoppers are removed examine what will ensue prove the following results if the barometer is suffici high: (i.) If k, the whole length of the outside arm, ex h, the whole length of the immersed arm, the me will flow outwards and the water will follow it. (ii.) If h>k, the ends of the immersed tube mus at a depth below the free surface of the water excee (h-k) p/o in order that the mercury may not flow 1 into the basin. M. Tripos. 18 ercury? c centi constant 1887. On tube osed at When e, and ciently Exceeds ercury ust be eding back :885. Definition. CHAPTER III HEAT-EXPANSION 1. Expansion of Solids The coefficient of linear expansion of a substance is the increase in length produced in unit length of the substance by a rise in temperature of 1 degree centigrade. APPROXIMATE COEFFICIENTS OF LINEAR EXPANSION. These values may be assumed in solving the examples in this chapter. All temperatures are expressed on the centigrade scale. Let a denote the coefficient of expansion of a body whose length at o° is : on heating to to the expansion produced will be lat. The length of the body at t° will be given by the equation 1. Find the length at 200° of a zinc rod whose length at o° is 128 cm. If the length at 200° be denoted by 1200, then 1200 = (1 + 200α) = 128(1 + 200 x 0.000029) = 128 x 1.0058 = 128.7424 cm. |