SUMMARY. The efficiency of a machine is the ratio of its useful output of work to the total input of work (work done on the machine); that is, Efficiency = output The efficiency of the simple machines varies greatly: in the case of a simple lever it may be almost 100 per cent and in the jackscrew as low as 25 per cent. QUESTIONS AND PROBLEMS * 1. (1) Distinguish between the mechanical advantage of a machine and its efficiency. (2) How is each calculated? 2. Why is the efficiency of the jackscrew low and that of the lever high? 3. Find the efficiency of a machine in which an effort of 12 lb. moving 5 ft. raises a weight of 25 lb. 2 ft. 4. Compute the efficiency of a machine which is run by a 2-horse-power motor if the machine can do but 50,000 ft. lb. of work per minute. 5. A wheel and axle was used to hoist 200 lb. from a well 50 ft. deep. The hands of the workman at the wheel made 100 complete revolutions of 8 ft. circumference each with an average force of 15 lb. Find (1) the efficiency of the machine; (2) the number of foot pounds of energy not utilized. 6. A steam shovel driven by a 5-horse-power engine lifts 200 t. of gravel to a height of 15 ft. in an hour. What is the efficiency of the steam shovel? What percentage of the power is lost because of friction? 7. (1) At what average horse power does a man work who loads 30 bricks a minute from the ground to a wagon 4 ft. from the ground? Each brick weighs 7 lb. (2) What is his efficiency if he tosses them 1 ft. higher than necessary? 8. What amount of work was done on a block and tackle having an efficiency of 60 per cent when by means of it a weight of 750 lb. was raised 50 ft.? *Supplementary questions and problems for Chapter VII are given in the Appendix. CHAPTER VIII THERMOMETRY; EXPANSION COEFFICIENTS* THERMOMETRY 161. Meaning of temperature. When a body feels hot to the touch we are accustomed to say that it has a high temperature; when it feels cold we say that it has a low temperature. Thus the word "temperature" is used to denote the condition of hotness or coldness of the body whose state is being described. 162. Measurement of temperature. So far as we know, up to the time of Galileo no one had ever used any special instrument for the measurement of temperature. People knew how hot or how cold it was from their feelings only. But under some conditions this sense of temperature is a very unreliable guide. For example, if the hand has been in hot water, tepid water will feel cold; if it has been in cold water, the same tepid water will feel warm; a room may feel warm to one who has been running, whereas it will feel cool to one who has been sitting still. Difficulties of this sort have led to the introduction in modern times of mechanical devices called thermometers, for measuring temperature. These instruments depend for their operation upon the fact that almost all bodies expand as they grow hot. 163. Galileo's thermometer. It was in 1592 that Galileo, at the University of Padua, constructed the first thermometer. He was familiar with the facts of expansion of solids, liquids, and gases; and since gases expand more than solids or liquids, * It is recommended that this chapter be preceded by laboratory measurements on the expansions of a gas and a solid. See, for example, Experiments 20 and 21 of "Exercises in Laboratory Physics," by Millikan, Gale, and Davis. he chose a gas as his expanding substance. His device was that shown in Fig. 148. Let a bulb of air B be connected with a water manometer m, as in Fig. 148. If the bulb is warmed by holding a Bunsen burner beneath it, or even by placing the hand upon it, the water at m will at once begin to descend, showing that the pressure exerted by the air contained in the bulb has been increased by the increase in its temperature. If Bis cooled with ice or ether, the water will rise at m. B m FIG. 148. Ex 164. Significance of temperature from the standpoint of the kinetic theory. If, as was stated in § 64, gas pressure is due to the bombardment of the walls by the molecules of the gas, then, since the number of molecules in the bulb can scarcely have been changed by slightly heating it, we are forced to conclude that the increase in pressure is caused by an increase in the velocity of the molecules which are already there. pansion of air From the standpoint of the kinetic theory the pressure exerted by a given number of molecules of a gas is determined by the kinetic energy of bombardment of these molecules against the containing walls. To increase the temperature is to increase the average kinetic energy of the molecules; to diminish the temperature is to diminish this average kinetic energy. The kinetic theory thus furnishes a very simple and natural explanation of the fact of the expansion of gases with a rise in temperature. by heat 165. Construction of a centigrade mercury thermometer. Forty years after Galileo invented his air thermometer, Jean Rey, a Frenchman, made water instead of air the thermometric substance by inverting Galileo's thermometer and filling the bulb and part of the stem with this liquid. Thermometer tubes were not sealed at the top until a quarter of a century later. It was not until about 1700 that mercury thermometers were invented. On account of their much greater convenience these have now replaced all others for practical purposes. The meaning of a degree of temperature change as measured by a mercury thermometer is best understood from a description of the method of making and graduating the thermometer. A bulb is blown at one end of a piece of thick-walled glass tubing of small, uniform bore. Bulb and tube are filled with mercury at a temperature slightly above the highest temperature for which the thermometer is to be used, and the tube is sealed off in a hot flame. As the mercury cools it contracts and falls away from the top of the tube, leaving a vacuum above it. -0° FIG. 149. Method ---100° The bulb is next surrounded with melting snow or ice, as in Fig. 149, and the point at which the mercury stands in the tube is marked 0°. Then the bulb and tube are placed in the steam rising from boiling water under a pressure of 76 cm., as in Fig. 150, and the new position of the mercury is marked 100°. The space between these two marks on the stem is then divided into 100 equal parts, and divisions of the same length are extended above the 100° mark and below the 0° mark. of finding the 0° FIG. 150. Method of finding the 100° point in calibrating a thermometer Therefore one degree of change in temperature, measured on such a thermometer, means such a temperature change as will cause the mercury in the stem to move over one of these divisions; that is, it is such a temperature change as will cause mercury contained in a glass bulb to expand of the amount which it expands in passing from the temperature of melting ice to that of steam under a pressure of 76 cm. Τ A thermometer in which the scale is divided in this way is called a centigrade thermometer. Thermometers graduated on the centigrade scale are used almost exclusively in scientific work, and also for ordinary purposes in most countries which have adopted the metric system. This scale was first devised in 1742 by Celsius, of Upsala, Sweden. For this reason it is sometimes called the Celsius scale instead of the centigrade. According to the kinetic theory an increase in temperature in a liquid, as in a gas, means an increase in the mean kinetic energy of the molecules; conversely, a decrease in temperature means a decrease in this average kinetic energy. C F 166. Fahrenheit thermometers. The common household thermometer in England and the United States differs from the centigrade only in the manner of its graduation. In its construction the temperature of melting ice is marked 32° instead of 0°, and that of boiling water 212° instead of 100°. The intervening stem is then divided into 180 parts. The zero of this scale is the temperature obtained by mixing equal weights of sal ammoniac (ammonium chloride) and snow. In 1714, when Fahrenheit devised this scale, he chose this zero because he thought it represented the lowest possible temperature that could be obtained in the laboratory. 100° 212° 90° 194° 80° 176° 70° 158° 60° 140° 50° 122° 40° 104° 30° 86° 20° 68° 167. Comparison of centigrade and Fahrenheit thermometers. From the methods of graduation of the Fahrenheit and centigrade thermometers it will be seen that 100° on the centigrade scale denotes the same difference of temperature as 180° on the Fahrenheit scale (Fig. 151). Hence a temperature difference of five centigrade degrees is equal to nine Fahrenheit degrees. Hence, to change from C. to F. tem FIG. 151. The centigrade and Fahrenheit scales |