PRACTICAL PHYSICS MILLIKAN AND GALE GIFT OF Agriculture education PEACTICAL PHYSICS BY EGBERT ANDREWS MILLIKAN, PH.D., Sc.D. DIRECTOR OF THE NORMAN BRIDGE LABORATORY OF PHYSICS, PASADENA, CALIFORNIA AND HENRY GORDON GALE, PH.D. PROFESSOR OF PHYSICS IN THE UNIVERSITY OF CHICAGO BEING A REVISION OF THE AUTHORS' "A FIRST COURSE IN PHYSICS" DONE IN COLLABORATION WITH WILLARD R. PYLE, B.S. HEAD OF THE DEPARTMENT OF PHYSICS, MORRIS HIGH SCHOOL NEW YORK CITY GTNN AND COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1906, 1913, BY ROBERT A. MILLIKAN AND HENRY G. GALE COPYRIGHT, 1920, 1922, BY GINN AND COMPANY ENTERED AT STATIONERS' HALL ALL RIGHTS RESERVED B424.1 tflfb A6RIO, OFPT. , GINN AND COMPANY PRO PRIETORS BOSTON U.S.A. PREFACE The chief aim of this book in all of its editions has been to present elementary physics in such a way as to stimulate the pupil to do some thinking on his own account about the hows and whys of the physical world in which he lives. To this end such subjects, and only such subjects, have been included as touch most closely the everyday life of the average pupil. In a word, the endeavor has been to make this book represent the practical, everyday physics which the average person needs to help him to adjust himself to his surroundings and to interpret his own experiences correctly. But the conditions of modern life are changing at an aston- ishing rate and calling for the continuous revision of any text which would keep pace with them. For example, within the past ten years the internal-combustion engine has not only taken its place as an agent of equal importance with the steam engine in the world's industries but, what is more im- portant, it has also come more fully into the daily life of the average man and woman than even the dynamo and motor have ever begun to do. The automobile is accordingly given fuller treatment in this new book than it has ever received before in any elementary physics text. Again, man's conquest of the air, after centuries of failure, is not only the most significant advance, on the practical side, of the twentieth century, but the airplane now attracts the attention and excites the interest of almost every man, woman, and child. Accordingly, the principles underlying this advance, and the methods by which it was brought about,, find as full treatment in this volume as is consistent iii 55752S iv PKEFACE with the simplicity and clearness demanded in a beginning course in physics. The book may be used, if desired, even in the second year of the high school. Further, the World War was responsible not only for extraordinary developments in physics but also for demon- strating, both to the American youth and to the leader of American industry, the necessity of the more intensive cul- tivation of physical science. These developments and these new demands, with which the authors came into the closest touch because of their service in the army both in this country and in France, have been fully reflected in this book, the emphasis, however, being placed upon develop- ments which make for peace rather than for war. As in preceding editions, the full-page inserts, though a very vital part of the book, are not a necessary and integral part of the course. They have been inserted, in more than double their former number, in order to add human and historic in- terest and to stimulate the pupil to look farther into a sub- ject than his immediate assignment requires him to do. It is thought that they will be found to be an invaluable adjunct to the course. Both the order and the treatment are in many places markedly different from those of preceding editions, and reflect the experience of the tens of thousands of teachers who have used this course, many of whom have assisted the authors with their suggestions. Especially in the problems have important improvements been made. For the sake of indicating in what directions omissions may be made, if necessary, without interfering with con- tinuity, paragraphs have here and there, as in former editions, been thrown into fine print. These paragraphs will be easily distinguished from the classroom experiments, which are in the same type. They are for the most part descriptions of physical appliances. PREFACE v The authors are under great obligation to all of their friends who have assisted them in this work, particularly to their collaborator, Mr. W. R. Pyle ; also to Mr. J. R. Towne of East High School, Minneapolis, Mr. C. F. Button, of the West High School of Commerce, Cleveland, Mr. E. Waite Elder, of the Eastside High School, Denver, Mr. C. E. Harris, of the East High School, Rochester, N. Y., Mr. Walter L. Barnum and Mr. Robert E. Hughes, of the Evanston High School, Evanston, 111., and Dr. George de Bothezat, aeronaut- ical expert of the Advisory Commission for Aeronautics. R. A. MILLIKAN H. G. GALE CONTENTS CHAPTER PAGE I. MEASUREMENT 1 Fundamental Units. Density II. PRESSURE IN LIQUIDS 11 Liquid Pressure beneath a Free Surface. Pascal's Law. The Principle of Archimedes III. PRESSURE IN AIR 26 Barometric Phenomena. Compressibility and Expansibility of Air. Pneumatic Appliances IV. MOLECULAR MOTIONS 49 Kinetic Theory of Gases. Molecular Motions in Liquids. Molecular Motions in Solids V. FORCE AND MOTION 57 Definition and Measurement of Force. Composition and Reso- lution of Forces. Gravitation. Falling Bodies. Newton's Laws VI. MOLECULAR FORCES 90 Elasticity. Capillary Phenomena. Absorption of Gases VII. WORK AND MECHANICAL ENERGY 105 Definition and Measurement of Work. Work and the Pulley. Work and the Lever. The Principle of Work. Power and Energy VIII. THERMOMETRY ; EXPANSION COEFFICIENTS .... 128 Thermometry. Expansion Coefficient of Gases. Expansion of Liquids and Solids. Applications of Expansion IX. WORK AND HEAT ENERGY 144 Friction. Efficiency. Mechanical Equivalent of Heat. Specific Heat X. CHANGE OF STATE 161 Fusion. Properties of Vapors. Hygrometry. Boiling. Artifi- cial Cooling. Industrial Applications vii viii CONTENTS CHAPTER PAGE XI. THE TRANSFERENCE OF HEAT 203 Conduction. Convection. Radiation. Heating and Ventilating XII. MAGNETISM 214 General Properties of Magnets. Terrestrial Magnetism XIII. STATIC ELECTRICITY 225 General Facts of Electrification. Distribution of Charge. Potential and Capacity XIV. ELECTRICITY IN MOTION 244 Detection of Electric Currents. Chemical Effects. Magnetic Effects. Measurement of Currents. Electric Bell and Tele- graph. Resistance and Electromotive Force. Primary Cells. Secondary Cells. Heating Effects XV. INDUCED CURRENTS 290 The Principle of the Dynamo and Motor. Dynamos. The Principle of the Induction Coil and Transformer XVI. NATURE AND TRANSMISSION OF SOUND 319 Speed and Nature. Reflection, Reenf orcement, and Interfer- ence XVII. PROPERTIES OF MUSICAL SOUNDS 337 Musical Scales. Vibrating Strings. Fundamentals and Overtones. Wind Instruments XVIII. NATURE AND PROPAGATION OF LIGHT 357 Transmission of Light. The Nature of Light XIX. IMAGE FORMATION 378 Images formed by Lenses. Images in Mirrors. Optical Instruments XX. COLOR PHENOMENA 402 Color and Wave Length. Spectra XXI. INVISIBLE RADIATIONS 417 Radiation from a Hot Body. Electrical Radiations. Cathode and Rontgen Rays. Radioactivity APPENDIX 447 INDEX . 465 PORTRAITS OF PHYSICISTS AND ILLUSTRATIONS OF RECENT ACHIEVEMENTS IN PHYSICS PAGE 1. The Navy-Curtiss Hydroplane, NC-4 (In colors) . . . Frontispiece 2. Archimedes 22 3. The Details of a Submarine . . . 23 4. Otto von Guericke 32 5. The Mercury-Diffusion Air Pump 33 6. British Dirigible Airship R-34 Arriving in America 44 7. The United States Army Observation Balloon 45 8. Galileo 72 9. French 340-mm. Gun in Action 73 10. Sir Isaac Newton 84 11. The Cream Separator 85 12. James Clerk-Maxwell 102 13. Heinrich Rudolph Hertz 102 14. A Gas Mask 103 15. James Prescott Joule 122 16. James Watt 122 17. The Rocket and the Virginian Mallet 123 18. Lord Kelvin (Sir William Thomson) 134 19. The Clermont and the Leviathan 135 20. A United States Dreadnaught in the Panama Canal 152 21. The Vickers-Vimy Airplane 153 22. A Tank 190 23. The Liberty Motor 191 24. Section of a Modern Automobile 198 25. The Carburetor and an Ignition System 199 26. William Gilbert 222 27. The Sperry Gyrocompass 223 28. Benjamin Franklin < 230 29. Franklin's Kite Experiment 231 30. Count Alessandro Volta 240 31. A Modern High-Tension Tower 241 32. Hans Christian Oersted 246 33. Joseph Henry 246 34. Electromagnets 247 ix X LIST OF ILLUSTRATIONS PAGE 35. Andre" Marie Ampere 256 36. Huge Rotor 257 37. Samuel F. B. Morse 260 38. Diagrams of Morse Telegraph 261 39. Georg Simon Ohm 268 40. The Electric Iron and Fuses 269 41. Michael Faraday 290 42. Induction Motor 291 43. Alexander Graham Bell 316 44. Thomas A. Edison 316 45. Guglielmo Marconi 316 46. Orville Wright 316 47. The Original Wright Glider and the First Power-Driven Airplane 317 48. Sound Waves of Spoken Words . 346 49. Sound Ranging Record of the End of the War 347 50. A. A. Michelson 358 51. Lord Rayleigh (John William Strutt) 358 52. Henry A. Rowland .358 53. Sir William Crookes 358 54. X-Ray Picture of the Human Thorax 359 55. Christian Huygens 364 56. The Great Telescope of the Yerkes Observatory 365 57. Section of a " Movie " Film ' 386 58. Arthur L. Foley's Sound-Wave Photographs 387 59. Three-Color Printing (In colors) 408 60. The Wireless Telephone utilized in Aviation 424 61. Cinematograph Film of a Bullet fired through a Soap Bubble . . 425 62. Alexanderson High-Frequency Alternator 428 63. Interior of Radio Broadcasting Station 429 64. Sir Joseph Thomson 440 65. Amplifier, and Diagram of Receiving and Amplifying Set .... 441 66. William Conrad Rontgen 446 67. Antoine Henri Becquerel 446 68. Madame Curie 446 69. E. Rutherford 446 70. X-Ray Spectra 447 PRACTICAL PHYSICS CHAPTER I MEASUREMENT FUNDAMENTAL UNITS 1. Introductory. A certain amount of knowledge about familiar things comes to us all very early in life. We learn almost unconsciously, for example, that stones fall and bal- loons rise, that the teakettle stops boiling when removed from the fire, that telephone messages travel by electric cur- rents, etc. The aim of the study of physics is to set us to thinking about how and why such things happen, and, to a less degree, to acquaint us with other happenings which we may not have noticed or heard of previously. Most of our accurate knowledge about natural phenomena has been acquired through careful measurements. We can measure three fundamentally different kinds of quantities, length, mass, and time, and we shall find that all other measurements may be reduced to these three. Our first prob- lem in physics is, then, to learn something about the units in terms of which all our physical knowledge is expressed. 2. The historic standard of length. Nearly all civilized nations have at some time employed a unit of length the name of which bore the same significance as does foot in English. There can scarcely be any doubt, therefore, that in each country this unit has been derived from the length of l 2 MEASUREMENT the human foot. It is probable that in England, after the yard (a unit which is supposed to have represented the length of the arm of King Henry I) became established as a standard, the foot was arbitrarily chosen as one third of this standard yard. In view of such an origin it will be clear why no agree- ment existed among the units in use in different countries. 3. Relations between different units of length. It has also been true, in general, that in a given country the different units of length in common use (such, for example, as the inch, the hand, the foot, the fathom, the rod, the mile, etc.) have been derived either from the lengths of different mem- bers of the human body or from equally unrelated magni- tudes, and in consequence have been connected with one another by different, and often by very awkward, multipliers. Thus, there are 12 inches in a foot, 3 feet in a yard, 5J yards in a rod, 1760 yards in a mile, etc. 4. Relations between units of length, area, volume, and mass. A similar and even worse complexity exists in the rela- tions of the units of length to those of area, capacity, and mass. Thus, there are 272| square feet in a square rod ; 57| cubic inches in a quart, and 31J gallons in a barrel. Again, the pound, instead of being the mass of a cubic inch or a cubic foot of water, or of some other common substance, is the mass of a cylinder of platinum, of inconvenient dimensions, which is preserved in London. 5. Origin of the metric system. At the time of the French Revolution the extreme inconvenience of existing weights and measures, together with the confusion arising from the use of different standards in different localities, led the National Assembly of France to appoint a commission to devise a more logical system. The result of the labors of this commission was the present metric system, which was introduced in France in 1793 and has since been adopted by the governments of most civilized nations except those of Great Britain and the FUNDAMENTAL UNITS 3 United States ; and even in these countries its use in scientific work is practically universal. The World War has done much to speed its adoption in these countries. 6. The standard meter. The standard length in the metric system is called the meter. It is the distance, at the freezing temperature, between two transverse parallel lines ruled on a bar of platmum-iridium (Fig. 1), which is kept at the International Bureau of Weights and Measures at Sevres, near Paris. This distance is 39.37 inches. In order that this standard length might be reproduced if lost, the commission attempted to make it one ten-millionth Exact size of the cross section FIG. 1. The standard meter of the distance from the equator to the north pole, measured on the meridian of Paris. But since later measurements have thrown some doubt upon the exactness of the commission's determination of this distance, we now define the meter, not as any particular fraction of the earth's quadrant, but simply as the distance between the scratches on the bar mentioned above. On account of its more convenient size, the centi- meter, one one-hundredth of a meter, is universally used, for scientific purposes, as the fundamental unit of length. 7. Metric standard capacity. The standard unit of capacity is called the liter. It is the volume of a cube which is one tenth of a meter (about 4 inches) on a side. The liter is therefore 4 MEASUKEMEXT equal to 1000 cubic centimeters (cc.). It is equivalent to 1.057 quarts. A liter and a quart are therefore roughly equivalent measures. 8. The metric standard of mass. In order to establish a connection between the unit of length and the unit of mass, the commission directed a committee of the French Academy to prepare a cylinder of platinum which should have the same weight as a liter of water at its temperature of greatest density, namely, 4 Centigrade (39 Fahrenheit). An exact equivalent of this cylinder, made of platinum-iridium and kept at Sevres with the standard meter, now represents the standard of mass in the metric system. It is called the standard kilogram and is equivalent to about 2.2 pounds. One one-thousandth of this mass was adopted as the fundamental unit of mass and was named the gram. For practical purposes, therefore, the gram may be taken as equal to the mass of one cubic centimeter of water, 9. The other metric units. The three standard units of the metric system the meter, the liter, and the gram have decimal multiples and submultiples, so that every unit of length, volume, or mass is connected with the unit of next higher denomination by an invariable multiplier, namely, ten. The names of the multiples are obtained by adding the Greek prefixes, deka (ten), hecto (hundred), kilo (thousand) ; while the submultiples are formed by adding the Latin prefixes, deci (tenth), centi (hundredth), and milli (thousandth). Thus : 1 dekameter = 10 meters 1 decimeter = J_. meter 1 hectometer = 100 meters 1 centimeter = -j-L meter 1 kilometer = 1000 meters 1 millimeter = 10 1 00 meter The most common of these units, with the abbreviations which will henceforth be used for them, are the following: meter (m.) millimeter (mm.) gram (g.) kilometer (km.) liter (1.) kilogram (kg.) centimeter (cm.) cubic centimeter (cc.) milligram (mg.) FUNDAMENTAL UNITS 5 10. Relations between the English and metric units. The following table, which is inserted for reference, gives the relation between the most common English and metric units. 1 inch (in.) = 2.54 cm. 1 cm. = .3937 in. 1 foot (ft.) = 30.48 cm. 1 m, = 1.094 yd. = 39.37 in. 1 mile (mi.) = 1.609 km. 1 km. = .6214 mi. 1 grain = 64.8 mg. 1 g. = 15.44 grains 1 oz. av. = 28.35 g. 1 g. = .0353 oz. 1 Ib. av. = .4536 kg. 1 kg. = 2.204 Ib. The relations 1 in. = 2.54 cm., 1 m. = 39.37 in., 1 kilo (kg.) = 2.2 Ib., 1 km. = .62 mi. should be memorized. Portions of a centimeter and of an inch scale are shown together in Fig. 2. 11. The standard unit of time. The second is taken among all civilized nations as the standard unit of time. It is 86400 part of the time from noon to noon. 12. The three fundamental units. It is evident that meas- urements of both area and volume may be reduced simply CENTIMETER 0123456 UUU I llll IIJllJU 1 1] 1 1 I) I I PI 1 1 1 I] I] If] I IT INCH 1 2 FIG. 2. Centimeter and inch scales to measurements of length; for an area is expressed as the product of two lengths, and a volume as the product of three lengths. For these reasons the units of area and volume are looked upon as derived units, depending on one fundamental unit, the unit of length. Now it is found that just as measurements of area and of volume can be reduced to measurements of length, so the determination of any measurable quantities, such as the pressure in a steam boiler, the velocity of a moving train. 6 MEASUREMENT the amount of electricity consumed by an electric lamp, the amount of magnetism in a magnet, etc., can be reduced simply to measurements of length, mass, and time. Hence the centimeter, the gram, and the second are considered the three fundamental units. Whenever any measurement has been reduced to its equivalent in terms of centimeters, grams, and seconds, it is said, for short, to be expressed in C.G.S. (Centimeter-Gram-Second) units. 13. Measurement of length. Measuring the length of a body consists simply in comparing its length with that of the standard meter bar kept at the International Bureau. In order that this may be done conveniently, great numbers of rods of the same length as this standard meter bar have been made and scattered all over the world. They are our common meter sticks. They are divided into 10, 100, or 1000 equal parts, great care being taken to have all the parts of exactly the same length. The method of making a measurement with such a bar is more or less familiar to everyone. 14. Measurement of mass. Similarly, measuring the mass of a body consists in comparing its mass with that of the standard kilogram. In order that this might be done con- veniently, it was first necessary to construct bodies of the same mass as this kilogram, and then to make a whole series of bodies whose masses were |, -j^-, y^-, 10 1 00 , etc. of the mass of this kilogram; in other words, to construct a set of standard masses commonly called a set of weights. With the aid of such a set of standard masses the deter- mination of the mass of any unknown body is made by first placing the body upon the pan A (Fig. 3) and counterpoising with shot, paper, etc., then replacing the unknown body by as many of the standard masses as are required to bring the pointer back to again. The mass of the body is equal to the sum of these standard masses. This rigorously correct method of weighing is called the method of substitution. FUNDAMENTAL UNITS If a balance is well constructed, however, a weighing may usually be made with sufficient accuracy by simply placing the unknown body upon one pan and finding the sum of the standard masses which must then be placed upon the other pan to bring the pointer again to 0. This is the usual method of weighing. It gives correct results, how- ever, only when the knife-edge is exactly midway between the points of support m and n of the two pans. The method of substitution, on the other hand, is independent of the position of the knife-edge. It is customary to consider that the mass of a body determined as here indicated is a measure of the quantity of matter which it contains. FIG. 3. The simple balance QUESTIONS AND PROBLEMS 1. The 200-meter run at the Olympic games corresponds to the 220- yard run in our local games. Which is the longer and how much ? 2. The French 75-mm. guns have what diameter in inches ? 3. The Twentieth Century Limited runs from New York to Chicago (967 mi.) in 20 hr. Find its average speed in miles per hour. 4. Name as many advantages as you can which the metric system has over the English system. Can you think of any disadvantages ? 5. What must you do to find the capacity in liters of a box when its length, breadth, and depth are given in meters ? to find the capacity in quarts when its dimensions are given in feet? 6. Find the number of millimeters in 6 km. Find the number of inches in 4 mi. Which is the easier? 7. With a Vickers-Vimy biplane Captain Alcock and Lieutenant Brown completed, on June 15, 1919, the first nonstop transatlantic flight of 1890 miles frojn Newfoundland to Ireland in 15 hr. 57 min. How many miles per hour? How many kilometers per hour? 8. Find the capacity in liters of a box .5 m. long, 20 cm. wide, and 100 mm. deep. 8 MEASUREMENT DENSITY 15. Definition of density. When equal volumes of different substances, such as lead, wood, iron, etc., are weighed in the manner described above, they are found to have widely differ- ent masses. The term " density " is used to denote the mass, or quantity of matter, per unit volume. Thus, for example, in the English system the cubic foot is the unit of volume and the pound the unit of mass. Since 1 cubic foot of water is found to weigh 62.4 pounds, we say that in the English system the density of water is 62.4 pounds per cubic foot. In the C.G.S. system the cubic centimeter is taken as the unit of volume and the gram as the unit of mass. Hence we say that in this system the density of water is 1 gram per cubic centimeter, for it will be remembered that the gram was taken as the mass of 1 cubic centimeter of water. Unless otherwise expressly stated, density is now universally under- stood to mean density in C.G.S. units; that is, the density of a substance is the mass in grams of 1 cubic centimeter of that sub- stance. For example, if a block of cast iron 3 cm. wide, 8 cm. long, and 1 cm. thick weighs 177.6 g., then, since there are 24 cc. in the block, the mass of 1 cc., that is, the density, is equal to |^-, or 7.4 g. per cubic centimeter. The density of some of the most common substances is given in the following table : DENSITIES OF SOLIDS (In grams per cubic centimeter) Aluminium 2.58 Nickel 8.9 Brass 8.5 Oak 8 Copper 8.9 Pine 5 Cork 24 Platinum 21.4 Glass 2.6 Silver 10.5 Gold 19.3 Tin 7.3 Iron (cast) 7.4 Tungsten 19.6 Lead . 11.3 Zinc . 7.1 DENSITY 9 DENSITIES OF LIQUIDS (In grams per cubic centimeter) Alcohol 79 Hydrochloric acid . . 1.27 Carbon bisulphide . . . 1.29 Mercury 13.6 Glycerin ..'.'. 1.26 Gasoline 75 16. Relation between mass, volume, and density. Since the mass of a body is equal to the total number of grams which it contains, and since its volume is the number of cubic centi- meters which it occupies, the mass of 1 cubic centimeter is evidently equal to the total mass divided by the volume. Thus, if the mass of 100 cubic centimeters of iron is 740 grams, the density of iron must equal 740 -f-100 = 7.4 grams to the cubic centimeter. To express this relation in the form of an equa- tion, let M represent the mass of a body, that is, its total number of grams ; V its volume, that is, its total number of cubic centimeters ; and D its density, that is, the number of grams in 1 cubic centimeter; then -f m This equation merely states the definition of density in algebraic form. 17. Distinction between density and specific gravity. The term " specific gravity " is used to denote the ratio between the weight of a body and the weight of an equal volume of water.* Thus, if a certain piece of iron weighs 7.4 times as much as an equal volume of water, its specific gravity is 7.4. But since the density of water in C.G.S. units is 1 gram per cubic centimeter, the density of iron in that system is 7.4 grams per cubic centimeter. It is clear, then, that density in 0. G-.S. units is numerically the same as specific gravity. * For the present purpose the terms "weight" and "mass" may be used interchangeably. They are in general numerically equal, although an impor- tant distinction between them will be developed in 73. Weight is in reality a force rather than a quantity of matter. 10 MEASUREMENT Specific gravity is the same in all systems, since it simply expresses how many times as heavy as an equal volume of water a body is. Density, however, which we have defined as the mass per unit volume, is different in different systems. Thus, in the English system the density of iron is 462 pounds per cubic foot (7.4 x 62.4), since we have found that water weighs 62.4 pounds per cubic foot and that iron weighs 7.4 times as much as an equal volume of water.* QUESTIONS AND PROBLEMS t 1. A liter of milk weighs 1032 grams. What is its density and its specific gravity ? 2. A ball of yarn was squeezed into of its original bulk. What effect did this produce upon its mass, its volume, and its density ? 3. If a wooden beam is 30 X 20 x 500 cm. and has a mass of 150 kg., what is the density of wood ? 4. Would you attempt to carry home a block of gold the size of a peck measure? (Consider a peck equal to 8 1. See table, p. 8.) 5. What is the mass of a liter of alcohol? 6. How many cubic centimeters in a block of brass weighing 34 g.? 7. What is the weight in metric tons of a cube of lead 2 m. on an edge ? (A metric ton is 1000 kilos, or about 2200 Ib.) 8. Find the volume in liters of a block of platinum weighing 45.5 kilos. 9. One kilogram of alcohol is poured into a cylindrical vessel and fills it to a depth of 8 cm. Find the cross section of the cylinder. 10. Find the length of a lead rod 1 cm. in diameter and weighing 1 kg. * Laboratory exercises on length, mass, and density measurements should accompany or follow this chapter. See, for example, Experiments 1, 2, and 3 of the authors' Manual. t Questions and problems to supplement this chapter and all following chapters are given in the Appendix, page 447. CHAPTER II PRESSURE IN LIQUIDS LIQUID PRESSURE BEKEATH A FREE SURFACE 18. Force beneath the surface of a liquid. We are all conscious of the fact that in order to lift a kilogram of mass we must exert an upward pull. Experience has taught us that the greater the mass, the greater the force which we must exert. The force is commonly taken as numerically equal to the mass lifted. This is called the weight measure of a force. A. push or pull which is equal to that required to sup- port a gram of mass is called a gram of force. Thus, five grams of force are needed to lift a new five-cent piece. To investigate the nature of the forces beneath the free surface of a liquid we shall use a pressure gauge of the form shown in Fig. 4. If the rubber diaphragm which is stretched across the mouth of a thistle tube A is pressed in lightly with the finger, the drop of ink B will be observed to move forward in the tube T, but it will return again to its first position as soon as the finger is removed. If the pressure of the finger is increased, the drop will move forward a greater distance than before. We may therefore take the amount of motion of the drop as a measure of the force acting on the diaphragm. Now let A be pushed down first 2 cm., then 4 cm., then 8 cm. below the surface of the water *(Fig. 4). The motion of the index B will show that the upward force continually increases as the depth increases. Careful measurements made in the laboratory will show that the force is directly proportional to the depth.* * It is recommended that quantitative laboratory work on the law of depths and on the use of manometers accompany this discussion. See, for example, Experiments 4 and 5 of the authors' Manual. 11 12 PKESSUKE IN LIQUIDS Let the diaphragm A (Fig. 4) be pushed down to some convenient depth (for example, 10 centimeters) and the position of the index noted. Then let it be turned sidewise so that its plane is vertical (see a, Fig. 4), and adjusted in position until its center is exactly 10 centimeters beneath the surface, that is, until the average depth of the diaphragm is the same as before. The position of the index will show that the force is also exactly Fm 4 Qauge for measuri liquid the same as before. pressure Let the diaphragm then be turned to the position Z>, so that the gauge measures the downward force at a depth of 10 centimeters. The index will show that this force is again the same. We conclude, therefore, that at a given depth a liquid presses up and down and sidewise on a given surface with exactly the same force. 19. Magnitude of the force. If a vessel like that shown in Fig. 5 is filled with a liquid, the force against the bottom is obviously equal to the weight of the column of liquid resting upon the bottom. Thus, if F repre- sents this force in grams, A the area in square centi- meters, h the depth in centimeters, and d the density in grams per cubic centimeter, we shall have F=Ahd. (1) FIG. 5 Since, as was shown by the experiment of the preceding section, the force is the same in all directions at a given depth, we have the following general rule : Tlie force which a liquid exerts against any surface is equal to the area of the surface times its average depth times the density of the liquid. It is important to remember that " average depth " means the vertical distance from the level of the free surface to the center of the area in question. PRESSURE BENEATH A FREE SURFACE 13 20. Pressure in liquids. Thus far attention has been con- fined to the total force exerted by a liquid against the whole of a given surface. It is often more convenient to imagine the surface divided into square centimeters or square inches, and then to consider the force on one of these units of area. In physics the word " pressure " is used exclusively to denote the force per unit area. Pressure is thus a measure of the intensity of the force acting on a surface, and does not de- pend at all on the area of the surface. Since, by 19, F= Ahd, and since by definition the pressure p is equal to the force per unit area, we have (2) ~ Therefore the pressure at a depth of h centimeters below the surface of a liquid of density d is hd grams per square centimeter. If the height is given in feet and the density in pounds per cubic foot, then the product hd gives pressure in pounds per square foot. Dividing by 144 gives the result in pounds per square inch. 21. Levels of liquids in connecting vessels. It is a perfectly familiar fact that when water is poured into a teapot it stands at exactly the same level in the spout as in the body of the teapot ; or if it is poured into a number of connected vessels like those shown in Fig. 6, the surfaces of the liquid in the various vessels lie in the same horizontal plane. Now the pressure at c (Fig. 7) was shown by the experiment of 18 to be equal to the density of the liquid times the depth eg. The pressure at o in the opposite direction must be equal to FIG. 6. Water level in communi- cating vessels 14 PBESSUKE IN LIQUIDS that at c, since the liquid does not tend to move in either direction. Hence the pressure at o must be ks times the density. If water is poured in at s so that the height Jcs is increased, the pressure to the left at o becomes greater than the pressure to the right at ody of confined liquid is transmitted undiminished to every portion of the surface of the containing vessel. 23. Multiplication of force by the transmission of pressure by liquids. Pascal himself pointed out that with the aid of the principle stated above we ought to be able to transform a very small force into one of un- 1 J^2M limited magnitude. Thus, if the !? . area of the cylinder ab (Fig. 11) a w> is 1 sq. cm., while that of the cylin- F- der AB is 1000 sq. cm., a force of FIG. 11. Multiplication of 1 kg. applied to ab would be trans- force by transmission of , , pressure mitted by the liquid so as to act with a force of 1 kg. on each square centimeter of the surface AB. Hence the total upward force exerted against the piston AB by the 1 kg. applied at ab would be 1000 kg. Pascal's own words are as follows : " A vessel full of water is a new prin- ciple in mechanics, and a new machine for the multiplication of force to any required extent, since one man will by this means be able to move any given weight." 24, The hydraulic press. The experimental proof of the correctness of the conclusions of the preceding paragraph is furnished by the hydraulic press, an instrument now in common use for subjecting to enormous pressures paper, cotton, etc. and for punching holes through iron plates, testing the strength of iron beams, extracting oil from PASCAL'S LAW 17 seeds, making dies, embossing metal, etc. Hydraulic presses of great power have been designed for use in steel works to replace huge steam hammers. Compressing forces of 10,000 tons or more are thus obtained. Much cold steel, as well as hot, is now pressed instead of hammered. Such a press is represented in section in Fig. 12. As the small piston p is raised, water from the cistern C enters the piston chamber through the valve v. As soon as the downstroke begins, the valve v closes, the valve v' opens, and the pres- sure applied on the piston p is trans- mitted through the tube K to the large reservoir, where it acts on the large cylinder P. The force exerted upon P is as many times that applied to p as the area of P is times the area OJ P- FIG. 12. Diagram of a hydraulic press 25. No gain in the product of force times distance. It should be noticed that, while the force acting on AB (Fig. 11) is 1000 times as great as the force acting on ab, the distance through which the piston AB is pushed up in a given time is but YoVo ^ the distance through which the piston ab moves down. For forcing ab down a distance of 1 centimeter crowds but 1 cubic centimeter of water over into the large cylinder, and this additional cubic centimeter can raise the level of the water there but 10 1 00 centimeter. We see, therefore, that the product of the force acting by the distance moved is precisely the same at both ends of the machine. This important conclusion will be found in our future study to apply to all machines. 18 PEESSUEE IN LIQUIDS 26. The hydraulic elevator. Another very common application of the principle of transformation of pressure by liquids is found in the hydraulic elevator. The simplest form of such an elevator is shown in Fig. 13. The cage A is borne on the top of a long piston P which runs in a cylindrical pit C of the same depth as the height to which the carriage must ascend. Water enters the pit either directly from the water mains, m, of the city's supply or, if this does not fur- nish sufficient pres- sure, from a special reservoir on top of the building. When the elevator boy pulls up on the cord cc, the valve v opens so as to make connection from m into C. The elevator then ascends. When cc is pulled down, v turns so as to permit the water in C to escape into the sewer. The elevator then descends. Where speed is re- quired the motion of the piston is com- municated indirectly to the cage by a sys- tem of pulleys like that shown in Fig. 14. With this arrangement a foot of upward motion of the piston P causes the counterpoise D of the cage to descend 2 feet, for it is clear from the figure that when the piston goes up 1 foot, enough rope must be pulled over the fixed pulley p to lengthen each of the two strands a and b 1 foot. Similarly, when the counterpoise descends 2 feet, the cage ascends 4 feet. Hence the cage moves four times as fast and four times FIG. 14 Diagrams of hydraulic elevators PASCAL'S LAW 19 as far as the piston. The elevators in the Eiffel Tower in Paiis are of this sort. They have a total travel of 420 feet and are capable of lifting 50 people 400 feet per minute. The cylinder C and piston P are often not in a pit but lie in a horizontal position. Most modern eleva- tors are electric rather than hydraulic. 27. City water supply. Fig. 15 illustrates the method by which a city is often supplied with water from a distant source. The aqueduct from the lake a passes under a road r, a brook ^ and a hill //, and into a reservoir e, from which it is forced by the pump p into the standpipe P, whence it is distributed to the houses of the city. If a static condition prevailed in FIG. 15. City water supply from lake the whole system, then the water level in e would of neces- sity be the same as that in a, and the level in the pipes of the building B would be the same as that in the standpipe P. But when the water is flowing, the friction of the mains causes the level in e to be somewhat less than that in a, and that in B less than that in P. It is on account of the friction both of the air and of the pipes that the fountain / does not rise nearly as high as the ideal limit shown in the figure. QUESTIONS AND PROBLEMS 1. A jug full of water may often be burst by striking a blow on the cork. If the surface of the jug is 200 sq. in. and the cross section of the cork 1 sq. in., what total force acts on the interior of the jug when a 10-lb. blow is struck on the cork? 2. How does your city get its water? How is the pressure in the pipes maintained? 20 PRESSUEE IN LIQUIDS 3. If the water pressure in the city mains is 70 Ib. to the square inch, how high above the town is the top of the water in the standpipe ? 4. The cross-sectional areas of the pistons of a hydraulic press were 3 sq. in. and 60 sq. in. How great a weight would the large piston sustain if 75 Ib. were applied to the small one ? 5. The diameters of the pistons of a hydraulic press were 2 in. and 20 in. What force would be produced upon the large piston by 50 Ib. on the small one ? 6. The water pressure in the city mains is 80 Ib. to the square inch. The diameter of the piston of a hydraulic elevator of the type shown in Fig. 13 is 10 in. If friction could be disregarded, how heavy a load could the elevator lift? If 30% of the ideal value must be allowed for frictional loss, what load will the elevator lift? 7. Suppose a tube 5 mm. square and 200 cm. long is inserted into the top of a box 20 cm. on a side and filled with water ; what will be the total force on the bottom of the box ? on the top ? THE PRINCIPLE OF ARCHIMEDES* 28. Apparent loss of weight of a body in a liquid. The preceding experiments have shown that an upward force acts against the bottom of any body immersed in a liquid. If the body is a boat, cork, piece of wood, or any body which floats, it is clear that, since it is in equilibrium, this upward force must be equal to the weight of the body. Even if the body does not float, everyday observation shows that it still loses a portion of its natural weight, for it is well known that it is easier to lift a stone under water than in air, or, again, that a man in a bathtub can support his whole weight by pressing lightly against the bottom with his fingers. It was indeed this very observation which first led the old Greek philosopher Archimedes (287-212 B.C.) (see opposite page 22) to the discovery of the exact law which governs the loss of weight of a body in a liquid. * A laboratory exercise on the experimental proof of Archimedes' princi- ple should either precede or accompany this discussion. See, foi example, Experiment 6 of the authors 1 Manual. THE PRINCIPLE OF ARCHIMEDES 21 Hiero, the tyrant of Syracuse, had ordered a gold crown made, but suspected that the artisan had fraudulently used silver as well as gold in its construction. He ordered Archi- medes to discover whether or not this were true. How to do so without destroying the crown was at first a puzzle to the old philosopher. While in his daily bath, noticing the loss of weight of his own body, it suddenly occurred to him that any body immersed in a liquid must apparently lose a loeight equal to the weight of the displaced liquid. He is said to have jumped at once to his feet and rushed through the streets of Syracuse crying, " Eureka ! Eureka ! " (I have found it ! I have found it!) 29. Theoretical proof of Archimedes* principle. It is prob- able that Archimedes, with that faculty which is so common among men of great genius, saw the truth of his conclusion without going . through any logical process of proof. Such a proof, however, can easily be given. Thus, since the upward force on the bottom of the block abed (Fig. 16) is equal to the weight of the column of liquid obce, and since the downward force on the top of this block is equal to the weight of the column of liquid oade, it is clear that FlG : 16 ' Proof that an immersed body the upward force must exceed the down- j s buoyed up by a ward force by the weight of the column force equal to the of liquid abed. Archimedes' principle may wei ht of the dis - J placed liquid be stated thus: The buoyant force exerted by a liquid is exactly equal to the weight of the displaced liquid. The reasoning is exactly the same, no matter what may be the nature of the liquid in which the body is immersed, nor how far the body may be beneath the surface. Further, if the body weighs more than the liquid which it displaces, it must 22 PKESSUKE IN LIQUIDS sink, for it is urged down with the force of its own weight, and up with the lesser force of the weight of the displaced liquid. But if it weighs less than the dis- placed liquid, then the upward force due to the displaced liquid is greater than its own weight, and consequently it must rise to the surface. When it reaches the surface, the downward force on the top of the block,, due to the liquid, becomes zero. The body must, however, continue to rise until the upward force on its bottom is equal to its own weight. But this upward force is al- FIG. 17. Proof that a floating body is buoyed up by a force equal to the weight of the dis- placed liquid ways equal to the weight of the displaced liquid, that is, to the weight of the column of liquid mbcn (Fig. 17). Hence A floating body must displace its own weight of the liquid in which it floats. This statement is embraced in the state- ment of Archimedes' principle, for a body which floats has lost its whole weight. 30. Specific gravity of a heavy solid. The specific gravity of a body is by definition the ratio of its weight to the weight of an equal volume of water ( 17). Since a submerged body displaces a volume of water equal to its own volume, how- ever irregular it may be, o -a --L t, i Weight of body Specific gravity of body TTT . 1 ^- - Weight of water displaced Making application of Archimedes' principle, we have i. j Weight of body Specific gravity of body = -- Loss of weight in water Fig. 18 shows a common method of weighing under water. FIG. 18. Method of weighing a body under water ARCHIMEDES (287-212 B.C.) (Bust in Naples Museum) The celebrated geometrician of antiquity; lived at Syracuse, Sicily; first made a determination of IT and computed the area of the circle ; discovered the laws of the lever and was author of the famous saying, " Give me where I may stand and I will move the world"; discovered the laws of flotation; invented various devices for repelling the attacks of the Romans in the siege of Syracuse ; on the capture of the city, while in the act of drawing geometrical figures in a dish of sand, he was killed by a Roman soldier to whom he cried out, " Don't spoil my circle " THE DETAILS OF A SUBMARINE The submarine, one of the newest of marine inventions, is a simple application of the principle of Archimedes, one of the oldest principles of physics. In order to submerge, the submarine allows water to enter her ballast tanks until the total weight of the boat and contents becomes nearly as great as that of the water she is able to displace. The boat is then almost submerged. When she is under head- way in this condition, a proper use of the horizontal, or diving, rudders sends her beneath the surface, or, if submerged, brings her to the surface, so that she can scan the horizon with her periscope. The whole operation takes but a few seconds. When the submarine wishes to come to the surface for recharging her batteries or for other purposes, she blows compressed air into her ballast tanks, thus driv- ing the water out of them. Submarines are propelled on the surface by Diesel oil engines ; underneath the surface, by storage batteries and electric motors THE PRINCIPLE OF ARCHIMEDES 23 FIG. 19. Method of finding specific gravity of a light solid 31. Specific gravity of a solid lighter than water. If the body is too light to sink of itself, we may still obtain the weight of the equal volume of water by forcing it beneath the surface with a sinker. Thus, suppose u\ represents the weight on the right pan of the balance when the body is in air and the sinker in water, as in Fig. 19, while w z is the weight on the right pan when both body and sinker are under water. Then w^ w 2 is obviously the buoyant effect of the water on the body alone and is therefore equal to the weight of the displaced water. 32. Specific gravity of liquids by the hydrometer method. The commercial hydrometer such as is now in common use for testing the specific gravity of alcohol, milk, acids, sugar solutions, etc. is of the form shown in Fig. 20. The stem is calibrated by trial so that the specific gravity of any liquid may be read upon it directly. The principle involved is that a floating body sinks until it displaces its own weight. By making the stem very slender the sensitiveness of the instrument may be made very great. Why ? 33. Specific gravity of liquids by "loss of weight " method. If any suitable solid be weighed, first in air, then in water, and then in a liquid of unknown specific gravity, by the principle of Archimedes the loss of weight in the liquid is equal to the weight of the liquid displaced, and the loss in water is equal to the weight of the water FIG. 20. Con- stant-weight hydrometer 24 PRESSURE IN LIQUIDS displaced. If we divide the loss of weight in the liquid by the loss of weight in water, we are dividing the weight of a given volume of liquid by the weight of an equal volume of water. Therefore^ To find the specific gravity of a liquid, divide the loss of weight of some solid in it by the loss of weight of the same body in water.* QUESTIONS AND PROBLEMS 1. Let a vessel of water, together with an object heavier than water, be counterpoised as in Fig. 21 (position a). Now if the object be placed inside the vessel of water (position 6), will the scales remain balanced? Predict the result and then try the experiment. 2. Does the weight ap- parently lost by a submerged body depend upon its volume or its weight ? Explain. 3. A brick lost 1 Ib. when submerged 1 ft. deep; how much would it lose if sus- pended 3 ft. deep ? 4. Will a boat rise or sink deeper in the water as it passes from a river to the ocean ? 5. A fish lies perfectly FIG. 21 motionless near the center of an aquarium. What is the average density of the fish? Explain. 6. Where do the larger numbers appear on hydrometers, toward the bottom or toward the top of the stem ? Explain. 7. A 150-lb. man can just float. What is his volume? 8. Describe fully how you would proceed to find the specific gravity of an irregular solid heavier than water, showing in every case why you proceed as you do. 9. A body loses 25 g. in water, 23 g. in oil, and 20 g. in alcohol. Find the specific gravity of the oil and of the alcohol. * Laboratory experiments on the determination of the densities of solids and liquids should follow or accompany the discussion of this chapter. See, for example, Experiments 7 and 8 of the authors' Manual. THE PRINCIPLE OF ARCHIMEDES 25 10. A platinum ball weighs 330 g. in air, 315 g. in water, and 303 g. in sulphuric acid. Find the volume of the ball and the specific gravity of the platinum and of the acid. 11. A piece of paraffin weighed 178 g. in air, and a sinker weighed 30 g. in water. Both together weighed 8 g. in water. Find the specific gravity of the paraffin. 12. A cube of iron 10 cm. on a side weighs 7500 g. What will it weigh in alcohol of density .82 ? 13. What fraction of the volume of a block of wood will float above water if its density is .5 ? if its density is .6 ? if its density is .9 ? State in general what fraction of the volume of a floating body is under water. 14. If a rectangular iceberg rises 100 ft. above water, how far does it extend below water? (Assume the density of the ice to be .9 that of sea water.) 15. A barge 30 it. by 15 ft. sank 4 in. when an elephant was taken aboard. What was the elephant's weight ? 16. A cubic foot of stone weighed 110 Ib. in water. Find its specific gravity. 17. Steel is three times as heavy as aluminum. When equal volumes of each are submerged in water, how do their apparent losses of weight compare ? 18. The density of cork is .25 g. per cubic centimeter. What force is required to push a cubic centimeter of cork beneath the surface of water? 19. A block of wood 15 cm. by 10 cm. by 4 cm. floats in water with 1 cm. in the air. Find the weight of the wood and its specific gravity. 20. The specific gravity of milk is 1.032. How is its specific gravity affected by removing part of the cream ? by adding water ? May these two changes be made so as not to alter its specific gravity at all? 21. A piece of sandstone having a specific gravity of 2.6 weighs 480 g. in water. Find its weight in air. 22. The density of stone is about 2.5. If a boy can lift 120 Ib., how heavy a stone can he lift to the surface of a pond ? 23. The hull of a modern battleship is made almost entirely of steel, its walls being of steel plates from 6 to 18 in. thick. Explain how it can float. CHAPTER III PRESSURE IN AIR BAROMETRIC PHENOMENA 34. The weight of air. To ordinary observation air is scarcely perceptible. It appears to have no weight and to offer no resist- ance to bodies passing through it. But if a bulb is balanced as in Fig. 22, and then removed and filled with air under pressure by a few strokes of a bicycle pump, it will be found, when placed on the balance again, to be heavier than it was before. On the other hand, if the bulb is connected with an air pump and exhausted, it will be found to have lost weight.* Evi- dently, then, air can be put into and taken out of a vessel, weighed, and handled, just like a liquid or a solid. We are accustomed to say that bodies are "as light as air "; yet careful measurement shows that it takes but 12 cubic feet of air to weigh a pound, so that a single large room contains more air than an ordinary man can lift. Thus, the air in a room 60 feet by 30 feet by 15 feet weighs more than .a ton. The exact weight of air at the freezing temperature and un- der normal atmospheric conditions is .001298 gram per cubic centimeter, that is, 1.293 grams per liter. A given volume of air therefore weighs y^- as much as an equal volume of water. * Another experiment is to weigh an electric-light bulb, then puncture it with a blowpipe and weigh again. 26 FIG. 22. Proof that air has weight BAROMETRIC PHENOMENA 27 35. Proof that air exerts pressure. Since air has weight, it is to be inferred that air, like a liquid, exerts force against any surface immersed in it. The following experiments prove this. Let a rubber membrane be stretched over a glass vessel, as in Fig. '23. As the air is exhausted from beneath the membrane the latter will be observed to be more and more depressed until it will finally burst under the pressure of the air above. Again, let a tin can be partly filled with water and the water boiled. The air will be expelled by the escaping steam. While the boiling is FIG. 23. Rubber mem- brane stretched by weight of air FIG. 24. Gallon can crushed by atmospheric pressure still going on, let the can be tightly corked, then placed in a sink or tray and cold water poured over it. The steam will be condensed and the weight of the air outside will crush the can (see Fig. 24). 36. Cause of the rise of liquids in exhausted tubes. If the lower end of a long tube be dipped into water and the air exhausted from the upper end, water will rise in the tube. We prove the truth of this statement every time we draw lemonade through a straw. The old Greeks and Romans explained such phenomena by saying that " nature abhors a vacuum," and this explanation was still in vogue in Galileo's time. But in 1640 the Duke of Tuscany had a deep well dug near Florence, and found to his surprise that no water pump which could be obtained would raise the water higher than about 32 feet above the level in the well. When he applied to the aged 28 PRESSURE AIR (1) Galileo (see opposite p. 72) for an explanation, the latter replied that evidently " nature's horror of a vacuum did not extend beyond 32 feet." It is quite likely that Galileo sus- pected that the pressure of the air was responsible for the phenomenon, for he had himself proved before that air had weight; and, furthermore, he at once devised another experi- ment to test, as he said, the " power of a vacuum." He died in 1642 before the experiment was performed, but suggested to his pupil Torricelli that he con- tinue the investigation. 37. Torricelli's experiment. Tor- ricelli argued that if water would rise 32 feet, then mercury, which is about 13 times as heavy as water, ought to rise but -^ as high. To test this inference he performed, in 1643, the following famous experiment : Let a tube about 4 ft. long, which is sealed at one end, be completely filled with mercury, as in Fig. 25, (J), then closed with the thumb and inverted, and the bottom immersed in a dish of mer- cury, as in Fig. 25, (2). When the thumb is removed from the bottom of the tube, the mercury will fall away from the upper end of the tube, in spite of the fact that in so doing it will leave a vacuum above it; and its upper surface will, in fact, stand about ^ of 32 ft., that is, between 29 and 30 in., above the mercury in the dish. Torricelli concluded from this experiment that the rise of liquids in exhausted tubes is due to an outside pressure exerted by the atmosphere on the surface of the liquid, and not to any mysterious sucking power created by the vacuum as is popularly believed even to-day. FIG. 25. Torricelli's experiment BAKOMETKIC PHENOMENA 29 FIG. 26. Barometer falls when air pres- sure on the mercury surface is reduced 38. Further decisive tests. An unanswerable argument in favor of this conclusion will be furnished if the mercury in the tube falls as soon as the air is removed from above the surface of the mercury in the dish. To test this point, let the dish and tube be placed on the table of an air pump, as in Fig. 26, the tube passing through a tightly fitting rubber stop- per A in the bell jar. As soon as the pump is started the mercury in the tube will, in fact, be seen to fall. As the pumping is continued it will fall nearer and nearer to the level in the dish, although it will not usually reach it, for the reason that an ordinary vacuum pump is not capable of producing as good a vacuum as that which exists in the top of the tube. As the air is allowed to return to the bell jar the mercury will rise in the tube CJIil * mid to its former level. 39. Amount of the atmospheric pressure. Torricelli's experiment shows exactly how great the atmospheric pressure is, since this pressure is able to balance a column of mercury of definite length. As the pres- sures along the same level ac (Fig. 27) are equal, the downward pressure exerted by the atmosphere on the surface of the mercury at c is equal to the downward pressure of the column of mercury at a. But the downward pressure at this point within the tube is equal to hd, where d is the density of mercury and li is the depth below the surface b. Since the average height of this FIG. 27. Air column to top of atmosphere balances the mercury column ab 30 PRESSURE IN AIR column at sea level is found to be 76 centimeters, and since the density of mercury is 13.6 grams per cubic centimeter, the downward pressure inside the tube at a is equal to 76 times 13.6 grams, or 1033.6 grams per square centimeter. Hence the atmospheric pressure acting on the surface of the mercury at c is 1033.6 grams, or, roughly, 1 kilogram per square centimeter. The pressure of one atmosphere is, then, about 15 pounds per square inch. 40. Pascal's experiment. Pascal thought of another way of testing whether or not it were indeed the weight of the outside air which sustains the column of mercury in an exhausted tube. He reasoned that, since the pressure in a liquid diminishes on ascending toward the surface, atmos- pheric pressure ought also to diminish on passing from sea level to a mountain top. As there was no mountain near Paris, he carried Torricelli's apparatus to the top of a high tower and found, indeed, a slight fall in the height of the column of mercury. He then wrote to his brother-in-law, Perrier, who lived near Puy de DOme, a mountain in the south of France, and asked him to try the experiment on a larger scale. Perrier wrote back that he was " ravished with admiration and astonishment" when he found that on ascending 1000 meters the mercury sank about 8 centimeters in the tube. This was in 1648, five years after Torricelli's discovery. At the present clay geological parties actually ascertain dif- ferences in altitude by observing the change in the barometric pressure as they ascend or descend. A fall of 1 millimeter in the barometric height corresponds to an ascent of about 12 meters. 41. The barometer. The modern barometer (Fig. 28) is essentially nothing more nor less than Torricelli's tube. Tak- ing a barometer reading consists simply in accurately measur- ing the height of the mercury column. This height varies from 73 to 76.5 centimeters in localities which are not far above BAROMETKIC PHENOMENA 31 sea level, the reason being that disturbances in the atmosphere affect the pressure at the earth's surface in the same way in which eddies and high waves in a tank of water would affect the liquid pressure at the bottom of the tank. The barometer does not directly foretell the weather, but it has been found that a low or rapidly falling pressure is usually accompanied, or soon followed, by stormy conditions. Hence the barometer, although not an infallible weather prophet, is never- theless of considerable assistance in fore- casting weather conditions some hours ahead. Further, by comparing at a central station the telegraphic reports of barometer readings made every few hours at stations all over the country, it is possible to deter- mine in what direction the atmospheric eddies which cause barometer changes and stormy conditions are traveling and hence to forecast the weather even a day or two in advance. 42. The first barometers. Torricelli actually constructed a barometer not essentially different from that shown in Fig. 28 and used it for observing changes in the atmospheric pressure; but perhaps the most interesting of the early barometers was that set up about 1650 by Otto von Guericke of Magdeburg (1602-1686) (see opposite p. 32). He used for his barometer a water column the top of which passed through the roof of his house. A wooden image which floated on the upper surface of the water appeared above the housetop in fair weather but retired from sight in foul, a circumstance which led his neighbors to charge him with being in league with Satan. 43. The aneroid barometer. Since the mercurial barometer is some- what long and inconvenient to carry, geological and surveying parties FIG. 28. The Fortin barometer 32 PRESSURE IN AIR commonly use an instrument called the aneroid barometer. It consists essentially of an air-tight cylindrical box the top of which is a metallic diaphragm which bends slightly under the influence of change in the atmospheric pressure. This motion of the top of the box is multiplied by a delicate system of levers and communicated to a hand which moves over a dial whose readings are made to correspond to the readings of a mercury barometer. These instruments are made so sensitive as to TIG. 29. The aneroid barometer indicate a change in pressure when they are moved no farther than from a table to the floor. In the self-recording aneroid barometer, or baro- graph, used by the United States Weather Bureau (Fig. 29), several of the air-tight boxes are superposed for greater sensitiveness, and the pressures are recorded in ink upon paper wound about a drum. Clockwork inside the drum makes it revolve once a week. A somewhat different form of the instrument is used by aviators to record altitude. QUESTIONS AND PROBLEMS 1. Why does not the ink run out of a pneumatic inkstand like that shown in Fig. 30 ? 2. If a tumbler is filled, or partly filled, with water, and a piece of writing paper is placed over the top, it may be inverted, as in Fig. 31, . without spilling the water. Explain. What is the function of the paper ? OTTO VON GUERICKE (1602-1686) German physicist, astronomer, and man of affairs ; mayor of Mag- deburg; invented the air pump in 1650, and performed many new experiments with liquids and gases ; discovered electrostatic repulsion ; constructed the famous Magdeburg hemispheres which four teams of horses could not pull apart (see p. 33) Water^Outlet THE MERCURY-DIFFUSION AIR PUMP The latest development of the air pump is shown in the accompanying diagram. It is over a million times more effective than an air pump of the mechanical kind invented by Von Guericke. The principle is as follows: The jet of water pouring out through J^ from an ordinary water tap T entrains the air in the chamber C and thus pulls the pressure in C" down to from 10 to 15 mm. of mer- cury. Next, the mercury jet ,7 2 > produced by boiling violently the mercury above the electric furnace F, entrains the air in the chamber (J" and thus lowers the pressure in this chamber to, say, .01 mm. of mercury. Again, the stream of mer- cury vapor pouring out of y a force which is equal to the weight of the displaced air. The body will therefore rise if its own weight is less than the weight of the air which it displaces. A balloon is a large silk bag (see opposite page 45) impregnated with rubber and filled either with hydrogen or with common illumi- nating gas. The former gas weighs about .09 kilogram per cubic meter, and common illumi- nating gas weighs about .75 kilogram per cubic meter. It will be remembered that ordinary air weighs about 1.20 kilograms per cubic meter. It will be seen, therefore, that the lifting force of hydrogen per cubic meter namely, 1.20 .09 = 1.11, is more than twice the lifting force of illuminating gas, 1.20 .75 = .45. Ordinarily a balloon is not completely filled at the start ; for if it were, since the outside pressure is continually diminishing as it ascends, the pressure of the inside gas would subject the bag to enormous strain and would surely burst it before it reached any considerable altitude. But if it is but partially inflated at the start, it can increase in volume as it ascends by simply inflating to a greater extent. Thus, a balloon which ascends until the pressure is but 7 centi- meters of mercury should be only about one fourth inflated when it is at the surface. The parachute (Fig. 46) is a huge, umbrella-like affair with which the aeronaut may descend in safety to the earth. After opening, it descends very slowly on account of the enormous surface exposed to the air. A hole in the top allows air to escape slowly, and thus keeps the parachute upright. FIG. 40. The parachute PNEUMATIC APPLIANCES 45 57. Helium balloons. One of the striking results of the World War was the development of the helium balloon. Helium is a noninflam- mable gas twice as dense as hydrogen and having a lifting power .92 as great. It is so rare an element that before the war not over 100 cu. ft. had been collected by anyone. Its pre-war price was $1700 per cu. ft. At the close of the war 147,000 cu. ft., extracted at a cost of ten cents a cubic foot from the gas wells of Texas and Okla- homa, were ready for shipment to France, and plans were under way for producing it at the rate of 50,000 cu. ft. per day. The production of a balloon gas that assures safety from fire opens up a new era for the dirigible balloon (see opposite page 44). 58. The diving bell. The diving bell (Fig. 47) is a heavy, bell-shaped body with rigid walls, which sinks of its own weight. Formerly the workmen who went down in the bell had at their disposal only the amount of air confined within it, and the water rose to a certain height within the bell on account of the compression of the air. But in modern practice the air is forced in from the surface through a connecting tube a (Fig. 48) by means of a force pump h. This arrangement, in addi- tion to furnishing a con- tinual supply of fresh air, makes it possible to force the water down to the level of the bottom of the i n T ,. . . FIG. 48. Laying foundations of piers bell. In practice a contm- with the diving bell ual stream of bubbles is kept flowing out from the lower edge .of the bell, as shown in Fig. 48, which illustrates subaqueous construction. 46 PRESSURE IN AIR, The pressure of the air within the bell must, of course, be the pressure existing within the water at the depth of the level of the water inside the bell; that is, in Fig. 47 at the depth AC. Thus, at a depth of 34 feet the pressure is 2 atmospheres. Diving bells are used for putting in the founda- tions of bridge piers, doing subaqueous excavating, etc. The so-called caisson, much used in bridge building, is simply a huge stationary diving bell, which the workmen enter through compartments provided with air-tight doors. Air is pumped into it precisely as in Fig. 48. 59. The diving suit. For most purposes except those of heavy engi- neering the diving suit (Fig. 49) has now replaced the diving bell. This suit is made of rubber and has a metal helmet. The diver is sometimes connected with the surface by a tube through which air is forced down to him. It passes out into the water through the valve V in his suit. But more commonly the diver is entirely independent of the surface, carrying air under a pressure of about 40 at- mospheres in a tank on his back. This air is allowed to escape gradually through the suit and out into the water through the valve V as fast as the diver needs it. When he wishes to rise to the surface, he simply admits enough air to his suit to make him float. In all cases the diver is subjected to the pressure ex- isting at the depth at which the suit or bell commu- nicates with the outside water. Divers seldom work at depths greater than 60 feet, and 80 feet is usually considered the limit of safety. But Chief Gunner's Mate Frank Crilley, investigating the sunken U. S. submarine f-4 at Honolulu in 1915, descended to a depth of 304 feet. The diver experiences pain in the ears and above the eyes when he is ascending or descending, but not when at rest. This is because it requires some time for the air to penetrate into the interior cavities of the body and establish equal pressure in both directions. 60. The gas meter. Gas from tjie city supply enters the meter through P (Fig. 50) and passes through the openings o and o x into the compart- ments B and B l of the meter. Here its pressure forces in the diaphragms FIG. 49. The div- ing suit PNEUMATIC APPLIANCES 47 d and d r The gas already contained in A and A 1 is therefore pushed out to the burners through the openings o f and o^ and the pipe P r As soon as the diaphragm d has moved as far as it can to the right, a lever which is worked by the movement of d causes the slide valve u to move to the left, thus closing o and shutting off con- nection between P and B, but at the same time opening o' and allowing the gas from P to enter compartment A through o'. A quarter of a cycle later u^ moves to the right and connects A l with P and B l with P r If u and u^ were set so as to work exactly together, there would fye slight fluctuations in the gas pressure at P r The movement of the diaphragms is recorded by a clockwork device, the dials of which in- dicate the number of cubic feet of gas which have passed through the meter. FIG. 50. The gas meter QUESTIONS AND PROBLEMS 1. A water tank 8 ft. deep, standing some distance above the ground, closed everywhere except at the top, is to be emptied. The only means of emptying it is a flexible tube, (a) What is the most convenient way of using the tube, and how could it be set into operation ? (b) How long must the tube be to empty the tan.fc completely? 2. Kerosene has a specific gravity of .8. Over what height can it be siphoned at normal pressure ? 3. Let a siphon of the form shown in Fig. 51 be made by filling a flask one third full of water, closing it with a cork through which pass two pieces of glass tubing, as in the figure, and then inverting so that the lower end of the straight tube is in a dish of water. If the bent arm is of consid- erable length, the fountain will play forcibly and continu- ously until the dish is emptied. Explain. 4. Diagram a lift pump on upstroke. What causes the water to rise in the suction pipe ? What happens on downstroke ? 5. Diagram a force pump with air dome on down- stroke. What happens on upstroke ? FIG. 51 6. If the cylinder of an air pump is of the same size as the receiver, what fractional part of the air is removed by one complete stroke? What fractional part is left after 3 strokes? after 10 strokes ? 48 PRESSURE IN AIR CUBIC FEET CUBIC FEET 7. If the cylinder of an air pump is one third the size of the receiver, what fractional part of the original air will be left after 5 strokes? What will be the reading of a barometer within the receiver, the outside pressure being 76 ? 8. Theoretically, can a vessel ever be completely exhausted by an air pump, even if mechanically perfect ? 9. Explain by reference to atmospheric pressure why a balloon rises. 10. How many of the laws of liquids and gases do you find illustrated in the experiment of the Cartesian diver? 11. Pneumatic dispatch tubes are now used in many large stores for the transmission of small pack- ages. An exhaust pump is attached to one end of the tube in which a tightly fitting carriage moves, and a compression pump to the other. If the air is half exhausted on one side of the carriage and has twice its normal density on the other, find the propelling force acting on the carriage when the area of its cross section is 50 sq. cm. 12. What determines how far a balloon will ascend? Under what conditions will it begin to descend ? Explain these phenomena by the principle of Archimedes. 13. If a diving bell (Fig. 47) is sunk until the level of the water within it is 1033 cm. beneath the surface, to what fraction of its initial volume has the inclosed air been reduced ? (1033 g. per sq. cm. = 1 atmosphere.) 14. If a diver's tank has a volume of 2 cu. ft. and contains air under a pressure of 40 atmospheres, to what volume will the air expand when it is released at a depth of 34 ft. under water ? 15. A submarine weighs 1800 tons when its submerging tanks are empty, and in that condition 10 per cent by volume of the submarine is above water. What weight of water must be let into the tanks to just submerge the boat? 16. (a) The upper figure shows a reading of 84,600 cu. ft. of gas. The lower figure shows the reading of the meter a month later. What was the amount of the bill for the month at $.80 per 1000 cu. ft. ? (b) Diagram the meter dials to represent 49,200 cu. ft. FIG. 52. The dials of a gas meter CHAPTER IV MOLECULAR MOTIONS KINETIC THEORY OF GASES 61. Molecular constitution of matter. In order to account for some of the simplest facts in nature for example, the fact that two substances often apparently occupy the same space at the same time, as when two gases are crowded together in the same vessel or when sugar is dissolved in water it is now universally assumed that all substances are composed of very minute particles called molecules. Spaces are supposed to exist between these molecules, so that when one gas enters a vessel which is already full of another gas the molecules of the one scatter themselves about among the molecules of the other. Since molecules cannot be seen with the most powerful microscopes, it is evident that they must be very minute. The number of them contained in a cubic centi- meter of air is 27 billion billion (27 x 10 18 ). It would take as many as a thousand molecules laid side by side to make a speck long enough to be seen with the best microscopes. 62. Evidence for molecular motions in gases. Certain very simple observations lead us to the conclusion that the mole- cules of gases, even in a still room, must be in continual and quite rapid motion. Thus, if a little chlorine, or ammonia, or any gas of powerful odor is introduced into a room, in a very short time it will have become perceptible in all parts of the room. This shows clearly that enough of the molecules of the gas to affect the olfactory nerves must have found their way across the room. 49 50 MOLECULAR MOTIONS FIG. 53. Illustrat- ing the diffusion of gases Again, chemists tell us that if two globes, one containing hydrogen and the other carbon dioxide gas, be connected as in Fig. 53, and the stopcock between them opened, after a few hours chemical analysis will show that each of the globes contains the two gases in exactly the same proportions, a result which is at first sight very surprising, since carbon dioxide gas is about twenty-two times as heavy as hydrogen. This mixing of gases in apparent violation of the laws of weight is called diffusion. We see, then, that such simple facts as the transference of odors and the diffusion of gases furnish very convincing evidence that the molecules of a gas are not at rest but are continually moving about. 63. Molecular motions and the indefinite expansibility of a gas. Perhaps the most striking property which we have found gases to possess is the property of indefinite or unlimited expansibility. The exist- ence of this property was demonstrated by the fact that we were able to attain a high degree of exhaustion by means of an air pump. No matter how much air was removed from the bell jar, the remainder at once expanded and filled the entire vessel. The motions of the molecules furnish a thoroughly satisfactory explanation of the phenomenon. The fact that, however rapidly the piston of the air pump is drawn up, gas always appears to follow it instantly, leads us to the conclusion that the natural velocity possessed by the molecules of gas must be very great. 64. Molecular motions and gas pressures. How are we to account for the fact that gases exert such pressures as they do against the walls of the vessels which contain them? We have found that in an ordinary room the air presses against the walls with a force of 15 pounds to the square KINETIC THEORY OF GASES 51 inch. Within an automobile tire this pressure may amount to as much as 100 pounds, and the steam pressure within the boiler of an engine is often as high as 240 pounds per square inch. Yet in all these cases we may be certain that the mole- cules of the gas are separated from each other by distances which are large in comparison with the diameters of the mole- cules ; for when we reduce steam to water, it shrinks to 16 1 00 of its original volume, and when we reduce air to the liquid form, it shrinks to about -g-i-^ of its ordinary volume. The explanation is at once apparent when we reflect upon the motions of the molecules. For just as a stream of water particles from a hose exerts a continuous force against a wall on which it strikes, so the blows which the innumerable molecules of a gas strike against the walls of the containing vessel must constitute a continuous force tending to push out these walls. In this way we account for the fact that vessels containing only gas do not collapse under the enor- mous external pressures to which we know them to be subjected. A soap bubble 6i inches in diameter is, at normal atmospheric pressure, under a total crushing force of one ton. 65. Explanation of Boyle's law. It will be remembered that it was discovered in the last chapter that when the den- sity of a gas is doubled, the temperature remaining constant, the pressure is found to double also ; when the density was trebled, the pressure was trebled ; etc. This, in fact, was the assertion of Boyle's law. Now this is exactly what would be expected if the pressure which a gas exerts against a given surface is due to blows struck by an enormous number of swiftly moving molecules ; for doubling the number of mole- cules in the given space, that is, doubling the density, would simply double the number of blows struck per second against that surface, and hence would double the pressure. The kinetic theory of gases which is here presented accounts in this simple way for Boyle's law. 52 MOLECULAR MOTIONS 66. Brownian movements and molecular motions. It has recently been found possible to demonstrate the existence of molecular motions in gases in a very direct and striking way. It is found that very minute oil drops suspended in perfectly stagnant air, instead of being them- selves at rest, are ceaselessly dancing about just as though they were endowed with life. In 1913 it was definitely proved that these mo- tions, which are known as the Brownian movements, are the direct result of the bombardment which the droplets receive from the flying molecules of the gas with which they are surrounded; for at a given instant this bombardment is not the same on all sides, and hence the suspended particle, if it is minute enough, is pushed hither and thither according as the bombardment is more intense first in one direction, then in another. There can be no doubt that what the oil drops are here seen to be doing, the molecules themselves are also doing, only in a much more lively way. 67. Molecular velocities. From the known weight of a cubic centi- meter of air under normal conditions, and the known force which it exerts per square centimeter (namely, 1033 grams), it is possible to calculate the velocity which its molecules must possess in order that they may produce by their collisions against the walls this amount of force. The result of the calculation gives to the air molecules under normal conditions a velocity of about 445 meters per second, while it assigns to the hydrogen molecules the enormous speed of 1700 meters (a mile) per second. The speed of a projectile is seldom greater than 800 meters (2500 feet) per second. It is easy to see, then, since the molecules of gases are endowed with such speeds, why air, for example, expands instantly into the space left behind by the rising piston of the air pump, and why any gas always fills completely the vessel which contains it (see mercury-diffusion air pump, opposite page 33). 68. Diffusion of gases through porous walls. Strong evi- dence for the correctness of the above views is furnished by the following experiment : Let a porous cup of unglazed earthenware be closed with a rubber stopper through which a glass tube passes, as in Fig. 54. Let the tube be dipped into a dish of colored water, and a jar containing hydrogen placed over the porous cup ; or let the jar simply be held in the position shown in the figure, and let illuminating gas be passed into it by means of a rubber tube connected with a gas jet. The rapid passage of bubbles out through the water will show that the gaseous pressure inside the MOLECULAR MOTIONS IN LIQUIDS 53 cup is rapidly increasing. Now let the bell jar be lifted, so that the hydrogen is removed from the outside. Water will at once begin to rise in the tube, showing that the inside pressure is now rapidly decreasing- The explanation is as follows : We have learned that the molecules of hydrogen have about four times the velocity of the mole- cules of air. Hence, if there are as many hydrogen molecules per cubic centimeter outside the cup as there are air molecules per cubic centimeter inside, the hydrogen molecules will strike the outside of the wall four times as frequently as the air molecules will strike the inside. Hence, in a given time the number of hydrogen molecules which pass into the interior of the cup through the little holes in the porous mate- rial is four times as great as the num- ber of air particles which pass out ; hence the pressure within increases. When the bell jar is removed, the hydrogen which has passed inside begins to pass out faster than the outside air passes in, and hence the inside pressure is diminished. MOLECULAR MOTIONS IN LIQUIDS 69. Molecular motions in liquids and evaporation. Evidence that the molecules of liquids as well as those of gases are in a state of perpetual motion is found, first, in the familiar facts of evaporation. We know that the molecules of a liquid in an open vessel are continually passing off into the space above, for it is only a matter of time when the liquid completely disappears and the vessel becomes dry. Now it is hard to imagine a way in which the molecules of a liquid thus pass out of the liquid into the space above, unless these molecules, while in the liquid condition, FIG. 54. Diffusion of hydrogen through porous cup 54 MOLECULAR MOTIONS are in motion. As soon, however, as such a motion is assumed, the facts of evaporation become perfectly intelligible. For it is to be expected that in the jostlings and collisions of rapidly moving liquid molecules an occasional molecule will acquire a velocity much greater than the average. This molecule may then, because of the unusual speed of its motion, break away from the attraction of its neighbors and fly off into the space above. This is indeed the mechanism by which we now believe that the process of evaporation goes on from the surface of any liquid. 70. Molecular motions and the diffusion of liquids. One of the most convincing arguments for the motions of molecules in gases was found in the fact of diffusion. But precisely the same sort of phenomena are observable in liquids. Let a few lumps of blue litmus be pulverized and dissolved in water. Let a tall glass cylinder be half filled with this water anc|, a few drops of ammonia added. Let the remainder of the litmus solution be turned red by the addition of one or two cubic centi- meters of nitric acid. Then let this acidulated water be introduced into the bottom of the jar through a FIG. 55. Diffusion thistle tube (Fig. 55). In a few minutes the line of of liquids separation between the acidulated water and the blue solution will be fairly sharp ; but in the course of a few hours, even though the jar is kept perfectly quiet, the red color will be found to have spread considerably toward the top, showing that the acid molecules have gradually found their way up. Certainly, then, the molecules of a liquid must be endowed with the power of independent motion. Indeed, every one of the arguments for molecular motions in gases applies with equal force to liquids. Even the Brownian movements can be seen in liquids, though they are here so small that high- power microscopes must be used to make them apparent. MOLECULAR MOTIONS IN SOLIDS 55 MOLECULAR MOTIONS IN SOLIDS 71. Molecular motions and the diffusion of solids. It has recently been demonstrated that if a layer of lead is placed upon a layer of gold, molecules of gold may in time be de- tected throughout the whole mass of the lead. This diffusion of solids into one another at ordinary temperature has been shown only for these two metals, but at higher temperatures (for example, 500 C.) all of the metals show the same char- acteristics to quite a surprising degree. The evidence for the existence of molecular motions in solids is, then, no less strong than in the case of liquids. 72. The three states of matter. Although it has been shown that, in accordance with current belief, the molecules of all substances are in very rapid motion, yet differences exist in the kind of motion which the molecules in the three states possess. Thus, in the solid state it is probable that the mole- cules oscillate with great rapidity about certain fixed points, always being held by the attractions of their neighbors, that is, by the cohesive forces (see 112), in very nearly the same positions with reference to other molecules in the body. In rare instances, however, as the facts of diffusion show, a molecule breaks away from its constraints. In liquids, on the other hand, while the molecules are, in general, as close together as in solids, they slip about with perfect ease over one another and thus have no fixed positions. This assump- tion is necessitated by the fact that liquids adjust themselves readily to the shape of the. containing vessel. In gases the molecules are comparatively far apart, as is evident from the fact that a cubic centimeter of water occupies about 1600 cubic centimeters when it is transformed into steam ; and, furthermore, they exert almost no cohesive force upon one another, as is shown by the indefinite expansibility of gases. 56 MOLECULAK MOTIONS QUESTIONS AND PROBLEMS 1. If a vessel with a small leak is filled with hydrogen at a pressure of 2 atmospheres, the pressure falls to 1 atmosphere about four times as fast as when the same experiment is tried with air. Can you see a reason for this? 2. What is the density of the air within an automobile tire that is inflated to a pressure of 80 Ib. per square inch ? (1 atmosphere = 14.7 Ib. per sq. in.) 3. A liter of air at a pressure of 76 cm. is compressed so as to occupy 400 cc. What is the pressure against the walls of the containing vessel? 4. If an open vessel contains 250 g. of air when the barometric height is 750 mm., what weight will the same vessel contain at the same tem- perature when the barometric height is 740 mm.? 5. Find the pressure to which the diver was subjected who descended to a depth of 304 ft. Find the density of the air in his suit, the density at the surface being .00128 g. per cubic centimeter and the temperature being assumed to remain constant. Take the pressure at the surface as 30 in. 6. A bubble of air which escaped from this diver's suit would increase to how many times its volume on reaching the surface? 7. Salt is heavier than water. Why does not all the salt in a mixture of salt and water settle to the bottom? CHAPTER V FORCE AND MOTION DEFINITION AND MEASUREMENT OF FORCE 73. Distinction between a gram of mass and a gram of force. If a gram of mass is held in the outstretched hand, a down- ward pull upon the hand is felt. If the mass is 50,000 g. in- stead of 1, this pull is so great that the hand cannot be held in place. The cause of this pull we assume to be an attractive force which the earth exerts on the matter held in the hand, and we define the gram of force as the amount of the earths pull at its surface upon one gram of mass. Unfortunately, in ordinary conversation we often fail alto- gether to distinguish between the idea of mass and the idea of force, and use the same word " gram " to mean sometimes a certain amount of matter and at other times the pull of the earth upon this amount of matter. That the two ideas are, how- ever, wholly distinct is evident from the consideration that the amount of matter in a body is always the same, no matter where the body is in the universe, while the pull of the earth upon that amount of matter decreases as we recede from the earth's surface. It will help to avoid confusion if we reserve the simple term " gram " to denote exclusively an amount of matter (that is, a mass) and use the full expression " gram of force " wherever we have in mind the pull of the earth upon this mass. 74. Method of measuring forces. When we wish to com- pare accurately the pulls exerted by the earth upon different masses, we find such sensations as those described in the 57 58 FORCE AND MOTION preceding paragraph very untrustworthy guides. An accurate method, however, of comparing these pulls is that furnished by the stretch produced in a spiral spring. Thus, the pull of the earth upon a gram of mass at its sur- face will stretch a given spring a given distance, ab (Fig. 56) ; the pull of the earth upon 2 grams of mass is found to stretch the spring a larger distance, ac\ upon 3 grams, a still larger distance, ad; etc. In order to graduate a spring balance (Fig. 57) so that it will thenceforth measure the values of any pulls exerted upon it, no matter how these pulls may arise, we have only to place a fixed -p IG 55 Method of surface behind the pointer and make lines measuring forces upon it corresponding to the points to which it is stretched by the pull of the earth upon different masses. Thus, if a man stretch the spring so that the pointer is opposite the mark corresponding to the pull of the earth upon 2 grams of mass, we say that he exerts 2 grams of force ; if he stretch it the distance corresponding to the pull of the earth upon 3 grams of mass, he exerts 3 grams of force ; etc. The spring balance thus becomes an instrument for measuring forces. 75. The gram of force varies slightly in differ- ent localities. With the spring balance it is easy to verify the statement made above, that the force of the earth's pull decreases as we recede from the earth's surface ; for upon a high moun- tain the stretch produced by a given mass is indeed found to be slightly less than at sea level. Furthermore, if the balance is simply carried from point to point over the earth's surface, the stretch is still found to vary slightly. For ex- ample, at Chicago it is about one part in 1000 less than it FIG. 57. The spring balance COMPOSITION AND RESOLUTION OF FORCES 59 is at Paris, and near the equator it is five parts in 1000 less than it is near the pole. This is due in part to the earth's rotation and in part to the fact that the earth is not a perfect sphere and that in going from the equator toward the pole we are coming nearer and nearer to the center of the earth. We see, therefore, that the weight of one gram of mass is not an absolutely definite unit of force. One gram of force is, strictly speaking, the weight of one gram of mass in latitude 45 at sea level. COMPOSITION AND RESOLUTION or FORCES 76. Graphic representation of force. A force is completely described when its magnitude, its direction, and the point at which it is applied are given. Since the three characteristics of a straight line are its length, its direction, and the point at which it starts, it is obviously possible to represent forces by means of straight lines. A ^ Thus, if we wish to represent the fact that FlG - 58< Gra P hic r f. -. . . , representation of a force ot 8 pounds, acting in an easterly a gingle force direction, is applied at the point A (Fig. 58), we draw a line 8 units long, beginning at the point A and extending to the right. The length of this line then repre- sents the magnitude of the force ; the direction of the line, the direction of the force ; and the starting point of the line, the point at which the force is applied. 77. Resultant of two forces acting in the same line. The resultant of two forces is defined as that single force which will produce the same effect upon a body as is produced by the joint action of the two forces. If two spring balances are attached to a small ring and pulled in the same direction until one registers 10 g. of force and the other 5, it will be found that a third spring balance attached to the same point and pulled in the opposite direc- tion will register exactly 15 g. when there is equilibrium ; 60 FORCE AND MOTION that is, the resultant of two parallel forces acting in the same direction is equal to the sum of the two forces. Similarly, the resultant of two oppositely directed forces applied at the same point is equal to the difference between them, and its direction is that of the greater force. 78. Equilibrant. In the last experiment the pull in the spring balance which registered 15 g. was not the resultant of the 5 g. and 10 g. forces ; it was rather a force equal and opposite to that resultant. Such a force is called an equilibrant. The equilibrant of a force or forces is that single force ivhich will just prevent the motion which the given forces tend to produce. It is equal and opposite to the resultant and has the same point of application. 2~" 79. The resultant of forces acting at an FI G . 59. Direction angle (concurrent forces). If a body at A of resultant of two is pulled toward the east with a force of equal f ^ g at ' 10 Ib. (represented in Fig. 59 by the line AC) and toward the north with a force of 10 Ib. (repre- sented in the figure by the line AIT), the effect upon the motion of the body must, of course, be the same as though some single force acted somewhere between AC and AB. If the body moves under the action of the two equal forces, it may be seen from symmetry that it must move along a line midway between AC and AB, that is, along the line AR. This line, therefore, indicates the direction as well as the point of appli- cation of the resultant of the forces AC and AB. If the two forces are not equal, as in Fig. 60, then the resultant will lie nearer the larger force. The following experiment will show the relation between the two forces and their resultant, 6Q The resultant Iies nearer the larer force COMPOSITION AND RESOLUTION OF FORCES 61 FIG. 61. Experimental proof of parallelogram law Let the rings of two spring balances be hung over nails B and C in the rail at the top of the blackboard (Fig. 61), and let a weight W be tied near the middle of the string joining the hooks of the two balances. The weight W is not supported by the pull of the balance E or by that of F; it is supported by their resultant, which evidently must act vertically up- ward, since the only single force capable of supporting the weight W is one that is equal and opposite to W. Let the lines OA and OD be drawn upon the black- board behind the string, and upon these lines lay off the distances Oa and Ob, which contain as many units of length as there are units of force indicated by the balances E and F respectively. Simi- larly, on a vertical line from lay off the exact distance OR required to represent the force that supports the weight. This, as noted above, represents the resultant. Now let a parallelogram be constructed upon Oa and Ob as sides. The line OR already drawn will be the diagonal. Hence, to find graphically the resultant of two concurrent forces, (.?) represent the concurrent forces, (2) construct upon them as sides a parallelogram, and (3) draw a diagonal from the point of application. This diagonal represents the point of application, direction, and magnitude of the resultant. 80. Component of a force. When- ever a force acts upon a body in some direction other than that in which the body is free to move, it is clear that the full effect of the force cannot be spent in producing motion. For ex- ample, suppose that a force is applied in the direction OR (Fig. 62) to a car on an elevated track. Evidently OR produces two distinct effects upon the car : on the one hand, it moves the car along the track; and, on the other, it presses it down against the rails. These two effects R FIG. 62. Component of a force 62 FORCE AND MOTION might be produced just as well by two separate forces acting in the directions OA and OB respectively. The value of the single force which, acting in the direction OA, will produce the same motion of the car on the track as is produced by OR, is called the component of OR in the direction OA. Simi- larly, the value of the single force which, acting in the direc- tion OB, will produce the same pressure against the rails as is produced by the force OR, is called the component of OR in the direction OB. In a word, the component of a force in a given direction is the effective value of the force in that direction. 81. Magnitude of the component of a force in a given direc- tion. Since, from the definition of component just given, the two forces, one to be applied in the direction OA and the other in the direction OB, are together to be exactly equivalent to OR in their effect on the car, their magnitudes must be represented by the sides of a par- allelogram of which OR is the diagonal. For in 79 it was shown that if any one force is to have the same effect upon a body as two forces acting simultaneously, it must be repre- sented by the diagonal of a parallelogram the sides of which represent the two forces. Hence, conversely, if two forces are to be equivalent in their joint effect to a single force, they must be sides of the parallelogram of which the single force is the diagonal. Hence the following rule : To find the com- ponent of a force in any given direction, represent the force by a line; then, using the line as a diagonal, construct upon it a rectangle the sides of which are respectively parallel and perpen- dicular to the direction of the required component. The length of the side which is parallel to the given direction represents the FIG. 63. Horizontal component of pull on a sled COMPOSITION AND RESOLUTION OF FORCES 63 magnitude of the component which is sought. Thus, in Fig. 62 the line Om completely represents the component of OR in the direction OA, and the line On represents the component of OR in the direction OB. Again, when a boy pulls on a sled with a force of 10 Ib. in the direction OR (Fig. 63), the force with which the sled is urged forward is represented by the length of Om, which is seen to be but 9.3 Ib. instead of 10 Ib. The component which tends to lift the sled is represented by On. To apply the test of experiment to the conclusions of the preceding paragraph, let a wagon be placed upon an inclined plane (Fig. 64), the height of which, be, is equal to one half its length ab. In this case the force acting on the wagon is the weight of the wagon, and its direction is downward. Let this force be represented by the line OR. Then, by the construction of the preceding paragraph, the line Om will represent the value of the force which is pulling the carriage down the plane, and the line On the value of the force which is producing pressure against the plane. Now, since the triangle ROm is similar to the triangle abc (for ZmOR = Z abc, Z RmO = Z acb, and Z ORm = we have Om _ bc^ OR ~ ab' FIG. 64. Component of that is, in this case, since be is equal to one weight parallel to an in- half of ab, Om is one half of OR. Therefore clined plane the force which is necessary to prevent the wagon from sliding down the plane should be equal to one half its weight. To test this conclusion let the wagon be weighed on the spring balance and then placed on the plane in the manner shown in the figure. The pull indicated by the balance will, indeed, be found to be one half the weight of the wagon. The equation Om/OR = bc/ab gives us the following rule for finding the force necessary to prevent a body from moving down an inclined plane, namely, the force which must be applied to a body to hold it in place upon an inclined plane bears the same ratio to the weight of the body as the height of the plane bears to its length. 64 FORCE A]S T D MOTION 82. Component of gravity effective in producing the motion of the pendulum. When a pendulum is drawn aside from its position of rest (Fig. 65), the force acting on the bob is its weight, and the direction of this force is vertical. Let it be represented by the line OR. The component of this force in the direction in which the bob is free to move is On, and the component at right angles to this direction is Om. The second component Om simply produces stretch in the string and pressure upon the point of suspension. The first compo- nent On is alone responsible for the motion of the bob. A consid- eration of the figure shows that this component becomes larger and larger the greater the dis- placement of the bob. When the bob is directly beneath the point of support, the component producing motion is zero. Hence a pendulum can be per- manently at rest only when its bob is directly beneath the point of suspension.* FIG. 65. Force acting on dis- placed pendulum QUESTIONS AND PROBLEMS 1. The engines of a steamer can drive it 12 mi. per hour. How fast can it go up a stream in which the current is 3 mi. per hour ? How fast can it come down the same stream ? 2. The wind drives a steamer east with a force which would carry it 12 mi. per hour, and its propeller is driving it south with a force which would carry it 15 mi. per hour. What distance will it actually travel in an hour ? Draw a diagram to represent the exact path. * It is recommended that the study of the laws of the pendulum be intro- duced into the laboratory work at about this point (see Experiment 12, authors' Manual). COMPOSITION AND RESOLUTION OF FORCES 65 3. A barge is anchored in a river during a storm. If the wind acts eastward on it with a force of 3000 lb. and the tide northward with a force of 4000 lb., what is the direction and magnitude of the equilibrant ; that is, the pull of the anchor cable upon the barge? 4. A picture weighing 20 lb. hangs upon a cord whose parts make an. angle of 120 with each other. Find the tension (pull) upon each part of the cord. 5. If the barrel of Fig. 66 weighs 200 lb., with what force must a man push par- allel to the skid to keep the barrel in place if the skid is 9 ft. long and the platform 3 ft. high? 6. A cake of ice weighing 200 lb. is held at rest upon an inclined plane 12 ft. long and F IG> 66. Force necessary to prevent a bar- 3 ft. high. By the resolution- re l from rolling down an inclined plane and-proportion method find the component of its weight that tends to make the ice slide down the incline. With what force must one push to keep the ice at rest ? How great is the component that tends to break the incline ? 7. A tight-rope 20 ft. long is depressed 1 ft. at the center when a man weighing 120 lb. stands upon it. Determine graphically the tension in the rope. 8. The anchor rope of a kite balloon makes an angle of 60 with the surface of the earth. If the lifting power of the balloon is 1000 lb., find the pull of the balloon on the rope and the horizontal force of the wind against the balloon. 9. A canal boat and the engine towing it move in parallel paths which are 50 ft. apart. The tow rope is 130 ft. long, and the force (effort) applied to the end of the rope is 1300 lb. Find what component of the loOOlb. acts parallel to the path of the boat. 10. In Fig. 6 7 the line on represents F IG . 67. Forces acting on a kite the pull of gravity on a kite, and the line om represents the pull of the boy on the string. What is the name given to the force represented by the line oRI FORCE AND MOTION FIG. 68. Forces acting on an aeroplane in flight 11. If the force of the wind against the kite is represented by the line AB, and it is considered to be applied at o, what must be the relation between the force oR and the component of AB parallel to n s ,, , ., . . .,., Direction of Flight oR when the kite is in equilib- rium under the action of the existing forces? 12. If the wind increases, why does the kite rise higher ? 13. Show from Fig. 68 what force supports an aeroplane in flight. (Remember that oR, the component of the wind pressure AB perpendicular to the plane, is the only acting force out of which a support for the aeroplane can be derived.) (See frontispiece and opposite pp. 153, 316, and 317.) GBAVITATION 83. Newton's law of universal gravitation. In order to ac- count for the fact that the earth pulls bodies toward itself, and at the same time to account for the fact that the moon and planets are held in their respective orbits about the earth and the sun, Sir Isaac Newton (16421727) (see opposite p. 84) first announced the law which is now known as the law of universal gravitation. This law asserts first that every body in the universe attracts every other body with a force which varies inversely as the square of the distance between the two bodies. This means that if the distance between the two bodies con- sidered is doubled, the force will become only one fourth as great; if the distance is made three, four, or five times as great, the force will be reduced to one ninth, one sixteenth, or one twenty-fifth of its original value ; etc. The law further asserts that if the distance between two bodies remains the same, the force with which one body attracts the other is proportional to the product of the masses of the two bodies. Thus we know that the earth attracts 3 cubic centi- meters of water with three times as much force as it attracts GRAVITATION 67 1, that is, with a force of 3 grams. We know also, from the facts of astronomy, that if the mass of the earth were doubled, its diameter remaining the same, it would attract 3 cubic cen- timeters of water with twice as much force as it does at pres- ent, that is, with a force of 6 grams (multiplying the mass of one of the attracting bodies by 3 and that of the other by 2 multiplies the forces of attraction by 3 x 2, or 6). In brief, then, Newton's law of universal gravitation is as follows : Any two bodies in the universe attract each other with a force which is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Two masses of 1 gram each at a distance apart of 1 cm. attract each other with a force of about 1 6 , 00,000.000 g ram - The masses of the sun and the earth are so great that even though 93,000,000 miles apart, they attract each other with a force of about 4,000,000,000,000,000,000 tons. A body weighing 100 pounds on the earth would weigh about 2700 pounds on the sun. A freely falling body on the earth drops 16 feet the first second, while on the sun it would fall 27 times that far in the first second, or 432 feet. On the moon we should weigh 1 of what we do on the earth; we could jump 6 times as high and should fall i as fast. 84. Variation of the force of gravity with distance above the earth's surface. If a body is spherical in shape and of uniform density, it attracts external bodies with the same force as though its mass were concentrated at its center. Since, there- fore, the distance from the surface to the center of the earth is about 4000 miles, we learn from Newton's law that the earth's pull upon a body 4000 miles above its surface is but one fourth as much as it would be at the surface. It will be seen, then,. that if a body be raised but a few feet or even a few miles above the earth's surface, the decrease in its weight must be a very small quantity, for the reason that a few feet or a few miles is a small distance compared with 68 FORCE AND MOTION 4000 miles. As a matter of fact, at the top of a mountain 4 miles high 1000 grams of mass is attracted by the earth with 998 grams instead of 1000 grams of force. 85. Center of gravity. From the law of universal gravita- tion it follows that every particle of a body upon the earth's surface is pulled toward the earth. It is evident that the sum of all these little pulls on the particles of which the body is composed must be equal to the total pull of the earth upon the body. Now it is always possible to find one single point in a body at which a single force, equal in magnitude to the weight of the body and directed upward, can be applied so that the body will remain at rest in whatever position it is placed. This point is called the center of gravity of the body. Since this force counteracts entirely the earth's pull upon the body, it must be equal and opposite to the resultant of all the small forces which gravity is exerting upon the different particles of the body. Hence the center of gravity may be de- fined as the point of application of the resultant of all the little downward forces of gravity acting upon the parts of the body ; that is, the center of gravity of a body is the point at which the entire weight of the body may be considered an concentrated. The earth's attraction for a body is therefore always considered not as a multitude of little forces but as one single force F (Fig. 69) equal to the pull FIG. 69. Center of of gravity upon the body and applied at its center of gravity G. It is evident, then, that under the influence of the earths pull, every body tends to assume the position in which its center of gravity is as low as possible. 86. Method of finding center of gravity experimentally. From the above definition it will be seen that the most direct way of finding the center of gravity of any flat body, like that shown in Fig. 70, is to find the point upon which it will balance. GRAVITATION 69 Let an irregular sheet of zinc be thus balanced on the point of a pencil or the head of a pin. Let a small hole be punched through the zinc at the point of balance, and let a needle be thrust through this hole. When the needle is held hor- izontally, the zinc will be found to remain at rest, no matter in what position it is turned. To illustrate another method of finding the center of gravity of the zinc, let it be supported from a pin stuck through a hole near its edge, that is, b (Fig. 70). Let a plumb line be hung from the pin, and let a line' In be drawn through 1} on the surface of the zinc parallel to and directly behind the plumb line. Let the zinc be hung from another point a, and let another line am be drawn in a similar way. FIG. 70. Locating center of gravity Since the attraction of the earth for a body may be con- sidered as a single force applied at the center of gravity, a suspended body (for example, the sheet of zinc) can remain at rest only when the center of gravity is directly beneath the point of support (see 85). It must therefore lie somewhere on the line am. For the same reason it must lie on the line bn. But the only point which lies on both of these lines is their point of inter- section G. The point of inter- section, then, of any two vertical lines dropped through two different points of suspension locates the center of gravity of a body. 87. Stable equilibrium. A body is said to be in stable equi- librium if it tends to return to its original position when very slightly tipped, or rotated, out of that position. A pendulum, A B c D FIG. 71. Illustration of varying degrees of stability 70 FOKCE AND MOTION a chair, a cube resting on its side, a cone resting on its base, a boat floating quietly in still water, are all illustrations. In general, a body is in stable equilibrium whenever tip- ping it slightly tends to raise its center of gravity. Thus, in Fig. 71 all of the bodies A, B, (7, Z>, are in stable equilibrium, for in order to overturn any one of them its center of gravity FIG. 72. Quebec bridge G must be raised through the height ai. If the weights are all alike, that one will be most stable for which ai is greatest. In building cantilever bridges such as the large one over the St. Lawrence River at Quebec (Fig. 72) the engineers build out the cantilever arms equally in opposite directions, so as to keep their centers of gravity constantly over the piers until the parts either meet at the center or are close enough to receive the central span, which is hoisted to place. The condition of stable equilibrium for bod- ies which rest upon a horizontal plane is that a vertical line through the center of gravity shall fall within the base, the base being defined as the polygon formed by connecting the points at which the body touches the plane, as ABC (Fig. 73) ; for it is clear that in such a case a slight displacement must raise the center of gravity along the arc of which OG is the radius. If the vertical line drawn through the center of gravity fall outside the base, as in Fig. 74, the body must always fall. B FIG. 73. Body in stable equilibrium GRAVITATION 71 The condition of stable equilibrium for bodies supported from a single point, as in the case of a pendulum, is that the point of support be above the center of gravity. For example, the beam of a balance cannot be in stable equilibrium, so that it will return to the horizontal position when slightly displaced, un- less its center of gravity g (Fig. 3, p. 7) is below the knife-edge C. (The pans are not to be con- sidered, since they are not rigidly connected to the beam.) 88. Neutral and unstable equilibrium. , . -i ^ i 7 -TT - FIG 74. Body not in A body is said to be in neutral equilibrium equilibrium when, after a slight displacement, it tends neither to return to its original position nor to move farther from it. Examples of neutral equilibrium are a spherical ball lying on a smooth plane, a cone lying on its side, a wheel free to rotate about a fixed axis through its center, or any body supported at its center of gravity. In general, a body is in neutral equilibrium when a slight displacement neither raises nor lowers its center of gravity. A body is in unstable equilibrium when, after a slight tip- ping, it tends to move farther from its original position. A cone balanced on its point or an egg on its end are examples. In all such cases a slight tipping lowers the center of gravity, and the motion then continues until the center of gravity is as low as circumstances will permit. The condition for unstable equilibrium in the case of a body supported by a point is that the center of gravity shall be above the point of support. QUESTIONS AND PROBLEMS 1. Explain why the toy shown in Fig. 75 will not lie upon its side, but rises to the vertical position. Does the center of gravity rise? 2. Where is the center of gravity of a hoop? of a cubical box? Is the latter more stable when empty or when full ? Why ? 3. Where must the center of gravity of the beam of a balance be with reference to the supporting knife-edge C? (Fig. 3, p. 7.) Why? Could you make a weighing if C and g coincided? Why? 72 FORCE AND MOTION 4. What is the object of ballast in a ship? 5. What is the most stable position of a brick? the least stable? Why? 6. In what state of equilibrium is a pendulum at rest? Why? 7. What purpose is served by the tail of a kite? 8. Do you get more sugar to the pound in ; \ Calcutta than in Aberdeen when using a beam *>j.( balance? when using a spring balance? Explain. 9. What change would there be in your weight if your mass were to become four times as great and that of the earth three times, the radius of the earth remaining the same ? 10. The pull of the earth on a body at its sur- face is 100 kg. Find the pull on the same body 4000 mi. above the surface ; 1000 mi. above the surface ; 3 mi. above 'the surface. (Take the earth's radius as 4000 mi.) FALLING BODIES 89. Galileo's early experiments. Many of the familiar and important experiences of our lives have to do with falling bodies. Yet when we ask ourselves the simplest question which involves quan- titative knowledge about gravity, such as, for example, Would a stone and a piece of lead dropped from the same point reach the ground at the same time or at different times ? most of us are uncertain as to the answer. In fact, it was the asking and the answering of this very question by Galileo, about 1590, which may be considered as the starting point of modern science. Ordinary observation teaches that light bodies like feathers fall slowly and heavy bodies like stones fall rapidly, and up to Galileo's time it was taught in the schools that bodies fall with " velocities proportional to their weights." Not content with book knowledge, however, FIG. 76. Leaning tower of Pisa, from which were performed some of Gali- leo's famous experiments on falling bodies GALILEO (1564-1642) Great Italian physicist, astronomer, and mathematician; "founder of experi- mental science"; was son of an impoverished nobleman of Pisa; studied medi- cine in early youth, but forsook it for mathematics and science ; was professor of mathematics at Pisa and at Padua ; discovered the laws of falling bodies and the laws of the pendulum ; was the creator of the science of dynamics ; constructed the first thermometer; first used the telescope for astronomical observations; discovered Jupiter's satellites and the spots on the sun. Modern physics begins with Galileo FALLING BODIES 73 Galileo tried it himself. In the presence of the professors and students of the University of Pisa he dropped balls of different sizes and materials from the top of the tower of Pisa (Fig. 76), 180 feet high, and found that they fell in practically the same time. He showed that even very light bodies like paper fell with velocities which approached more and more nearly those of heavy bodies the more compactly they were wadded together. From these experiments he inferred that all bodies, even the lightest, would fall at the same rate if it were not for the resistance of the air. That the air resistance is indeed the chief factor in the slowness of fall of feathers and other light objects can be shown by pumping the air out of a tube containing a feather (or some small pieces of tissue paper) and a coin (Fig. 77). The more com- plete the exhaustion the more nearly do the feather and the coin fall side by side when the tube is inverted. The air pump, however, was not invented until sixty years after Galileo's time. 90. Exact proof of Galileo's conclusion. We can demon- strate the correctness of Galileo's conclusion in still another way, one which he himself used. Let balls of iron and wood, for example, be started together down the inclined plane of Fig. 78. They will be found to keep together all the C I FIG 77. Feather and coin fall to- gether in a vacuum FIG. 78. Spaces traversed and velocities acquired by falling bodies in one, two, three, etc. seconds way down. (If they roll in a groove, they should have the same diame- ter ; otherwise, size is immaterial.) The experiment differs from that 74 FORCE AND MOTION of the freely falling bodies only in that the resistance of the air is here more nearly negligible because the balls are moving more slowly. In order to make them move still more slowly and at the same time to eliminate completely all possible effects due to the friction of the plane, let us follow Galileo and suspend the different balls as the bobs of pen- dulums of exactly the same length, two meters long at least, and start them swinging through equal arcs. Since now the bobs, as they pass through any given position, are merely moving very slowly down identi- cal inclined planes (Fig. 65), it is clear that this is only a refinement of the last experiment. We shall find that the times of fall, that is, the periods, of the pendulums are exactly the same. From the above experiment we conclude with Galileo and with Newton, who performed it with the utmost care a hundred years later, that in a vacuum the velocity acquired per second by a freely falling body is exactly the same for all bodies. 91. Relation between distance and time of fall. Having found that, barring air resistance, all bodies fall in exactly the same way, we shall next try to find what relation exists between distance and time of fall ; and since a freely falling body falls so rapidly as to make direct measurements upon it difficult, we shall adopt Galileo's plan of studying the laws of falling bodies through observing the motions of a ball rolling down an inclined plane. Let a grooved board 17 or 18 ft. long be supported as in Fig. 78, one end being about a foot above the other. Let the side of the board be divided into feet, and let the block B be set just 16 ft. from the start- ing point of the ball A. Let a metronome or a clock beating seconds be started, and let the marble be released at the instant of one click of the metronome. If the marble does not hit the block so that the click pro- duced by the impact of the ball coincides exactly with the fifth click of the metronome, alter the inclination until this is the case. (This adjust- ment may well be made by the teacher before class.) Now start the marble again at some click of the metronome, and note that it crosses the 1-ft. mark exactly at the end of the first second, the 4-ft. mark at the end of the second second, the 9-ft. mark at the end of the third second, and hits B at the 16-ft. mark at the end of the fourth second. This can be tested more accurately by placing B successively at the FALLING BODIES 75 9-ft., the 4-ft., and the 1-ft. mark and noting that the click produced by the impact coincides exactly with the proper click of the metronome. We conclude, then, with Galileo, that, the, distance traversed by a falling body in any number of seconds is the distance traversed the first second times the square of the number of seconds ; that is, if D represents the distance traversed the first second, S the total space, and t the number of seconds, S = Dt*. 92. Relation between velocity and time of fall. In the last paragraph we investigated the distances traversed in one, two, three, etc. seconds. Let us now investigate the velocities acquired on the same inclined plane in one, two, three, etc. seconds. Let a second grooved board Jlf be placed at the bottom of the incline, in the manner shown in Fig. 78. To eliminate friction it should be given a slight slant, just sufficient to cause the ball to roll along it with uniform velocity. Let the ball be started at a distance D up the incline, D being the distance which in the last experiment it was found to roll during the first second. It will then just reach the bottom of the incline at the instant of the second click. Here it will be freed from the influ- ence of gravity, and will therefore move along the lower board with the velocity which it had at the end of the first second. It will be found that when the block is placed at a distance exactly equal to 2 D from the bottom of the incline, the ball will hit it at the exact instant of the third click of the metronome, that is, exactly two seconds after starting ; hence the velocity acquired in one second is 2 D. If the ball is started at a distance 4 D up the incline, it will take it two seconds to reach the bottom, and it will roll a distance 4 D in the next second ; that is, in two seconds it acquires a velocity 4 D. In three seconds it will be found to acquire a velocity 6 Z>, etc. The experiment shows, first, that the gain in velocity each second is the same; second, that the amount of this gain is numerically equal to twice the distance traversed the first second. Motion, like the above, in which velocity is gained at a constant rate is called uniformly accelerated motion. In uniformly accelerated motion the gain each second in the velocity is called the acceleration. It is numerically equal to twice the distance traversed the first second. 76 FORCE AND MOTION 93. Formal statement of the laws of falling bodies. Put- ting together the results of the last two paragraphs, we obtain the folloAving table, in which D represents the distance trav- ersed the first second in any uniformly accelerated motion. NUMBER OF SECONDS (t) VELOCITY AT THE END OF EACH SECOND (v) GAIN IN VELOCITY EACH SECOND (a) TOTAL DISTANCE TRAVERSED (S) 1 2 D 21) ID 2 4D 2D 4D 3 QD 2D 91) 4 SD 2D 16 D t 2tD 2D PD Since D was shown, in 92, to be equal to one half of the acceleration a, we have at once, by substituting J a for D in the last line of the table, v = at, (1) S = %aP. (2) These formulas are simply the algebraic statement of the facts brought out by our experiments, but the reasons for these facts may be seen as follows : Since in uniformly accelerated motion the acceleration a is the velocity in centimeters per second gained each second, it follows at once that when a body starts from rest, the velocity which it has at the end of t seconds is given by v at. This is formula (1). To obtain formula (2) we have only to reflect that distance traversed is always equal to the average velocity multiplied by the time. When the initial velocity is zero, as in this case, and the final velocity is at, average velocity = (0 + at) -s- 2 = 1 at. Hence 5 = $ at\ This is formula (2). These are the fundamental formulas of uniformly accelerated motion, but it is sometimes convenient to obtain the final velocity v directly from the total distance of fall S, or vice versa. This may of course be done by simply substituting in (2) the value of t obtained from (1), namely, - This gives v = V2 a5. (3) FALLING BODIES 77 Distances in ft. per sec. in feet Ot To illustrate the use of these formulas, sup- pose we wish to know with what velocity a body will hit the earth if it falls from a height of 200 meters, or 20,000 centimeters. From (6) we get v - V2 x 980 x 20,000 = 6261 cm. per second. 95. Height of ascent. If we wish to find the height S to which a body projected vertically upward will rise, we reflect that the time of ascent must be the initial velocity divided by the upward velocity which the body loses per second, that is, t = - ; and the height reached 9 32.16 94. Acceleration of a freely falling body. If in the above experiment the slope of the plane be made steeper, the results will obviously be precisely the same, ex- velocities cept that the acceleration has a larger value. If the board is tilted until it be- comes vertical, the body becomes a freely falling body (Fig. 79). In this case the distance traversed the first second is found to be 490 centimeters, or 16.08 feet. Hence the acceleration, expressed in centimeters, is 980 ; in feet, 32.16. This acceleration of free fall, called the acceleration of gravity, is usually denoted by the letter g. For freely falling bodies, then, the three formulas of the preceding paragraph become v = ff t, (4) 64.32 128.64 (16.08) 16.08 > (48.24) 64.32 (80.40) 144.72 (112.56) 257.28 FIG. 79. A freely fall- ing body 78 FOKCE AND MOTION must be this multiplied by the average velocity ; that is, (7) FIG. 80. Path of a projectile Since (7) is the same as (6), we learn that in a vacuum the speed with which a body must be projected upward to rise to a given height is the same as the speed which it acquires in falling from the same height. 96. Path of a projectile. Imagine a projectile to be shot along the line ab (Fig. 80). If it were not for gravity and the resistance of the air, the projectile would travel with uniform velocity along the line ab, arriving at the points 1, 2, 3, etc. at the end of the successive seconds. Because of gravity, how- ever, the projectile would be vertically below these points by the distances 16.08 ft., 64.32 ft., 144.72 ft., etc. Hence it would follow the path indi- cated by the dotted curve (a parabola). But because of air resistance the height of flight and range are diminished, and the general shape of the trajectory is similar to the continuous curved line. 97. The airplane. The principles underlying stability, as well as those having to do with the resolution of forces, are well illustrated by the modern airplane, which grew out of a study of the laivs of air resistance and the properties of gliders. When a plate of area A moves in still air in a direction perpendicular to its plane, with a velocity V (see Fig. 81, (1)), the air resistance R is found by experiment to be given by the equation R = KAY*, (8) where R is the force in kilograms, A the area in square meters, V the speed in meters per second, and K a constant which has the value .08. Thus, when an automobile is going 40 miles FALLING BODIES 79 per hour (18 meters per second), the force of the air against .5 square meter of wind-shield is .08 x .5 x (18) 2 = 13 kg. When the plate moves so that the direction of its motion makes a small angle i (between and 10) (Fig. 81, (2)) with its plane, the air resistance R is perpendicular to the plate and is given by the empirical formula R = kAV% (9) where J?, A, and V have the same significance as above, i is the angle in degrees, and k is very near to .005. As i, which is called the angle of attack or of incidence, decreases, the center of pressure C (Fig. 81, (2)) moves R (1) FIG. 81. Forces acting on a glider toward the front edge and tends toward a certain definite limiting position C Q as the angle i becomes smaller and smaller. When a flat object like a sheet of paper is allowed to fall, it is acted upon by two forces, one W, acting at its center of gravity g, which is always vertical and equal to the weight, and the other R, which is due to the air resistance acting at the center of pressure C and perpendicular to the plane. If the plane is to fall without acceleration and with- out rotation, that is, if it is to glide, it is clear that these two forces must act at the same point and be equal and opposite. Hence any gliding plane must be horizontal and must move with a speed V at an angle i (see Fig. 81 (3)), given by the equation (10) 80 FORCE AND MOTION Since the plane must be horizontal, and since there is only one angle of attack which will bring the center of pressure and the center of gravity together, it will be seen that the gliding angle i is the same for all values of the weight W, but that the speed V will be proportional to the square root of the weight (see equation 10). FIG. 82. A stabilized glider The foregoing theory of gliding may be nicely illustrated with paper gliders thus : Fold a sheet of writing paper lengthwise, exactly along the middle. Refold the upper half twice on itself so as to make it \ its original width; then fasten it down to the lower half with paste or light gummed paper. The center of gravity will now be -^ of the new width behind the back edge of the folded portion. When started slowly with the folded edge forward, the paper will glide as described. Heavier paper will glide at the same angle but with greater speed. If started thin edge foremost, the forces at once turn the glider over, and it glides with the heavier edge in front. To in- crease the lateral stability it is sufficient to give the paper the shape shown in Fig. 82. (See opposite p. 317.) When the motor of an airplane stops, the plane glides safely to FlG> 8 3. Forces acting on an earth under the laws of equation airplane in flight 10. If the airplane propeller is pulling forward with a horizontal force Q, and the wings are set back at an angle i, R and W no longer balance each other, but their resultant is equal and opposite to Q; that is, the forces R, W, and Q form, a system in equilibrium, as shown in Fig. 83. The plane moves forward horizontally with a speed V. If the angle i or the force Q is increased, the plane rises ; if i or Q is diminished, the plane descends. FALLING BODIES 81 98. The laws of the pendulum. The first law of the pendu- lum was found in 90, namely, (1) The periods of pendulums of equal lengths swinging through short arcs are independent of the weight and material of the bobs. Let the two pendulums of 90 be set swinging through arcs of lengths 5 centimeters and 25 centimeters respectively. We shall thus find the second law of the pendulum, namely, (2) The period of a pendulum swinging through a short arc is independent of the amplitude of the arc. Let pendulums ^ and ^ as long as the above be swung with it. The long pendulum will be found to make only one vibration while the others are making two and three respectively. The third law of the pendulum is therefore (3) The periods of pendulums are directly proportional to the square roots of their lengths. The accurate determination of g is never made by direct measure- ment, for the laws of the pendulum just established make this instru- ment by far the most accurate one obtainable for this determination. It is only necessary to measure the length of a long pendulum and the time t between two successive passages of the bob across the mid-point, and then to substitute in the formula t = *\l- in order to obtain g with a iff high degree of precision. The deduction of this formula is not suitable for an elementary text, but the formula itself may well be used for checking the value of g, given in 94. QUESTIONS AND PROBLEMS 1. If a body starts from rest and travels with a constant acceleration of 10 ft. per second each second, how fast will it be going at the close of the fifth second? What is its average velocity during the 5 sec., and how far did it go in this time? 2. A body starting from rest and moving with uniformly accelerated motion acquired a velocity of 60 ft. per second in 5 sec. Find the acceler- ation. What distance did it traverse during the first second ? the fifth ? 3. A body moving with uniformly accelerated motion traversed 6 ft. during the first second. Find the velocity at the end of the fourth second. 82 FORCE AND MOTION 4. A ball thrown across the ice started with a velocity of 80 ft. per second. It was retarded by friction at the rate of 2 ft. per second each second. How long did it roll? How far did it roll ? 5. A bullet was fired with a velocity of 2400 ft. per second from a rifle having a barrel 2 ft. long. Find (a) the average velocity of the bullet while moving the length of the barrel; (b) the time required to move through the barrel ; (c) the acceleration of the bullet while in the barrel. 6. A ball was thrown vertically into the air with a velocity of 160 ft. per second. How long did it re- main in the air? (Take g=32 ft. per sec 2 .) 7. A baseball was thrown upward. It remained in the air 6 sec. With what velocity did it leave the hand ? How high did it go ? 8. A ball dropped from the top of the Woolworth Building in New York City, 780 ft. above Broadway, would require how many seconds to fall ? With what velocity would it strike ? (Take g = 32 ft. per sec 2 .) 9. How high was an airplane from which a bomb fell to earth in 10 sec. ? 10. W T ith what speed does a bullet strike the earth if it is dropped from the Eiffel Tower, 335 m. high ? 11. If the acceleration of a marble rolling down an inclined plane is 20 cm. per second, what velocity will it have at the bottom, the plane being 7 m. long ? 12. If a man can jump 3 ft. high on the earth, how high could he jump on the moon, where g isj as much? 13. The brakes were set on a train running 60 mi. per hour, and the train stopped in 20 sec. Find the acceleration in feet per second each second and the distance the train ran after the brakes were applied. 14. How far will a body fall from rest during the first half second ? 15. With what velocity must a ball be shot upward to rise to the height of the Washington Monument (555 ft.)? How long before it will return ? 16. Fig. 84 represents the pendulum and escapement of a clock. The escapement wheel D is urged in the direction of the arrow by the clock weights or spring. The slight pushes communicated by the teeth of the wheel keep the pendulum from dying down. Show how the length of the pendulum controls the rate of the clock. 17. What force supports an airplane in flight? What is "gliding"? FIG. 84 NEWTON'S LAWS OF MOTION 83 18. A pendulum that makes a single swing per second in New York City is 99.3 cm., or 39.1 in., long. Account for the fact that a seconds pendulum at the equator is 39 in. long, while at the poles it is 39.2 in. long. 19. How long is a pendulum whose period is 3 sec.? 2 sec.? \ sec.? sec.? 20. A man was let down over a cliff on a rope to a depth of 500 ft. What was his period as a pendulum ? NEWTON'S LAWS OF MOTION 99. First law inertia. It is a matter of everyday observa- tion that bodies in a moving train tend to move toward the forward end when the train stops and toward the rear end when the train starts ; that is, bodies in motion seem to want to keep on moving, and bodies at rest to remain at rest. Again, a block will go farther when driven with a given blow along a surface of glare ice than when knocked along an asphalt pavement. The reason which everyone will assign for this is that there is more friction between the block and the asphalt than between the block and the ice. But when would the body stop if there were no friction at all? Astronomical observations furnish the most convincing answer to this question^ for we cannot detect any retardation at all in the motions of the planets as they swing around the sun through empty space. Furthermore, since mud flies off tangentially from a rotating carriage wheel, or water from a whirling grindstone, and since, too, we have to lean inward to prevent ourselves from falling outward in going around a curve, it appears that bodies in motion tend to maintain not only the amount but also the direction of their motion (see gyrocompass opposite p. 223). In view of observations of this sort Sir Isaac Newton, in 1686, formulated the following statement and called it the first law of motion. 84 FOKCE AND MOTION Every body continues in its state of rest or uniform motion in a straight line unless impelled by external force to change that state. This property, which all matter possesses, of resisting any at- tempt to start it if at rest, to stop it if in motion, or in any way to change either the direction or amount of its motion, is called inertia. 100. Centrifugal force. It is inertia alone Xyhich prevents the planets from falling into the sun, which causes a rotating sling to pull hard on the hand until the stone is released, and which then causes the stone to fly off tangentially. It is inertia which makes rotating liquids move out as far as possi- ble from the axis of rotation (Fig. 85), which makes flywheels sometimes burst, which makes the equatorial diameter of the earth greater than the polar, which makes the heavier milk , ,. , ! ,, ,, ,. , . FIG. 85. Illustrating centrifugal move out farther than the lighter f orce cream in the dairy separator (see opposite p. 85), etc. Inertia manifesting itself in this tendency of the, parts of rotating systems to move away from the center of rotation is called centrifugal force. 101. Momentum. The quantity of motion possessed by a moving body is defined as the product of the mass and the velocity of the body. It is commonly called momentum. Thus, a 10-gram bullet moving 50,000 centimeters per second has 500,000 units of momentum ; a 1000-kg. pile driver moving 1000 centimeters per second has 1,000,000,000 units of mo- mentum ; etc. We shall always express momentum hi C.G.S. units, that is, as a product of grams by centimeters per second. 102. Second law. Since a 2-gram mass is pulled toward the earth with twice as much force as is a 1-gram mass, and since both, when allowed to fall, acquire the same velo'city in SIR ISAAC NEWTON (1642-1727) English mathematician and physicist, "prince of philosophers" ; professor of mathematics at Cambridge University; formulated the law of gravitation ; discovered the binomial theorem ; invented the method of the calculus ; announced the three laws of motion which have become the basis of the science of mechanics ; made important discoveries in light; is the author of the celebrated " Principia " (Principles of Natural Philosophy) , published in 1687 Skim-milk Outlet Cream Outlet Skim-milk Outlet THE CREAM SEPARATOR (2) The milk is poured into a central tube (see 1, a) at the top of a nest of disks (see 1 and 4} situated within a steel bowl. The milk passes to the bottom of the cen- tral tube, then rises through three series of holes (see 1, b, b, b, etc.) in the nest of disks, and spreads outward into thin sheets between the slightly separated disks. By means of a system of gears (see 3) the disks and bowl are made to revolve from 6000 to 8000 revolutions per minute. The separation of cream from skim-milk is quickly effected in these thin sheets ; the heavier skim-milk (water, casein, and sugar) is thrown outward by centrifugal force against the under sur- faces of the bowl disks (see 5), then passes downward and outward along these under surfaces to the periphery of the bowl (see 1, d, d, d, etc.), and finally rises to the skim-milk outlet. The lighter cream is thereby at the same time displaced inward and upward along the upper surfaces of the bowl disks (see 5), then passes over the inner edges of the disks to slots (see 1, c, c, c, etc.) on the outside of the central tube, finally rising to the cream outlet, which is above the outlet for the skim-milk (see 1 and 2) NEWTON'S LA\VS OF MOTION 85 a second, it follows that in this case the momentums produced in the two bodies by the two forces are exactly proportional to the forces themselves. In all cases in which forces simply over- come inertias this rule is found to hold. Thus, a 3000-pound pull on an automobile on a level road, where friction may be neglected, imparts in a second just twice as much velocity as does a 1500-pound pull. In view of this relation Newton's second law of motion was stated thus: Rate of change of momentum is proportional to the force acting, and the change takes place in the direction in which the force acts. 103. The third law. When a man jumps from a boat to the shore, we all know that the boat experiences a backward thrust ; when a bullet is shot from a gun, the gun recoils, or " kicks " ; when a billiard ball strikes another, it loses speed, that is, is pushed back while the second ball is pushed forward. The following experiment will show how effects of this sort may be studied quantitatively. Let a sceel ball A (Fig. 80) be allowed to fall from a position C against another exactly similar ball B. In the impact A will lose practically all of its velocity, and B will ^ ^ move to a position D, which is at the same FIG. 86. Illustration of height as C. Hence the velocity acquired third law by B is almost exactly equal to that which A had before impact. These velocities would be exactly equal if the balls were perfectly elastic. It is found to be true experimentally that the momentum acquired by B plus that retained by A is exactly equal to the momentum which A had before the impact. The momentum acquired by B is therefore exactly equal to that lost by A. Since, by the second law, change in momentum is proportional to the force acting, this experiment shows that A pushed forward on B with precisely the same force with which B pushed back on A. Now the essence of Newton's third law is the assertion that in the case of the man jumping from the boat the mass 86 FORCE AND MOTION of the man times his velocity is equal to the mass of the boat times its velocity, and that in the case of the bullet and gun the mass of the bullet times its velocity is equal to the mass of the gun times its velocity. The truth of this assertion has been established by a great variety of experiments. Newton stated his third law thus: To every action there is an equal and opposite reaction. Since force is measured by the rate at which momentum changes, this is only another way of saying that whenever a body acquires momentum some other body acquires an equal and opposite momentum. It is not always easy to see at first that setting one body into motion involves imparting an equal and opposite momen- tum to another body. For example, when a gun is held against the earth and a bullet shot upward, we are conscious only of the motion of the bullet; the other body is in this case the earth, and its momentum is the same as that of the bullet. On account of the greatness of the earth's mass, however, its velocity is infinitesimal. 104. The dyne. Since the gram of force varies somewhat with locality, it has been found convenient for scientific purposes to take the second law as the basis for the definition of a new unit of force. It is called an absolute, or C.G.S., unit because it is based upon the fundamental units of length, mass, and time, and is therefore independent of gravity. It is named the dyne and is defined as the force which, acting for one second upon any mass, imparts to it one unit of momentum ; or the force which, act- ing for one second upon a one-gram mass, produces a change in its velocity of one centimeter per second. 105. A gram of force equivalent to 980 dynes. A gram of force was defined as the pull of the earth upon 1 gram of mass. Since this pull is capable of imparting to this mass in 1 second a velocity of 980 centi- meters per second, that is, 980 units of momentum, and since a dyne is the force required to impart in 1 second 1 unit of momentum, it is clear that the gram of force is equivalent to 980 dynes of force. The dyne is therefore a very small unit, about equal to the force with which the earth attracts a cubic millimeter of water. NEWTON'S LAWS OF MOTION 87 106. Algebraic statement of the second law. If a force F acts for t seconds on a mass of m grams, and in so doing increases its velocity v- centimeters per second, then, since the total momentum imparted in a time t is mv, the momentum imparted per second is ; and since force in dynes is equal to momentum imparted per second, we have But since - is the velocity gained per second, it is equal to the acceler- ation a. Equation (8) may therefore be written F = ma. (9) This is merely stating in the form of an equation that force is measured by rate of change of momentum. Thus, if ap engine, after pull- ing for 30 sec. on a train having a mass of 2,000,000 kg., has given it a velocity of 60 cm. per second, the force of the pull is 2,000,000,000 x | = 4,000,000,000 dynes. To reduce this force to grams we' divide by 980, and to reduce it to kilos we divide further by 1000. Hence the pull exerted by the engine on the train = 4 ' Sgo t ggo 000 = 4081 k S'-> or 4 - 08 l metric tons. QUESTIONS AND PROBLEMS 1. What principle is applied when one tightens the head of a hammer by pounding on the handle? 2. Why does not the car C of Fig. 87 fall? What carries it from BtoDl 3. Why does a flywheel cause machinery to run more steadily? 4. Balance a calling card on the finger and place a coin upon it. Snap out the card, leaving the coin balanced on the finger. What principle is illustrated? 5. Is it any easier to walk toward the rear than toward the front of a rapidly moving B train? Why? Fl( . 87 A very anc i en t loop the loops 6. Suspend a weight by a string. Attach a piece of the same string to the bottom of the weight. If the lower string is pulled with a sudden jerk, it breaks ; but if the pull is steady, the upper string will break. Explain. 7. Where does a body weigh the more, at the poles or at the equator ? Give two reasons. 88 FORCE AND MOTION 8. If the trains A, B, and C (Fig. 88) are all running 60 mi. per hour, what is the velocity of A with reference to 5? to C? 9. If a weight is dropped from the roof to the floor of a moving- car, will it strike the point on the floor which was di- rectly beneath its starting point? 10. Why is a running track banked at the turns? C< 11. If the earth were to cease rotating, would bodies p IG- g8 on the equator weigh more or less than now ? Why ? 12. How is the third law involved in rotary lawn sprinklers? 13. The modern way of drying clothes is to place them in a large cylinder with holes iu the sides, and then to set it in rapid rotation. Explain. 14. Explain how reaction pushes the ocean liner and the airplane forward. 15. If one ball is thrown horizontally from the top of a tower and another dropped at the same instant, which will strike the earth first ? (Remember that the acceleration produced by a force is in the direction in which the force acts and proportional to it, whether the body is at rest or in motion. See second law.) If possible, try the experiment with an arrangement like that of Fig. 89- 16. If a rifle bullet were fired horizontally from a tower 19.6 m. high with a speed of 300 m., how far from the base of the tower would it strike the earth if there were no air resistance ? \ FIG. 89. Illustrating New- ton's second law FIG. 90. Hydraulic ram 17. The hydraulic ram (Fig. 90) is a practical illustration of the principle of inertia. With its aid water from a pond P can be raised NEWTON'S LAWS OF MOTION FIG. 91 into a tank that stands at a higher level than the pond. With the aid of Fig. 91 explain how it works, remembering that the valve V will not close until the stream of water flowing around it acquires suffi- cient speed. 18. If two men were together in the middle of a perfectly smooth (frictionless) pond of ice, how could they get off? Could one man get off if he were there alone ? 19. If a 10-g. bullet is shot from a 5-kg. gun with a speed of 400 m. per second, what is the backward speed of the gun ? 20. If a team of horses pulls 500 Ib. in drawing a wagon, with what force does the wagon pull backward upon the team ? ^hy do the wheels turn before the hoofs of the horses slide ? 21. Why does a falling mass, on striking, exert a force in excess of its weight? 22. A pull of a dyne acts for 3 sec. on a mass of 1 g. What velocity does it impart ? 23. How long must a force of 100 dynes act on a mass of 20 g. to impart to it a velocity of 40 cm. per second? 24. A force of 1 dyne acts on 1 g. for 1 sec. How far has the gram been moved at the end of the second ? A laboratory exercise on the composition of forces should be performed during the study of this chapter, See, for example, Experiment 11 of the authors' Manual. CHAPTER VI MOLECULAR FORCES* MOLECULAR FORCES IN SOLIDS. ELASTICITY 107. That the molecules of solids cling together with forces of great magnitude is proved by some of the simplest facts of nature ; for solids not only do not expand indefinitely like gases, but it often requires enormous forces to pull their molecules apart. Thus, a rod of cast steel 1 centimeter in diameter may be loaded with a weight of 7.8 tons before it will be pulled in two. The following are the weights in kilograms necessary to break drawn wires of different materials, 1 square millimeter in cross section, the so-called relative tenacities of the wires. Lead, 2.6 Platinum, 43 Iron, 77 Silver, 37 Copper, 51 Steel, 91 108. Elasticity. We can obtain additional infor- mation about the molecular forces existing in different substances by studying what happens when the weights ap- plied are not large enough to break the wires. Thus, let a long steel wire (for ex- P ample, No. 26 piano wire) be suspended FIG. 92. Elasticity of a steel wire from a hook in the ceiling, and let the * This chapter should be preceded by a laboratory experiment in which Hooke's law is discovered by the pupil for certain kinds of deformation easily measured in the laboratory. See, for example, Experiment 13 of the authors' Manual. 90 MOLECULAR FORCES IN SOLIDS 91 lower end be wrapped tightly about one end of a meter stick, as in Fig. 92. Let a fulcrum c be placed in a notch in the stick at a distance of about 5 cm. from the point of attachment to the wire, and let the other end of the stick be provided with a knitting needle, one end of which is opposite the vertical mirror scale S. Let enough weights be applied to the pan P to place the wire under slight tension ; then let the reading of the pointer p on the scale S be taken. Let three or four kilogram weights be added successively to the pan and the correspond- ing positions of the pointer read. Then let the readings be taken again as the weights are successively removed. In this last operation the pointer will probably be found to come back exactly to its first position. This characteristic which the steel has shown in this experi- ment, of returning to its original length when the stretching weights are removed, is an illustration of a property possessed to a greater or less extent by all solid bodies. It is called elasticity. 109. Limits of perfect elasticity. If a sufficiently large weight is applied to the end of the wire of Fig. 92, it will be found that the pointer does not return exactly to its original position when the weight is removed. We say, therefore, that steel is perfectly elastic only so long as the distorting forces are kept within certain limits, and that as soon as these limits are overstepped it no longer shows perfect elasticity. Different substances differ very greatly in the amount of distortion which they can sustain before they show this failure to return completely to the original shape. 110. Hooke's law. If we examine the stretches produced by the successive addition of kilogram weights in the experiment of 108, Fig. 92, we shall find that these stretches are all equal, at least within the limits of observational error. Very carefully conducted experiments have shown that this law, namely, that the successive application of equal forces pro- duces a succession of equal stretches, holds very exactly for all sorts of elastic displacements so long, and only so long, as the limits of perfect elasticity are not overstepped. This 92 MOLECULAR FORCES law is known as Hooke's law, after the Englishman Robert Hooke (1635-1703). Another way of stating this law is the folloAving : Within the limits of perfect elasticity elastic deforma- tions of any sort, be they twists or bends or stretches, are directly proportional to the forces producing them. The common spring balance (Fig. 57) is an application of Hooke's law. 111. Cohesion and adhesion. The preceding experiments have brought out the fact that, in the solid condition at least, molecules of the same kind exert attractive forces upon one another. That molecules of unlike substances also exert mutually attractive forces is equally true, as is proved by the fact that glue sticks to wood with tremendous tenacity, mortar to bricks, nickel plating to iron, etc. The forces which bind like kinds of molecules together are commonly called cohesive forces ; those which bind together molecules of. unlike kind are called adhesive forces. Thus, we say that mucilage sticks to wood because of adhesion, while wood itself holds together because of cohesion. Again, adhe- sion holds the chalk to the blackboard, while cohesion holds together the particles of the crayon. 112. Properties of solids depending on cohesion. Many of the physical properties in which solid substances differ from one another depend on differences in the cohesive forces existing between their molecules. Thus, we are accustomed to classify solids with relation to their hardness, brittleness, ductility, malleability, tenacity, elasticity, etc. The last two of these terms have been sufficiently explained in the preceding para- graphs; but since confusion sometimes arises from failure to understand the first four, the tests for these properties are here given. We test the relative hardness of two bodies by seeing which will scratch the other. Thus, the diamond is the hardest of all substances, since it scratches all others and is scratched by none of them. MOLECULAR FORCES IN LIQUIDS 93 We test the relative brittleness of two substances by seeing which will break the more easily under a blow from a hammer. Thus, glass and ice are very brittle substances ; lead and copper are not. We test the relative ductility of two bodies by seeing which can be drawn into the thinner wire. Platinum is the most ductile of all substances. It has been drawn into wires only .00003 inch in diameter. Glass is also very ductile when sufficiently hot, as may be readily shown by heating it to softness in a Bunsen flame, when it may be drawn into threads which are so fine as to be almost invisible. We test the relative malleability of two substances by seeing which can be hammered into the thinner sheet. Gold, the most malleable of all substances, has been hammered into sheets 3 0*0 o o mcn ni thickness. QUESTIONS AND PROBLEMS 1. Tell how you may, by use of Hooke's law and a 20-lb. weight, make the scale for a 32-lb. spring balance. 2. A broken piece of wrought iron or steel may be welded by heating the broken ends white hot and pounding them together. Gold foil is welded cold in the process of filling a tooth. Explain welding. 3. A piece of broken wood may be mended with glue. What does the glue do? 4. Why are springs made of steel rather than of copper? 5. If a given weight is required to break a given wire, how much force is required to break two such wires hanging side by side ? to break one wire of twice the diameter ? MOLECULAR FORCES IN LIQUIDS. CAPILLARY PHENOMENA 113. Proof of the existence of molecular forces in liquids. The facility with which liquids change their shape might lead us to suspect that the molecules of such substances exert almost no force upon one another, but a simple experiment will show that this is far from true. 94 MOLECULAR FORCES FIG. 93. Illustrating cohesion of water By means of sealing wax and string let a glass plate be suspended horizontally from one arm of a balance, as in Fig. 93. After equilibrium is obtained, let a surface of water be placed just beneath the plate and the beam pushed down until contact is made. It will be found necessary to add a considerable weight to the opposite pan in order to pull the plate away from the water. Since a layer of water will be found to cling to the glass, it is evident that the added force applied to the pan has been expended in pulling water molecules away from water molecules, not in pulling glass away from water. Similar experiments may be performed with all liquids. In the case of % mercury the glass will not be found to be wet, showing that the co- hesion of mercury is greater than the adhesion of glass and mercury. 114. Shape assumed by a free liquid. Since, then, every molecule of a liquid is pulling on every other molecule, any body of liquid which is free to take its natural shape, that is, which is acted on only by its own cohesive forces, must draw itself together until it has the smallest possible surface com- patible with its volume ; for, since every molecule in the surface is drawn toward the interior by the attraction of the molecules within, it is clear that molecules must continually move toward the center of the mass until the whole has reached the most compact form possible. Now the geometrical figure which has the smallest area for a given volume is a sphere. We conclude, therefore, that if we could relieve a body of liquid from the action of gravity and other outside forces, it would at once take the form of a perfect sphere. This conclusion may be easily verified by the following experiment: Let alcohol be added to water until a solution is obtained in which a drop of common lubricating oil will float at any depth. Then with a pipette insert a large globule of oil beneath the surface. The oil will be seen to float as a perfect sphere within the body of the liquid (Fig. 94). (Unless the drop is viewed from above, the vessel should have flat rather MOLECULAR FORCES IN LIQUIDS 95 FIG. 94. Spherical globule of oil, freed from action of gravity than cylindrical sides, otherwise the curved surface of the water will act like a lens and make the drop appear flattened.) The reason that liquids are not more commonly observed to take the spherical form is that ordinarily the force of gravity is so large as to be more influential in deter- _ = mining their shape than are the cohesive forces. As verification of this statement we have only to observe that as a body of liquid becomes smaller and smaller that is, as the gravitational forces upon it be- come less and less it does indeed tend more and more to take the spherical form. Thus, very small globules of mercury on a table will be found to be almost perfect spheres, and raindrops or minute floating particles of all liquids are quite accurately spherical. 115. Contractility of liquid films; surf ace tension. The tend- ency of liquids to assume the smallest possible surface fur- nishes a simple explanation of the contractility of liquid films. Let a soap bubble 2 or 3 inches in diameter be blown on the bowl of a pipe and then allowed to stand. It will at once begin to shrink in size and in a few minutes will disappear within the bowl of the pipe. FIG. 95 'FiG. 96 FIG. 97 Illustrating the contractility of soap films The liquid of the bubble is simply obeying the tendency to reduce its surface to a minimum, a tendency which is due only to the mutual at- tractions which its molecules exert upon one another. A candle flame 96 MOLECULAE FORCES held opposite the opening in the stem of the pipe will be deflected by the current of air which the contracting bubble is forcing out through the stem. Again, let a loop of fine thread be tied to a wire ring, as in Fig. 95. Let the ring be dipped into a soap solution so as to form a film across it, and then let a hot wire be thrust through the film inside the loop. The tendency of the film outside the loop to contract will instantly snap out the thread into a perfect circle (Fig. 96). The reason that the thread takes the circular form is that, since the film outside the loop is striving to assume the smallest possible surface, the area inside the loop must of course become as large as possible. The circle is the figure which has the largest possible area for a given perimeter. Let a soap film be formed across the mouth of a clean 2-inch funnel, as in Fig. 97. The tendency of the film to contract will be sufficient to lift its weight against the force of gravity. The tendency of a liquid to reduce its exposed surface to a minimum, that is, the tendency of any liquid surface to act like Fig. 98. Some of the stages through which a slowly forming drop passes a stretched elastic membrane, is called surface tension. The elas- tic nature of a film is illustrated in Fig. 98, which is from a motion-picture record of some of the stages through which a slowly forming drop passes. 116. Ascension and depression of liquids in capillary tubes. It was shown in Chapter II that, in general, a liquid stands at the same level in any number of communicating vessels. The following experiments will show that this rule ceases to hold in the case of tubes of small diameter. MOLECULAR FORCES IN LIQUIDS 9T Let a series of capillary tubes of diameter varying from 2 mm. to .1 inm. be arranged as in Fig. 99. When water or ink is poured into the vessel it will be found to rise higher in the tubes than in the vessel, and it will be seen that the smaller the tube the greater the height to which it rises. If the water is replaced by mercury, however, the effects will be found to be just inverted. The mercury is depressed in all the tubes, the depression being greater in proportion as the tube is smaller (Fig. 100, (1)). This depression is most easily ob- served with a U-tube like that shown in Fig. 100, (2). Experiments of this sort have established the following laws: 1. Liquids rise in capillary tubes when they FlG - " Rise of 7 7 />,,, 7 T -i liquids in capillary are capable oj wetting them, but are depressed tubes in tubes which they do not wet. 2. The elevation in the one case and the depression in the oilier are inversely proportional to the diameters of the tubes. It will be noticed, too, that when a liquid rises, its surface within the tube is concave upward, and when it is depressed its surface is convex upward. 117. Cause of curvature of a liquid surface in a capillary tube. All of the effects pre- sented in the last paragraph can be explained by a consider- ation of cohesive and adhesive forces. However, throughout the explanation we must keep in mind two familiar facts : first, that the surface of a body of water at rest, for example a pond, is at right angles to the resultant force, that is, gravity, which acts upon it; and, second, that the force of gravity acting on a minute amount of liquid is negligible in comparison with its own cohesive force (see 114). FIG. 100. Depression of mercury in capillary tubes 98 MOLECULAR FORCES Consider, then, a very small body of liquid close to the point o (Fig. 101), where water is in contact with the glass wall of the tube. Let the quantity of liquid considered be so minute that the force of gravity act- ing upon it may be disregarded. The force of adhesion si the wall will pull the liquid particles FIG. 101 FIG. 102 Condition for elevation of a liquid near a wall at o in the direction oE. The force of cohesion of the liquid will pull these same particles in the direction oF. The resul- tant of these two pulls on the liquid at o will then be repre- sented by oR (Fig. 101), in accordance with the parallelogram law of Chapter V. If, then, the resultant oR of the adhesive force oE and the cohesive force oF lies to the left of the vertical om (Fig. 102), since the surface of a liquid always assumes a position at right angles to the resultant force, the liquid must rise up against the wall as water does against glass (Fig. 102). If the cohesive force 0^(Fig. 103) is strong in comparison with the adhesive force oE, the resultant oR will fall to the right of the vertical, in which case the liquid must be depressed about o. Whether, then, a liquid will rise against a solid wall or be depressed by it will depend only on the relative strengths of the adhesion of the wall for the liquid and the cohesion of the liquid for itself. Since mercury does not wet glass, we know that cohesion is here relatively strong, and we should expect, therefore, that the mercury F FIG. 103. Condition for the depression of a liquid near a wall MOLECULAR FORCES IN LIQUIDS 99 would be depressed, as indeed we find it to be. The fact that water will wet glass indicates that in this case adhesion is relatively strong, and hence we should expect water to rise against the walls of the containing vessel, as in fact it does. It is clear that a liquid which is depressed near the edge of a vertical solid wall must assume within a tube a surface which is convex upward, while a liquid which rises against a wall must within such a tube be concave upward. 118. Explanation of ascension and depression in capillary tubes. As soon as the curvatures just mentioned are pro- duced, the concave surface aob (Fig. 104) tends, by virtue of FIG. 104 FIG. 105 A concave meniscus causes a rise in a capillary tube FIG. 106 FIG. 107 A convex meniscus causes a fall surface tension, to straighten out into the flat surface ao'b. But it no sooner thus begins to straighten out than adhesion again elevates it at the edges. It will be seen, therefore, that the liquid must continue to rise in the tube until the weight of the volume of liquid lifted, namely amnb (Fig. 105), balances the tendency of the surface aob to flatten out. That the liquid will rise higher in a small tube than in a large one is to be expected, since the weight of the column of liquid to be supported in the small tube is less. The convex mercury surface aob (Fig. 106) falls until the upward pressure at 0, due to the depth h of mercury (Fig. 107), balances the tendency of the surface aob to flatten. 100 MOLECULAK FOBCES 119. Capillary phenomena in everyday life. Capillary phe- nomena play a very important part in the processes of nature and of everyday life. Thus the rise of oil in wicks of lamps, the complete wetting of a towel when one end of it is allowed to stand in a basin of water, the rapid absorption of liquid by a lump of sugar when one corner of it only is immersed, the taking up of ink by blotting paper, are all illustrations of pre- cisely the same phenomena which we observe in the capillary tubes of Fig. 99. 120. Floating of Small Objects On water. Let a needle be laid very carefully on the surface of a dish of water. In spite of the fact that it is nearly eight times as dense as water it will be found to float. If the needle has been previously magnetized, it may be made to move about in any direction over the surface in obedience to the pull of a magnet held, for example, underneath the table. FIG. 108. Cross section of a To discover the cause ot this apparently floating needle impossible phenomenon, examine closely the surface of the water in the immediate neighborhood of the needle. It will be found to be depressed in the manner shown in Fig. 108. This furnishes at once the explanation. So long as the needle is so small that its own weight is no greater than the upward force exerted upon it by the tendency of the depressed (and therefore concave) liquid surface to straighten out into a flat surface, the needle cannot sink in the liquid, no matter how great its density. If the water had wet the needle, that is, if it had risen about the needle instead of being depressed, the tendency of the liquid surface to flatten out would have pulled it down into the liquid instead of forcing it upward. Any body about which a liquid is depressed will therefore float on the surface of the liquid if its mass is not too great. Even if the liquid FIG. 109. Insect walking on the surface of water MOLECULAK FORCES IN LIQUIDS 101 tends to rise about a body when it is perfectly clean, an im- perceptible film of oil upon the body will cause it to depress the liquid, and hence to float. The above experiment explains the familiar phenomenon of insects walking and running on the surface of water (Fig. 109) in apparent contradiction to the law of Archimedes, in ac- cordance with which they should sink until they displace their own weight of the liquid. QUESTIONS AND PROBLEMS 1. Explain how capillary attraction comes usefully into play in the steel pen, camel's-hair brushes, lamp wicks, and sponges. 2. Candle grease may be removed from clothing by covering it with blotting paper and then passing a hot flatiron over the paper. Explain. 3. Why will a piece of sharp-cornered glass become rounded when heated to redness in a Bunsen flame ? 4. The leads for pencils are made by subjecting powdered graphite to enormous pressures produced by hydraulic machines. Explain how the pressure changes the powder to a coherent mass. 5. Float two matches an inch apart. Touch the water between them with a hot wire. The matches will spring apart. What does this show about the effect of temperature on surface tension ? 6. Repeat the experiment, touching the water with a wire moistened with alcohol. What do you infer as to the relative surface tensions of alcohol and water? 7. Fasten a bit of gum camphor to one end of half a toothpick and lay it upon the surface of a large vessel of clean still water. Explain the motion. 8. Shot are made by pouring molten lead through a sieve on top of a tall tower and catching it in water at the bottom. Why are the shot spherical? 9. Explain how capillary attraction makes an irrigation system successful. 10. In building a macadam road coarse stones are placed at the bottom, on top of them smaller stones, and finally little granules tightly rolled together by means of a steam roller. Explain how this arrange- ment of material keeps the road dry. 11. What force is mainly responsible for the return of the water that has gravitated into the soil? Would the looseness of the soil make any difference (dry farming) ? 102 MOLECULAR FORCES ABSORPTION OF GASES BY SOLIDS AND LIQUIDS 121 . Absorption Of gases by solids. Let a large test tube be filled with ammonia gas by heating aqua ammonia and causing the evolved gas to displace mercury in the tube, as in Fig. 110. Let a piece of charcoal an inch long and nearly as wide as the tube be heated to redness and then plunged be- neath the mercury. When it is cool, let it be slipped underneath the mouth of the test tube and allowed to rise into the gas. The mercury will be seen to rise in the tube, as in Fig. 111. Why? FIG. 110. Filling tube with ammonia This property of absorbing gases is possessed to a notable degree by porous substances, especially coconut and peach-pit charcoal. It is not improbable that all solids hold, closely adhering to their surfaces, thin layers of the gases with which they are in contact, and that the prominence of the phenomena of absorption in porous substances is due to the great extent of sur- face possessed by such substances. That the same substance exerts widely different attractions upon the molecules of different gases is shown by the fact that char- coal will absorb 90 times its own volume of ammonia gas, 35 times its volume of carbon dioxide, and only 1.7 times its volume of hydrogen. The usefulness of charcoal as a deodorizer is due to its enormous ability to absorb certain kinds of gases. This property made it available for use in gas masks (see opposite p. 103) during the World War. If a little spongy platinum is suspended in a vessel above wood alcohol, it will glow brightly because of the absorption into the platinum of both vapor of alcohol and oxygen. The rapid FIG. 111. Absorp- tion of ammonia gas by charcoal JAMES CLERK-MAXWELL (1831-1879) One of the greatest of mathemati- cal physicists ; born in Edinburgh, Scotland ; professor of natural philosophy at Marischal College, Aberdeen, in 1856, of physics and astronomy in Kings College, Lon- don, in I860, and of experimental physics in Cambi'idge University from 1871 to 1879 ; one of the most prominent figures in the develop- ment of the kinetic theory of gases and the mechanical theory of heat ; author of the electro- magnetic theory of light a the- ory which has become the basis of nearly all modern theoretical work in electricity and optics (see p. 426) HEINRICH RUDOLPH HERTZ (1857-1894) One of the most brilliant of Ger- man physicists, who, in spite of his early death at the age of thirty- seven, made notable contributions to theoretical physics, and left be- hind the epoch-making experimen- tal discovery of the electromagnetic waves predicted by Maxwell. Wire- less telegraphy is but an applica- tion of this discovery of so-called " Hertzian " waves (see p. 422). The capital discovery that ultra-violet light discharges negatively electri- fied bodies is also due to Hertz A GAS MASK U. S. Official A great variety of poisonous gases having a density greater than air were set free and carried by the wind against the Allied armies in the World War. and others were fired in explosive shells. Until gas masks were devised these gases, settling into the trenches, wrought frightful havoc among the troops. The absorptive power of charcoal, especially when impregnated with certain chemicals, proved an effec- tive barrier against the deadly fumes, since all of the air entering the lungs of the soldiers had to be inhaled through the charcoal within a canister carried in the bag designed to hold the gas mask. The illustration shows an American gas mask adjusted to the head of an American soldier MOLECULAR FORCES IX LIQUIDS 103 rise in temperature is due to the increased rate of oxidation of the alcohol brought about by this more intimate mixture. This property of platinum is utilized in the platinum-alcohol cigar lighter (Fig. 112). 122. Absorption of gases in liquids. Let a beaker containing cold water be slowly heated. Small bubbles of air will be seen to Sponau collect in great numbers upon the walls and Platinum^ to rise through the liquid to the surface. Platinun That they are indeed bubbles of air and not Wick of steam is proved, first, by the fact that they WoodAlcohqi appear when the temperature is far below inCotton boiling, and, second, by the fact that they do not condense as they rise into the higher and FIG. 112. The platinum- cooler layers of the water. alcohol dgar Hghter The experiment shows two things : first, that water ordinarily contains considerable quantities of air dissolved in it ; and, second, that the amount of air which water can hold decreases as the temperature rises. The first point is also proved by the existence of fish life ; for fishes obtain the oxygen which they need to support life from air which is dissolved in the water. The amount of gas which will be absorbed by water varies greatly with the nature of the gas. At C. and a pressure of 76 centimeters 1 cubic centimeter of water will absorb 1050 cubic centimeters of ammonia, 1.8 cubic centimeters of carbon dioxide, and only .04 cubic centimeter of oxygen. Commercial aqua ammonia is simply ammonia gas dissolved in water. The following experiment illustrates the absorption of ammonia by water : Let the flask F (Fig. 113) and tube b be filled with ammonia by passing a current of the gas in at a and out through b. Then let a be corked up and b thrust into G, a flask nearly filled with water which has been colored slightly red by the addition of litmus and a drop or two of acid. As the ammonia is absorbed the water will slowly rise in b, and as soon 104 MOLECULAR FORCES FIG. 113. Absorp- tion of ammonia by water as it reaches F it will rush up very rapidly until the upper flask is nearly full. At the same time the color will change from red to blue because of the action of the ammonia upon the litmus. Experiment shows that in every case of ab- sorption of a gas by a liquid or a solid the quantity of gas absorbed decreases with an in- crease in temperature, a result which was to have been expected from the kinetic theory, since increasing the molecular veloc- ity must of course increase the difficulty which the adhesive forces have in retaining the gaseous molecules. 123. Effect of pressure upon absorption. Soda water is ordinary water which has been made to absorb large quantities of carbon dioxide gas. This impregnation is accom- plished by bringing the water into contact with the gas under high pressure. As soon as the pressure is relieved, the gas passes rapidly out of solution. This is the cause of the characteristic effervescence of soda water. These facts show clearly that the amount of carbon dioxide which can be absorbed by water is greater for high pressures than for low. As a matter of fact, careful experiments have shown that the amount of any gas absorbed is directly proportional to the pres- sure, so that if carbon dioxide under a pressure of 10 atmos- pheres is brought into contact with water, ten times as much of the gas is absorbed as if it had been under a pressure of 1 atmosphere. This is known as Henry's law. QUESTIONS AND PROBLEMS 1. Why do fishes in an aquarium die if the w r ater is not frequently renewed ? 2. Explain the apparent generation of ammonia gas when aqua ammonia is heated. 3. Why, in the experiment illustrated in Fig. 113, was the flow so much more rapid after the water began to run over into -F? CHAPTER VII WORK AND MECHANICAL ENERGY* DEFINITION AND MEASUREMENT OF 124. Definition of work. Whenever a force moves a body on which it acts, it is said to do work upon that body, and the amount of the work accomplished is measured by the product of the force acting and the distance through which it moves the body. Thus, if i gram of mass is lifted 1 centi- meter in a vertical direction, 1 gram of force has acted, and the distance through which it has moved the body is 1 centi- meter. We say, therefore, that the lifting force has accom- plished 1 gram centimeter of work. If the gram of force had lifted the body upon which it acted through 2 centimeters, the work done would have been 2 gram centimeters. If a force of 3 grams had acted and the body had been lifted through 3 centimeters, the work done would have been 9 gram centimeters, etc. Or, in general, if W represent the work accomplished, F the value of the acting force, and s the dis- tance through which its point of application moves, then the definition of work is given by the equation W=Fxs. (1) In the scientific sense no work is ever done unless the force succeeds in producing motion in the body on which it * It is recommended that this chapter be preceded by an experiment in which the student discovers for himself the law of the lever, that is, the principle of moments (see, for example, Experiment 16, authors' Manual), and that it be accompanied by a study of the principle of work as exempli- fied in at least one of the other simple machines (see, for example, Experi- ment 17, authors' Manual). 105 106 WORK AND MECHANICAL ENERGY acts. A pillar supporting a building does 110 work ; a man tugging at a stone, but failing to move it, does no work. In the popular sense we sometimes say that we are doing work when we are simply holding a weight or doing anything else which results in fatigue; but in physics the word "work" is used to describe not the effort put forth but the effect ac- A tA/ complished, as represented in equation (1). 125. Units of work. There are two common units of work in the metric system, the gram centimeter and the kilogram meter. As the names imply, the gram centimeter is the work done by a force of 1 gram when it moves the point on which it acts 1 centimeter. The kilogram meter is the work done by a kilogram of force -./hen it moves the point on which it acts 1 meter. The gram meter also is sometimes used. Corresponding to the English unit o^ force, the pound, is the unit of work, the foot pound. It is the work done by a " pound of force " when it moves the point on which it acts 1 foot. Thus, it takes a foot pound of work to lift a pound of mass 1 foot high. In the absolute system of units the dyne is the unit of force, and the dyne centimeter, or erg, is the corresponding unit of work. The erg is the amount of work done by a force of 1 dyne when it moves the point on which it acts 1 centimeter. To raise 1 liter of water from the floor to a table 1 meter high would require 1000 x 980 X 100 = 98,^00,000 ergs of work. It will be seen, therefore, that the erg is an exceedingly small unit. For this reason it is customary to employ a unit which is equal to 10,000,000 ergs. It is called a joule, in honor of the great English physicist James Prescott Joule (1818-1889). The work done in lifting a liter of water 1 meter is therefore 9.8 joules. QUESTIONS AND PROBLEMS 1. To drag a trunk weighing 120 Ib. required a force of 40 Ib. How much work would be required to drag this trunk 2 yd. ? to lift it 2 yd. vertically ? 2. A carpenter pushed 5 Ib. on his plane while taking off a shaving 4 ft. long. How much work was done? WOEK AND THE PULLEY 10T 3. How many foot pounds of work does a 150-lb. man do in climbing to the top of Mt. Washington, which is 6300 ft. high ? 4. A horse pulls a metric ton of coal to the top of a hill 30 m. high. Express the work accomplished in kilogram meters (a metric ton = 1000 kg.). 5. If the 20,000 inhabitants of a city use an average of 20 liters of water per capita per day, how many kilogram meters of work must the engines do per day if the water has to be raised to a height of 75 m. ? WORK EXPENDED UPON AND ACCOMPLISHED BY SYSTEMS OF PULLEYS 126. The single fixed pulley. Let the force of the earth's attrac- tion upon a mass R be overcome by pulling upon a spring balance S, in the manner shown in Fig. 114, until R moves slowly upward. If R is 100 grams, the spring balance will also be found to t f register a force of 100 grams. /HFT Experiment therefore shows that in the use of the single fixed pulley the acting force, or effort, E, which is producing the motion, is equal to the resisting force, or resistance, JK, which is opposing the motion. Again, since the length of the string is always \ A constant, the distance s through which the point E A, at which E is applied, must move is always FIG. 114. The equal to the distance s' through which the weight sm s le fixed R is lifted. Hence, if we consider the work put into the system at A, namely, E x , and the work accomplished by the system at R, namely, R x s', we find, obviously, since R = E and s = *', that Exs = fixs'; (2) that is, in the case of the single fixed pulley, the work done by the acting force E (the effort) is equal to the work done against the resisting force R (the resistance), or the work put into the machine at A is equal to the work accomplished by the machine at R. 108 WOBK AND MECHANICAL ENEKGY 127. The single movable pulley. Now let the force of the earth's attraction upon the mass R be overcome by a single movable pulley, as shown in Fig. 115. Since the weight of R (R repre- senting in this case the weight of both the pulley and the suspended mass) is now supported half by the strand C and half by the strand B, the force E which must act at A to hold the weight in place, or to move it slowly upward if there is no friction, should be only one half of R. A reading of the balance will show that this is indeed the case. Experiment thus shows that in the case of the single movable pulley the effort E is just one half as great as the resistance R. But when we again consider the work which the force E must clo to lift the weight R a dis- tance *', we see that A must move upward 2 FIG. 115. The single movable pulley inches in order to raise R 1 inch; for when R moves up 1 inch, both of the strands B and C must be shortened 1 inch. As before, therefore, since R = 2 E and that is, in the case of the single movable pulley, as in the case of the fixed pulley, the work put into the machine by the effort E is equal to the work accomplished by the machine against the resistance R. 128. Combinations of pulleys. Let a weight R be lifted by means of such a system of pulleys as is shown in Fig. 116, either (1) or (2). Here, since R is supported by 6 strands of the cord, it is clear that the force which must be applied at A in order to hold R in place, or to make it move slowly upward if there is no friction, should be but i of R. The experiment will show this to be the case if the effects of friction, which are often very considerable, are eliminated 'by taking the mean of the forces which must be applied at E to cause it to move first slowly upward and then slowly down- ward. The law of any combination of movable pulleys may WORK AND THE PULLEY 109 then be stated thus : If n represents the number of strands between which the weight is divided, E = R/n. (3) But when we again consider the work which the force E must do in order to lift the weight R through a distance *', we see that, in order that the weight R may be moved up through 1 inch, each of the strands must be short- ened 1 inch, and hence the point A must move through n inches ; that is, s' = s/n. Hence, ignoring friction, in this case also we have (i) E X s = R x s f ; that is, although the effort E is only - of the resistance R, the work put n FIG. 116. Combinations of pulleys into the machine by the effort E is equal to the work accomplished by the machine against the resistance R. 129. Mechanical advantage. The above experiments show that it is sometimes possible by applying a small force E to overcome a much larger resisting force R. The ratio of the resistance R to the effort E (ignoring friction) is called the mechanical advan- tage of the machine. Thus, the mechanical advantage of the single fixed pulley is 1, that of the single movable pulley is 2, that of the system of pulleys shown in Fig. 116 is 6, etc. If the acting force is applied at R instead of at E the me- chanical advantage of the systems of pulleys of Fig. 116 is i ; for it requires an application of 6 pounds at R to lift 1 pound at E. But it will be observed that the resisting force at jE'now moves six times as fast and six times as far as the acting force at R. We can thus either sacrifice speed to gain force, or 110 WOEK AND MECHANICAL ENERGY sacrifice force to gain speed ; but in every case, whatever we gain in the one we lose in the other. Thus in the hydraulic elevator shown in Fig. 13, p. 18, the cage moves only as fast as the piston ; but in that shown in Fig. 14 it moves four times as fast. Hence the force applied to the piston in the latter case must be four times as great as in the former if the same load is to be lifted. This means that the diameter of the latter cylinder must be twice as great. QUESTIONS AND PROBLEMS 1. Although the mechanical advantage of the fixed pulley is only 1, it is extensively used in connection with clothes lines, awnings, open wells, and flags. Explain. 2. If the hydraulic elevator of Fig. 14, p. 18, is to carry a total load of 20,000 lb., what force must be applied to the piston? If the water pressure is 70 lb. per square inch, what must be the area of the piston? 3. Draw a diagram of a set of pulleys by which a force of 50 lb. can support a load of 200 lb. 4. Draw a diagram of a set of pulleys by which a force of 50 lb. can support 250 lb. What would be the mechanical advantage of this arrangement ? 5. Two men, pulling 50 lb. each, lifted 300 lb. by a system of pulleys. Assuming no friction, how many feet of rope did they pull down in raising the weight 20 f t. ? WORK AND THE LEVER 130. The law of the lever. The lever is a rigid rod free to turn about some point P called the fulcrum (Fig. 117). First let a meter stick be balanced as in the figure, and then let a mass of, say, 300 g. be hung by a thread from a ' ' point 15 cm. from FIG. 117. The simple lever the fulcrum. Then let a point be found on the other side of the fulcrum at which a weight of 100 g. will just support the 300 g. This point will be found to be I L__L J I 1 I J WORK AND THE LEVER 111 45 cm. from the fulcrum. Jt will be seen at once that the product of 300 x 15 is equal to the product of 100 x 45. Next let the point be found at which 150 g. just balance the 300 g. This will be found to be 30 cm. from the fulcrum. Again, the products 300 x 15 and 150 x 30 are equal. No matter where the weights are placed, or what weights are used on either side of the fulcrum, the product of the effort E by its distance I A p B from the fulcrum (Fig. 118) will be found to be equal to the product of the re- sistance R by its distance I' s from the fulcrum. Now the ^IG. 118. Illustrating the law of moments, namely, El=Rl' perpendicular distances I and I' from the fulcrum to the line of action of the forces are called the lever arms of the forces E and R, and the product of a force by its lever arm is called the moment of that force. The above experiments on the lever may then be generalized in the following law : The moment of the effort is equal to the moment of the resistance. Algebraically stated, it is El = Rl'. (4) It will be seen that the mechanical advantage of the lever, namely R/E, is equal to l/l' ; that is, to the lever arm of the effort divided by the lever arm of the resistance. 131. General laws of the lever. If parallel forces are applied at several points on a lever, as in Figs. 119 and 120, it will be found, in the particular cases illustrated, that for equilibrium . 200 x 30 - 100 x 20 + 100 x 40 and 300 x 20 + 50 x 40 = 100 x 15 + 200 x 32.5 ; that is, the sum of all the moments which are tending to turn the beam in one direction is equal to the sum of all the moments tending to turn it in the opposite direction. 112 WORK AND MECHANICAL ENERGY If, further, we support the levers of Figs. 119 and 120 by spring balances attached at P, we shall find, after allowing for the weight of the stick, that the two forces indicated by the balances are respectively 200 + 100 + 100 = 400 Q and 300 + 100 + 200 50 = 550 ; that is, the sum of all the forces acting in one direction on the lever is equal to the sum of all the forces act' ing in the opposite direction. 200 FIG. 119 FIG. 120 Condition of equilibrium of a bar acted upon by several forces These two laws may be combined as follows : If we think of the force exerted by the spring balance as the equilibrant of all the other forces acting on the lever, then we find that the resultant of any number of parallel forces is their algebraic sum, and its point of application is the point about which the algebraic sum of the moments is zero. 132. The couple. There is one case, however, in which paral- lel forces can have no single force as fheir resultant, namely, the case represented in Fig. 121. Such a pair of equal F and opposite forces acting at different points on a lever is called a couple and can be neutralized * g only by another couple tending to produce rotation in the opposite direction. The moment of such a * FlG ' 12L The cou P le couple is evidently F 1 X oa + F 2 x ob F l X ab ; that is, it is one of the forces times the total distance between them. The forces applied to the steering wheel of an automobile illustrate the couple. WORK AND THE LEVEE 113 133. Work expended upon and accomplished by the lever. We have just seen that when the lever is in equilibrium that is, when it is at rest or is moving uniformly the relation between the effort E and the resistance R is expressed in the equation of moments, namely El = Rl f . Let us now suppose, precisely as in the case of the pulleys, that the force E raises the weight R through a small distance s f . To ac- complish this, the point A to E which E is attached must ,, , ,. , FIG. 122. Showing that the equation move through a distance 8 O f moments, 1W = r, is equivalent to (Fig. 122). From the simi- Es = Ks' larity of the triangles APn and BPm it will be seen that l/l' is equal to s/s'. Hence equation (4), which represents the law of the lever, and which may be written E/ R = I' /I, may also be written in the form E/R = *'/, or Es *= Rs 1 . Now Es represents the work done by the effort E, and Rs 1 tlie work done against the resistance R. Hence the law of u. moments, which has just been found by experiment to be the law of the lever, is equivalent to the statement that whenever work is accomplished l>y the use of the lever, the work expended upon the lever by the effort E is equal to the work accomplished by the lever against the resistance R. 134. The three classes of levers. Although the law stated in 133 applies to all forms of the lever, it is customary to divide them into three classes, as follows : 1. In levers of the first class the fulcrum P is between the acting force E and the resisting force R (Fig. 123). The mechanical advantage of levers of this class is greater or less than unity according as the lever arm I of the effort is greater or less than the lever arm V of the resistance. 114 WORK AND MECHANICAL ENERGY 2. In levers of the second class the resistance R is between the effort E and the fulcrum P (Fig. 124). Here the level- arm of the effort, that is, the distance from E to P, is neces- sarily greater than the lever arm of the resistance, that is, the I ' S7 (1) (2) FIG. 123. Levers of first class FIG. 124. Levers of second class FIG. 125. Levers of third class distance from R to P. Hence the mechanical advantage of levers of the second class is always greater than 1. 3. In levers of the third class the acting force is between the resisting force and the fulcrum (Fig. 125). The mechanical advantage is then obviously less than 1, that is. in this type of lever force is always sacrificed for the sake of gaining speed. QUESTIONS AND PROBLEMS 1. In which of the three classes of levers does the wheelbarrow belong? grocer's scales? pliers? sugar tongs? a claw hammer? a pump handle ? 2. Explain the principle of weighing by the steelyards (Fig. 126). What must be the weight of the bob P if at a distance of 40 cm. from the fulcrum it balances a weight of 10 kg. placed at a distance of 2 cm. from ? 3. If you knew your own weight, how could you determine the weight of a companion if you had only a teeter board and a foot rule ? 4. How would you arrange a crowbar to use it as a lever of the first class in overturning a heavy object? as a lever of the second'class ? 5. Why do tinners' shears have long handles and short blades and tailors' shears just the opposite ? WORK AND THE LEVER 115 FIG. 126. Steelyards 6. By reference to moments explain (a) why a door can be closed more easily by pushing- at the knob than at a point close to the hinges ; (b) why a heavier load can be lifted on a wheelbarrow having long- handles than on one with short han- dles; (c) why a long-handled shovel generally has a smaller blade than one with a shorter handle. 7. Two boys carry a load of 60 Ib. on a pole between them. If the load is 4 ft. from one boy and 6 ft. from the other, how many pounds does each boy carry? (Consider the force ex- erted by one of the boys as the effort, the load as the resistance, and the second boy as the fulcrum.) 8. Where must a load of 100 Ib. be placed on a stick 10 ft. long if the man who holds one end is to support 30 Ib. while the man at the other end supports 70 Ib. ? 9. One end of a piano must be raised to remove a broken caster. The force required is 240 Ib. Make a diagram to show how a 6-foot steel bar may be used as a second-class lever to raise the piano with an effort of 40 Ib. 10. When a load is carried on a stick over the shoulder, why does the pressure on the shoulder become greater as the load is moved farther out on the stick? 11. A safety valve and weight are arranged as in Fig. 127. If ab is 1 in. and be 101 in., what effective steam pressure per square inch is required on the valve to unseat it, if the area of the valve is \ sq. in. and the weight of the ball 4 Ib. ? 12. The diameters of the piston and cylinder of a hydraulic press are re- spectively 3 in. and 30 in. The piston rod is attached 2 ft. from the fulcrum of a lever 12 ft. long (Fig. 12, p. 17). What force must be applied at the end of the lever to make the press exert a force of 5000 Ib. ? 13. Three boys sit on a seesaw as follows : A (= 75 Ib.), 4 ft. to the right of the fulcrum; B (= 100 Ib.), 7 ft. to the right of the fulcrum; C (= x Ib.), 7 ft. to the left of the fulcrum. Equilibrium is produced by a man, 12 ft. to the right of the fulcrum, pushing up with a force of 25 Ib. Find C's weight. FIG. 127 116 WOEK AND MECHANICAL ENERGY THE PRINCIPLE OF WORK 135. Statement of the principle of work. The study of pulleys led us to the conclusion that in all cases where such machines are used the work done by the effort is equal to the work done against the resistance, provided always that friction may be neglected and that the motions are uniform so that none of the force exerted is used in overcoming inertia. The study of levers led to precisely the same result. In Chapter II the study of the hydraulic press showed that the same law applied in this case also, for it was shown that the force on the small piston times the distance through which it moved was equal to the force on the large piston times the distance through which it moved. Similar experi- ments upon all sorts of machines have shown that the follow- ing is an absolutely general law : In all mechanical devices of whatever sort, in all cases where friction may ~be neglected, the work expended upon the machine is equal to the work accom- plished by it. This important generalization, called " the principle of work," was first stated by Newton in 1687. It has proved to be one of the most fruitful principles ever put forward in the history of physics. By its application it is easy to deduce the relation between the force applied and the force overcome in any sort of machine, provided only that friction is negligible and that the motions take place slowly. It is only necessary to produce, or imagine, a displacement at one end of the machine, and then to measure or calculate the corresponding displacement at the other end. The ratio of the second displacement to the first is the ratio of the force acting to the force overcome. 136. The wheel and axle. Let us apply the work principle to discover the law of the wheel and axle (Fig. 128). When the large wheel has made one revolution, the point A on the rope THE PRINCIPLE OF WORK 117 moves down a distance equal to the circumference of the wheel. During this time the weight R is lifted a distance equal to the circumference of the axle. Hence the equa- tion Es Rs J becomes E x 2 7rR w = R X 2wr a , where R w and r a are the radii of the wheel and axle respectively. This equation may be writ- ten in the form Rjr a ', (5) s as FIG. 128. The wheel and axle that is, the weight lifted on the axle many times the force applied to the wheel as the radius of the wheel is times the radius of the axle. Otherwise stated, the mechanical advantage of the wheel and axle is equal to the radius of the ivheel divided Iry the radius of the axle. The capstan (Fig. 129) is a spe- cial case of the wheel and axle, the length of the lever arm taking the place of the radius of the wheel, and the radius of the barrel corre- sponding to the radius of the axle. 137. The work principle applied to the inclined plane. The work done against gravity in lifting a weight R (Fig. 130) from the bottom to the top of a plane is evidently equal to R times the height h of the plane. But the work done by the acting force E while the carriage of weight R is being pulled from the bottom tp the top of the plane is equal to E times the length I of the plane. Hence the principle of work gives m = R j^ or R / E = FIG. 129. The capstan FlG m The indined pljme 118 WORK AND MECHANICAL ENERGY FIG. 131. The jackscrew that is, the mechanical advantage of the inclined plane, or the ratio of the weight lifted to the force acting parallel to the plane, is the ratio of the length of the plane to the height of the plane. This is precisely the con- clusion at which we arrived in another way in Chapter V, p. 63. 138. The screw. The screw (Fig. 131) is a combination of the inclined plane and the lever. Its law is easily obtained from the prin- ciple of work. When the force which acts on the end of the lever has moved this point through one complete revolution, the weight R, which rests on top of the screw, has evidently been lifted through a vertical distance equal to the distance between two adjoining threads. This distance d is called the pitch of the screw. Hence, if we represent by / the length of the lever, the work principle gives that is, the mechanical advantage of the screw, or the ratio of the-iveight lifted to FIG. 132. The letter press the force applied, is equal to the ratio of the circumference of the circle moved over by the end of the lever to the distance between the threads of the screw. In actual practice the friction in such an arrangement is always very great, so that the effort exerted must always be considerably greater than that given by equation 7. The common jackscrew just described (and used chiefly for raising build- ings), the letter press (Fig. 132), and the vise FIG. 133. The vise (Fig. 133) are all familiar forms of the screw. 139. A train of gear wheels. A form of machine capable of very high mechanical advantage is the train of gear wheels shown in Fig. 134. THE PRINCIPLE OF WORK 119 Let the student show from the principle of work, namely Es = Rs', that the mechanical advantage, that is, -^, of such a device is circum. of a no. cogs in d no. cogs in / circum. of e no. cogs in c no. cogs in b (8) 140. The worm wheel.* Another device of high mechanical advantage is the worm wheel (Fig. 135). Show that if I is the length of the crank arm C, n the number of teeth in the cogwheel / W, and r the radius of the axle, the mechanical advantage is given by 2-Trln _ I 2 TIT r (9) FIG. 134. Train of gear wheels FIG. 135. The worm gear This device is used most frequently when the primary object is to decrease speed rather than to multiply force. It will be seen that the crank handle must make n turns while the cogwheel is making one. The worm-gear " drive " is generally used in the rear axles of auto trucks. 141. The differential pulley. In the differential pulley (Fig. 136) an endless chain passes first over the fixed pulley A, then down and around the movable pulley C, then up again over the fixed pul- ley B, which is rigidly attached to A, but differs slightly from it in diameter. On the circumfer- ence of all the pulleys are projec- tions which fit between the links, and thus keep the chains from slip- ping. When the chain is pulled down at E, as in Fig. 136, (2), until the upper rigid system of pulleys has made one complete revolution, the chain between the upper and lower pulleys has been shortened by the difference be- tween the circumferences of the FIG. 136. The differential pulley 120 WORK AND MECHANICAL ENERGY pulleys A and B, for the chain has been pulled up a distance equal to the circumference of the larger pulley and let down a distance equal to the circumference of the smaller pulley. Hence the load R has been lifted by half the difference between the circumferences of A and B. The mechan- ical advantage is therefore equal to the circumference of A divided by one half the difference between the circumferences of A and B. QUESTIONS AND PROBLEMS 1. A 1500-pound safe must be raised 5 ft. The force which can be applied is 250 Ib. What is the shortest inclined plane which can be used for the purpose? 2. A 300-pound barrel was rolled up a plank 12 ft. long into a doorway 3 ft. high. What force was applied parallel to the plank ? 3. A force of 80 kg. on a wheel whose diameter is 3 m. balances a weight of 150 kg. on the axle. Find the diameter of the axle. 4. If the capstan of a ship is 12 in. in diameter and the levers are 6 ft. long, what force must be exerted by each of 4 men in order to raise an anchor weighing 2000 Ib. ? 5. If, in the compound lever of Fig. 137, AC = 6 ft., BC=l ft., DF = 4 ft., FG = 8 in., HJ = 5 ft., and.// = 2 ft., what force applied at E will support a weight of 2000 Ib. at R ? B I H R E FIG. 137. The com- pound lever FIG. 138. Hay scales FIG. 139. Windlass with gears 6. The hay scales shown in Fig. 138 consist of a compound lever with fulcrums at F, F', F", F'". If Fo and F'o' are lengths of 6 in., FE and F'E 5 ft., F"n 1 ft., F"m 6 ft., rF'" 2 in., and F'"S 20 in., how many pounds at W will be required to balance a weight of a ton on the platform? POWER AND ENERGY 121 7. In the windlass of Fig. 139 the crank handle has a length of 2 ft., and the barrel a diameter of 8 in. There are 20 cogs in the small cogwheel and 60 in the large one. What is the mechanical advantage of the arrangement? 8. If in the crane of Fig. 140 the crank arm has a length of J m., and the gear wheels A, B, C, and D have- respectively 12, 48, 12, and 60 cogs, while the axle over which the chain runs has a radius of 10 cm., what is the mechanical advantage of the crane ? 9. If a worm wheel (Fig. 135) has 30 teeth, and the crank is 25 cm. long, while the radius of the axle is 3 cm., what is the mechanical advantage of the arrangement? 10. A small jackscrew has 20 threads to the inch. Using a lever 3^ in. long will give what mechanical FIG. 140. The crane advantage? (Use 3.1416.) 11. The screw of a letter press has 5 threads to the inch, and the diameter of the wheel is 12 in. If there were no friction, what pres- sure would result from a rotating force of 20 Ib. applied to the wheel ? 12. Eight jackscrews, each of which has a pitch of in. and a lever arm of 18 in., are being worked simultaneously to raise a building weigh- ing 100,000 Ib. What force would have to be exerted at the end of each lever if there were no friction ? What if 75 % were wasted in friction ? 13. What is gained by using a machine whose mechanical advantage is -^ ? Name two or three household appliances whose mechanical advan- tage is less than 1. POWER AXD ENERGY 142. Definition of power. When a given load has been raised a given distance a given amount of work has been done, whether the time consumed in doing it is small or great. Time is therefore not a factor which enters into the deter- mination of work ; but it is often as important to know the rate at which work is done as to know the amount of work 122 WORK AND MECHANICAL ENERGY accomplished. The rate of doing work is called power, or activity. Thus, if P represent power, W the work done, and t the time required to do it, P = -^- (10) {/ 143. Horse power. James Watt (1736-1819), the inventor of the steam engine, considered that an average horse could do 33,000 foot pounds of work per minute, or 550 foot pounds per second. The metric equivalent is 76.05 kilogram meters per second. This number is probabty considerably too high, but it has been taken ever since, in English-speaking countries, as the unit of power, and named the horse power (H.P.). The power of steam engines has usually been rated in horse power. The horse power of an ordinary railroad locomotive is from 500 to 1000. Stationary engines and steamboat engines of the largest size often run from 5000 to 20,000 H.P. The power of an average horse is about -| H.P., and that of an ordinary man about ^ H.P. 144. The kilowatt. In the metric system the erg has been taken as the absolute unit of work. The corresponding unit of power is an erg per second. This is, however, so small that it is customary to take as the practical unit 10,000,000 ergs per second ; that is, one joule per second (see 125, p. 106). This unit is called the watt, in honor of James Watt. The power of dynamos and electric motors is almost always expressed in kilowatts, a kilowatt representing 1000 watts ; and in modern practice even steam engines are being increasingly rated in kilowatts rather than in horse pow r er. A horse power is equiva- lent to 746 watts, or about | of a kilowatt. A kilowatt is almost exactly equal to 102 kilogram meters per second. 145. Definition of energy. The energy of a body is denned as its capacity for doing work. In general, inanimate bodies possess energy only because of work which has been done upon them at some previous time. Thus, suppose a kilogram weight JAMES PRESCOTT JOULE (1818-1889) English physicist, born at Man- chester ; most prominent figure in the establishment of the doctrine of the conservation of energy ; studied chemistry as a boy under John Dalton, and became so inter- ested that his father, a prosperous Manchester brewer, fitted out a laboratory for him at home ; con- ducted mostof his researches either in a basement of his own house or in a yard adjoining his brewery ; discovered the law of heating a conductor by an electric current ; carried out, in connection with Lord Kelvin, epoch-making re- searches upon the thermal prop- erties of gases; did important work in magnetism; first proved experi- mentally the identity of various forms of energy JAMES WATT (1736-1819) The Scotch instrument maker at the University of Glasgow, who may properly be considered the inventor of the steam engine ; for, although a crude and inefficient type of steam engine was known before his time, he left it in essen- tially its present form. The mod- ern industrial era may be said to begin with Watt = |j a I If 83 II O rfl U ^ "C '^ *O r^ -4-> d K I s H S < S POWER AXD ENERGY 123 FIG. 141. tion of Illustra- potential energy is lifted from the first position in Fig. 141 through a height of 1 m. and placed upon the hook H at the end of a cord which passes over a frictionless pulley p and is attached at the other end to a-second kilogram weight B. The operation of lifting A from position 1 to position 2 has required an expenditure upon it of 1 kg. m. (100,000 g. cm., or 98,000,000 ergs) of work. But in position 2, A is itself possessed of a certain capacity for doing work which it did not have before ; for if it is now started down- ward by the application of the slightest con- ceivable force, it will, of its own accord, return to position 1, and will in so doing raise the kilogram \veight B through a height of 1 m. In other words, it will do upon B exactly the same amount of work that was originally done upon it. 146. Potential and kinetic energy. A body may have a capacity for doing work not only because it has been given an elevated position but also because it has in some way acquired velocity ; for example, a heavy flywheel will keep machinery running for some time after the power has been shut off, and a bullet shot upward will lift itself a great distance against gravity because of the velocity which has been imparted to it. Similarly, any body which is in motion is able to rise against gravity, or to set other bodies in motion by colliding with them, or to overcome resistances of any conceivable sort. Hence, in order to distinguish between the energy which a body may have because of an advantageous position, and the energy which it may have because it is in motion, the two terms "potential energy" and "kinetic energy" are used. Potential energy includes the energy of lifted weights, of coiled or stretched springs, of bent bows, etc., in a word, potential energy is energy of position, while kinetic energy is energy of motion. 124 WORK AND MECHANICAL ENERGY 147. Transformations of potential and kinetic energy. The swinging of a pendulum and the oscillation of a weight attached to a spring illustrate well the way in which energy which has once been put into a body may be transformed back and forth between the potential and kinetic varieties. When the pendulum bob is at rest at the bottom of its arc, it possesses no energy of either type, since, on the one hand, it is as low as it can be, and, on the other, it has no velocity. When we pull it up the arc to the posi- tion A (Fig. 142), we do an amount of work upon it which is equal in gram centimeters to its weight in grams times the distance AD in centimeters ; that is, we store up in it this amount of potential energy. As now the bob falls to C this potential energy is com- pletely transformed into kinetic en- ergy. That this kinetic energy at C is exactly equal to the potential energy FIG. 142. Transformation at A is proved by the fact that if f ric- of P tential aild kinetic GiiGr*v tion is completely eliminated, the bob rises to a point B such that BE is equal to AD. We see, therefore, that at the ends of its swing the energy of the pendulum is all potential, while in the middle of the swing its energy is all kinetic. In intermediate positions the energy is part potential and part kinetic, but the sum of the two is equal to the original potential energy. 148. General statement of the law of frictionless machines. In our development of the law of machines, which led us to the conclusion that the work of the acting force is always equal to the work of the resisting force, we were careful to make two important assumptions : first, that friction was negligible ; second, that the motions were all either uniform or so slow that no appreciable velocities were imparted. In other words, POWER AND ENERGY 125 we assumed that the work of the acting force was expended simply in lifting weights or compressing springs, that is, in storing up potential energy. If now we drop the second assumption, a very simple experiment will show that our con- clusion must be somewhat modified. Suppose, for instance, that instead of lifting a 500-gram weight slowly by means of a balance, we jerk it up suddenly. We shall now find that the initial pull indicated by the balance, instead of being 500 g., will be considerably more, perhaps as much as several thou- sand grams if the pull is sufficiently sudden. This is obviously because the acting force is now overcoming not merely the 500 g. which represents the resistance of gravity, but also the inertia of the body, since velocity is being imparted to it. Now work done in imparting velocity to a body, that is, in over- coming its inertia, always appears as kinetic energy, while work done in overcoming gravity appears as the potential energy of a lifted weight. Hence, whether the motions produced by machines are slow or fast, if friction is negligible the law for all devices for transforming work may be stated thus: The work of the acting force is equal to the sum of the potential and kinetic energies stored up in the mass acted upon. In machines which work against gravity the body usually starts from rest and is left at rest, so that the kinetic energy resulting from the whole operation is zero. Hence in such cases the work done is the weight lifted times the height through which it is lifted, whether the motion is slow or fast. The kinetic energy im- parted to the body in starting is all given up by it in stopping. 149. The measure of potential energy. The measure of the potential energy of any lifted body, such as a lifted pile driver, is equal to the work which has been spent in lifting the body. Thus, if h is the height in centimeters and M the weight in grams, then the potential energy P.E. of the lifted mass is P.E. = Mh gram centimeters. 126 WORK AND MECHANICAL ENERGY Similarly, if h is the height in feet, and M the weight in pounds, p E = m foot pounds . 150. The measure of kinetic energy. Since the force of the earth's attraction for M grams is Mg dynes, if we wish to express the potential energy in ergs instead of in gram centimeters, we have P.E, = Mffh ergs. (12) Since this energy is all transformed into kinetic energy when the mass falls the distance h, the product Mgh also represents the number of ergs of kinetic energy which the moving weight has when it strikes the pile. If we wish to express this kinetic energy in terms of the velocity with which the weight strikes the pile, instead of the height from which it has fallen, we have only to substitute for h its value in terms of g and the velocity acquired (see equation (3), p. 76), namely h = i? 2 /2 g. This gives the kinetic energy K.E. in the form K.E. = -i J/y 2 ergs. (13) Since it makes no difference how a body has acquired its velocity, this represents the general formula for the kinetic energy in ergs of any moving body, in terms of its mass and its velocity. Thus, the kinetic energy of a 100-gram bullet moving with a velocity of 10,000 cm. per second is K.E. = x 100 x (10,000) 2 = 5,000,000,000 ergs. Since 1 g. cm. is equivalent to 980 ergs, the energy of this bullet is 5 ' oo y 8 ' 000 = 5,102,000 g. cm., or 51.02 kg. m. We know, therefore, that the powder pushing on the bullet as it moved through the rifle barrel did 51.02 kg. m. of work upon the bullet in giving it the velocity of 100 m. per second. In general terms, if M is in grams and v in centimeters per second, K.E. = ^ - j^jr. g. cm. ; if M is in pounds and v in feet per second, QUESTIONS AND PROBLEMS 1. A stick of dynamite has great capacity for doing work. Before the explosion occurs, is the energy in the potential or the kinetic form ? 2. Explain the use of the sand blast in cleaning castings, making frosted glass, cutting figures on glassware, cleaning off the walls of stone buildings, etc. POWER AND ENERGY 127 3. How much work is required to lift the 500-pound weight of a pile driver 30 ft.? How much potential energy is then stored in it? How much work does it do when it falls ? If the falling mass drives the pile into the earth i ft., what is its average force upon the pile ? 4. A man weighing 198 Ib. walked to the top of the stairway of the Washington Monument (500 ft. high) in 10 min. At what horse-power rate did he work? 5. A farm tractor drew a gang plow at the rate of 2^ mi. per hour, maintaining an average drawbar pull of 1500 Ib. At what average H.P. was the tractor working ? 6. In the course of a stream there is a waterfall 22 ft. high. It is shown by measurement that 450 cu. ft. of water per second pours over it. How many foot pounds of energy per second could be obtained from it? What horse power? 7. How many gallons of water (8 Ib. each) could a 10-horse-power engine raise in one hour to a height of 60 f t. ? 8. A certain airplane using three 400-horse-power motors flew 80 mi. per hour. With how many pounds backward force did the propellers push against the air ? 9. If a rifle bullet can just pass through a plank, how many planks will it pass through if its speed is doubled ? 10. A steel ball dropped into a pail of moist clay from a height of a meter sinks to a depth of 2 cm. How far will it sink if dropped 4 in. ? 11. Neglecting friction, find how much force a boy would have to exert to pull a 100-pound wagon up an incline which rises 5 ft. for every 100 ft. of length traversed on the incline. In addition to giving the numerical solution of the problem, state why you solve it as you do and how you know that your solution is correct. CHAPTER VIII THERMOMETRY ; EXPANSION COEFFICIENTS* THERMOMETBY 151. Meaning of temperature. When a body feels hot to the touch we are accustomed to say that it has a high temperature, and when it feels cold that it has a low temperature. Thus the word " temperature " is used to denote the condition of hot- ness or coldness of the body whose state is being described. 152. Measurement of temperature. So far as we know, up to the time of Galileo no one had ever used any special instru- ment for the measurement of temperature. People knew how hot or how cold it was from their feelings only. But under some conditions this temperature sense is a very unreliable guide. For example, if the hand has been in hot water, tepid water will feel cold; while if it has been in cold water, the same tepid water will feel warm ; a room may feel hot to one who has been running, while 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. 153. Galileo's thermometer. It was in 1592 that Galileo, at the University of Padua in Italy, constructed the first * It is recommended that this chapter be preceded by laboratory measure- ments on the expansions of a gas and a solid. See, for example, Experiments 14 and 15 of the authors' Manual. 128 THERMOMETRY 129 thermometer. He was familiar with the faots of expansion of solids, liquids, and gases ; and since gases expand more than solids or liquids, he chose a gas as his expanding substance. His device was that shown in Fig. 143. Let a bulb of air B be connected with a water manometer m, as in Fig. 143. 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 B is cooled with ice or ether, the water will rise at m. FIG. 143. Expansion of air by heat 154. Significance of temperature from the standpoint of the kinetic theory. Now if, as was stated in 64, gas pressure is due to the bombardment of the walls by the mole- cules of the gas, since the number of mole- cules in the bulb can scarcely have been changed by slightly heating it we are forced to conclude that the increase in pressure is due to an increase in the velocity of the molecules which are already there. 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 tem- perature is to increase the average kinetic energy of the mole- cules, and 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 expan- sion of gases with a rise in temperature. 155. The construction of a centigrade mercury thermometer. It was not until about 1700 that mercury thermometers were invented. On account of their extreme convenience these have now replaced all others for practical purposes. 130 THERMOMETRY; EXPANSION COEFFICIENTS The meaning of a degree of temperature change as measured by a mercury thermometer is best understood from a descrip- tion 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 thermom- eter 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. The bulb is next sur- rounded with melting snow or ice, as in Fig. 144, and the point at which the mer- cury stands in the tube is marked 0. Then the bulb and tube are placed in the steam rising from boiling FlG 144< Method water under a pressure of of finding the 76 cm., as in Fig. 145, and P oint in calibrat - . . ing a thermometer the new position or the mercury is marked 100. The space between these two marks 011 the stem is then divided into 100 equal parts, and divisions of the same length are extended above the 100 mark and below the mark. One degree of change in temperature, measured on such a thermometer, means, then, 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 y^ of the amount which it expands in passing from the temperature FIG. 145. Method of finding the 100 point in calibrat- ing a thermometer THERMOMETRY 131 of melting ice to that of steam under a pressure of 76 cm. 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 instead of the centigrade scale. 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 ; and, conversely, a decrease in tem- perature means a decrease in this average kinetic energy. 156. Fahrenheit thermometers. The com- mon 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 boil- ing water 212 instead of 100. The inter- vening stem is then divided into 180 parts. The zero of this scale is the temperature ob- tained by mixing equal weights of sal ammo- niac (ammonium chloride) and snow. In 1714, when Fahrenheit devised this scale, he chose this zero because he thought it repre- sented the lowest possible temperature that could be obtained in the laboratory. 157. Comparison of centigrade and Fah- renheit 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 temper- ature as 180 on the Fahrenheit scale (Fig. 146). Hence five FIG. 146. The cen- tigrade and Fahren- heit scales 132 THEEMOMETRY; EXPANSION COEFFICIENTS centigrade degrees are equal to nine Fahrenheit degrees. In Fig. 147, C represents the number of degrees in the centigrade reading, while F represents the number in the Fahrenheit reading. Since five centigrade degrees cover the same space on the stem as nine of the smaller Fahrenheit degrees, it is evident that iooHllh2i2 C ^5 T^T GO Q By this expression of the relation of the two scales it is very easy to reduce the readings of one thermometer to the scale of the other. For example, to find what Fahrenheit reading corresponds to 20 C. we have FIG. 147. Compari- 158. The range of the mercury thermom- son of centigrade and C1 . OAO _ Fahrenheit scales eter. Since mercury freezes at 39 C., temperatures lower than this are very often measured by means of alcohol thermometers, for the freezing point of alcohol is 130 C. Similarly, since the boiling point of mercury is about 360 C., mercury thermometers cannot be used for measuring very high temperatures. For both very high and very low temperatures in fact, for all temperatures a gas thermometer is the standard instrument. 159. The standard hydrogen thermometer. The modern gas thermometer (Fig. 148) is, however, widely different from that devised by Galileo (Fig. 143). It is not usually the increase in the volume of a gas kept under constant pressure which is taken as the measure of temperature change, but rather the in- crease in pressure which the molecules of a confined gas exert against the walls of a vessel whose volume is kept constant. The essential features of the method of calibration and use THERMOMETRY 133 100C oc P373A D273A of the standard hydrogen thermometer at the International Bureau of Weights and Measures at Paris are as follows: The bulb B (Fig. 148) is first filled with hydrogen and the space above the mercury in the tube a made as nearly a perfect vacuum as possible. B is then surrounded with melting ice (as in Fig. 144) and the tube a raised or lowered until the mercury in the arm b stands exactly opposite the fixed mark c on the tube. Now, since the space above D is a vacuum, the pressure exerted by the hydrogen in B against the mercury surface at c just supports the mercury column ED. The point D is marked on a strip of metal behind the tube a. The bulb B is then placed in a steam bath like that shown in Fig. 145. The increased pressure of the gas in B at once begins to force the mercury down at c and up at D. But by raising the arm a the mercury in b is forced back again to c, the increased pressure of the gas on the surface of the mercury at c being balanced by the increased height of the mercury column supported, which is now EF instead of ED. When the gas in B is thoroughly heated to the temperature of the steam, the arm a is very carefully adjusted so that the mercury in b stands very exactly at c, its original level. A second mark is then placed on the metal strip exactly opposite the new level of the mercury, that is, at F. Then D is marked C., and F is marked 100 C. The vertical distance between these marks is di- vided into 100 exactly equal parts. Divisions of exactly the same length are carried above the 100 mark and below 'the mark. One degree of change in tempera- ture is then defined as any change in temperature which will cause the pressure of the gas in B to change by the amount represented by the distance between any two of these divisions. This distance is found to be 2^ of the height ED. -K O'Aor FIG. 148. The stand- ard gas thermometer In other words, one degree of change in temperature on the centigrade scale is such a temperature change as ivill cause the 134 XHEBMOMETBY; EXPANSION COEFFICIENTS pressure exerted by a confined volume of hydrogen to change ly ~-$ of its pressure at the temperature of melting ice (0 C.). 160. Absolute temperature. Since, then, cooling the hydro- gen through 1 C., as denned above, reduces the pressure -^-^ of its value at C/, it is clear that cooling it 278 below C. would reduce its pressure to zero. But from the stand- point of the kinetic theory this would be the temperature at which all motions of the hydrogen molecules would cease. This temperature is called the absolute zero, and the temper- ature measured from this zero is called absolute temperature. Thus, if A is used to denote the absolute scale, we have C. - 273 A., 100 C. = 373 A., 15 C. = 288 A., etc. It is customary to indicate temperatures on the centigrade scale by , and on the absolute scale by T. We have, then, T= + 273. (1) 161. Comparison of gas and mercury thermometers. Since an inter- national committee has chosen the hydrogen thermometer described in 159 as the standard of temperature measurement, it is important to know whether mercury thermometers, graduated in the manner described in 155, agree with gas thermometers at temperatures other than and 100 (where, of course, they must agree, since these temperatures are in each case the starting points of the graduation). A careful comparison has shown that although they do not agree exactly, yet fortunately the disagreements at ordinary temperatures are small, not amounting to more than .2 anywhere between and 100. At 300 C., however, the differ- ence amounts to about 4. (Mercury thermometers are actually used for measuring temperatures above the boiling point of mercury, 360C. They are then filled with nitrogen, the pressure of which prevents boiling.) Hence for all ordinary purposes mercury thermometers are sufficiently accurate, and no special standardization of them is necessary. But in all scientific work, if mercury thermometers are used at all, they must first be compared with a gas thermometer and a table of corrections obtained. The errors of an alcohol thermometer are considerably larger than those of a mercury thermometer. 162. Low temperatures. The absolute zero of temperature can, of course, never be attained, but in recent years rapid SIR WILLIAM THOMSON, LORD KELVIN (1824-1907) One of the best known and most prolific of nineteenth-century physicists ; born in Belfast, Ireland ; professor of physics in Glasgow University, Scotland, for more than fifty years ; especially renowned for his investigations in heat and electricity ; originator of the absolute thermodynamic scale of temperature ; formulator of the second law of thermodynamics; inventor of the electrometer! the mirror galvanometer, and many other important electrical devices THERMOMETRY 135 strides have been made toward it. Forty years ago the low- est temperature which had ever been measured was 110 C., the temperature attained by Faraday in 1845 by causing a mixture of ether and solid carbon dioxide to evaporate in a vacuum. But in 1880 air was first liquefied and found, by means of a gas thermometer, to have a temperature of -190C. When liquid air evaporates into a space which is kept exhausted by means of an air pump, its temperature falls to about 220 C. Recently hydrogen has been lique- fied and found to have a temperature at atmospheric pressure of 243 C. All of these temperatures have been measured by means of hydrogen thermometers. By allowing liquid hydrogen to evaporate into a space kept exhausted by an air pump, Dewar in 1900 attained a temperature of 260. In 1911 Kamerlingh Onnes liquefied helium and attained a temperature of 271.3 C., only 1.7 above absolute zero (see 217). QUESTIONS AND PROBLEMS 1. Define 0C. and 100 C. What is 1C.? 1F.? 2. From a study of the behavior of gases we conclude that there is a temperature at which the molecules are at rest and at which bodies there- fore contain no heat. Give the reasoning that leads to this conclusion. 3. Normal room temperature is 68 F. What is it centigrade? 4. The normal temperature of the human body is 98. 6 F. What is it centigrade ? 5. What temperature centigrade corresponds to 0F. ? 6. Mercury freezes at about 40 F. What is this centigrade ? 7. The temperature of liquid air is 190 C. What is it Fahrenheit ? 8. The lowest temperature attainable by evaporating liquid helium is - 271.3 C. What is it Fahrenheit ? 9. What is the absolute zero of temperature on the Fahrenheit scale ? 10. Why is a fever thermometer made with a very long cylindrical bulb instead of a spherical one ? 11. When the bulb of a thermometer is placed in hot water, it at first falls a trifle and then rises. Why ? 12. How does the distance between the mark and the 100 mark vary with the size of the bore, the size of the bulb remaining the same ? 13. What is meant by the absolute zero of temperature? 136 THEEMOMETRY; EXPANSION COEFFICIENTS 14. Why is the temperature of liquid air lowered if it is placed under the receiver of an air pump and the air exhausted ? 15. Two thermometers have bulbs of equal size. The bore of one has a diameter twice that of the other. What are the relative lengths of the stems between and 100 ? EXPANSION COEFFICIENTS 163. The laws of Charles and Gay-Lussac. When, as in the experiment described in 159, we keep the volume of a gas constant and observe the rate at which the pressure increases with the rise in temperature, we obtain the pressure coefficient of expansion, which is denned as the ratio between the increase in pressure per degree and the value of the pressure at C. This was first done for different gases by a Frenchman, Charles, in 1787, who found that the pressure coefficients of expansion of all gases are the same. This is known as the law of Charles. When we arrange the experiment so that the gas can expand as the temperature rises, the pressure remaining constant, we obtain the volume coefficient of expansion, which is defined as the ratio between the increase in volume per degree and the total vol- ume of the gas at C. This was first done for different gases in 1802 by another Frenchman, Gay-Lussac, who found that all gases have the same volume coefficient of expansion, this coefficient being the same as the pressure coefficient, namely, 1/273. This is known as the law of G-ay-Lussac. From the definition of absolute temperature and from Charles's law we learn that, for all gases at constant volume, pressure is proportional to absolute temperature ; that is, P T =^ = ' (2) ^ 2 T , Also, from Gay-Lussac's law we learn that, for all gases at constant pressure, volume is proportional to absolute temperature ; that is, EXPANSION COEFFICIENTS 137 If pressure, temperature, and volume all vary,* we have P V T fili-fi. (4) P V T a 2 2 Any one of these six quantities may be found if the other five are known. If the volume remains constant, that is, if V 1 = V$ equation (4) reduces to (2), that is, to Charles's law. If the pressure remains constant, P I = P 2 and equation (4) reduces to (3), that is, to Gay-Lussac's law. If the temperature does not change, T^ = T 2 and equation (4) reduces to P^V^ = P 2 V 2 <> that is, to Boyle's law. If the ratio of densities instead of volumes is y j) sought, it is only necessary to replace in (3) and (4) by J . QUESTIONS AND PROBLEMS 1. Why is it unsafe to let a pneumatic inkstand like that of Fig. 30, p. 33, remain in the sun ? 2. To what temperature must a cubic foot of gas initially at C. be raised in order to double its volume, the pressure remaining constant? 3. If the volume of a quantity of air at 30 C. is 200 cc., at what temperature will its volume be 300 cc., the pressure remaining the same? 4. If the air within a bicycle tire is under a pressure of 2 atmospheres, that is, 152 cm. of mercury, when the temperature is 10 C., what pressure will exist within the tube when the temperature changes to 35 C.? 5. If the pressure to which 15 cc. of air is subjected changes from 76 cm. to 40 cm., the temperature remaining constant, what does its volume become ? (See Boyle's law, p. 36.) If, then, the temperature of the same gas changes from 15 C. to 100 C., the pressure remaining constant, what will be the final volume ? 6. The air within a half-inflated balloon occupies a volume of 100,000 1. The temperature is 15 C. and the barometric height 75 cm. What will be its volume after the balloon has risen to the height of Mt. Blanc, where the pressure is 37 cm. and the temperature 10 C.? * If this is not clear to the student, let him recall that if the speeds of two runners are the same, then their distances are proportional to their times, that is, -Dj/Da = t^/t z ; but if their times are the same and the speeds different, Dj/D-j = Sj/Sg. If now one runs both twice as fast and twice as long, he evi- dently goes 4 times as far; that is, if time and speed both vary, D^D^ t 138 THERMOMETRY; EXPANSION COEFFICIENTS EXPANSION OF LIQUIDS AND SOLIDS 164. The expansion of liquids. The expansion of liquids differs from that of gases in that 1. The coefficients of expansion of liquids are all consider- ably smaller than those of gases. 2. Different liquids expand at wholly different rates ; for example, the coefficient of alcohol between and 10 C. is .0011 ; of ether it is .0015 ; of petroleum, .0009 ; of mercury, .000181. 3. The same liquid often has different coefficients at dif- ferent temperatures ; that is, the expansion is irregular. Thus, if the coefficient of alcohol is obtained between and 60 C., instead of between and 10 C., it is .0013 instead of .0011. The coefficient of mercury, however, is very nearly constant through a wide range of temperature, which indeed might have been inferred from the fact that mercury thermometers agree so well with gas thermometers. 165. Method of measuring the expansion coeffi- cients of liquids. One of the most convenient and common methods of measuring the coeffi- cients of liquids is to place them in bulbs of known volume, provided with capillary necks of known diameter, like that shown in Fig. 149, and then to watch the rise of the liquid in the neck for a given rise in temperature. A certain allowance must be made for the expansion of the bulb, but this can readily be done if the coefficient of expansion of the substance of which the bulb is made is known. 166. Maximum density of water. When water is treated in the way described in the preceding para- graph, it reaches its lowest position in the stem at 4 C. As FIG. 149. Bulb for investigat- ing expansions of liquids EXPANSION OF LIQUIDS AND SOLIDS 139 the temperature falls from that point down to C., water exhibits the peculiar property of expanding with a decrease in temperature. We learn, therefore, that water has its maximum density at a temperature of 4 C. 167. The cooling of a lake in winter. The preceding para- graph makes it easy to understand the cooling of any large body of water with the approach of winter. The surface layers are first cooled and contract. Hence, being then heavier than the lower layers, they sink and are replaced by the warmer water from beneath. This process of cooling at the surface, and sinking, goes on until the whole body of water has reached a temperature of 4 C. After this condi- tion has been reached, further cooling of the surface layers makes them lighter than the water beneath, and they now remain on top until they freeze. Thus, before any ice what- ever can form on the surface of a lake, the T^hole mass of water to the very bottom must be cooled to 4 C. This is why it requires a much longer and more severe period of cold to freeze deep bodies of water than shallow ones. Further, since the circulation described above ceases at 4 C., practically all of the unfrozen water will be at 4 C. even in the coldest .weather. Only the water which is in the immediate neighborhood of the ice will be lower than 4 C. This fact is of vital importance in the preservation of aquatic life. 168. Expansion of solids. The proofs of expansion of solids with an increase in temperature may be seen on every side. Railroad rails are laid with spaces between their ends so that they may expand during the heat of summer without crowd- ing each other out of place. Wagon tires are made smaller 1 than the wheels which they are to fit. They are then heated until they become large enough to be driven on, and in cooling they shrink again and thus grip the wheels with 140 THERMOMETRY; EXPANSION COEFFICIENTS immense force. A common lecture-room demonstration of expansion is the following: Lei the ball B, which when cool just slips through the ring 7?, be heated in a Bunsen flame. It will now be found too large to pass through the ring ; but if the ring is heated, or if the ball is again cooled, it will pass through easily (see Fig. 150). If the expansion of gases and liquids is due to the increase in the average kinetic energy ., ,. .-, . i -, ,-, . FIG. 150. Expansion of agitation of their molecules, the foregoing of so]icls experiments with solids must clearly be given a similar interpretation. In a word, then, the temperature of a given substance, be it solid, liquid, or gas, is determined by the average kinetic energy of agitation of its molecules. 169. Linear coefficients of expansion of solids. It is often more convenient to measure the increase in length of one edge of an Expanding solid than to measure its increase in volume. The ratio between the increase in length per degree rise in temperature and the total length is called the linear coeffi- cient of expansion of the solid. Thus, if l l represent the length of a bar at ^, and 1 2 its length at 2 , the equation which defines the linear coefficient k is L - 1 L-l _2 ',(,-*,) (5) The linear coefficients of a few common substances are given in the following table : Aluminium . .000023 Gold 000014 Silver . . . .000019 Brass . . . .000019 Iron 000012 Steel 000013 Copper . . . .000017 Lead 000029 Tin 000023 Glass . .000009 Platinum . . .000009 Zinc 000030 APPLICATIONS OF EXPANSION 141 APPLICATIONS OF EXPANSION 170. Compensated pendulum. Since a long pendulum vi- brates more slowly than a short' one, the expansion of the rod which carries the pendulum bob causes an ordinary clock to run too slowly in summer, and its contraction causes it to run too fast in winter. For this reason very accurate clocks are provided with compensated pen- dulums, which are so constructed that the distance of the bob beneath the point of support is independent of the temperature. This is accomplished by suspending the bob, by means of two sets of rods of different material, in such a way that the expansion of one set raises the bob, while the expan- sion of the other set lowers it. Such a pendulum is shown in Fig. 151. The ex- FlG pansion of the iron rods 6, c?, e, and i tends pensated pendulum to lower the bob, while that of the copper rods c tends to. raise it. In order to produce complete com- pensation it is only necessary to make the total lengths of iron and copper rods inversely proportional to the coefficients of expansion of iron and copper. 171. Compensated balance wheel. In the balance wheel of an accurate watch (Fig. 152) another application of the unequal expansion of metals is made. Increase in temperature both increases the radius of the wheel and weakens _, FIG. 152. The compen- the elasticity of the spring which con- sated balance wheel trols it. Both of these effects tend to make the watch lose time. This tendency may be counter- acted by bringing the mass of the rotating parts in toward The com _ 142 THEBMOMETRY; EXPANSION COEFFICIENTS the center of the wheel. This is accomplished by making the arcs be of metals of different expansion coefficients, the inner FIG. 153 FIG. 154 Unequal expansion of metals metal, shown in black in the figure, having the smaller coeffi- cient. The free ends of the arcs are then sufficiently pulled in by a rise in temperature to counteract the retarding effects. The principle is precisely the same as that which finds simple illustration in the compound bar shown in Fig. 153. This bar consists of two strips, one of brass and one of iron, riveted to- gether. When the bar is placed edgewise in a Bunsen flame, so that both metals are heated equally, it will be found to bend in such a way that the more expansible metal, namely, the brass, is on the outside of the curve, as shown in Fig. 154. When it is cooled with snow or ice, it bends in the opposite direction. The common thermostat (Fig. 155) is precisely such a bar, which is arranged so as to open the drafts by closing an electri- cal circuit at a when it is too cold, and to close the drafts by making contact at b when it is too warm. FIG. 155. The thermostat QUESTIONS AND PROBLEMS FIG. 156 1. Why is the water at the bottom of a lake usually colder than that at the top ? Why is the water at the bottom of very deep mountain lakes in some instances observed to be at 4 C. the whole year round, while that at the top varies from C. to quite warm ? 2. Give three reasons why mercury is a better liquid to use in ther- mometers than water. 3. Why is a thick tumbler more likely to break when hot water is poured into it than a thin one ? APPLICATIONS OF EXPANSION 143 4. Pendulums are often compensated by using cylinders of mercury, as in Fig. 156. Explain. 5. The steel cable from which Brooklyn Bridge hangs is more than a mile long. By how many feet does a mile of its length vary between a winter day when the temperature is 20 C. and a summer day when it is 30 C. ? 6. If a surveyor's steel tape is exactly 100 ft. long at 20 C., how much too short would it be at C. ? 7. A certain glass flask is graduated to hold 1000 cc. at 15 C. How many cubic centimeters will the same flask hold at 40 C., the coefficient of cubical expansion of glass being .000025 ? 8. The dial thermometer is a compound bar (Fig. 157) with iron on the outside and brass on the inside. A thread t is wound about the central cylinder c. Explain the action. 9. Why may a glass stopper sometimes be loosened by pouring hot water on the neck of a bottle ? 10. A metal rod 230 cm. long expanded 2.75 mm. in being raised from 0C. to 100C. Find its coefficient of linear expansion. 11. The changes in temperature to which long lines of steam pipes are subjected make it necessary to introduce "expansion joints." These joints consist of brass collars fitted tightly by means of packing over the separated ends of two adjacent lengths of pipe. If the pipe is of iron, and such a joint is inserted every 200 ft., and if the range of tem- perature which must be allowed for is from 30 C. to 125 C., what is the minimum play which must be allowed for at each expansion joint ? 12. Show from equation 5, p. 140, that linear coefficient of expansion may be defined as increase in length per unit length per degree. CHAPTER IX WORK AND HEAT ENERGY FRICTION 172. Friction always results in wasted work. All of the experiments mentioned in Chapter VII were so arranged that friction could be neglected or eliminated. So long as this condition was fulfilled it was found that the result of universal experience could be stated thus : The work done by the acting force is equal to the sum of the kinetic and potential energies stored up. But wherever friction is present this law is found to be inexact, for the work of the acting force is then always somewhat greater than the sum of the kinetic and potential energies stored up. If, for example, a block is pulled over the horizontal surface of a table, at the end of the motion no velocity has been imparted to the block, and hence no kinetic energy has been stored up. Further, the block has not been lifted nor put into a condition of elastic strain, and hence no potential energy has been communicated to it. We cannot in any way obtain from the block more work after the motion than we could have obtained before it was moved. It is clear, therefore, that all of the work which was done in moving the block against the friction of the table was wasted work. Experience shows that, in general, where work is done against friction it can never be regained. Before considering what becomes of this wasted work we shall consider some of the factors on which friction depends and some of the laws which are found by experiment to hold in cases in which friction occurs. 144 FRICTION 145 173. Coefficient of friction. It is found that if F represents the force parallel to a plane which is necessary to maintain uniform motion in a body which is pressed against the plane with a force F', then, for small F=300 velocities, the ratio depends only on the nature of the surfaces in contact, and not at all on the FIG. 158. The ratio of FtoF' is area or on the velocity of the the coefficient of friction 77T motion. The ratio is called the coefficient of friction for F the given materials. Thus (Fig. 158), if F is 300 g. and F' goo ' The coeffi- is 800 g., the coefficient of friction is MS = -375. cient of iron on iron is about .2 ; of oak on oak, about .4. 174. Rolling friction. The chief cause of sliding friction is the inter- locking of minute projections. When a round solid rolls over a smooth surface, the frictional resistance is generally much less than when it slides ; for example, the coefficient of friction of cast-iron wheels rolling on iron rails may be as low as .002, that is, y^ of the sliding friction (1) (2) FIG. 159. Friction in bearings (1) Common bearing ; (2) ball bearing of iron on iron. This means that a pull of 1 pound will keep a 500- pound car in motion. Sliding friction is not, however, entirely dis- pensed with in ordinary wheels, for although the rim of the wheel rolls on the track, the axle slides continuously at some point c (Fig. 159, (1)) upon the surface of the journal. Journals are frequently lined with brass or Babbitt metal, since this still further lowers the coefficient. The great advantage of the ball bearing (Fig. 159, (2)) is that the sliding friction in the hub is almost completely replaced by rolling friction. The manner in which ball bearings are used in a bicycle 146 WORK AND HEAT ENERGY pedal is illustrated in Fig. 160. The free-wheel ratchet ia shown in Fig. 161. The pawls a and b enable the pedals and chain wheel W to stop while the rear axle continues to revolve. Roller bearings are showD in Fig. 162. Oils and greases prevent rapid wear of bearings by lessening friction. FIG. 160. The bicycle pedal FIG. 161. Free-wheel ratchet 175. Fluid friction. When a solid moves through a fluid, as when a bullet moves through the air or a ship through the water, the resistance encountered is not at all independent of velocity, as in the case of solid friction, but increases for slow speeds nearly as the square of the velocity, and for high speeds at a rate considerably greater. This explains why it is so expensive to run a fast train ; for the re- sistance of the air, which is a small part of the total resistance so long as the train is moving slowly, becomes the predominant factor at high speeds. The resistance offered to steamboats running at high speeds is usually considered to increase as the cube of the velocity. Thus, the Cedric, of the White Star Line, having a speed of 17 knots, has a horse power of 14,000 and a total weight, when loaded, of about 38,000 tons, while the Mauretania, of the Cunard Line, having a speed of 25 knots, has engines of 70,000 horse power, although the total weight when loaded is only 32,500 tons. QUESTIONS AND PROBLEMS 1. Mention three ways of lessening friction in machinery. 2. In what respects is friction an advantage, and in what a disadvan- tage, in everyday life ? Could we get along without it ? 3. Why is a stream swifter at the center than at the banks ? 4. Why does a team have to keep pulling after a load is started? FIG. 162. Roller bear- ings of automobile front wheel EFFICIENCY 147 5. Why is sand often placed on a track in starting a heavy train ? 6. In what way is friction an advantage in lifting buildings with a jackscrew ? In what way is it a disadvantage ? 7. A smooth block is 10 x 8 x 3 in. Compare the distances which it will slide when given a certain initial velocity on smooth ice if resting first, on a 10 x 8 face ; second, on a 10 x 3 face ; third, on an 8 x 3 face. 8. What is the coefficient of friction of brass on brass if a force of 25 Ib. is required to maintain uniform motion in a brass block weighing 200 Ib. when it slides horizontally on a brass bed ? 9. The coefficient of friction between a block and a table is .3. What force will be required to keep a 500-gram block in uniform motion ? EFFICIENCY 176. Definition of efficiency. Since it is only in an ideal machine that there is no friction, in all actual machines the work clone by the acting force always exceeds, by the amount of the work done against friction, the amount of potential and kinetic energy stored up. We have seen that the former is wasted work in the sense that it can never be regained. Since the energy stored up represents work which can be regained, it is termed useful work. In most machines an effort is made to have the useful work as large a fraction of the total work expended as possible. The ratio of the useful work to the total work done by the acting force is called the EFFICIENCY of the machine. Thus Useful work accomplished Efficiency = - r- ^n C 1 ) lotal work expended Thus, if in the system of pulleys shown in Fig. 116 it is necessary to add a weight of 50 g. at E in order to pull up slowly an added weight of 240 g. at 7t, the work done by the 50 g. while E is moving over 1 cm. will be 50 x 1 g. cm. The useful work accomplished in the same time 1 240 x 1 4 is 240 x g. cm. Hence the efficiency is equal to * = = 80%. o 50 x 1 o 177. Efficiencies of some simple machines. In simple levers the friction is generally so small as to be negligible ; hence the efficiency of such machines is approximately 100%. When 148 WORK AND HEAT ENERGY inclined planes are used as machines, the friction is also small, so that the efficiency generally lies between 90% and 100%. The efficiency of the commercial block and tackle (Fig. 116), with several movable pulleys, is usually considerably less, varying between 40% and 60%. In the jackscrew there is necessarily a very large amount of friction, so that although the mechanical advantage is enormous, the efficiency is often as low as 25%. The differential pulley of Fig. 136 has also a very high mechanical advantage with a very small efficiency. Gear wheels such as those shown in Fig. 134, or chain gears such as those used in bicycles, are machines of comparatively high efficiency, often utilizing between 90% and 100% of the energy expended upon them. 178. Efficiency of overshot water wheels. The overshot water wheel (Fig. 163) utilizes chiefly the potential energy of the water at S, for the wheel is turned by the weight of the water in the buckets. The work expended on the wheel per second, in foot pounds or gram centimeters, is the product of the weight of the water which passes over it per second by the distance through which it falls. The efficiency is the work which the wheel can accomplish in a second divided by this quantity. Such wheels are very common in mountainous regions, where it is easy to obtain considerable fall but where the streams carry a small volume of water. The efficiency is high, being often between 80 % and 90 %. The loss is due not only to the friction in the bearings and gears (see C) but also to the fact that some of the water is spilled from the buckets or passes over without entering them at all. This may still be regarded as a frictional loss, since the energy disappears in internal friction when the water strikes the ground. 179. Efficiency of undershot water wheels. The old-style undershot wheel (Fig. 164) so common in flat countries, where there is little fall but an abundance of water utilizes only the kinetic energy of the water FIG. 163. Overshot water wheel EFFICIENCY 149 FIG. 164. The undershot wheel running through the race from ,-1. It seldom transforms into useful work more than 25% or 30% of the potential energy of the water above the dam. There are, however, certain mod- ern forms of undershot wheel. which are extremely efficient. For example, the Pelton icheel (Fig. 165), developed since 1880 and now very commonly used for small-power purposes in cities supplied with waterworks, sometimes has an efficiency as high as 83 %. The water is delivered from a nozzle against cup-shaped buckets arranged as in the figure. At the Big Creek development in California, Pelton wheels 94 inches in diameter are driven by water coming with a velocity of 350 feet per second (how many miles per hour?) through nozzles 6 inches in diameter. The head of water is here 1900 ft. 180. Efficiency of water turbines. The turbine wheel was invented in France in 1833 and is now used more than any other form of w r ater wheel. It stands completely under water in a case at the bottom of a turbine pit, rotating in a hori- zontal plane. Fig. 166 shows the method of installing a turbine at Niagara. C is the outer case into which the water enters from the penstock p. Fig. 167, (1), shows the outer case with contained turbine ; Fig. 167, (2), is the inner case, in which are the fixed guides G, which direct the water at the most advantageous angle against the blades of the wheel inside ; Fig. 167, (3), is the wheel itself ; and Fig. 167, (4), is a section of wheel and inner case, showing how the water enters through the guides and impinges upon the blades W. The spent water simply falls down from the blades into the tailrace T (Fig. 166). The amount of water which passes through the turbine can be controlled by means of the rod P (Fig. 167, (1)), which can be turned so as to increase or decrease the size of the openings between the guides G (Fig. 167, (2)). The energy expended upon the turbine per second is the product of the mass of water which passes through it by the height of the turbine pit. Efficiencies as high as 90 % have been attained with such wheels. FIG. 165. The Pelton water wheel 150 WORK AND HEAT ENERGY One of the largest turbines in existence is operated by the Puget Sound Power Co. It develops 25,000 horse power under a 440-foot head of water. FIG. 166. A turbine installed FIG. 167. The turbine wheel (1) Outer case ; (2) inner case ; (3) rotating part ; (4) section QUESTIONS AND PROBLEMS 1 . Why is the efficiency of the jackscrew low and that of the lever high ? 2. Find the efficiency of a machine in which an effort of 12 Ib. moving 5 ft. raises a weight of 25 Ib. 2 ft. 3. What amount of work was done on a block and tackle having an efficiency of 60 % when by means of it a weight of 750 Ib. was raised 50 ft. ? 4. A force pump driven by a 1-horse-power engine lifted 4 cu. ft. of water per minute to a height of 100 ft. What was the efficiency of the pump? 5. If it is necessary to pull on a block and tackle with a force of 100 Ib. in order to lift a weight of 300 Ib., and if the force must move 6 ft. to raise the weight 1 ft., what is the efficiency of the system ? MECHANICAL EQUIVALENT OF HEAT 151 6. If the efficiency had been 65%, what force would have been necessary in the preceding problem ? 7. The Niagara turbine pits are 136 ft. deep, and their average horse power is 5000. Their efficiency is 85%. How much water does each turbine discharge per minute? MECHANICAL EQUIVALENT OF HEAT * 181. What becomes of wasted work ? In all the devices for transforming work which, we have considered we have found that on account of frictional resistances a certain per cent of the work expended upon the machine is wasted. The question which at once suggests itself is, What becomes of this wasted work ? The following familiar facts suggest an answer. When two sticks are vigorously rubbed together, they become hot ;. augers and drills often become too hot to hold ; matches are ignited by friction ; if a strip of lead is struck a few sharp blows with a hammer, it is appreciably warmed. Now, since we learned in Chapter VIII that, according to modern notions, increasing the temperature of a body means simply increasing the average velocity of its molecules, and therefore their average kinetic energy, the above facts point strongly to the conclu- sion that in each case the mechanical energy expended has been simply transformed into the energy of molecular motion. This view was first brought into prominence in 1798 by Benjamin Thompson, Count Rumford, an American by birth, who was led to it by observing that in the boring of cannon heat was continuously developed. It was first carefully tested by the English physicist James Prescott Joule (see opposite p. 122) (1818-1889) in a series of epoch-making experiments extend- ing from 1842 to 1870. In order to understand these experi- ments we must first learn how heat quantities are measured. * This subject should be preceded by a laboratory experiment upon the "law of mixtures," and either preceded or accompanied by experiments upon specific heat and mechanical equivalent. See authors' Manual, Exper- iments 18, 19, and 20. 152 WORK AND HEAT ENERGY 182. Units of heat ; the calorie and the British thermal unit. The calorie is the amount of heat that is required to raise the temperature of 1 gram of water through 1 (,"'., and the British thermal unit {B. T. Z7.) is the amount of heat that is required to raise the temperature of 1 pound of water through 1 F. (One B.T.U. = 252 cal.) Thus, when a hun- dred grams of water has its temperature raised 4 C. we say that four hundred calories of heat have entered the water. Similarly, when a hundred grams of water has its temperature lowered 10 C. we say that a thousand calories have passed out of the water. If, then, we wish to measure, for instance, the amount of heat developed in a lead bullet when it strikes against a target, we have only to let the spent bullet fall into a known weight of water and to measure the number of degrees through which the temperature of the w^ater rises. The product of the number of grams of w r ater by its rise in temperature is, then, by definition, the number of calories of heat which have passed into the water. It will be noticed that in the above definition we maive no assumption whatever as to what heat is. Previous to the nineteenth century physicists generally held it to be an invisible, weightless fluid, the passage of which into or out of a body caused it to grow hot or cold. This view accounts well enough for the heating which a body experiences when it is held in contact with a flame or other hot body, but it has difficulty in explaining the heating produced by rubbing or pounding. Rumford's view accounts easily for this, as we have seen, while it accounts no less easily for the heating of cold bodies by contact with hot ones; for we have only to think of the hotter and therefore more energetic molecules of the hot body as communicating their energy to the mole- cules of the colder body in much the same way in which a rapidly moving billiard ball transfers part of its kinetic energy to a more slowly moving ball against which it strikes. Underwood & Underwood THE ViCKERS-ViMY AIRPLANE The first nonstop transatlantic airplane flight was made on June 14, 1919, from St. John's, Newfoundland, to Clifden, Ireland, a distance of 1890 miles. This historic flight the longest ever made was accomplished in fifteen hours and fifty-seven minutes, through fog and sleet, at an average speed of 118.5 miles per hour, a feat which won the $50,000 prize which had been offered for nearly five years by the London Daily Mail. The plane was driven by two 360-horse-power Rolls-Royce motors and carried 865 gallons of gasoline. It was piloted by Capt. John Alcock and navigated by Lieut. Arthur W. Brown. This airplane had a wing spread of 67 feet and a length of 42 feet 8 inches MECHANICAL EQUIVALENT OF HEAT 153 183. Joule's experiment on the heat developed by friction. Joule argued that if the heat produced by friction etc. is indeed merely mechanical energy which has been transferred to the molecules of the heated body, then the same number of calories must always be produced by the disappearance of a given amount of mechanical energy. And this must be true, no matter whether the work is expended in overcoming the friction of wood 011 wood, of iron on iron, in percussion, in compression, or in any other conceivable way. To see whether or not this was so he caused mechanical energy to disappear in as many ways as possible and measured in every case the amount of heat developed. In his first experiment he caused paddle wheels to rotate in a vessel of water by means of falling weights W (Fig. 168). The amount of work done by gravity upon the weights in causing them to descend through any distance d was equal to their weight W times this dis- tance. If the weights descended slowly and uniformly, this work was all expended in overcoming the resistance of the water to the motion of the paddle wheels through it ; that is, it was wasted in eddy currents in the water. Joule measured the rise in the temperature of the water and found that the mean of his three FlG . 168> Jou i e > s first experiment on best trials gave 427 gram meters the mechanical equivalent of heat as the amount of work required to develop enough heat to raise a gram of water one degree. This value, confirmed by modern experiments, is now generally accepted as correct. He then repeated the experiment, substituting mercury for water, and obtained 425 gram meters as the work necessary to produce a calorie of heat. The difference between these numbers is less than was to have been expected from the unavoidable errors in the observations. He then devised an arrangement in which the heat was developed by the friction of iron on iron, and again obtained 425. 154 WORK AND HEAT ENERGY x/ 184. Heat produced by collision. A Frenchman named Hirn was the first to make a careful determination of the relation between the heat developed by collision and the kinetic energy which disappears. He allowed a steel cylinder to fall through a known height and crush a lead ball by its impact upon it. The amount of heat developed in the lead was meas- ured by observing the rise in temperature of a small amount of water into which the lead was quickly plunged. As the mean of a large number of trials he also found that 425 gram meters of energy disappeared for each calorie of heat that appeared. 185. Heat produced by the compression of a gas. Another way in which Joule measured the relation between heat and work was by compressing a gas and comparing the amount of work done in the compression with the amount of heat developed. Every bicyclist is aware of the fact that when he inflates his tires the pump grows hot. This is due partly to the friction of the piston against the walls, but chiefly to the fact that the downward motion of the piston is transferred to the molecules which come in contact with it, so that the velocity of these molecules is increased. The principle is precisely the same as that involved in the velocity communicated to a ball by a bat. If the bat is held rigidly fixed and a ball thrown against it, the ball rebounds with a certain velocity ; but if the bat is moving rapidly forward to meet the ball, the latter rebounds with a much greater velocity. So the molecules which in their natural motions collide with an advancing piston rebound with greater velocity than they would if they had impinged upon a fixed wall. This increase in the molecular velocity of a gas on compression is so great that when a mass of gas at C. is compressed to one half its volume, the temperature rises to 87 C. The effect may be strikingly illustrated by the fire syringe (Fig. 169). Let a few drops of carbon bisulphide be placed on a small bit of cotton, dropped to the bottom of the tube A, and then removed; then let the MECHANICAL EQUIVALENT OF HEAT 155 FIG. 169. The fire syringe piston B be inserted and very suddenly depressed. Sufficient heat will be developed to ignite the vapor, and a flash will result. (If the flash does not result from the first stroke, withdraw the piston completely, then reinsert, and compress again.) To measure the heat of compression Joule surrounded a small compression purnp with water, took 300 strokes on the pump, and meas- ured the rise in temperature of the water. As the result of these measurements he obtained 444 gram meters as the mechanical equivalent of the calorie. The experiment, however, could not be performed with great exactness. Joule also measured the converse effect, namely, the cooling produced in a gas which is pushing forward a piston and thus doing work. He obtained 437 grain meters. 186. Significance of Joule's experiments. Joule made three other determinations of the relation between heat and work by methods involving electrical measurements. He published as the mean of all his determinations 426.4 gram meters as the mechanical equivalent of the calorie. But the value of his experiments does not lie primarily in the accuracy of the final results, but rather in the proof which they for the first time furnished that whenever a given amount of work is tvasted, no matter in what particular way this waste occurs, the same definite amount of heat always appears. The most accurate determination of the mechanical equiva- lent of heat was made by Rowland (see opposite p. 358) (1848- 1901) in 1880. He obtained 427 gram meters (4.19 x 10 7 ergs). We shall generally take it as 42,000,000 ergs. The mechan- ical equivalent of 1 B. T. U. is 778 foot pounds. 187. The conservation of energy. We are now in a position to state the law of all machines in its most general form, that is, in such a way as to include even the cases where friction 156 WORK AND HEAT ENERGY is present. It is : The work done by the acting force is equal to the sum of the kinetic and potential energies stored up plus the mechanical equivalent of the heat developed. In other words, whenever energy is expended on a machine or device of any kind, an exactly equal amount of energy always appears either as useful work or as heat. The useful work may be represented in the potential energy of a lifted mass, as when water is pumped up to a reservoir; or in the kinetic energy of a moving mass, as when a stone is thrown from a sling; or in the potential energies of molecules whose posi- tions with reference to one another have been changed, as when a spring has been bent; or in the molecular potential energy of chemically separated atoms, as when an electric current separates a compound substance. The ivasted work always appears in the form of increased molecular motion, that is, in the form of heat. This important generalization has received the name of the Principle of the Conservation of Energy. It may be stated thus: Energy may be transformed, but it can never be created or destroyed. 188. Perpetual motion. In all ages there have been men who have spent their lives in trying to invent a machine out of which work could be continually obtained, without the ex- penditure of an equivalent amount of work upon it. Such devices are called perpetual-motion machines. The possibility of the existence of such a device is absolutely denied by the statement of the principle of the conservation of energy. For only in case there is no heat developed, that is, in case there are no frictional losses, can the w r ork taken out be equal to the work put in, and in no case can it be greater. Since, in fact, there are always some frictional losses, the principle of the conservation of energy asserts that it is impossible to make a machine which will keep itself running forever, even though it does no useful work ; for no matter how much kinetic or poten- tial energy is imparted to the machine to begin with, there MECHANICAL EQUIVALENT OF HEAT 157 must always be a continual drain upon this energy to overcome frictional resistances, so that as soon as the wasted work has become equal to the initial energy, the machine must stop. The principle of the conservation of energy has now gained universal recognition and has taken its place as the corner stone of all physical science. 189. Transformations of energy in a power plant. The transforma- tions of energy which take place in any power plant, such as that at Niagara, are as follows : The energy first exists as the potential energy of the water at the top of the falls. This is transformed in the turbine pits into the kinetic energy of the rotating wheels. These turbines drive dynamos in which there is a transformation into the energy of electric currents. These currents travel on wires as far as Syracuse, 150 miles away, where they run street cars and other forms of motors. The principle of conservation of energy asserts that the work which gravity did upon the water in causing it to descend from the top to the bottom of the turbine pits is exactly equal to the work done by all the motors, plus the heat developed in all the wires and bearings and in the eddy currents in the water. Let us next consider where the water at the top of the falls obtained its potential energy. Water is being continually evaporated at the sur- face of the ocean by the sun's heat. This heat imparts sufficient kinetic energy to the molecules to enable them to break away from the attrac- tions of their fellows and to rise above the surface in the form of vapor- The lifted vapor is carried by winds over the continents and precipitated in the form of rain or snow. Thus the potential energy of the water above the falls at Niagara is simply transformed heat energy of the sun. If in this way we analyze any available source of energy at man's disposal, we find in almost every case that it is directly traceable to the sun's heat as its source. Thus, the energy contained in coal is simply the energy of separation of the oxygen and carbon which were separated in the processes of growth. This separation was effected by the sun's rays. The earth is continually receiving energy from the sun at the rate of 342,000,000,000,000 horse power, or about a quarter of a million horse power per inhabitant. We can form some conception of the enormous amount of energy that the sun radiates in the form of heat by reflecting that the amount received by the earth is not more than ,0 0,0 158 WORK AND HEAT ENERGY of the total given out. Of the amount received by the earth not more than 1 1 OQ part is stored up in animal and vegetable life and lifted water. This is practically all of the energy which is available on the earth for man's use. QUESTIONS AND PROBLEMS 1. Show that the energy of a waterfall is merely transformed solar energy. 2. Analyze the transformations of energy which occur when a bullet is fired vertically upward. 3. Meteorites are small, cold bodies moving about in space. Why do they become luminous when they enter the earth's atmosphere ? 4. The Niagara Falls are 1GO ft. high. How much warmer is the water at the bottom of the falls than at the top? 5. How many B. T. U. are required to warm 10 Ib. of water from freezing to boiling ? 6. Two and a half gallons of water ( = 20 Ib.) were warmed from, 68F. to 212F. If the heat energy put into the water could all have beei* made to do useful work, how high could 10 tons of coal have been hoisted ? SPECIFIC HEAT 190. Definition of specific heat. When we experiment upon different substances, we find that it requires wholly different amounts of heat energy to produce in one gram of mass one degree of change in temperature. Let 100 g. of lead shot be placed in one test tube, 100 g. of bits of iron wire in another, and 100 g. of aluminium wire in a third. Let them all be placed in a pail of boiling water for ten or fifteen minutes, care being taken not to allow any of the water to enter any of the tubes. Let three small vessels be provided, each Of which contains 100 g. oi: water at the temperature of the room. Let the heated shot be poured into the first beaker, and after thorough stirring let the rise in the temperature of the water be noted. Let the same be done with the other metals. The aluminium will be found to raise the temperature about twice as much as the iron, and the iron about three times as much as the lead. Hence, since the three metals have cooled through approximately the same number of degrees, we must conclude that about six times as much heat has passed out of the aluminium as out of the lead; that is; each' gram of aluminium in cooling 1C. gives out about six times as many calories as a gram of lead. SPECIFIC HEAT 159 The number of calories taken up by 1 gram of a substance when its temperature rises through 1 (7., or given up when it falls through 1 (7., is called the specific heat of that substance. It will be seen from this definition, and the definition of the calorie, that the specific heat of water is 1. 191. Determination of specific heat by the method of mix- tures. The preceding experiments illustrate a method for measuring accurately the specific heats of different substances ; for, in accordance with the principle of the conservation of energy, when hot and cold bodies are mixed, as in these ex- periments, so that heat energy passes from one to the other, the gain in the heat energy of one must be just equal to the loss in the heat energy of the other. This method is by far the most common one for determin- ing the specific heats of substances. It is known as the method of mixtures. Suppose, to take an actual case, that the initial temperature of the shot used in 190 was 95 C. and that of the water 19.7, and that, after mixing, the temperature of the water and shot was 22. Then, since 100 g. of water has had its temperature raised through 22 19.7 = 2.3, we know that 230 calories of heat have entered the water. Since the temperature of the shot fell through 95 22 = 73, the number of calories given up by the 100 g. of shot in falling 1 was ^f- = 3.15. Hence the specific heat of lead, that is, the number of calories of heat given Q -| T up by 1 gram of lead when its temperature falls 1C., is - = .0315. Or, again, we may work out the problem algebraically as follows : Let x equal the specific heat of lead. Then the number of calories which come out of the shot is its mass times its specific heat times its change in temperature,ihat is, 100 X a: x (95 22) ; and, similarly, the number which enter the water is the same, namely, 100 x 1 x (22 19.7). Hence we have 100 (95 - 22) x = 100 (22 - 19.7), or x = .0315. By experiments of this sort the specific heats of some of the common substances have been found to be as follows: 160 WOEK AND HEAT ENEKGY TABLE OF SPECIFIC HEATS Aluminium 218 Iron 113 Brass 094 Lead 0315 Copper 095 Mercury 0333 Glass 2 Platinum 032 Gold 0316 Silver 0568 Ice . .504 Zinc .0935 QUESTIONS AND PROBLEMS 1. A barrelful of tepid water, when poured into a snowdrift, melts much more snow than a cupful of boiling water does. Which has the greater quantity of heat ? 2. Why is a liter of hot water a better foot warmer than an equal volume of any substance in the preceding table ? 3. The specific heat of water is much greater than that of any other liquid or of any solid. Explain how this accounts for the fact that an island in mid-ocean undergoes less extremes of temperature than an inland region. 4. How many calories are required to heat a laundry iron weighing 3 kg. from 20 C. to 130 C.? 5. How many B. T. U. are required to warm a 6-pound laundry iron from 75 F. to 250 F. ? 6. If 100 g. of mercury at 95 C. are mixed with 100 g. of water at 15 C., and if the resulting temperature is 17.6 C., what is the specific heat of mercury? 7. Tf 200 g. of water at 80 C. are mixed with 100 g. of water at 10 C., what will be the temperature of the mixture? (Let x equal the final temperature ; then 100 (x 10) calories are gained by the cold water, while 200 (80 a:) calories are lost by the hot water.) 8. What temperature will result if 400 g. of aluminium at 100 C. are placed in 500 g. of water at 20 C. ? 9. Eight pounds of water were placed in a copper kettle weighing 2.5 Ib. How many B. T. U. are required to heat the water and the kettle from 70 F. to 212 F.? If 4.3 cu. ft. of gas was used to do this, and if each cubic foot of gas on being burned yields 625 B. T. U., what is the efficiency of the heating apparatus? 10. If a solid steel projectile were shot with a velocity of 1000 m. (3048 ft.) per second against an impenetrable steel target, and all the heat generated were to go toward raising the temperature of the projectile, what would be the amount of the increase ? CHAPTER X CHANGE OF STATE FUSION * 192. Heat Of fusion. If on a cold day in winter a quantity of snow is brought in from out of doors, where the temperature is below 0C. and placed over a source of heat, a thermometer plunged into the snow will be found to rise slowly until the temperature reaches 0C., when it will become stationary and remain so during all the time the snow is melting, provided only that the contents of the vessel are continu- ously and vigorously stirred. As soon as the snow is all melted, the temperature will begin to rise again. Since the temperature of ice at C. is the same as the temperature of water at C., it is evident from this experiment that when ice is being changed to water, the entrance of heat energy into it does not produce any change in the average kinetic energy of its molecules. This energy must therefore all be expended in pulling apart the molecules of the crystals of which the ice is composed, and thus reducing it to a form in which the molecules are held together less intimately, that is, to the liquid form. In other words, the energy which existed in the flame as the kinetic energy of molecular motion has been transformed, upon passage into the melting solid, into the potential energy of molecules which have been pulled apart against the force of their mutual attraction. The number *This subject should be preceded by a laboratory exercise on the curve of cooling through the point of fusion, and followed by a determination of the heat of fusion of- ice. See, for example, Experiments 21 and 22 of the authors 1 Manual. 161 162 CHANGE OF STATE of calories of heat energy required to melt one gram of any substance without producing any change in its temperature is called the heat of fusion of that substance. 193. Numerical value of heat of fusion of ice. Since it is found to require about 80 times as long for a given flame to melt a quantity of snow as to raise the melted snow through 1 C., we conclude that it requires about 80 calories of heat to melt 1 g. of snow or ice. This constant is, however, much more accurately determined by the method of mixtures. Thus, suppose that a piece of ice weighing 131 g. is dropped into 500 g. of water at 40 C., and suppose that after the ice is all melted the temperature of the mixture is found to bo 15 C. The number of calories which have come out of the water is 500 x (40 - 15) = 12,500. But 131 x 15 = 1965 calories of this heat must have been used in raising the ice from C. to 15 C. after the ice, by melting, became water at 0. The remainder of the heat, namely, 12,500 1965 = 10,535, must have been used in melting the 131 g. of ice. Hence the number of calories required to melt 1 g. of ice is 1 " j \ 5 = 80.4. To state the problem algebraically, let x = the heat of fusion of ice. Then we have 131 x + 1965 - 12,500 ; that is, x = 80.4. According to the most careful determinations the heat of fusion of ice is 80.0 calories. 194. Energy transformation in fusion. The heat energy that goes into a body to change it from the solid state to the liquid state no longer exists as heat within the liquid. It has ceased to exist as heat energy at all, having been transformed into molecular potential energy ; that is, the heat which disap- pears represents the work that was done in effecting the change of state ) and it is, therefore, the exact equivalent of the potential energy gained by the rearranged molecules. This is strictly in accord with the law of conservation of energy. FUSION 163 195. Heat given out when water freezes. Let snow and salt be added to a beaker of water until the temperature of the liquid mixture is as low as 10 C. or 12 C. Then let a test tube containing a ther- mometer and a quantity of pure water be thrust into the cold solution. If the thermometer is kept very quiet, the temperature of the water in the test tube will fall four or five or even ten degrees below C. without producing solidification. But as soon as the thermometer is stirred, or a small crystal of ice is dropped into the neck of the tube, the ice crystals will form with great suddenness, and at the same time the thermometer will rise to C., where it will remain until all the water is frozen. The experiment shows in a very striking way that the proc- ess of freezing is a heat-evolving process. This was to have been expected from the principle of the conservation of energy ; for since it takes 80 calories of heat energy to turn a gram of ice at C. into water at (7., this amount of energy must reappear when the water turns back to ice. 196. Use made of energy transformations in melting and freezing. A refrigerator (Fig. 170) is a box constructed with double walls so as to make it difficult for heat to pass in from the outside. Ice is kept in the upper part of one compartment so as to cool the air at the top, which, because of its greater density when cool, settles and causes a circulation as indicated by the arrows. To melt each gram of ice 80 calories must be taken from the air and food within the refrigera- tor. If the ice did not melt, it would be worthless for use in refrigerators. The heat given off by the freezing of water is often turned to practical account ; for example, tubs of water are sometimes placed in vegetable cellars to prevent the vegetables from freezing. The effectiveness of this procedure is due to the fact that the temperature at which the vegetables freeze is slightly FIG. 170. A refrigerator 164 CHANGE OF STATE lower than C. As the temperature of the cellar falls the water therefore begins to freeze first, and in so doing evolves enough heat to prevent the temperature of the room from falling as far below C. as it otherwise would. It is partly because of the heat evolved by the freezing of large bodies of water that the temperature never falls so low in the vicinity of large lakes as it does in inland localities. , 197. Melting points of crystalline substances. If a piece of ice is placed in a vessel of boiling water for an instant and then removed and wiped, it will not be found to be in the slightest degree warmer than a piece of ice which has not been exposed to the heat of the warm water. The melting point of ice is therefore a perfectly fixed, definite temperature, above which the ice can never be raised so long as it remains ice, no matter how fast heat is applied to it. All crystalline sub- stances are found to behave exactly like ice in this respect, each substance of this class having its characteristic melting point. The following table gives the melting points of some of the commoner crystalline substances : Mercury . Ice . Benzine . -39C. 0C. 7C. Sulphur . . Tin . . . Lead . 114 C. 233C. 330C. Silver . . Copper Cast iron . . 954C. . 1100C. . 1200 C. Acetic acid . 17C. Zinc . . . 433C. Platinum . . 1775C. Paraffin . . 54C. Aluminium . 650C. Iridium . . 1950C. We may summarize the experiments upon melting points of crystalline substances in the two following laws : 1. The temperatures of solidification and fusion are the same. 2. The temperature of the melting or solidifying substance remains constant from the moment at which melting or solidi- fication begins until the process is completed. 198. Fusion of noncrystalline, or amorphous, substances. Let the end of a glass rod be held in a Bunsen flame. Instead of changing suddenly from the solid to the liquid state, it will gradually grow softer FUSION 165 and softer until, if the rod is not too thick and the flame is sufficiently hot, a drop of molten glass will finally fall from the end of the rod. If the temperature of the rod had been measured during this process, it would have been found to be continually rising. This behavior, so completely unlike that of crystalline sub- stances, is characteristic of tar, wax, resin, glue, gutta-percha, alcohol, carbon, and in general of all amorphous substances. Such substances cannot be said to have any definite melting points at all, for they pass through all stages of viscosity both in melting and in solidifying. It is in virtue of this property that glass and other similar substances can be heated to soft- ness and then molded or rolled into any desired shape. 199. Change of volume on solidifying. One has only to reflect that ice floats, or that bottles or crocks of water burst when they freeze, in order to know that water expands upon solidifying. In fact, 1 cubic foot of water becomes 1.09 cubic feet of ice, thus expanding more than one twelfth of its initial volume when it freezes. This may seem strange in view of the fact that the molecules are certainly more closely knit together in the solid than in the liquid state ; but the strange- ness disappears when we reflect that the molecules of water in freezing group themselves into crystals, and that this operation presumably leaves comparatively large free spaces between different crystals, so that, although groups of individual mole- cules are more closely joined than before, the total volume occupied by the whole assemblage of molecules is greater. But the great majority of crystalline substances contract upon solidifying and expand upon liquefying. Water, anti- mony, bismuth, cast iron, and a few alloys containing antimony or bismuth are the chief exceptions. It is only from substances which expand, or which change in volume very little on solidi- fying, that sharp castings can be made ; for it is clear that contracting substances cannot retain the shape of the mold. It is for this reason that gold and silver coins must be stamped 166 CHANGE OF STATE rather than cast. Any metal from which type is to be cast must be one which expands upon solidifying, for it need scarcely be said that perfectly sharp outlines are indispensable to good type. Ordinary type metal is an alloy of lead, anti- mony, and copper, which fulfills these requirements. 200. Effect of the expansion which water undergoes on freezing. If water were not unlike most substances in that it expands on freezing, many, if not all, of the forms of life which now exist on the earth would be impossible ; for in winter the ice would sink in ponds and lakes as fast as it froze, and soon our rivers, lakes, and perhaps our oceans also would become solid ice. The force exerted by the expansion of freezing water is very great. Steel bombs have been burst by filling them with water and exposing them on cold winter nights. One of the chief agents in the disintegration of rocks is the freezing and conse- quent expansion of water which has percolated into them. 201. Pressure lowers the melting point of substances which expand on solidifying. Since the outside pressure acting on the surface of a body tends to prevent its expansion, we should expect that any increase in the outside pressure would tend to prevent the solidification of substances which expand upon freezing. It ought therefore to require a lower temperature to freeze ice under a pressure of two atmospheres than under a pressure of one. Careful experiments have verified this conclusion and have shown that the melting point of ice is lowered .0075 C. for an increase of one atmosphere in the outside pressure. Although this lowering is so small a quantity, its existence may be shown as follows : Let two pieces of ice be pressed firmly together beneath the surface of a vessel full of warm water. When taken out they will be found to be frozen together, in spite of the fact that they have been immersed in a medium much warmer than the freezing point of water. The explanation is as follows : FUSION 167 At the points of contact the pressure reduces the freezing point of the ice below C., and hence it rnelts and gives rise to a thin film of water the temperature of which is slightly below C. When this pressure is released, the film of water at once freezes, for its tempera- ture is below the freezing point corresponding to ordinary atmospheric pressure. The same phenomenon may be even more strikingly illus- trated by the following experiment : Let two weights of from 5 to 10 kg. be hung by a wire over a block of ice as in Fig. 171. In half an hour or less the wire will be found to have cut completely through the block, leaving the ice, however, as solid as at first. The explanation is as follows : Just below the wire the ice melts be- cause of the pressure ; as the wire sinks through the layer of water thus formed, the pressure on the water is relieved and it immediately freezes again above the wire. Geologists believe that the con- tinuous flow of glaciers is partly ElG> 171> Reg elation due to the fact that the ice melts at points where the pressures become large, and freezes again when these pressures are relieved. This process of melting under pressure and freezing again as soon as the pressure is relieved is known as regelation. Substances ivhich expand on solidifying have their melting points lowered by pressure, and those which contract on solidify- ing have their melting points raised by pressure. QUESTIONS AND PROBLEMS 1. What is the meaning of the statement that the heat of fusion of mercury is 2.8 ? 2. Explain how the presence of ice keeps the interior of a refrig- erator from becoming warm. 3. How many times as much heat is required to melt any piece of ice as to warm the resulting water 1 C. ? 1 F. ? How many B. T. U. are required to melt 1 Ib. of ice? How many foot pounds of energy are required to do the work of melting 1 Ib. of ice ? 168 CHANGE OF STATE 4. If the heat of fusion of ice were 40 instead of 80, how would this affect the quantity of ice that would have to be bought for the refriger- ator during the summer? 5. Five pounds of ice melted in 1 hr. in an unopened refrigerator. How many B. T. U. caine through the walls of the refrigerator in the hour? 6. Just what will occur if 1000 calories be applied to 20 g. of ice at 0C.? 7. How many grains of ice must be put into 200 g. of water at 40 C. to lower the temperature 10 C. ? 8. How many grams of ice must be put into 500 g. of water at 50 C. to lower the temperature to 10 C.? 9. Why will snow pack into a snowball if the snow is melting, but not if it is much below C. ? EVAPORATION AND THE PROPERTIES OF VAPORS 202. Evaporation and temperature. If it is true that in- crease in temperature means increase in the mean velocity of molecular motion, then the number of molecules which chance in a given time to acquire the velocity necessary to carry them into the space above the liquid ought to increase as the temperature increases ; that is, evaporation ought to take place more rapidly at high temperatures than at low. Common observation teaches that this is true. Damp clothes become dry under a hot flatiron but not under a cold one ; the sidewalk dries more readily in the sun than in the shade ; we put wet objects near a hot stove or radiator when we wish them to dry quickly. 203. Evaporation of solids, sublimation. That the mole- cules of a solid substance are found in a vaporous condition above the surface of the solid, as well as above that of a liquid, is proved by the often-observed fact that ice and snow evaporate even though they are kept constantly below the freezing point. Thus, wet clothes dry in winter after freezing. An even better proof is the fact that the odor of camphor can be detected many feet away from the camphor EVAPORATION 169 crystals. The evaporation of solids may be rendered visible by the following striking experiment: Let a few crystals of iodine be placed on a watch glass and heated gently with a Bunsen flame. The visible vapor of iodine will appear above the crystals, though none of the liquid is formed. A great many substances at high temperatures pass from the solid to the gaseous condition without passing through the liquid state. The vaporization of a solid is called sublimation. 204. Saturated vapor. If a liquid is placed in an open vessel, there ought to be no limit to the number of molecules which can be lost by evaporation, for as fast as the molecules emerge from the liquid they are carried away by air currents. As a matter of fact, experience teaches that water left in an open dish does waste away until the dish is completely dry. But suppose that the liquid is evaporating into a closed space, such as that shown in Fig. 172. Since the molecules which leave the liquid cannot escape from the space S, it is clear that as time goes on the number of molecules which have passed off from the liquid into this space must contin- ually increase ; in other words, the density of the vapor in S must grow greater and greater. But there is an absolutely definite limit to the density which the vapor can attain ; for as soon as it reaches a certain value, depending on the tem- perature and on the nature of the liquid, the number of molecules returning per second to the liquid surface will be exactly equal to the number escaping. The vapor is then said to be saturated. If the density of the vapor is lessened temporarily by in- creasing the size of the vessel S, more molecules will escape from the liquid per second than return to it, until the density of the vapor has regained its original value. 170 CHANGE OF STATE If, on the other hand, the density of the vapor has been increased by compressing it, more molecules return to the liquid per second than escape, and the density of the vapor falls quickly to its saturated value. We learn, then, that the density of the saturated vapor of a liquid depends on the tem- perature alone and cannot be affected by changes in volume. 205. Pressure of a saturated vapor. Just as a gas exerts a pressure against the walls of the containing vessel by the blows of its moving molecules, so also does a confined vapor. But at any given temperature the density of a saturated vapor can have only a definite value ; that is, there can be only a definite number of molecules per cubic centimeter. It fol- lows, therefore, that just as at any temperature the saturated vapor can have only one density, so also it can have only one pressure. This pressure is called the pressure of the saturated vapor correspond- ing to the given temperature. Let two Torricelli tubes be set up as in Fig. 173, and with the aid of a curved pipette (Fig. 173) let a drop of ether be introduced into the bottom of tube 1. This drop will at once rise to the top, and a portion of it will evaporate into the vacuum which exists above the mer- cury. The pressure of this vapor will push down the mercury column, and the number of centimeters of this depres- sion will of course be a measure of the pressure of the vapor. It will be observed that the mercury will fall almost in- stantly to the lowest level which it will ever reach, a fact which indi- cates that it takes but a very short time for the condition of saturation to be attained. The pressure of the saturated ether vapor at the temperature of the room will be found to be as much as 40 centimeters. FIG. 173. Vapor pressure of a saturated vapor EVAPORATION 1T1 Let a Bunsen flame be passed quickly across the tubes of Fig. 173^ near the upper level of the mercury. The vapor pressure will increase rapidly in tube 1, as shown by the fall of the mercury column. The experiment proves that both the pressure and the density of a saturated vapor increase rapidly with the tem- perature. This was to have been expected from our theory, for increasing the temperature of the liquid increases the mean velocity of its molecules and hence increases the num- ber which attain each second the velocity necessary for escape. How rapidly the density and pressure of saturated water vapor increase with temperature may be seen from the fol- lowing table: TABLE OF CONSTANTS OF SATURATED ABATER VAPOR The table shows the pressure P, in millimeters of mercury, and the density D of aqueous vapor saturated at temperatures t C. t. P. D. t. P. D. t. P. D. - 10 2.2 .0000023 4 6.1 .0000064 18 15.3 .0000152 - 9 2.3 .0000025 5 6.5 .0000068 19 16.3 .0000162 - 8 2.5 .0000027 6 7.0 .0000073 20 17.4 .0000172 - 7 2.7 .0000029 7 7.5 .0000077 21 18.5 .0000182 - 6 2.9 .0000032 8 8.0 .0000082 22 19.6 .0000193 - 5 3.2 .0000034 9 8.5 .0000087 23 20.9 .0000204 40 3.4 .0000037 10 9.1 .0000093 24 22.2 .0000216 - 3 3.7 ,0000040 11 9.8 .0000100 25 23.5 .0000229 - 2 3.9 .0000042 12 10.4 .0000106 26 25.0 .0000242 JO 4.2 .0000045 13 11.1 .0000112 27 26.5 .0000256 4.6 .0000049 14 11.9 .0000120 28 28.1 .0000270 1 4.9 .0000052 15 12.7 .0000128 30 31.5 .0000301 2 5.3 .0000056 16 13.5 .0000135 35 41.8 .0000393 3 5.7 .0000060 17 14.4 .0000144 40 54.9 .0000509 206. The influence of air on evaporation. We observed that when ether was inserted into a Torricelli tube the mercury fell very suddenly to its final position, showing that in a vacuum the condition of saturation is reached almost instantly. 172 CHANGE OF STATE This was to have been expected from the great velocities which we found the molecules of gases and vapors to possess. Let air be introduced into tube 2 (Fig. 173) until the mercury col- umn stands at a height of from 45 to 55 cm. Measure the height of the mercury column. In order to see what effect the presence of air has upon evaporation, let a drop of ether be introduced into the tube. The mercury will not be found to sink instantly to its final level as it did before ; but although it will fall rapidly at first, it will continue to fall slowly for several hours. At the end of a day, if the temperature has remained constant, it will show a depression which indicates a vapor pressure of the ether just as great as that existing in a tube which contains no air. The experiment leads, then, to the rather remarkable con- clusion that just as much liquid will evaporate into a space which is already full of air as into a vacuum. The air has no effect except to retard greatly the rate of evaporation. 207. Explanation of the retarding influence of air on evapo- ration. This retarding influence of air on evaporation is easily explained by the kinetic theory ; for while in a vacuum the molecules which emerge from the surface fly at once to the top of the vessel, when air is present the escaping molecules collide with the air molecules before they have gone any appreciable distance away from the surface (probably less than .00001 centimeter), and only work their way up to the top after an almost infinite number of collisions. Thus, while the space immediately above the liquid may become saturated very quickly, it requires a long time for this condition of saturation to reach the top of the vessel. QUESTIONS AND PROBLEMS 1. Account for the evaporation of naphthaline moth balls at ordi- nary room temperatures. 2. Why do clothes dry more quickly on a windy day than on a quiet day? 3. If the inside of a barometer tube is wet when it is filled with mer- cury, will the height of the mercury be the same as in a dry tube ? HYGROMETRY 173 4. How many grams of water will evaporate at 20 C. into a closed room 18 x 20 x 4 m. ? (See table, p. 171.) 5. At a temperature of 15 C. what will be the error in the baro- metric height indicated by a barometer which contains moisture? 6. At 20 C. how great was the error in reading due to the presence of water vapor in Otto von Guericke's barometer ? HYGROMETRY, OR THE STUDY OF MOISTURE CONDITIONS IN THE ATMOSPHERE * 208. Condensation of water vapor from the air. Were it not for the retarding influence of air upon evaporation we should be obliged to live in an atmosphere which would be always completely saturated with water vapor, for the evaporation from oceans, lakes, and rivers would almost instantly saturate all the regions of the earth. This condition one in which moist clothes would never dry, and in which all objects would be perpetually soaked in moisture would be exceedingly uncomfortable if not altogether unendurable. But on account of the slowness with which, as the last ex- periment showed, evaporation into air takes place, the water vapor which always exists in the atmosphere is usually far from saturated, even in the immediate neighborhood of lakes and rivers. Since, however, the amount of vapor which is necessary to produce saturation rapidly decreases with a fall in temperature, if the temperature decreases continually in some unsaturated locality it is clear that a point must soon be reached at which the amount of vapor already existing in a cubic centimeter of the atmosphere is the amount correspond- ing to saturation. Then, if the temperature still continues to fall, the vapor must begin to condense. Whether it condenses as dew or cloud or fog or rain will depend upon how and where the cooling takes place. * It is recommended that this subject be preceded by a laboratory deter- mination of dew point, humidity, etc. See, for example, Experiment 10 of the authors' Manual. 174 CHANGE OF STATE 209. The formation of dew and frost. If the cooling is due to the natural radiation of heat from the earth at night after the sun's warmth is withdrawn, the atmosphere itself does not fall in temperature nearly as rapidly as do solid objects on the earth, such as blades of grass, trees, stones, etc. The layers of air which come into immediate contact with these cooled bodies are themselves cooled, and as they thus reach a temperature at which the amount of moisture which they already contain is in a saturated condition, they begin to deposit this mois- ture, in the form of dew or frost, upon the cold objects. The drops of moisture which collect on an ice pitcher in summer illustrate perfectly the formation of dew. If condensation takes place upon a surface colder than the freezing temper- ature, frost is formed, as is observed, for instance, on grass and on windowpanes. 210. The formation of fog. If the cooling at night is so great as not only to bring the grass and trees below the tem- perature at which the vapor in the air in contact with them is in a state of saturation, but also to lower the whole body of air near the earth below this temperature, then the condensa- tion takes place not only on the solid objects but also on dust particles suspended in the atmosphere. This constitutes a fog. 211. The formation of clouds, rain, sleet, hail, and snow. When the cooling of the atmosphere takes place at some dis- tance above the earth's surface, as when a warm current of air enters a cold region, if the resultant temperature is below that at which the amount of moisture already in the air is sufficient to produce saturation, this excessive moisture im- mediately condenses about floating dust particles and forms a cloud. If the cooling is sufficient to free a considerable amount of moisture, the drops become large and fall as rain. If this falling rain freezes before it reaches the ground, it is called sleet. If the temperature at which condensation begins is be- low freezing, the condensing moisture forms into snouflakes. HYGROMETRY 175 When the violent air currents . which accompany thunder- storms cany the condensed moisture up and down several times through alternate regions of snow and rain, hailstones are formed. 212. The dew point. The temperature to which the atmosphere must be cooled in order that condensation of the water vapor within it may begin is called the dew point." This temperature may be found by partly filling with water a brightly polished vessel of 200 or 300 cubic centimeters capacity and dropping into it little pieces of ice, stirring thoroughly at the same time with a thermometer. The dew point is the temperature indicated by the thermometer at the instant a film of moisture ap- pears upon the polished surface. In winter the dew point is usually below freezing, and it will there- fore be necessary to add salt to the ice and water in order to make the film appear. The experi- ment may be performed equally well by bubbling a current of air through ether contained in a polished tube (Fig. 174). 213. Humidity of the atmosphere. From the dew point and table given in 205, p. 171, we can easily find what is com- monly known as the relative humidity or the degree of satura- tion of the atmosphere. Relative humidity is defined as the ratio between the amount of moisture per cubic centimeter actu- ally present in the air and the amount which ivould be present if the air were completely saturated. This is precisely the same as the ratio between the pressure which the water vapor pres- ent in the air exerts and the pressure which it would exert if it were present in sufficient quantity to be in the saturated condition. An example will make clear the method of find- ing the relative humidity. FIG. 174. Apparatus for deter- mining dew point 176 CHANGE OF STATE Suppose that the dew point were found to lye 15 C. on a day on whicli the temperature of the room was 25 C. The amount of moisture actu- ally present in the air then saturates it at 15 C. We see from the P column in the table that the pressure of saturated vapor at 15 C. is 12.7 millimeters. This is, then, the pressure exerted by the vapor in the air at the time of our experiment. Running down the table, we see that the amount of moisture required to produce saturation at the tempera- ture of the room, that is, at 25, would exert a pressure of 23.5 millimeters. Hence at the time of the experiment the air contains 12.7/23.5, or .54, as much water vapor as it might hold. AVe say, therefore, that the air is 54% saturated, or that the relative humidity is 54%. 214. Practical value of humidity determinations. From hu- midity determinations it is possible to obtain much information regarding the likelihood of rain or frost. Such observations are continually made for this purpose at all meteorological stations. They are also made in greenhouses, to see that the air does not become too dry for the welfare of the plants, and in hospitals and public buildings and even in private dwellings, in order to insure the maintenance of hygienic liv- ing conditions. For the most healthful conditions the relative humidity should be kept at from 50% to 60%. Low relative humidity in the home causes discomfort and colds, and leads to waste of fuel estimated at from 10% to 25%. The average home heated to 72 F. by steam or hot water is estimated by health authorities to have a relative hu- midity of 30%, and even as little as 25% with hot-air heat- ing. This is less than the average humidity of extensive desert regions. Higher humidity in the home would diminish the cooling effect due to rapid evaporation of the perspiration from the body, and would make us feel comfortable if a lower temperature were maintained (see 215). 215. Cooling effect of evaporation. Let three shallow dishes be partly filled, the first with water, the second with alcohol, and the third with ether, the bottles from which these liquids are obtained having stood in the room long enough to acquire its temperature. Let three students HYGKOMETKY 177 carefully read as maiiy thermometers, first before their bulbs have been immersed in the respective liquids and then after. In every case the temperature of the liquid in the shallow vessel will be found to be somewhat lower than the temperature of the air, the difference being greatest in the case of ether and least in the case of water. It appears from this experiment that an evaporating liquid assumes a temperature somewhat lower than its surroundings, and that the substances which evaporate the most readily assume the lowest temperatures. In dry, hot climates where ice is not readily obtained drink- ing water is frequently kept in canvas bags or unglazed earth- enware. The slow evaporation of the water from the outside of the porous container keeps the water within quite cool. Another way of establishing the same truth is to place a few drops of each of the above liquids in succession on the bulb of the arrange- ment shown in Fig. 143 and observe the rise of water in the stem ; or, more simply still, to place a few drops of each liquid on the back of the hand and notice that the order in which they evaporate namely, ether, alcohol, water is the order of greatest cooling. In twenty-four hours a healthy person perspires from a pint to a quart, while one who exercises violently may perspire a gallon in that time. 216. Explanation of the cooling effect of evaporation. The kinetic theory furnishes a simple explanation of the cooling effect of evaporation. We saw that, in accordance with this theory, evaporation means an escape from the surface of those molecules which have acquired velocities considerably above the average. But such a continual loss of the most rapidly moving molecules involves, of course, a continual diminution of the average velocity of the molecules left behind, and this means a decrease in the temperature of the liquid. Again, we should expect the amount of cooling to be pro- portional to the rate at which the liquid is losing molecules. Hence, of the three liquids studied, ether should cool most rapidly, since it evaporates most rapidly. 178 CHANGE OF STATE 217. Freezing by evaporation. In 206 it was shown that a liquid will evaporate much more quickly into a vacuum than into a space containing air. Hence, if we place a liquid under the receiver of an air pump and exhaust rapidly, we ought to expect a much greater fall in temperature than when the liquid evaporates into air. This conclusion may be strikingly verified as follows: Let a thin watch glass be filled with ether and placed upon a drop of cold water, preferably ice water, which rests upon a thin glass plate. Let the whole arrangement be placed underneath the receiver of an air pump and the air rapidly exhausted. After a few minutes of pumping the watch glass will be found frozen to the plate. By evaporating liquid helium in this way Professor Kara- erlingh Onnes of Leyden, in 1911, attained the lowest tem- perature that had ever been reached, namely, 271.3 C. ( - 456.3 F.), less than 2 C. above absolute zero. 218. Effect of air currents upon evaporation. Let four ther- mometer bulbs, the first of which is dry, the second wetted with water, the third with alcohol, and the fourth with ether, be rapidly fanned and their respective temperatures observed. The results will show that in all of the wetted thermometers the fanning will considerably augment the cooling, but the dry thermometer will be wholly unaffected. The reason why fanning thus facilitates evaporation, and therefore cooling, is that it removes the saturated layers of vapor which are in immediate contact with the liquid and re- places them by unsaturated layers into which new evaporation may at once take place. From the behavior of the dry-bulb thermometer, however, it will be seen that fanning produces cooling only when it can thus hasten evaporation. A dry body at the temperature of the room is not cooled in the slightest degree by blowing a current of air across it. 219. The wet- and dry-bulb hygrometer. In the wet- and dry-bulb hygrometer (Fig. 175) the principle of cooling by evaporation finds a very useful application. This instrument HYGKOMETRY 179 consists of two thermometers, the bulb of one of which is dry, while that of the other is kept continually moist by a wick dipping into a vessel of water. Unless the air is saturated the wet bulb indicates a lower tempera- ture than the dry one, for the reason that evaporation is continually taking place from its surface. How much lower will depend on how rapidly the evaporation proceeds, and this in turn will depend upon the relative humidity of the atmos- phere. Thus, in a completely saturated atmosphere no evaporation whatever takes place at the wet bulb, and it consequently indicates the same temperature as the dry one. By comparing the indications of this instrument with those of the dew-point hygrometer (Fig. 1 74) tables have been constructed which enable one to deter- mine at once from the readings of the two thermometers both the relative humidity and the dew point. On account of their convenience instruments of this sort are used almost exclusively in practical work. They are not very reliable unless the air is made to circulate about the wet bulb before the reading is taken. In scientific work this is always done. 220. Effect of increased surface upon evap- oration. Let a small test tube containing a few drops of water be dipped into a larger tube or a small glass containing ether, as in Fig. 176, and let a current of air be forced rapidly through the ether with an aspirator in the manner shown. The water within the tube will be frozen in a few minutes, if the aspirator is operated vigor- ously. The experiment works most successfully if the walls of the test tube are quite thin and the walls of the outer vessel fairly thick. Why ? FIG. 175. Wet- and dry- bulb hygrometer FIG. 176. Freezing water by the evap- oration of ether 180 CHANGE OF STATE The effect of passing bubbles through the ether is simply to increase enormously the evaporating surface, for the ether molecules which could before escape only at the upper sur- face can now escape into the air bubbles as well. 221. Factors affecting evaporation. The above results may be summarized as follows: The rate of evaporation depends (1) on the nature of the evaporating liquid ; (2) on the temperature of the evaporating liquid ; (3) on the degree of saturation of the space into which the evaporation takes place (4) on the density of the air or other gas above the evaporating surface ; (5) on the rapidity of the circulation of the air above the evaporating surface ; (6) on the extent of the exposed surface of the liquid. QUESTIONS AND PROBLEMS 1. Why do spectacle lenses become coated with mist on entering a warm house on a cold winter day? 2. Does dew "fall"? 3. Why are icebergs frequently surrounded with fog? 4. Dew will not usually collect on a pitcher of ice water standing in a warm room on a cold winter day. Explain. 5. The dew point in a room was found to be 8 C. What was the relative humidity if the temperature of the air was 10 C.? 20 C.? 30 C. ? (Consult table, p. 171.) 6. What weight of water is contained in a room 5x5x3 m. if the relative humidity is 60% and the temperature 20 C. ? (See table, p. 171.) 7. If a glass beaker and a porous earthenware vessel are filled with equal amounts of water at the same temperature, in the course of a few minutes a noticeable difference of temperature will exist between the two vessels. Which will be the cooler, and why ? Will the difference in temperature between the two vessels be greater in a dry or in a moist atmosphere ? 8. Why will an open, narrow-necked bottle containing ether not show as low a temperature as an open shallow dish containing the same amount of ether? 9. Why is the heat so oppressive on a very damp day in summer? 10. A morning fog generally disappears before noon. Explain the reason for its disappearance. BOILING 181 11. What becomes of the cloud which you see about a blowing loco- motive whistle ? Is it steam ? 12. Explain why it is necessary in winter to add moisture to the air of our homes to maintain proper relative humidity, but not necessary in the summer. 13. What factors affecting evaporation are illustrated by the follow- ing : (1) a wet handkerchief dries faster if spread out, (2) clothes dry best on a windy day, (3) clothes do not dry rapidly on a cold day, (4) clothes dry slowly on humid days ? Explain each fact. BOILING * 222. Heat of vaporization defined. The experiments per- formed in Chapter IV, Molecular Motions, led us to the conclusion that, at the free surface of any liquid, molecules frequently acquire velocities sufficiently high to enable them to lift themselves beyond the range of attraction of the mole- cules of the liquid and to pass off as free gaseous molecules into the space above. They taught us, further, that since it is only such molecules as have unusually high velocities which are able thus to escape, the average kinetic energy of the mole- cules left behind is continually diminished by this loss from the liquid of the most rapidly moving molecules, and conse- quently the temperature of an evaporating liquid constantly falls until the rate at which it is losing heat is equal to the rate at which it receives heat from outside sources. Evapora- tion, therefore, always takes place at the expense of the heat energy of the liquid. The number of calories of heat which dis- appear in the formation of one gram of vapor is called the heat of vaporization of the liquid. 223. Heat due to condensation. When molecules pass off from the surface of a liquid, they rise against the downward * It is recommended that this subject be accompanied by a laboratory determination of the boiling point of alcohol by the direct method and by the vapor-pressure method, and that it be followed by an experiment upon the fixed points of a thermometer and the change of boiling point with pressure. See, for example, Experiments 23 and 24 of the authors 1 Manual. 182 CHANGE OF STATE forces exerted upon them by the liquid, and in so doing ex change a part of their kinetic energy for the potential energy of separated molecules in precisely the same way in which a ball thrown upward from the earth exchanges its kinetic energy in rising for the potential energy which is represented by the separation of the ball from the earth. Similarly, just as when the ball falls back it regains in the descent all of the kinetic energy lost in the ascent, so when the molecules of the vapor reenter the liquid they must regain all of the kinetic energy which they lost when they passed out of the liquid. We may expect, therefore, that every gram of steam which condenses will generate in this process the same number of calories as was required to vaporize it. This is the prin- ciple of the steam heating of buildings, by which the heat energy that disappears in converting the water in the boilers into steam is generated again when the steam condenses to water within the radiators. 224. Measurement of heat of vaporization. To find accurately the number of calories expended in the vaporization, or released in the condensation, of a gram of water at 100 C., we pass steam rapidly for two or three minutes from an arrangement like that shown in Fig. 177 into a vessel containing, say, 500 g. of water. We ob- serve the initial and final temperatures and the initial and final weights of the water. If, for example, the gain in weight of the water is 16.5 g., we know that 16.5 g. of steam have been condensed. If the rise in temperature of the water is from 10 C. to 30 C., we know that 500 x (30- 10) = 10,000 calories of heat have entered the water. If x represents the number of calories given up by 1 g. of steam in condensing, then the total heat imparted to the water by the con- densation of the steam is 16.5 x calories. This condensed steam is at first water at 100 C., which is then cooled to 30 C. In this cooling FIG. 177. Heat of vaporiza- tion of water BOILING 183 process it gives up 16.5 x (100 30) = 1155 calories. Therefore, equat- ing the heat gained by the water to the heat lost by the steam, we have 10,000 = 16.5 x + 1155, or x = 536. This is the method usually employed for finding the heat of vaporization. The now accepted value of this constant is 536. 225. Boiling temperature defined. If a liquid is heated by means of a flame, it will be found that there is a certain tem- perature above which it cannot be raised, no matter how rapidly the heat is applied. This is the temperature which exists when bubbles of vapor form at the bottom of the vessel and rise to the surface, growing larger as they rise. This temperature is commonly called the boiling temperature. But a second and more exact definition of the boiling point may be given. It is clear that a bubble of vapor can exist within the liquid only when the pressure exerted by the vapor within the bubble is at least equal to the atmospheric pressure pushing down on the surface of the liquid; for if the pres- sure within the bubble were less than the outside pressure, the bubble would immediately collapse. Therefore the boiling point is the temperature at which the pressure of the saturated vapor first becomes equal to the pressure existing outside. 226. Variation of the boiling point with pressure. Since the pressure of a saturated vapor varies rapidly with the temper- ature, and since the boiling point has been defined as the temperature at which the pressure of the saturated vapor is equal to the outside pressure, it follows that the boiling point must vary as the outside pressure varies. Thus let a round-bottomed flask be half filled with water and boiled. After the boiling has continued for a few minutes, so that the steam has driven out most of the air from the flask, let a rubber stopper be inserted and the flask removed from the flame and inverted as shown in Fig. 178. The temperature will fall rapidly below the boiling point ; but if cold water is poured over the flask, the water will again begin to boil vigorously, for the cold water, by condensing the steam, lowers the 184 CHANGE OF STATE FIG. 178. Lowering the boiling point by dimin- ishing the pressure pressure within the flask, and thus enables the water to boil at a temper- ature lower than 100 C. The boiling will cease, howe^ 7 er, as soon as enough vapor is formed to restore the pressure. The operation may be repeated many times without reheating. <* At the city of Quito, Ecuador, water boils at 90 C. ; on the top of Mt. Blanc it boils at 84 C. ; and on Pikes Peak, at 89 C. On the other hand, in the boiler of a locomotive on which the gauge records a pressure of 250 pounds, as is frequently the case, the boiling point of the water is 208 C. (406 F.). Closed boilers provided with safety valves (see (7, Fig. 179) and known as digesters are used for more rapid cooking in mountainous regions. Indeed, a temperature only a few de- grees above 100 C. causes starch grains to burst open much more rapidly than does a temperature of 100 C. Large digesters are used in ex- tracting gelatin from bones and in reclaim- ing valuable fatty substances at garbage plants. In the cold-pack method of pre- serving fruits and vegetables the final sterilizing is done by placing the jars or cans in closed boilers known as steam- FIG. 179. A closed boiler for family use pressure canners.* 227. Evaporation and boiling. The only essential difference between evaporation and boiling is that the former consists in the passage of molecules into the vaporous condition from the free surface only, while the latter consists in the passage of the molecules into the vaporous condition both at the free surface and at * Farmers' 1 Bulletin No. 839, on steam-pressure canning, may be obtained from the United States Department of Agriculture, Washington, D. C. BOILING 185 the surface of bubbles which exist within the body of the liquid. The only reason why vaporization takes place so much more rapidly at the boiling temperature than just below it is that the evaporating surface is enormously increased as soon as the bubbles form. The reason why the temperature cannot be raised above the boiling point is that the surface always increases, on account of the bubbles, to just such an extent that the loss of heat because of evaporation is exactly equal to the heat received from the fire. 228. Distillation. Let water holding in solution some aniline dye be boiled in B (Fig. 180). The vapor of the liquid will pass into the tube T, where it will be condensed by the cold water which is kept in continual circulation through the jacket /. The condensed water col- lected in P will be seen to be free from all traces of the color of the dissolved aniline. We learn, then, that when solids are dissolved in liquids, the vapor which rises from the solu- tion contains none of the dissolved substance. Sometimes it is the pure liquid in P which is desired, as in the manufacture of alcohol, and sometimes the solid which remains in B, as in the manufacture of sugar. In the white-sugar industry it is necessary that the evaporation take place at a low temperature, so that the sugar may not be scorched. Hence the boiler is kept par- tially exhausted by means of an air pump, thus enabling the solution to boil at considerably reduced temperatures. 229. Fractional distillation. When bo'th of the constituents of a solution are volatile, as in the case of a mixture of alcohol and water, the vapor of both will rise from the liquid. But the one which has the lower boiling point, that is, the higher FIG. 180. Distillation 186 CHANGE OF STATE vapor pressure, will predominate. Hence, if we have in B (Fig. 180) a solution consisting of 50% alcohol and 50% water, it is clear that we can obtain in P, by evaporating and condensing, a solution containing .a much larger percent- age of alcohol. By repeating this operation a number of times we can increase the purity of the alcohol. This process is called fractional distillation. The boiling point of the mixture lies between the boiling points of alcohol and water, being higher the greater the percentage of water in the solution. Gasoline and kerosene are separated from crude oil, and different grades of gasoline are separated from each other by fractional distillation. QUESTIONS AND PROBLEMS 1. A fall of 1 C. in the boiling point is caused by rising 960 ft. How hot is boiling water at Denver, 5000 ft. above sea level? 2. How may we obtain pure drinking water from sea water ? 3. After water has been brought to a boil, will eggs become hard any quicker when the flame is high than when it is low ? 4.. The hot water which leaves a steam radiator may be as hot as the steam which entered it. How, then, has the room been warmed ? 5. In a vessel of water which is being heated fine bubbles rise long before the boiling point is reached. Why is this so ? How can you dis- tinguish between this phenomenon and boiling ? 6. When water is boiled in a deep vessel, it will be noticed that the bubbles rapidly increase in size as they approach the surface. Give two reasons for this. 7. Why are burns caused by steam so much more severe than burns caused by hot water of the same temperature ? 8. How many times as much heat is required to convert any body of boiling water into steam as to warm an equal weight of water 1 C.? 9. How many B. T. U. are liberated within a radiator when 10 Ib. of steam condense there ? 10. In a certain radiator 2 kg. of steam at 100 C. condensed to water in 1 hr. and the water left the radiator at 90 C. How many calories were given to the room during the hour? 11. How many calories are given up by 30 g. of steam at 100 C. in condensing and then cooling to 20 C. ? How much water will this steam raise from 10 C. to 20 C.? ARTIFICIAL COOLING 187 ARTIFICIAL COOLING 230. Cooling by solution. Let a handful of common salt be placed in a small beaker of water at the temperature of the room and stirred with a thermometer. The temperature will fall several degrees. If equal weights of ammonium nitrate and water at 15 C. are mixed, the tem- perature will fall as low as 10 C. If the water is nearly at C. when the ammonium nitrate is added, and if the stirring is done with a test tube partly filled with ice-cold water, the water in the tube will be frozen. These experiments show that the breaking up of the crystals of a solid requires an expenditure of heat energy, as well when this operation is effected by solution as by fusion. The reason for this will appear at once if we consider the analogy between solution and evaporation ; for just as the molecules of a liquid tend to escape from its surface into the air, so the molecules at the surface of the salt are tending, because of their velocities, to pass off, and are only held back by the attractions of the other molecules in the crystal to which they belong. If, how- ever, the salt is placed in water, the attraction of the water molecules for the salt molecules aids the natural velocities of the latter to carry them beyond the attraction of their fellows. As the molecules escape from the salt crystals two forces are acting on them, the attraction of the water molecules tending to increase their velocities, and the attraction of the remaining salt molecules tending to diminish these velocities. If the latter force has a greater resultant effect than the former, the mean velocity of the molecules after they have escaped will be diminished and the solution will be cooled. But if the attraction of the water molecules amounts to more than the backward pull of the dissolving molecules, as when caustic potash or sulphuric acid is dissolved, the mean molecular velocity is increased and the solution is heated. 231. Freezing points of solutions. If a solution of one part of common salt to ten of water is placed in a test tube and immersed in a " freezing mixture " of water, ice, and salt, the 188 CHANGE OF STATE temperature indicated by a thermometer in the tube will not be zero when ice begins to form, but several degrees below zero. The ice which does form, however, will be found, like the vapor which rises above a salt solution, to be free from salt, and it is this fact which furnishes a key to the explanation of why the freezing point of the salt solution is lower than that of pure water. For cooling a substance to its freezing point simply means reducing its temperature, and therefore the mean ve- locity of its molecules, sufficiently to enable the cohesive forces of the liquid to pull the molecules together into the crystalline form. Since in the freezing of a salt solution the cohesive forces of the water are obliged to overcome the attractions of the salt molecules as well as the molecular motions, the motions must be rendered less, that is, the temperature must be made lower, than in the case of pure water in order that crystallization may occur. From this reasoning we should ex- pect that the larger the amount of salt in solution the lower would be the freezing point. This is indeed the case. The lowest freezing point obtainable with common salt in water is 22 C., or 7.6 F. This is the freezing point of a saturated solution. 232. Freezing mixtures. If snow or ice is placed in a vessel of water, the water melts it, and in so doing its temperature is reduced to the freezing point of pure water. Similarly, if ice is placed in salt water, it melts and reduces the temperature of the salt water to the freezing point of the solution. This may be one, or two, or twenty-two degrees below zero, according to the concentration of the solution. Therefore, whether we put the ice in pure water or in salt water, enough of it always melts to reduce the whole mass to the freezing point of the liquid, and each gram of ice which melts uses up 80 calories of heat. The efficiency of a mixture of salt and ice in producing cold is therefore due simply to the fact that the freezing point of a salt solution is lower than that of pure water. INDUSTRIAL APPLICATIONS 189 The best proportions are three parts of snow or finely shaved ice to one part of common salt. If three parts of calcium chloride are mixed with two parts of snow, a tem- perature of 55 C. may be produced. This is low enough to freeze mercury. QUESTIONS AND PROBLEMS 1. When salt water freezes, the ice formed is free from salt. What effect, then, does freezing have on the concentration of a salt solution ? 2. A partially concentrated salt solution which has a freezing point of 5 C. is placed in a room which is kept at 10 C. Will it all freeze ? 3. Explain why salt is thrown on icy sidewalks on cold winter days. 4. Give two reasons why the ocean freezes less easily than the lakes. 5. Why does pouring H 2 SO 4 into water produce heat, while pouring the same substance upon ice produces cold? 6. Why will a liquid which is unable to dissolve a solid at a low temperature often do so at a higher temperature ? (See 230.) 7. When the salt in an ice-cream freezer unites with the ice to form brine, about how many calories of heat are used for each gram of ice melted ? Where does it come from ? If the freezing point of the salt solution were the same as that of the cream, would the cream freeze ? INDUSTRIAL APPLICATIONS 233. The modern steam engine. Thus far in our study of the transformations of energy we have considered only cases in which mechanical energy was transformed into heat energy. In all heat engines we have examples of exactly the reverse operation, namely, the transformation of heat energy back into mechanical energy. 'How this is done may best be understood from a study of various modern forms of heat engines. The invention of the form of the steam engine which is now in use is due to James Watt, who, at the time of the invention (1768), was an instrument maker in the University of Glasgow. The operation of such a machine can best be understood from the ideal diagram shown in Fig. 181. Steam generated in the boiler B by the fire F passes through the pipe S into 190 CHANGE OF STATE the steam chest V, and thence through the passage N into the cylinder (7, where its pressure forces the piston P to the left. It will be seen from the figure that as the driving rod R moves toward the left the so-called eccentric rod R', which controls the valve P 7 , moves toward the right. Hence, when the piston has reached the left end of its stroke, the passage FIG. 181. Ideal diagram of a steam engine N will have been closed, while the passage M will have been opened, thus throwing the pressure from the right to the left side of the piston, and at the same time putting the right end of the cylinder, which is full of spent steam, into connection with the exhaust pipe E. This operation goes on continually, the rod R' opening and closing the passages M and ^Vat just the proper moments to keep the piston moving back and forth through- out the length of the cylinder. The shaft carries a heavy flywheel W, the great inertia of which insures constancy in speed. The motion of the shaft is communicated to any i! ill THE LIBERTY MOTOR This 400-horse-power motor, one of America's important contributions to the World War, was developed for use on the larger types of bombing airplanes. It makes 1700 revolutions per minute and has twelve cylinders, which are water-cooled. It weighs 806 pounds, or about 2 pounds per horse power. The NC-4, which made the first transatlantic flight, was equipped with three of these motors INDUSTEIAL APPLICATIONS 191 desired machinery by means of a belt which passes over the pulley W. Within the boiler the steam is at high pressure and high temperature ( 226). The steam falls in temperature within the cylinder while doing the work of pushing the piston. A steam engine is a mechanical device ivhich accomplishes useful work by transforming heat energy into mechanical energy. 234. Condensing and noncondensing engines. In most sta- tionary engines the exhaust E leads to a condenser which consists of a chamber Q, into which plays a jet of cold water T, and in which a partial vacuum is maintained by means of an air pump. In the best engines the pressure within Q is not more than from 3 to 5 centimeters of mercury, that is, not more than a pound to the square inch. Hence the con- denser reduces the back pressure against that end of the piston which is open to the atmosphere from 15 pounds down to 1 pound, and thus increases the effective pressure which the steam on the other side of the piston can exert. 235. The eccentric. In practice the valve rod R' is not attached as in the ideal engine indicated in Fig. 181, but motion is communicated to it by a so-called ec- centric. This consists of a circular disk K (Fig. 182) rigidly at- tached to the axle but so set that its center does not coincide with the center of the axle A. The disk # rotates inside the collar C and thus communicates to FTG> 182 . The eccentric the eccentric rod R' a back-and-forth motion which operates the valve V in such a way as to admit steam alternately through M and N at the proper time. 236. The boiler. When an engine is at work, steam is being removed very rapidly from the boiler ; for example, a railway locomotive consumes from 3 to fi tons of water per hour. Tt is therefore necessary to have 192 CHANGE OF STATE the fire in contact with as large a surface as possible. In the tubular boiler this end is accomplished by causing the flames to pass through a large number of metal tubes immersed in water. The arrangement FIG. 183. Diagram of locomotive of the furnace and the boiler may be seen from the diagram of a loco- motive shown in Fig. 183. (See early and modern types opposite p. 123.) 237. The draft. In order to force the flames through the tubes B of the boiler a powerful draft is required. In locomotives this is obtained by running the exhaust steam from the cylinder C (Fig. 183) into the smokestack E through the blower F. The strong current through F draws with it a portion of the air from the smoke box Z), thus producing within D a partial vacuum into which a powerful draft rushes from the furnace through the tubes B. The coal consumption of an ordinary locomotive is from one-fourth ton to one ton per hour. In stationary engines a draft is obtained by mak- ing the smokestack very high. Since in this case the pressure which is forcing the air through the furnace is equal to the difference in the weights of columns of air of unit cross section inside and outside the chimney, it is evident that this pressure will be greater the greater the height of the smokestack. This is the reason for the immense heights given to chimneys in large power plants. 238. The governor. Fig. 184 shows an ingenious device of Watt's, called a governor, for automatically regulating the speed with which a stationary engine runs. If it runs too fast, the heavy rotating balls B FIG. 184. The governor INDUSTRIAL APPLICATIONS 193 move apart and upward and in so doing operate a valve which reduces the speed by partially shutting off the supply of steam from the cylinder. 239. Compound engines. In an engine which has but a single cylin- der the full force of the steam has not been spent when the cylinder is opened to the exhaust. The waste of energy which this entails is obviated in the compound engine (see Fig. 311) by allowing the partially spent steam to pass into a second cylinder of larger area than the first. The most efficient of modern engines have three and sometimes four cylin- ders of this sort, and the en- gines are accordingly called triple or quadruple expansion engines. FIG. 185. Cross-compound engine cylinders Fig. 185 shows the relation be- tween any two successive cylin- ders of a cross-compound engine. By automatic devices not differing in principle from the eccentric, valves C 1 , D 2 , and E 2 open simultaneously and thus permit steam from the boiler to enter the small cylinder A, while the partially spent steam in the other end of the same cylinder passes through D 2 into B, and the more fully exhausted steam in the upper end of B passes out through E 2 . At the upper end of the stroke of the pistons P and P', C 1 , D 2 , and E 2 automatically close, while C 2 , D\ and E 1 simultaneously open and thus reverse the direction of motion of both pistons. These pistons are attached to the same shaft. 240. Efficiency of a steam engine. We have seen that it is possible to transform completely a given amount of mechani- cal energy into heat energy. This is done whenever a moving body is brought to rest by means of a frictional resistance. But the inverse operation, namely, that of transforming heat energy into mechanical energy, differs in this respect, that it is only a comparatively small fraction of the heat developed by combustion which can be transformed into work. For it is not difficult to see that in every steam engine at least a part of the heat must of necessity pass over with the exhaust steam into the condenser or out into the atmosphere. This loss is so 194 CHANGE OF STATE great that even in an ideal engine not more than about 23% of the heat of combustion could be transformed into work. In practice the very best condensing engines of the quadruple- expansion type transform into mechanical work not more than 17% of the heat of combustion. Ordinary locomotives utilize at most not more than 8%. The efficiency of a heat engine is defined as the ratio between the heat utilized, or transformed into work, and the total heat expended. The efficiency of the best steam engines is therefore about -|^, or 75%, of that of an ideal heat engine, while that of the ordinary locomotive is only about ^-, or 26%, of the ideal limit. 241. Principle of the internal-combustion engine. Let two iron or steel wires be pushed through a cork stopper and their ends * brought near together (1/32 inch will do) (Fig. 186). With an atomizer spray into the bottle a small amount of benzine or gasoline (the amount to use can be determined by trial), insert the stopper, and bring the tips of the heavily insulated wires leading from an induction coil to the underside of the wires a, b. A spark will pass at s ; and, if the mix- ture is not too " rich " or too " lean," a violent explosion will occur, throwing the stopper as high as the ceiling. (A heavy round bottle must be used for safety. Wrap it well in wire gauze.) Within the last two decades gas engines have become quite as important a factor in modern life as steam engines. (See opposite pp. 190, 191, and 198.) Such engines are driven by properly timed explosions of a mixture of gas and air occurring within the cylinder. Fig. 187 is a diagram illustrating the four stages into which it is convenient to divide the complete cycle of opera- tions which goes on within such an engine. Suppose that the heavy flywheel W has already been set in motion. As the piston p moves down in the first stroke (see 1) the valve D FIG. 186. A mixture of gasoline vapor and air will explode INDUSTRIAL APPLICATIONS 195 FIG. 187. Principle of the gas engine opens and an explosive mixture of gas and air is drawn into the cylinder through D. As the piston rises (see 2) valve D closes, and the mixture of gas and air is compressed into a small space in the upper end of the cylinder. An electric spark ignites the explosive mixture, and the force of the explosion drives the piston violently down (see 3}. At the besrirmintj \ y o o of the return stroke (see 4) the exhaust valve E opens, ; \ w } \w j \ w and as the piston moves up, the spent gaseous prod- ucts of the explosion are forced out of the cylinder. The initial condition is thus restored and the cycle begins over again. Since it is only during the third stroke that the engine is receiving energy from the exploding gas, the flywheel is always made very heavy so that the energy stored up in it in the third stroke may keep the machine running with little loss of speed during the other three parts of the cycle. The efficiency of the gas engine is often as high as 25%, or nearly double that of the best steam engines. Furthermore, it is free from smoke, is very compact, and may be started at a moment's notice. On the other hand, the fuel (gas or gasoline) is comparatively expensive. Most automobiles are run by gasoline engines, chiefly because the lightness of the engine and of the fuel to be carried are here considera- tions of great importance. It has been the development of the light and efficient gas engine which has made possible man's recent conquest of the air through the use of the airplane and airship. 242. The automobile. The plate opposite page 198 shows the principal mechanical features of the automobile in their relation to one another. It will be seen that the cylinders 196 CHANGE OF STATE of the engine are surrounded by water jackets which form part of a circulating system. The heat of the engine is car- ried by convection currents in this water to the radiator, where it is lost to the atmos- phere through the air currents produced in part by a revolving fan (10). Unless some means were provided for cooling a gas engine, it would become so over- heated that the pistons would stick fast. The power of the engine is transmitted to the rear axle through the clutch (11), the transmission (12), and the differential gearing. 243. The clutch and the transmis- sion. Since a gas engine develops its power by a series of violent ex- plosions within the cylinders, it is clear that it cannot start with a load as does the steam engine. In start- ing an automobile it is first necessary that the engine acquire a reasonable speed and that the power be applied gradually to the rear axle by the use of a friction clutch (11); otherwise the engine will stall. The shaft of the engine has upon its rear .end a flywheel which, in the cone clutch, is turned to a conical shape inside. Close to this but attached to the transmission shaft is the clutch plate, a heavy disk faced with leather, which Rear (Transmission Shaft] l|f Countershaft Neutral Firxt (Low Speed) 2 Second (Intermediate Speed) FIG. 188. Automobile transmission INDUSTRIAL APPLICATIONS 197 fits the inside of the flywheel and is pressed into it by a spring suffi- ciently strong to prevent any slipping when the clutch is engaged. The driver throws out the clutch by depressing a lever with his foot. In the disk clutch the bearing surfaces are two series of disks, one revolving with the engine shaft, the other with the transmission. The amount of work done by a gas engine in a minute depends upon the work done by each explosion multiplied by the number of explosions per minute. Therefore it can develop its full power only while revolving rapidly. In hill climbing, for example, the speed of the engine must be great while that of the car is comparatively small. To meet this require- ment a system of reduction gears called the transmission (12) is used to make the number of revolutions of the driving shaft less than that of the crank shaft (4) of the engine. In Fig. 188, (1), the gears are in neutral, gears 1 and # being always in mesh. By use of the gear-shift lever (14) gears 3 and 5 (Fig. 188) are made to slide upon a square shaft. Before shifting the gears the clutch is released to disconnect the power of the motor from the driving shaft ; and, to avoid a clash when meshing the gears on the transmission shaft with those on the counter- shaft, care should be taken that they revolve at about the same speed. Fig. 188, (2), shows the low-speed connection. In shifting to second speed (Fig. 188, (#)) the clutch is released, gear 5 is thrown into neutral, and finally gear 3 is meshed with 4, after which the clutch is allowed to grip. In going to high speed (Fig. 188, (4)) gear 3 is shifted through neutral to engagement with gear 1. This connects the crank shaft of the engine directly to the driving shaft so that the two revolve at the same speed. For the reverse (Fig. 188, (5)) an eighth gear is thrown up from beneath so as simultaneously to engage 5 and 7. Such an interposition of a third gear wheel between 5 and 7 obviously reverses the direction of rotation of the driving shaft. 244. The differential. An automobile is driven by .power applied to the rear axle. This requires the axle to be in two parts with a differential between, so that in turning corners the outer wheel may revolve faster than the inner. It will be seen from the large drawing opposite page 198, and from Fig. 189, that the pinion FIG. 189. The differential 198 CHANGE OF STATE attached to the driving shaft rotates the main bevel gear B, to which are rigidly attached the differential gears 1 and 2. The left axle is directly connected to gear 3, and only indirectly connected to the main bevel gear B through gears 1 and 2. In running straight both rear wheels revolve at the same rate; therefore, while gears 3 and 4 and the main bevel gear are revolving at the same speed they carry around with them pinions 1 and 2, which are now, however, not revolving on their bearings. When the car is turning a corner, gears 3 and 4 are turning at different rates ; hence pinions 1 and 2 are not only carried around by the main bevel gear but at the same time are revolved in opposite directions on their bearings. 245. The carburetor. The carburetor is a device for converting liquid gasoline, kerosene, etc. into -vapor and mixing it with air in proper proportions for complete combustion. ,The simple principle of carburetion is shown in the upper diagram opposite page 199. Liquid gasoline comes through the supply pipe and enters the float chamber through the valve V. By acting on the levers L the float closes the valve V when the gasoline reaches a certain level. From the float chamber the gasoline is drawn to the spray nozzle O. While the engine is running, the downward movement of the pistons in stroke 1 (Fig. 187) sucks air violently past the spray nozzle into the region called the venturi, where the jet of gasoline is emerging from 0. The spray of fuel thus formed intermingles with air in the mixing chamber and passes by the throttle to the engine as a highly explosive mixture. 246. The ignition. The lower diagram opposite page 199 illustrates the principle of high-tension magneto ignition which is widely used on automobiles. A rolling contact R is mounted on the cam shaft, which revolves at half crank-shaft speed and is carried around the interior of the stationary fiber ring D. When the switch S is closed and the roller R passes across the metal segment G, a current of electricity passes from the magneto through the rolling contact to the central shaft C, and from there through the iron work of the car to the magneto by way of the primary coil of the induction coil. While the roller is in contact with the segment G the induction coil produces a shower of sparks between the points P of the spark plug, thus igniting the explosive mixture in the cylinder of the engine. Since the power stroke of the piston occurs but once in two revolutions of the crank shaft, it is necessary that the crank shaft revolve twice while the contact revolves but once. This, as shown in the diagram, is accomplished by having the crank shaft geared to the cam shaft in a velocity ratio of 2 to 1. To Engine Throttle -Mixing chamber Venturi -Spray nozzle Needle valve Gasoline supply THE C A 11 B u RBTO R Insulating Fibre Ring Magneto ro0s"i / . \ SmtchAS Cam Sraft Gear (42 Teeth) Lever Jor moving ring D -Metal Segment Rolling Contact ^-Crank Shaft Gear 3 (ZITeeth) 'rank Handle Ax IGNITION SYSTEM INDUSTRIAL APPLICATIONS 199 The explosive mixture requires a very short but measurable time for combustion ; hence the full force of the explosion occurs a short time after the spark ignites the mixture. Therefore, at high speed the spark should occur a little earlier with reference to the position of the piston than at low speed. The spark is advanced or retarded by a spark lever L which changes the position of the segment G by pulling around slightly the movable fiber ring to which it is attached. The diagram applies to a one-cylinder engine. In case the engine has four cylinders, three additional segments must be added, as indicated by the clear spaces, together with three additional induction coils and spark plugs.* 247. The steam turbine. The r steam turbine represents the M : W latest development of the heat engine. In principle it is very much like the common wind- mill, the chief difference being that it is steam instead of air which is driven at a high veloc- ity against a series of blades arranged radially about the cir- cumference of the wheel that is set into rotation. The steam, however, unlike the wind, is FlG 190< The principle of the always directed by nozzles at steam turbine the angle of greatest efficiency against the blades (see Fig. 190). Furthermore, since the energy of the steam is far from spent after it has passed through one set of blades (such as that shown in Fig. 190), it is in practice always passed through a whole series of such sets (Fig. 191), every alternate row of which is rigidly attached to the rotating shaft, while the intermediate rows are fastened to the immovable outer jacket of the engine and only serve as guides to redirect the steam at the most favorable angle against the next row of movable blades. In this way the steam is kept alternately bounding from fixed to movable blades until its energy is expended. The number of rows of blades is often as high as sixteen. * The pupil may well consult the more extended treatises for actual details of the many different systems of ignition used on automobile and airplane engines. 200 CHANGE OF STATE Turbines are at present coming rapidly into use, chiefly for large- power purposes. Their advantages over the reciprocating steam engine lie first in the fact that they run with almost no jarring, and therefore require much lighter and less expensive foundations, and second in the fact that they occupy less than one tenth the floor space of ordinary engines of the same capacity. Their efficiency is fully as high as that Exhaust Revolving Stationary Revolving Nozzle FIG. 191. Path of steam in Curtis's turbine of the best reciprocating engines. The highest speeds attained by ves- sels at sea, namely, about 40 miles per hour, have been made with the aid of steam turbines. One of the largest vessels which have thus far been launched, the Berengaria, 919 feet long, 98 feet wide, 100 feet high (from the keel to the top of her ninth deck), having a total displace- ment of 57,000 tons and a speed of 221 knots, is driven by four steam turbines having a total horse power of 61,000. One of the immense rotors contains 50,000 blades and develops 22,000 horse power. The United States Shipping Board, on July 24, 1919, announced plans for INDUSTRIAL APPLICATIONS 201 building two gigantic ocean liners swifter and larger than any afloat. They are to be 1000 feet long and are to have a horse power of 110,000 and a speed of 30 knots. (See opposite p. 135.) 248. Manufactured ice. In the great majority of modern ice plants the low temperature required for the manufacture of the ice is produced by the rapid evaporation of liquid ammonia. At ordinary temperatures ammonia is a gas, but it may be liquefied by pressure alone. At 80 F. a pressure of 155 pounds per square inch, or about 10 atmospheres, is required to produce its liquefaction. Fig. 192 shows the essential parts of an ice plant. The compressor, which is usually run by a steam engine, Low Pressure Gau HighPressure Gauge FIG. 192. Compression system of ice manufacture forces the gaseous ammonia under a pressure of 155 pounds into the con- denser coils shown on the right, and there liquefies it. The heat of con- densation of the ammonia is carried off by the running water which constantly circulates about the condenser coils. From the condenser the liquid ammonia is allowed to pass very slowly through the regulat- ing valve V into the coils of the evaporator, from which the evaporated ammonia is pumped out so rapidly that the pressure within the coils does not rise above 34 pounds. It will be noted from the figure that the same pump which is there labeled the compressor exhausts the ammonia from the evaporating coils and compresses it in the condensing coils, for the valves are so arranged that the pump acts as an exhaust pump on one side and as a compression pump on the other. The rapid evaporation of the liquid ammonia under the reduced pressure existing 202 CHANGE OF STATE within the evaporator cools these coils to a temperature of about 5 F. The brine with which these coils are surrounded has its temperature thus reduced to about 16 or 18 F. This brine is made to circulate about the cans containing the water to be frozen. The heat of vapori- zation of ammonia at 5 F. is 314 calories. Many thousands of feet of circulating saltwater pipe are laid horizon- tally and covered with water to be frozen for large indoor skating rinks. 249. Cold storage. The artificial cooling of factories and cold-storage rooms is accomplished in a manner exactly similar to that employed in the manufacture of ice. The brine is cooled precisely as described above, and is then pumped through coils placed in the rooms to be cooled. In some systems carbon dioxide is used instead of ammonia, but the principle is in no way altered. Sometimes, too, the brine is dispensed with, and the air of the rooms to be cooled is forced by means of fans directly over the cold coils containing the evaporating ammonia or carbon dioxide. It is in this way that theaters and hotels are cooled. QUESTIONS AND PROBLEMS 1. Why is a gas engine called an internal-combustion engine ? 2. Why do gasoline engines have flywheels ? Why is a one-cylinder engine of the four-cycle type especially in need of a flywheel ? 3. How does the temperature of the steam within a locomotive boiler compare with its temperature at the moment of exhaust ? Explain. 4. On the drive wheels of locomotives there is a mass of iron op- posite the point of attachment of the drive shaft. Why is this necessary ? 5. Why does not the water in a locomotive boil at 100 C. ? 6. If liquid oxygen is placed in an open vessel, its temperature will not rise above 182 C. Why not? Suggest a way in which its tem- perature could be made to rise above 182 C., and a way in which it could be made to fall below that temperature. 7. How many foot pounds of energy are there in 1 Ib. of coal con- taining 14,000 B. T. U. per pound ? How many pounds of iron must be held at a height of 150 ft. to have as much energy as this pound of coal? 8. The average locomotive has an efficiency of abdut 6%. What horse power does it develop when it is consuming 1 ton of coal per hour? (See Problem 7, above.) 9. What amount of useful work did a gasoline engine working at an efficiency of 25% do in using 100 Ib. of gasoline containing 18,000 B.T.U. per pound? 10. What pull does a 1000 H.P. locomotive exert when it is running at 25 mi. per hour and exerting its full horse power? CHAPTER XI THE TRANSFERENCE OF HEAT CONDUCTION 250. Conduction in solids. If one end of a short metal bar is held in the fire, the other end soon becomes too hot to hold ; but if the metal rod is replaced by one of wood or glass, the end away from the flame is not appreciably heated. This experiment and others like it show that nonmetallic substances possess much less ability to conduct heat than do metallic substances. But although all metals are good conductors as compared with nonmetals, they differ widely among themselves in their con- ducting powers. Let copper, iron, and German-silver wires 50 cm. long and about 3 mm. in diameter be twisted together at one end as in Fig. 193, and let a Bunsen flame be applied to the twisted ends. Let a match be slid slowly from the cool end of each wire toward the hot end, until the heat from the wire ignites it. The .copper will be found to be the best conductor and the German silver the poorest. In the following table some common substances are arranged in the order of their heat conductivities. The measurements have been made by a method not differing in principle from that just described. For convenience, silver is taken as 100. FIG. 193. Differences in the heat conductivities of metals Silver . . . . 100 Tin . . 15 Mercury . 1.5 Copper . . Aluminium . 74 . 48 Iron . Lead . . . . 12 8.5 Ice .... Glass .... .21 .15 Brass . . 27 German silver . 6.3 Hard rubber . .04 203 204 THE TRANSFERENCE OF HEAT FIG. 194. Water a nonconductor 251. Conduction in liquids and gases. Let a small piece of ice be held by means of a glass rod in the bottom of a test tube full of ice water. Let the upper part of the tube be heated with a Bunsen burner as in Fig. 194. The upper part of the water may be boiled for some time without melting the ice. Water is evidently, then, a very poor conductor of heat. The same thing may be shown more strikingly as follows : The bulb of an air ther- mometer is placed only a few milli- meters beneath the surface of water contained in a large funnel arranged as in Fig. 195. If now a spoonful of ether is poured on the water and set on fire, the index of the air thermometer will show scarcely any change, in spite of the fact that the air thermometer is a very sensitive indicator of changes in temperature. Careful measurements of the conductivity of water show that it is only about 12 1 QO of that of silver. The conductivity of gases is even less, not amounting on the average to more than ^ that of water. 252. Conductivity and sensation. It is a fact of common observation that on a cold day in winter a piece of metal feels much colder to the hand than a piece of wood, notwithstanding the fact that the tempera- ture of the wood must be the same as that of the metal. On the other hand, if the same two bodies had been lying in the hot sun in midsummer, the wood might be handled without discomfort, but the metal would be uncomfortably hot. The explanation of these phenomena is found in the fact that the iron,|being a much better conductor than the wood, removes heaF from FIG. 195. Burning ether on the water does not affect the air thermometer CONDUCTION 205 the hand much more rapidly in winter, and imparts heat to the hand much more rapidly in summer, than does the wood. In general, the better a conductor the hotter it will feel to a hand colder than itself, and the colder to a hand hotter than itself. Thus, in a cold room oilcloth, a fairly good conductor, feels much colder to the touch than a carpet, a comparatively poor conductor. For the same reason linen clothing feels cooler to the touch in winter than woolen goods. 253. The role of air in nonconductors. Feathers, fur, felt, etc. make very warm coverings, because they are very poor conductors of heat and thus prevent the escape of heat from the body. Their poor conductivity is due in large measure to the fact that they are full of minute spaces containing air, and gases are the best nonconductors of heat. It is for this reason that freshly fallen snow is such an efficient prqtection to vegetation. Farmers always fear for their fruit trees and vines when there is a severe cold snap in winter, unless there is a coating of snow on the ground to prevent a deep freezing. 254. The Davy safety lamp. Let a piece of wire gauze be held above an open gas jet and a match applied above the gauze. The flame will be found to burn above the gauze as in Fig. 196, (1), but it will not pass through to the lower side. If it is ignited below the gauze, the flame will not pass through to the upper side but will burn as shown in Fig. 196, (2). The explanation is found in , T r> ,T ,i T FIG. 196. A flame will not pass the fact that the gauze conducts through wire gauze the heat away from the flame so rapidly that the gas on the other side is not raised to the temperature of ignition. Safety lamps used by miners are completely incased in gauze, so that if the mine is full of inflammable gases, they are not ignited outside of the gauze by the lamp. 206 THE TKANSFERENCE OF HEAT FIG. 197. A lireless cooker QUESTIONS AND PROBLEMS 1. With the aid of Fig. 197, which represents a fireless cooker, ex- plain the principle on which fireless, cooking is done. 2. Why do firemen wear flannel shirts in summer to keep cool and in winter to keep warm ? 3. If a package of ice cream is put inside a paper bag, it will not melt so fast on a hot day. Explain. 4. If the ice in a refrigerator is wrapped up in blankets, what is the effect on the ice ? on the refrigerator ? 5. If a piece of paper is wrapped tightly around a metal rod and held for an instant in a Bunsen flame, it will not be scorched. If held in a flame when wrapped around a wooden rod, it will be scorched at once. Explain. 6. If one touches the pan contain- ing a loaf of bread in a hot oven, he receives a much more severe burn than if he touches the bread itself, although the two are at the same temperature. Explain. 7. Why are plants often covered with paper on a night when frost is expected? 8. Why will a moistened finger or the tongue freeze instantly to a piece of iron on a cold winter's day, but not to a piece of wood ? 9. Does clothing ever afford us heat in winter? How, then, does it keep us warm ? 10. Why is the outer pail of an ice-cream freezer made of thick wood and the inner can of thin metal ? CONVECTION 255. Convection in liquids. Although the conducting power of liquids is so small, as was shown in the experiment of 251, they are yet able, under certain circumstances, to transmit heat much more effectively than solids. Thus, if the ice in the experiment of Fig. 194 had been placed at the top and the flame at the bottom, the ice would have been melted very quickly. This shows that heat is transferred very much CONVECTION 207 more readily from the bottom of the tube toward the top than from the top toward the bottom. The mechanism of this heat transference will be evident from the following experiment : Let a round-bottomed flask (Fig. 198) be half filled with water and a few crystals of magenta dropped into it. Then let the bottom of the flask be heated with a Bunsen burner. The magenta will reveal the fact that the heat sets up currents the direction of which is upward in the region im- mediately above the flame but downward at the sides of the vessel. It will not be long before the whole of the water is uniformly colored. This shows how thorough is the mixing accomplished by the heating. The explanation of the phenomenon is as follows: The water nearest the flame be- came heated and expanded. It was thus ren- dered less dense than the surrounding water, and was accordingly forced to the top by the pressure transmitted from the colder and therefore denser water at the sides which then came in to take its place. It is obvious that this method of heat transfer' is applicable only to fluids. The essential difference between it and con- duction is that the heat is not transferred from molecule to molecule throughout the whole mass, but is rather transferred by the bodily movement of comparatively large masses of the heated liquid from one point to another. This method of heat transference is known as convection. 256. Winds and ocean currents. Winds are convection cur- rents in the atmosphere caused by unequal heating of the earth by the sun. Let us consider, for example, the land and sea breezes so familiar to all dwellers near the coasts of large bodies of water. During the daytime the land is heated more rapidly than the sea, because the specific heat of water is much greater than that of earth. Hence the hot air over the FIG. 198. Convec- tion currents 208 THE TRANSFERENCE OF HEAT land expands and is forced up by the colder and denser air over the sea which moves in to take its place. This con- stitutes the sea breeze, which blows during the daytime, usually reaching its maximum strength in the late afternoon. At night the earth cools more rapidly than the sea and hence the direction of the wind is reversed. The effect of these breezes is seldom felt more than twenty-five miles from shore. Ocean currents are caused partly by the unequal heating of the sea and partly by the direction of the prevailing winds. In general, both winds and currents are so modified by the configuration of the continents that it is only over broad expanses of the ocean that the direction of either can be predicted from simple considerations. RADIATION 257. A third method of heat transference. There are certain phenomena in connection with the transfer of heat for which conduction and convection are wholly unable to account, For example, if one sits in front of a hot grate fire, the heat which he feels cannot come from the fire by convection, because the Currents of air are moving toward the fire rather than away from it. It cannot be due to conduction, because the conductivity of ah- is extremely small and the colder currents of air moving toward the fire would more than neutralize any transfer outward due to conduction. There must therefore be some way in which heat travels across the intervening space other than by conduction or convection. It is still more evident that there must be a third method of heat transfer when we consider the heat which comes to us from the sun. Conduction and convection take place only through the agency of matter; but we know that the space between the earth and the sun is not filled with ordinary matter, or else the earth would be retarded . in its motion through space. Radiation is the name given to this third RADIATION 209 method by which heat travels from one place to another, and which is illustrated in the passing of heat from a grate fire to a body in front of it, or from the sun to the earth. 258. The nature of radiation. The nature of radiation will be discussed more fully in Chapter XXI. It will be sufficient here to call attention to the following differences between conduction, convection, and radiation. First, while conduction and convection are comparatively slow processes, the transfer of heat by radiation takes place with the enormous speed with which light travels, namely 186,000 miles per second. That the two speeds are the same is evident from the fact that at the time of an eclipse of the sun the shutting off of heat from the earth is observed to take place at the same time as the shutting off of light. Second, radiant heat travels in straight lines, while conducted or convected heat may follow the most circuitous routes. The proof of this statement is found in the familiar fact that ra- diation may be cut off by means of a screen placed directly between a source and the body to be protected. Third, radiant heat may pass through a medium without heating it. This is shown by the fact that the upper regions of the atmosphere are very cold, even in the hottest days in summer, or that a hothouse may be much warmer than the glass through which the sun's rays enter it. 259. The Dewar flask and the thermos bottle. For the retention of extremely cold liquids, such, for example, as liquefied air, whose boiling point is -190 C. (=- 310 F.), Dewar invented a double-walled vessel. The space between the walls is a. vacuum, and the inner surface of the outer vessel and the outer surface of the inner vessel are silvered. There are three ways in which heat may pass inward through the double wall conduction, convection, and radiation. The vacuum prevents almost entirely the first two, while the silver- ing eliminates passage of heat by radiation. The well-known 210 THE TRANSFERENCE OF HEAT glass part of the thermos bottle (Fig. 199) is simply a cylindrical Dewar flask for keeping liquids either hot or cold, since it is as difficult for heat to pass outward through the walls as to pass inward. The glass flask is provided with a cork stopper, and a strong outside metal case for its protection. Hot liquids, as well as /) those that are cold, may be kept for several hours in a thermos bottle with only a few degrees change in temperature. THE HEATING AND VENTILATING OF BUILDINGS 260. The principle of ventilation. The heating and ventilating of buildings are accomplished chiefly through the agency of convection. FIG. 199. The in- ner glass flask of a thermos bottle To illustrate the principle of ventilation let a candle be lighted and placed in a vessel containing a layer of water (Fig. 200). When a lamp chimney is placed over the candle so that the bottom of the chimney is under the water, the flame will slowly die down and will finally be extinguished. This is because the oxygen, which is essential to combustion, is gradually used up and no fresh supply is possible with the arrangement described. If the chimney is raised even a very little above the water, the dying flame will at once brighten. Why? If a metal or cardboard. partition is inserted in the chimney, as in Fig. 200, the flame will burn con- tinuously, even when the bottom of the chimney is under water. The reason will be clear if a piece of burning touch paper (blotting paper soaked in a solu- tion of potassium nitrate and dried) is held over the chimney. The smoke will show the direction of the air currents. If the chimney is a large one, in order that the first part of the above experiment may succeed, it may be necessary to use two candles ; for too small a heated area permits the formation of downward currents at the sides. FIG. 200. Con- vection currents in air HEATING AND VENTILATING 211 261. Ventilation of houses. In order to secure satisfactory ventilation it is estimated that a room should be supplied with 2000 cubic feet of fresh air per hour for each occupant (a gas burner is equivalent in oxygen consumption to four persons). A current of air mov- ing with a speed great enough to be just perceptible has a velocity of about 3 feet per second. Hence the area of opening required for each person when fresh air is entering at this speed is about 25 or 30 square inches. The manner of supply- ing this requisite amount of fresh air in dwelling houses depends upon the particular method of heating employed. If a house is heated by stoves or fireplaces, no special provision for ventilation is needed. The foul air is drawn up the chimney with the smoke, and the fresh air which replaces it finds entrance through cracks about the doors and windows and through the walls. 262. Hot-air heating. In houses heated by hot-air furnaces an air duct ought always to be supplied for the entrance of fresh cold air, in the manner shown in Fig. 201 (see " cold-air inlet "). This cold air from out of doors is heated by passing in a circuitous way, FIG. 202. Princi- as shown by the arrows, over the outer jacket of iron p] e o f hot-water which covers the fire box. It is then delivered to the heating FIG. 201. Ho^-air heating 212 THE TRANSFERENCE OF HEAT rooms. Here a part of it escapes through windows and doors, and the rest returns through the cold-air register to be reheated, after being mixed with a fresh supply from out of doors. When the fire is first started, in order to gain a strong draft the damper C is opened so that the smoke may pass directly up the chimney. After the fire is under way the damper C is closed so that the smoke and hot Cold water Copper heating coils FIG. 203. A gas heating coil FIG. 204. Hot-water heater gases from the furnace must pass, as indicated by the dotted arrows, over a roundabout path, in the course of which they give up the major part of their heat to the steel walls of the jacket, which in turn pass it on to the air which is on its way to the- living rooms. 263. Hot-water heating. To illustrate the principle of hot-water heating let the arrangement shown in Fig. 202 be set up, the upper vessel being filled with colored water, and then let a flame be applied to the lower vessel. The colored water will show that the current moves in the direction of the arrows. HEATING AND VENTILATING 213 This same principle is involved in the gas heating coil used in connection with the kitchen boiler (Fig. 203). Heat from the flame passes through the copper coil to the water, and convection begins as indicated by the arrows. When hot water is drawn from the top of the boiler, cold water enters near the bottom so as not to mingle with the hot water that is being used. The principle is still further illustrated by the cooling systems used for keeping automobile engines from becoming overheated. Heat passes from the engine into the water, which loses heat in circulating through the coils of the radiator. The actual arrangement of boiler and radiators in one system of hot- water heating is shown in Fig. 204. The water heated in the furnace rises directly through the pipe A to a radiator R, and returns again to the bottom of the furnace through the pipes B and D. The circulation is maintained because the column of water in A is hotter and therefore lighter than the water in the return pipe B. By eliminating the expansion tank and partly filling the boiler with water the system could be converted into a steam-heating plant. QUESTIONS AND PROBLEMS 1. If we attempt to start a fire in the kitchen range when the chimney is cold and damp, the range " smokes." Explain. 2. Why is a hollow wall filled with sawdust a better nonconductor of heat than the same wall filled with air alone ? 3. In a system of hot-water heating why does the return pipe always connect at the bottom of the boiler, while the outgoing pipe connects with the top ? 4. Which is thermally more efficient, a cook stove or a grate? Why? 5. When a room is heated by a fireplace, which of the three methods of heat transference plays the most important role ? 6. Why do you blow on your hands to, warm them in winter and fan yourself for coolness in summer? 7. If you open a door between a warm and a cold room, in what direction will a candle flame be blown which is placed at the top of the door? Explain. 8. Why is felt a better conductor of heat when it is very firmly packed than when loosely packed? 9. If 2 metric tons of coal are burned per month in your house, and if your furnace allows one third of the heat to go up the chimney, how many calories remain to be used per day ? (Take 1 g. as yielding 6000 calories. A metric ton = 1000 kg.) CHAPTER XII MAGNETISM* GENERAL PROPERTIES OF MAGNETS 264. Magnets. It has been known for many centuries that some specimens of the ore known as magnetite (Fe 3 O 4 ) have the property of attracting small bits of iron and steel. This ore probably received its name from the fact that it was first observed in the province of Magnesia, in Thessaly. Pieces of this ore which exhibit this attractive property are known as natural magnets. It was also known to the ancients that artificial magnets may be made by stroking pieces of steel with natural magnets, but it was not until about the twelfth century that the dis- covery was made that ta suspended magnet tvill assume a north- and-south position. Because of this latter property natural magnets became known as lodestones (leading stones), and magnets, either artificial or natural, began to be used for deter- mining directions. The first mention of the use of the compass in Europe is in 1190. It is thought to have been introduced from China. (See opposite p. 223 for the gyrocompass.) Magnets are now made either by stroking bars of steel in one direction with a magnet, or by passing electric currents about the bars in a manner to be described later. The form shown in Fig. 205 is called a bar magnet, that shown in Eig. 206 a horseshoe magnet. The latter form is the more common, and is the better form for lifting. *Tliis chapter should be either accompanied or preceded by laboratory experiments on magnetic fields and on the molecular nature ef magnetism. See, for example, Experiments 25 and 26 of the authors' Manual. 214 GENERAL PROPERTIES OF MAGNETS 215 FIG. 206. A horseshoe magnet If a magnet is dipped into iron filings, the filings will be seen to cling in tufts near the ends. but scarcely at all near the middle (Fig. 207). These places near the ends of a magnet at which its strength seems to be concentrated are called the poles of the magnet. The end of a freely swing- ing magnet which points to the north is designated as the north-seeking pole, or simply the north pole (jV) ; and the other end as the south-seeking pole, or the south pole ($) The direction in ivhich a compass needle points is called the magnetic meridian. 265. The laws of magnetic attraction and repulsion. In the experiment with the iron filings no particular difference was observed between the action of the two poles. That there is a difference, however, may be shown by experi- menting with two magnets, either of which may be suspended (see Fig. 208). If two N poles are brought near one another, they are found to repel each other. The S poles likewise are found to repel each other. But the N pole of one magnet is found to be attracted by the S pole of another. The results of these experiments may be summarized in a general law: Magnet poles of like kind repel each other, while poles of unlike kind attract. The force which any two poles exert upon each other in air is equal to the product of the pole strengths divided by the square of the distance between them. FlG - 208 ' Ma S netic at - , . 7.-I/.T 17.7 tractions and repulsions A unit pole is defined as a pole which, when placed at a distance of 1 centimeter from an exactly similar pole, in air, repels it with a force of 1 dyne. FIG. 207. Iron filings cling- ing to a bar magnet 216 MAGNETISM 266. Magnetic materials. Iron and steel are the only substances which exhibit magnetic properties to any marked degree. Nickel and cobalt are also attracted appreciably by strong magnets. Bismuth, antimony, and a number of other substances are actually repelled instead of attracted, but the effect is very small. It has recently been found possible to make quite strongly magnetic alloys out of certain nonmag- netic materials. For example, a mixture of 65% copper, 27% manganese, and 8% aluminium is quite strongly magnetic. These are called Heusler alloys. For practical purposes, how- ever, iron and steel may be considered as the only magnetic materials. 267. Magnetic induction. If a small un- magnetized nail is suspended from one end of a bar magnet, it is found that a second nail may be suspended from this first nail, which itself acts like a magnet, a third from the second, etc., as shown in Fig. 209. But if the bar magnet is carefully pulled away from the first nail, the others will instantly fall away from each other, thus showing that the nails were strong magnets only so long as they were in contact with the bar magnet. Any piece of soft iron may be thus magnetized temporarily by holding it in contact with a permanent magnet. In- deed, it is not necessary that there be actual contact, for if a nail is simply brought near to the permanent magnet it is found to become a magnet. This may be proved by presenting some iron filings to one end of a nail held near a magnet in the manner shown in Fig. 210. Even inserting a plate of glass, or of copper, or of any other material except iron between S and N will not change appreciably the number of filings which cling FIG. 209. Magnetism induced by contact FIG. 210. Magnet- ism induced with- out contact GENERAL PROPERTIES OF MAGNETS 217 to the end of $', a fact which shows that nonmagnetic mate- rials are transparent to magnetic forces. But as soon as the permanent magnet is removed, most of the filings will fall. Magnetism produced by the mere presence of adjacent magnets, unth or without contact, is called induced magnetism. If the induced magnetism of the nail in Fig. 210 is tested with a compass needle, it is found that the remote induced pole is of the same kind as the inducing pole, while the near pole is of unlike kind. This is the general law of magnetic induction. Magnetic induction explains the fact that a magnet attracts an unmagnetized piece of iron, for it first magnetizes it by induction, so that the near pole is unlike the inducing pole, and the remote pole like the inducing pole ; and then, since the two unlike poles are closer together than the like poles, the attraction overbalances the repulsion and the iron is drawn toward the magnet. Magnetic induction also explains the formation of the tufts of iron filings shown in Fig. 207, each little filing becoming a temporary magnet such that the end which points toward the inducing pole is unlike this pole, and the end which points away from it is like this pole. The bushlike appearance is due to the repulsive action which the outside free poles exert upon each other. 268. Retentivity and permeability. A piece of soft iron will very easily become a strong temporary magnet, but when removed from the influence of the magnet it loses practically all of its magnetism. On the other hand, a piece of steel will not be so strongly magnetized as the soft iron, but it will retain a much larger fraction of its magnetism after it is removed from the influence of the permanent magnet. This quality of resisting either magnetization or demagnetiza- tion is called retentivity. Thus steel has a much greater reten- tivity than wrought iron, and, in general, the harder the steel the greater its retentivity. 218 MAGNETISM A substance which has the property of becoming strongly magnetic under the influence of a permanent magnet, whether it has a high retentivity or not, is said to possess permeability in large degree. Thus iron is much more permeable than nickel. 269. Magnetic lines of force. If we could separate the N and S poles of a small magnet so as to get an independent N pole, and were to place this N pole near the N pole of a bar magnet, it would move over to the S pole along some curved path similar to that shown in Fig. 211. The reason it would FlG " 21 ^ A line of force set up by the magnet AB move in a curved path is that it would be simultaneously repelled by the N pole of the bar magnet and attracted by its S pole, and the relative strengths of these two forces would continually change as the relative distances of the moving pole from these two poles changed. To verify this conclusion let a strongly magnetized sewing needle be floated in a small cork in a shallow dish of water, and let a bar or horseshoe magnet be placed just above or just beneath the dish (see Fig. 212). The cork and needle will then move as would an independent pole, since the remote pole of the needle is so much farther from the magnet than the near pole that its influence on the motion is very small. The cork will actually be found to move in a curved path from N to S. Any path which an independent N pole would take in going from N to S is called a line of force. The simplest way of finding the direction of this path at any point near a magnet is to hold a short compass needle at the point con- sidered. The needle sets itself along the line in which its poles would move if independent, that is, along the line of force which passes through the given point (see C, Fig. 211). FIG. 212. Showing direction of motion of an isolated pole near a magnet GENERAL PROPERTIES OF MAGNETS 219 270. Fields of force. The region about a magnet in which its magnetic forces can be detected is called its field of force. The easiest way of gaining an idea of the way in which the \ ! / /V""^,\ UN \ '. ' / // ,.-* , N \\ \ \ \ \ i / , x^^ \\ (( 'iff' ''" ~^ N \\V< : \$M& FIG. 213. Arrangement of iron filings about a bar magnet FIG. 214. Ideal diagram of field of a bar magnet lines of force are arranged in the magnetic field about any magnet is to sift iron filings upon a piece of paper placed immediately over the magnet. Each little filing becomes a temporary magnet by induction, and therefore, like the com- pass needle, sets itself in the direction of the line of force at the point where it is. Fig. 213 shows how the filings arrange themselves about a bar magnet. Fig. 214 is the corresponding ideal diagram showing the lines of force emerging from the JV" pole and passing about in curved paths to the S pole. It is custom- ary to imagine these lines as re- turning through the magnet from S to N in the manner shown, so that each line is thought of as a closed curve. This conven- tion was introduced by Faraday, and has been found of great assistance in correlating the facts of magnetism. A magnetic field of unit strength is defined as a field in which a unit magnet pole experiences 1 dyne of force. It is customary FIG. 215. The strength of a mag- netic field is represented by the number of lines of force per square centimeter 220 MAGNETISM to represent graphically such a field by drawing one line per square centimeter through a surface such as ABCD (Fig. 215) taken at right angles to the lines of force. If a unit N pole be- tween N and S (Fig. 215) were pushed toward S with a force of 1000 dynes, the strength of the field would be 1000 units and it would be represented by 1000 lines per square centimeter. 271. Molecular nature of magnetism. If a small test tube full of iron filings be stroked from end to end with a magnet, it will be found to have become itself a magnet ; but it will lose its magnetism as soon as the filings are shaken up. If a magnetized knitting needle is heated red-hot, it will be found to have lost its magnetism completely. Again, if such a needle is jarred, or hammered, or twisted, the strength of its poles, as measured by their ability to pick up tacks or iron filings, will be found to be greatly diminished. These facts point to the conclusion that magnetism has something to do with the arrangement ~of the molecules, since causes which violently dis- turb the molecules of a mag- net weaken its magnetism. Jf Jja ff Again, if a magnetized needle & I % $ 4 is broken, each part will be FIG. 210. Effect of breaking a magnet found to be a complete mag- net; that is, two new poles will appear at the point of breaking. a new N pole on the part which has the original S pole, and a new S pole on the part which has the original N pole. The subdivision may be continued indefinitely, but always with the same result, as indicated in Fig. 216. This suggests that the molecules of a magnetized bar may themselves be little magnets arranged in rows with their opposite poles in contact. If an unmagnetized piece of hard steel is pounded vigorously while it lies between the poles of a magnet, or if it is heated to redness and then allowed to cool in this position, it will be found to have become magnetized. This suggests that the GENEEAL PKOPERTIES OF MAGNETS 221 molecules of the steel are magnets even when the bar as a whole is not magnetized, and that magnetization may consist in causing them to arrange themselves in rows, end to end, just as the magnetization of the tube of iron filings mentioned above was due to a special arrangement of the filings. 272. Theory of magnetism. In an unmagnetized bar of iron or steel it is probable, then, that the molecules themselves are tiny magnets which are arranged either haphaz- ard or in little closed groups or chains, as in Fig. 217, SO that, 011 the FlG - 21L Arrangement of molecules in an unmagnetized iron bar whole, opposite poles neutralize each other throughout the bar. But when the bar is brought near a magnet, the molecules are swung around by the outside magnetic force into an arrangement somewhat like the one shown in Fig. 218, where the op- posite poles completely neutralize each other only in the middle of FIG 21g Arrangement of molec ules in a the bar. According to magnetized iron bar this view, heating and jarring weaken the magnet because they tend to shake the molecules out of alignment. On the other hand, heating and jarring facilitate magnetization when the bar is between the poles of a magnet because they assist the magnetizing force in breaking up the molecular groups and chains and getting the molecules into alignment. Soft iron has higher permea- bility than hard steel, because the molecules of the former substance are much easier to swing into alignment than those of the latter substance. Steel has a very much greater re- tentivity than soft iron, because its molecules are not so easily moved out of position when once they have been aligned. \ 222 MAUNETISM 273. Saturation. Strong evidence for the correctness of the above view is found in the fact that a piece of iron or steel cannot be magnet- ized beyond a certain limit, no matter how strong the magnetizing cm CM cm cm cm cm CM CM cm cm cm am cm annum ami an as an am an am am cm am cm an cm nm cm cannon am aicmizmcMcnaBCMcraca cm en an ami OB an cm cm ami a" rm OB axemen am cm I force is. This limit ,, FIG. 219. Arrangement of molecules m a probably corresponds to saturated magnet the condition in which the axes of all the molecules are brought into parallelism, as in Fig. 219. The magnet is then said to be saturated, since it is as strong as it is possible to make it. TERRESTRIAL MAGNETISM 274. The earth's magnetism. The fact that a compass needle always points north and south, or approximately so, indicates that the earth itself is a great magnet having an 8 pole near the geographic north pole and an N pole near the geographic south pole ; for the magnetic pole of the earth which is near the geographic north pole must of course be unlike the pole of a suspended magnet which points toward it, and the pole of the suspended magnet which points toward the north is the one which, by convention, it has been decided to call the JVpole. The magnetic pole of the earth which is near the north geo- graphic pole was found in 1831 by Sir James Ross in Boothia Felix, Canada, latitude 70 30' N., longitude 95 W. It was located again in 190.5 by Captain Amundsen (the dis- coverer of the geographic south pole, 1912) at a point a little farther west. Its approximate location is 70 5' N. and 96 46' W. It is probable that it shifts its position slowly. 275. Declination. The earliest users of the compass were aware that it did not point exactly north; but it was Columbus who, on his first voyage to America, made the discovery, much to the alarm of his sailors, that the direction of the compass WILLIAM GILBERT (1540-1603) English physician and physicist; first Englishman to appreciate fully the value of experimental observations; first to discover through careful experimentation that the compass points to the north not because of some influence of the stars, but because the earth is itself a great magnet ; first to use the word " electricity " ; first to discover that electrification can be produced by rub- bing a great many different kinds of substances ; author of the epoch-making book entitled "De Magnete, etc.," published in London in 1600 TEKKESTRIAL MAGNETISM 223 needle changes as one moves about over the earth's surface. The chief reason for this variation is found in the fact that the magnetic poles do not coincide with the geographic poles ; but there are also other causes, such as the existence of large deposits of iron ore, which produce local effects upon the needle. The number of degrees by which at a given point on the earth the needle varies from a true north-and-south line is called its declination at that point. Lines drawn over the earth through points of equal declination are called isogonic lines. 276. The dipping needle. Let an umnagnetized knitting needle a (Fig. 220) be thrust through a cork, and let a second needle b be passed through the cork at right angles to a and as close to it as possible. Let a pin c be adjusted until the system is in neutral equilibrium about b as an axis, when a is pointing east and west. Then let a be carefully magnetized by stroking one end of it, from the middle out, FJG 22Q Arrangement with the N pole of a strong magnet, and the for s } low i n g dip other end, from the middle out, with the S pole of the same magnet. If now the needle is replaced on its supports and turned into a north-and-south position, its N pole will be found to dip so as to cause the needle to make an angle of from 60 to 70 with the horizontal. The experiment shows that in this latitude the earth's mag- netic lines make a large angle with the horizontal. This angle between the earth's surface and the direction of the magnetic lines is called the dip, or inclination, of the needle. At Wash- ington it is 71 5' and at Chicago 72 50V At the magnetic pole it is of course 90, and at the so-called magnetic equator, which is an irregular curved line near the geographic equator, the dip is 0. 277. The earth's inductive action. That the earth acts like a great magnet may be very strikingly shown in the following way : Hold a steel rod (for example, a tripod rod) parallel to the earth's magnetic lines (the north end slanting down at an angle of about 70 or 75) and strike it a few sharp blows with a hammer. The rod will 224 MAGNETISM be found to have become a magnet with its upper end an S pole, like the north pole of the earth, and its lower end an N pole. If the rod is reversed and tapped again with the hammer, its magnetism will be re- versed. If held in an east-and-west position and tapped, it will become demagnetized, as will be shown by the fact that either end of it will attract either end of a compass needle. In some respects a soft-iron rod is more satisfactory for this experiment than a steel rod, on account of the smaller retentivity. QUESTIONS AND PROBLEMS 1. Make a diagram to show the general shape of the lines of force between unlike poles of two bar magnets ; between like poles. 2. Devise an experiment which will show that a piece of iron attracts a magnet just as truly as the magnet attracts the iron. 3. In testing a needle with a magnet to see if the needle is magnet ized why must you get repulsion before you can be sure it is magnetized? 4. A nail lies with its head near the N pole of a bar magnet. Diagram the nail and magnet, and draw from the N pole through the nail a closed curve to represent one line of force. 5. Explain, on the basis of induced magnetization, the process by which a magnet attracts a piece of soft iron. 6. Do the facts of induction suggest to you any reason why a horse- shoe magnet retains its magnetism better when a bar of soft iron (a keeper, or armature) is placed across its poles than when it is not so treated? (See Fig. 218.) 7. Why should the needle used in the experiment of 276 be placed east and west, when adjusting for neutral equilibrium, before it is magnetized ? 8. How would an ordinary compass needle act if placed over one of the earth's magnetic poles ? How would a dipping needle act at these points ? 9. Why are the tops of steam radiators magnetic poles, as proved by their invariable repulsion of the 5 pole of a compass ? 10. Give two proofs that the earth is a magnet. 11. A magnetic pole of 80 units' strength is 20 cm. distant from a similar pole of 30 units' strength. Find the force between them. CHAPTER XIII STATIC ELECTRICITY GENERAL FACTS OF ELECTRIFICATION 278. Electrification by friction. If a piece of hard rubber or a stick of sealing wax is rubbed with flannel or cat's fur and then brought near some dry pith balls, bits of paper, or other light bodies, these bodies are found to jump toward the rod. This sort of attraction, so familiar to us from the behavior of our hair in winter when we comb it with a rubber comb, was observed as early as 600 B. c., when Thales of Greece com- mented upon the fact that rubbed amber draws to itself threads and other light objects. It was not, however, until A. D. 1600 that Dr. William Gilbert, physician to Queen Elizabeth, and sometimes called the father of the modern science of electricity and magnetism, discovered that the effect could be produced by rubbing together a great variety of other substances besides amber and silk, such, for example, as glass and silk, sealing wax and flannel, hard rubber and cat's fur, etc. Gilbert (see opposite p. 222) named the effect which was produced upon these various substances by friction electrifi- cation, after the Greek name electron, meaning " amber." Thus, a body ivhich, like rubbed amber, has been endowed with the property of attracting light bodies is said to have been electrified, or to have been given a charge of electricity. In this statement nothing whatever is said about the nature of electricity. We simply define an electrically charged body as one which has been put into the condition in which it acts toward light bodies like the rubbed amber or the rubbed sealing wax. To 225 226 STATIC ELECTRICITY this day we do not know with certainty what the nature of electricity is, but we are fairly familiar with the laws which govern its action. The following sections deal with these laws. 279. Positive and negative electricity. Let a pith ball suspended by a silk thread, as in Fig. 221, be touched to a glass rod which has been rubbed with silk ; the ball will thus be put into the condition in which it is strongly repelled by this rod. Next let a stick of sealing wax or an ebonite rod which has been rubbed with cat's fur or flannel be brought near the charged ball. It will be found that it is not repelled but, on the contrary, is very strongly at- tracted. Similarly, if the pith ball has touched the sealing wax so that it is repelled by it, it is found to be FIG. 221. Pith-ball electroscope strongly attracted by the glass rod. Again, two pith balls both of which have been in contact with the glass rod are found to repel each other, while pith balls one of which has been in contact with the glass rod and the other with the sealing wax attract each other. Evidently, then, the electrifications which are imparted to glass by rubbing it with silk and to sealing wax by rubbing it with flannel are opposite in the sense that an electrified body that is attracted by one is repelled by the other. We say, therefore, that there are two kinds of electrification, and we arbitrarily call one positive and the other negative. Thus, a positively electrified body is one which acts with respect to other electrified bodies like a glass rod which has been rubbed with silk, and a negatively electrified body is one which acts like a piece of sealing wax which has been rubbed with flannel. These facts and definitions may be stated in the following general law: Electrical charges of like kind repel each other, while charges of unlike kind attract each other. The forces of attrac- tion or repulsion are found, like those of gravitation and magnetism, to decrease as the square of the distance increases. GENERAL FACTS OF ELECTRIFICATION 227 280. Measurement of electrical quantities. The fact of attraction and repulsion is taken as the basis for the definition and measurement of so-called quantities of electricity. Thus, a small charged body is said to contain 1 unit of electricity when it will repel an exactly equal and similar charge placed 1 centimeter away with a force of 1 dyne. The number of units of electricity on any charged body is then measured by the force which it exerts upon a unit charge placed at a given distance from it; for example, a charge w r hich at a distance of 10 centimeters repels a unit charge with a force of 1 dyne contains 100 units of elec- tricity, for this means that at a distance of 1 centimeter it would repel the unit charge w r ith a force of 100 dynes (see 279). 281. Conductors and nonconductors. Let an electroscope E (Fig. 222), consisting of a pair of gold leaves a and ft, suspended from an insulated metal rod r and protected from air currents by a case /, be connected with the metal ball B by means of a wire. Now let an ebonite rod be electrified and rubbed over B. The immediate divergence of the gold leaves will show that a portion of the electric charge placed upon B has been carried by the wire to the gold leaves, where it causes them to diverge in accordance with the law that bodies charged with the same kind of electricity repel FIG. 222. Illustrating conduction each other. Let the experiment be repeated when E and B are connected with a thread of silk or a long rod of wood instead of the metal wire. No divergence of the leaves will be observed. If a moistened thread con- nects E and B, the leaves will be seen to diverge slowly when the ball B is charged, showing that a charge is carried slowly by the moist thread. These experiments .make it clear that while electric charges pass with perfect readiness from one point to another in a wire, they are quite unable to pass along dry silk or wood, and pass with difficulty along moist silk. We are therefore accustomed to divide substances into two classes, conductors and noncon- ductors, or insulators, according to their ability to transmit A 228 STATIC ELECTRICITY electrical charges from point to point. Thus, metals and solutions of salts and acids in water are all conductors of electricity, while glass, porcelain, rubber, mica, shellac, wood, silk, vaseline, turpentine, paraffin, and oils are insulators. No hard-aid-fast line, however, can be drawn between conduc- tors and nonconductors, since all so-called insulators conduct to some slight extent, while the so-called conductors differ greatly in the facility with which they transmit charges. The fact of conduction brings out sharply one of the most essential distinctions between electricity and magnetism. Mag- netic poles exist only in iron and steel, while electrical charges may be communicated to any body whatever, provided it is insulated. These charges pass over conductors and can be transferred by contact from one body to any other, while mag- netic poles remain fixed in posi- tion and are wholly uninfluenced by contact with other bodies, unless these bodies themselves are magnets. FIG. 223. Illustrating induction 282. Electrostatic induction. Let the ebonite rod be electrified by friction and slowly brought toward the knob of the gold-leaf electroscope (Fig. 223). The leaves will be seen to diverge, even though the rod does not approach to within a foot of the electroscope. This makes it clear that the mere influence which an electric eharge exerts upon a conductor placed in its neighborhood is able to produce electrification in that conductor. This method of producing electrification is called electrostatic induction. As soon as the charged rod is removed, the leaves will be seen to collapse completely. This shows that this form of elec- trification is only a temporary phenomenon which is due simply to the presence of the charged body in the neighborhood. GENEKAL FACTS OF ELECTRIFICATION 229 283. Nature of electrification produced by induction. Let a metal ball A (Fig. 224) be strongly charged by rubbing it with a charged rod, and let it then be brought near an insulated* metal body B which is provided with pith balls or strips of paper a, b, c, as shown. The di- vergence of a and c will show that the ends of B have received electrical charges because of the presence of A, while the failure of b to diverge /" "N o. . _ _ will show that the middle of is ( + ) Cl _ ? _ t) uncharged. Further, the rod which charged A will be found to repel c FIG. 224. Nature of induced but to attract a. charges We conclude, therefore, that when a conductor is brought near a charged body, the end away from the inducing charge is electrified with the same kind of electricity as that on the in- ducing body, while the end toward the inducing body receives electricity of the opposite kind. 284. The electron theory of electricity. The atoms of all substances are now known to contain as constituents both positive and negative electricity, the latter existing in the form of minute corpuscles, or electrons, each of which has a mass 1 8 * of that of the hydrogen atom. These electrons are probably grouped in some way about the positive electricity as a nucleus. The sum of the negative charges of these elec- trons is supposed to be just equal to the positive charge of the nucleus, so that in its normal condition the whole atom is neutral, or uncharged. But in conductors electrons are con- tinually getting loose from the atoms and reentering other atoms, so that at any given instant there are in every con- ductor a number of free negative electrons and a correspond- ing number of atoms which have lost electrons and which are therefore positively charged. Such a conductor would, as a- whole, show no charge of either positive or negative electricity. * Sulphur is practically a perfect insulator in all weathers, wet or dry. Metal conductors of almost any shape resting upon pieces of sulphur will serve the purposes of this experiment in summer or winter. 230 STATIC ELECTRICITY But as soon as a body charged, for example, positively (Fig. 224) is brought near su^h a conductor, the negatively charged electrons are attracted to the near end, leaving behind them the positively charged atoms, which are not free to move from their positions. On the other hand, if a negatively charged body is brought near the conductor, the negative electrons stream away and the near end is left with the immovable plus atoms. As soon as the inducing charge is removed, the con- ductor becomes neutral again, because the little negative cor- puscles return to their former positions under the influence of the attraction of the positive atoms. This is the present-day picture of the mechanism of electrification by induction. The charge of one electron is called the elementary electrical charge. Its value has recently been accurately measured. There are 2.095 billion of them in one of the units denned in 280. Every electrical charge consists of an exact number of these ultimate electrical atoms. 285 . Charging by induction. Let two metal balls or two eggshells, A and B, which have been gilded or covered with tin foil be suspended by silk threads and touched together, as in Fig. 225. Let a positively charged body C be brought near them. As described above, A and B will at once exhibit evidences of electrification ; that is, A will repel a positively charged pith ball, while B will attract it. If C is re- moved while A and B are still in contact, the separated charges reunite and A and B cease to exhibit electrification. But if FIG. 22o. Obtaining a A and B are separated from each other plug ftnd a mmug chftrge while C is in place, A will be found to bv i n( j uc ti O n remain positively charged and B nega- tively charged. This may be proved either by the attractions and repul- sions which they show for charged rods brought near them or by the effects which they produce upon a charged electroscope brought into their vicinity, the leaves of the latter falling together when it is brought near one and spreading farther apart when brought near the other. BENJAMIN FRANKLIN (1706-1790) Celebrated American statesman, philosopher, and scientist; born at Boston, the sixteenth child of poor parents ; printer and pub- lisher by occupation; pursued scientific studies in electricity as a diversion rather than as a profession ; first proved that the two coats of a Leyden jar are oppositely charged; introduced the terms positive and negative electricity; proved the identity of lightning and frictional electricity by flying a kite in a thunder- storm and drawing sparks from the insulated lower end of the kite string ; invented the lightning rod ; originated the one-fluid theory of electricity which regarded a positive charge as indi- cating an excess, a negative charge a deficiency, in a certain normal amount of an all-pervading electrical fluid FRANKLIN'S KITE EXPERIMENT In June, 1752, Franklin demonstrated the identity of the electric spark and light- ning. To prevent his kite from being torn in the rain he made it of a silk handker- chief. The lower end of the kite string and a silk ribbon were tied to the ring of a key, and, to prevent any charge that might appear upon the string and the key from escaping through his body to the earth, he held the kite by grasping the insulating silk ribbon. Standing under a shed to keep the ribbon dry, Franklin, by presenting his knuckle to the key, obtained sparks similar to those produced by his electric machine. With these sparks he charged his Leyden jar and used it to give a shock. Indeed, he performed with lightning all the experiments which he had previously performed with sparks from his frictional machine. The experiment is dangerous and should not be attempted by inexperienced persons GENERAL FACTS OF ELECTRIFICATION 231 We see, therefore, that if tve cut in two, or separate into two parts, a conductor while it is under the influence of an electric charge, ive obtain two permanently charged bodies, the remoter part having a charge of the same sign as that of the inducing charge, and the near part having a charge of unlike sign. Under the influence of the positive charge on C the negative electrons moved out of A into B, which act made A positive and B negative. Let the conductor R (Fig. 226) be touched at a by -the finger while a charged rod C is near it. Then let the finger be removed and after it the rod C. If now a negatively charged pith ball is brought near B, it will be repelled, showing that B has become negatively charged. In (+ B -a) this experiment the body of the experimenter corresponds to the egg A of the preceding experiment^ ^ m A fe d and removing the finger from B ti(m hag ft charge Qf ^ Qpposite corresponds to separating the two that of the inducing charge eggshells. Let the last experi- ment be repeated with only this modification, that B is touched at b rather than at a. When B is again tested with the pith ball, it will still be found to have a negative charge, exactly as when the finger was touched at a. We conclude, therefore, that no matter where the body B is touched, the sign of the charge left upon it is always opposite to that of the inducing charge. This is because the negative electricity, that is, the electrons, can under no circumstances escape from b so long as C is present, for they are bound by the attraction of the positive charge on C. Indeed, the final negative charge on B is due merely to the fact that the positive charge on C pulls electrons into B from the finger, no matter where B is touched. In the same way, if C had been negative, it would have pushed electrons off from B through the finger and thus have left B positively charged. 232 STATIC ELECTRICITY 286. Charging the electroscope by induction. Let an ebonite rod which has been rubbed with catskin be brought near the knob of the electroscope (Fig. 223). The leaves at once diverge. (Make a dia- gram of the electroscope with the negatively charged ebonite rod near the knob. By use of + and signs explain the electrical condition of both the knob and the leaves.) Let the knob be touched with the finger while the rod is held in place. The leaves will fall together. (Explain by a diagram as before.) Let the finger be removed and then the rod. The leaves will fly apart again. (By a diagram explain the final elec- trical condition of both the knob and the leaves.) The electroscope has been charged by induction, and since the charge on the ebonite rod was negative, the charge on the electroscope must be positive. If this conclusion is tested by bringing the charged ebonite rod near the electroscope, the leaves will fall together as the rod approaches the knob. How does this prove that the charge on the electroscope is positive ? If the empty neutral hand approaches the knob, the leaves diverge less. Explain. 287. Plus and minus electricities always appear simultane- ously and in equal amounts. Let an ebonite rod be completely discharged by passing it quickly through a Bunsen flame. Let a flannel cap having a silk thread attached be slipped over the rod, as in Fig. 227, and twisted rapidly around a immber of times. When rod and cap together are held near a charged electroscope, no effect will be observed ; but if the cap is pulled off, it will be found to be positively charged, while the rod will be found to have a negative charge. FIG. 22 7. Plus and Since the two together produce no effect, minus electricities the experiment shows that the plus and alwavs developed in equal amounts minus charges were equal in amount. This experiment confirms the view already brought forward in connection with induction, that electrification always consists in a separation of plus and minus charges which already exist in equal amounts within the bodies in which the electrification is developed. DISTRIBUTION OF CHARGE 233 QUESTIONS AND PROBLEMS 1. If pith balls, or any light figures, are placed between two plates (Fig. 228), one of which is connected to earth and the other to one knob of an electrical machine in operation, the figures will bound back and forth between the two plates as long as the machine is operated. Explain. 2. Given a gold-leaf electroscope, a glass rod, and a piece of silk, how, in general, would you proceed to test the sign of the electrification of an unknown charge? 3. Charge a gold-leaf electroscope by induction from a glass rod. Warm a piece of paper and stroke it on the clothing. Hold it over the charged electroscope. If the divergence of the gold leaves is increased, is the charge on the paper + or ? If the divergence is decreased, what is the sign of the charge on the paper ? 4. If you are given a positively charged insulated sphere, how could you charge two other spheres, one positively and the other negatively, without diminishing the charge on the first sphere? 5. If you bring a positively charged glass rod near the knob of an electroscope and then touch the knob, why do you not remove the nega- tive electricity which is on the knob ? 6. In charging an electroscope by induction, why must the finger be removed before the removal of the charged body? 7. If you hold a brass rod in the hand and rub it with silk, the rod will show no sign of electrification ; but if you hold the brass rod with a piece of sheet rubber and then rub it with silk, you will find it elec- trified. Explain. 8. State as many differences as you can between the phenomena of magnetism and those of electricity. 9. If an electrified rod is brought near to a pith ball siispended by a silk thread, the ball is first attracted to the rod and then repelled from it. Explain this. DISTRIBUTION OF ELECTRIC CHARGE UPON CONDUCTORS 288. Electric charges reside only upon the outside surface of conductors. Let a deep tin cup (Fig. 229) be placed upon an insulating stand and charged as strongly as possible either from an ebonite rod or from an electrical machine. If now a smooth metal ball suspended by a silk thread is touched to the outside of the charged cup and then brought near the knob of a charged electroscope, it will show a strong charge ; but if it is touched to the inside of the cup, it will show no charge at all. 234 STATIC ELECTRICITY These experiments show that an electric charge resides entirely on the outside surface of a conductor. This is a result which might have been inferred from the fact that all the little electrical charges of which the total charge is made up repel each other and there- fore move through the conductor until they are, on the average, as far apart as possible. 289. Density of charge greatest where curvature of surface is greatest. Since all of the parts of an electric charge tend, because of their mutual repulsions, to get as far apart as possible, we should infer that if a charge of either sign is placed upon an oblong conductor like that of Fig. 230, (1), it will distribute itself so that the electrification at the ends will be stronger than that at the middle. FIG. 229. Proof that charge resides on surface (1) To test this inference let a proof plane a flat metal disk (for example, a cent) provided with an insulating handle be touched to one end of such a charged body, the charge conveyed to a gold-leaf electroscope, and the amount of separation of the leaves noted. Then let the experiment be repeated when the proof plane touches the middle of the body. The separation of the leaves in the latter case will be found to be very much less than in the former. If we should test the distribu- tion on a pear-shaped body (Fig. 230, (2)) in the same way, we should find the density of electrification considerably greater on the small end than on the large one. By density of electrification is meant the quantity of electricity on unit area of the surface. 290. Discharging effect of points. The above experiments indicate that if one end of a pear-shaped body is made more and more pointed, then, when the body is charged, the electric FIG. 230. Distribution of charge over oblong bodies DISTRIBUTION OF CHARGE 235 density on this end will become greater and greater. The fol- lowing experiment will show what happens when the conductor is provided with a sharp point. Let a very sharp needle be attached to any smooth insulated metal body provided with paper or pith-ball indicators, as in Fig. 224, p. 229. If the body is now charged either with a rubbed rod or with an electric machine, as soon as the supply of electricity is stopped the paper indi- cators will immediately fall, showing that the body is losing its charge. To show that this is certainly due to the effect of the point, remove the needle and repeat. The indicators will fall very slowly if at all. The experiment shows that the electrical density upon the point is so great that the charge escapes from it into the air. This is because the intense charge on the point causes many of the adjacent molecules of the air to lose an electron. This leaves these molecules positively charged. The free electrons attach themselves to neutral molecules, thus charging them negatively. One set of these electrically charged molecules (called ions) is attracted to the point and the other repelled from it. The former set move to the conductor, give up their charges to it, and thus neutralize the charge upon it. The effect of points may be shown equally well by charging the gold- leaf electroscope and holding a needle in the hand within a few inches of the knob. The leaves will fall together rap- idly. In this case the needle point becomes elec- trified by induction and discharges to the knob electricity of the opposite kind to that on the knob, thus neutralizing its charge. An entertain- ing variation of the last experiment is to attach a tassel of tissue paper to an insulated conductor j, 031 Dischare 1 and electrify it strongly. The paper streamers i ng effect of points under their mutual repulsions will stand out in all directions, but as soon as a needle point is held in the hand near them, they will fall together (Fig. 231), being discharged as described above. 291 . The electric whirl. Let an electric whirl (Fig. 232) be bal- anced upon a pin point and attached to one knob of an electric machine. As soon as the machine is started, the whirl will rotate rapidly in the direction of the arrows. 236 STATIC ELECTRICITY FIG. 232. The electric whirl FIG. 233. The elec- tric wind The explanation is as follows : The air close to each point is ionized, as explained in 290. The ions of sign unlike that of the charge on the point are drawn to the point and discharged. The other set of ions is repelled. But since this repulsion is mu- tual, the point is pushed back with the same force with which these ions are pushed forward ; hence the rotation. The repelled ions in their turn drag the air with them in their forward motions and thus produce the " electric wind," which may be detected easily by the hand or by a candle flame (Fig. 233). 292. Lightning and lightning rods. It was in 1752 that Franklin (see opposite p. 230), during a thunderstorm, sent up his historic kite (see opposite p. 231). This kite was pro- vided with a pointed wire at the top. As soon as the hempen kite-string had become wet he succeeded in drawing ordinary electric sparks from a key attached to the lower end. This experiment demonstrated for the first time that thunderclouds carry ordinary electrical charges which may be drawn from them by points, just as the charge was drawn from the tassel in the experiment of 290. It also showed that lightning is nothing but a huge electric spark. Franklin applied this dis- covery in the invention of the lightning rod. The way in which the rod discharges the cloud and protects the building is as follows : As the charged cloud approaches the building it induces an opposite charge in the rod. This induced charge escapes rapidly and quietly from the sharp point in the manner explained above and thus neutralizes the charge of the cloud. To illustrate, let a metal plate C (Fig. 234) be supported above a metal ball E, and let C and E be attached to the two knobs of an electri- cal machine. When the machine is started, sparks will pass from C to E. POTENTIAL AND CAPACITY 237 But if a point p is connected to E, the sparking will cease ; that is, the point will protect E from the discharges, even though the distance Cp be considerably greater than CE. The lower end of a lightning rod should be buried deep enough so that it will always be surrounded by moist earth, since dry earth is a poor conductor. It will be seen, therefore, that lightning rods protect buildings not because they conduct the lightning to earth, but because they prevent the formation of powerful charges in the neighborhood of the buildings on which they are placed. Flashes of lightning over a mile long have frequently been observed. Thunder is due to the violent expansion of heated air along the path of discharge. The roll of thunder is due to reflections from clouds, hills, etc.* FIG. 234. Illustrating the action of a lightning rod POTENTIAL AND CAPACITY 293. Potential difference. There is a very instructive anal- ogy between the use of the word " potential " in electricity and " pressure " in hydrostatics. For ex- ample, if water will flow from tank A to tank B through the connecting pipe R (Fig. 235), we infer that the hydrostatic pressure at a must be greater than that at J, and we attribute the flow directly to this difference in pressure. In exactly the same way, if, when two bodies A and B (Fig. 236) are connected by a conducting wire r, a charge of -f- electricity R * A laboratory exercise on static electrical effects should follow the discus- sion of this section. See, for example, Experiment 27 of the authors' Manual. 238 STATIC ELECTRICITY is found to pass from A to B (that is, if electrons are found to pass from B to A) we say that the electrical potential is higher at A than at B, and we assign this difference of poten- tial as the cause of the flow.* Thus, just as water tends to flow from points of higher hydrostatic pressure to points of lower hydrostatic pressure, so elec- ^_^ ^_^ tricity tends to flow from points of ( 269 ' The sounder a comparatively large current. 315. Plan of a telegraphic system. The actual arrangement of the various parts of a telegraphic system is shown in the drawings on the opposite page. When an operator at Chicago wishes to send a message to New York, he first opens the switch which is connected to his key, and which is always kept closed except when he is sending a message. He then begins to operate his key, thus controlling the clicks of both his own sounder and that at New York. When the Chicago switch is closed and the one at New York open, the New York operator is able to send a message back over the same line. In practice a message is not usually sent as far as from Chicago to New York over a single line, save in the case of transoceanic cables. Instead it is automatically transferred, say at Cleveland, to a second line, which carries it on to Buffalo, where it is again transferred to a third line, which carries it on to New York. The transfer is made in precisely the same way as the transfer from the main circuit to the sounder circuit. If, for example, the sounder circuit at Cleveland is lengthened so as to extend to Buffalo,. and if the sounder itself is replaced by a relay (called in this case a repeater), and the local battery by a line battery, then the sounder circuit has been transformed into a repeater circuit, and all the conditions are met for an automatic transfer of the message at Cleveland. QUESTIONS AND PROBLEMS 1. Draw a diagram showing how an electric bell works. 2. Draw a diagram of a short two-station telegraph line which has only one instrument at each station. 3. Draw a diagram showing how the relay and sounder operate in a telegraphic circuit. Why is a relay used? 262 . ELECTRICITY IN MOTION RESISTANCE AND ELECTROMOTIVE FORCE 316. Electrical resistance.* Let the circuit of a galvanic cell be connected through a lecture-table ammeter, or any low-resistance gal- vanometer, and, for example, 20 feet of No. 30 copper wire, and let the deflection of the needle be noted. Then let the copper wire be replaced by an equal length of No. 30 German-silver wire. The deflection will be found to be a very small fraction of what it was at first. A cell, therefore, which is capable of developing a certain fixed electrical pressure is able to force very much more current through a given wire of copper than through an exactly similar wire of German silver. We say, therefore, that German silver offers a higher resistance to the passage of electricity than does copper. Similarly, every particular substance has its own characteristic power of transmitting electrical currents. Since silver is the best conductor known, resistances of different substances are commonly referred to it as a standard, and the ratio between the resistance of a. given wire of any substance and the resistance of an exactly similar silver wire is called the specific resistance of that sub- stance. The specific resistances of some of the commoner metals in terms of silver are given below: Silver . . . 1.00 Soft iron . . G.OO German silver 18.1 Copper . . . 1.11 Platinum . . 7.20 Mercury. . . 63.1 Aluminium. . 1.87 Hard steel . . 13.5 Nichrome . . 66.6 The resistance of any conductor is directly proportional to its length and inversely proportional to the area of its cross section or to the square of its diameter. The unit of resistance is the ohm, named after Georg Ohm (see opposite p. 268). A length of 9.35 feet of No. 30 copper * This subject should be accompanied and followed by laboratory experi- ments on Ohm's law, on the comparison of wire resistances, and on the measurement of internal resistances. See, for example, Experiments 32, 33, and 34 of the authors' Manual. RESISTANCE AND ELECTROMOTIVE FORCE 263 wire, or 6.2 inches of No. 30 German-silver wire, has a resistance of about one ohm. The legal definition of the ohm is a resistance equal to that of a column of mercury 106.3 centimeters long and 1 square millimeter in cross section, at 0. 317. Resistance and temperature. Let the circuit of a galvanic cell be closed through a galvanometer of very low resistance and about 10 feet of No. 30 iron wire wrapped about a strip of asbestos. Let the deflection of the galvanometer be observed as the wire is heated in a Bunsen flame. As the temperature rises higher and higher the current will be found to fall continually. The experiment shows that the resistance of iron increases with rising temperature. This is a general law which holds for all metals. In the case of liquid conductors, on the other hand, the resistance usually decreases with increasing temperature. Carbon and a few other solids show a similar behavior, the filament in the early form of incandescent electric lamp having only about half the resistance when hot which it has when cold. 318. Electromotive force and its measurement.* The poten- tial difference which a galvanic cell or any other generator of electricity is able to maintain between its terminals when these terminals are not connected by a wire that is, the total electrical pressure which the generator is capable of exerting is commonly called its electromotive force, usually abbreviated to E.M.F. TJie E.M.F. of an electrical generator may be de- fined as its capacity for producing electrical pressure, or P.D. This P.D. might be measured, as in 294, by the deflection produced in an electroscope when one terminal is connected to the case of the electroscope and the other terminal to the knob. Potential differences are, in fact, measured in this way in all so-called electrostatic voltmeters. * This subject should be preceded or accompanied by laboratory work on E.M.F. See, for example, Experiment 31 of the authors' Manual. 264 ELECTKICITY IN MOTION #1- FIG. 270. Hydrostatic analogy of the action of a galvanic cell The more common type of potential-difference measurer consists, however, of an instrument made like a galvanometer (Fig. 268), except that the coil of wire is made of very many turns of extremely fine wire, so that it carries a very small current. The amount of current which it does carry, however, is proportional to the difference in elec- trical pressure existing between its ends when these are touched to the two points whose P.D. is sought. The principle un- derlying this type of voltmeter will be better understood from a consideration of the following water analogy. If the stopcock K (Fig. 270) in the pipe con- necting the water tanks C and D is closed, and if the water wheel A is set in motion by applying a weight IF, the wheel will turn until it creates such a difference in the water levels between C and D that the back pressure against the left face of the wheel stops it and brings the weight Wto rest. In precisely the same way the chemical action within the galvanic cell whose terminals are not joined (Fig. 271) develops positive and negative charges upon these terminals ; that is, creates a P.D. between them until the back electrical pressure through the cell due to this P.D. is sufficient to put a stop to further chemical action. The seat of the E.M.F. is at the sur- faces of contact of the metals with the acid, where the chemical actions take place. Now, if the water reservoirs (Fig. 270) are put in communication by opening the stopcock K, the differ- ence in level between C and D will begin to fall, and the wheel will begin to build it up again. But if the carrying FIG. 271. Meas- urement of P.D. between the ter- minals of a gal- vanic cell RESISTANCE AND ELECTROMOTIVE FORCE 265 capacity of the pipe ab is small in comparison with the capac- ity of the wheel to remove water from D and supply it to (7, then the difference of level which permanently exists between C and D when K is open will not be appreciably smaller than when it is closed. In this case the current which flows through ab may obviously be taken as a measure of the difference in pressure which the pump is able to maintain between C and D when K is closed. In precisely the same way, if the terminals C and D of the cell (Fig. 271) are connected by attaching to them the terminals a and b of any conductor, they at once begin to discharge through this conductor, and their P.D. therefore begins to fall. But if the chemical action in the cell is able to recharge C and D very rapidly in comparison with the ability of the wire to discharge them, then the P.D. between C and D will not be appreciably lowered by the presence of the connecting conductor. In this case the current which flows through the conducting coil, and therefore the deflec- tion of the needle at its center, may be taken as a measure of the electrical pres- sure developed by the cell, that is, of the P.D. between its unconnected terminals. FIG. 272. Lecture- Ihe common voltmeter (Fig. 272) is, table voltmeter then, exactly like an ammeter, save that it offers so high a resistance to the passage of electricity through it that it does not appreciably reduce the P.D. between the points to which it is connected. 319. The commercial voltmeter. Fig. 273 shows the con- struction of the common form of commercial voltmeter. It dif- fers from the ammeter (Fig. 264) in that the shunt is omitted, and a high-resistance coil R is put in series with the moving coil c. The resistance of a voltmeter may be many thousand ohms. The current that passes through it is very small. 266 ELECTRICITY IN MOTION 320. The electromotive forces of galvanic cells. Let a voltmeter of any sort be connected to the terminals of a simple galvanic cell, like that of Fig. 242. Then let the distance between the plates and the amount of their immersion be changed through wide limits. It will be found that the deflection produced is altogether independent of the shape or size . of the plates or their distance apart. But if the nature of the plates is changed, the deflection changes. Thus, while copper and zinc in dilute sulphuric acid have an E.M.F. of one volt, carbon and zinc show an E.M.F. of at least 1.5 volts, while carbon and copper will show an E.M.F. of very much less than a volt. Sim- ilarly, by changing the nature of the liquid in which the plates are immersed, we can produce changes in the deflection of the voltmeter.* FIG. 273. Principle of commercial voltmeter We learn, therefore, that the E.M.F. of a galvanic cell depends simply upon the materials of which the cell is composed, and not at all upon the shape, size, or distance apart of the plates. 321. Fall of potential along a conductor carrying a current. Not only does a P.D. exist between the terminals of a cell on open circuit, but also between any two points on a conductor through which a current is passing. For example, in the electrical circuit shown in Fig. 274 the potential at the point a is higher than that at m, that at m higher than that at n, etc., just as, in the water circuit shown in Fig. 275, the * A vertical lecture-table voltmeter (Fig. 272) and a similar ammeter are desirable for this and some of the following experiments, but homemade high- and low-resistance galvanometers, like those described in the authors' Manual, are thoroughly satisfactory, save for the fact that one student must take the readings for the class. RESISTANCE AND ELECTROMOTIVE FORCE 26T FIG. 274. Showing method of connecting voltmeter to find P.D. between any two points m and n on an electrical circuit hydrostatic pressure at a is greater than that at m, that at m greater than that at n, etc. The fall in the water pressure between m and n (Fig. 275) is measured by the water head n's. If we wish to measure the fall in electrical potential between m and n (Fig. 274), we touch the terminals of a voltmeter to these points in the manner shown in the figure. Its reading gives us at once the P.D. between m and n in volts, provided always that its own current- carrying capacity is so small that it does not appreciably lower the P.D. between the points m and n by being touched across them ; that is, provided the current which flows through it is neg- ligible in comparison with that which flows through the conductor which already joins the points m and n. 322. Ohm's law. In 1826 Ohm announced the discovery that the currents furnished ly different gal- vanic cells, or combinations of cells, are alivays directly proportional to the E.M.FSs existing in the circuits in which the currents flow, and inversely proportional to the total resistances of these circuits ; that is, if / represents the current in amperes, E the E.M.F. in volts, and R the resistance of the circuit in ohms, then Ohm's law as applied to the complete cir- m R rt FIG. 275. Hydrostatic anal- ogy of fall of potential in an electrical circuit cuit is TT /= ; that is, current = electromotive force resistance 268 ELECTRICITY IN MOTION As applied to any portion of an electrical circuit, Ohm's law is J= . that is , curren t = Potential difference ^ R resistance where P.D. represents the difference of potential in volts be- tween any two points in the circuit, and R the resistance in ohms of the conductor connecting these two points. This is one of the most important laws in physics. Both of the above statements of Ohm's law are included in the equation volts , ON amperes = (3) ohms 323. Internal resistance of a galvanic cell. Let the zinc and copper plates of a simple galvanic cell be connected to an ammeter, and the distance between the plates then increased. The deflection of the needle will be found to decrease, or if the amount of immersion is decreased, the current also will decrease. Now, since the E.M.F. of a cell was shown in 320 to be wholly independent of the area of the plates immersed or of the distance between them, it will be seen from Ohm's law that the change in the current in these cases must be due to some change in the total resistance of the circuit. Since the wire which constitutes the outside portion of the circuit has remained the same, we must conclude that the liquid within the cell, as well as the external wire, offers resistance to the pas- sage of the current. This internal resistance of the liquid is directly proportional to the distance between the plates, and inversely proportional to the area of the immersed portion of the plates. If, then, we represent the external resistance of the circuit of a galvanic cell by R e and the internal by R^ Ohm's law as applied to the entire circuit takes the form (4) GEORG SIMON OHM (1787-1854) German physicist and discoverer of the famous law in physics which bears his name. He was born and educated at Erlangen. It was in 1826, while he was teaching mathematics at a gym- nasium in Cologne, that he published his famous paper on the experimental proof of his law. At the time of his death he was professor of experimental physics in the university at Munich. The ohm, the practical unit of resistance, is named in his honor -. o "*"* O gj "^ ^ "^ 5 bx) .c "^ *c B I a &1? |1 o t o & 5 * 'o ^ c t . ^ < c -^ * M ^ O O PH Q QJ ,_! ^ 0^ si rt 05 T5 ^ p i .2! r w w ^ G S % o>^ 2^ e C ,- o> a G U S o 5 fi " g g -C g .2 1 sme^i /TBTr^ FIG. 277. Series connections through each of them in succession, they are said to be connected in series. The total resistance of a number of conductors so connected is the sum of the several resistances. Thus, in the case shown in the figure the total resistance between a and b is 10 ohms. When n exactly similar conductors are joined in the manner shown in Fig. 278, that is, in parallel or multiple, the total resistance between a and b is \/n of the resistance of one of them ; for obviously, with a given P.D. between the points a and b, four conductors will carry four times as much current as one, and n conductors will carry n times as much current as one. Therefore the resistance, which is inversely proportional to the carrying capacity (see 322), is \/n as much as that of one. 327. Shunts. A wire connected in parallel with another wire is said to be a shunt to that wire. Thus, the conductor X (Fig. 279) is said to be shunted across the resistance R. Under such conditions the currents carried by R and X will be inversely proportional to their resistances, so that if X is 1 ohm and R is 10 ohms, R will carry -^ as much current as X, or -i. of the whole current. In other words, since the carrying power, or conductance, of X is ten times that of R, ten times as much current will flow through X as through R, or -j-^- of the whole current will pass through the shunt. The ammeter (Fig. 264) uses a shunt of exceedingly small resistance. FIG. 278. Parallel con- nections FIG. 279. A shunt RESISTANCE AND ELECTKOMOTIVE FORCE 271 QUESTIONS AND PROBLEMS 1. How can you prove that the E.M.F. of a cell does not depend upon the size or nearness of the plates ? 2. How can you prove that the internal resistance of a cell becomes smaller when the plates are made larger ? when placed closer together ? 3. If the potential difference between the terminals of a cell on open circuit is to be measured by means of a galvanometer, why must the galvanometer have a high resistance ? 4. Why are iron wires used on telegraph lines but copper wires on trolley systems? 5. A voltmeter which has a resistance of 2000 ohms is shunted across the terminals A and B of a wire which has a resistance of 1 ohm. What fraction of the total current flowing from A to B will be carried by the voltmeter ? 6. In a given circuit the P.D. across the terminals of a resistance of 19 ohms is found to be 3 volts. What is the P.D. across the termi- nals of a 3-ohm wire in the same circuit ? 7. The resistance of a certain piece of German-silver wire is 1 ohm. What will be the resistance of another piece of the same length but of twice the diameter ? 8. How much current will flow between two points whose P.D. is 2 volts, if they are connected by a wire having a resistance of 12 ohms ? 9. What P.D. exists between the ends of a wire whose resistance is 100 ohms when the wire is carrying a current of .3 ampere ? 10. If a voltmeter attached across the terminals of an incandescent lamp shows a P.D. of 110 volts, while an ammeter connected in series with the lamp indicates a current of .5 ampere, what is the resistance of the incandescent filament ? 11. A certain storage cell having an E.M.F. of 2 volts is found to furnish a current of 20 amperes through an ammeter whose resistance is .05 ohm. Find the internal resistance of the cell. 12. The E.M.F. of a certain battery is 10 volts and the strength of the current obtained through an external resistance of 4 ohms is 1.25 amperes. What is the internal resistance of the battery ? 13. Consider the diameter of No. 20 wire to be three times that of No. 30. A certain No. 30 wire 1 meter long has a resistance of 6 ohms. What would be the resistance of 4 meters of No. 20 wire made of the same metal? 14. Three wires, each having a resistance of 15 ohms, were joined in parallel and a current of 3 amperes sent through them. How much was the E.M.F. of the current? 272 ELECTRICITY IN MOTION r-*H FIG. 280. Chemical actions in the vol- taic cell PRIMARY CELLS 328. Study of the action of a simple cell. If the simple cell already described, that is, zinc and copper strips in dilute sulphuric acid, is carefully observed, it will be seen that, so long as the plates are not connected by a conductor, fine bubbles of gas are slowly formed at the zinc plate, but none at the copper plate. As soon, however, as the two strips are put into metallic connection, bubbles appear in great numbers about the copper plate (Fig. 280), and at the same time a current manifests itself in the connecting wire. These are bubbles of hydro- gen. Their appearance on the zinc may be pre- vented either by using a plate of chemically pure zinc or by amalgamating impure zinc, that is, by coating it over with a thin film of mercury. But the bubbles on the copper cannot be thus disposed of. They are an invariable accompaniment of the current in the circuit. If the current is allowed to run for a considerable time, it will be found that the zinc wastes away, even though it has been amalgamated, but that the copper plate does not undergo any change. We learn, therefore, that the electrical current in the simple cell is accompanied by the eating up of the zinc plate by the liquid, and by the evolution of hydrogen bubbles at the copper plate. In every type of galvanic cell, actions similar to these two are always found ; that is, one of the plates is always eaten up, and upon the other plate some element is deposited. The zinc, which is eaten, is the one which we found to be negatively charged when tested ( 300), so that when the terminals are connected through a wire, the negative electrons flow through this wire from the zinc plate to the copper plate. This means, in accordance with the convention mentioned in the footnote to 293, that the direction of the current thrmtgh the external circuit is always from the uneaten to the eaten plate. 329. Local action and amalgamation. The cause of the appearance of the hydrogen bubbles at the surface of im- pure zinc when dipped in dilute sulphuric acid is that little PEIMARY CELLS 273 electrical circuits are set up between the zinc and the small impurities in it (carbon or iron particles) in the manner indicated in Fig. 281. If the zinc is pure, these little local currents cannot, of course, be set up, and consequently no hydrogen bubbles appear. Amalgamation stops this so-called local action, because the mercury dissolves the zinc, while it does not dissolve the carbon, iron, or other impurities. The zinc-mercury amalgam formed is a homogeneous substance which spreads over the whole surface and covers up the impurities. It is important, therefore, to amalgamate the zinc in a battery, in order to prevent the consumption of the zinc when the cell is on open circuit. The zinc is under all circumstances eaten up when the current is flowing, amalgamation serving only to prevent its consumption when the circuit is open. FIG. 281. Local action 330. Theory of the action of a simple cell. A simple cell may be made of any two dissimilar metals immersed in a solution of any acid or salt. For simplicity let us examine the action of a cell cbmposed of plates of zinc and copper immersed in a dilute solution of hydrochloric acid. The chemical formula for hydrochloric acid is HC1. This means that each molecule of the acid consists of one atom of hydrogen combined with one atom of chlorine. As was explained under electrolysis ( 302), the acid dissociates into positively and negatively charged ions (Fig. 282). When a zinc plate is placed in such a solu- tion, the acid attacks it and pulls zinc atoms into solution. Now, whenever a metal dis- solves in an acid, its atoms, for some unknown reason, go into solution bearing little positive charges. The corresponding negative charges must be left on the zinc plate in precisely the same way in which a negative charge is left on silk when positive electrification is produced on a glass rod by rubbing it the silk. It is in this way, then, that we account for the negative FIG. 282. Showing disso- ciation of hydrochloric- acid molecules in water 274 ELECTRICITY IN MOTION charge which we found upon the zinc plate in the experiment which was performed with the galvanic cell and the electroscope (see 300). The passage of positively charged zinc ions into solution gives a posi- tive charge to the solution about the zinc plate, so that the hydrogen ions tend to be repelled away from this plate. When these repelled hydrogen ions reach the copper plate, some of them give up their charges to it and then collect as bubbles of hydrogen gas. It is in this way that we account for the positive charge which we found on the copper plate in the experiment of 300. If the zinc and copper plates are not connected by an outside con- ductor, this passage of positively charged zinc ions into solution con- tinues but a very short time, for the zinc soon becomes so strongly charged negatively that it pulls back on the + zinc ions with as much force as the acid is pulling them into solution. In precisely the same way the copper plate soon ceases to take up any more positive electricity from the hydrogen ions, since it soon acquires a large enough + charge to repel them from itself with a force equal to that with which they are being driven out of solution by the positively charged zinc ions. It is in this way that we account for the fact that on open circuit no chemi- cal action goes on in the simple galvanic cell, the zinc and copper plates simply becoming charged to a definite difference of potential which is called the E.M.F. of the cell. When, however, the copper and zinc plates are connected by a wire, a current at once flows from the copper to the zinc, and the plates thus begin to lose their charges. This allows the acid to pull more zinc into solution at the zinc plate, and allows more hydrogen to go out of solution at the copper plate. These processes, therefore, go on continuously so long as the plates are connected. Hence a continuous current flows through the connecting wire until the zinc is all eaten up or the hydrogen ions have all been driven out of the solution, that is, until either the plate or the acid has become exhausted. 331. Polarization. If the simple galvanic cell described is con- nected to a lecture-table ammeter through two or three feet of No. 30 German-silver wire, the deflection of the needle will decrease slowly ; but if the hydrogen is removed from the copper plate (this can be done completely only by removing and thoroughly drying the plate), the deflection will be found to return to its first value. The experiment shows clearly that the observed falling off in current was due to the collection of hydrogen about the PKIMAKY CELLS 275 copper plate. This phenomenon of the weakening of the cur- rent from a galvanic cell is called the polarization of the celL 332. Causes of polarization. The presence of the hydrogen bubbles on the positive plate causes a diminution in the strength of the current for two reasons : first, since hydrogen is a nonconductor, by collecting on the plate it diminishes the effective area of the plate and therefore increases the internal resistance of the cell; second, the collection of the hydrogen about the copper plate lowers the E.M.F. of the cell, because it virtually substitutes a hydrogen plate for the copper plate, and we have already seen ( 320) that a change in any of the materials of which a cell is composed changes its E.M.F. That there is a real fall in E.M.F. as well as a rise in internal resistance when a cell polarizes may be directly proved in the following way: Let the deflection of a lecture-table voltmeter whose terminals are attached to the freshly cleaned plates of a simple cell be noted. Then let the cell's terminals be short-circuited through a coarse wire for half a minute. As soon as the wire is removed, the E.M.F., indicated by the voltmeter, will be found to be much lower than at first. It will, however, gradually creep back toward its old value as the hydrogen disappears- from the plate, but a thorough cleaning and drying of the plate will be necessary to restore completely the original E.M.F. The different forms of galvanic cells in common use differ chiefly in different devices employed either for disposing of the hydrogen bubbles or for preventing their formation. The most common types of such cells are described in the following sections. 333. The Daniell cell. The Daniell cell consists of a zinc plate immersed in zinc sulphate and a copper plate immersed in copper sul- phate, the two liquids being kept apart either by means of an unglazed earthen cup, as in the type shown in Fig. 283,* or else by gravity. * To set up, fill the battery jar with a saturated solution of copper sul- phate. Fill the porous cup with water and add a handful of zinc sulphate crystals. 276 ELECTEICITY IK MOTION FIG. 283. The Daniell cell In this cell polarization is almost entirely avoided, for the reason that no opportunity is given for the formation of hydrogen bubbles; for, just as the hydrochloric acid solution described in 330 consists of positive hydrogen ions and negative chlorine ions in water, so the zinc sulphate (ZnSO 4 ) solution consists of positive zinc ions and negative SO 4 ions, and the copper sulphate solution of positive copper ions and negative SO 4 ions. Now the zinc of the zinc plate goes into solution in the zinc sulphate in precisely the same way that it goes into solution in the hydro- chloric acid of the simple cell described in 330. This gives a positive charge to the solution about the zinc plate, and causes a movement of the positive ions between the two plates from the zinc toward the copper, and of negative ions in the opposite direction, both the Zn and the SO 4 ions being able to pass through the porous cup. Since the positive ions about the copper plate consist of atoms of copper, it will be seen that the material which is driven out of solution at the copper plate, instead of being hydrogen, as in the simple cell, is metallic copper. Since, then, the element which is deposited on the copper plate is the same as that of which it already consists, it is clear that neither the E.M.F. nor the resistance of the cell can be changed because of this deposit; that is, the cause of the polarization of the simple cell has been removed. The great advantage of the Daniell cell lies in the relatively high degree of constancy in its E.M.F. (1.08 volts). It has a comparatively high internal resistance (one to six ohms) and is therefore incapable of producing very large currents, about one ampere at most. It will fur- nish a very constant current, however, for a great length of time, in fact, until all of the copper is driven out of the copper sulphate solution. In order to keep a constant supply of the copper ions in the solution, copper- sulphate crystals are kept in the compartment S of the cell of Fig. 283 or in the bottom of the gravity cell. These dissolve as fast as the solution loses its strength through the deposition of copper on the copper plate. The Daniell is a so-called " closed-circuit " cell ; that is, its circuit should be left closed (through a resistance of thirty or forty ohms) PRIMARY CELLS 277 FIG. 284. The Western normal cell whenever the cell is not in use. If it is left on open circuit, the copper sulphate diffuses through the porous cup, and a brownish muddy deposit of copper or copper oxide is formed upon the zinc. Pure copper is also deposited in the pores of the porous cup. Both of these actions damage the cell. When the circuit is closed, however, since the electrical forces always keep the copper ions moving toward the copper plate, these damaging effects are to a large extent avoided. 334. The Western normal cell ; the volt. This cell consists of a positive electrode of mercury in a paste of mercurous sul- phate, and a negative electrode of cad- mium amalgam in a saturated solution of cadmium sulphate (Fig. 284). It is so easily and exactly reproducible and has an E.M.F. of such extraordinary constancy that it has been taken by international agreement as the standard in terms of which all E.M.F.'s and P.D.'s are rated. Thus the E.M.F. of a Weston normal cell at 20 C. is taken as 1.0183 volts. The legal definition of the volt is then " an electrical pressure equal to 1 * 8 3 of that produced by a Weston normal celV 335. The Leclanche cell. The Leclanche cell (Fig. 285) consists of a zinc rod in a solution of ammonium chloride (150 grams to a liter of water) and a carbon plate placed inside of a porous cup which is packed full of manganese dioxide and powdered graphite or carbon. As in the simple cell, the zinc dis- solves in the liquid, and hydrogen is liberated at the carbon, or positive, plate. Here it is slowly attacked by the manganese dioxide. This chemical action is, however, not quick enough to prevent rapid polarization when large currents are taken from the cell. The cell slowly recovers when allowed to stand for a while FIG. 285. The Leclanche" cell 278 ELECTRICITY IN MOTION on open circuit. The E.M.F. of a Leclanche cell is about 1.5 volts, and its initial internal resistance is somewhat less than an ohm. It therefore furnishes a momentary current of from one to three amperes. The immense advantage of this type of cell lies in the fact that the zinc is not at all eaten by the ammonium chlo- ride when the circuit is open, and that therefore, unlike the Daniell cell, it can be left for an indefinite time on open circuit without deterioration. Leclanche cells are used almost exclusively where momentary currents only are needed, as, for example, on doorbell circuits. The cell requires no attention for years at a time, other than the oc- casional addition of water to replace loss by evaporation, and the occa- sional addition of ammonium chlo- ride (NH 4 C1) to keep positive NH 4 and negative Cl ions in the solution. 336. The dry cell. The dry cell (Fig. 286) is a modified form of Leclanche cell. It is not really dry, since the mixture within is a moist paste. The ordinary dry cell contains approximately 100 grams of water. The zinc plate is in the form of a cylindrical can and holds the moist black mixture in which the carbon plate is embedded. This mixture consists of ammonium chloride, black oxide of manganese, zinc chloride, powdered petroleum coke, and a small amount of graphite. As in the Leclanche cell, it is the action of the ammonium chloride upon the zinc which produces the current. The manganese dioxide overcomes the polarization due to hydrogen. The function of the ZnCl. 2 is to overcome the polar- ization due to ammonia. The graphite diminishes internal resistance, which, in a fresh cell of ordinary size, may be less - Pitch Sand ?-$( Carbon rod Moist Hack mixture Pulp board lining Zinc plate FIG. 286. The dry cell PEIMAKY CELLS 279 FIG. 287. Cells con- nected in series than ^o of an ohm. Because of the low internal resistance these cells will deliver 30 or more amperes on momentary short circuit, and on account of their great convenience they are manufactured by the million annually, one firm alone making as high as 30,000 a day. 337. Combinations of cells. There are two ways in which cells may be combined : first, in series-, and, second, in parallel. When they are connected in series, the zinc of one cell is joined to the copper of the second, the zinc of the second to the copper of the third, etc., the copper of the first and the zinc of the last being joined to the ends of the external resist- ance (see Fig. 287). The E.M.F. of such a combination is the sum of the E.M.F.'s of the single cells. The internal resistance of the combina- tion is also the sum of the internal resist- ances of the single cells. Hence, if the external resistances are very small, the current furnished by the combination will be no larger than that furnished by a single cell, since the total resistance of the circuit has been increased in the same ratio as the total E.M.F. But if the external resist- ance is large, the current produced by the combination will be very much greater than that produced by a single cell. Just how much greater can always be deter- mined by applying Ohm's law ; for if there are n cells in series, and E is the E.M.F. of each cell, the total E.M.F. of the cir- cuit is nE. Hence, if R e is the external resistance and R { the internal resistance of a single cell, then Ohm's law gives ll >. . . ^ I R i 3 b e . ]' m FIG. 288. Water anal- ogy of cells in series T nE 280 ELECTEICITY IN MOTION FIG. 289. Cells in parallel FIG. 290. Water analogy of cells in parallel If the n cells are connected in parallel, that is, if all the coppers are connected together and all the zincs, as in Fig. 289, the E.M.F. of the combination is only the E.M.F. of a single cell, while the internal resistance is 1/n of that of a single cell, since connecting the cells in this way is simply equivalent to multiplying the area of the plates n times. The current furnished by such a combination will be given by the formula /= If, therefore, R e is negligibly small, as in the case of a heavy copper wire, the current flowing through it will be n times as great as that which could be made to flow through it by a single cell. Figs. 288 and 290 show by means of the water analogy why the E.M.F. of cells in series is the sum of the several E.M.F.'s, and why the E.M.F. of cells in parallel is no greater than that of a single cell. These considerations show that the rules which should govern the combination of cells are as follows : Connect in series when R e is large in comparison with jft . ; connect in parallel when R { is large in comparison with R e . QUESTIONS AND PROBLEMS 1. A certain dry cell having an E. M. F. of 1.5 volts delivered a current of 30 amperes through an ammeter having a negligible resist- ance. Find the internal resistance of the cell. 2. Why is a Leclanche" cell better than a Daniell cell for ringing doorbells ? 3. Diagram three wires in series and three cells in series. If each wire has a resistance of .1 ohm, what is the resistance of the series ? If each cell has a resistance of .1 ohm, what is the internal resistance of the series? SECONDARY CELLS 281 4. Diagram three wires in parallel or multiple, and three cells in multiple. If each wire has a resistance of 6 ohms, what is the joint resistance of the three ? If each cell has an internal resistance of 6 ohms, what is the resistance of the group ? 5. With the aid of Figs. 288 and 290 discuss the water analogies of the rules at the end of 337. 6. If the internal resistance of a Daniell cell of the gravity type is 4 ohms, and its E.M. F. 1.08 volts, how much current will 40 cells in series send through a telegraph line having a resistance of 500 ohms ? What current will 40 cells joined in parallel send through the same circuit? What current will one such cell send through the same circuit? 7. W r hat current will the. 40 cells in parallel send through an am- meter which has a resistance of .1 ohm ? What current would the 40 cells in series send through the same ammeter? What current would a single cell send through the same ammeter ? 8. Under what conditions \vill a small cell give practically the same current as a large one of the same type ? 9. How many cells, each of E.M.F. 1.5 volts and internal resistance .2 ohm, will be needed to send a current of at least 1 ampere through an external resistance of 40 ohms ? 10. Why is it desirable that a galvanometer which is to be used for measuring currents have as low a resistance as possible? 11. Ordinary No. 9 telegraph wire has a resistance of 20 ohms to the mile. What current will 100 Daniell cells in series, each of E.M.F. of 1 volt, send through 100 miles of such wire, if the two relays have a resistance of 150 ohms each and the cells an internal resistance of 4 ohms each? 12. If the relays of the preceding problem had each 10,000 turns of wire in their coils, how many ampere turns were effective in magnetizing their electromagnets ? 13. If, on the above telegraph line, sounders having a resistance of 3 ohms each and 500 turns were to be put in the place of the relays, how many ampere turns would be effective in magnetizing their cores ? Why, then, does the electromagnet of the relay have a high resistance ? SECONDARY CELLS 338. Lead storage batteries. Let two 6 by 8 inch lead plates be screwed to a half-inch strip of some insulating material, as in Fig. 291, and immersed in a solution consisting of one part of sulphuric acid to ten parts of water. Let a current from two storage or three dry cells in series, C, be sent through this arrangement, an ammeter A or any 282 ELECTEICITY IX MOTION low-resistance galvanometer being inserted in the circuit. As the current flows, hydrogen bubbles will be seen to rise from the cathode (the plate at which the current leaves the solution), while the positive plate, or anode, will begin to turn dark brown. At the same time the reading of the ammeter will be found to decrease rap- idly. The brown coating is a compound of lead and oxygen, called lead peroxide (PbO 2 ), which is formed by the action upon the plate of the oxygen which is ^ ^ The Q Q liberated, precisely as .in the experiment the gtorage battery on the electrolysis of water ( 302). Now let the batteries be removed from the circuit by opening the key K v and let an electric bell B be inserted in their place by closing the key K 2 . The bell will ring and the ammeter A will indicate a current flowing in a direction opposite to that of the original current. This current will decrease rapidly as the energy which was stored in the cell by the original current is expended in ringing the bell. This experiment illustrates the principle of the storage bat- tery. Properly speaking, there has been no storage of electricity, but only a storage of chemical energy. Two similar lead plates have been changed by the action of the current into two dissimilar plates, one of lead and one of lead peroxide ; in other words, an ordinary galvanic cell has been formed, for any two dissimilar metals in an electrolyte constitute a primary galvanic cell. In this case the lead per- oxide plate corresponds to the copper of an ordinary cell, and the lead plate to the zinc. This cell tends to create a current opposite in direction to that of the charging current ; that is, its E.M.F. pushes back against the E.M.F. of the charging cells. It was for this reason that the ammeter reading fell. When the charging current is removed, this cell acts exactly like a primary galvanic cell and furnishes a current until the thin coating of peroxide is used up. The only important differ- ence between a commercial storage cell (Fig. 292) and the one which we have here used is that the former is provided in SECONDARY CELLS 283 Fig. 292. Lead-plate storage cell the making with a. much thicker coat of the " active material " (lead peroxide on the positive plate and a porous, spongy lead on the negative) than can be formed by a single charging such as we used. This material is pressed into interstices in the plates, as shown in Fig. 292. The E.M.F. of the lead storage cell is about 2 volts. Since the plates are always very close together and may be given any desired size, the internal resistance is usually small, so that the currents furnished may be very large. The usual efficiency of the lead stor- age cell is about 75% ; that is, only about ^ as much electrical energy can be obtained from it as is put into it. 339. Nickel-iron storage batteries. Thomas A. Edison (see opposite p. 316) developed and perfected the nickel-iron caustic-potash storage cell. The electrolyte is a 21% solu- tion of caustic potash in water. The negative plates contain iron powder securely retained in perforated flat rectangular capsules, while the positive plates contain oxide of nickel in perforated cylindrical containers. For equal capacities the Edison cell weighs about half as much as the lead cell, and it will stand a remarkable amount of electrical and mechan- ical abuse. The E.M.F. is about 1.2 volts. In efficiency it is a little below the lead cell. Caustic potash is now replaced by caustic soda. QUESTIONS AND PROBLEMS 1. In charging a storage battery is it better to say that the current passes into the cell or through it ? What is " stored " ? 2. The lead peroxide plate and the nickel oxide plate are both called " the positives." What is the relation of the charging current to these plates ? 284 ELECTRICITY IN MOTION HEATING EFFECTS OF THE ELECTRIC CURRENT 340. Heat developed in a wire by an electric current. Let the terminals of two or three dry cells in series be touched to a piece of No. 40 iron or German-silver wire and the length of wire between these terminals shortened to -^ inch or less. The wire will be heated to incan- descence and probably melted. The experiment shows that in the passage of the current through the wire the energy of the electric current is trans- formed into heat energy. The electrical energy expended when a current flows between points of given P.D. may be spent in a variety of ways. For example, it may be spent in producing chemical separation, as in the charging of a storage cell; it may be spent in doing mechanical work, as is the case when the current flows through an electric motor ; or it may be spent wholly in heating the wire, as was the case in the experiment. It will always be expended in this last way when no chemical or mechanical changes are pro- duced by it. (See drawings opposite p. 269 for uses made of heating effects.) 341. Energy relations of the electric current. We found in Chapter IX that energy expended on a water turbine is equal to the quantity of water passing through it times the difference in level through which the water falls ; or, that the power (rate of doing work) is the product of the fall in level and the current strength. In just the same way it is found that when a current of electricity passes through a conductor, the power, or rate of doing work, is equal to the fall in potential between the ends of 'the conductor times the strength of the electric current. If the P.D. is expressed in volts and the current in amperes, the power is given in watts, and we have yoltg x ampereg = wattg _ The energy of the electric current is usually measured in kilowatt hours. HEATING EFFECTS 285 A kilowatt hour is the quantity of energy furnished in one hour ly a current ivhose rate of expenditure of energy is a kilowatt. 342. Incandescent lamps. The ordinary incandescent lamp (Fig. 293) consists of a tungsten filament heated to incan- descence by an electric current. Since the filament would burn up in a few seconds in air, it is placed in a highly exhausted bulb. When in use it slowly vaporizes, depositing a dark, mirror- like coating of metal upon the inner surface of the bulb. The lead-in wires are sold- ered one to the base A of the socket and the other to its rim J5, these being the elec- trodes through which the current enters and leaves the lamp. The wires ^#, w, sealed into the w^alls of the bulb, must have the same coefficient of expansion as the glass to prevent leakage of air. Incandescent lamps are usually grouped in parallel or multiple, on a circuit that maintains a potential of something over 100 volts between the terminals of the lamps (Fig. 318). The rate of consump- tion of energy is about 1.25 watts per candle power for the ordinary sizes. Tungsten filaments, being operated at a much higher temperature than is possible with the now almost obsolete carbon filament, have an efficiency nearly three times as great. A customer usually pays for his light by the kilowatt hour ( 341). The rate at which energy is consumed by a lamp carrying ^ ampere at 100 volts is 25 watts. Two such lamps running for 4 hours would, therefore, consume 2 x 4 x 25 = 200 watt hours = .200 kilowatt hour. The energy is measured and recorded on a recording watt-hour meter (Fig. 321). FIG. 293. The tung- sten vacuum lamp 286 ELECTRICITY IX MOTION By filling the bulb with nitrogen a very efficient form of the tungsten lamp is obtained. The long filament is wound into an exceedingly fine spiral to minimize heat radiation. As we have already learned ( 207), the presence of gas retards evaporation; hence, because of the nitrogen the fila- ment may be raised to a higher temperature than is permis- sible in a vacuum. A greatly increased candle power results from the slight increase in current. Moreover, the convection currents in the gas-filled lamp cause the mir- ror due to vaporization to form near the top of the globe, where it does not obscure the intensity of the light. The larger sizes of gas-filled lamps consume only .6 watt per candle power. 343. The arc light. When two carbon rods are placed end to end in the circuit of a powerful elec- tric generator,, the carbon about the point of contact is heated red-hot. If, then, the ends of the carbon rods are separated one-fourth inch or so, the current will still continue to flow, for a conducting layer of incandescent vapor, called an electric arc, is produced between the poles. The appearance of the arc is shown in Fig. 294. At the + pole a hollow, or crater, is formed in the carbon, while the carbon becomes cone-shaped, as in the figure. The carbons are con- sumed at the rate of about an inch an hour, the + carbon wasting away about twice as fast as the one. The light comes chiefly from the + crater, where the temperature is about 3800 C., the highest attainable by man. All known substances are volatilized in the electric arc. The open arc requires a current of 10 amperes and a P.D. between its terminals of about 50 volts. Such a lamp produces about 500 * candle power, and therefore consumes energy at the rate of about 1 watt per candle power. The light of the arc lamp is due to the intense heat developed on account of resistance, not to actual combustion, or burning. Nevertheless, in the open arc the oxygen of the air unites so rapidly with * This is the so-called "mean spherical" candle power. The candle power in the direction of maximum illumination is from 1000 to 1200. FIG. 294. The arc light HEATING EFFECTS 287 the carbon at the hot tips that in a few hours the rods are consumed. To overcome this difficulty the inclosed arc (Fig. 295) is used. Shortly after the arc is " struck " the oxygen in the inner globe is used up and then the hot carbon tips are surrounded by an atmosphere of carbon dioxide and nitrogen. Under these conditions the carbons last 130 to 150 hours. The inclosed arc is much longer than the open arc, and therefore in this lamp the P.D. between the tips is greater, usually about 80 volts, while the rest of the P.D. of the line is taken up in the resistance coils of the lamp. The recently invented flaming arc, produced between carbons which have a composite core con- sisting chiefly of carbon and fluoride of calcium, sometimes reaches an efficiency as high as .27 watt per candle power. It gives an excellent yellow light, which penetrates fog well. 344. The arc light auto- matic feed. Since the two carbons of the arc gradually waste away, they would soon become so far separated that the arc could no longer be maintained were it not for an automatic feeding device which keeps the distance be- tween the carbon tips very nearly constant. Fig. 296 shows the essential fea- tures of one form of this device. When no current flows through the lamp, gravity holds the carbon tips at e together; but as soon as the current is thrown on, it energizes the magnet coils m, m, which draw up the U-shaped iron core, thus striking the arc at e. As the carbons slowly waste away, the arc becomes longer, the resistance greater, and the cur- rent less; hence the upward magnetic pull weakens and the upper carbon descends, and vice versa. From time to time the upper carbon slips down through the friction clutch c. It is clear, therefore, that this automatic device will maintain that particular length of arc for which FIG. 295. Mechanism of a direct-current in- closed arc lamp FIG. 296. Feeding device for arc lamp 288 ELECTRICITY IN MOTION equilibrium exists between the effect of gravity pulling down and mag- netism pulling up. A dashpot d, containing a stationary piston, prevents the magnetic pull from suddenly drawing the tips at e too far apart. 345. The Cooper-Hewitt mercury lamp. The Cooper-Hewitt mercury lamp (Fig. 297) differs from the arc lamp in that the incandescent body is a long column of mercury vapor instead of an incandescent solid. The lamp consists of an exhausted tube three or four feet long, the positive electrode at the top consi sting of a plate of iron, while the negative electrode at the bottom is a small quantity of mer- cury. Under a suf- ficient difference of potential between these terminals a long mercury-vapor arc is formed, which stretches from terminal to terminal in the tube. This arc emits a very brilliant light, but it is almost entirely wanting in red rays. The strength of its actinic rays makes it especially valuable in photography. Its commercial efficiency is about .6 watt per candle power. Cooper- Hewitt lamps having quartz tubes are used for sterilizing purposes because of the powerful ultra-violet rays which the quartz transmits. FIG. 297. The Cooper-Hewitt mercury-vapor arc lamp QUESTIONS AND PROBLEMS 1. What is meant by a 104-volt lamp? What would happen to such a lamp if the P.D. at its terminals amounted to 500 volts? Trolley cars are usually furnished with current at about 500 volts ; how would you use 100-volt lamps on such a circuit? 2. A very common electric lamp used in our homes is marked 25 watts and carries about ^ ampere. One fresh dry cell on short circuit will deliver 30 or more amperes. Will the cell light the lamp? 3. A 50-vol6 carbon lamp carrying 1 ampere has about the same candle power as a 100-volt carbon lamp carrying ampere. Explain why. I FIG. 298 HEATING EFFECTS 289 4. If a storage cell has an E.M.F. of 2 volts and furnishes a cur- rent of 5 amperes, what is its rate of expenditure of energy in watts? 5. Fig. 298 shows the connections for a lamp L which can be turned on or off at two different points a or b. Explain how it works. 6. How many 100-volt lamps each carrying ^ ampere may be main- tained on a circuit where the total power may not exceed 600 watts ? 7. What will it cost to use an electric laundry iron for 6 hours if it takes 3.5 amperes on a 104-volt circuit, the cost of current being $.09 per kilowatt hour ? 8. A certain electric toaster takes 5 amperes at 110 volts. It will make two pieces of toast at once in 3 minutes. At what horse-power rate does the toaster convert electrical energy into heat energy? At $.08 per kilowatt hour what does it cost to make 12 pieces of toast? 9. How many lamps, each of resistance 20 ohms and requiring a current of .8 ampere, can be lighted by a dynamo that has an output of 4000 watts? 10. If one of the wire loops in a tungsten lamp is short-circuited, what effect will this have on the amount of current flowing through the lamp? on the brightness of the filament? 11. How many cells working as in problem 4 would be equivalent to 1 H.P. ? (See 144, p. 122.) 12. Since one calorie is equal to 42,000,000 ergs, 1 watt (10,000,000 ergs per second) develops in one second .24 calories. Therefore the number of calories, H, developed in t seconds by a current of / amperes between two points whose P.D. is V volts is expressed by the equation H = I x V x t x .24. How many calories per minute are given out by the electric toaster of problem 8? 13. From the equation of problem 12 show that H = PR x t x .24. 14. How many minutes are required to heat 600 g. of water from 15 C. to 100 C. by passing 5 amperes through a 20-ohm coil immersed in the water? 15. Why is it possible to get a much larger current from a storage cell than from a Daniell cell ? 16. If an automobile is equipped with 6-volt lamps, how many lead storage cells must be on the car? Are these cells in series or multiple? 17. A small arc lamp requires a current of 5 amperes and a difference of potential between its terminals of 45 volts. What resistance must be connected in series with it in order to use it on a 110-volt circuit? CHAPTER XY INDUCED CURRENTS THE PRINCIPLE OF THE DYNAMO AND MOTOR 346. Current induced by a magnet. Let 400 or 500 turns of No. 22 copper wire be wound into a coil C (Fig. 299) about two and a half inches in diameter. Let this coil be connected into circuit with a lecture-table galvanometer (Fig. 263), or even a simple detector made by suspending in a box, with No. 40 copper wire, a coil of 200 turns of No. 30 copper wire (see Fig. 299). Let the coil C be S[ thrust suddenly over the N pole of a strong horse- shoe magnet. The deflec- FlG 299 Induction of electric currents tion of the pointer p of by ma gnets the galvanometer will in- dicate a momentary current flowing through the coil. Let the coil be held stationary over the magnet. The pointer will be found to come to rest in its natural position. Now let the coil be removed suddenly from the pole. The pointer will move in a direction opposite to that of its first deflection, showing that a reverse current is now being generated in the coil. We learn, therefore, that a current of electricity may be induced in a conductor by causing the latter to move through a magnetic field, while a magnet has no such influence upon a conductor which is at rest with respect to the field. This dis- covery, one of the most important in the history of science, was announced by the great Faraday in 1831. From it have sprung directly most of the modern industrial developments of electricity. 290 MICHAEL FARADAY (1791-1867) Famous English physicist and chemist ; one of the most gifted of experimenters son of a poor blacksmith ; apprenticed at the age of thirteen to a London book- binder, with whom he worked nine years ; applied for a position in Sir Humphry Davy's laboratory at the Royal Institution in 1813 ; became director of this labo- ratory in 1825; discovered electromagnetic induction in 1831; made the first dynamo; discovered in 1833 the laws of electrolysis, now known as Faraday's laws ; the farad, the practical unit of electrical capacity, is named in his honor INDUCTION MOTOR One of the most familiar of the more recent applications of the great principle of induction discovered by Faraday is the induction motor, which has come into extensive use in both large and small sizes. The particular one here shown is known as the squirrel-cage form, in which there is no electrical connection between the stator (the stationary part) and the rotor (the revolving part) . The stator is wound on a laminated core like the stator of a dynamo, while the rotor consists of copper bars laid in a slotted laminated core, their ends being joined to copper rings, one at each end. The bars are therefore in parallel. The alternating current applied to the stator windings develops a magnetic field which rotates around the iron ring of the stator. This is equivalent to a set of magnetic poles mechanically rotated around the rotor. The magnetic lines of force which there- fore cut across the copper bars of the rotor generate in them an E.M.F. which causes a current to flow in the copper system of the rotor. The rotating field reacts with the field produced by the current in the conductors of the rotor so as to cause the rotor to be dragged around after the rotating field PRINCIPLE OF THE DYNAMO AND MOTOR 291 347. Direction of induced current. Lenz's law. In order to find the direction of the induced current, let a very small P.D. from a galvanic cell be applied to the terminals A and B (Fig. 299), and note the direction in which the pointer moves when the current enters, say, at A. This will at once show in what direction the current was flow- ing in the coil C when it was being thrust over the N pole. By a simple application to C of the right-hand rule ( 308) we can then tell which was the N and which the face of the coil when the induced current was flowing through it. In this way it will be found that if the coil w T as being moved past the N pole of the magnet, the current induced in it was in such a direction as to make the lower face of the coil an N pole during the downward motion and an S pole during the upward motion. In the first case the repulsion of the N pole of the magnet and the N pole of the coil tended to oppose the motion of the coil while it was moving from a to b, and the attraction of the N pole of the magnet and the S pole of the coil tended to oppose the motion while it was moving from b to c. In the second case the repulsion of the two N poles tended to oppose the motion between b and c, and the attraction between the N pole of the magnet and the S pole of the coil tended to oppose the upward motion from b to a. In every case, therefore) the motion is made against an opposing force. From these experiments, and others like them, we arrive at the following law : Wfenever a current is induced by the rela- tive motion of a magnetic field and a conductor, the direction of the induced current is always such as to set up a magnetic field which opposes the motion. This is Lenz's law. This law might have been predicted at once from the principle of the conservation of energy ; for this principle tells us that since an electric current possesses energy, such a current can appear only through the ex- penditure of mechanical work or of some F IG . 300. other form of energy. 348. Condition necessary for an induced E.M.F. Let the coil be held in the position shown in Fig. 300, and moved back and forth parallel to the magnetic field, that is, parallel to the line NS. No current will be induced. Currents induced only when conductor cuts lines of force 292 INDUCED CURRENTS FIG. 301. E.M.F. induced when a straight conductor cuts magnetic lines By experiments of this sort it is found that an E.M.F. is induced in a coil only when the motion takes place in such a way as to change the total number of magnetic lines of force which are inclosed by the coil. Or, to state this rule in more general form, an E.M.F. is induced in any element of a con- ductor when, and only when, that element is moving in such a way as to cut magnetic lines of force.* It will be noticed that the first statement of the rule is included in the second, for whenever the number of lines of force which pass through a coil changes, some lines of force must cut across the coil from the inside to the outside, or vice versa. 349. The principle of the electric motor. Let a vertical wire ab be rigidly attached to a horizontal wire gh, and let the latter be supported by a ring or other metallic support, in the manner shown in Fig. 302, so that ab is free to oscillate about gh as an axis. Let the lower end of ab dip into a trough of mercury. When a magnet is held in the position shown and a current from a dry cell is sent down through the wire, the wire will in- stantly move in the direction indicated by the arrow f, namely, at right angles to the direction of the lines of magnetic force. Let the direction of the current in the wire be reversed. The direction of the force acting on the wire will be found to be reversed also. We learn, therefore, that a wire carrying a current in a magnetic field tends to move in * If a strong electromagnet is available, these experiments are more instruc- tive if performed, not with a coil, as in Fig. 300, but with a straight rod (Fig. 301) to the ends of which are attached wires leading to a galvanometer. Whenever the rod moves parallel to the lines of magnetic force there will be no deflection, but whenever it moves across the lines the galvanometer needle will move at once. FIG. 302. The prin- ciple of the electric motor PRINCIPLE OF THE DYNAMO AND MOTOR 293 a direction at right angles both to the direction of the field and to the direction of the current. This fact underlies the opera- tion of all electric motors. 350. The motor and dynamo rules. A convenient rule for determining whether the wire ab (Fig. 302) will move forward or back in a given case may be obtained as follows: If the field of a magnet alone is represented by Fig. 303, and that due to the current * alone by Fig. 304, then the resultant field when the current-bearing wire is placed between the poles of the magnet is that shown in Fig. 305 ; for the strength of the FIG. 303. Field of magnet alone FIG. 304. Field of current alone FIG. 305. Field of magnet and current field above the wire is now the sum of the two separate fields, while the strength below it is their difference. Now Faraday thought of the lines of force as acting like stretched rubber bands. This would mean that the wire in Fig. 305 would be pushed down. Whether the lines of force are so conceived or not, the motor rule may be stated thus : A current in a magnetic field tends to move away from the side* on which its lines are added to those of the field. The dynamo rule follows at once from the motor rule and Lenz's law. Thus, when a wire is moved through a magnetic field the current induced in it must be in such a direction as * The cross in the conductor of Fig. 304, representing the tail of a retreat- ing arrow, is to indicate that the current flows away from the reader. A dot, representing the head of an advancing arrow, indicates a current flowing toward the reader. 294 INDUCED CURRENTS to oppose the motion ; therefore the induced current will be in such a direction as to increase the number of lines on the side toward which it is moving. 351. Strength of the induced E.M.F. The strength of an induced E.M.F. is found to depend simply upon the number of lines of force cut per second by the conductor, or, in the case of a coil, upon the rate of change in the number of lines of force which pass through the coil. The strength of the current which flows is then given by Ohm's law ; that is, it is equal to the induced E.M.F. divided by the resistance of the circuit. The number of lines of force which the conductor cuts per second may always be determined if we know the velocity of the conductor and the strength of the magnetic field through which it moves. For it will be remembered that, according to the convention of 270, a field of unit strength is said to con- tain one line of force per square centimeter, a field of 1000 units strength 1000 lines per square centimeter, etc. In a conductor which is cutting lines at the rate of 100,000,000 per second there is an induced E.M.F. of 1 volt. * The reason why we used a coil of 500 turns instead of a single turn in the experiment of 346 was that by thus making the conductor in which the current was to be induced cut the lines of force of the magnet 500 times instead of once, we obtained 500 times as strong an induced E.M.F., and therefore 500 times as strong a current for a given resistance in the circuit. 352. Currents induced in rotating coils. Let a 400- or 500-tum coil of No. 28 copper wire be made small enough to rotate between the poles of a horseshoe magnet, and let it be connected into the circuit of a galvanometer, precisely as in 346. Starting with the coil in the posi- tion of Fig. 306, (1), let it be rotated suddenly clockwise (looking down from above) through 180. A strong deflection of the galvanometer will be observed. Let it be rotated through the next 180 back to the starting point. An opposite deflection will be observed. * This may be considered as the scientific definition of the volt, convenience alone having dictated the legal definition given in 334. PRINCIPLE OF THE DYNAMO AND MOTOR 295 (l) FIG. 306. Direction of cur- rents induced in a coil rotat- ing in a magnetic field The arrangement is a dynamo in miniature. During the first half of the revolution (see Fig. 306, (2)) the wires on the right side of the loop were cutting the lines of force in one direction, while the wires on the left side were cutting them in the oppo- site direction. A current was being generated down on the right side of the coil and up on the left side (see dynamo rule). It will be seen that both currents flow around the coil in the same direction. The in- duced current is strongest when the coil is in the position shown in Fig. 306, (2), because there the lines of force are being cut most rapidly. Just as the coil is moving into or out of the position shown in Fig. 306, (1), its edges are moving parallel to the lines of force, and hence no current is induced, since no lines of force are being cut As the coil moves through the last 180 of its revolution both sides are cutting the same lines of force as before, but they are cutting them in an opposite direction ; hence the current generated during this last half is opposite in direction to that of the first half.* QUESTIONS AND PROBLEMS 1. Can the number of lines of force within a closed coil of wire be increased or decreased without the lines being cut by the wire ? Explain. 2. Under what conditions may an electric current be produced by a, magnet ? 3. How many lines of force must be cut per second to induce 10 volts? 4. If a coil of wire is rotated about a vertical axis in the earth's field> an alternating current is set up in it. In what position is the coil when the current changes direction? *A laboratory experiment on the principles of induction should be performed at about this point. See, for example, Experiment 36 of the authors' Manual. 296 INDUCED CURRENTS 5. State Lenz's law, and show how it follows from the principle of the conservation of energy. 6. A coil is thrust over the S pole of a magnet. Is the direction of the induced current clockwise or counterclockwise as you look down upon the pole ? 7. A ship having an iron mast is sailing east. In what direction is the E.M.F. induced in the mast by the earth's magnetic field? If a wire is brought from the top of the mast to its bottom, no current will flow through the circuit. Why? 8. A current is flowing from top to bottom in a vertical wire. In what direction will the wire tend to move on account of the earth's magnetic field? DYNAMOS 353. A simple alternating-current dynamo. The simplest form of commercial dynamo consists of a coil of wire so arranged as to rotate continuously between the poles of a powerful electromagnet (Fig. 307). In order to make the mag- netic field in which the con- ductor is moved as strong as possible, the coil is wound upon an iron core C. This greatly increases the total number of lines of magnetic force which pass between JV and S, for instead of an air path the core offers an iron path, as shown in Fig. 308. The rotating part, consisting of the coil with its core, is called the armature. One end of the coil is attached to the insulated metal ring R, which is attached rigidly to the shaft of the armature and therefore rotates with it, while the other end of the coil is attached to a second ring R'. The brushes b and V, which constitute the terminals of the external circuit, are always in contact with these rings. FIG. 307. Drum-wound armature DYNAMOS 297 As the coil rotates, an induced alternating current passes through the circuit. This current reverses direction as often as the coil passes through the position shown in Fig. 308, that is, the position in which the con- ductors are moving parallel to the lines of force ; for at this instant the conductors which were moving up begin to move down, and those which were moving down begin to move up. The cur- rent reaches its maximum value when the coils are moving through a position 90 farther on, for then the lines of force are being cut most rapidly by the conductors OR- both sides of the coil. These facts are graphi- cally represented by the curve of E.M.F.'s (Fig. 309). 354. The multipolar alternator. For most commercial purposes it is found desirable to have 120 or more alternations of current per second. This could not be attained easily with two-pole machines like those FIG. 308. End view of drum armature 90 270 etc. FIG. 309. Curve of alternating electromotive force sketched in Figs. 307 and 308. Hence commercial alternators are usually built with a large number of poles alternately N and S, arranged around the circumference of a circle in the manner shown in Fig. 310. These poles are excited by a direct current. The dotted lines represent the direction of the lines of force through the iron. It will be seen that the coils which are passing beneath N poles have induced currents set up in them the direction of which is opposite to that of the currents which are induced in the conductors which are passing beneath the S poles. Since, however, the direction of winding of the armature coils changes between each two poles, all the inductive effects of all the poles are added in the coil and constitute at any instant one single current 298 INDUCED CURRENTS flowing around the complete circuit in the manner indicated by the arrows in the diagram. This current reverses direction at the instant at which all the coils pass the midway points between the JV and S poles. The number of alternations per second is equal to the number of poles multiplied by the number of revolutions per second. The field magnets ,/Vand S of such a dynamo are usually excited by a direct current from some other source. Fig. 311 represents an alternating- current dynamo with revolving field and stationary armature con- nected directly to a tandem com- pound engine. Alternators of 5000- kilowatt capacity (nearly 7000 FIG. 310. Diagram of alternating- horse power) have been built to current dynamo run at the unusually high speed of 3600 revolutions per minute. Alternators of lower speed but of very much greater capacity are common (see huge rotor opposite p. 257). 355. The principle of the commutator. By the use of a so- called commutator it is possible to transform a current which FK; .311. Alternating-current dynamo is alternating in the coils of the armature to one which always flows in the same direction through the external portion of the circuit. The simplest possible form of such a commutator DYNAMOS 299 FIG. 312. The simple commutator is shown in Fig. 312. It consists of a single metallic ring which is split into two equal insulated semicircular segments a and c. One end of the ro- tating coil is soldered to one of these semicircles, and the other end to the other semi- circle. Brushes b and b ! are set in such positions that they lose contact with one semicircle and make contact with the other at the instant at which the current changes direction in the armature. The current, therefore, always passes out to the external circuit through the same brush. While a cur- rent from such a coil and com- mutator as that shown in the figure would always flow in the same direction through the ex- ternal circuit, it would be of a pulsating rather than a steady character, for it would rise to a FIG. 313. Two-pole direct-current maximum and fall again to zero dynamo with ring armature twice during each complete revo- lution of the armature. This effect is avoided in the com- mercial direct-current dynamo by building a commutator of a large number of segments instead of two, and connecting 90 180 270 360 etc. FIG. 314. Curve of commutated electromotive force each to a portion of the armature coil in the manner shown in Fig. 313. The result of using a simple split-ring com- mutator is shown graphically in Fig. 314. 300 INDUCED CURRENTS FIG. 315. The direct-current dynamo, drum winding 356. The drum-armature direct-current dynamo. Fig. 315 is a diagram illustrating the construction of a commercial two-pole direct-current dynamo of the drum-armature type. At a given instant currents are being induced in the same direction in all the conductors on the left half of the armature. The cross on these conduc- tors, representing the tail of a retreat- ing arrow, is to indicate that these currents flow away from the reader. No E.M.F.'s are induced in the con- ductors at the top and bottom of the armature, where the motion is parallel to the magnetic lines. On the right half of the ring, on the other hand, the induced currents are all in the opposite direction, that is, toward the reader, since the conductors are here all moving up instead of down. The dot in the middle of these conductors represents the head of an approaching arrow. It will be seen, how- ever, in tracing out the con- nections 1, 1 1? 2, 2 P 3, 3 r etc., of Fig. 315 (the dotted lines representing connections at the back of the drum), that the coil is so wound about the drum that the currents in both halves are always flowing toward one brush b, from which they are led to the external circuit and back at T)'. This condition always exists, no matter how fast p IG the rotation; for it will be seen that as each loop ro- tates into the position where the direction of its current reverses, it passes a brush and therefore at once becomes a part of the circuit on the other half of the drum where the currents are all flowing in the opposite direction. Fig. 316 shows a typical modern four-pole generator, and Fig. 317 the corresponding drum-wound armature. Fig. 326 (p. 310) A four-pole direct-current generator DYNAMOS 301 Main Circuit illustrates nicely the method of winding such an armature, each coil beginning on one segment of the commutator and ending on the adjacent segment. 357. Dynamo lighting circuit. The type of circuit generally used in B.C. incandescent lighting is shown in Fig. 318. The lamps are arranged in parallel between the mains. The field magnets are excited partly by FIG. 31 7. A modern drum armature a few series turns which carry the whole current going to the lamps, and partly by a shunt coil consisting of many turns of fine wire (Fig. 318). This combination of series and shunt winding maintains the P.D. across the mains constant for a great range of loads. Such a machine is called a compound wound dynamo, to distinguish it from a series wound machine, for example, which dispenses with the shunt coil. In all self-exciting machines there is enough residual magnetism left in the iron cores after stopping to start feeble induced currents when started up again. These cur- rents immediately increase the strength of the magnetic field, and so tjie machine quickly builds up its current until the limit of mag- FlG - 318 - The compound- netization is reached. wound d y namo 358. The electric motor. In construction the electric motor differs in no essential respect from the dynamo. To analyze the operation as a motor of such a machine as that shown in Fig. 313, suppose a current from an outside source is 1 first sent around the coils of the field magnets and then into the arma- ture at b f . Here it will divide and flow through all the con- ductors on the left half of the ring in one direction, and through all those on the right half in the opposite direction. Hence, in accordance with the motor rule, all the conductors on the left side are urged upward by the influence of the field, and all those on the right side are urged downward. The armature will therefore begin to rotate, and this rotation 302 INDUCED CURRENTS will continue as long as the current is sent in at b r and out at b; for as fast as coils pass either b or b r the direction of the current flowing through them changes, and therefore the direction of the force acting on them changes. The left half is therefore always urged up and the right half down. The greater the strength of the current, the greater the force acting to produce rotation. If the armature is of the drum type (Fig. 315), the con- ditions are not essentially different; for, as may be seen by following out the windings, the current entering at b' will flow through all the conductors on the left half in one direction and through those on the right half in the opposite direc- tion. The commu- tator keeps these conditions always fulfilled. The induc- tion motor is pictured and described oppo- FIG. 319. Railway motor, upper field raised site page 291. The electric motor is a device which receives electrical energy and converts it into mechanical energy. The dynamo is a device which re- ceives mechanical energy from a steam engine, water wheel, or other source and converts it into electrical energy, 359. Street-car motors. Electric street cars are nearly all operated by direct-current series-wound motors placed under the cars and attached by gears to the axles. Fig. 319 shows a typical four-pole street-car motor. The two upper field poles are raised with the case when the motor is opened for inspection, as in the figure. The current is generally supplied by compound-wound dynamos which maintain a constant potential of about 500 volts between the trolley or third rail and the track which is used as the return circuit. The cars are always operated in parallel, as shown in Fig. 320. In a few instances street cars are operated upon DYNAMOS 303 alternating, instead of upon direct-current, circuits. In such cases the motors are essentially the same as direct-current series-wound motors ; for since in such a machine the current must reverse in the field magnets at the same time that it reverses in the armature, it will be seen that Trolley Wire or 3d Rail 1 nnnnnnnnj 1 DDDDDI at Power V > 1 Station >C y=m rm \ L) O ' Track FIG. 320. Street-car circuit the armature is always impelled to rotate in one direction, whether it is supplied with a direct or with an alternating current. Other types of A.C. motors are not well adapted to starting with full load. 360. Back E.M.F. in motors. When an armature is set into rotation by sending a current from some outside source through it, its coils move through a magnetic field as truly as if the rotation were produced by a steam engine, as is the case in running a dynamo. An induced E.M.F. is therefore set up by this rotation. In other words, while the machine is acting as a motor it is also acting as a dynamo. The direction of the induced E.M.F. due to this dynamo effect will be seen, from Lenz's law or from a consideration of the dynamo and motor rules, to be opposite to the outside P.D., which is causing current to pass through the motor. The faster the motor rotates, the faster the lines of force are cut, and hence the greater the value of this so-called lack E.M.F. If the motor were doing no work, the speed of rotation would increase until the back E.M.F. reduced the current to a value simply sufficient to overcome friction. It will be seen, therefore, that, in general, the faster the motor goes, the less the current which passes through its armature, for this current is always due to the difference between the P.D. applied at the brushes 500 volts in the case of trolley cars and the back E.M.F. When the 304 INDUCED CURRENTS motor is starting, the back E.M.F. is zero ; and hence, if the full 500 volts were applied to the brushes, the current sent through would be so large as to ruin the armature through overheating. To prevent this motors are furnished with a starting box, con- sisting of resistance coils which are thrown into series with the motor on starting, and thrown out again gradually as the speed increases and the back E.M.F. rises.* Trolley cars are usually run by two motors which, on starting, work in series, so that each supplies a part of the starting resistance for the other. After speed is acquired, they work in parallel. 361. The recording watt-hour meter. The recording watt- hour meter (Fig. 321) is the instrument which fixes our electric-light bills. It is essentially an electric motor containing no iron, so that the current through the armature A is proportional to the P.D. between the mains, while the current through the field mag- nets F is the current flowing into the house. Therefore the force act- ing between A and F, or the turning power on A (torque), is propor- tional to the product of volts by amperes ; that is, it is proportional to the watts consumed. The rate of rotation is made slow by the magnetic drag due to the reaction between the magnets M and the current induced in the rotating aluminium disk D which rotates between the poles of the magnets. The record- ing dials have therefore a speed which is proportional to the 'watts used, and their total rotation is proportional to the total energy, or watt hours, consumed. * This discussion should be followed by a laboratory experiment on the study of a small electric motor or dynamo. See, for example, Experiment No. 87 of the authors' Manual. FIG. 321. Interior of watt- hour meter INDUCTION COIL AND TBANSFOKMER 305 QUESTIONS AND PROBLEMS 1. What is the function (use) of the field magnet of a dynamo? Wood is cheaper than iron ; why are not the field cores made of wood? 2. How would it affect the voltage of a dynamo to increase the speed of rotation of its armature ? Why ? to increase the number of turns of wire in the armature coils? Why? to increase the strength of the magnetic field? Why? 3. When a wire is cutting lines of force at the rate of 100,000,000 per second, there is induced in it an E.M.F. of one volt. A certain dynamo armature has 50 coils of 5 loops each and makes 600 revolutions per minute. Each wire cuts 2,000,000 lines of force twice in a revolution. What is the E.M.F. developed? 4. What does the commutator of a dynamo do? What is the pur- pose of the commutator of a motor ? 5. Explain the process of "building up " in a dynamo. 6. Explain how an alternating current in the armature is trans- formed into a unidirectional current in the external circuit. 7. Explain why a series-wound motor can run on either a direct or an alternating circuit. 8. If a current is sent into the armature of Fig. 313 at V, and taken out at &, which way will the armature revolve ? 9. Will it take more work to rotate a dynamo armature when the circuit is closed than when it is open ? Why ? 10. Single dynamos often operate as many as 10,000 incandescent lamps at 110 volts. If these lamps are all arranged in parallel and each requires a current of .5 ampere, what is the total current furnished by the dynamo ? What is the activity of the machine in kilowatts and in horse power? 11. How many 110-volt lamps like those of Problem 10 can be lighted by a 12,000-kilowatt generator ? 12. Why does it take twice as much work to keep a dynamo running when 1000 lights are on t]ie circuit as when only 500 are turned on? PRINCIPLE OF THE INDUCTION COIL AND TRANSFORMER 362. Currents induced by varying the strength of a magnetic field. Let about 500 turns of No. 28 copper wire be wound around one end of an iron core, as in Fig. 322, and connected to the circuit of a galvanometer G. Let about 500 more turns be wrapped about another portion of the core and connected into the circuit of two dry cells. When the key K is closed, the deflection of the galvanometer will indicate that 306 INDUCED CUBKENTS a temporary current has been induced in one direction through the coil s ; and when it is opened, an equal but opposite deflection will indicate an equal current flow- ing in the opposite x-^ -, p ^ direction. V^J ^ ' '""", ,'ui\u _^ rrn FIG. 322. Induction of current by magnetizing experiment and demagnetizing an iron core illustrates the prin- ciple of the induction coil and the transformer. The coil jo, which is connected to the source of the current, is called the primary coil, and the coil s, in which the currents are induced, is called the secondary coil. Causing lines of force to spring into existence inside of s (in other words, magnetizing the space inside of s) has caused an induced current to flow in s ; and demagnetizing the space inside of s has also induced a current in s in accordance with the general principle stated in 348, that any change in the number of magnetic lines of force which thread through a coil induces a current in the coil. We may think of the lines as always existing as closed loops (see Fig. 258, p. 255) which collapse upon demagnetization to mere double lines at the axis of the coil. Upon magnetization one of these two lines springs out, cutting the encircling conductors and inducing a current. 363. Direction of the induced current. Lenz's law, which, it will be remembered, followed from the principle of conser- vation of energy, enables us to predict at once the direction of the induced currents in the above experiments ; and an observation of the deflections of the galvanometer enables us to verify the correctness of the predictions. Consider first the case in which the primary circuit is made and the core thus magnet- ized. According to Lenz's law the current induced in the sec- ondary circuit must be in such a direction as to oppose the change which is being produced by the primary current, that is, in such a direction as to tend to magnetize the core oppositely to the direction in which it is being magnetized by the primary. This INDUCTION COIL AND TRANSFORMER 307 means, of course, that the induced current in the secondary must encircle the core in a direction opposite to the direction in which the primary current encircles it. We learn, therefore, that on making the current in the primary the current induced in the secondary is opposite in direction to that in the primary. When the current in the primary is broken, the magnetic field created by the primary tends to die out. Hence, by Lenz's law, the current induced in the secondary must be in such a direction as to tend to oppose this process of demagnetization, that is, in such a direction as to magnetize the core in the same direction in which it is magnetized by the decaying current in the primary. Therefore, at break the current induced in the secondary is in the same direction as that in the primary. 364. E.M.F. of the secondary. If half of the 500 turns of the secondary s (Fig. 322) are unwrapped, the deflection will be found to be just half as great as before. Since the resistance of the circuit has not been changed, we learn from this that the E.M.F. of the secondary is proportional to the number of turns of wire upon it, a result which followed also from 351. If, then, we wish to develop a very high E.M.F. in the secondary, we have only to make it of a very large number of turns of fine wire. 365. Self-induction. If, in the experiment illustrated in Fig. 322, the coil % had been made a part of the same circuit as p, the E.M.F.'s induced in it by the changes in the magnetism of the core would of course have been just the same as above. In other words, when a current starts in a coil, the magnetic field which it itself produces tends to induce a current oppo- site in direction to that of the starting current, that is, tends to oppose the starting of the current; and when a current in a coil stops, the collapse of its own magnetic field tends to induce a current in the same direction as that of the stopping current, that is, tends to oppose the stopping of the current. This means merely that a current in a coil acts as though it had 308 INDUCED CURRENTS inertia, and opposes any attempt to start or stop it. This inertia- like effect of a coil upon itself is called self-induction. Let a few dry cells be inserted into a circuit containing a coil of a large number of turns of wire, the circuit being closed at some point by touching two bare copper wires together. Holding the bare wire in the fingers, break the circuit between the hands and observe the shock due to the current which the E. M. F. of self-induction sends through your body. Without the coil in circuit you will obtain no such shock, though the current stopped when you break the circuit will be many times larger. 366. The induction coil. The induction coil, as usually made (Fig. 323), consists of a soft iron core C composed of a bundle of soft iron wires ; a primary coil p wrapped around this core and consisting of, say, 200 turns of coarse copper wire 0) I d (2) FIG. 323. Induction coil (for example, No. 16), which is connected into the circuit of a battery through the contact point at the end of the screw d\ a secondary coil s surrounding the primary in the manner indicated in the diagram and consisting generally of between 30,000 and 1,000,000 turns of No. 36 copper wire, the termi- nals of which are the points t and t' ; and a hammer b, or other automatic arrangement for making and breaking the circuit of the primary. (See ignition system opposite p. 199.) Let the hammer b be held away from the opposite contact point by means of the finger, then touched to this point, then pulled quickly away. A spark will be found to pass between t and i' at break only never at make. This is because, on account of the opposing influence at make of self- induction in the primary, the magnetic field about the primary rises INDUCTION COIL AND TRANSFORMER 309 very gradually to its full strength, and hence its lines pass into the sec- ondary coil comparatively slowly. At break, however, by separating the contact points very quickly we can make the current in the primary fall to zero in an exceedingly short time, perhaps not more than .00001 second ; that is, we can make all of its lines pass out of the coil in this time. Hence the rate at which lines thread through or cut the secondary is perhaps 10,000 times as great at break as at make, and therefore the E.M.F. is also something like 10,000 times as great. In the normal use of the coil the circuit of the primary is automatically made and broken at I by means of the magnet and the spring r, precisely as in the case of the electric bell. Let the student analyze this part of the coil for him- self. The condenser shown in the diagram, with its two sets of plates connected to the conductors on either side of the spark gap between r and d, is not an essential part of a coil, but when it is introduced it is found that the length of the spark which can be sent across between t and t' is considerably increased. The reason is as follows : When the circuit is broken at b, the inertia (that is, the self-induction) of the primary current tends to make a spark jump across from d to b ; and if this happens, the current con- tinues to flow through this spark (or arc) until the terminals have become separated through a con- siderable distance. This' makes the current die down gradually instead of suddenly, as it ought to do to produce a high E.M.F ; but when a condenser is inserted, as soon as b begins to leave d the current begins to flow into the condenser, and this gives the hammer time to get so far away from d that an arc cannot be formed. This means a sudden break and a high E. M. F. Since a spark passes between t and if only at break, it must always pass in the same direction. Coils which give 24-inch sparks (perhaps 500,000 volts) are not uncommon. Such coils usually have hundreds of miles of wire upon their secondaries. 367. Laminated cores ; Foucault currents. The core of an induction coil should always be made of a bundle of soft-iron wires insulated from one an- other by means of shellac or varnish (see Fig. 324) ; for whenever a current is started or stopped in the primary p of a coil furnished with a solid iron core (see Fig. 325), the change in the magnetic field of the primary induces a current in the FIG. 324. Core of insulated iron wire FIG. 325. Diagram showing eddy cur- rents in solid core 310 INDUCED CURRENTS FIG. 326. Laminated drum-armature core with commutator, showing one coil wound on the core conducting core C, for the same reason that it induces one in the second- ary s. This current flows around the body of the core in the same direction as the induced current in the secondary, that is, in the direc- tion of the arrows. The only effect of these so-called eddy or Foucault currents is to heat the core. This is obviously a waste of energy. If we can prevent the appearance of these currents, all of the energy which they would waste in heat- ing the core may be made to appear in the current of the secondary. The core is therefore built of varnished iron wires, which run parallel to the axis of the coil, that is, perpendicular to the direction in which the cur- rents would be induced. The induced E.M.F. therefore finds no closed circuits in which to set up a current (Fig. 324). It is for the same rea- son that the iron cores of dynamo and motor armatures, instead of being solid, consist of iron disks placed side by side, as shown in Fig. 326, and insulated from one another by films of oxide. A core of this kind is called a laminated core. It will be seen that in all such cores the spaces or slots between the laminae must run at right angles to the direction of the induced E.M.F., that is, perpendicular to the conductors upon the core. 368. The transformer. The commercial transformer is a modified form of the induction coil. The chief difference is that the core R (Fig. 327), instead of being straight, is bent into the form of a ring or is given some other shape such that the magnetic lines of force have a continuous iron path instead of being obliged to push out into the air, as in the induction coil. always an alternating instead of an inter- mittent current which is sent through the primary A. Send- ing such a current through A is equivalent to first magnetiz- ing the core in one direction, then demagnetizing it, then magnetizing it in the opposite direction, etc. The result of TT< , i , . FIG. 327. Diagram of Furthermore, it .s transform ' er INDUCTION COIL AND TRANSFORMER 311 Main Conductor FlG ' m Alternating-current light- ing circuit with transformers these changes in the magnetism of the core is of course an induced alternating current in the secondary B. 369. The use of the transformer. The use of the transformer is to convert an alternating current from one voltage to another which, for some rea- son, is found to be more convenient. For example, in electric lighting where an al- ternating current is used, the E.M.F. generated by the dy- namo is usually either 1100 or 2200 volts, a voltage too high to be introduced safely into private houses. Hence transformers are connected across the main conductors in the manner shown in Fig. 328. The current which passes into the houses to supply the lamps does not come directly from the dynamo. It is an induced current generated in the transformer. Through the use of small transformers the voltage of the current of the house lighting circuit is further reduced and made available for the ringing of doorbells. 370. Pressure in primary and secondary. If there are a few turns in the primary and a large number in the secondary, the transformer is called a step-up transformer, because the P.D. produced at the terminals of the secondary is greater than that applied at the terminals of the primary. In electric lighting, transformers are mostly of the step-down type ; that is, a high P.D. (say, 2200 volts) is applied at the terminal of the primary, and a lower P.D. (say, 110 volts) is obtained at the terminals of the secondary. In such a transformer the primary will have twenty times as many turns as the secondary. In general, the ratio between the voltages at the terminals of the primary and secondary is the ratio of the number of turns of wire upon the two. 312 INDUCED CURRENTS 371. Efficiency of the transformer. In a perfect transformer the efficiency would be unity. This means that the electrical power, or watts, put into the primary (that is, the volts applied to its terminals times the amperes flowing through it) would be exactly equal to the power, or watts, taken out in the secondary (that is, the volts generated in it times the strength of the in- duced current) ; and, in fact, in actual transformers the latter product is often more than 97% of the former (that is, there is less than 3% loss of energy in the transformation). This lost energy appears as heat in the transformer. This transfer, which goes on in a big transformer, of huge quantities of power from one circuit to another entirely independent circuit, with- out noise or motion of any sort and almost without loss, is one of the most wonderful phenomena of modern industrial life. 372. Commercial transformers. Fig. 329 illustrates a common type of transformer used in electric lighting. The core is built up of sheet-iron laminae about ^ millimeter thick. Fig. 330 shows a section of the same FIG. 329. The core type of transformer FIG. 330. Cross sec- tion of transformer, showing shape of magnetic field FIG. 331. Trans- former case transformer. The closed magnetic circuit of the core is indicated by the dotted lines. The primaries and the secondaries are indicated by the letters P and S. Fig. 331 is the case in which the transformer is placed. Such cases may be seen, attached to poles outside of houses wherever alternating currents are used for electric lighting (Fig. 332). 373. Electrical transmission of power. Since the rate of production of electrical energy by a dynamo is the product of the E.M.F. generated by the current furnished, it is evident that in order to transmit from INDUCTION COIL AND TRANSFORMER 313 Transformer one point to another a given number of watts, say, 10,000, it is pos- sible to have either an E.M.F. of 100 volts and a current of 100 amperes or an E.M.F. of 1000 volts and a current of 10 amperes. In the two cases, however, the loss of energy in the wire which carries the current from the place where it is generated to the place where it is used will be widely different. For, watts amperes x volts ; but, from Ohm's law, volts = amperes x ohms. Therefore watts = amperes 2 x ohms = I 2 R. If, then, R represents the resistance of this transmitting wire, the so- called "line resistance," and / the current flowing through it, the heat developed in it will be proportional to I' 2 R. Hence the energy wasted in heating the line will be but ^-^ as much in the case of the 1000 volt, 10-ampere current as in the case of the 100 volt, 100-ampere current. Hence for long-distance transmission, where line losses are considerable, it is important to use the highest possible voltages. On account of the difficulty of insulating the commutator segments from one another, voltages higher than 1200 or 1500 cannot be obtained with direct-current dynamos of the kind that have been described. With alternators, however, the difficulties of insulation are very much less on account of the absence .of a commutator. The large 10,000- horse-power alternating-current dynamos on the Canadian side of Niagara Falls generate directly 12,000 volts. This is the highest volt- age thus far produced by generators. In all cases where these high pressures are employed they are transformed down at the receiving end of the line to a safe and convenient voltage (from 50 to 500 volts) by means of step-down transformers. It will be seen from the above facts that alternating currents are best suited for long-distance transmission. The Big Creek plant in California transmits power 241 miles at a pressure of 150,000 volts. (See opposite p. 241.) The Southern Sierras Power Company of FIG. 332. Transformer on electric- light pole 814 INDUCED CURRENTS California sends current 830 miles across the desert. Transmission at 220,000 volts is now under consideration for a line to extend the length of California, over 1100 miles. In all such cases step-up transformers, situated at the power house, transfer the electrical energy developed by 2.200 volts 22.000 volts 22,000 2,200 Alternator m Transformer Step-down wv( Transformer [\uo l \o\uo Motcr Lamps Power House Distant City FIG. 333. High- voltage long-distance transmission line the generator to the line, and step-down transformers, situated at the receiving end, transfer it to the motors or lamps which are to be sup- plied (Fig. 333). The generators used on the American side of Niagara Falls produce a pressure of 2300 volts. For transmis- sion to Buffalo, 20 miles away, this is transformed up to 22,000 volts. At Buffalo it is transformed down to the voltages suitable for operating the street cars, lights, and factories of the city. On the Canadian side the generators produce currents at 12,000 volts, as stated, and these are transformed up, for long-distance transmission, to 22,000, 40,000, and 60,000 volts (see Fig. 166, p. 150). 374. The tungar rectifier. Negative electrons are found to escape from a filament that is heated to in- candescence, and if this filament is then made more than, say, 25 volts negative with respect to a near-by anode any gas that surrounds the filament is found to be ionized (split into positively and negatively charged parts) by the violence of the blows which the electrons strike against its molecules. It is thus rendered con- ducting. These facts are utilized in the tungar rectifier of the alternating current. The bulb (Fig. 334) is filled with argon to a pressure of 3 to 8 cm. The anode is a small cone of graphite or tungsten, FIG. 334. Tun- gar bulb INDUCTION COIL AND TRANSFORMER 315 FIG. 335. Principle of opera- tion of the tungar rectifier and the cathode is a coiled tungsten filament. When the rectifier is in operation, the cone and the filament are alternately + and , one being + while the other is . When the cone is -f and the filament , the negative electrons from the filament are forced across the space from the fila- ment to the cone, and the argon, which is thereby ionized, carries the current from the cone to the filament. When the cone is and the filament +, the nega- tive electrons cannot escape from the filament ; hence the gas does not become conducting. The principle of operation can be understood from Fig. 335. The rectifier is connected to the alternating-current line at C and D. The alternating current in the primary coil P of the transformer T causes an induced current in S, which keeps the filament F incandes- cent. Under the action of the current, A and F are alternately + and . When ,P is , the electrons escape and ionize -the gas, permitting the current to pass. When .F is + the negative electrons are driven back into the filament and cannot escape to ionize the gas. Hence no current passes. In this way a unidirectional pulsating current passes through the storage batteries or other load. This rectifier is used largely for charging storage batteries for small-power purposes. 375. Principle of the carbon microphone. Let a dry cell, an ammeter, and two pieces 'of electric-arc carbon be arranged in series (Fig. 336). Press the carbons very gently and observe the reading of the. ammeter. Press , gradually harder, then gradually less, watch- ing the instrument. The current increases with increase in pressure, and decreases with decrease in pressure. FIG. 336. The principle of the carbon transmitter This peculiar behavior of carbon in offering a variable resistance with varia- tion in pressure is taken advantage of in constructing the carbon transmitter of the telephone. In the modern trans- mitter, however, the current is made to traverse many particles of granular carbon, which, lying loosely together, furnish a very great number of loose contacts (see Fig. 339). 316 INDUCED CURRENTS 376. Principle of the telephone. The telephone was invented in 1875 by Alexander Graham Bell of Washington (see on opposite page) and Elisha Gray of Chicago. The simple local-battery system is shown in Fig. 337. The current from the battery B (Fig. 337) is led first to the back of the diaphragm E, whence it passes through a little chamber C, filled with granular carbon, to the conducting back d of the transmitter, and thence through the primary p of the induction coil, and back to the battery. When a sound is made in front of the microphone, the vibrations produced by the sounding body are transmitted by Receiver Receiver B B FIG. 337. The telephone circuit (local-battery system) the air to the diaphragm, thus causing the latter to vibrate back and forth. These vibrations of the diaphragm vary the pressure upon the many contact points of the granular carbon through which the primary current flows. This produces con- siderable variation in the resistance of the primary circuit, so that as the diaphragm moves forward, that is, toward the carbon, a comparatively large current flows through p, and as it moves back a much smaller current. These changes in the current strength in the primary p produce changes in the magnetism of the soft-iron core of the induction coil. Currents are therefore induced in the secondary s of the induction coil, and these currents pass over the line and affect the receiver at the other end. A step-up induction coil is used to get sufficient potential to work through the high resistance of a long line. Clinedinet ALEXANDER GRAHAM BELL, WASHINGTON, D.C. Inventor of the telephone, 1875 Underwood & Underwood THOMAS A. EDISON, ORANGE, NEW JERSEY Inventor of the phonograph, the incan- descent lamp, etc. GUGLIELMO MARCONI (ITALY) ORVILLE WRIGHT, DAYTON, OHIO Inventor of commercial wireless Inventor, with his brother Wilbur, of telegraphy the airplane A GROUP OF MODERN INVENTORS THE WRIGHT AIRPLANE The most significant and far-reaching of the advances of the twentieth century, namely, man's conquest of the air after centuries of failure, was made when the Wright brothers first introduced the principle upon which all successful flight by heavier-than-air machines now depends, namely, control of stability by the warp- ing of wings, or by ailerons (hinged attachments to wings), in connection with the use of a rudder. The ripper panel shows one of the original gliders (Wilbur Wright inside) with which the Wrights first mastered the art of gliding (1900- 1903) and made more than a thousand gliding flights, some of them GOO feet long, following in this work the principles of gliding flight first demonstrated by Lilienthal and a little later, much more completely, by Chanute of Chicago (1895- 1897) . The lower panel shows " the first successful power flight in the history of the world " (Orville Wright in the machine, Wilbur running beside it as it rose from the track) . Four such flights were made on the morning of December 17, 1903, the longest of which lasted 59 seconds and covered a distance of 852 feet against a 20-mile wind INDUCTION COIL AND TRANSFORMER 317 FIG. 338. The modern receiver A modern telephone receiver is shown in Fig. 338. It consists of a permanently magnetized U-shaped piece of steel in front of whose poles is a soft-iron diaphragm which almost touches the ends of the mag- net. Wound in opposite directions upon the two poles are coils of fine insulated wire in series with each other and the line wire. G- is the earpiece, E the diaphragm, A the U-shaped magnet, and B the coils, consisting of many turns of fine wire and having soft-iron cores. When the rapidly alternating current from the secondary coil s (Fig. 337) flows through the coils of the receiver, the poles of the permanent magnet are thereby alternately strengthened and weakened in synchronism with the sound waves falling upon the diaphragm of the transmitter. The variations in the magnetic pull upon the diaphragm of the receiver cause it to send out sound waves exactly like those which fell upon the diaphragm of the trans- mitter. Telephonic conversation can be car- ried on over great distances as rapidly as if the parties sat on opposite sides of the same table. An electrical im- pulse passes over the telephone wires from New York to San Francisco in about one fifteenth of a second. The cross section of a complete long-distance transmitter is shown in Fig. 339. The current traverses granular carbon held between solid blocks of carbon. FIG . 339. Cross section of a long-distance telephone transmitter 318 INDUCED CUKKENTS QUESTIONS AND PROBLEMS 1. Draw a diagram of an induction coil and explain its action. 2. Does the spark of an induction coil occur at make or at break ? Why? 3. Explain why an induction coil is able to produce such an enor- mous E.M.F. 4. Why could not an armature core be made of coaxial cylinders of iron running the full length of the armature, instead of flat disks, as shown in Fig. 326 ? 5. What relation must exist between the number of turns on the primary and secondary of a transformer which feeds 110-volt lamps from a main line whose conductors are at 1100 volts P.D.? 6. Name two uses and two disadvantages of mechanical friction ; of electrical resistance. 7. A transformer is so wound as to step the voltage of the lighting circuit from 2200 volts down to 110. Sketch the transformer and its connections, marking the primary and the secondary, and state the relative number of turns in each. If the house circuit uses 40 amperes, what current must flow in the primary ? 8. Why does a " tungar " rectify an alternating current? 9. The same amount of power is to be transmitted over two lines from a power plant to a distant city. If the heat losses in the two lines are to be the same, what must be the ratio of the cross sections of the two lines if one current is transmitted at 100 volts and the other at 10,000 volts? (Power = IE; heat loss is proportional to 7 2 72.) 10. In telephoning from New York to San Francisco how far do you think the sound goes ? What passes along the telephone wire ? CHAPTER XVI* NATURE AND TRANSMISSION OF SOUND SPEED AND NATURE OF SOUND 377. Sources of sound. If a sounding tuning fork provided with a stylus is stroked across a smoked-glass plate, it produces a wavy line, as shown in Fig. 340 ; if a light suspended ball is brought into contact with it, the latter is thrown off with considerable violence. If we look about for the source of any sudden ^^^r ' Wr^S*~~~- / ' inpitm^Q*-*' noise, we find that some object has FJG m Trace ma B . Q . out condensations which add their effects as before ("see C'\ In the middle of , , FIG. 353. Graphical illustration of beats this second, however, the two forks are vibrating in opposite directions (see .ZT); that is, one is sending out rarefactions while the other sends out condensations. At the ear of the observer the union of the rarefaction (backward motion of the air particles) produced by one fork with the condensation (forward motion) pro- duced by the. other results in no motion at all, provided the two motions have the same energy ; that is, in the middle of the second the two sounds have united to produce silence (see B'). It will be seen from the above that the number of beats per second is equal to the difference in the vibration numbers of the two forks. To test this conclusion, let more wax or a heavier coin be added to the weighted prong ; the number of beats per second will be increased. Diminishing the weight will reduce the number of beats per second. In tuning a piano the double and triple strings are brought into unison by tuning so as to eliminate beats. 397. Interference of sound waves by reflection. Let a thin cork about an inch in diameter be attached to one end of a brass rod from one to two meters long. Let this rod be clamped firmly in the middle, as in Fig. 354. Let a piece of glass tubing a meter or more long and from an inch to an inch and a half in diameter be slipped over the cork, as shown. Let the end of the rod be stroked longitudinally with a well-resined cloth. A loud, shrill note will be produced. 334 NATURE AND TRANSMISSION OF SOUND This note is due to the fact that the slipping of the resined cloth over the surface of the rod sets the latter into longitudinal vibrations, so that its ends impart alternate condensations and rarefactions to the layers of air in contact with them. As soon as this note is started FIG. 354. Interference of advancing and retreating trains of sound waves the cork dust inside the tube will be seen to be intensely agitated. If the effect is not marked at first, a slight slipping of the glass tube forward or back will bring it out. Upon examination it will be seen that the agitation of the cork dust is not uniform, but at regular intervals throughout the tube there will be regions of complete rest, n A , rc 2 , n s , etc., separated by regions of intense motion. The points of rest correspond to the positions in which the reflected train of sound waves returning from the end of the tube neutralizes the effect of the advancing train passing down the tube from the vibrating rod. The points of rest are called nodes, the intermediate ^ a 3 a z % portions loops or anti- *==QR I I I II j TU/ v 4. n * n & n * n * nodes. 1 ne distance between these nodes is FlG ' 355 ' ^tance between nodes is one-half 1. 1 1 ^ WaVG len th one-half wave length, for at the instant that the first wave front 1 (Fig. 355) reaches the end of the tube it is reflected and starts back toward R. Since at this instant the second wave front # 2 is just one wave length to the left of a^ the two wave fronts must meet each other at a point w , just one-half wave length from the end of the tube. The exactly equal and opposite motions of the particles in the two wave fronts exactly neutralize each other. Hence the point n^ is a point of no motion, that is, a node. Again, at the in- stant that the rejected wave front a^ met the advancing wave front # 2 at 1? the third wave front a 8 was just one wave length to the left of n Hence, as the first wave front a^ continues REFLECTION AND KEENFOKCEMENT 335 to travel back toward R it meets a g at n^ just one-half wave length from n^ and produces there a second node. Similarly, a third node is produced at n^ one-half wave length to the left of n^ etc. Thus the distance between two nodes must always be just one lialf the wave length of the waves in the train. In the preceding discussion it has been tacitly assumed that the two oppositely moving waves are able to pass through each other without either of them being modified by the presence of the other. That two opposite motions are, in fact, transferred in just this manner through a medium consisting of elastic particles may be beautifully shown by the following experiment with the row of balls used in 380. Let the ball at one end FIG. 356. Nodes and loops in a cord of the row be raised a Black line denotes advancing tram ; dotted line, distance of, say, 2 inches reflected train and the ball at the other end raised a distance of 4 inches. Then let both balls be dropped simultaneously against the row. The two opposite motions will pass through each other in the row altogether without modification, the larger motion appearing at the end opposite to that at which it started, and the smaller likewise. Another and more complete analogy to the condition existing within the tube of Fig. 354 may be had by simply vibrating one end of a two- or three-meter rope, as in Fig. 356. The trains of advancing and reflected waves which continuously travel through each other up and down the rope will unite so as to form a series of nodes and loops. The nodes at c and e are the points at which the advancing and reflected waves are always urging the cord equally in opposite directions. The distance between them is one half the wave length of the train sent down the rope by the hand. QUESTIONS AND PROBLEMS 1. Account for the sound produced by blowing across the mouth of an empty bottle. The bottle may be tuned to different pitches by add- ing more or less water. Explain. 2. Explain the roaring sound heard when a sea shell, a tumbler, or an empty tin caa is held to the ear. 336 NATURE AND TRANSMISSION OF SOUND 3. Find the number of vibrations per second of a fork which produces resonance in a closed pipe 1 ft. long ; in an open pipe 1 ft. long*. (Take the speed of sound as 1120 ft. per second.) 4. A gunner hears an echo 5^ sec. after he fires. How far away was the reflecting surface, the temperature of the air being 20 C.? 5. The shortest closed air column that gave resonance with a tuning fork was 32 cm. Find the rate of the fork if the velocity of sound was 340 meters per second. - 'i - 6. A tuning fork gives strong resonance when held on its flat side or on its edge, but when held cornerwise over the air column the resonance ceases. Explain. 7. What is meant by the phenomenon of beats in sound? How may it be produced, and what is its cause ? 8. What is the length of the shortest closed tube that will act as a resonator to a fork whose rate is 427 per second ? (Temperature = 20 C.) 9. A fork making 500 vibrations per second is found to produce resonance in an air column like that shown in Fig. 349, first when the water is a certain distance from the top, and again when it is 34 cm. lower. Find the velocity of sound. 10. Show why an open pipe needs to be twice as long as a closed pipe if it is to respond to the same note. CHAPTER XVII PROPERTIES OF MUSICAL SOUNDS MUSICAL SCALES 398. Physical basis of musical intervals. Let a metal or card board disk 10 or 12 inches in diameter be provided with four concentric rows of equidistant holes, the successive rows containing respectively 24, 30, 36, and 48 holes (Fig. 357). The holes should be about J inch in diameter, and the rows should be about \ inch apart. Let this disk (a siren) be placed in the rotating apparatus and a constant speed imparted. Then let a jet of air be directed, as in 385, against each row of holes in succession. It will be found that the musical sequence do, mi, sol, do" results. If the speed of rotation is in- creased, each note will rise in pitch, but the sequence will remain unchanged. We learn, therefore, that the musical FIG. 357. Siren for produc- sequence do, mi, sol, do' consists of notes in S musical^sequence do, mi, whose vibration numbers have the ratios of 24, 30, 36, and 48, that is, 4, 5, 6, 8, and that this sequence is independent of the absolute vibration numbers of the tones. Furthermore, when two notes an octave apart are sounded together, they form the most harmonious combination which it is possible to obtain. These characteristics of notes an octave apart were recognized in the earliest times, long before any- thing whatever was known about the ratio of their vibration numbers. The preceding experiment showed that this ratio is the simplest possible, namely, 24 to 48, or 1 to 2. Again, the next easiest musical interval to produce, and the next 337 338 PROPERTIES OF MUSICAL SOUNDS most harmonious combination which can be found, corre- sponds to the two notes commonly designated as do, sol Our experiment showed that this interval corresponds to the next simplest possible vibration ratio, namely, 24 to 36, or 2 to 3. When sol is sounded with do', the vibration ratio is seen to be 36 to 48, or 3 to 4. We see, therefore, that the three simplest possible ratios of vibration numbers, namely, 1 to 2, 2 to 3, and 3 to 4, are used in the production of the three no tea do, sol, do'. Again, our experiment shows that another har- monious musical interval, do, mi, corresponds to the vibration ratio 24 to 30, or 4 to 5. We learn, therefore, that harmonious musical intervals correspond to very simple vibration ratios. 399. The major diatonic scale. When the three notes do, mi, sol, which, as seen above, have the vibration ratios 4, 5, 6, are all sounded together, they form a remarkably pleasing combination of tones. This combination was picked out and used very early in the musical development of the race. It is now known as the major chord. The major diatonic scale is built up of three major chords in the manner shown in the following table, where the first major chord is denoted by 1, the second by 2, and the third by 3. Syllables do re mi fa sol la, si do re Letters CD E F G A B C' D' Relative vibration numbers . . 24 27 30 32 36 40 45 48 54 111 22 2 333 The chords do-mi-sol (the tonic), sol-si-re (the dominant), and fa-la-do (the subdominant) occur frequently in all music. Standard middle C forks made for physical laboratories all have the vibration number 256, which makes A in the physical scale 426J. In the so-called international pitch A has 435 vibrations, and in the widely adopted American Federation of Musicians' pitch, 440. VIBRATING STRINGS 339 400. The even-tempered scale. If G- is taken as do, and a scale built up as above, it will be found that six of the above notes in each octave can be used in this new key, but that two additional ones are required (see table below). Similarly, to build up scales, as above, in all the keys demanded by modern music would require about fifty notes in each octave. Hence a compromise is made by dividing the octave into twelve equal intervals represented by the eight white arid five black keys of a piano. How much this so-called even-tempered scale differs from the ideal, or diatonic, scale is shown below. Note C D E F G A R C' D' E' F' G' Diatonic .... 256 288 320 341 384 426| 480 512 576 640 682.2 768 Diatonic key of G 384 433 480 512 576 640 720 768 Tempered ... 256 287.4 322.7 341.7 383.8 430.7 483.5 512 574.8 645.4 683.4 767.6 VIBRATING STRINGS* 401 . Laws of vibrating strings. Let two piano wires be stretched over a box or a board with pulleys attached so as to form a sonometer (Fig. 358). Let the weights A and B be adjusted until the two wires emit exactly the same note. The phenomenon of beats will make it possible to do this with Afi great accuracy Then let ^ ^ ^ The gonometer the bridge D be inserted ^exactly at the middle of one of the wires, and the two wires plucked in succession. The interval will be recognized at once as do, do'. Next let the bridge be inserted so as to make one wire two thirds as long as the other, and let the two be plucked again. The interval will be recognized as do, s#L Now it was shown in 398 that do 1 has twice as many vibrations per second as do, and sol has three halves as many. Hence, since the length corresponding to do' is one half as great as the first length, and that corresponding to sol two thirds *This discussion should be followed by a laboratory experiment on the laws of vibrating strings. See, for example, Experiment 41 of the authors 1 Manual. 340 PROPERTIES OF MUSICAL SOUNDS as great, we conclude from this experiment that, other things being equal, the vibration numbers of strings are inversely proportional to their lengths. Again, let the two wires be tuned to unison, and then let the weight A be increased until the pull which it exerts on the wire is exactly four times as great as that exerted by B. The note given out by the A wire will again be found to be an octave above that given out by the B wire. We learn, then, that the vibration numbers of similar strings of equal length are proportional to the square roots of their tensions. In stringed instruments, for example the piano, the differ- ent pitches are obtained by using strings of different length, tension, and mass per unit length. 402. Nodes and loops in vibrating strings. Let a string a meter long be attached to one of the prongs of a large tuning fork which makes in the neighborhood of 100 vibrations per second. Let the other end be at- tached as in the figure and the fork set into vi- ^ FlG . 359. string vibrating as a whole bration. If the fork is not electrically driven, which is much to be preferred, it may be bowed with a violin bow or struck with a soft mallet. By making the tension of the thread, for example, proportional to the numbers 9, 4, and 1 it will be found possible to make it vibrate either as a whole, as in Fig. 359, or in two or three parts (Fig. 360), This effect is due, as explained in 397, to FlG * 36 - String vibrating in three G^ segments the interference of the direct and reflected waves sent down the string from the vibrating fork. But we shall show in the next paragraph that in considering the effects of the vibrating string on the surrounding air we shall make no mistake if we think of it as clamped at each node, and as actually vibrating in two or three or four separate parts, as the case may be. FUNDAMENTALS AND OVERTONES 341 FUNDAMENTALS AND OVERTONES 403. Fundamentals and overtones. If the assertion just made be correct, then a string which has a node in the middle communicates to the air twice as many pulses per second as the same string when it vibrates as a whole. This may be conclusively shown as follows : Let the sonometer wire (Fig. 358) be plucked in the middle and the pitch of the corresponding tone carefully noted. Then let the finger be touched to the middle of the wire, and the latter plucked midway between this point and the end.* . The octave of the original note will be distinctly heard. Next let the finger be touched at a point one third of the wire length from one end, and the wire again plucked. The note will be recognized as .so/'. Since we learned in 399 that sol' has three halves as many vibrations as do', it must have three times as many vibrations as the original note. Hence a wire which is vibrating in three segments sends out three times as many vibrations as when it is vibrating as a whole. When a wire vibrates simply as a whole, it gives forth the lowest note which it is capable of producing. This note is called the fundamental or first partial of the wire. When the wire is made to vibrate in two parts, it gives forth, as has just been shown, a note an octave higher than the fundamental. This is called the first overtone or second partial. When the wire is made to vibrate in three parts it gives forth a note cor- responding to three times the vibration number of the funda- mental, namely, sol'. This is called the second overtone or third partial. When the wire vibrates in four parts, it gives forth the third overtone, which is two octaves above the fundamental. The overtones of wires are often called harmonics. They bear the vibration ratios 2, 3, 4, 5, 6, 7, etc. to the fundamental.! * It is well to remove the finger almost simultaneously with the plucking. t Some instruments, such as bells, can produce higher tones whose vibra- tion numbers are not exact multiples of the fundamental. These notes are still called overtones, but they are not called harmonics, the latter term being reserved for the multiples. Strings produce harmonics only. 342 PROPERTIES OF MUSICAL SOUNDS 404. Simultaneous production of fundamentals and overtones. Thus far we have produced overtones only by forcing the wire to remain at rest at properly chosen points during the bowing. Now let the wire be plucked at a point one fourth of its length from one end, without being touched in the middle. The tone most distinctly heard will be the fundamental ; but if the wire is now touched very lightly exactly in the middle, the sound, instead of ceasing altogether, will continue, but the note heard will be an octave higher than the fundamental, showing that in this case there was superposed upon the vibration of the wire as a whole a vibration in two segments also (Fig. 361). By touching the wire in the middle the vibra- tion as a whole was destroyed, but that in two parts re- mained. Let the experiment be repeated, with this differ- FIG. 361. A wire simultaneously emitting ence, that the wire is now its fundamental and first overtone plucked in the middle instead of one fourth its length from one end. If it is now touched in the middle, the sound will entirely cease, showing that when a wire is plucked in the middle there is no first overtone superposed upon the fundamental. Let the wire be plucked again one fourth of its length from one end and careful attention given to the compound note emitted. It will be found possible to recognize both the fundamental and the first overtone sounding at the same time. Similarly, by plucking at a point one sixth of the length of the wire from one end, and then touching it at a point one third of its length from the end, the second overtone may be made to appear distinctly, and a trained ear will detect it in the note given off by the wire, even before the fundamental is suppressed by touching at the point indicated. The experiments show, therefore, that in general the note emitted by a string plucked at random is a complex one, consist- ing of a fundamental and several overtones, and that just what overtones are present in a given case depends on where and how the wire is plucked. 405. Quality. Let the sonometer wire be plucked first in the middle and then close to one end. The two notes emitted will have exactly the same pitch, and they may have exactly the same loudness, FUNDAMENTALS AND OVERTONES 343 but they will be easily recognized as different in respect to somethmg which we call quality. The experiment of the last paragraph shows that the real physical difference in the tones is a difference in the sorts of overtones which are mixed with the fundamental in the two cases. Again, let a mounted C f fork be sounded simultaneously with a mounted C fork. The resultant tone will sound like a rich, full C, which will change into a hollow C when the C' is quenched with the hand. Everyone is familiar with the fact that when notes of the same pitch and londness are sounded upon a piano, a violin, and a cornet, the three tones can be readily distinguished. The last experiments suggest that the cause of this difference lies "in the fact that it is only the fundamental which is the same in the three cases, while the overtones are different. In other words, the characteristic of a tone which we call its qual- ity is determined simply by the number and prominence of the overtones which *are present. If the overtones present are few and weak, while the fundamental is strong, the tone is, as a rule, soft and mellow, as when a sonometer wire is plucked in the middle, or a closed organ pipe is blown gently, or a tuning fork is struck with a soft mallet. The presence of comparatively strong overtones up to the fifth adds fullness and richness to the resultant tone. This is illustrated by the ordinary tone from a piano, in which several if not all of the first five overtones have a prominent place. When overtones higher than the sixth are present, a sharp metallic quality begins to appear. This is illustrated when a tuning fork is struck, or a wire plucked, with a hard body. It is in order to avoid this quality that the hammers which strike against piano wires are covered with felt. 406. Analysis of tones by the manometric flame. A very simple and beautiful way of showing the complex character of most tones is furnished by the so-called manometric flames. This device consists of the following parts : a chamber in the block B (Fig. 362), through which gas is led by way of the 344 PROPERTIES OF MUSICAL SOUNDS tubes C and D to the flame F ; a second chamber in the block A, separated from the first chamber by an elastic diaphragm made of very thin sheet rubber or paper, and communicating with the source of sound through the tube E and trumpet G ; and a rotating mirror M by which the flame is observed. When a note is produced before the mouthpiece G> the vibra- tions of the diaphragm produce variations in the pressure of FIG. 362. Analysis of sounds with manometric flames the gas coming to the flame through the chamber in B, so that when condensations strike the diaphragm the height of the flame is increased, and when rarefactions strike it the height of the flame is diminished. If these up-and-down motions of the flame are viewed in a rotating mirror, the longer and shorter images of the flame, which correspond to successive intervals of time, appear side by side, as in Fig. 363. If a rotating mirror is not to be had, a piece of ordinary mirror glass held in the hand and oscillated back and forth about a vertical axis will be found to give satis factory results. FUNDAMENTALS AND OVERTONES 345 First let the mirror be rotated when no note is sounded before the mouthpiece. There will be no fluctuations in the flame, and its image, as seen in the moving mirror, will be a straight band, as shown in 2 (Fig. 363). Next let a mounted C fork be sounded, or some other simple tone produced in front of G. The image in the mirror will be that shown in 3. Then let another fork, C', be sounded in place of the C. The image will be that shown in 4. The images of the flame are now twice as close together as before, since the blows strike the diaphragm twice as often. Next let the open ends of the resonance boxes of the tuning forks C and C' be held together in front of G. The image of the flame will be as shown in 5. If the vowel o be sung in the pitch Bb before the mouthpiece, a figure exactly similar to 5 will be produced, thus showing that this last note is a com- plex, consisting of a fundamental and its first overtone. ; , * ? e C-B g a ^ . mi;s fli O ir m ^ I>,T! K89r||J _ O m OJ EC '3 -2 J2 o '3 ij'rflH "^ 3* C A ** ^ Illlisi FUNDAMENTALS AND OVEKTOKES 347 large resultant effect. In the same way a large number of very feeble pulls may set a heavy pendulum into vibrations of considerable amplitude if the pulls come at intervals exactly equal to the natural period of the pendulum. On the other hand, if the two sounding bodies have even a slight difference of period, the effect of the first pulses is neutralized by the ef- fect of succeeding pulses as soon as the two bodies, on account of their difference in period, get to swinging in opposite directions. Let notes of different pitches be sung into a piano when the dampers are lifted. The wire which has the pitch of the note sounded will in every case respond. Sing a little off the key and the response will cease. 409. Sympathetic vibrations produced by overtones. It is not essential, in order that a body may be set into sympathetic vibrations, that it have the same pitch as the sounding body, provided its pitch corresponds exactly with the pitch of one of the overtones of that body. Thus, if the damper is lifted from the C string of a piano and the octave below, C v is sounded loudly, C will be heard to sound after C has been quenched by the damper. In this case it is the first overtone of C l which is in exact tune with C, and which therefore sets it into sympathetic vibration. Again, if the damper is lifted from the G string while C\ is sounded, this note will be found to be set into vibration by the second overtone of C r A still more interesting case is obtained by removing the damper from E while C 1 is sounded. When C 1 is quenched, the note which is heard is not E, but an octave above E ; that is, ". This is because there is no overtone of C l which corresponds to the vi- bration of E] but the fourth overtone of C v which has five times the vibration number of C v corresponds exactly to the vibration number of E', the first overtone of E. Hence E is set into vibration not as a whole but in halves. 410. Physical significance of harmony and of discord. Let two pieces of glass tubing about an inch in diameter and a foot and a half long be supported vertically, as shown in Fig. 365. Let two gas jets (made by drawing down pieces of one-fourth inch glass tubing until, with full gas pressure, the flame is about an inch long) be thrust inside these tubes to a height of about three or four inches from the bottom. Let 348 PEOPEETIES OF MUSICAL SOUNDS the gas be turned down until the tubes begin to sing. Without attempt- ing to discuss the part which the flame plays in the production of the sound, we wish simply to call attention to the fact that the two tones are either quite in unison or so near it that only a few beats are produced per second. Now let the length of one of the tubes be slightly increased by slipping the paper cylinder S up over its end. The number of beats will be rapidly increased until they will become indistinguishable as separate beats and will merge into a jarring, grating discord. The experiment teaches that discord is simply a phenomenon of beats. If the vibra- tion numbers do not differ by more than five or six, that is, if there are not more than five or six beats per second, the effect is not particularly unpleasant. From this point on, however, as the difference in the vibration numbers, and therefore in the num- ber of beats per second, increases, the un- pleasantness increases, and becomes worst at FIG. 365. Illustrat- a difference of about thirty. Thus, the notes ing ^ d ?^f ion B and C", which differ by about thirty-two beats per second, produce about the worst possible discord. When the vibration numbers differ by as much as seventy, which is about the difference between C and E, the effect is again pleasing, or harmonious. Moreover, in order that two notes may harmonize well, it is necessary not only that the notes themselves shall not produce an unpleasant number of beats, but also that such beats shall not arise from their overtones. Thus, C and B are very discordant, although they differ by a large number of vibrations per second. The discord in this case arises between B and C", the first overtone of C. Again, there are certain classes of instruments, of which bells are a striking example, which produce insufferable discords when even such notes as do, sol, do', are sounded simultaneously WIND INSTRUMENTS 349 upon them. This is because these instruments, unlike strings and pipes, have overtones which are not harmonics, that is, which are not multiples of the fundamental ; and these over- tones produce beats either among themselves or with one of the fundamentals. It is for this reason that in playing chimes the bells are struck in succession, not simultaneously. QUESTIONS AND PROBLEMS 1. In what three ways do piano makers obtain the different pitches? 2. What did Helniholtz prove by means of his resonators? 3. If middle C is struck on a piano while the key for G in the octave above is held down, G will be distinctly heard when C is silenced. Explain. 4. At what point must the G^ string be pressed by the finger of the violinist in order to produce the note C ? 5. If one wire has twice the length of another and is stretched by four times the stretching force, how will their vibration numbers compare ? 6. A wire gives out the note G. What is its fourth overtone? 7. If middle C had 300 vibrations per second, how many vibrations would F and A have ? 8. What is the fourth overtone of C? the fifth overtone? 9. There are seven octaves and two notes on an ordinary piano, the lowest note being A 4 and the highest one C"". If the vibration number of the lowest note is 27, find the vibration number of the highest. 10. Find the wave length of the lowest note on the piano; the wave length of the highest note. (Take the speed of sound as 1130 ft. per sec.) 11. A violin string is commonly bowed about one seventh of its length from one end. Why is this better than bowing in the middle ? 12. Build up a diatonic scale on C = 264. WIND INSTRUMENTS 411. Fundamentals of closed pipes. Let a tightly fitting rubber stopper be inserted in a glass tube a (Fig. 366), eight or ten inches long and about three fourths of an inch in diameter. Let the stopper be pushed along the tube until, when a vibrating C' fork is held before the mouth, resonance is obtained as in 391. (The length will be six or seven inches.) Then let the fork be removed and a stream of air blown 350 PEOPEETIES OF MUSICAL SOUNDS FIG. 366. Musical notes from pipes across the mouth of the tube through a piece of tubing b, flattened at one end as in the figure.* The pipe will be found to emit strongly the note of the fork. In every case it is found that a note which a pipe may be made to emit is always a note to which it is able to re- spond when used as a resonator. Since, in 392, the best resonance was found when the wave length given out by the fork was four times the length of the pipe, we learn that when a current of air is suitably directed across the mouth of a closed pipe, it will emit a note lohich has a wave length four times the length of the pipe. This note is called the fundamental of the pipe. It is the lowest note which the pipe can be made to produce. 412. Fundamentals of open pipes. Since we found in 393 that the lowest note to which a pipe open at the lower end can respond is one the wave length of which is twice the pipe length, we infer that an open pipe, when suitably blown, ought to emit a note the wave length of which is twice the pipe length. This means that if the same pipe is blown first when closed at the lower end and then when open, the first note ought to be an octave lower than the second. Let the pipe a (Fig. 366) be closed at the bottom with the hand and blown ; then let the hand be removed and the operation repeated. The second note will indeed be found to be an octave higher than the first. We learn, therefore, that the fundamental of an open pipe has a wave length equal to twice the pipe length. 413. Overtones in pipes. It was found in 392 that there is a whole series of pipe lengths which respond to a given * If the arrangement of Fig. 366 is not at hand, simply blow with the lips across the edge of a piece of ordinary glass tubing within which a rubber stopper may be pushed back and forth. WIND INSTEUMENTS 351 fork, and that these lengths bear to the wave length of the fork the ratios |, |, |, etc. This is equivalent to saying that a closed pipe of fixed length can respond to a whole series of notes whose vibration numbers have the ratios 1, 3, 5, 7, etc. Similarly, in 393, we found that in the case of an open pipe the series of pipe lengths which will respond to a given fork bear to the wave length of the fork the ratios J, J, |, |-, etc. This, again, is equivalent to saying that an open pipe can re- spond to a series of notes whose vibration numbers have the ratios 1, 2, 3, 4, 5, etc. Hence we infer that it ought to be possible to cause both open and closed pipes to emit notes of higher pitch than their fundamentals (that is, overtones), and that the first overtone of an open pipe should have twice the rate of vibration of the fundamental (that is, it should be do', the fundamental being considered as do) ; that the second overtone should vibrate three times as fast as the fundamental (that is, it should be sol'); that the third overtone should vibrate four times as fast (that is, it should be do") ; that the fourth overtone should vibrate five times as fast (that is, it should be mi") ; etc. In the case of the closed pipe, however, the first overtone should have a vibration rate three times that of the fundamental (that is, it should be sol') ; the second overtone should vibrate five times as fast (that is, it should be mi") ; etc. In other words, while an open pipe ought to give forth all the harmonics, both odd and even, a closed pipe ought to produce the odd harmonics but be entirely incapable of producing the even ones. Let the pipe of Fig. 366 be blown so as to produce the fundamental when the lower end is open. Then let the strength of the air blast be increased. The note will be found to spring to do'. By blowing still harder it will spring to sol', and a still further increase will probably bring out do". The odd and the even harmonics are, in fact, emitted by the open pipe, as our theory predicted. When the lower end is closed, however, the first overtone will be found to be sol', and the next one mi", just as our theory demands for the closed pipe. 352 PKOPERTIES OF MUSICAL SOUNDS 414. Mechanism of emission of notes by pipes. Blowing across the mouth of a pipe produces a musical note, because the jet of air vibrates back and forth across the lip in a period which is determined wholly by the natural resonance period of the pipe. Thus, suppose that the jet a (Fig. 367) first strikes just inside the edge, or Up, of the pipe. A con- densational pulse starts down the pipe. When it returns to the mouth after reflection at the closed end, it pushes the jet outside the lip. This starts a rarefaction down the pipe, which, after return from the lower end, pulls the jet in again. There are thus sent out into the room regu- larly timed puffs, the period of which is con- trolled by the reflected pulses coming back from the lower end, that is, by the natural resonance period of the pipe. By blowing more violently it is possible to create, by virtue of the friction of the walls, so great and so sudden a compression in the mouth of the pipe that the jet is forced FlG : 367 \ vibrat - out over the edge before the return of the first reflected pulse. In this case no note will be produced unless the blowing is of just the right intensity to cause the jet to swing out in the period corresponding to an overtone. In this case the reflected pulses will return from the end at just the right intervals to keep the jet swinging in this period. This shows why a current of a particular intensity is required to start any particular overtone. Another way of looking at the matter is to think of the pipe as being filled up with air until the pressure within it is great enough to force the jet outside the lip, upon which a period of discharge follows, to be succeeded in turn by another period of charge. These periods are controlled by the length of the pipe and the violence of the blowing, precisely as described above. WIND INSTRUMENTS 353 jm If ( I U FIG. 368. Organ pipes With open pipes the situation is in no way different save that the reflection of a condensation as a rarefaction at the lower end makes the natural period twice as high, since the pipe length is now one-half wave length instead of one-fourth wave length (see 393). 415. Vibrating air-jet instruments. The mechanism of the production of musical tones by the ordinary organ pipe, the flute, the fife, the piccolo, and all whistles is essen- tially the same as in the case of the pipe of Fig. 367. In all these instruments an air jet is made to play across the edge of an opening in an air chamber, and the reflected pulses returning from the other end of the chamber cause it to vibrate back and forth, first into the chamber and then out again. FIG. 369. Moutt net, showing the ton ue I which opens and closes the upper end of closed at the remote end. In the flute it is open, in the pipe whistles it is usually closed, and in organ pipes it may be either open or closed. Fig. 368 shows a cross section of two types of organ pipes. The jet of air from S vibrates across the lip L in obedience to the pressure exerted on it by waves reflected from O. Pipe organs are provided with a different pipe for each note, but the flute, piccolo, and fife are made to produce a whole series of notes, either by blowing overtones or by open- ing holes in the tube, an oper- ation which is equivalent to cutting the tube off at the hole. Although important orchestral instruments, the flute and pic- FIG. 370. The vibrating tongue of the colo are not rich in overtones. mouth organ, accordion, etc. of regularly timed puffg of air ig made to pass from the instrument to the ear of the observer precisely as in the case of the rotating disk air chamber may be either open or 354 PKOPERTIES OF MUSICAL SOUNDS 416. Vibrating reed instruments. In reed instruments the vibrating air jet is replaced by a vibrating reed, or tongue, which opens and closes, at absolutely regular intervals, an opening against which the performer is directing a current of air. In the clari- net, the oboe, the bassoon, etc. the reed is placed at the upper end of the tube (see /, Fig. 369), and the theory of its opening and closing the orifice so as to admit successive puffs of air to the pipe is identical with the theory of the fluctuation of the air jet into and out of the organ pipe. For in these instruments the reed has little rigidity and its vibrations are controlled largely by the reflected pulses but partly by the reed and by the lips of the performer. In other reed instruments, like the mouth organ, the common reed organ, or the accordion, it is the elasticity of the reed alone (see z, Fig. 370) which controls the emission of pulses. In such instruments there is no necessity for air chambers. The arrows of Fig. 370 in- dicate the direction of the air current which is inter- rupted as the reed vibrates between the positions z ^ ^-j T , an *2 reed-organ pipe In still other reed instruments, like the reed pipes used in large organs (Fig. 371), the period of the pulses is controlled partly by the elasticity of the reed and partly by the return of the reflected waves ; in other words, the natural period of the reed is more or less coerced by the period of the reflected pulses. Within certain limits, therefore, such in- struments may be tuned by changing the length of the vibrating reed / without changing the length of the pipe. This is done by push- ing the wire r up or down. 417. Vibrating lip in- struments. In instruments of the bugle and cornet -pio. 372. The cornet type the vibrating reed is replaced by the vibrating lips of the musician, the period of their vibra- tion being controlled, precisely as in the organ pipe or the clarinet, by the period of the returning pulses. In the bugle the pipe length is fixed, WIND INSTRUMENTS 355 FIG. 373 I Cover and because of the narrowness of the tube all bugle calls are played with overtones. In the cornet (Fig. 372) and in most forms of horns, valves o, />, c, worked by the fingers, vary the length of the pipe, and hence such instruments can produce as many series of fundamentals and overtones as there are possible tube lengths. In the trombone the variation of pitch is accomplished by blow- ing overtones and by changing the length of the tube by a sliding U-shaped portion 418. The phonograph. In the original form of the phonograph the sound waves, collected by the cone, are carried to a thin metallic disk C (Fig. 373), exactly like a telephone diaphragm, which takes up very nearly the vibration form of the wave which strikes it. This vibration form is permanently impressed on the w r ax-coated cylinder M by means of a stylus D which is attached to the back Glass Diaphragm C \Needle Point \D FIG. 374. Mechanism for form- ing gramophone records of the disk. When the stylus is run a second time over the groove which it first made in the wax, it receives again and imparts to the disk the vibration form which first fell upon it. This is the principle of the Vegetable- Tissue Diaphragm The diamond point FIG. 375. The Edison diamond reproducer 356 PROPERTIES OF MUSICAL SOUNDS dictaphone and the ediphone, used to replace stenographers in business offices. The typist writes the letter by listening to the reproduction of the dictation. In the most familiar of the modern forms of the phonograph (gramo- phone, etc.) the needle point D, instead of digging a groove in wax, vibrates back and forth (see Fig. 374) over a greased zinc disk. This wavy trace in the disk is etched out with chromic acid. Then a copper mold is made by the electeotyping process, and as many as a thousand facsimiles of the original wavy line are impressed on hard-rubber disks by heat and pressure. When the needle is again run over the disk, it follows along the wavy groove and transmits to the diaphragm C vibra- tions exactly like those which originally fell upon it. Spoken words and vocal and orchestral music are reproduced in pitch, loudness, and quality with wonderful exactness. This instrument is one of the many inventions of Thomas Edison (see opposite p. 316). The diamond-tip reproducer used with the hill-and-dale Edison disks is shown in Fig. 375. QUESTIONS AND PROBLEMS 1. What proves that a musical note is transmitted as a wave motion ? 2. What evidence have you that sound waves are longitudinal vibrations ? 3. Why is the pitch of a sound emitted by a phonograph raised by increasing the speed of rotation of the disk ? 4. AVhat will be the relative lengths of a series of organ pipes which produce the eight notes of a diatonic scale ? 5. Will the pitch of a pipe organ be the same in summer as on a cold day in winter? What could cause a difference? 6. Explain how an instrument like the bugle, which has an air column of unchanging length, may be made to produce several notes of different pitch, such as C, G, C\ E', G'. (C is not often used.) 7. Why is the quality of an open organ pipe different from that of a closed organ pipe ? 8. The velocity of sound in hydrogen is about four times as great as it is in air. If a C pipe is blown with hydrogen, what will be the pitch of the note emitted ? 9. What effect will be produced on a phonograph record made with the instrument of Fig. 374 if the loudness of a note is increased? if the pitch is lowered an octave? CHAPTER XVIII NATURE AND PROPAGATION OF LIGHT TRANSMISSION OF LIGHT 419. Speed of light. Before the year 1675 light was thought to pass instantaneously from the source to the observer. In that year, however, Olaus Romer, a young Danish astron- omer, made the following observations. He had observed accurately the instant at which one of Jupiter's satellites M (Fig. 376) passed into Jupiter's shadow when 1C the earth was at E, and predicted, from the known mean time be- tween such eclipses, the exact instant at which a given eclipse should occur six months later when the earth was at E'. It actually took place 16 minutes 36 seconds (996 seconds) later. He concluded that the 996 seconds' delay represented the time required for light to travel across the earth's orbit, a distance known to be about 180,000,000 miles. The most precise of modern determinations of the speed of light are made by laboratory methods. The generally accepted value, that of Michelson, of The University of Chicago, is 299,860 kilometers per second. It is sufficiently correct to remember it as 300,000 kilometers, or 186,000 miles. 357 FIG. 376. Illustrating Romer's determination of the velocity of light 358 NATURE AND PROPAGATION OF LIGHT Though this speed would carry light around the earth 1\ times in a second, yet it is so small in comparison with interstellar distances that the light which is now reaching the earth from the nearest fixed star, Alpha Centauri, started 4.4 years ago. If an observer on the pole star had a telescope powerful enough to enable him to see events on the earth, he would not have seen the battle of Gettysburg (which occurred in July, 1863) until January, 1918. Both Foucault in France and Michelson in America have measured directly the velocity of light in water and have found it to be only three fourths as great as in air. It will be shown later that in all transparent liquids and solids it is less than it is in air. 420. Reflection Of light.* Let a beam of sunlight be admitted to a darkened room through a narrow slit. The straight path of the beam will be rendered visible by the brightly illumined dust particles sus- pended in the air. Let the beam fall on the surface of a mirror. Its direction will be seen to be sharply changed, as shown in Fig. 377. Let the mirror be held so that it is per- pendicular to the beam. The beam will be seen to be reflected directly back on itself. Let the mirror be turned through an angle of 45. The reflected beam will move through 90. FIG. 377. Illustrating law of reflection of light The experiment shows roughly, therefore, that the angle IOP, be- tween the incident beam and the normal to the mirror, is equal to the angle FOR, between the re- flected beam and the normal to the mirror. The first angle, IOP, is called the angle of incidence, and the second, FOR, the angle of reflection. The angle of reflection is equal to the angle of incidence. * An exact laboratory experiment on the law of reflection should either precede or follow this discussion. See, for example, Experiment 42 of the authors' Manual. A. A. MICHELSON, CHICAGO Distinguished for extraordinarily accu- rate experimental researches in light. First American scientist to receive the Nobel prize LOUD RAYLEIGH (ENGLAND) Distinguished for the discovery of argon , for very accurate determinations in elec- tricity and sound and for profound theo- retical studies HENRY A. ROWLAND, JOHNS HOPKINS Distinguished for the invention of the concave grating and for epoch-making studies in heat and electricity SIB WILLIAM CROOKES, LONDON Distinguished for his pioneer work (1875) in the study and interpretation of cath- ode rays (pp.438 and 443) A GROUP OF MODERN PHYSICISTS X-RAY PICTURE OF THE HUMAN THORAX This figure is a remarkable picture of the human thorax with the apex of the heart showing clearly on the right of the spinal column and the base stretching across the column, part of it showing distinctly on the left side opposite the apex TRANSMISSION OF LIGHT 359 421. Diffusion of light. In the last experiment the light was reflected by a very smooth plane surface. Now let the beam be allowed to fall upon a rough surface like that of a sheet of unglazed white paper. No reflected beam will be seen; but instead the whole room will be brightened appreciably, so that the outline of objects before invisible may be plainly distinguished. The beam has evidently been scattered in all directions by the innumerable little reflecting surfaces of which the surface of the paper is composed. The effect will be' much more noticeable if the beam is allowed to fall alternately on a piece of dead- black cloth and on FIG. 378. Regular and irregular reflection the white paper. The light is largely absorbed by the cloth, while it is scattered or diffusely reflected by the paper. Illumination sufficiently strong for sewing on white material may be altogether too weak for working on black goods. The difference between a smooth reflector and a rough one is illustrated in greatly magnified form in Fig. 378. The air shafts of apartment houses are made white to get the maximum diffusion of day- light into rooms that might otherwise be very dark. 422. Visibility of nonluminous bodies. Everyone is familiar with the fact that certain classes of bodies, such as the sun, a gas flame, etc., are self-luminous (that is, visible on their own account), while other bodies, like books, chairs, tables, etc., can be seen only when they are in the presence of luminous bodies. The above experiment shows how such nonluminous, diffusing bodies become visible in the presence of luminous bodies. For, since a diffusing surface scatters in all directions the light which falls upon it, each small element of such a surface is sending out light in a great many directions, in much the same way in which each point on a luminous surface is sending 860 NATUKE AND PROPAGATION OF LIGHT out light in all directions. Hence we always see the outline of a diffusing surface as we do that of an emitting surface, no matter where the eye is placed. On the other hand, when light comes to the eye from a polished reflecting surface, since the form of the beam is wholly undisturbed by the reflection, we see the outline not of the mirror but rather of the source from which the light came to the mirror, whether this source is itself self-luminous or not. All bodies other than self- luminous ones are visible only by the light which they diffuse. Black bodies send no light to the eye, but their outlines can be distinguished by the light which comes from the back- ground. Any object which can be seen, therefore, may be re- garded as itself sending rays to the eye ; that is, it may be treated as a luminous body. 423. Refraction. Let a narrow beam of sunlight be allowed to fall on a thick glass plate with a polished front and whitened back* (Fig. 379). It will be seen to split into a re- flected and a transmitted portion. The transmitted portion will be seen to be bent toward the per- pendicular OP drawn into the glass. Upon emergence into the air it will be bent again, but this time away from the per- pendicular O'P' drawn into the air. Let the incident beam strike FIG. 379. Path of a ray through a medium bounded by parallel faces the surface at different angles. It will be seen that the greater the angle of incidence the greater the bending. At normal incidence there will be no bending at all. If the upper and lower faces of the glass are parallel, the bending at the two faces will always be the same, so that the emergent beam is parallel to the incident beam. *A11 of these experiments on reflection and refraction may be done effectively and conveniently by using disks of glass, like those used with the Hart! Optical Disk, through which the beam can be traced. TRANSMISSION OF LIGHT 361 Similar experiments made with other substances have brought out the general law that whenever light travels obliquely from one medium into another in which the speed is less it is lent toivard the perpendicular, and when itfia&sesfrom one medium to another in which the speed is greater it is bent away from the perpendic- ular, drawn into the second medium. 424. Total reflection ; critical angle. Since rays emerging from a medium like water into one of less density, like air, are always bent /row the perpendicular (see II A, ImB, etc., Fig. 380), it is clear that if the angle of incidence on the under surface of the water is made larger and larger, a point must be reached at which the refracted ray is parallel to the surface (see InC, Fig. 380). It is interesting to inquire what will happen to a ray lo which strikes the surface at a still greater angle of incidence IoP f . It will not be unnatural to suppose that since the ray nC just grazed the surface, the ray lo will not be able to emerge at all. The O following experiment will show that this is indeed the case. FIG. 380. Rays coming from a source I under water to the boundary between air and water at different angles of incidence Let a prism with three polished edges, a polished front, and a whitened back be held in the path of a narrow beam of sun- light, 1 as shown in Fig. 381. If the angle of incidence I OP is small, the beam will divide at into a reflected and a trans- mitted portion, the former going to 5', the latter to S (neglect the color for the present). Let the prism be rotated slowly in the direction of the arrow. A point will be reached at which the transmitted beam disappears completely, while at the same time the spot at S' shows an appreciable increase in brightness. Since FIG. 381. Transmission and reflection of light at surface AB of a right-angled prism 362 NATURE AND PROPAGATION OF LIGHT the transmitted ray OS has totally disappeared, the whole of the light incident at O must be in the reflected beam. The angle of incidence IOP at which this occurs is called the critical angle. This angle for crown glass is 42.5, for water 48.5, for diamond 23.7. The critical angle for any substance may be defined as the angle of incidence in that substance for which the angle of refraction into air is 90. We learn, then, that when a ray of light traveling in any medium meets another in which the speed is greater, it is totally reflected if the angle of incidence is greater than a certain angle called the critical angle. QUESTIONS AND PROBLEMS 1. In Fig. 382 the portion acdb of the shadow is called the umbra, aec and Idf the penumbra. What kind of source has no penumbra ? 2. The sun is much larger than the earth. Draw a diagram showing the shape of the earth's umbra and penumbra. 3. Will it ever be possible for the moon to totally eclipse the sun from the whole of the earth's sur- face at once ? 4. Sirius, the brightest star, is about 52,000,000,- 000,000 miles away. If it were suddenly annihilated, how long would it shine on for us ? 5. Why is a room with JT FlG - 383 ' Anti ~ glare " lens " for EIG. 382. Shadow from automobile head- a broad source light white walls much lighter than a similar room with black walls ? 6. If the word " white : be painted with white paint (or whiting moistened with alcohol) across the face of a mirror and held in the path of a beam of sunlight in a darkened room, in the middle of the spot on the wall which receives the reflected beam the word " white " will appear in black letters. Explain. 7. Compare the reflection of light from white blotting paper with, that from a plane mirror. Which of these objects is more easily seent from a distance ? Why ? TRANSMISSION OF LIGHT 363 8. Devise an arrangement of mirrors by means of which you could see over and beyond a high stone wall or trench embankment. This is a very simple form of periscope. 9. Draw diagrams to show in what way a beam of light is bent (a) in passing through a prism; (b) in passing obliquely through a plate-glass window. 10. Explain the effect of the anti-glare " lens " (Fig. 383) upon the light of the automobile. 11. The moon has practically no atmos- phere. We know this because when a star appears to pass behind the moon there is FIG. 384 no decrease or increase in its apparent velocity while disappearing or coming into view again. If the moon had an atmosphere like the earth, explain how this would affect the apparent velocity of the star at both these times. 12. If a penny is placed in the bottom of a vessel in such a position that the edge just hides it from view (Fig. 384), it will become visible as soon as water is poured into the vessel. Explain. FIG. 385 FIG. 386 FIG. 387. A diagonal eyepiece 13. A stick held in water appears bent, as shown in Fig. 385. Explain. 14. A glass prism placed in the position shown in Fig. 386 is the most perfect reflector known. Why is it better than an ordinary mirror? 15. Diagonal eyepieces containing a right-angle prism of crown glass (Fig. 387) are used on astronomical telescopes in viewing celestial objects at a high altitude. Explain. 16. Explain why a straight wire seen obliquely through a piece of glass appears broken, as in Fig. 388. 17. The earth reflects sixteen times as much light to the moon as the moon does to the earth. Trace from the sun to the eye of the ob- server the light by which he is able to see the dark part of the new moon. Why can we not see the dark part of a third-quarter moon? FIG. 388 364 NATUKE AND PKOPAGATION OF LIGHT THE NATURE OF LIGHT 425. The corpuscular theory of light. All of the properties of light which have so far been discussed are perhaps most easily accounted for on the hypothesis that light consists of streams of very minute particles, or corpuscles, projected with the enormous velocity of 300,000 kilometers per second from all luminous bodies. The facts of straight-line propagation and reflection are exactly as we should expect them to be if this were the nature of light. The facts of refraction can also be accounted for, although somewhat less simply, on this hypothesis. As a matter of fact, this theory of the nature of light, known as the corpuscular theory, was the one most generally accepted up to about 1800. 426. The wave theory of light. A rival hypothesis, which was first completely formulated by the great Dutch physicist Huygens (1629-1695), regarded light, like sound, as & form of wave motion. This hypothesis met at the start with two very serious difficulties. In the first place, light, unlike sound, not only travels with perfect readiness through the best vacuum which can be obtained with an air pump, but it travels without any apparent difficulty through the great interstellar spaces which are probably infinitely better vacua than can be obtained by artificial means. If, therefore, light is a wave motion, it must be a wave motion of some medium which fills all space and yet does not hinder the motion of the stars and planets. Huygens assumed such a medium to exist, and called it the ether. The second difficulty in the way of the wave theory of light was that it apparently failed to account for the fact of straight-line propagation. Sound waves, water waves, and all other forms of waves with which we are most familiar bend readily around corners, while light apparently does not. It was this difficulty chiefly which led many of the most CHRISTIAN HUYGENS (1629-1695) Great Dutch physicist, mathematician, and astronomer; dis- covered the rings of Saturn ; made important improvements in the telescope ; invented the pendulum clock (1656) ; developed with marvelous insight the wave theory of light; discovered in 1690 the "polarization " of light. (The fact of double refraction was discovered by Erasmus Bartholinus in 1669, but Huygens first noticed the polarization of the doubly refracted beams and offered an explanation of double refraction from the standpoint of the wave theory) THE GREAT TELESCOPE OF THE YERKES OBSERVATORY (UNIVERSITY OF CHICAGO) This is the largest refracting telescope in the world. The objective is an achro- matic lens (see 475) 40 inches in diameter, which is mounted in a tube 63 feet long. In order to follow the apparent motions of the heavenly bodies due to the rotation of the earth, the entire tube and counterpoises, weighing 21 tons, are driven by a giant clock. The speed of the clock is controlled by a governor, similar in principle to that of Fig. 184. By means of electric motors the telescope may be pointed in any direction. It fs then clamped to the clock, which keeps it pointed toward the same region of the sky as long as may be desired. The entire floor may be raised or lowered to accommodate the observer THE NATURE OF LIGHT 365 famous of the early philosophers, including the great Sir Isaac Newton, to reject the wave theory and to support the projected-particle theory. Within the last hundred years, however, this difficulty has been completely removed, and in addition other properties of light have been discovered for which the wave theory offers the only satisfactory expla- nation. The most important of these properties will be treated in the next paragraph. 427. Interference Of light. Let two pieces of plate glass about half an inch wide and four or five inches long be separated at one end by a thin sheet of paper in the manner shown in Fig. 389, while the other end is clamped or held firmly together, so that a very thin wedge of air exists between the plates. Let a piece of asbestos or blotting paper be soaked in a solution of common salt (sodium chlo- ride) and placed over the tube of a Bunsen burner so as to touch the flame in the manner shown. The flame will be colored a bright yellow by the sodium in the salt. When the eye looks at the reflection of the flame from the glass surfaces, a series of fine black and yellow lines will be seen to cross the plate. Paper FIG. 389. Interference of light waves The wave theory offers the fol- lowing explanation of these effects. Each point of the flame sends out light waves which travel to the glass plate and are in part reflected and in part transmitted at all the surfaces of the glass, that is, at A'B r y at AB, at CD, and at C'D' (Fig. 389). We will consider, how- ever, only those reflections which take place at the two faces of the air wedge, namely, at AB and CD. Let Fig. 390 repre- sent a greatly magnified section of these two surfaces. Let the wavy line as represent a light wave reflected from the surface AB at the point a, and returning thence to the eye.- 366 NATURE AND PROPAGATION OF LIGHT CA Let the dotted wavy line ir represent a light wave reflected from the surface CD at the point i, and returning thence to the eye. Similarly, let all the continuous wavy lines of the figure represent light waves reflected from different points on AB to the eye, and let all the dotted wavy lines represent waves reflected from corresponding points on CD to the eye. Now, in precisely the same way in which two trains of sound waves from two tun- ing forks were found, in the experiment il- lustrating beats (see 396), to interfere with each other so as to produce silence whenever the two waves corresponded to motions of the air particles in opposite directions, so in this experiment the two sets of light waves from A B and CD inter- fere with each other so as to produce darkness wherever these two waves corre- spond to motions of the light-transmitting medium in opposite directions. The dark bands, then, of our experiment are sim- ply the places at which the two beams reflected from the two surfaces of the air film neutralize or destroy each other, while the light bands correspond to the places at which the two beams reenforce each other and thus produce illumination of double intensity. The position of the second dark band c must of course be determined by the fact that the distance from c to k and back (see Fig. 390) is a wave length more than from a to i and back, and so on down the wedge. This Interference Reinforcement Interference Re enforcement Interference Reenforcement Interference Reenforcement FIG. 390. Explanation of formation of dark and light bands by interference of light waves THE NATUKE OF LIGHT 367 phenomenon of the interference of light is met with in many different forms, and in every case the wave theory furnishes at once a wholly satisfactory explanation of the observed effects, while the corpuscular theory, on the other hand, is unable to account for any of these interference effects with- out the most fantastic and violent assumptions. Hence the corpuscular theory is now practically abandoned, and light is universally regarded by physicists as a form of wave motion. 428. The ether. We have already indicated that if the wave theory is to be accepted, we must conceive, with Huy- gens, that all space is filled with a medium, called the ether, in which the waves can travel. This medium cannot be like any of the ordinary forms of matter; for if any of these forms existed in interplanetary space, the planets and the other heavenly bodies would certainly be retarded in their motions. As a matter of fact, in all the hundreds of years during which astronomers have been making accurate obser- vations of the motions of heavenly bodies no such retarda- tion has ever been observed. The medium which transmits light waves must therefore have a density which is infinitely small even in comparison with that of our lightest gases. Further, in order to account for the transmission of light through transparent bodies, it is necessary to assume that the ether penetrates not only all interstellar spaces but all inter- molecular spaces as well. 429. Wave length of yellow light. Although light, like sound, is a form of wave motion, light waves differ from sound waves in several important respects. In the first place, an analysis of the preceding experi- ment, which seems to establish so conclusively the correctness of the wave theory, shows that the wave length of light is extremely minute in comparison with that of ordinary sound waves. The wave length of the yellow light used in that experiment is .00006 centimeter (about 40,000 incn > The number of vibrations per second made by the little particles which send out the light waves may be found, as in the case of sound, by 868 NATUKE AND PKOPAGATION OF LIGHT dividiug the velocity by the wave length. Since the velocity of light is 30,000,000,000 centimeters per second and the wave length is .00006 centimeter, the number of vibrations per second of the particles which emit yellow light has the enormous value 500,000,000,000,000. 430. Wave theory explanation of refraction. Let one look ver- tically down upon a glass or tall jar full of water and place his finger on the side of the. glass at the point at which the bottom appears to be, as seen through the water (Fig. 391). In every case it will be found that the point touched by the finger will be about one fourth of the depth of the water above the bottom. According to the wave theory this effect is due to the fact that the speed of light is less in water than in air. Thus, consider a wave which originates at any point P (Fig. 392) beneath a surface of water and spreads from that point with equal speed in all directions. At the instant at which the front of this wave first touches the surface at o it will, of course, FIG. 391. Appar- be of spherical form, having P as its center, ent elevation of Let aob be a section of this sphere. An in- th , bo " om of a body of water stant later, if the speed had not changed in passing into air, the wave would have still had P as its center, and its form would have coincided with the dotted line cOjd, so drawn that ac, oo^ and bd are all equal. But if the velocity in air is greater than in water, then at the instant considered the disturbance will have reached some point o 2 instead of o^ and hence the emerging wave will actually have the form of the heavy line co 2 d instead of the dotted line co^d. Now this wave co^d is more curved than the old wave aob, and hence it has its center at some point P' above P. In other words, the wave has bulged upward in passing from water into air. Therefore, when a section of this wave enters the eye at E, the wave appears to originate not at P but at P', for the light actually comes to the eye from P 1 as a center THE NATURE OF LIGHT 369 rather than from P. We conclude, therefore, that if light travels more slowly in water than in air, all objects beneath the surface of water ought to appear nearer to the eye than they actually are. This is precisely what we found to be the case in our experiment. Furthermore, since when the eye is in any position other than E, for example E\ the light travels to it over the broken path PdE\ the construction shows that light is always bent away from the perpendicu- lar when it passes obliquely into a medium in which the speed is greater. If it had passed into a medium of less speed, the point P would have appeared depressed below its natural position, because the wave, on emerg- ing into the slower medium, instead of bulging upward would be flattened, and therefore would have its center of curvature, or apparent point of origin, below P ; hence the oblique rays would have appeared to be bent toward the perpendicular, as we found in 423 to be the case. 431. The ratio of the speeds of light in air and water. The experiment with the tall jar of water in 430 not only indi- cates qualitatively that the speed of light in air is greater than in water, but it furnishes a simple means of determining the ratio of the two speeds. Thus, in Fig. 392 the line oo 2 represents just how far the wave travels in air while it is traveling the distance ac (= oo^) in water. Hence 2 is the \ ratio of the speeds of light in air and in water. FIG. 392. Representing a wave emerging from water into air 370 NATUKE AND PROPAGATION OF LIGHT Now the curvatures of the arcs cod and cod are measured by the reciprocals of their respective radii i 1 Curvature of that is, Curvature of co^d dP dP' (1) Now when the arcs are small, a condition which in general is realized in experimental work, their curvatures are propor- tional to the extent to which they bulge out from the straight line cod^ ; that is, Curvature of co^d Curvature of co^d From (1) and (2) we get Speed in air _ speed in air speed in water dP Speed in water dP' (2) (3) * Construct an angle of 45 (Fig. 393, (1)). Its arc contains 45 and the angle formed by the tangents , t' is 45. Now with a radius three times as great (Fig. 393, (2)) draw an arc whose length is equal to that of the arc in Fig. 393 (1). Since the radius is three times as great, this arc contains 15, and the angle formed by the tangents is 15. From this we see that the arc whose radius is three times as great curves, or changes its direction, one third as fast ; that is, the change in curvature of an arc of given length varies inversely with the radius. In gen- eral then, the curvature of an arc is measured by the reciprocal of its radius. t oc (Fig. 394) is a mean proportional between the two segments of the diameter ; hence ao x od = oc 2 . For very small arcs od is practically equal to the diameter r. Hence FIG. 393 ao = 2r oc 2 1 or ao = x - 2 r Therefore ao is proportional to _. That is, the distances to which two small arcs having a r common chord bulge out from the chord are proportional to the respective curvatures of the arcs. FIG. 394 THE NATURE OF LIGHT 371 But in looking vertically downward, as in the experiment dP , oP , with the iar of water, -- becomes ; hence, dP' oP' Speed in air _ oP _ real depth Speed in water oP' apparent depth But in our experiment we found that the bottom was appar- oP 4 ently raised one fourth of the depth ; that is, that - = - We conclude, therefore, that light travels three fourths as fast in water as in air. The fact that the value of this ratio, as determined by this indirect method, is exactly the same as that found by Foucault and Michelson (see opposite p. 358) by direct measurement (419) furnishes one of the strongest proofs of the correct- ness of the wave theory. 432. Index of refraction. The ratio of the speed of light in air to its speed in any other medium is called the index of refrac- tion of that medium. It is evident that the method employed iii the last paragraph for determining the index of refraction of water can be easily applied to any transparent medium whether liquid or solid.* The refractive indices of some of the commoner substances are as follows: Water 1.33 Crown glass 1.53 Alcohol 1.36 Flint glass 1.67 Turpentine 1.47 Diamond 2.47 433. Light waves are transverse. Thus far we have discov- ered but two differences between light waves and sound waves ; namely, the former are disturbances in the ether and are of vary short wave length, while the latter are disturbances in * To show the extreme beauty, simplicity, and accuracy of this method of getting index of refraction it is suggested that the teacher use the following method in his laboratory work. A very sharp pencil must be used for this exercise. Make a dot P on a sheet of paper. Put the glass plate (Fig. 395, (1)) on the sheet so that the 372 NATURE AND PROPAGATION OF LIGHT ordinary matter and are of relatively great wave length. There exists, however, a further radical difference which follows from a capital discovery made by Huygens (see opposite p. 364) in the year 1690. It is this : While sound waves consist, as we have already seen, of longitudinal vibrations of the particles of the transmitting medium, that is, vibrations back and forth in the line of propagation of the wave, light waves are like the water waves of Fig. 346, p. 324, in that they consist of transverse vibrations, that is, vibrations of the medium at right angles to the direction of the line of propagation. In order to appreciate the difference between the behavior of waves of these two types under certain conditions, conceive edge of the label pasted around the edge of the glass coincides with the dot (or in case a prism (Fig. 395, (2)) is used, let the apex P coincide with the dot). Draw the base line ef and the other sides of the glass, holding it firmly down meanwhile. Be sure that at no time during the exercise does the glass slip the slightest from its first position. Lay a ruler upon the paper in a slantwise position cd (not touching the glass), and, with one eye closed, make its edge point ex- actly at the apparent position of P as seen through the glass. If you are now sure that your ruler did not push the glass out of position, draw a line cd with the sharp pencil. Similarly, draw another line ab about as far to the right of the center as cd is to the left. Remove your glass and complete the drawing indicated in the diagram. P' is the apparent position of P. As you have already learned from your text, the ratio of the velocities of light in air and glass is found by dividing dP by dP'. Measure these distances very carefully to 0.1 mm., and calculate the index of refraction to two decimal places. Make two or three more trials and compare results. FIG. 395. Index of refraction THE NATURE OF LIGHT 373 of transverse waves in a rope being made to pass through two gratings in succession, as in Fig. 396. So long as the slits in both gratings are parallel to the plane of vibration of the hand, as in Fig. 396, (1), the waves can pass through them with perfect ease ; but if the slits in the first grating P are parallel to the direction of vibra- tion, while those of the second grating Q are turned at right angles to this direction, as in _,. nf .~ x ~ x ., . . FIG. 396. Transverse waves passing Kg. 896, (2), it is evi- through site dent that the waves will pass readily through P, but will be stopped completely by Q, as shown in the figure. In other words, these gratings P and Q will let through only such vibrations as are parallel to the direction of their slits. If, on the other hand, a longitudinal instead of a transverse W ave such, for example, as a sound wave had approached such a grating, it would have been as much transmitted in one position of the grating as in another, since a to-and-fro motion of the particles can evidently pass through the slits with exactly the same ease, no matter how they are turned. Now two crystals of tourmaline are found to behave with respect to light waves just as the two gratings behave with respect to the waves on the rope. Let one such crystal a (Fig. 397) be held in front of a small hole in a screen through which a beam of sunlight is FlG> 39L Tourmaline tongs passing to a neighboring wall ; or, if the sun is not shining, simply let the crystal be held between the eye and a source of light. The light will be readily transmitted, although some- what diminished in intensity. Then let a second crystal b be held in line with the first. The light will still be transmitted, provided the axes of 374 NATURE AND PROPAGATION OF LIGHT the crystals are parallel, as shown in Fig. 308. When, however, one of the crystals is rotated in its ring through 90 (Fig. 399), the light is cut off. This shows that a crystal of tourma- line is capable of transmitting only FIG. 398. Light pass- ing through tourmaline FIG. 399. Light cut off by crossed tourmaline crystals light which is vibra- ting in one particu- lar plane. From this ex- periment, there- fore, we are forced to conclude that light waves are transverse rather than longitudinal vibrations. The experiment illustrates what is technically known as the polarization of light, and the beam which, after passage through , is unable to pass through b if the axes of a and b are crossed, is known as a polarized beam. It is, then, the phenomenon of the polariza- tion of light upon which we base the conclusion that light waves are transverse. 434. Intensity of illumination. Let four candles be set as close together as possible in such a position B as to cast upon a white screen C, placed in a well-darkened room, a shadow of an opaque object O (Fig. 400). Let one single candle be placed in a position A such that it will cast another shadow of upon the screen. Since light from A falls on the shadow cast by B, and light from B falls on the shadow cast by A, it is clear that the two shadows will appear equally dark only when light of equal intensity falls on each ; that is, when A and B produce equal illumination upon the screen. Let the positions of A and B be shifted until this condition is fulfilled. Then let the distances from B to C and from A to C be measured. If all five candles are burning with flames of the same size, the first distance will be found to be just twice as great as the second. Hence the illumination produced upon the screen by each one of the candles at B is but one fourth as great as that produced on the screen by one candle at A, one half as far away. FIG. 400. Rumford's photometer THE NATURE OF LIGHT 375 The above is the direct experimental proof that the intensity of illumination varies inversely as the square of the distance from the source. The theoretical proof of the law is furnished at once by Fig. 401, for since all the light which falls from the candle L on A is spread over four times as large an area when it reaches B, twice as far away, and over nine times as large an area FIG. 401. Proof of law of inverse squares when it reaches (7, three times as far away, obviously the in- tensities at B and at C can be but one fourth and one ninth as great as at A. The above method of comparing experimentally the inten- sities of two lights was first used by Count Rumford. The arrangement is therefore called the Rumford photometer (light measurer). 435. Candle power. The last experiment furnishes a method of comparing the light-emitting powers of various sources of light. For example, suppose that the four candles at B are replaced by a gas flame, and that for the condition of equal illumination upon the screen the two distances BC and AC are the same as above, namely, 2 to 1. We should then know that the gas flame, which is able to produce the same illumination at a distance of two feet as a candle at a distance of one foot, has a light-emitting power equal to four candles. In general, then, the candle powers of any tivo sources which produce equal illumination on a given screen are directly proportional to the squares of the distances of the sources from the screen. It is customary to express the intensities of all sources of light in terms of candle power, one candle power being denned as the amount of light emitted by a sperm candle J- inch in 376 NATURE AND PROPAGATION OF LIGHT diameter and burning 120 grains (7.776 grams) per hour. The candle power of an ordinary gas flame burning 5 cubic feet per hour is from 16 to 25, depending on the quality of the gas. A standard candle at a distance of 1 foot gives an intensity of illumination called & foot-candle. A 100-candle-power lamp, for example, at a distance of 1 foot gives an intensity of illu- mination of 100 foot-candles ; at 2 feet, of 25 foot-candles ; at 5 feet, of 4 foot-candles ; and at 10 feet, of 1 foot-candle. 436. Bunsen's photometer. Let a drop of oil or melted paraffin be placed in the middle of a sheet of unglazed white paper to render it translucent. Let the paper be held near a window and the side away from the window observed. The oiled spot will appear lighter than the remainder of the paper. Then let the paper be held so that the side nearest the window may be seen. The oiled spot will appear darker than the rest of the paper. We learn, therefore, that when the paper is viewed from the side of greater illumination, the oiled spot appears dark ; but when it is viewed from the side of lesser illumination, the spot appears light. If, then, the two sides of the paper are equally illuminated, the spot ought to be of the same brightness when viewed from either side. Let the room be darkened and the oiled paper placed between two gas flames, two electric lights, or any two equal sources of light. It will be observed that when the paper is held closer to one than the other, the spot will appear dark when viewed from the side next the closer light; but if it is then moved until it is nearer the other source, the spot will change from dark to light when viewed always from the same side. It is always pos- ^IG. 402. Bunsen's photometer sible to find some position for the oiled paper at which the spot either disappears altogether or at least appears the same when viewed from either side. This is the position at which the illuminations from the two sources are equal. Hence, to find the candle power of any unknown source it is only necessary to set up a candle on one side and the unknown source on the other, as in Fig. 402, and to move the spot A to the position of equal illumination. The can- dle power of the unknown source at C will then be the square of the distance from C to A, divided by the square of the distance from B to A. This arrangement is known as the Bunsen photometer. THE NATURE OF LIGHT 377 QUESTIONS AND PROBLEMS 1. Distinguish between candle power, intensity of light, and inten- sity of illumination. 2. How many candles will be required to produce the same intensity of illumination at 2 m. that is produced by 1 candle at 30 cm. ? 3. A 500-candle-power lamp is placed 50 m. from a darkly shaded place along the street. At what distance would a 100-candle-power lamp have to be to produce the same intensity of illumination ? 4. If a 2-candle-power light at a distance of 1 ft. gives enough illumination for reading, how far away must a 3 2-candle-power lamp be placed to make the same illumination? How strong a lamp should be used at a distance of 8 ft. from the book ? 5. A Bunsen photometer placed between an arc light and an incan- descent light of 32 candle power is equally illuminated on both sides when it is 10 ft. from the incandescent light and 36 ft. from the arc light. What is the candle power of the arc ? 6. A 5-candle-power and a 30-candle-power source of light are 2 m. apart. Where must the oiled disk of a Bunsen photometer be placed in order to be equally illuminated on the two sides by them ? 7. If the sun were at the distance of the moon from the earth, in- stead of at its present distance, how much stronger would sunlight be than at present? The moon is 240,000 mi. and the sun 93,000,000 mi. from the earth. 8. If a gas flame is 300 cm. from the screen of a Rumford photom- eter, and a standard candle 50 cm. away gives a shadow of equal inten- sity, what is the candle power of the gas flame ? 9. Will a beam of light going from water into flint glass be bent toward or away from the perpendicular drawn into the glass ? 10. When light passes obliquely from air into carbon bisulphide it is bent more than when it passes from air into water at the same angle. Is the speed of light in carbon bisulphide greater or less than in water? 11. If light travels with a velocity of 186,000 miles per sec. in air, what is its velocity in water, in crown glass, and in diamond? (See table of indices of refraction, p. 371.) CHAPTER XIX IMAGE FORMATION IMAGES FORMED BY LENSES 437. Focal length of a convex lens. Let a convex lens be tfeld in the path of a beam of sunlight which enters a darkened room, where it is made plainly visible by means of chalk dust or smoke. The beam will be found to converge to a focus F, as shown in Fig. 403. The explanation is as follows: The waves from the sun or any distant object are without any appreciable curvature when they strike the lens; that is, they are so-called plane waves (see Fig. 403). Since the speed of light is less in glass than in air, the cen- Fic " 403 ' prhlci P al focusFand focal length CF of a convex lens tral portion of these waves is retarded more than the outer portions in passing through the lens. Hence, on emerging from the lens the waves are con- cave instead of plane, and close in to a center or focus at F. A second way of looking at the phenomenon is to think of the " rays " which strike the lens as being bent by it, in accordance with the laws given in 423, so that they all pass through the point F. The line through the point C (the optical center) of the lens, perpendicular to its faces, is called the principal axis. The point F at which rays parallel to the principal axis (incident plane waves) are brought to a focus is called the principal focus. The distance CF from the center of the lens to the prin- cipal focus is called the focal length (/) of the lens. 378 IMAGES FORMED BY LENSES 379 FIG. 404. Focal plane of a convex lens The plane F'FF" (Fig. 404) m which plane waves (parallel rays) coming to the lens from slightly different directions, as from the top and bottom of a distant house, all have their foci F', F", etc. is //UmUWmffiqttttffl -F called the focal plane of the lens. Since the curvature of any arc is defined as the reciprocal of its radius (see footnote, p. 370), the curvature which a lens impresses on an incident plane wave is equal to - Moreover, no matter what the J curvature of an incident wave may be, the lens will ahvays change the curvature by the same amount, - c/ Let the focal length of a convex lens be accurately determined by measuring the distance from the middle of the lens to the image of a distant house. 438. Conjugate foci. If a point source of light is placed at F (Fig. 403), it is obvious that the light which goes through the lens must exactly retrace its former path ; that is, its FIG. 405. Conjugate foci waves will be rendered plane or its rays parallel by the lens. But if the point source is at a distance D greater than / (Fig. 405), then the waves upon striking the lens have a curvature (sin<3e the curvature of an arc is defined as the reciprocal of its radius), which is less than their former curvature, -. Since the lens was able to subtract all the 380 IMAGE FORMATION curvature from waves coming from F and render them plane, by subtracting the same curvature from the flatter waves from P it must render them concave ; that is, the rays after passing through the lens are converging and intersect at P'. If the source is placed at P', obviously the rays will meet at P. Points such as P and P', so related that one is the image of the other, are called conjugate foci. 439. Formula for conjugate foci ; secondary foci. Since in Fig. 405 the curvature of the wave when it emerges from the lens is opposite in direction to its curvature when it reaches the lens, the sum of these curvatures, -- 1 -- , repre- sents the power of the lens to change the curvature of the incident wave, which by 437 is Hence ' that is, the sum of the reciprocals of the distances of the conju- gate foci from the lens is equal to the reciprocal of the focal length. If D = D f , then the equation shows that both D and D { are equal to 2/. The two conjugate foci S and S 1 which are at equal dis- tances from the lens are called the secondary foci, and their distance from the lens is twice the focal distance. FIG. 406. Formation of a real image by a lens 440. Images of Objects. Let a candle or electric-light bulb be placed between the principal focus F and the secondary focus S at PQ (Fig. 406), and let a screen be placed at P'Q'. An enlarged inverted image will be seen upon the screen. IMAGES FORMED BY LENSES 381 This image is formed as follows: All the light which strikes the lens from the point P is brought together at a point P'. The location of this image P' must be on a straight line drawn from P through C ; for any ray passing through C will remain parallel to its original direction, since the portions of the lens through which it enters and leaves may be regarded as small parallel planes (see 423). The image P'Q' is therefore always formed between the lines drawn from P and Q through C. If the focal length / and the distance of the object D are known, the distance of the image D. may be obtained easily from formula (1). The position of the image may also be found graphically as follows : Of the cone of rays passing from P to the lens, that FIG. 407. Ray method of constructing an image ray which is parallel to the principal axis must, by 43 7, pass through the principal focus F . The intersection of this line with the straight line through C locates the image P r (see Fig. 407). Q', the image of Q, is located similarly. 441. Size of image. Since the image and object are always between the intersecting straight lines PP f and QQ', the similar triangles PCQ and P'CQ' show that , . Length of object _ distance of object from lens Length of image distance of image from lens It may be seen from Fig. 407, as well as from formulas (1) and (2), that 1. When the object is at S the image is at ', and image and object are of the same size. 382 IMAGE FORMATION FIG. 408. Virtual image formed by a convex lens 2. As the object moves out from S to a great distance the image moves from S f up to F' 9 becoming smaller and smaller. 3. As the object moves from S up to F the image moves out to a very great distance to the right, becoming larger and larger. 4. When the object is at F the emerging waves are plane (the emerging rays are parallel), and no real image is formed. 442. Virtual image. We have seen that when the object is at F the waves after passing through the lens are plane. If, then, the object is nearer to the lens than F, the emerg- ing waves, although reduced in curvature, will still be con- vex, and, if received by an - , eye at E, will appear to come / . *, r ,. . FIG. 409. Ray method of locating from a point P* (Fig. 408). a virtual image in a convex lens Since, however, there is actu- ally no source of light at P', this sort of image is called a virtual image. Such an image cannot be projected upon a screen as a real image can, but must be observed by an eye. The graphical location of a virtual image may be accomplished precisely as in the case of a real image ( 440). It will be seen that in this case (Figs. 408 and 409) the image is enlarged and erect. 443. Image in concave lens. When a plane wave strikes a concave lens, it must emerge as a divergent wave, since the middle of the wave is retarded by the glass less than the edges (Fig. 410). The point F from which plane waves appear to come after passing through such a lens is the principal focus of the lens. For the same FIG. 410. Virtual focus of a con- cave lens IMAGES IN MIRRORS 383 reason .as in the case of the convex lens the centers of the transmitted waves from P and Q (Fig. 411), that is, the images P 1 and Q', must lie upon the lines PC and QC; and since the FIG. 411. Image in a concave lens FIG. 412. Ray method of locating an image in a concave lens curvature is increased by the lens, they must lie closer to the lens than P and Q. Fig. 411 shows the way in which such a lens forms an image. This image is always virtual, erect, and diminished. .The graphical method of locating the image is shown in Fig. 412. IMAGES IN MIRRORS 444. Image of a point in a plane or a curved mirror. We are all familiar with the fact that to an eye at E (Fig. 413), looking into a plane mirror mn, a pen- cil point at P appears to be at some point P' behind the mirror. We are able in the laboratory to find experi- mentally the exact location of this image P' with respect to P and the mirror, but we may also obtain this location from theory as follows : Con- sider a light wave which originates in the point P (Fig. 413) and spreads in all directions. Let aob be a section of the wave at the instant at which it reaches the reflecting surface mn. An instant later, if there were no reflecting surface, the wave would have reached the position of the dotted line cofl. FIG. 413. Wave reflected from a plane surface 384 IMAGE FORMATION Since, however, reflection took place at win, and since the reflected wave is propagated backward with exactly the same velocity with which the original wave would have been prop- agated forward, at the proper instant the reflected wave must have reached the position of the line co 2 d, so drawn that oo^ is equal to oo z . Now the wave co^d has its center at some point P', and it will be seen that P' must lie just as far below mn as P lies above it, for cofl and co^d are arcs of equal circles JXlM FIG. 414. Wave reflected from a convex surface FIG. 415. Wave reflected from a concave surface having the common chord cd. For the same reason, also, P' must lie on the perpendicular drawn from P through mn. When, then, a section of this reflected wave co 2 d enters the eye at E, the wave appears to have originated at P' and not at P, for the light actually comes to the eye from P' as a center rather than from P. Hence P 1 is the image of P. We learn, therefore, that the image of a point in a plane mir- ror lies on the perpendicular drawn from the point to the mirror and is as far back of the mirror as the point is in front of it. Precisely the same construction applied to curved mirrors shows at once (Fig. 414 and Fig. 415) that the image of a point in any mirror, plane or curved, must lie on the perpen- dicular drawn from the point to the mirror ; but if the mirror IMAGES IN MIRRORS 385 is convex, the image is nearer to it than is the point, while if it is concave, the image, if formed behind the mirror at all (that is, if it is virtual), is farther from the mirror than is the point. 445. Construction of image of object in a plane mirror. The image of an object in a plane mirror (Fig. 416") may be located by applying the law proved above for each of its points, that is, by drawing from each point a perpendicular to the reflecting surface and extending it an equal distance on the other side. To find the path of the rays which come to an eye placed at FIG. 416. Construction of image of ,, ,, , , . object in a plane mirror E from any point of the object, such as A, we have only to draw a line from the image A' of this point to the eye and connect the point of intersection of this line with the mirror, namely C, with the original point A. ACE is then the path of the ray. Let a candle (Fig. 417) be placed exactly as far in front of a pane of window glass as a bottle full of water is behind it, both objects being on the same perpendicular drawn through the glass. The candle will appear to be burning inside the water. This explains a large class of familiar optical illusions, such as the w figure suspended in mid-air," the "bust of a person without a trunk," the ** stage ghost," etc. In the last case the illu- sion is produced by causing the audience to look at the actors obliquely through a sheet of very clear plate glass, the edges of which are concealed by draperies. Images of strongly illuminated figures at one side then appear to the audience to be in the midst of the actors. 446. Focal length of a curved mirror half its radius of curva- ture. The effect of a convex mirror on plane waves incident upon it is shown in Fig. 418. The wave which would at a FIG. 417. Position of image in a plane mirror 386 IMAGE FORMATION given instant have been at co^d is at co z d, where oo l = oo 2 . The center F from which the waves appear to come to the eye E is the focus of the mirror. Now so long as the arc cod is small its curva- ture may, without appreciable error, be measured by o^p (see footnote, p. 370); that is, by the departure of the curved line cod from the straight line co^d. FIG. 418. Reflection of a plane wave from a convex mirror Since o o was made equal to oo , we have 0^ = 2 o^o ; that is, the curvature of the reflected wave is equal to twice the curvature of the mirror, or In other words, the focal length of 1 1 ' , R - = 2x-; hence /=- a mirror is equal to one half its radius. 447. Image of an object in a convex mirror. We are all familiar with the fact that a convex mirror always forms behind the mirror a virtual, erect, and diminished image. The reason for this is shown clearly in Fig. 419. The image of the point P lies, as in plane mirrors (see 444), always on the perpendicular to the mirror, but now neces- sarily nearer to the mirror than the focus F, since, as the point P is moved from a position very close to the mirror, where SECTION OF A ft MOVIE" FILM SHOWING SECRETARY or WAR BAKER TURNING HIS HEAD TO SPEAK TO GENERAL PERSHING The moving-picture camera makes a series of snapshots upon a film, usually at the rate of 10 per second. The film is drawn past the lens with a jerky movement, being held at rest during the instant of exposure and moved forward while the shutter is closed. The pictures are |-inch high and 1 inch wide. Since 1 foot of film per second is drawn past the lens, a reel of film 1000 feet long (the usual length) contains 16,000 pictures. From the reel of negatives a reel of positives is printed for use in the projection apparatus. The optical illusion of " moving " pictures is made possible by a peculiarity of the eye called persistence of vision. To illustrate this let a firebrand be rapidly whirled in a circle. The spot of light appears drawn into a luminous arc. This phenomenon is due to the fact that we continue to see an object for a small fraction of a second after the image of it disappears from the retina. The period of time varies somewhat with different individuals. The so-called "moving" pictures do not move at all. In normal projection 16 brilliant stationary pictures per second appear in succession upon the screen, and during the interval between the pictures the screen is perfectly dark. It is during this period of darkness that the film is jerked forward to get the next picture into position for projection. The eye, however, detects no period of ,dark- ness, for on account of persistence of vision it continues to see the stationary picture not only during this period of darkness but dimly for an instant even after the next picture appears upon the screen. This causes the successive station- ary pictures, which differ but slightly, to blend smoothly into each other and thus give the effect of actual motion r 5 6 PHOTOGRAPHS OF SOUND WAVES HAVING THEIR ORIGIN IN AN ELECTRIC SPARK BEHIND THE MIDDLE OF THE BLACK DISK 1. A spherical sound wave. 2. The same wave .00007 second later. 3. A wave re- flected from a plane surface, curvature unchanged. 4. A wave reflected from a convex surface, curvature increased. 5. The source at the focus of a SO 2 lens. The photograph shows first, the original wave on the right ; second, the reflected wave, with its increased curvature ; and third, the transmitted plane wave. 6. Source at focus of a concave mirror ; the reflected wave is plane. (Taken by Professor A. L. Foley and Wilmer H. Souder, of the University of Indiana) IMAGES IN MIRKORS 387 its image is just behind it, out to an infinite distance, its image moves back only to the focal plane through F. Hence the image must lie somewhere between F and the mirror. The image P'Q' of an object PQ is always diminished, because it lies between the converging lines PC and QC. It can be located by the ray method (Fig. 419) exactly as in the case of concave lenses. In fact, a convex mirror and a concave lens have exactly the same opti- cal properties. This is be- cause each always increases the curvature of the incident waves by an amount - 448. Images in concave mirrors. Let the images ob- tainable with a concave mirror be studied precisely as were those obtainable from a convex lens. It will be found that exactly the same series of images is obtained : when the object is between the mirror and the principal focus, the image is virtual, enlarged, and erect; when it is at the focus the reflected waves are plane, that is, the rays from each point are a parallel bundle ; when it is between the FIG. 420. Real image of candle formed by a concave mirror FIG. 421. Method of formation of a real image by a concave mirror principal focus and the center of curvature, the image is inverted, en- larged, and real (Figs. 420 and 421) ; when it is at a distance R (= oC) from the mirror, the image is also at a distance R and of the same size as the object, though inverted ; when the object is moved from R out to 388 IMAGE FORMATION a great distance, the image moves from C up to F, and is always real, inverted, and diminished. The most convenient way of finding the focal length is to find where the image of a distant object is formed. We learn, then, that a con- cave mirror has exactly the optical properties of a con- vex lens. This is because, like the convex lens, it always FlG ' 422 ' Ra 7 method of locating rea ' J image in a concave mirror diminishes the curvature of the waves. The same formulas hold throughout, and the same constructions are applicable (see Fig. 422). 449. Summary for lenses and spherical mirrors.* 1. Real images, inverted ; virtual images, erect. The length of all images is given by L n D n where L and L t - denote the length of object and image respec- tively, and D and D f their distances from the lens or mirror. 2. Convex lenses and concave mirrors have the same optical proper- ties (always diminish the curvature of the waves). a. If object is more distant than principal focus, image is real and (1) enlarged when object is between principal focus and twice focal length ; (2) diminished when object is beyond two focal lengths. b. If object is less distant than principal focus, image is virtual and always enlarged. 3. Concave lenses and convex mirrors have the same optical prop- erties (always increase the curvature of the waves). Image always virtual and diminished for any position of object. 4. - DO A / ( 439) * Laboratory experiments on the formation of images by concave mirrors and by lenses should follow this discussion. See, for example, Experiments 45 and 46 of the authors' Manual. IMAGES IN MIKKORS 389 This formula may be used in all cases if the following points are borne in mind : a. D is always to be taken as positive. b. D { is to be taken as positive for real images and negative for virtual images. c. f is to be taken as positive for converging systems (convex lenses and concave mirrors) and negative for diverging systems (con- cave lenses and convex mirrors). QUESTIONS AND PROBLEMS 1. Show from a construction of the image that a man cannot see his entire length in a vertical mirror unless the mirror is half as tall as he is. Decide from a study of the figure whether or not the distance of the man from the mirror affects the case. 2. A man is standing squarely in front of a plane mirror which is very much taller than he is. The mirror is tipped toward him until it makes an angle of 45 with the horizontal. He still sees his full length. What position does his image occupy ? 3. How tall is a tree 200 ft. away if the image of it formed by a lens of focal length 4 in. is 1 in. long? (Consider the image to be formed in the focal plane.) 4. How long an image of the same tree will be formed in the focal plane of a lens having a focal length of 9 in.? 5. What is the difference between a real and a virtual image? 6. When does a convex lens form a real, and when a virtual, image ? When an enlarged, and when a diminished, image? When an erect, and when an inverted, one ? 7. When a camera is adjusted to photograph a distant object, what change in the length of the bellows must be made to photograph a near object? Explain clearly why this adjustment is necessary. 8. Rays diverge from a point 20 cm. in front of a converging lens whose focal length is 4 cm. At what point do the rays come to a focus? 9. An object 2 cm. long was placed 10 cm. from a converging lens and the image was formed 40 cm. from the lens on the other side. Find the focal length of the lens and the length of the image. 10. An object is 15 cm. in front of a convex lens of 12 cm. focal length. What will be the nature of the image, its size, and its distance from the lens ? 11. Why does the nose appear relatively large in comparison with the ears when the face is viewed in a convex mirror? 12. Can a convex mirror ever form an inverted image? Why? 390 IMAGE FORMATION FIG. 423. Image formed small opening by a OPTICAL INSTRUMENTS 450. The photographic camera. A fairly distinct, though dim, image of a candle flame can be obtained with nothing more elaborate than a pinhole in a piece of cardboard (Fig. 423). If the receiving screen is replaced by a photographic plate, the ar- rangement becomes a pinhole camera, with which good pictures may be taken if the exposure is sufficiently long. If we try to increase the brightness of the image by enlarging the hole, the image becomes blurred, because the narrow pencils a^' ^ <*$ $ etc. become cones whose bases a' ^ a' z , overlap and thus destroy the distinctness of the outline. It is possible, without sacrific- ing distinctness of outline, to gain the increased brightness due to the larger hole by placing a lens in the hole (Fig. 424). If the receiving screen is now a sensitive plate, the arrangement becomes a photographic camera (Fig. 425). But while with the pinhole camera the screen may be at any distance from the hole, with a lens the plate and the object must be at conjugate foci of the lens. Let a lens of, say, 4 feet focal length be placed in front of a hole in the shutter of a darkened room, and a semitransparent screen (for example, architect's tracing paper) placed at the focal plane. A per- fect reproduction of the opposite landscape will appear FIG. 424. Principle of the photo- graphic camera FIG. 425. The photographic camera OPTICAL INSTRUMENTS 391 451. The projecting lantern. The projecting lantern is essen- tially a camera in which the position of object and image have been interchanged ; for in the use of the camera the object is at a considerable distance, and a small inverted image is formed on a plate placed somewhat farther from the lens than the focal distance. In the use of the projecting lantern the object P (Fig. 426) is placed a trifle farther from the lens L 1 than its focal length, and an enlarged inverted image is formed on Fiu. 426. The projecting lantern (stereopticon) a distant screen S. In both instruments the optical part is simply a convex lens, or a combination of lenses which is equivalent to a convex lens. The object P, whose image is formed on the screen, is usu- ally a transparent slide which is illuminated by a powerful light A. The image is as many times larger than the object as the distance from L' to S is greater than the distance from L' to P. The light A is usually either an incandescent lamp or an electric arc. The moving-picture projector employs a long film of small " positives " which moves swiftly between the condensing lens L and the projecting lens L 1 (see opposite p. 386). The above are the only essential parts of a projecting lantern. In order, however, that the slide may be illuminated as brilliantly as pos- sible, a so-called condensing lens L is always used. This concentrates light upon the transparency and directs it toward the screen. 392 IMAGE FORMATION In order to illustrate the principle of the instrument, let a beam of sunlight be reflected into the room and fall upon a lantern slide. When a lens is placed a trifle more than its focal distance in front of the slide, a brilliant picture will be formed on the opposite wall. 452. The eye. The eye is essentially a camera in which the cornea C (Fig. 427), the aqueous humor I, and the crystalline lens o act as one single lens which forms an inverted image P'Q' on the retina, an expan- sion of the optic nerve covering the inside of the back of the eyeball. In the case of the camera the images of objects at different distances are obtained by placing the plate nearer to or farther from the lens. In the eye, however, the distance from the retina to the lens remains constant, and the adjustment for different distances is effected by changing the focal length of the lens system in such a way as always to keep the image upon the retina. Thus, when the normal eye is perfectly Q FIG. 427. The human eve FIG. 428. The pupil dilates when the light is dim and contracts when it is intense relaxed, the lens has just the proper curvature to focus plane waves upon the retina, that is, to make distant objects dis- tinctly visible. But by directing attention upon near objects we cause the muscles which hold the lens in place to contract OPTICAL INSTRUMENTS 393 in such a way as to make the lens more convex, and thus bring into distinct focus objects which may be as close as eight or ten inches. This power of adjustment or accommodation, however, varies greatly in different individuals. The iris, or colored part of the eye, is a diaphragm which varies the amount of light which is admitted to the retina (Figs. 428, (1) and (2)). 453. Nearsightedness and farsightedness. In a normal eye, provided the lens is re- laxed and resting, parallel rays come to a focus OR the retina (Fig. 429, (1)) ; in a near- sighted eye they focus in front of the retina (Fig. 429, (2)) ; and in a far sighted eye they reach the retina before coming to a focus (Fig. 429, (3)). Those who are nearsighted can see distinctly only those objects which are near. The usual reason for nearsighted- ness is that the retina is too far from the lens. The diverging lens corrects this defect of vision because it makes the rays from a distant object enter the eye as if they had come from an object near by ; that is, it partially counteracts the converging effect of the eye (Fig. 429 (2)). Those who are farsighted cannot when the lens is relaxed see distinctly even a very distant object. The usual reason for farsightedness is that the eyeball is too short from lens to retina. The rays from near objects are converged, or focused, towards f behind the retina in spite of all effort at accom- modation. A converging lens gives distinct vision because FIG. 429. Defects of vision 394 IMAGE FORMATION it supplements the converging effect of the eye (Fig. 429, (3)). In old age the lens loses its power of accommodation, that is, the ability to become more convex when looking at a near object; hence, in old age a normal eye requires the same sort of lens as is used in true farsightedness. 454. The apparent size of a body. The apparent size of a body depends simply upon the size of the image formed upon the retina by the lens of the eye, and hence upon the size of the visual angle pCq (Fig. 430). The size of this angle evidently increases as the object is brought nearer to the eye (seeP(7$). Thus, the image formed on the retina when a man is 100 feet from the eye is in reality only one tenth as large as the image formed of the same man when he is but 10 feet away. We do not actually interpret the larger image as representing a larger man simply because we have been taught by lifelong experience to take account of the known distance of an object in forming our estimate of its actual size. To an infant who has not yet formed ideas of distance the man 10 feet away doubtless appears ten times as large as the man 100 feet away. 455. Distance of most distinct vision. When we wish to examine an object minutely, we bring it as close to the eye as possible in order to increase the size of the image on the retina. But there is a limit to the size of the image which can be pro- duced in this way ; for when the object is brought nearer to the normal eye than about 10 inches, the curvature of the incident wave becomes so great that the eye lens is no longer able, without too much strain, to thicken sufficiently to bring the image into sharp focus upon the retina. Hence a person with normal eyes holds an object which he wishes to see as distinctly as possible at a distance of about 10 inches. FIG. 430. The visual angle OPTICAL INSTRUMENTS 395 Although this so-called distance of most distinct vision varies somewhat with different people, for the sake of having a standard of comparison in the determination of the magnify- ing powers of optical instruments some exact distance had to be chosen. The distance so chosen is 10 inches, or 25 centimeters. 456. Magnifying power of a convex lens. If a convex lens is placed immediately before the eye, the object may be brought much closer than 25 centimeters without loss of distinctness, for the curvature of the wave is partly or even wholly overcome by the lens before the light en- ters the eye. If we wish to use a lens as a magnifying glass to the best advantage, we place the eye as close to it as we can, so as to gather as large a cone of rays as possible, and then place the object at a distance from the lens equal to its focal length, so that the waves after passing through it are plane. They are then focused by the eye with the least possible effort. The visual angle in such a case is PcQ (Fig. 431, (1)) ; for, since the emergent waves are plane, the rays which pass through the center of the eye from P and Q are parallel to the lines through PC and Qc. But if the lens were not present, and if the object were 25 centimeters from the eye, the visual angle would be the small angle pcq (Fig. 431, (2)). The magnify- ing power of a simple lens is due, therefore, to the fact that by its use an object can be viewed distinctly when held closer to the eye than is otherwise possible. This condition gives a visual angle that increases the size of the image on the retina. FIG. 431. Magnifying power of a lens 396 IMAGE FORMATION Tl}e less the focal length of the lens, the nearer to it may the object be placed, and therefore the greater the visual angle, or magnifying power. The ratio of the two angles PcQ and pcq is approximately 25/f, where / is the focal length of the lens expressed in centimeters. Now the magnifying power of a lens or microscope is defined as the ratio of the angle actually subtended by the image when viewed through the instrument, to the angle subtended by the object when viewed with the unaided eye at a distance of 25 centi- meters. Therefore the magnifying power of a simple lens is 25/f. Thus, if a lens has a focal length of 2.5 centimeters, it produces a magnification of 10 diameters when the object is placed at its principal focus. If the lens has a focal length of 1 centimeter, its magnifying power is 25, etc. 457. Magnifying power of an astronomical telescope. In the astronom- ical telescope the objective, or forward lens, forms at its principal foe UK an image P'Q' of an object PQ which is usually very distant. This image Q FIG. 432. The magnifying power of a telescope objective is F/25 may be viewed by the unaided eye at a distance of 25 cm. (Fig. 432). The focal length of the objective is usually very much longer than 25 cm. (about 2000 cm. in the case of the great Yerkes telescope shown opposite p. 365), so that the visual angle P'EQ' is increased by means of the objective alone, the increase being F/'25*, that is, in direct proportion to its focal length. In practice, however, the image is not viewed with the unaided eye, but with a simple magnifying glass called an eyepiece (Fig. 433), placed so that the image is at its focus. Since we have seen in 456 that the simple magnifying glass increases the visual angle 25// times, / being the focal length of the eyepiece, it is clear that the total magnification *The angle PoQ = angle P'oQf. Consider the short line Q'P' as an arc, and the angles Q'EP' and Q'oP' are inversely proportional to their radii, F and 25. OPTICAL INSTRUMENTS 397 produced by both lenses, used as above, is F/25 x 25/f=F/f, The. magnifying power of an astronomical telescope is therefore the focal length of the objective divided by the focal length of the eyepiece. It will be seen, therefore, that to get a high mag- nifying power it is necessary Objective ^-\^ Eye- to use an objec- Top - MIMllliniinnn, ^P l f ce l tive of as great To Q focal length as possible and an eyepiece of as short focal length as possible. The focal length of FIG. 433. The magnifying power of a telescope is F/f eyepiece the great lens at the Yerkes Observatory is about C2 feet, and its diam- eter 40 inches. The great diameter enables it to collect a very large amount of light, which makes celestial objects more plainly visible. Eyepieces often have focal lengths as small as -| inch. Thus, the Yerkes telescope, when used with a ^-inch eyepiece, has a magnifying power of 2976. 458. The magnifying power of the com- pound microscope. The compound micro- scope is like the telescope in that the front lens, or objective, forms a real image of the object at the focus of the eyepiece. The size of the image P'Q' (Fig. 434) formed by the objective is" as many times the size of the object PQ as v f the dis- tance from the objective to the image, is times u, the distance from the objec- tive to the object (see 441). Since the eyepiece magnifies this image 25// times, the total magnifying power of a com- , . . . t-25 pound microscope is Ordinarily v FIG. 434. The compound microscope is practically the length L of the micro- scope tube, and u is the focal length F of the objective. Wherever this is the case, then, the magnifying power of the compound micro- . 25 L scope is- 398 IMAGE FORMATION The relation shows that in order to get a high magnifying power with a compound microscope the focal length of both eyepiece and objective should be as short as possible, while the tube length should be as long as possible. Thus, if a microscope has both an eyepiece and an objective of 6 millimeters focal length and a tube 15 centimeters long, its magni- fying power will be = 1042. Magnifications as high as 2500 or .5 x .0 3000 are sometimes used, but it is impossible to go much farther, for the reason that the image becomes too faint to be seen when it is spread over so large an area. 459. The opera glass. On account of the large number of lenses which must be used in the terrestrial telescope, it is too bulky and awk- ward for many purposes, and hence it is often replaced by the opera glass (Fig. 435). This instrument consists of an objective like that of FIG. 435. The opera glass the telescope, and an eyepiece which is a concave lens of the same focal length as the eye of the observer. The effect of the eyepiece is there- fore to just neutralize the lens of the eye. Hence the objective, in effect, forms its image directly upon the retina. It will be seen that the size of the image formed upon the retina by the objective of the opera glass is as much greater than the size of the image formed by the naked eye as the focal length CR of the objective is greater than the focal length cR of the eye. Since the focal length of the eye is the same as that of the eyepiece, the magnifying power of the opera glass, like that of the astro- nomical telescope, is the ratio of the focal lengths of the objective and eyepiece. Objects seen with an opera glass appear erect, since the image formed on the retina is inverted, as is the case with images formed by the lens of the eye unaided. 460. The stereoscope. Binocular vision. When an object is seen with both eyes, the images formed on the two retinas differ slightly, because of the fact that the two eyes, on account of their lateral separation, are viewing the object from slightly different angles. It is this difference OPTICAL INSTRUMENTS 399 in the two images which gives to an object or landscape viewed with two eyes an appearance of depth, or solidity, which is wholly wanting when one eye is closed. The stereoscope is an in- strument which reproduces in photographs this effect of binocular vision. Two photographs of the same object are taken from slightly different points of view. These photographs are mounted at A and B (Fig. 436), where they are simultaneously viewed by the two eyes through the two prismatic lenses m and n. These two lenses superpose the two images at C because of their action as prisms, and at the same time magnify them because of their action as simple magnifying lenses. The result is that the observer is conscious of viewing but one photograph ; but this differs from ordinary photographs in that, FIG. 436. Principle instead of being flat, it has all of the characteristics of the stereoscope of an object actually seen with both eyes. The opera glass has the advantage over the terrestrial telescope of affording the benefit of binocular vision ; for while telescopes are usually constructed with one tube, opera glasses always have two, one for each eye. 461. The Zeiss binocular. The greatest disadvantage of the opera glass is that the field of view is very small. The terrestrial telescope has a larger field but is of inconvenient length. An instrument called the Zeiss binocular (Fig. 437) has recently come into use, which combines the compact- ness of the opera glass with the wide field of view of the ter- restrial telescope. The compact- ness is gained by causing the light to pass back and forth through total reflecting prisms, as in the figure. These reflec- tions also perform the function of reinverting the image, so that the real image which is formed at the focus of the eye- piece is erect. It will be seen, therefore, that the instrument is essen- tially an astronomical telescope in which the image is reinverted by reflection, and in which the tube is shortened by letting the light pass back and forth between the prisms. FIG. 437. The Zeiss binocular 400 IMAGE FORMATION A further advantage which is gained by the Zeiss binocular is due to the fact that the two objectives are separated by a distance which is greater than the distance between the eyes, so that the stereoscopic effect is more prominent than with the unaided eye or with the ordinary opera glass.* 462. The periscope. A periscope is a sort of double-jointed telescope which makes use of total reflection twice, at the top and at the bottom. The system of lenses gives a magnification of about 1^ diameters, as FIG. 438. A parabolic reflector this has been found best to make ships appear at their true distances from the submarine. There is no stereoscopic effect, since the periscope is not double like a binocular. 463. Parabolic reflectors. For the projection of a more nearly cylin- drical beam than is possible with spherical mirrors, it is customary to use parabolic reflectors, as in automobile headlights (Fig. 438, (1) and (2)). The light is placed a little closer to the reflector than the princi- pal focus, so that the reflected light may spread somewhat. The same principle is employed in searchlights, except that the source of light (usually a powerful arc) is kept more nearly at the principal focus of the reflector. The Sperry 60-inch searchlight, the most powerful in the world, has a beam candle power of approximately two thirds that of the sun, and its light is plainly visible at a distance of one hundred miles. * Laboratory experiments on the magnifying powers of lenses and on the construction of microscopes and telescopes should follow this chapter. See for example, Experiments 47, 48, and 49 of the authors' Manual. OPTICAL INSTRUMENTS 401 QUESTIONS AND PROBLEMS 1. Why is it necessary for the pupils of your eyes to be larger in a dim cellar than in the sunshine? Why does the photographer use a large stop on dull days in photographing moving objects ? 2. If a photographer wishes to obtain the full figure on a plate of cabinet size, does he place the subject nearer to or farther from the camera than if he wishes to take the head only? Why? 3. A child 3 ft. in height stood 15 ft. from a camera whose lens had a focal length of 18 in. What was the distance from the lens to the photographic plate and the length of the child's photograph? 4. If 20 sec. is the proper length of exposure when you are printing photographs by a gas light 8 in. from the printing frame, what length of exposure would be required in printing from the same negative at a distance of 16 in. from the same light? 5. If a 20-second exposure is correct at a distance of 6 in. from an 8-candle-power electric light, w T hat is the required time of exposure at a distance of 12 in. from a 32-candle-power electric light? 6. The image, on the retina, of a book held a foot from the eye is larger than that of a house on the opposite side of the street. Why do we not judge that the book is actually larger than the house? 7. W^hat sort of lenses are necessary to correct shortsightedness? longsightedness? Explain with the aid of a diagram. 8. What is the magnifying power of a J-in. lens used as a simple magnifier? 9. If the length of a microscope tube is increased after an object has been brought into focus, must the object be moved nearer to or farther from the lens in order that the image may again be in focus ? 10. Explain as well as you can how a telescope forms the image that you see when you look into it. 11. Is the image on the retina erect or inverted? CHAPTER XX COLOR PHENOMENA COLOR AND WAVE LENGTH 464. Wave lengths of different colors. Let a soap film be formed across the top of an ordinary drinking glass, care being taken that both the solution and the glass are as clean as possible. Let a beam of sun- light or the light from a projecting lantern pass through a piece of red glass at A, fall upon the soap film F, and be reflected from it to a white screen S (see Fig. 439). Let a convex lens L of from 6 to 12 inches focal length be placed in the path of the re- flected beam in such a posi- tion as to produce an image of the film upon the screen S, that is, in such a position that film and screen are at conjugate foci of the lens. The system of red and black bands upon the screen is formed precisely as in 427, by the interference of the two beams of light coming from the front and back sur- FIG. 439. Projection of soap-film fringes faces of the wedge-shaped film. Now let the red glass be held in one half of the beam and a piece of green glass in the other half, the two pieces being placed edge to edge, as shown at A. Two sets of fringes will be seen side by side on the screen. The fringes will be red and black on one side of the image, and green and black on the other ; but it will be noticed at once that the dark bands on the green side are closer together than the dark bands on the other side ; in 402 COLOR AND WAVE LENGTH 403 fact, seven fringes on the side of the film which is covered by the green glass will be seen to cover about the same distance as six fringes on the red side.* Since it was shown in Fig. 390 that the distance between two dark bands corresponds to an increase of one-half wave length in the thickness of the film, we conclude, from the fact that the dark bands on the red side are farther apart than those on the green side, that red light must have a longer wave length than green light. The wave length of the central portion of each colored region of the spectrum is roughly as follows : Red. . . ... .000068cm. Green . . . .000052cm. Yellow . . . . . .000058cm. Blue 000046cm. Violet 000042cm. Let the red and green glasses be removed from the path of the beam. The red and green fringes will be seen to be replaced by a series of bands brilliantly colored in different hues. These are due to the fact that the lights of different wave length form interference bands at different places on the screen. Notice that the upper edges of the bands (lower edges in the inverted image) are reddish, while the lower edges are bluish. We shall find the explanation of this fact in 473. 465. Composite nature of white light. Let a beam of sunlight pass through a narrow slit and fall on a prism, as in Fig. 440. The beam which enters the prism as white light is dispersed into red, yellow, green, blue, and violet lights, although each color merges, by insensible gradations, into the next. This band of color is called a spectrum. We conclude from this experiment that white light is a mix- ture of all the colors of the spectrum, from red to violet inclusive. FIG. 440. White light decom- posed by a prism * The experiment may be performed at home by simply looking through red and green glasses at a soap film so placed as to reflect white light to the eye. 404 COLOR PHENOMENA 466. Color of bodies in white light. Let a piece of red glass be held in the path of the colored beam of light in the experiment of the preceding section. All the spectrum except the red will disappear, thus showing that all the wave lengths except red have been absorbed by the glass. Let a green glass be inserted in the same way. The green portion of the spectrum will remain strong, while the other portions will be greatly enfeebled. Hence green glass must have a much less absorbing effect upon wave lengths which correspond to green than upon those which correspond to red and blue. Let the green and red glasses be held one behind the other in the path of the beam. The spectrum will almost completely vanish, for the red glass has absorbed all except the red rays, and the green glass has absorbed these. We conclude, therefore, that the color wHich a body has in ordinary daylight is determined by the wave lengths which the body has not the power of absorbing. Thus, if a body appears white in daylight, it is because it diffuses or reflects all wave lengths equally to the eye, and does not absorb one set more than another. For this reason the light which comes from it to the eye is of the same composition as daylight or sunlight. If, however, a body appears red in daylight, it is because it absorbs the red rays of the white light which falls upon it less than it absorbs the others, so that the light which is diffusely reflected contains a larger proportion of red wave lengths than is contained in ordinary light. Similarly, a body appears yellow, green, or blue when, it absorbs less of one of these colors than of the rest of the colors contained in white light, and therefore sends a preponderance of some particular wave length to the eye. 467. Color of bodies placed in colored lights. Let a body which appears w T hite in sunlight be placed in the red end of the spectrum. It will appear to be red. In the blue end of the spectrum it will appear to be blue, etc. This confirms the conclusion of the last paragraph, that a white body has the power of diffusely reflecting all the colors of the spectrum equally. Next let a skein of red yarn be held in the blue end of the spec- trum. It will appear nearly black. In the red end of the spectrum COLOR AND WAVE LENGTH 405 it will appear strongly red. Similarly, a skein of blue yarn will appear nearly black in all the colors of the spectrum except blue, where it will have its proper color. These effects are evidently due to the fact that the red yarn, for example, has the power of diffusely reflecting red wave lengths copiously, but of absorbing, to a large extent, the others. Hence, when held in the blue end of the spectrum, it sends but little color to the eye, since no red light is falling upon it. Soak a handful of asbestos or cotton batting in a saturated salt solu- tion ; squeeze out most of the brine ; pour over the material a quantity of strong alcohol. When ignited, this will produce a large flame of al- most pure-yellow light. In a darkened room allow the yellow light to fall strongly upon a spectrum chart of six colors. The only color on the chart that appears natural is the yellow. 468. Compound colors. It must not be inferred from the preceding paragraphs that every color except white has one definite wave length, for the same effect may be produced on the eye by a mix- ture of several different wave lengths as is produced by a single wave length. This statement may be proved by the use of an apparatus known as Newton's color disk (Fig. 441). The arrangement makes it possible to rotate differently colored sectors so rapidly before the eye that the effect is precisely the same as though the colors came to the eye simul- taneously. If one half of the disk is red and the other half green, the rotat- ing disk will appear yellow, the color being very similar to the yellow of the spectrum. If green and violet are mixed in the same way, the result will be light blue. Although the colors produced in this way are not distinguishable by the eye FIG. 441. Newton's color disk 406 COLOK PHENOMENA from spectral colors, it is obvious that their physical constitu- tion is wholly different ; for while a spectral color consists of waves of a single wave length, the colors produced by mix- ture are compounds of several wave lengths. For this reason the spectral colors are called pure and the others compound. In order to tell whether the color of an object is pure or com- pound, it is only necessary to observe it through a prism. If it is compound, the colors will be separated, giving an image of the object for each color. If it is pure, the object will appear through the prism exactly as it does without the prism. By compounding colors in the way described above we can produce many which are not found in the spectrum. Thus, mixtures of red and blue give purple or crimson ; mixtures of black with red, orange, or yellow give rise to the various shades of brown. Lavender may be formed by adding three parts of white to one of blue ; lilac, by adding to fifteen parts of white four parts of red and one of blue ; olive, by adding one part of black to two parts of green and one of red. 469. Complementary colors. Since white light is a combi- nation of all the colors from red to violet inclusive, it might be expected that if one or several of these colors L were subtracted from a frrTrrrrl 0fi(^ . white light, the residue would be colored light. To test this experimentally let a beam of sunlight be passed through a slit s, a prism P, and a lens L, to a r IG< 442. Recombination of spectral colors screen S, arranged as in into white light Fig. 442. A spectrum will be formed at R V, the position conjugate to the slit s, and a pure white spot will appear on the screen when it is at the position which is conju- gate to the prism face ab. Let a card be slipped into the path of the COLOR AND WAVE LENGTH 407 beam at E, so as to cut off the red portion of the light. The spot on S will appear a brilliant shade of greenish blue. This is the compound color left after red is taken from the white light. This shade of blue is therefore called the complementary color of the red which has been subtracted. Two complementary colors are therefore denned as any two colors which produce white when added to each other. Let the card be slipped in from the side of the blue rays at V. The spot will first take on a yellowish tint when the violet alone is cut out; and as the card is slipped farther in, the image will become a deep shade of red when violet, blue, and part of the green are cut out. Next let a lead pencil be held vertically between R and V so as to cut off the middle part of the spectrum ; that is, the yellow and green rays. The remaining red, blue, and violet will unite to form a brilliant purple. In each case the color on the screen is the complement of that which is cut out. 470. Retinal fatigue. Let the gaze be fixed intently for not less than twenty or thirty seconds on a point at the center of a block of any brilliant color for example, red. Then look off at a dot on a white wall or a piece of white paper, and hold the gaze fixed there for a few seconds. The brilliantly colored block will appear on the white wall, but its color will be the complement of that first looked at. The explanation of this phenomenon, due to so-called " ret- inal fatigue," is found in the fact that although the white sur- face is sending waves of all colors to the eye, the nerves which responded to the color first looked at have become fatigued, and hence fail to respond to this color when it comes from the white surface. Therefore the sensation produced is that due to white light minus this color ; that is, to the complement of the original color. A study of the spectral colors by this method shows that the following colors are complementary. Red Orange Yellow Violet Green Bluish green Greenish blue Blue Greenish yellow Crimson 471. Color of pigments. When yellow light is added to the proper shade of blue, white light is produced, since these colors are complementary. But if a yellow pigment is added to a blue one, the color of the mixture will be green. This is 408 COLOR PHENOMENA because the yellow pigment removes the blue and violet by absorption, and the blue pigment removes the red and yellow, so that only green is left. When pigments are mixed, therefore, each one subtracts cer- tain colors from white light, and the color of the mixture is that color which escapes absorption by the different ingredients. Adding pigments and adding colors, as in 468, are therefore wholly dissimilar processes and produce widely different results. 472. Three-color printing. It is found that all colors can be produced by suitably mixing with the color disk (Fig. 441) three spectral colors, namely red, green, and blue-violet. These are therefore called the three primary colors. The so- called primary pigments are simply the complements of these three primary colors. They are, in order, peacock blue, crim- son, and light yellow. The three primary colors when mixed yield white. The three primary pigments when mixed yield black, because together they subtract all the ingredients from white light. The process of three-color printing consists in mixing on a white background, that is, on white paper, the three primary pigments in the following way: Three differ- ent photographs of a given-colored object are taken, each through a filter of gelatin stained the color of one of the primary colors. From these photographs halftone " blocks " are made in the usual way. The colored picture is then made by carefully superposing prints from these blocks, using with each an ink whose color is the complement of that of the " filter " through which the original negative was taken. The plate on the opposite page illustrates fully the process. It will be interesting to examine differently colored portions with a lens of moderate magnifying power. 473. Colors of thin films. The study of complementary colors has furnished us with the key to the explanation of the fact, observed in 464, that the upper edge of each colored band produced by the water wedge is reddish, while the lower edge THREE-COLOR PRINTING 1, yellow impression (negative made through a blue-violet filter) ; 2, crimson im- pression (negative made through a green filter) ; 3, crimson on yellow ; 4, blue impression (negative made through a red filter) ; 5, yellow, crimson, and blue combined (the final product). The circles at the right show the colors of ink used in making each impression. Notice the different colors in 5, which are made by combining yellow, crimson, and blue COLOE AND WAVE LENGTH 409 is bluish. The red on the upper edge is due to the fact that there the shorter blue waves are destroyed by interference and a complementary red color is left; while on the lower edge of each fringe, where the film is thicker, the longer red waves interfere, leaving a complementary blue. In fact, each wave length of the incident light produces a set of fringes, and it is the superposition of these different sets which gives the result- ant colored fringes. Where the film is too thick the overlapping is so complete that the eye is unable to detect any trace of color in the reflected light. In films which are of uniform thickness, instead of wedge- shaped, the color is also uniform, so long as the observer does not change the angle at which the film is viewed. With any change in this angle the thickness of film through which the light must pass in coming to the observer changes also, and hence the color changes. This explains the beautiful play of iridescent colors seen in soap bubbles, thin oil films, mother of pearl, etc. 474. Chromatic aberration. It has heretofore been assumed that all the waves which fall on a lens from a given source are brought to one and the same focus. But since blue rays are bent more than red ones in passing through a prism, it is clear that in passing through a lens the blue light must be brought to a focus at some point v (Fig. 443) nearer to the lens than r, where the red light is focused, and that the foci for intermediate colors must fall in intermediate positions. It is for this reason that an image formed by a simple lens is always fringed with color. Let a card be held at the focus of a lens placed in a beam of sunlight (Fig. 443). If the card is slightly nearer the lens than the focus, the spot of light will be surrounded by a red fringe, for the red rays, being FIG. 443. Chromatic aberration in a lens 410 COLOB, PHENOMENA least refracted, are on the outside. If the card is moved out beyond the focus, the red fringe will be found to be replaced by a blue one ; for after crossing at the focus it will be the more refrangible rays which will then be found outside. This dispersion of light produced by a lens is called chromatic aberration. 475. Achromatic lenses. The color effect caused by the chromatic aberration of a simple lens greatly impairs its use- fulness. Fortunately, however, it has been found possible to eliminate this effect almost completely by combining into one lens a convex lens of crown glass and a concave lens of flint glass (Fig. 444). The first lens then pro- duces both bending and dispersion, while FIG. 444. An achro- the second almost completely overcomes maticlens the dispersion without entirely overcoming the bending. Such lenses are called achromatic lenses. The first one was made by John Dollond in London in 1758. They are used in the construction of all good telescopes and microscopes. QUESTIONS AND PROBLEMS 1. What determines the color of an opaque body? a transparent body? What is the appearance of a bunch of green grass when seen by pure red light? Explain. 2. What is w white " ? What is " black " ? Explain why a block of ice is transparent while snow is opaque and white. 3. Why do white bodies look blue when seen through a blue glass ? 4. What color would a yellow object appear to have if looked at through a blue glass? (Assume that the yellow is a pure color.) 5. A gas flame is distinctly yellow as compared with sunlight. What wave lengths, then, must be comparatively weak in the spectrum of a gas flame ? 6. Why does dark blue appear black by candle light? 7. Certain blues and greens cannot be distinguished from each other by candle light. Explain. 8. Does blue light travel more slowly or faster in glass than red light ? How do you know? SPECTRA 411 SPECTRA 476. The rainbow. There is formed in nature a very beau- tiful spectrum with which everyone is familiar the rainbow. Let a spherical bulb F (Fig. 445) 1J or 2 inches in diameter be filled with water and held in the path of a beam of sunlight which enters the room through a hole in a piece of cardboard AB. A miniature rainbow will be formed on the screen around the open- ing, the violet edge of the bow being toward the cen- ter of the circle and the red outside. A beam of light which enters the flask at C is there both refracted and dispersed ; at D it is totally reflected ; and at E it is again re- fracted and dispersed on passing out into the air. Since in both of the re- fractions the violet is bent more than the red, it is obvious that it must return nearer to the direction of the incident beam than the red rays. If the flask were a perfect sphere, the angle included between the inci- dent ray OC and the emergent red ray ER would be 42 ; and the angle between the incident ray and the emergent violet ray E V would be 40. The actual rainbow seen in the heavens is due to the refraction and reflection of light in the drops of water in the air, which act exactly as did the flask in the preceding experiment. If the observer is standing at E with his back to the sun, the light which comes from the drops so as to make an angle of 42 (Fig. 446) with the line drawn from the observer to the sun must be red light ; while the light which comes from drops which are at an angle of 40 from the eye must be violet light. In direct sunshine the pris- matic color seen in a dewdrop changes to another color when the head is shifted side wise. It is clear that those drops FIG. 445. Artificial rainbow 412 COLOR PHENOMENA Sal whose direction from the eye makes any particular angle with the line drawn from the eye to the sun must lie on a circle whose center is on that line. Hence we see a circular arc of light which is violet on the inner edge and red on the outer edge. A sec- ond bow having the n . -i FIG. 446. Primary and secondary rainbows red on the inside and the violet on the outside is often seen outside of the one just described, and concentric with it. This bow arises from rays which have suffered two internal reflections and two refractions, in the manner shown in Fig. 446. 477 . Continuous Spectra. Let a Bunsen burner arranged to produce a white flame be placed behind a slit ,s (Fig. 447). Let the slit be viewed through a prism P. The spectrum will be a continuous band of color. If now the air is admitted f. at the base of the burner, and if a clean platinum wire is held in the flame directly in front of the slit, the white- hot platinum will also give a continuous spectrum.* All incandescent solids and liquids are found to give spectra of this type which contain all the wave lengths from the extreme red to the extreme violet. O The continuous spectrum of a luminous gas flame is due to *By far the most satisfactory way of performing these experiments with spectra is to provide the class with cheap plate-glass prisms, like those used in Experiment 50 of the authors' Manual, rather than to attempt to project line spectra. FIG. 447. Arrangement for viewing spectra SPECTRA 413 the incandescence of solid particles of carbon suspended in the flame. The presence of these solid particles is proved by the fact that soot is deposited on bodies held in a white flame. 478. Bright-line Spectra. Let a bit of asbestos or a platinum wire be dipped into a solution of common salt (sodium chloride) and held in the flame, care being taken that the wire itself is held so low that the spectrum due to it cannot be seen. The continuous spectrum of the preceding paragraph will be replaced by a clearly defined yellow image of the slit which occupies the position of the yellow portion of the spectrum. This shows that the light from the sodium flame is not a compound of a number of wave lengths, but is rather of just the wave length which corresponds to this particular shade of yellow. The light is now coming from the incandescent sodium vapor and not from an incandescent solid, as in the preceding experiments. Let another platinum wire be dipped in a solution of lithium chloride and held in the flame. Two distinct images of the slit, &' and s" (Fig. 447), will be seen, one in red and one in yellow. Let calcium chloride be intro- duced into the flame. One distinct image of the slit will be seen in the green and another in the red. Strontium chloride will give a blue and a red image, etc. (The yellow sodium image will probably be present in each case, because sodium is present as an impurity in nearly all salts.) These narrow images of the slit in the different colors are called the characteristic spectral lines of the substances. The experiments show that incandescent vapors and gases give rise to bright-line spectra, and not continuous spectra like those pro- duced by incandescent solids and liquids (see on opposite page). The method of analyzing compound substances through a study of the lines in the spectra of their vapors is called spectrum analysis. It was first used by Bunsen in 1859. 479. The solar spectrum. Let a beam of sunlight pass first through a narrow slit S (Fig. 448), not more than i millimeter in width, then through a prism P, and finally let it fall on a screen S', as shown in Fig. 448. Let the position of the prism be changed until a beam of white light is reflected from one of its faces to that portion of the screen which was previously occupied by the central portion of the spectrum, 414 COLOR PHENOMENA Then let a fens L be placed between the prism and the slit, and moved back and forth until a perfectly sharp white image of the slit is formed on the screen. This adjustment is made in order to get the slit S and the screen S' in the positions of conjugate foci of the lens. Now let the prism be turned to its original position. The spectrum on the screen will then consist of a series of colored images of the slit arranged side by side. This is called a pure spectrum, to dis- tinguish it from the spectrum shown in Fig. 440, in which no lens was used to bring the rays of each particular color to a FlG 448> Arrangement for obtaining a particular point, and in which pure spectrum there was therefore much over- lapping of the different colors. If the slit and screen are exactly at con- jugate foci of the lens, and if the slit is sufficiently narrow, the spectrum will be seen to be crossed vertically by certain dark lines. These lines were first observed by the Englishman Wol- laston in 1802, and were first studied carefully by the German Fraunhofer in 1814, who counted and mapped out as many as seven hundred of them. They are called, after him, the Fraunhofer lines. Their existence in the solar spectrum shows that certain wave lengths are absent from sunlight, or, if not entirely absent, are at least much weaker than their neighbors. When the experiment is performed as described above, it will usually not be possible to count more than five or six distinct lines. 480. Explanation of the Fraunhofer lines. Let the solar spec- trum be projected as in 479. Let a few small bits of metallic sodium be laid upon a loose wad of asbestos which has been saturated with alcohol. Let the asbestos so prepared be held to the left of the slit, or between the slit and the lens, and there ignited. A black band will at once ap- pear in the yellow portion of the spectrum, in the place where the color is exactly that of the sodium flame itself ; or, if the focus was sufficiently sharp so that a dark line could be seen in the yellow before the sodium SPECTRA 416 was introduced, this line will grow very much blacker when the sodium is burned. Evidently, then, this dark line in the yellow part of the solar spectrum is in some way due to sodium vapor through which the sunlight has somewhere passed. The experiment at once suggests the ex- planation of the Fraurihofer lines. The white light which is emitted by the hot nucleus of the sun, and which contained all wave lengths, has had certain wave lengths weakened by absorption as it passed through the vapors and gases surrounding the sun and the earth. For it is found that every gas or vapor ivill absorb exactly those wave lengths which it is itself ca- pable of emitting when incandescent. This is for precisely the same reason that a tuning fork will respond to, that is, absob, only vibrations which have the same period as those which it is itself able to emit. Since, then, the dark line in the yellow portion of the sun's spectrum is in exactly the same place as the bright yellow line produced by incandescent sodium vapor, or the dark line which is produced whenever white light shines through sodium vapor, we infer that sodium vapor must be contained in the sun's atmosphere. By comparing in this way the positions of the lines in the spectra of different elements with the positions of various dark lines in the sun's spectrum, many of the elements which exist on the earth have been proved to exist also in the sun. For example, Kirchhoff showed that the four hundred sixty bright lines of iron which were known to him were all exactly matched by dark lines in the solar spectrum. Fig. 449 shows a copy of a and iron spectra 416 COLOR PHENOMENA photograph of a portion of the solar spectrum in the middle,, and the corresponding bright-line spectrum of iron each side of it. It will be seen that the coincidence of bright and dark lines is perfect. 481. Doppler's principle applied to light waves. We have seen (see The Doppler effect, 387, p. 326) that the effect .of the motion of a sounding body toward an observer is to shorten slightly the wave length of the note emitted, and the effect of motion away from an observer is to increase the wave length. Similarly, when a star is moving toward the earth, each particular wave length emitted will be slightly less than the wave length of the corresponding light from a source on the earth's surface. Hence in this star's spectrum all the lines will be displaced slightly toward the violet end of the spectrum. If a star is moving away from the earth, all its lines will be displaced toward the red end. From the direction and amount of displacement, therefore, we can cal- culate the velocity with which a star is moving toward or receding from the solar system. Observations of this sort have shown that some stars are moving through space toward the solar system with a velocity of 150 miles per second, while others are moving away with almost equal velocities. The whole solar system appears to be sweeping through space with a velocity of about 12 miles per second ; but even at this rate it would be at least 70,000 years before the earth would come into the neighborhood of the nearest star, even if it were moving directly toward it. QUESTIONS AND PROBLEMS .1. From the table on page 403 calculate how many waves of red and of violet light there are to an inch. 2. In what part of the sky will a rainbow appear if it is formed in the early morning ? 3. Why do we believe that there is sodium in the sun? 4. What sort of spectrum should moonlight give? (The moon has no atmosphere.) 5. If you were given a mixture of a number of salts, how would you proceed, with a Bunsen burner, a prism, and a slit, to determine whether or not there was any calcium in the mixture ? 6. Draw a diagram of a slit, a prism, and a lens, so placed as to- form a pure spectrum. 7. How can you show that the wave lengths of red and green lights are different, and how can you determine which one is the longer ? CHAPTER XXI INVISIBLE RADIATIONS RADIATION FKOM A HOT BODY 482. Invisible portions of the spectrum. When a spectrum is photographed, the effect on the photographic plate is found to extend far beyond the limits of the shortest visible violet rays. These so-called ultra-violet rays have been photographed and measured at the Ryerson Physical Laboratory, University of Chicago, down to a wave length of .00000273 centimeter, which is only one fifteenth the wave length of the shortest violet waves. The longest rays visible in the extreme red have a wave length of about .00008 centimeter, but delicate thermoscopes reveal a so-called infra-red portion of the spectrum, the investiga- tion of which was carried, .in 1912, by Rubens and von Baeyer of Berlin, to wave lengths as long as .03 centimeter, 400 times as long as the longest visible rays. The presence of these long heat rays may be detected by means of the radiometer (Fig. 450), an instrument perfected by E. F. Nichols at Dartmouth. In its common form it consists of a partially exhausted bulb, within which is a little aluminium wheel carrying four vanes blackened on one face and polished on the other. When the instrument is held in sunlight or before a lamp, the vanes rotate in such a way that the blackened faces always move away from the source of radiation, because they absorb ether waves better than do the polished faces, and thus become hotter. The heated air in contact with these faces then exerts a greater pressure against them than does the air in contact with the polished faces. 417 FIG. 450. The Crookes radi- ometer 418 INVISIBLE RADIATIONS FIG. 451. A simple thermoscope A still simpler way of studying these long heat waves was devised in 1912 by TrowJbridge of Princeton. A rubber band AC (Fig. 451) a millimeter wide is stretched to double its length over a glass plate FGHI, and the thinnest possible glass staff ED, carrying a light mirror E about 2 millimeters square, is placed under the rubber band at its middle point B. When the spectrum is thrown upon the portion A B of the band, the change in its length produced by the heating causes ED to roll, and a spot of light reflected from E to the wall to shift its position by an amount proportional to the heating. Let either the radiometer or the thermoscope described above be placed just beyond the red end of the spectrum. It will indicate the presence here of heat rays of even greater energy than those in the visible spectrum. Again, let a red-hot iron ball and one of the detectors be placed at conjugate foci of a large mirror (Fig. 452). The invisible heat rays will be found to be reflected and focused just as are light rays. Next let a .flat bottle filled with water be inserted between the detector and any source of heat. It will be found that water, although transparent to light rays, absorbs nearly all of the infra-red rays. But if the water is replaced by carbon bisulphide, the infra-red rays will be freely transmitted, even though the liquid is rendered opaque to light waves by dissolv- ing iodine in it. 483. Radiation and temperature. All bod- dies, even such as are at ordinary tempera- tures, are continually radiating energy in the form of ether waves. This is proved by the fact that even if a body is placed in the best vacuum obtainable, it continually falls in temperature when surrounded by a colder body, for example, liquid air. The ether waves emitted at ordinary temperatures are doubtless very long as compared with light waves. As the temperature is raised, FIG. 452. Reflection of infra-red rays RADIATION FROM A HOT BODY 419 more and more of these long waves are emitted, but shorter and shorter waves are continually added. At about 525 C. the first visible waves, that is, those of a dull red color, begin to appear. From this temperature on, owing to the addition of shorter and shorter waves, the color changes continuously, first to orange, then to yellow, and, finally, between 800 C. and 1200 C., to white. In other words, all bodies get "red-hot" at about 525 C. and "white-hot" at from 800 C. to 1200 C. Some idea of how rapidly the total radiation of ether waves increases with increase of temperature may be obtained from the fact that a hot platinum wire gives out thirty-six times as much light at 1400 C. as it does at 1000 C., although at the latter temperature it is already white-hot The radi- ations from a hot body are sometimes classified as heat rays, light rays, and chemical, or actinic, rays. The classifi- cation is, however, misleading, since all ether waves are heat waves in the sense that, when absorbed by matter, they pro- duce heating effects, that is, molecular motions. Radiant heat is, then, the radiated energy of ether waves of any and all wave lengths. 484. Radiation and absorption. Although all substances begin to emit waves of a given wave length at approximately the same temperature, the total rate of emission of energy at a given temperature varies greatly with the nature of the radiating surface. In general, experiment shows that surfaces which are good absorbers of ether radiations are also good radiators. From this it follows that surfaces which are good reflectors, like the polished metals, must be poor radiators. Thus, let two sheets of tin, 5 or 10 centimeters square, one brightly polished and the other covered on one side with lampblack, be placed in vertical planes about 10 centimeters apart, the lampblacked side of one facing the polished side of the other. Let a small ball be stuck with a bit of wax to the outer face of each. Then let a hot metal 420 INVISIBLE KADIATIONS plate or ball (Fig. 453) be held midway between the two. The wax on the tin with the blackened face will melt and its ball will fall first, showing that the lampblack ab- sorbs the heat rays faster than does the polished tin. Now let two blackened glass bulbs be connected, as in Fig. 454, through a U-tube containing colored water, and let a well- polished tin can, one side of which has been blackened, be FlG>453< Goodre _ FlG . 4 54. Goodab- filled with boiling water and flectorg ^ pQor ^^ ^ ^ placed between them. The mo- absorbers radiators tion of the water in the U-tube will show that the blackened side of the can is radiating heat much more rapidly than the other, although the two are at the same temperature. QUESTIONS AND PROBLEMS 1. The atmosphere is transparent to most of the sun's rays. Why are the upper regions of the atmosphere so much colder than the lower regions ? 2. When one is sitting in front of an open-grate fire, does he receive most heat by conduction, by convection, or by radiation ? 3. Sunlight in coming to the eye travels a much longer air path at sunrise and sunset than it does at noon. Since the sun appears red or yellow at these times, what rays are absorbed most by the atmosphere ? 4. Glass transmits all the visible waves, but does not transmit the long infra-red rays. From this fact explain the principle of the hotbed. 5. Which will be cooler on a hot day, a white hat or a black one? 6. Will tea cool more quickly in a polished or in a tarnished metal vessel ? 7. Which emits the more red rays, a white-hot iron or the same iron when it is red-hot ? 8. Liquid-air flasks and thermos bottles are double-walled glass vessels with a vacuum between the walls. Liquid air will keep many times longer if the glass walls are silvered than if they are not. Why ? Why is the space between the walls evacuated ? ELECTRICAL RADIATIONS 421 ELECTRICAL RADIATIONS 485. Proof that the discharge of a Leyden jar is oscillatory. We found in 408, p. 346, that the sound waves sent out by a sounding tuning fork will set into vibration an adjacent fork, provided the latter has the same natural period as the former. Following is the complete electrical analogy of this experiment. Let the inner and outer coats of a Leyden jar A (see Fig. 455) be connected by a loop of wire cdef, the sliding crosspiece de being arranged so that the length of the loop may be altered at will. Also let a strip of tin foil be brought over the edge of this jar from the inner coat to within about 1 millimeter of the outer coat at C. Let the two coats of an exactly similar jar B be connected with the knobs n and n by a second similar wire loop of fixed length. Let the two jars be placed side by side with their loops par- FlG 455 Sympathetic electrical vibrations allel, and let the jar B be successively charged and discharged by connecting its coats with a static machine or an induction coil. At each discharge of jar B through the knobs n and n a spark will appear in the other jar at C, provided the crosspiece de is so placed that the areas of the two loops are equal. When de is slid along so as to make one loop considerably larger or smaller than the other, the spark at C will disappear. The experiment therefore demonstrates that two electrical circuits, like two tuning forks, can be tuned so as to respond to each other sympathetically, and that just as the tuning forks will cease to respond as soon as the period of one is slightly altered, so this electric resonance disappears when the exact symmetry of the two circuits is destroyed. Since, obviously, this phenomenon of resonance can occur only between systems which have natural periods of vibration, the experiment proves that the discharge of a Leyden jar is a vibratory, that is, an 422 INVISIBLE RADIATIONS oscillatory, phenomenon. As a matter of fact, when such a spark is viewed in a rapidly revolving mirror, it is actually found to consist of from ten to thirty flashes following each other at equal intervals. Fig. 456 is a photograph of such a spark. In spite of these oscillations the whole discharge may be made to take place in the incredibly short time of I , OO Q, OOO of a second. This fact, coupled with the extreme brightness of the spark, has made possible the surprising results of so-called instantaneous electric-spark pho- -, r~, , , ., FIG. 456. Oscillations of the tography. The plate opposite electric spark page 425 shows the passage of a bullet through a soap bubble. The film was rotated continu- ously instead of intermittently, as in ordinary moving-picture photography. The illuminating flashes, 5000 per second, were so nearly instantaneous that the outlines are not blurred. 486. Electric waves. The experiment of 485 demonstrates not only that the discharge of a Leyden jar is oscillatory but also that these electrical oscillations set up in the surrounding medium disturbances, or waves of some sort, which travel to a t neighboring circuit and act upon it precisely as the air waves acted on the second tuning fork in the sound experiment. Whether these are waves in the air, like sound waves, or dis- turbances in the ether, like light waves, can be determined by measuring their velocity of propagation. The first determina- tion of this velocity was made by Heinrich Hertz (see oppo- site p. 102) in 1888. He found it to be precisely the same as that of light, that is, 300,000 kilometers per second. This result shows, therefore, that electrical oscillations set up ivaves in the ether. These waves are now known as Hertzian waves. The length of the waves emitted by the oscillatory spark of instantaneous photography is evidently very great, namely, about VoVoV.AV = 30 meters > since the velocity of light is ELECTRICAL RADIATIONS 423 300,000,000 meters per second, and since there are 10,000,000 oscillations per second ; for we have seen in 382, p. 323, that wave length is equal to velocity divided by the number of oscillations per second. By diminishing the size of the jar and the length of the circuit the length of the waves may be greatly reduced. By causing the electrical discharges to take place between two balls only a fraction of a millimeter in diameter, instead of between the coats of a condenser, elec- trical waves have been obtained as short as .3 centimeter, only ten times as long as the longest measured heat waves. 487. Detection of electric waves. In the experiment of 485 we detected the presence of the electric waves by means of a small spark gap C in a circuit almost identical with that in which the oscillations were set up. The visible spark may be employed for the detection of waves many feet away from the source, but for detecting the feeble waves which come in from a source hundreds or thousands of miles away we must depend upon sounds produced in an extremely sensitive tele- phone receiver, as explained in the next section. 488. Wireless telegraphy. Commercial wireless telegraphy was realized in 1896 by Marconi (see opposite p. 316), eight years after the discovery of Hertzian waves. The essential elements of a tuned wave-train, or " spark," system of wireless telegraphy are as follows: The key K at the transmitting station (Fig. 457, (1) ) is depressed to allow a current from the alternator A to pass through the primary coil P of a transformer T" 13 the frequency of the alternations in practice being usually about 500 cycles per second. The high-voltage current- induced in the secondary S charges the condenser C l until its potential rises high enough to cause a spark discharge to take place across the gap s. This discharge of C^ is oscillatory ( 485), and the oscillations thus produced in the condenser circuit containing C v s, and L 1 may, in a low-power sfiort-wave transmitting set, have a frequency as high as 1,000,000 per second. An oscillation frequency much lower than this is generally used and is subject to the control of the operator through 424 INVISIBLE RADIATIONS the sliding contact c, precisely as in the case illustrated in Fig. 455. The oscillations in the condenser circuit induce oscillations in the aerial- wire system, which is tuned to resonance with it through the sliding contact 6-'. (1) (2) FIG. 457. Transmitting and receiving stations for wireless telegraphy As long as the key K is kept closed (assuming a 500-cycle alternator to be used), 1000 sparks per second occur at s, and therefore a regular series of 1000 wave trains (Fig. 458) pass off from the aerial every second and move away with the velocity of light. If the oscillations which produce a wave train have a frequency of, say, 500,000 per second, each wave in the It , 300,000,000 wave train has a length of Direction of Propagation 500,000 600 meters ; and if these wave trains are produced at the rate of 1000 per second, they follow each other, at regular distances of 300,000 meters, that is, nearly 200 miles. The waves sent out by the aerial system of the transmitting station induce like os- dilations in the distant aerial system of the receiving station (Fig. 457, (2) ), which is tuned to resonance with it. In case the receiving aerial must be tuned to respond to very long waves, the switch is closed to cut out the condenser C 2 , and the inductance, or loading coil, B^ is used ; whereas, to tune to very short waves, the switch is opened and the variable FIG. 458. One wave train from oscillatory discharge itih WIKELESS TELEPHONE UTILIZED IN AVIATION One of the most notable developments of the war was the directing of a squadron of airplanes in intricate maneuvers by wireless telephone either from the ground or by the commander in the leading plane. The upper panel shows the pilot and the observer conversing with special apparatus designed to eliminate plane noises, and the lower panel shows President Wilson talking by wireless to airplanes CINEMATOGRAPH FILM OF A BULLET FIRED THROUGH A SOAP BUBBLE The flight of the missile may be followed easily. It will be seen that the bubble breaks, not when the bullet enters, but when it emerges. (From " Moving Pictures," by F. A. Talbot. Courtesy of J. B. Lippincott Company) ELECTRICAL RADIATIONS 425 condenser C' is brought into use, the loading coil not being utilized. 1 The oscillations in the aerial circuit of the receiving station induce exactly similar ones in the detector circuit, which is tuned to resonance with the receiving aerial by means of L 2 , B 2 , and C 3 . The so-called detector of these oscillations may be simply a crystal of galena D in series with the telephone receivers R. This crystal, like the tungar rectifier of 374, has the property of transmitting a current in one direction only. 2 Were it not for this property the telephone could not be used as a detector, because its diaphragm cannot vibrate with a fre- quency of the order of a million; and even if it could, it would produce sound waves far above the limit of hearing. Because of this rectifying property of the crystal the receiver diaphragm is drawn in only once while the oscillations produced by a given wave train last, this effect being- due to the rectified pulsating current which passes in one direction through the receivers and then ceases until the oscillations due to the next spark ar- FlG 459 United gtates navy rive. Since 1000 of the intermittent standard radio receivers wave trains strike upon the aerial each second, the operator at the receiving station hears a continuous musical note of this pitch as long as the key K is depressed. The working of the key, however, as in ordinary telegraphy, breaks the regular series of wave trains into groups of wave trains, so that the short and long notes heard in the receivers (Fig. 459) correspond to the dots and dashes of telegraphy. The receiving circuit, when tuned as shown in Fig. 457, (2), is highly selective ; that is, it will not pick up waves of other periods. The loading coils B l and 5 2 , as well as the two variable condensers C 2 and C 3 , are usually omitted from small amateur receiving sets ; but when this is done, the receiving set is less selective and less sensitive. The resist- ance of the receivers is so high, usually from 1000 to 4*000 ohms, that 1 In the diagram an arrow drawn diagonally across a condenser indicates that, for the sake of tuning, the condenser is made adjustable. Similarly, an arrow across two circuits coupled inductively, like the primary and secondary of the "oscillation transformer " J" , indicates that the amount of interaction of the two circuits can be varied, as, for example, by sliding one coil a longer or shorter distance inside the other. 2 Crystal detectors have been largely superseded by the "audion " for both wireless telegraphy and wireless telephony. 426 INVISIBLE RADIATIONS they do not interfere with the oscillations of the condenser system across which they are placed. The receiving station shown in Pig. 457, (2), may also be used for receiving tvireless- telephone messages. The simplified circuit of an audion receiving station is shown opposite page 441. Although the spark, or wave-train, system of wireless teleg- raphy is still widely used, the " continuous wave " system is rapidly displacing it. Just as sound waves differing slightly in frequency combine to produce the phenomenon of beats ( 890), so electrical oscillations differing in frequency give, when combined, a "beat effect." For instance, if electrical oscillations of, say, 30,000 per second and 31,000 per second combine, beats will occur at the rate of 1000 per second, which is a frequency within the limit of hearing. The elec- trical oscillations mentioned above have a frequency beyond the limit of hearing and hence are said to have radio fre- quency ; but the beats being within the range of hearing have an audio frequency. Now let us assume that there is at the transmitting station an alternating-current generator which throws into the aerial powerful undamped oscillations of 30,000 per second; and suppose further that at the receiv- ing station there is an oscillation generator which maintains relatively weak oscillations of 31,000 per second in the local receiving aerial. These weak oscillations produced in the receiving aerial by the local generator make no sound in the receiver, being above the limit of hearing ; but whenever, and as long as, the operator at the transmitting station depresses his key, waves come in at the rate of 30,000 per second, strike against the receiving aerial and develop therein weak oscillations which combine with those already present to make 1000 beats per second. These beat effects are rectified by a crystal or by a vacuum tube and passed through the receiver. The listener, therefore, hears long and short musical sounds just as he does when receiving by the spark system. The beat method of receiving is called the heterodyne system. ELECTKICAL RADIATIONS 427 489. Modulated continuous waves.* The vibrations consti- tuting articulate speech are exceedingly complex, as may be seen from an inspection of the full-page halftone opposite page 346. Because of this complexity it is impossible to trans- mit speech by means of discontinuous waves (Fig. 460) such as are employed in the system of spark telegraphy described in the preceding section. The parts of the voice lost because Direction of propagation > -4 4 4 -4 4 -4 -* -4- FIG. 460. A series of wave trains of the gaps between the wave trains would render the language unintelligible. Theoretically the voice could be transmitted by continuous electromagnetic waves having the frequencies of voice vibrations, but such a method is entirely impracti- cable on account of the enormous length of aerial needed to produce such long waves and the tremendous amount of power which would be required. Therefore, the only satis- factory method thus far developed is to transmit speech FIG. 461. Continuous, or carrier, waves of radio frequency on continuous, or " carrier," waves (Fig. 461) having a frequency (radio frequency) above the limit of hearing. At the sending station the continuous waves (Fig. 461) are " modulated " by the voice at the transmitter ; that is, the sound waves of the voice act upon the apparatus in such a way as to alter the otherwise uniform amplitude of the series of continuous waves (Fig. 462). These "modulated" con- tinuous waves on reaching the aerial of the receiving station produce corresponding oscillatory currents in the wires of the * The pupil should master 374, 375, 376, 485, 486, 487, and 488 before reading the six sections following. 428 INVISIBLE RADIATIONS aerial. By means of a crystal or a vacuum tube, the oscilla- tory currents are rectified into a series of unidirectional elec- trical currents, or pulses, somewhat after the mann-er indicated FIG. 462. Modulated radio-frequency waves in Fig. 463. These variable pulses of radio frequency, on reaching the telephone receivers of the listener, produce dia- phragm vibrations of low frequencies (audio frequencies), which (Innnnnfl Innnflnnr FIG. 463. Rectified oscillations rarely go outside the limits of 100 and 3000 vibrations per second. They are represented by the irregular line in Fig. 464. The vibrations of the diaphragms of the receivers, therefore, FIG. 464. Audio-frequency variations correspond to the vibrations of the voice of the speaker at the distant transmitting station. 490. Method of producing continuous waves. One of the most important of the different means of producing high- power continuous waves is by use of the Alexanderson high- frequency alternator (see on opposite page). This is an alternating-current dynamo made in various powers up to 200 kilowatts (= 268 horse power), the rotor in some of the machines having the very high speed of 20,000 revolutions per minute. For transoceanic telegraphy these machines cause currents of from 600 to 1200 amperes to oscillate in the sending aerial. This powerful sustained oscillation of electrons in an aerial produces continuous electromagnetic waves (Fig. 461). 1 >> 3 -2 2 11s 3 w S ill 111 I III w ^s.t? > >-^ +^ 'Z* & " M S I pT ^ ,2 ^5 O g 2 73 ; H* 2; o 02 H 3 O) O t-i tfs t: s ^ r2 g '3 * oa g III ! i PH 2 5 6 e J * r. pi - T * " r- _~ o -2 -5 5 g IS .2 be o a 4J a ~ 1 1 APPENDIX SUPPLEMENTARY QUESTIONS AND PROBLEMS CHAPTER I. 1. A new lead pencil is 7 in. long. How many centi- meters long is it? 2. From the bed rock upon which the Woolworth Building in New York rests to the top of the tower is 278.3 m. How many feet is it? 3. The wing spread of the NC-4 is 126 ft. How many meters is it? 4. How many kilograms are there in the 16-pound shot? 5. Name three uses made of lead because of its great density, and two uses of cork due to its small density. 6. A flask held 2520 g. of glycerin when filled. What was the capac- ity of the flask in liters? (See table of densities, p. 9.) CHAPTER II. 1. A standpipe 100ft. high is filled with water. Find the pressure at the bottom in pounds per square foot and in pounds per square inch. ca 2. Deep-sea fish have been caught in nets at a depth of a mile. How many pounds pressure are there to the square inch at this depth? (Specific gravity of sea water = 1.026.) 3. If the pressure at a tap on the first floor reads 80 Ib. per square inch, and at a tap two floors above, 68 Ib., what is the difference in feet between the levels of the two taps? 4. Find the total force against the gate of a lock if its width is 60 ft. and the depth of the water 20 ft. Will it have to be made stronger if it holds back a lake than if it holds back a small pond ? 5. Fig. 477 represents an instrument commonly known as the hydrostatic bellows. If the base C is 20 in. square and the tube is filled with water to a depth of 5 ft. above the top of C, what is the value of the weight which the bellows can support ? 6. A hydraulic press having a piston 1 in. in diameter exerts a force of 10,000 Ib. when 10 Ib. are applied to this piston. What is the diam- eter of the large piston ? 447 FIG. 477 Hydrostatic bellows 448 APPENDIX B = 7. A floating dock is shown in Fig. 478. When the chambers c are filled with water, the dock sinks until the water line is at A. The vessel is then floated into the dock. As soon as it is in place, the water is pumped from the chambers until the water line is as low as B. Work- men can then get at all parts of the bottom. If each of the chambers is 10 ft. high and 10 ft. wide, what must be the length of the dock if it is to be available for the Beren- garia (Cunard Line), of 50,000 tons' weight? 8. If each boat of a pontoon bridge is 100 ft. long and 75 ft. wide at the water line, how much will it sink when a locomotive weighing 100 tons passes over it? 9. What must be the specific gravity of a liquid in which a body having a specific gravity of 6.8 will float with half its volume submerged V 10. A block of wood 10 in. high sinks 6 in. in water. Find the density of the wood. 11. If this block sank 7 in. in oil, what would be the density of the oil '( 12. A graduated glass cylinder contains 190 cc. of water. An egg weighing 40 g. is dropped into the glass ; it sinks to the bottom and xaises the water to the 225-cc. mark. Find the density of the egg. FIG. 478. Floating dock CHAPTER III. 1. Explain the process of making air enter the lungs; of making lemonade rise in a straw. 2. If a circular piece of wet leather having a string attached to the middle is pressed down on a flat, smooth stone, as in Fig. 479, the latter may often be lifted by pulling on the string. Is it pulled up or pushed up ? Explain. 3. Make a labeled drawing of a simple Torricellian barometer, naming all the parts in the diagram. 4. The body of the average man has 15 sq. ft. of sur- face. What is the total force of the atmosphere upon Mm ? Why is he unconscious of this crushing force ? 5. If the variation of the height of a mercury barometer is 2 in., how far did the image rise and fall in Otto von Guericke's water barometer ? (See 42.) 6. What is Boyle's law ? A mass of air 3 cc. in volume is introduced into the space above a barometer column which originally stands at 760 mm. The column sinks until it is only 570 mm. high. Find the Tolume now occupied by the air. FIG. 479 QUESTIONS AND PEOBLEMS 449 7. There is a pressure of 80 cm. of mercury on 1000 cc. of gas. What pressure must be applied to reduce the volume to 600 cc. if the temperature is kept constant? 8. Pressure tests for boilers or steel tanks of any kind are always made by filling them with water rather than with air. Why ? 9. If the water within a diving bell is at a depth of 1033 cm. beneath the surface of a lake, what is the density of the air inside if at the sur- face the density of air is .0013 and its pressure 76 cm. ? What would be the reading of a barometer within the bell ? 10. If a diver descends to a depth of 100 ft., what is the pressure to which he is subjected? What is the density of the air in his suit, the density at the surface where the pressure is 75 cm. being .0012? (Assume the temperature to remain unchanged.) 11. How many of the laws of liquids and gases do you find illustrated in the experiment of the Cartesian diver? 12. Pascal proved by an experiment that a siphon would not run if the bend in the arm were more than 34 ft. above the upper water level. He made it run, however, by inclining it sidewise until the bend was less than 34 ft. above this level. Explain. 13. How high will a lift pump raise water if it is located upon the side of a mountain where the barometer reading is 71 cm.? 14. Find the lifting power of a kite balloon whose capacity is 37,000 cu. ft., the lifting power of the gas being 64.4 Ib. per 1000 cu. ft. and the weight of the balloon, cordage, car, and observer being 1300 Ib. CHAPTER IV. 1. W T hy does a confined body of gas exert pressure inversely proportional to its volume ? 2. A lump of copper sulphate placed at the bottom of a graduate filled with water will dissolve and very slowly pass upward, although a copper-sulphate molecule is many times heavier than a water molecule. Explain. CHAPTER V. 1. An airplane which flies in still air with a velocity of 120 mi. per hour is flying in a wind whose velocity is 60 mi. per hour toward the east. Find the actual velocity of the airplane and the direction of its motion when headed north ; east ; south ; west. 2. Represent graphically a force of 30 Ib. acting southeast and a force of 40 Ib. acting southwest at the same point. What will be the magnitude of the resultant, and what will be its approximate direction? 3. Two concurrent forces, each of 50 Ib., act at an angle of 60 with each other. Find the resultant graphically. 450 APPENDIX 4. A child weighing 100 Ib. sits in a swing. The swing is drawn aside and held in equilibrium by a horizontal force of 40 Ib. Find the tension in each of the two ropes of the swing. 5. Four clothes posts were arranged to form a square. A clothes- line was drawn around the outside of the posts with a force of GO Ib. With what force is each post drawn toward the center of the square ? 6. A man weighing 150 Ib. stood at the middle of a tight-rope whose two parts were each 50 ft. long. What was the tension on the parts of the rope, the weight of the man depressing the center of the rope 1 ft.? 7. A boy pulls a loaded sled weighing 200 Ib. up a hill which rises 1 ft. in 5 measured along the slope. Neglecting friction, how much force must he exert? 8. A cask weighing 100 Ib. is held at rest upon an inclined plank 8 ft. long and 3 ft. high. By the resolution-and-proportion method find the component of its weight that tends to break the plank. 9. What force will be required to support a 50-lb. ball on an inclined plane of which the length is 10 times the height? 10. A boy is able to exert a force of 75 Ib. Neglecting friction, how long an inclined plane must he have in order to push a truck weighing 350 Ib. up to a doorway 3 ft. above the ground? 11. Could a kite be flown from an automobile when there is no wind? Explain. 12. Why is it unsafe to stand up in a canoe? 13. If a lead pencil is balanced on its point on the finger, it will be in unstable equilibrium, but if two knives -are stuck into it, as in Fig. 480, it will be in stable equilibrium. Why? 14. Why does a man lean forward when he climbs a hill? 15. A boy dropped a stone from a bridge and noticed that it struck the water in just 3 sec. How fast was it going when it struck? How high was the bridge above the water ? 16. If a body sliding without friction down an inclined plane moves 40 cm. during the first second of its descent, and if the plane is 500 cm. long and 40.8 cm. high, what is the value of Determination of focal length of a Fig. 487. Prove that when convex mirror the diameter of the re- flected beam rq is twice the diameter of the hole np, the distance mo from the mirror to the screen is equal to the focal length oF of the mirror. 6. If a rose R is pinned up- side down in a brightly illumi- nated box, a real image may be formed in a glass of water W by a concave mirror C (Fig. 488). Where must the eye be placed to see the image ? 7. How far is the rose from FlG 48g> Image of object at center the mirror in the arrangement of curvature of Fig. 488 ? 8. A candle placed 20 cm. in front of a concave mirror has its image formed 50 cm. in front of the mirror. Find the radius of the mirror. 9. The parabolic mirror used as an objective in one of the telescopes at the Mount Wilson observatory is 100 in. in diameter and has a focal length of about 50 ft. What magnification is obtained when it is used with a 2-inch eyepiece; with a 1-inch eyepiece? What is gained by the use of a mirror of such enormous diameter ? 464 APPENDIX 10. A compound microscope has a tube length of 8 in., an objective of focal length ^ in., and an eyepiece of focal length 1 in. What is its magnifying power ? 11. If the focal length of the eye is 1 in., what is the magnifying power of an opera glass whose objective has a focal length of 4 in.? 12. Explain as well as you can how a telescope forms the image which you see when you look into it. 13. The magnifying power of a microscope is 1000, the tube length is 8 in., and the focal length of the eyepiece is ^ in. What is the focal length of the objective ? CHAPTER XX. 1. If a soap film is illuminated with red, green, and yellow strips of light, side by side, how will the distance between the yellow fringes compare with that between the red fringes? with that between the green fringes? (See table on page 403.) 2. What will be the apparent color of a red body when it is in a room to which only green light is admitted ? 3. Will a reddish spot on an oil film be thinner or thicker than an adjacent bluish portion? 4. Explain the ghastly appearance of the face of one who stands under the light of a Cooper-Hewitt mercury-vapor arc lamp. 5. Draw a figure to show how a spectrum is formed by a prism, and indicate the relative positions of the red, the yellow, the green, and the blue in this spectrum. 6. Why is a rainbow never seen during the middle part of the day? 7. If you look at a broad sheet of white paper through a prism, it will appear red at one edge and blue at the other, but white in the middle. Explain why the middle appears uncolored. 8. Can you see any reason why the vibrating molecules of an incan- descent gas might be expected to give out a few definite wave lengths, while the particles of an incandescent solid give out all possible wave lengths ? 9. Can you see any reason why it is necessary to have the slit narrow and the slit and screen at conjugate foci of the lens in order to show the Fraunhof er lines in the experiment of 480 ? CHAPTER XXI. 1. How are ultra-violet waves detected? What apparatus is used to reveal infra-red waves? 2. Explain how the heat of the sun warms the earth. 3. What is electric resonance? How may it be demonstrated? 4. Describe the construction of an X-ray tube. Describe as well as you can the action within it when in use. INDEX Aberration, chromatic, 409 Absolute temperature, 134 Absolute units, 6 Absorption of gases, 102 ff. ; of light waves, 414 ; and radiation, 419 Acceleration, defined, 75 ; of gravity, 77 Achromatic lens, 410 Adhesion, 92 ; effects of, 98 Aeronauts, height of ascent of, 37 Air, weight of, 26 ; pressure of, 27 ; compressibility of, 34 ; expansi- bility of, 34 Air pump, 33, 41 Airplane, frontispiece ; principle of gliding of, 78-80; principle of flight of, 80 ; Vickers-Vimy, 153 ; Liberty motor in, 191 ; Wright, 317 Airship, 44 Alternator, 298 Amalgamation of zinc plate, 272 Ammeter, 257 Ampere, portrait of, 256 Ampere, definition of, 251, 257 Ampere turns, 255 Amplifier, 431 Amundsen, 222 Anode, 248 Arc light, 286; automatic feed for, 287 Archimedes, principle of, 21 ; por- trait of, 22 Armature, ring type, 255, 299 ; drum type, 297, 300, 301, 310 Atmosphere, pressure of, 29 ; extent and character of, 36 ; humidity of, 175 Atoms, energy in, 445 Audion, 429 Automobile, 195, 198 ; clutch and transmission, 196; differential, 197 ; carburetor, 198, 199 ; ignition sys- tem, 198, 199; anti-glare "lens," 362 Back E. M. F. in motors, 303 Baeyer, von, 417 Balance, 7 Balance wheel, 141 Ball bearings, 145, 146 Balloon, kite, 44, 45 ; dirigible, 44 ; helium, 45 Barometer, mercury, 30 ; von Guer- icke's, 31; the aneroid, 31; the self- registering, 32, 38 Batteries, primary, 272 ff. ; storage, 281, 283 Battleship, 152 Bearings, ball, 145, 146 ; roller, 146 Beats, 332, 348 Becquerel, 441 ; portrait of, 446 Bell, Alexander Graham, 316 ; por- trait of, 316 Bell, electric, 259 Bicycle pedal, 146 Binocular vision, 398 Boiler, steam, 191 Boiling points, definition of, 183 ; effect of pressure on, 183 Boyle's law, stated, 36; explained, 51 British thermal unit, 152 Brittleness, 92 Brooklyn Bridge, 143 Brownian movements, 52 Bunsen, 376 Caisson, 46 Calories, 152 ; developed by electric currents, 289 Camera, pinhole, 390 ; photographic, 390 Candle power, of incandescent lamps, 285 ; of arc lamps, 286 ; defined, 375 Canner, steam-pressure, 184 Capacity, electric, 240 Capillarity, 96 ff. Capstan, 117 Carburetor, 198, 199 466 466 INDEX Cartesian diver, 43 Cathode, defined, 248 Cathode rays, 436 Cells, galvanic, 245 ; primary, 272 ff.; local action in, 272 ; theory of, 273 ; Daniell, 275 ; Weston, 277 ; Leclanche", 277 ; dry, 278 ; com- binations of, 279, 280 j storage, 281, 283 Center of gravity, 68 Centrifugal force, 84 Charcoal, absorption by, 102 Charles, law of, 136 Chemical effects of currents, 248 Cigar lighter, platinum-alcohol, 103 Ciwrnont, 135 Clouds, formation of, 174 Clutch, automobile, 196 Coefficient of expansion of gases, 136; of liquids, 138 ; of solids, 140 Coefficient of friction, 145 Cohesion, 92; properties of solids depending on, 92 ; in liquids, 93 ; in liquid films, 93 Coils, magnetic properties of, 252 ff. ; currents induced in rotating, 294 Cold storage, 202 Color, and wave length, 402; of bodies, 404 ; compound, 405 ; com- plementary, 406 ; of pigments, 40 7; of thin films, 408 Commutator, 298 Compass, 222. See also Gyrocompass Component, 61 ; magnitude of, 62 Concurrent forces, 60 Condensation of water vapor, 173 Condensers, 240 Conduction, of heat, 203 ; of electric- ity, 227 Conjugate foci, 379 Conservation of energy, 155 Convection, 206 ff . Cooling, of a lake, 139; by expansion, 155 ; and evaporation, 176 ; arti- ficial, by solution, 187 Cooper-Hewitt lamp, 288 Coulomb, 251 Couple, 112 Crane, 121 Cream separator, 85 Crilley, 46 Critical angle, 361, 362 Crookes, 358, 443 ; portrait of, 358 Curie, 441, 442, 444 ; portrait of, 446 Currents, wind and ocean, 207 ; elec- tric, defined, 245 ; effects of elec- tric, 248 ff . ; magnetic fields about, 252 ; measurement of electric, 256 ff. ; induced electric, 290 ff. Curvature, of a liquid surface, 97 ; of waves, 369 ; defined, 370 ; of a mirror, 386 ; center of, 463 Daniell cell, 275 Davy safety lamp, 205 Declination, 222 Densities, table of, 8, 9 Density, defined, 8; formula for, 9; of air, 26 ; maximum, of water, 138 ; of saturated vapor, 171 ; of electric charge, 234 Descartes, 43 Dew, formation of, 174 Dew point, 175 Dewar flask, 209 Differential, automobile, 197 Diffusion, of gases, 50, 52 ; of liquids, 54 ; of solids, 55 ; of light, 359 Digester, 184 Dipping needle, 223 Discord, 347 Dispersion, 403 Dissociation, 249, 273 Distillation, 185 Diving bell, 45 Diving suit, 46 Doppler effect, in sound, 326 ; in light, 416 Dry cell, 278 Ductility, 92 Dynamo, principle of, 290 ; rule for, 293; alternating-current, 296; four- pole direct-current, 300 ; series- wound, shunt-wound, and com- pound-wound, 301 ; defined, 302 Dyne, 86 Eccentric, 191 Echo, 327 Edison, 356 ; portrait of, 316 Efficiency, defined, 147 ; of simple machines, 147 ; of water motors, 148, 149 ; of steam engines; 193 j of electric lights, 285 ff. ; of trans- formers, 312 Elasticity, 90 ; limits of, 91 INDEX 467 Electric charge, unit of, 227 ; distri- bution of, 233 ; density of, 234 Electric iron, 269 Electric motor, principle of, 292 ; construction of, 301 ; defined, 302 Electricity, static, 225 ff. ; electron theory of, 229, 438 ff. ; current of, 244 ff. Electrolysis of water, 248 Electromagnet, 247, 255 Electromotive force, defined, 263 ; of galvanic cells, 266 ; induced, 291 ; strength of induced, 294 ; curve of alternating, 297; curve of commutated, 299; back, in motors, 303 ; in secondary circuit, 307 ; at make and break, 308 Electron theory, 229, 438 ff . Electrophorus, 242 Electroplating, 249 Electroscope, 227, 232 Electrostatic voltmeter, 239 Electrotyping, 250 Energy, denned, 122 ; potential and kinetic, 123 ; transformations of, 124, 157, 162, 163 ; formulas for, 125, 126; conservation of, 155; from sun, 157 ; expenditure of electric, 284 ; stored in atoms, 445 Engine, steam, 189 ; steam, defined, 191 ; compound steam, 193, 298 ; gas, 191, 194 English equivalent of metric units, 5 Equilibrant, 60 Equilibrium, stable, 69 ; neutral, 71 ; unstable, 71 Erg, 106 Ether, 367 Evaporation, 53 ; effect of tempera- ture on, 168 ; of solids, 168 ; effect of air on, 171, 172 ; cooling effect of, 176 ; freezing by, 178 ; effect of air currents on, 178 ; effect of sur- face on, 179 ; and boiling, 184 Expansion, of gases, 136 ; of liquids, 138 ; of solids, 139 ; unequal, of metals, 142 ; cooling by, 155 ; on solidifying, 165 Eye, 392 ; pupil of, 392 ; nearsighted, 393 ; f arsighted, 393 Fahrenheit, 131 Falling bodies, 72-78 Faraday, 251, 290 ; portrait of, 290 Fields, magnetic, 219 Films, contractility of, 95 ; color of, 408 Fire syringe, 155 Tireless cooker, 206 Float valve, 452 , Floating dry dock, 448 Floating needle, 100 Flotation, law of, 22 Focal length, of convex lens, 378; of convex mirror, 385, 463 Fog, formation of, 174 Foley, 387 Foot-candle, 376 Force, beneath liquid, 11 ; definition of, 57 ; method of measuring, 57 ; composition of, 59; resultant of, 59 ; component of, 61, 62 ; centrifu- gal, 84 ; lines of, 218 ; fields of, 219 Formula for lenses and mirrors, 388 Foucault, 358 Foucault currents, 309 Franklin, 236 ; portrait of , 230 ; kite experiment of, 231 Fraunhofer lines, 414 Freezing mixtures, 188 Freezing points, table of, 164 ; of solutions, 187 Friction, 144 ff. Frost, formation of, 174 Fundamentals, denned, 341 ; in pipes, 349, 350 Fuse, electric, 269 Fusion, heat of, 161, 162 Galileo, 72, 73, 128, 132; portrait of, 72 Galvani, 245 Galvanic cell, 245 Galvanometer, 256, 257 Gas engine, 191, 194 Gas heating coil, 213 Gas mask, 103 Gas meter, 46 ; dials of, 48 Gay-Lussac, law of, 136 Geissler tubes, 437 Gilbert, 225 ; portrait of, 222 Gliding, principle of, 78-80 Governor, 192 Gram, of mass, 4 ; of force, 57 ; of force, variation of, 58 Gramophone, 355 468 INDEX Gravitation, law of, 66 Gravity, variation of, 58, 67 ; center of, 68 Guericke, Otto von, 31, 41 ; portrait of, 32 Gun, 354-mm., in action, 73 Gyrocom^ss, 83, 223 Hail, formation of, 174 Hardness, 92 Harmony, 347 Hay scales, 120 Headlight, automobile, 400 Heat, mechanical equivalent of, 151 ff. ; unit of, 152 ; produced by friction, 153 ; produced by colli- sion, 154 ; produced by compres- sion, 154 ; specific, 158 ; of fusion, 161 ; of vaporization, 181 ; trans- ference of, 203 Heating, by hot air, 211 ; by hot water, 212 ; by steam, 213 Heating effects of electric currents, 284, 289 Helium, 45, 445 Helmholtz, 345 Henry, Joseph, portrait of, 246 Henry's law, 104 Hertz, 422, 436 ; portrait of, 102 Heusler alloys, 216 Hiero, 21 Him, 154 Hooke's law, 91 Horse power, 122 Humidity, 175 Huygens, 364, 372 ; portrait of, 364 Hydraulic elevator, 18 Hydraulic press, 17 Hydraulic ram, 88, 89 Hydrogen thermometer, 132 Hydrometer, 23 Hydrostatic bellows, 447 Hydrostatic paradox, 14 Hygrometry, 173 Ice, manufactured, 201 Ignition, automobile system of, 198, 199 Images, by convex lenses, 378 ff. ; size of, 381 ; virtual, 382 ; by con- cave lenses, 382 ; in plane mirrors, 383 ; in convex mirrors, 384, 386 ; in concave mirrors, 384, 387 Imp, bottle, 43 Incandescent lighting, 285 Incidence, angle of, 358 Inclination, 223 Inclined plane, 63, 117 Index of refraction, 371 Induction, magnetic, 216; electro- static, 228 ; charging by, 230 ; of current, 290 Induction coil, 308 Induction motor, 291 Inertia, 83 Insect on water, 100 Insulators, 227 Intensity, of sound, 326 ; of illumi- nation, 374 Interference, of sound, 333 ; of light, 365 Ions, 235, 249, 273 Iron, electric, 269 Isoclinic lines, 457 Isogonic lines, 223 Jackscrew, 118 Joule, 106, 122, 151 ff.; portrait of, 122 Kelvin, portrait of, 134 Kilogram, the standard, 4 Kilowatt, 122 Kilowatt hour, 285 Kinetic energy, 123, 126 Kirchhoff, 415 Laminated cores, 310 Lamps, incandescent, 285 ; arc, 286 ; Cooper-Hewitt, 288 Lantern, projecting, 391 Leclanch<* cell, 277 Lenses, 378 ff . ; optical center of, 378 ; principal axis of, 378 ; principal focus of, 378 ; formula for, 380 ; magnifying power of, 395 ; achro- matic, 410 Lenz's law, 291 Level of water, 13 Lever, 110 ff. ; compound, 120 Leviathan, 135 Leyden jar, 241 Liberty motor, 191 Light, speed of, 357 ; reflection of, 358 ; diffusion of, 359 ; refraction of, 360; nature of, 364; corpuscular theory of, 364; wave theory of, INDEX 469 364; interference of, 365; wave length of, 367, 403 ; intensity of, 374; electromagnetic theory of, 436 Lightning, 236 Lightning rods, 236 Lines, of force, 218 ; isogonic, 223 Liquids, densities of, 9 ; pressure in, 13 ; transmission of pressure by, 15; incompressibility of, 33; ex- pansion of, 138 Liter, 3 Local action, 272 Locomotive, 192; Mallet, 123; Kocket, 123 Loudness of sound, 326 Machines, general law of, 116, 124, 156 ; efficiencies of, 147 Magdeburg hemispheres, 33 Magnet, natural, 214 ; laws of the, 215; poles of the, 215; lifting, 247 Magnetism, 214 ff. ; nature of, 220 ; theory of, 221 ; terrestrial, 222 ; residual, 301 Magnifying power, of lens, 395 ; of telescope, 396 ; of microscope, 397 ; of opera glass, 398 Malleability, 92 Manometric flames, 343 Marconi, 423 ; portrait of, 316 Mass, unit of, 4; measurement of, 6 Matter, three states of, 55 Maxwell, 436 ; portrait of, 102 Mechanical advantage, 109 Mechanical equivalent of heat, 153 ff. Melting points, table of, 164 ; effect of pressure on, 166 Meter, standard, 3 Michelson, 357 ; portrait of, 358 Microphone, 315 Microscope, 397 Mirrors, 383 ff.; convex, 384, 386; concave, 384, 387 ; formula for, 388 Mixtures, method of, 159 Molecular constitution of matter, 49 Molecular forces, in solids, 90 ; in liquids, 93 Molecular motions, in" gases, 49, 50 ; in liquids, 53 ; in solids, 55 Molecular nature of magnetism, 220 Molecular velocities, 52, 129 Moments of force, 111 Momentum defined, 84 Morse, 260 ; portrait of, 260 Motion, uniformly accelerated, 75 laws of, 76 ; perpetual, 156 Motor, Liberty, 191 ; electric-induc- tion, 291 ; street-car, 302. See also Electric motor Motor rule, 293 Moving pictures, 386 Newton, law of gravitation, 66 ; laws of motion, 83-87 ; portrait of, 84 ; principle of work, 116; corpuscular theory, 364 Niagara, 157 Nichols, E. F., 417 Nodes, in pipes, 334 ; in strings, 340 Noise and music, 325 Nonconductors, of heat, 205 ; of elec- tricity, 227 North magnetic pole, 222 Ocean currents, 207 Oersted, 246 ; portrait of, 246 Ohm, 263 ; portrait of, 268 Ohm's law, 267 Onnes, Kamerlingh, 135, 178 Opera glass, 398 Optical instruments, 890 ff . Organ pipes, 353, 354 Oscillatory discharge, 422 Overtones, 341 ; in pipes, 350 Parabolic reflector, 400 Parachute, 44 Parallel connections, 270, 280 Parallelogram law, 61 Pascal, 15, 16, 30 Pendulum, force moving, 64 ; laws of, 81 ; compensated, 141 Periscope, 400 Permeability, 217 Perpetual motion, 156 Perrier, 30 Phonograph, 355 Photometers, 374, 376 Pisa, tower of, 72 Pitch, cause of, 325 Pneumatic inkstand, 33 Points, discharging effect of, 234 Polarization, of galvanic cells, 274 ; of light, 374 Potential, defined, 237 ; measure- ment of, 239, 265 j unit of, 277 470 INDEX Power, definition of, 121 ; horse, 122 ; electric, 284 Pressure, in liquids, 13 ; defined, 13 ; transmission of, by liquids, 15 ; in air, 27 ; amount of atmospheric, 29 ; coefficient of expansion, 133, 136; effect of, on freezing, 166; of saturated vapor, 1 70 ; in primary and secondary, 311 Projectile, path of, 78 Pulley, 108 ff.; differential, 119 Pump, air, 33, 41 ; compression, 41 ; lift, 42 ; force, 43 Quality of musical notes, 342 Quebec Bridge, 70 E-34, dirigible airship, 44 Radiation, thermal, 208; invisible, 417 ff.; and temperature, 418 ; and absorption, 419 ; electrical, 421 Radioactivity, 441 ff. Radiometer, 417 Radium, discovery of, 441 Rain r formation of, 174 Rainbow, 411 Ratchet wheel, 146 Rayleigh, portrait of, 358 Rays, infra-red, 417 ; ultra-violet, 417 ; cathode, 437 ; Rontgen, 439 ; Becquerel, 442 ; a:, /3, and 7, 442 ff. Rectifier, tungar, 314 ; crystal, 425 Reflection, of sound, 327 ; of light, 358; angle of, 358; total, 361, 462 Refraction, of light, 360 ; explana- tion of, 368 ; index of, 371 Refrigerator, 163 Regelation, 167 Relay, 260 Resistance, electric, defined, 262 ; specific, 262; laws of, 262; unit of, 263 ; internal, 268 ; measurement of, 269 Resistances, table of, 262 Resonance, acoustical, 328 ff.; elec- trical, 421 Resonators, 331 Resultant, 59 Retentivity, 217 Retinal fatigue, 407 Right-hand rule, 252, 254 Rise of liquids, in exhausted tubes, 27 ; in capillary tubes, 97 Roller bearings, 146 Romer, 357 Rontgen, 439 ; portrait of, 446 Ross, 222 Rotor, generator, 257 Rowland, 155 ; portrait of, 358 Rubens, 408 Rumford, 151, 374 Rutherford, 442 ; portrait of, 446 Saturation of vapors, 169 ; magnetic, 222 Scales, musical, 337 ; diatonic, 338 ; even-tempered, 339 Screw, 118 Sea breeze, 207 Searchlight, 400 Secondary cells, 281 ff. Self-induction, 307 Separator, cream, 85 Series connections, 270, 279 Shadows, 362 Shunts, 258, 270 Singing flame, 348 Siphon, explanation of, 40; inter- mittent, 40 Siren, 337 Sleet, formation of, 174 Snow, formation of, 174 Soap films, 95, 402 Solar spectrum, 414, 415 Sonometers, 339 Sound, sources of, 319; nature of, 319 ; speed of, 320 ; musical, 325 ; intensity of, 326 ; reflection of, 327 Sound foci, 328 Sound waves, interference of, 333 ; photographs of, 346, 387 Sounder, 260 Sounding boards, 331 Spark, oscillatory nature of the, 422 ; in vacuum, 436 Spark length and potential, 240 Spark photography, 422 Speaking tubes, 326 Specific gravity, 9 ; methods of find- ing, 22 ff. Specific heat, defined, 158; meas- ured, 159 Specific heats, table of, 160 INDEX 4T1 Spectra, 411 ff. ; continuous, 412 ; bright-line, 413 ; pure, 414 ; solar, 414; X-ray, 447 Spectrum, 403 ; invisible portions of, 417 Spectrum analysis, 413 Speed, of sound, 320 ; of light, 357 ; of light in water, 369 ; of electric waves, 423 ff. Spinthariscope, 443 Starting box, 304 Steam engine, 189 ff. Steam turbine, 199 Steelyards, 115 Stereoscope, 398 Storage cells, 281, 283 Strings, laws of, 339 Sublimation, 168 Submarine, 23, 44 Sun, energy derived from, 157 ; spec- trum of, 414 Surface tension, 95 Sympathetic electrical vibrations, 421 Sympathetic vibrations of sound, 346 ff. Tank, British, 190 Telegraph, 259 ff. ; wireless, 423 ff. Telephone, 316 ff. ; wireless, 427 ff. Telescope, astronomical, 396; Yerkes, 365, 396, 397 Temperature, measurement of, 128 ; absolute, 134 ; low, 134 Tenacity, 90 Thermometer, Galileo's, 128 ; mer- cury, 129 ; Fahrenheit, 131 ; gas, 132-134; alcohol, 132, 134; the dial, 143 Thermos bottle, 209 Thermoscope, 418 Thermostat, 142 Thomson, 438 ; portrait of, 440 Three-color printing, 408 Torricelli, experiment of, 28 Tower, high-tension, 241 Transformer, 312-314 Transmission, of pressure, 15 ; elec- trical, 312 ; of sound, 321 Transmission, automobile, 196 Transmitter, telephone, 316, 317 Trowbridge, 418 Tungar rectifier, 314 Turbine, water, 149 ; steam, 199 Units, of length, 2 ; of area, 2 ; of volume, 2 ; of mass, 4 ; of time, 5 ; three fundamental, 5 ; C. G. S., 6 ; of force, 57, 86 ; of work, 106 ; of power, 122, 284 ; of heat, 152 ; of magnetic pole, 215 ; of magnetic field, 219 ; of current, 251, 257 ; of resistance, 267 ; of potential, 277 ; of light, 375, 376 Vacuum, sound in, 320; spark in, 436 Vaporization, heat of, 181, 182 Velocity, of falling body, 75; of sound, 320 ; of light, 357 Ventilation, 210, 211 Vibration, forced, 331 ; of strings, 339 ; sympathetic, 346 ff. Vibration numbers, 337 Vision, distance of most distinct, 394 ; binocular, 398 Visual angle, 394 Volt, 239, 277, 294 Volta, 245 ; portrait of, 240 Voltmeter, electrostatic, 239; com- mercial, 265, 266 Watch, balance wheel of, 141 ; wind- ing mechanism of, 453 Water, density of, 4 ; city supply of, 19 ; maximum density of, 138 ; ex- pansion of, on freezing, 166 Water wheels, 148-150 Watt, 122, 284 Watt, James, 122, 189 ; portrait of, 122 Watt-hour meter, 304 Wave length, defined, 322 ; formula for, 323 ; of yellow light, 367 ; of other lights, 403 Wave theory of light, 364 Wave train, 322, 424, 425 Waves, condensational, 323 ; water, 324 ; longitudinal and transverse, 325 ; light, are transverse, 371 ; electric, 422 ; modulated, 427 Weighing by method of substitu- tion, 6 Welsbach mantle, 442 Weston cell, 277 Wet- and dry-bulb hygrometer, 178 Wheel, and axle, 116; gear, 119; worm, 119 ; water, 148-160 472 INDEX White light, nature of, 403 X-ray picture of human thorax, 359 Wind instruments, 349 X-ray spectra, 440, 447 Windlass, 120, 453 X-rays, 439 ff. Winds, 207 Wireless telegraphy, 423 ff. Yale lock, 452 Wireless telephony, 427 ff . Yard, 2 Work, defined, 105 ; units of, 106 ; Yerkes telescope, 365, 396, 397 principle of, 116, 125, 156 Wright, Orville, 317 ; portrait of, 316 Zeiss binocular, 399 RETURN CIRCULATION DEPARTMENT TO** 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Renewals and Recharges may be made 4 days prior to the due date. Books may be Renewed by calling 642-3405. IE AS STAMPED BELOW FORM NO. DD6 UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 BERKELEYLIBRAWES 2~ UNIVERSITY OF CALIFORNIA LIBRARY - *** *r.